LMFDB - Embedded newform 135.5.h.a.44.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,5,Mod(44,135)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(135, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5, 3]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("135.44");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	135.h (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$13.9549450163$
Analytic rank:	$0$
Dimension:	$44$
Relative dimension:	$22$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			44.13
Character	$\chi$	$=$	135.44
Dual form			135.5.h.a.89.13

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.547257 + 0.947877i) q^{2} +(7.40102 - 12.8189i) q^{4} +(17.4860 - 17.8673i) q^{5} +(-16.9207 + 9.76914i) q^{7} +33.7133 q^{8} +(26.5053 + 6.79660i) q^{10} +(100.405 - 57.9690i) q^{11} +(-158.728 - 91.6418i) q^{13} +(-18.5199 - 10.6925i) q^{14} +(-99.9665 - 173.147i) q^{16} -89.0232 q^{17} -52.2295 q^{19} +(-99.6252 - 356.388i) q^{20} +(109.895 + 63.4480i) q^{22} +(-232.002 + 401.839i) q^{23} +(-13.4788 - 624.855i) q^{25} -200.607i q^{26} +289.206i q^{28} +(1255.41 - 724.811i) q^{29} +(767.942 - 1330.11i) q^{31} +(379.121 - 656.657i) q^{32} +(-48.7186 - 84.3831i) q^{34} +(-121.327 + 473.149i) q^{35} -641.462i q^{37} +(-28.5830 - 49.5071i) q^{38} +(589.511 - 602.364i) q^{40} +(1195.31 + 690.111i) q^{41} +(-1776.09 + 1025.43i) q^{43} -1716.12i q^{44} -507.859 q^{46} +(589.129 + 1020.40i) q^{47} +(-1009.63 + 1748.73i) q^{49} +(584.909 - 354.732i) q^{50} +(-2349.50 + 1356.49i) q^{52} +64.4267 q^{53} +(719.940 - 2807.62i) q^{55} +(-570.451 + 329.350i) q^{56} +(1374.06 + 793.316i) q^{58} +(-2837.79 - 1638.40i) q^{59} +(929.409 + 1609.78i) q^{61} +1681.05 q^{62} -2369.02 q^{64} +(-4412.91 + 1233.59i) q^{65} +(7257.38 + 4190.05i) q^{67} +(-658.862 + 1141.18i) q^{68} +(-514.884 + 143.931i) q^{70} +4564.71i q^{71} -8077.75i q^{73} +(608.027 - 351.045i) q^{74} +(-386.551 + 669.527i) q^{76} +(-1132.62 + 1961.75i) q^{77} +(4328.70 + 7497.52i) q^{79} +(-4841.68 - 1241.52i) q^{80} +1510.67i q^{82} +(-1670.06 - 2892.62i) q^{83} +(-1556.66 + 1590.60i) q^{85} +(-1943.96 - 1122.34i) q^{86} +(3384.99 - 1954.33i) q^{88} +10168.3i q^{89} +3581.05 q^{91} +(3434.10 + 5948.04i) q^{92} +(-644.811 + 1116.84i) q^{94} +(-913.285 + 933.198i) q^{95} +(-11868.3 + 6852.15i) q^{97} -2210.10 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 162 q^{4} - 6 q^{5} + 28 q^{10} - 228 q^{11} - 282 q^{14} - 1058 q^{16} - 8 q^{19} + 2196 q^{20} - 148 q^{25} - 2370 q^{29} - 1112 q^{31} - 436 q^{34} - 850 q^{40} - 1830 q^{41} - 5668 q^{46} + 5396 q^{49}+ \cdots - 58746 q^{95}+O(q^{100}) $ 44 * q - 162 * q^4 - 6 * q^5 + 28 * q^10 - 228 * q^11 - 282 * q^14 - 1058 * q^16 - 8 * q^19 + 2196 * q^20 - 148 * q^25 - 2370 * q^29 - 1112 * q^31 - 436 * q^34 - 850 * q^40 - 1830 * q^41 - 5668 * q^46 + 5396 * q^49 - 13524 * q^50 - 3072 * q^55 + 8250 * q^56 + 27348 * q^59 + 2782 * q^61 + 8348 * q^64 + 24906 * q^65 + 4782 * q^70 - 64728 * q^74 - 1776 * q^76 + 7900 * q^79 + 6766 * q^85 - 9192 * q^86 + 1560 * q^91 - 18826 * q^94 - 58746 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/135\mathbb{Z}\right)^\times$.

$n$	$56$	$82$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.547257	+	0.947877i	0.136814	+	0.236969i	0.926289	−	0.376814i	$-0.122980\pi$
$2$	0.547257	+	0.947877i	0.136814	+	0.236969i	−0.789475	+	0.613783i	$0.789647\pi$
$3$		0			0
$3$		0			0
$4$	7.40102	−	12.8189i	0.462564	−	0.801184i
$5$	17.4860	−	17.8673i	0.699440	−	0.714691i
$5$	17.4860	−	17.8673i	0.699440	−	0.714691i
$6$		0			0
$7$	−16.9207	+	9.76914i	−0.345319	+	0.199370i	−0.662622	−	0.748954i	$-0.730556\pi$
$7$	−16.9207	+	9.76914i	−0.345319	+	0.199370i	0.317302	+	0.948324i	$0.397223\pi$
$8$	33.7133			0.526770
$9$		0			0
$10$	26.5053	+	6.79660i	0.265053	+	0.0679660i
$11$	100.405	−	57.9690i	0.829796	−	0.479083i	−0.0239867	−	0.999712i	$-0.507636\pi$
$11$	100.405	−	57.9690i	0.829796	−	0.479083i	0.853783	+	0.520629i	$0.174303\pi$
$12$		0			0
$13$	−158.728	−	91.6418i	−0.939220	−	0.542259i	−0.0495044	−	0.998774i	$-0.515764\pi$
$13$	−158.728	−	91.6418i	−0.939220	−	0.542259i	−0.889716	+	0.456515i	$0.849098\pi$
$14$	−18.5199	−	10.6925i	−0.0944893	−	0.0545534i
$15$		0			0
$16$	−99.9665	−	173.147i	−0.390494	−	0.676355i
$17$	−89.0232			−0.308039			−0.154019	−	0.988068i	$-0.549222\pi$
$17$	−89.0232			−0.308039			−0.154019	+	0.988068i	$0.549222\pi$
$18$		0			0
$19$	−52.2295			−0.144680			−0.0723400	−	0.997380i	$-0.523047\pi$
$19$	−52.2295			−0.144680			−0.0723400	+	0.997380i	$0.523047\pi$
$20$	−99.6252	−	356.388i	−0.249063	−	0.890970i
$21$		0			0
$22$	109.895	+	63.4480i	0.227056	+	0.131091i
$23$	−232.002	+	401.839i	−0.438567	+	0.759621i	−0.997579	−	0.0695388i	$-0.977847\pi$
$23$	−232.002	+	401.839i	−0.438567	+	0.759621i	0.559012	+	0.829160i	$0.311181\pi$
$24$		0			0
$25$	−13.4788	−	624.855i	−0.0215660	−	0.999767i
$26$		−	200.607i		−	0.296755i
$27$		0			0
$28$			289.206i			0.368886i
$29$	1255.41	−	724.811i	1.49276	−	0.861844i	0.492792	−	0.870147i	$-0.335976\pi$
$29$	1255.41	−	724.811i	1.49276	−	0.861844i	0.999966	+	0.00830339i	$0.00264308\pi$
$30$		0			0
$31$	767.942	−	1330.11i	0.799107	−	1.38409i	−0.121091	−	0.992641i	$-0.538639\pi$
$31$	767.942	−	1330.11i	0.799107	−	1.38409i	0.920198	−	0.391453i	$-0.128027\pi$
$32$	379.121	−	656.657i	0.370235	−	0.641266i
$33$		0			0
$34$	−48.7186	−	84.3831i	−0.0421441	−	0.0729957i
$35$	−121.327	+	473.149i	−0.0990423	+	0.386244i
$36$		0			0
$37$		−	641.462i		−	0.468562i	−0.972169	−	0.234281i	$-0.924726\pi$
$37$		−	641.462i		−	0.468562i	0.972169	−	0.234281i	$-0.0752736\pi$
$38$	−28.5830	−	49.5071i	−0.0197943	−	0.0342847i
$39$		0			0
$40$	589.511	−	602.364i	0.368444	−	0.376478i
$41$	1195.31	+	690.111i	0.711069	+	0.410536i	0.811457	−	0.584412i	$-0.198675\pi$
$41$	1195.31	+	690.111i	0.711069	+	0.410536i	−0.100388	+	0.994948i	$0.532008\pi$
$42$		0			0
$43$	−1776.09	+	1025.43i	−0.960568	+	0.554584i	−0.896348	−	0.443351i	$-0.853789\pi$
$43$	−1776.09	+	1025.43i	−0.960568	+	0.554584i	−0.0642203	+	0.997936i	$0.520456\pi$
$44$		−	1716.12i		−	0.886426i
$45$		0			0
$46$	−507.859			−0.240009
$47$	589.129	+	1020.40i	0.266695	+	0.461929i	0.968006	−	0.250926i	$-0.0807350\pi$
$47$	589.129	+	1020.40i	0.266695	+	0.461929i	−0.701311	+	0.712855i	$0.747402\pi$
$48$		0			0
$49$	−1009.63	+	1748.73i	−0.420503	+	0.728333i
$50$	584.909	−	354.732i	0.233964	−	0.141893i
$51$		0			0
$52$	−2349.50	+	1356.49i	−0.868898	+	0.501659i
$53$	64.4267			0.0229358			0.0114679	−	0.999934i	$-0.496350\pi$
$53$	64.4267			0.0229358			0.0114679	+	0.999934i	$0.496350\pi$
$54$		0			0
$55$	719.940	−	2807.62i	0.237997	−	0.928138i
$56$	−570.451	+	329.350i	−0.181904	+	0.105022i
$57$		0			0
$58$	1374.06	+	793.316i	0.408461	+	0.235825i
$59$	−2837.79	−	1638.40i	−0.815223	−	0.470669i	0.0335434	−	0.999437i	$-0.489321\pi$
$59$	−2837.79	−	1638.40i	−0.815223	−	0.470669i	−0.848766	+	0.528768i	$0.822654\pi$
$60$		0			0
$61$	929.409	+	1609.78i	0.249774	+	0.432621i	0.963463	−	0.267842i	$-0.0863104\pi$
$61$	929.409	+	1609.78i	0.249774	+	0.432621i	−0.713689	+	0.700463i	$0.752977\pi$
$62$	1681.05			0.437317
$63$		0			0
$64$	−2369.02			−0.578374
$65$	−4412.91	+	1233.59i	−1.04448	+	0.291974i
$66$		0			0
$67$	7257.38	+	4190.05i	1.61670	+	0.933404i	0.987765	+	0.155949i	$0.0498435\pi$
$67$	7257.38	+	4190.05i	1.61670	+	0.933404i	0.628938	+	0.777455i	$0.283490\pi$
$68$	−658.862	+	1141.18i	−0.142488	+	0.246796i
$69$		0			0
$70$	−514.884	+	143.931i	−0.105078	+	0.0293738i
$71$			4564.71i			0.905517i	0.891633	+	0.452759i	$0.149560\pi$
$71$			4564.71i			0.905517i	−0.891633	+	0.452759i	$0.850440\pi$
$72$		0			0
$73$		−	8077.75i		−	1.51581i	−0.652365	−	0.757905i	$-0.726223\pi$
$73$		−	8077.75i		−	1.51581i	0.652365	−	0.757905i	$-0.273777\pi$
$74$	608.027	−	351.045i	0.111035	−	0.0641060i
$75$		0			0
$76$	−386.551	+	669.527i	−0.0669237	+	0.115915i
$77$	−1132.62	+	1961.75i	−0.191030	+	0.330873i
$78$		0			0
$79$	4328.70	+	7497.52i	0.693590	+	1.20133i	0.970654	+	0.240482i	$0.0773054\pi$
$79$	4328.70	+	7497.52i	0.693590	+	1.20133i	−0.277063	+	0.960852i	$0.589361\pi$
$80$	−4841.68	−	1241.52i	−0.756512	−	0.193988i
$81$		0			0
$82$			1510.67i			0.224669i
$83$	−1670.06	−	2892.62i	−0.242424	−	0.419890i	0.718981	−	0.695030i	$-0.244609\pi$
$83$	−1670.06	−	2892.62i	−0.242424	−	0.419890i	−0.961404	+	0.275140i	$0.911276\pi$
$84$		0			0
$85$	−1556.66	+	1590.60i	−0.215455	+	0.220152i
$86$	−1943.96	−	1122.34i	−0.262839	−	0.151750i
$87$		0			0
$88$	3384.99	−	1954.33i	0.437112	−	0.252367i
$89$			10168.3i			1.28371i	0.766825	+	0.641856i	$0.221835\pi$
$89$			10168.3i			1.28371i	−0.766825	+	0.641856i	$0.778165\pi$
$90$		0			0
$91$	3581.05			0.432441
$92$	3434.10	+	5948.04i	0.405731	+	0.702746i
$93$		0			0
$94$	−644.811	+	1116.84i	−0.0729754	+	0.126397i
$95$	−913.285	+	933.198i	−0.101195	+	0.103401i
$96$		0			0
$97$	−11868.3	+	6852.15i	−1.26137	+	0.728255i	−0.973340	−	0.229365i	$-0.926335\pi$
$97$	−11868.3	+	6852.15i	−1.26137	+	0.728255i	−0.288034	+	0.957620i	$0.593002\pi$
$98$	−2210.10			−0.230123
$99$		0			0
$100$	−8109.73	−	4451.78i	−0.810973	−	0.445178i
$101$	−6147.82	+	3549.45i	−0.602668	+	0.347951i	−0.770091	−	0.637935i	$-0.779789\pi$
$101$	−6147.82	+	3549.45i	−0.602668	+	0.347951i	0.167422	+	0.985885i	$0.446456\pi$
$102$		0			0
$103$	4058.54	+	2343.20i	0.382557	+	0.220869i	0.678930	−	0.734203i	$-0.262444\pi$
$103$	4058.54	+	2343.20i	0.382557	+	0.220869i	−0.296373	+	0.955072i	$0.595777\pi$
$104$	−5351.25	−	3089.54i	−0.494753	−	0.285646i
$105$		0			0
$106$	35.2580	+	61.0686i	0.00313795	+	0.00543509i
$107$	20235.1			1.76741			0.883705	−	0.468044i	$-0.155041\pi$
$107$	20235.1			1.76741			0.883705	+	0.468044i	$0.155041\pi$
$108$		0			0
$109$	2264.65			0.190611			0.0953053	−	0.995448i	$-0.469617\pi$
$109$	2264.65			0.190611			0.0953053	+	0.995448i	$0.469617\pi$
$110$	3055.27	−	854.074i	0.252502	−	0.0705846i
$111$		0			0
$112$	3383.00	+	1953.17i	0.269690	+	0.155706i
$113$	5005.26	−	8669.36i	0.391985	−	0.678938i	−0.600726	−	0.799455i	$-0.705122\pi$
$113$	5005.26	−	8669.36i	0.391985	−	0.678938i	0.992711	+	0.120517i	$0.0384552\pi$
$114$		0			0
$115$	3122.98	+	11171.8i	0.236142	+	0.844749i
$116$		−	21457.4i		−	1.59463i
$117$		0			0
$118$		−	3586.50i		−	0.257577i
$119$	1506.33	−	869.680i	0.106372	−	0.0614138i
$120$		0			0
$121$	−599.679	+	1038.67i	−0.0409588	+	0.0709428i
$122$	−1017.25	+	1761.93i	−0.0683453	+	0.118378i
$123$		0			0
$124$	−11367.1	−	19688.4i	−0.739276	−	1.28046i
$125$	−11400.1	−	10685.4i	−0.729609	−	0.683865i
$126$		0			0
$127$			19581.7i			1.21406i	0.794677	+	0.607032i	$0.207640\pi$
$127$			19581.7i			1.21406i	−0.794677	+	0.607032i	$0.792360\pi$
$128$	−7362.40	−	12752.0i	−0.449365	−	0.778323i
$129$		0			0
$130$	−3584.29	−	3507.81i	−0.212088	−	0.207563i
$131$	24663.9	+	14239.7i	1.43721	+	0.829773i	0.997655	−	0.0684439i	$-0.0218034\pi$
$131$	24663.9	+	14239.7i	1.43721	+	0.829773i	0.439553	+	0.898216i	$0.355137\pi$
$132$		0			0
$133$	883.757	−	510.237i	0.0499608	−	0.0288449i
$134$			9172.14i			0.510812i
$135$		0			0
$136$	−3001.26			−0.162266
$137$	13118.2	+	22721.5i	0.698931	+	1.21058i	0.968837	+	0.247698i	$0.0796740\pi$
$137$	13118.2	+	22721.5i	0.698931	+	1.21058i	−0.269906	+	0.962887i	$0.586993\pi$
$138$		0			0
$139$	16654.1	−	28845.7i	0.861967	−	1.49297i	−0.00805999	−	0.999968i	$-0.502566\pi$
$139$	16654.1	−	28845.7i	0.861967	−	1.49297i	0.870027	−	0.493004i	$-0.164101\pi$
$140$	5167.33	+	5057.07i	0.263639	+	0.258014i
$141$		0			0
$142$	−4326.79	+	2498.07i	−0.214580	+	0.123888i
$143$	−21249.5			−1.03915
$144$		0			0
$145$	9001.71	−	35104.8i	0.428143	−	1.66967i
$146$	7656.72	−	4420.61i	0.359201	−	0.207385i
$147$		0			0
$148$	−8222.86	−	4747.47i	−0.375405	−	0.216740i
$149$	4558.66	+	2631.94i	0.205336	+	0.118551i	0.599142	−	0.800643i	$-0.295509\pi$
$149$	4558.66	+	2631.94i	0.205336	+	0.118551i	−0.393806	+	0.919194i	$0.628842\pi$
$150$		0			0
$151$	−5636.90	−	9763.41i	−0.247222	−	0.428201i	0.715532	−	0.698580i	$-0.246184\pi$
$151$	−5636.90	−	9763.41i	−0.247222	−	0.428201i	−0.962754	+	0.270379i	$0.912851\pi$
$152$	−1760.83			−0.0762131
$153$		0			0
$154$	−2479.33			−0.104542
$155$	−10337.3	−	36979.4i	−0.430271	−	1.53921i
$156$		0			0
$157$	17441.0	+	10069.5i	0.707573	+	0.408517i	0.810162	−	0.586207i	$-0.199379\pi$
$157$	17441.0	+	10069.5i	0.707573	+	0.408517i	−0.102589	+	0.994724i	$0.532713\pi$
$158$	−4737.82	+	8206.15i	−0.189786	+	0.328719i
$159$		0			0
$160$	−5103.35	−	18256.2i	−0.199350	−	0.713131i
$161$		−	9065.85i		−	0.349749i
$162$		0			0
$163$		−	15388.7i		−	0.579197i	−0.957148	−	0.289598i	$-0.906478\pi$
$163$		−	15388.7i		−	0.579197i	0.957148	−	0.289598i	$-0.0935218\pi$
$164$	17693.0	−	10215.1i	0.657830	−	0.379798i
$165$		0			0
$166$	1827.90	−	3166.02i	0.0663340	−	0.114894i
$167$	8523.96	−	14763.9i	0.305639	−	0.529382i	−0.671765	−	0.740765i	$-0.734463\pi$
$167$	8523.96	−	14763.9i	0.305639	−	0.529382i	0.977403	+	0.211383i	$0.0677967\pi$
$168$		0			0
$169$	2515.93	+	4357.72i	0.0880898	+	0.152576i
$170$	−2359.59	−	605.055i	−0.0816467	−	0.0209362i
$171$		0			0
$172$			30356.8i			1.02612i
$173$	9704.36	+	16808.4i	0.324246	+	0.561610i	0.981359	−	0.192181i	$-0.0615561\pi$
$173$	9704.36	+	16808.4i	0.324246	+	0.561610i	−0.657114	+	0.753792i	$0.728223\pi$
$174$		0			0
$175$	6332.36	+	10441.3i	0.206771	+	0.340940i
$176$	−20074.3	−	11589.9i	−0.648061	−	0.374158i
$177$		0			0
$178$	−9638.28	+	5564.66i	−0.304200	+	0.175630i
$179$			6659.35i			0.207838i	0.994586	+	0.103919i	$0.0331383\pi$
$179$			6659.35i			0.207838i	−0.994586	+	0.103919i	$0.966862\pi$
$180$		0			0
$181$	−15739.8			−0.480443			−0.240221	−	0.970718i	$-0.577220\pi$
$181$	−15739.8			−0.480443			−0.240221	+	0.970718i	$0.577220\pi$
$182$	1959.75	+	3394.39i	0.0591642	+	0.102475i
$183$		0			0
$184$	−7821.55	+	13547.3i	−0.231024	+	0.400145i
$185$	−11461.2	−	11216.6i	−0.334877	−	0.327731i
$186$		0			0
$187$	−8938.40	+	5160.59i	−0.255609	+	0.147576i
$188$	17440.6			0.493454
$189$		0			0
$190$	−1384.36	−	354.983i	−0.0383479	−	0.00983332i
$191$	−25333.0	+	14626.0i	−0.694417	+	0.400922i	−0.805265	−	0.592915i	$-0.797977\pi$
$191$	−25333.0	+	14626.0i	−0.694417	+	0.400922i	0.110847	+	0.993837i	$0.464643\pi$
$192$		0			0
$193$	−37232.0	−	21495.9i	−0.999543	−	0.577086i	−0.0914300	−	0.995812i	$-0.529144\pi$
$193$	−37232.0	−	21495.9i	−0.999543	−	0.577086i	−0.908113	+	0.418725i	$0.862477\pi$
$194$	−12990.0	−	7499.78i	−0.345148	−	0.199271i
$195$		0			0
$196$	14944.5	+	25884.7i	0.389019	+	0.673800i
$197$	48322.8			1.24514			0.622571	−	0.782563i	$-0.286088\pi$
$197$	48322.8			1.24514			0.622571	+	0.782563i	$0.286088\pi$
$198$		0			0
$199$	10246.4			0.258740			0.129370	−	0.991596i	$-0.458705\pi$
$199$	10246.4			0.258740			0.129370	+	0.991596i	$0.458705\pi$
$200$	−454.413	−	21065.9i	−0.0113603	−	0.526647i
$201$		0			0
$202$	−6728.88	−	3884.92i	−0.164907	−	0.0952093i
$203$	−14161.6	+	24528.5i	−0.343652	+	0.595223i
$204$		0			0
$205$	33231.6	−	9289.59i	0.790757	−	0.221049i
$206$			5129.34i			0.120872i
$207$		0			0
$208$			36644.4i			0.846996i
$209$	−5244.12	+	3027.69i	−0.120055	+	0.0693137i
$210$		0			0
$211$	−5566.99	+	9642.31i	−0.125042	+	0.216579i	−0.921749	−	0.387786i	$-0.873240\pi$
$211$	−5566.99	+	9642.31i	−0.125042	+	0.216579i	0.796707	+	0.604365i	$0.206573\pi$
$212$	476.823	−	825.882i	0.0106093	−	0.0183758i
$213$		0			0
$214$	11073.8	+	19180.4i	0.241807	+	0.418822i
$215$	−12735.2	+	49664.5i	−0.275504	+	1.07441i
$216$		0			0
$217$			30008.5i			0.637273i
$218$	1239.34	+	2146.61i	0.0260783	+	0.0451689i
$219$		0			0
$220$	−30662.4	−	30008.1i	−0.633520	−	0.620002i
$221$	14130.5	+	8158.24i	0.289316	+	0.167037i
$222$		0			0
$223$	41863.5	−	24169.9i	0.841833	−	0.486032i	−0.0160539	−	0.999871i	$-0.505110\pi$
$223$	41863.5	−	24169.9i	0.841833	−	0.486032i	0.857887	+	0.513839i	$0.171777\pi$
$224$			14814.7i			0.295256i
$225$		0			0
$226$	10956.7			0.214517
$227$	9830.70	+	17027.3i	0.190780	+	0.330441i	0.945509	−	0.325596i	$-0.105565\pi$
$227$	9830.70	+	17027.3i	0.190780	+	0.330441i	−0.754729	+	0.656037i	$0.772232\pi$
$228$		0			0
$229$	−41057.9	+	71114.4i	−0.782935	+	1.35608i	0.147290	+	0.989093i	$0.452945\pi$
$229$	−41057.9	+	71114.4i	−0.782935	+	1.35608i	−0.930225	+	0.366990i	$0.880388\pi$
$230$	−8880.43	+	9074.06i	−0.167872	+	0.171532i
$231$		0			0
$232$	42323.9	−	24435.7i	0.786340	−	0.453993i
$233$	−36907.6			−0.679837			−0.339918	−	0.940455i	$-0.610399\pi$
$233$	−36907.6			−0.679837			−0.339918	+	0.940455i	$0.610399\pi$
$234$		0			0
$235$	28533.3	+	7316.63i	0.516674	+	0.132488i
$236$	−42005.1	+	24251.7i	−0.754185	+	0.435429i
$237$		0			0
$238$	1648.70	+	951.878i	0.0291064	+	0.0168046i
$239$	−23140.0	−	13359.9i	−0.405105	−	0.233888i	0.283579	−	0.958949i	$-0.408478\pi$
$239$	−23140.0	−	13359.9i	−0.405105	−	0.233888i	−0.688684	+	0.725061i	$0.741811\pi$
$240$		0			0
$241$	−7637.82	−	13229.1i	−0.131503	−	0.227770i	0.792753	−	0.609543i	$-0.208647\pi$
$241$	−7637.82	−	13229.1i	−0.131503	−	0.227770i	−0.924256	+	0.381773i	$0.875314\pi$
$242$	−1312.71			−0.0224150
$243$		0			0
$244$	27514.3			0.462145
$245$	13590.6	+	48617.5i	0.226416	+	0.809955i
$246$		0			0
$247$	8290.29	+	4786.40i	0.135886	+	0.0784540i
$248$	25889.8	−	44842.5i	0.420946	−	0.729099i
$249$		0			0
$250$	3889.63	−	16653.6i	0.0622341	−	0.266457i
$251$		−	17504.6i		−	0.277847i	−0.990303	−	0.138923i	$-0.955636\pi$
$251$		−	17504.6i		−	0.277847i	0.990303	−	0.138923i	$-0.0443642\pi$
$252$		0			0
$253$			53795.8i			0.840441i
$254$	−18561.0	+	10716.2i	−0.287696	+	0.166101i
$255$		0			0
$256$	−10893.9	+	18868.8i	−0.166228	+	0.287915i
$257$	−28510.3	+	49381.3i	−0.431654	+	0.747647i	−0.997016	−	0.0771960i	$-0.975403\pi$
$257$	−28510.3	+	49381.3i	−0.431654	+	0.747647i	0.565362	+	0.824843i	$0.308737\pi$
$258$		0			0
$259$	6266.53	+	10854.0i	0.0934174	+	0.161804i
$260$	−16846.7	+	65698.7i	−0.249212	+	0.971874i
$261$		0			0
$262$			31171.2i			0.454099i
$263$	−29717.1	−	51471.5i	−0.429630	−	0.744142i	0.567210	−	0.823573i	$-0.308023\pi$
$263$	−29717.1	−	51471.5i	−0.429630	−	0.744142i	−0.996840	+	0.0794316i	$0.974689\pi$
$264$		0			0
$265$	1126.57	−	1151.13i	0.0160422	−	0.0163920i
$266$	967.285	+	558.462i	0.0136707	+	0.00789279i
$267$		0			0
$268$	107424.	−	62021.3i	1.49566	−	0.863518i
$269$		−	107707.i		−	1.48847i	−0.667919	−	0.744234i	$-0.732815\pi$
$269$		−	107707.i		−	1.48847i	0.667919	−	0.744234i	$-0.267185\pi$
$270$		0			0
$271$	−11971.8			−0.163012			−0.0815059	−	0.996673i	$-0.525973\pi$
$271$	−11971.8			−0.163012			−0.0815059	+	0.996673i	$0.525973\pi$
$272$	8899.33	+	15414.1i	0.120287	+	0.208344i
$273$		0			0
$274$	−14358.1	+	24869.0i	−0.191248	+	0.331251i
$275$	−37575.6	−	61957.4i	−0.496867	−	0.819271i
$276$		0			0
$277$	−23205.3	+	13397.6i	−0.302432	+	0.174609i	−0.643535	−	0.765417i	$-0.722533\pi$
$277$	−23205.3	+	13397.6i	−0.302432	+	0.174609i	0.341103	+	0.940026i	$0.389200\pi$
$278$	36456.2			0.471718
$279$		0			0
$280$	−4090.32	+	15951.4i	−0.0521725	+	0.203462i
$281$	−37711.0	+	21772.5i	−0.477591	+	0.275737i	−0.719412	−	0.694584i	$-0.755589\pi$
$281$	−37711.0	+	21772.5i	−0.477591	+	0.275737i	0.241821	+	0.970321i	$0.422255\pi$
$282$		0			0
$283$	−53849.1	−	31089.8i	−0.672366	−	0.388190i	0.124607	−	0.992206i	$-0.460233\pi$
$283$	−53849.1	−	31089.8i	−0.672366	−	0.388190i	−0.796972	+	0.604016i	$0.793566\pi$
$284$	58514.8	+	33783.5i	0.725486	+	0.418859i
$285$		0			0
$286$	−11629.0	−	20142.0i	−0.142170	−	0.246246i
$287$	−26967.2			−0.327395
$288$		0			0
$289$	−75595.9			−0.905112
$290$	38201.3	−	10678.8i	0.454236	−	0.126978i
$291$		0			0
$292$	−103548.	−	59783.6i	−1.21444	−	0.701159i
$293$	8608.02	−	14909.5i	0.100269	−	0.173672i	−0.811526	−	0.584316i	$-0.801363\pi$
$293$	8608.02	−	14909.5i	0.100269	−	0.173672i	0.911796	+	0.410644i	$0.134696\pi$
$294$		0			0
$295$	−78895.4	+	22054.5i	−0.906583	+	0.253427i
$296$		−	21625.8i		−	0.246825i
$297$		0			0
$298$			5761.40i			0.0648777i
$299$	73650.6	−	42522.2i	0.823822	−	0.475634i
$300$		0			0
$301$	20035.1	−	34701.8i	0.221135	−	0.383017i
$302$	6169.67	−	10686.2i	0.0676470	−	0.117168i
$303$		0			0
$304$	5221.20	+	9043.38i	0.0564967	+	0.0978551i
$305$	45014.1	+	11542.7i	0.483892	+	0.124082i
$306$		0			0
$307$		−	107387.i		−	1.13940i	−0.821854	−	0.569698i	$-0.807060\pi$
$307$		−	107387.i		−	1.13940i	0.821854	−	0.569698i	$-0.192940\pi$
$308$	16765.0	+	29037.9i	0.176727	+	0.306100i
$309$		0			0
$310$	29394.8	−	30035.7i	0.305877	−	0.312547i
$311$	118121.	+	68197.1i	1.22125	+	0.705091i	0.965185	−	0.261568i	$-0.0842395\pi$
$311$	118121.	+	68197.1i	1.22125	+	0.705091i	0.256068	+	0.966659i	$0.417573\pi$
$312$		0			0
$313$	−81900.2	+	47285.1i	−0.835981	+	0.482654i	−0.855896	−	0.517148i	$-0.826994\pi$
$313$	−81900.2	+	47285.1i	−0.835981	+	0.482654i	0.0199154	+	0.999802i	$0.493660\pi$
$314$			22042.5i			0.223564i
$315$		0			0
$316$	128147.			1.28332
$317$	−70757.2	−	122555.i	−0.704129	−	1.21959i	−0.967005	−	0.254758i	$-0.918004\pi$
$317$	−70757.2	−	122555.i	−0.704129	−	1.21959i	0.262876	−	0.964830i	$-0.415329\pi$
$318$		0			0
$319$	84033.2	−	145550.i	0.825790	−	1.43031i
$320$	−41424.7	+	42327.9i	−0.404538	+	0.413359i
$321$		0			0
$322$	8593.31	−	4961.35i	0.0828798	−	0.0478507i
$323$	4649.64			0.0445670
$324$		0			0
$325$	−55123.3	+	100417.i	−0.521878	+	0.950696i
$326$	14586.6	−	8421.57i	0.137252	−	0.0792424i
$327$		0			0
$328$	40297.7	+	23265.9i	0.374570	+	0.216258i
$329$	−19936.9	−	11510.6i	−0.184190	−	0.106342i
$330$		0			0
$331$	−41375.6	−	71664.7i	−0.377649	−	0.654108i	0.613071	−	0.790028i	$-0.289934\pi$
$331$	−41375.6	−	71664.7i	−0.377649	−	0.654108i	−0.990720	+	0.135921i	$0.956601\pi$
$332$	−49440.5			−0.448545
$333$		0			0
$334$	18659.2			0.167263
$335$	201767.	−	56402.3i	1.79788	−	0.502582i
$336$		0			0
$337$	72053.4	+	41600.0i	0.634446	+	0.366297i	0.782472	−	0.622686i	$-0.213959\pi$
$337$	72053.4	+	41600.0i	0.634446	+	0.366297i	−0.148026	+	0.988983i	$0.547292\pi$
$338$	−2753.72	+	4769.59i	−0.0241039	+	0.0417491i
$339$		0			0
$340$	8868.95	+	31726.8i	0.0767210	+	0.274453i
$341$		−	178067.i		−	1.53135i
$342$		0			0
$343$		−	86364.2i		−	0.734084i
$344$	−59877.8	+	34570.5i	−0.505998	+	0.292138i
$345$		0			0
$346$	−10621.6	+	18397.1i	−0.0887230	+	0.153673i
$347$	51814.1	−	89744.7i	0.430317	−	0.745332i	−0.566583	−	0.824005i	$-0.691735\pi$
$347$	51814.1	−	89744.7i	0.430317	−	0.745332i	0.996900	+	0.0786729i	$0.0250683\pi$
$348$		0			0
$349$	−75356.4	−	130521.i	−0.618685	−	1.07159i	−0.989726	−	0.142977i	$-0.954332\pi$
$349$	−75356.4	−	130521.i	−0.618685	−	1.07159i	0.371041	−	0.928616i	$-0.379001\pi$
$350$	−6431.61	+	11716.4i	−0.0525030	+	0.0956438i
$351$		0			0
$352$		−	87909.1i		−	0.709494i
$353$	11068.8	+	19171.8i	0.0888286	+	0.153856i	0.907016	−	0.421096i	$-0.138354\pi$
$353$	11068.8	+	19171.8i	0.0888286	+	0.153856i	−0.818188	+	0.574951i	$0.805021\pi$
$354$		0			0
$355$	81558.9	+	79818.6i	0.647165	+	0.633355i
$356$	130347.	+	75255.6i	1.02849	+	0.593798i
$357$		0			0
$358$	−6312.25	+	3644.38i	−0.0492513	+	0.0284353i
$359$			90926.7i			0.705509i	0.935716	+	0.352755i	$0.114755\pi$
$359$			90926.7i			0.705509i	−0.935716	+	0.352755i	$0.885245\pi$
$360$		0			0
$361$	−127593.			−0.979068
$362$	−8613.71	−	14919.4i	−0.0657314	−	0.113850i
$363$		0			0
$364$	26503.4	−	45905.2i	0.200032	−	0.346465i
$365$	−144327.	−	141248.i	−1.08334	−	1.06022i
$366$		0			0
$367$	17103.7	−	9874.80i	0.126986	−	0.0733156i	−0.435161	−	0.900353i	$-0.643309\pi$
$367$	17103.7	−	9874.80i	0.126986	−	0.0733156i	0.562148	+	0.827037i	$0.309975\pi$
$368$	92769.7			0.685032
$369$		0			0
$370$	4359.76	−	17002.2i	0.0318463	−	0.124194i
$371$	−1090.14	+	629.394i	−0.00792019	+	0.00457272i
$372$		0			0
$373$	94695.0	+	54672.2i	0.680627	+	0.392960i	0.800091	−	0.599878i	$-0.204784\pi$
$373$	94695.0	+	54672.2i	0.680627	+	0.392960i	−0.119464	+	0.992839i	$0.538118\pi$
$374$	−9783.21	−	5648.34i	−0.0699420	−	0.0403811i
$375$		0			0
$376$	19861.5	+	34401.1i	0.140487	+	0.243331i
$377$	−265692.			−1.86937
$378$		0			0
$379$	75560.0			0.526034			0.263017	−	0.964791i	$-0.415283\pi$
$379$	75560.0			0.526034			0.263017	+	0.964791i	$0.415283\pi$
$380$	5203.37	+	18614.0i	0.0360344	+	0.128906i
$381$		0			0
$382$	−27727.4	−	16008.4i	−0.190012	−	0.109704i
$383$	−108125.	+	187277.i	−0.737100	+	1.27670i	0.216695	+	0.976239i	$0.430472\pi$
$383$	−108125.	+	187277.i	−0.737100	+	1.27670i	−0.953796	+	0.300456i	$0.902861\pi$
$384$		0			0
$385$	15246.2	+	54539.9i	0.102858	+	0.367954i
$386$		−	47055.1i		−	0.315815i
$387$		0			0
$388$			202852.i			1.34746i
$389$	−48977.4	+	28277.1i	−0.323666	+	0.186868i	−0.653025	−	0.757336i	$-0.726500\pi$
$389$	−48977.4	+	28277.1i	−0.323666	+	0.186868i	0.329360	+	0.944205i	$0.393167\pi$
$390$		0			0
$391$	20653.6	−	35773.0i	0.135096	−	0.233993i
$392$	−34037.9	+	58955.3i	−0.221508	+	0.383664i
$393$		0			0
$394$	26445.0	+	45804.0i	0.170353	+	0.295061i
$395$	209652.	+	53759.8i	1.34371	+	0.344559i
$396$		0			0
$397$		−	254481.i		−	1.61464i	−0.590117	−	0.807318i	$-0.700918\pi$
$397$		−	254481.i		−	1.61464i	0.590117	−	0.807318i	$-0.299082\pi$
$398$	5607.39	+	9712.29i	0.0353993	+	0.0613134i
$399$		0			0
$400$	−106844.	+	64798.3i	−0.667777	+	0.404990i
$401$	−48697.9	−	28115.7i	−0.302846	−	0.174848i	0.340875	−	0.940109i	$-0.389277\pi$
$401$	−48697.9	−	28115.7i	−0.302846	−	0.174848i	−0.643721	+	0.765261i	$0.722610\pi$
$402$		0			0
$403$	−243788.	+	140751.i	−1.50107	+	0.866646i
$404$			105078.i			0.643797i
$405$		0			0
$406$	−31000.1			−0.188066
$407$	−37184.9	−	64406.2i	−0.224480	−	0.388811i
$408$		0			0
$409$	−51157.2	+	88606.9i	−0.305816	+	0.529689i	−0.977443	−	0.211201i	$-0.932263\pi$
$409$	−51157.2	+	88606.9i	−0.305816	+	0.529689i	0.671627	+	0.740890i	$0.265596\pi$
$410$	26991.6	+	26415.6i	0.160569	+	0.157142i
$411$		0			0
$412$	60074.7	−	34684.2i	0.353914	−	0.204332i
$413$	64023.0			0.375350
$414$		0			0
$415$	−80885.9	−	20741.1i	−0.469652	−	0.120430i
$416$	−120354.	+	69486.6i	−0.695465	+	0.401527i
$417$		0			0
$418$	−5739.76	−	3313.85i	−0.0328505	−	0.0189662i
$419$	50586.4	+	29206.1i	0.288142	+	0.166359i	0.637103	−	0.770778i	$-0.280132\pi$
$419$	50586.4	+	29206.1i	0.288142	+	0.166359i	−0.348962	+	0.937137i	$0.613466\pi$
$420$		0			0
$421$	117518.	+	203548.i	0.663043	+	1.14842i	0.979812	+	0.199921i	$0.0640685\pi$
$421$	117518.	+	203548.i	0.663043	+	1.14842i	−0.316770	+	0.948503i	$0.602598\pi$
$422$	−12186.3			−0.0684301
$423$		0			0
$424$	2172.04			0.0120819
$425$	1199.92	+	55626.6i	0.00664317	+	0.307967i
$426$		0			0
$427$	−31452.4	−	18159.1i	−0.172504	−	0.0995950i
$428$	149760.	−	259392.i	0.817540	−	1.41602i
$429$		0			0
$430$	−54045.3	+	15107.9i	−0.292295	+	0.0817084i
$431$		−	43601.5i		−	0.234718i	−0.993090	−	0.117359i	$-0.962557\pi$
$431$		−	43601.5i		−	0.234718i	0.993090	−	0.117359i	$-0.0374429\pi$
$432$		0			0
$433$		−	102452.i		−	0.546445i	−0.961951	−	0.273222i	$-0.911910\pi$
$433$		−	102452.i		−	0.546445i	0.961951	−	0.273222i	$-0.0880895\pi$
$434$	−28444.4	+	16422.4i	−0.151014	+	0.0871880i
$435$		0			0
$436$	16760.7	−	29030.4i	0.0881696	−	0.152714i
$437$	12117.3	−	20987.9i	0.0634519	−	0.109902i
$438$		0			0
$439$	51232.1	+	88736.5i	0.265835	+	0.460440i	0.967782	−	0.251789i	$-0.0810189\pi$
$439$	51232.1	+	88736.5i	0.265835	+	0.460440i	−0.701947	+	0.712229i	$0.747686\pi$
$440$	24271.5	−	94654.0i	0.125370	−	0.488915i
$441$		0			0
$442$			17858.6i			0.0914121i
$443$	123815.	+	214453.i	0.630906	+	1.09276i	0.987367	+	0.158451i	$0.0506500\pi$
$443$	123815.	+	214453.i	0.630906	+	1.09276i	−0.356461	+	0.934310i	$0.616017\pi$
$444$		0			0
$445$	181679.	+	177803.i	0.917457	+	0.897880i
$446$	45820.2	+	26454.3i	0.230350	+	0.132992i
$447$		0			0
$448$	40085.4	−	23143.3i	0.199724	−	0.115311i
$449$			79287.5i			0.393289i	0.980475	+	0.196645i	$0.0630045\pi$
$449$			79287.5i			0.393289i	−0.980475	+	0.196645i	$0.936995\pi$
$450$		0			0
$451$	160020.			0.786723
$452$	−74088.0	−	128324.i	−0.362636	−	0.628104i
$453$		0			0
$454$	−10759.8	+	18636.6i	−0.0522029	+	0.0904180i
$455$	62618.2	−	63983.5i	0.302467	−	0.309062i
$456$		0			0
$457$	101766.	−	58754.5i	0.487270	−	0.281325i	−0.236171	−	0.971711i	$-0.575893\pi$
$457$	101766.	−	58754.5i	0.487270	−	0.281325i	0.723441	+	0.690386i	$0.242559\pi$
$458$	−89876.9			−0.428467
$459$		0			0
$460$	166324.	+	42649.5i	0.786030	+	0.201557i
$461$	−317908.	+	183544.i	−1.49589	+	0.863653i	−0.999989	−	0.00472615i	$-0.998496\pi$
$461$	−317908.	+	183544.i	−1.49589	+	0.863653i	−0.495901	+	0.868379i	$0.665162\pi$
$462$		0			0
$463$	79996.4	+	46185.9i	0.373171	+	0.215451i	0.674843	−	0.737961i	$-0.264211\pi$
$463$	79996.4	+	46185.9i	0.373171	+	0.215451i	−0.301672	+	0.953412i	$0.597545\pi$
$464$	−250998.	−	144914.i	−1.16583	−	0.673090i
$465$		0			0
$466$	−20198.0	−	34983.9i	−0.0930114	−	0.161100i
$467$	82073.0			0.376328			0.188164	−	0.982138i	$-0.439746\pi$
$467$	82073.0			0.376328			0.188164	+	0.982138i	$0.439746\pi$
$468$		0			0
$469$	−163733.			−0.744372
$470$	8679.80	+	31050.2i	0.0392929	+	0.140562i
$471$		0			0
$472$	−95671.2	−	55235.8i	−0.429435	−	0.247934i
$473$	−118886.	+	205917.i	−0.531384	+	0.920384i
$474$		0			0
$475$	703.989	+	32635.8i	0.00312017	+	0.144646i
$476$		−	25746.1i		−	0.113631i
$477$		0			0
$478$		−	29245.2i		−	0.127997i
$479$	78567.0	−	45360.7i	0.342428	−	0.197701i	−0.318917	−	0.947783i	$-0.603319\pi$
$479$	78567.0	−	45360.7i	0.342428	−	0.197701i	0.661345	+	0.750082i	$0.269986\pi$
$480$		0			0
$481$	−58784.7	+	101818.i	−0.254082	+	0.440083i
$482$	8359.71	−	14479.4i	0.0359830	−	0.0623243i
$483$		0			0
$484$	8876.46	+	15374.5i	0.0378922	+	0.0656311i
$485$	−85099.6	+	331871.i	−0.361779	+	1.41086i
$486$		0			0
$487$		−	240214.i		−	1.01284i	−0.862287	−	0.506421i	$-0.830968\pi$
$487$		−	240214.i		−	1.01284i	0.862287	−	0.506421i	$-0.169032\pi$
$488$	31333.4	+	54271.1i	0.131573	+	0.227892i
$489$		0			0
$490$	−38645.9	+	39488.5i	−0.160958	+	0.164467i
$491$	198232.	+	114449.i	0.822264	+	0.474734i	0.851197	−	0.524847i	$-0.175877\pi$
$491$	198232.	+	114449.i	0.822264	+	0.474734i	−0.0289325	+	0.999581i	$0.509211\pi$
$492$		0			0
$493$	−111761.	+	64525.0i	−0.459827	+	0.265481i
$494$			10477.6i			0.0429345i
$495$		0			0
$496$	−307074.			−1.24819
$497$	−44593.3	−	77237.9i	−0.180533	−	0.312693i
$498$		0			0
$499$	−5232.25	+	9062.53i	−0.0210130	+	0.0363956i	−0.876341	−	0.481692i	$-0.840022\pi$
$499$	−5232.25	+	9062.53i	−0.0210130	+	0.0363956i	0.855328	+	0.518087i	$0.173356\pi$
$500$	−221348.	+	67054.9i	−0.885392	+	0.268220i
$501$		0			0
$502$	16592.2	−	9579.54i	0.0658412	−	0.0380134i
$503$	−87312.8			−0.345097			−0.172549	−	0.985001i	$-0.555200\pi$
$503$	−87312.8			−0.345097			−0.172549	+	0.985001i	$0.555200\pi$
$504$		0			0
$505$	−44081.9	+	171910.i	−0.172853	+	0.674092i
$506$	−50991.8	+	29440.1i	−0.199159	+	0.114984i
$507$		0			0
$508$	251016.	+	144924.i	0.972689	+	0.561582i
$509$	−211363.	−	122031.i	−0.815819	−	0.471013i	0.0331537	−	0.999450i	$-0.489445\pi$
$509$	−211363.	−	122031.i	−0.815819	−	0.471013i	−0.848972	+	0.528437i	$0.822778\pi$
$510$		0			0
$511$	78912.7	+	136681.i	0.302207	+	0.523439i
$512$	−259444.			−0.989700
$513$		0			0
$514$	−62409.9			−0.236226
$515$	112834.	−	31541.9i	0.425429	−	0.118925i
$516$		0			0
$517$	118303.	+	68302.6i	0.442605	+	0.255538i
$518$	−6858.81	+	11879.8i	−0.0255617	+	0.0442741i
$519$		0			0
$520$	−148774.	+	41588.4i	−0.550199	+	0.153803i
$521$		−	342391.i		−	1.26138i	−0.776034	−	0.630692i	$-0.782771\pi$
$521$		−	342391.i		−	1.26138i	0.776034	−	0.630692i	$-0.217229\pi$
$522$		0			0
$523$		−	313862.i		−	1.14746i	−0.819046	−	0.573728i	$-0.805497\pi$
$523$		−	313862.i		−	1.14746i	0.819046	−	0.573728i	$-0.194503\pi$
$524$	365076.	−	210777.i	1.32960	−	0.767645i
$525$		0			0
$526$	32525.8	−	56336.3i	0.117559	−	0.203618i
$527$	−68364.6	+	118411.i	−0.246156	+	0.426355i
$528$		0			0
$529$	32270.6	+	55894.3i	0.115318	+	0.199736i
$530$	1707.65	+	437.883i	0.00607922	+	0.00155886i
$531$		0			0
$532$		−	15105.1i		−	0.0533704i
$533$	−126486.	−	219080.i	−0.445234	−	0.771167i
$534$		0			0
$535$	353831.	−	361546.i	1.23620	−	1.26315i
$536$	244670.	+	141260.i	0.851631	+	0.491689i
$537$		0			0
$538$	102093.	−	58943.5i	0.352721	−	0.203644i
$539$			234109.i			0.805823i
$540$		0			0
$541$	81438.3			0.278249			0.139125	−	0.990275i	$-0.455571\pi$
$541$	81438.3			0.278249			0.139125	+	0.990275i	$0.455571\pi$
$542$	−6551.63	−	11347.8i	−0.0223024	−	0.0386288i
$543$		0			0
$544$	−33750.6	+	58457.7i	−0.114047	+	0.197535i
$545$	39599.6	−	40463.0i	0.133321	−	0.136228i
$546$		0			0
$547$	−345995.	+	199760.i	−1.15636	+	0.667627i	−0.950430	−	0.310939i	$-0.899357\pi$
$547$	−345995.	+	199760.i	−1.15636	+	0.667627i	−0.205934	+	0.978566i	$0.566023\pi$
$548$	388353.			1.29320
$549$		0			0
$550$	38164.5	−	69523.7i	0.126164	−	0.229830i
$551$	−65569.4	+	37856.5i	−0.215972	+	0.124692i
$552$		0			0
$553$	−146489.	−	84575.3i	−0.479020	−	0.276563i
$554$	−25398.6	−	14663.9i	−0.0827541	−	0.0477781i
$555$		0			0
$556$	−246514.	−	426975.i	−0.797430	−	1.38119i
$557$	464065.			1.49578			0.747892	−	0.663821i	$-0.231066\pi$
$557$	464065.			1.49578			0.747892	+	0.663821i	$0.231066\pi$
$558$		0			0
$559$	375888.			1.20291
$560$	94053.0	−	26291.7i	0.299914	−	0.0838383i
$561$		0			0
$562$	−41275.3	−	23830.3i	−0.130682	−	0.0754496i
$563$	282361.	−	489064.i	0.890816	−	1.54294i	0.0519165	−	0.998651i	$-0.483467\pi$
$563$	282361.	−	489064.i	0.890816	−	1.54294i	0.838899	−	0.544287i	$-0.183200\pi$
$564$		0			0
$565$	−67375.8	−	241023.i	−0.211061	−	0.755025i
$566$		−	68056.4i		−	0.212440i
$567$		0			0
$568$			153891.i			0.476999i
$569$	477380.	−	275615.i	1.47448	−	0.851292i	0.474895	−	0.880043i	$-0.342486\pi$
$569$	477380.	−	275615.i	1.47448	−	0.851292i	0.999587	+	0.0287503i	$0.00915278\pi$
$570$		0			0
$571$	107836.	−	186778.i	0.330744	−	0.572865i	−0.651914	−	0.758293i	$-0.726034\pi$
$571$	107836.	−	186778.i	0.330744	−	0.572865i	0.982658	+	0.185428i	$0.0593670\pi$
$572$	−157268.	+	272397.i	−0.480672	+	0.832549i
$573$		0			0
$574$	−14758.0	−	25561.6i	−0.0447923	−	0.0775825i
$575$	254218.	+	139551.i	0.768902	+	0.422083i
$576$		0			0
$577$			302487.i			0.908562i	0.890858	+	0.454281i	$0.150104\pi$
$577$			302487.i			0.908562i	−0.890858	+	0.454281i	$0.849896\pi$
$578$	−41370.4	−	71655.6i	−0.123832	−	0.214484i
$579$		0			0
$580$	−383384.	−	375203.i	−1.13967	−	1.11535i
$581$	56516.9	+	32630.0i	0.167427	+	0.0966641i
$582$		0			0
$583$	6468.79	−	3734.76i	0.0190321	−	0.0109882i
$584$		−	272327.i		−	0.798483i
$585$		0			0
$586$	18843.2			0.0548731
$587$	67877.0	+	117566.i	0.196991	+	0.341198i	0.947551	−	0.319603i	$-0.103550\pi$
$587$	67877.0	+	117566.i	0.196991	+	0.341198i	−0.750560	+	0.660802i	$0.770216\pi$
$588$		0			0
$589$	−40109.2	+	69471.2i	−0.115615	+	0.200251i
$590$	−64081.0	−	62713.7i	−0.184088	−	0.180160i
$591$		0			0
$592$	−111067.	+	64124.7i	−0.316915	+	0.182971i
$593$	−612873.			−1.74285			−0.871427	−	0.490525i	$-0.836805\pi$
$593$	−612873.			−1.74285			−0.871427	+	0.490525i	$0.836805\pi$
$594$		0			0
$595$	10800.9	−	42121.3i	0.0305089	−	0.118978i
$596$	67477.4	−	38958.1i	0.189962	−	0.109674i
$597$		0			0
$598$	80611.6	+	46541.1i	0.225421	+	0.130147i
$599$	152718.	+	88171.6i	0.425634	+	0.245740i	0.697485	−	0.716600i	$-0.254303\pi$
$599$	152718.	+	88171.6i	0.425634	+	0.245740i	−0.271851	+	0.962339i	$0.587636\pi$
$600$		0			0
$601$	−55051.5	−	95352.0i	−0.152412	−	0.263986i	0.779701	−	0.626152i	$-0.215371\pi$
$601$	−55051.5	−	95352.0i	−0.152412	−	0.263986i	−0.932114	+	0.362165i	$0.882038\pi$
$602$	43857.4			0.121018
$603$		0			0
$604$	−166875.			−0.457423
$605$	8072.28	+	28876.9i	0.0220539	+	0.0788932i
$606$		0			0
$607$	189924.	+	109653.i	0.515470	+	0.297607i	0.735079	−	0.677981i	$-0.237145\pi$
$607$	189924.	+	109653.i	0.515470	+	0.297607i	−0.219609	+	0.975588i	$0.570478\pi$
$608$	−19801.3	+	34296.8i	−0.0535656	+	0.0927784i
$609$		0			0
$610$	13693.2	+	48984.7i	0.0367999	+	0.131644i
$611$		−	215955.i		−	0.578471i
$612$		0			0
$613$			75985.1i			0.202212i	0.994876	+	0.101106i	$0.0322382\pi$
$613$			75985.1i			0.202212i	−0.994876	+	0.101106i	$0.967762\pi$
$614$	101790.	−	58768.3i	0.270002	−	0.155886i
$615$		0			0
$616$	−38184.2	+	66137.0i	−0.100629	+	0.174294i
$617$	242066.	−	419271.i	0.635863	−	1.10135i	−0.350468	−	0.936575i	$-0.613977\pi$
$617$	242066.	−	419271.i	0.635863	−	1.10135i	0.986331	−	0.164773i	$-0.0526892\pi$
$618$		0			0
$619$	−23051.5	−	39926.4i	−0.0601614	−	0.104203i	0.834376	−	0.551196i	$-0.185828\pi$
$619$	−23051.5	−	39926.4i	−0.0601614	−	0.104203i	−0.894537	+	0.446993i	$0.852495\pi$
$620$	−550543.	−	141173.i	−1.43221	−	0.367254i
$621$		0			0
$622$			149285.i			0.385866i
$623$	−99335.4	−	172054.i	−0.255934	−	0.443291i
$624$		0			0
$625$	−390262.	+	16844.5i	−0.999070	+	0.0431220i
$626$	−89640.9	−	51754.2i	−0.228748	−	0.132068i
$627$		0			0
$628$	258162.	−	149050.i	0.654595	−	0.377930i
$629$			57105.0i			0.144335i
$630$		0			0
$631$	240096.			0.603013			0.301507	−	0.953464i	$-0.402510\pi$
$631$	240096.			0.603013			0.301507	+	0.953464i	$0.402510\pi$
$632$	145935.	+	252766.i	0.365362	+	0.632826i
$633$		0			0
$634$	77444.8	−	134138.i	0.192670	−	0.333714i
$635$	349871.	+	342405.i	0.867681	+	0.849166i
$636$		0			0
$637$	320513.	−	185048.i	0.789890	−	0.456043i
$638$	183951.			0.451919
$639$		0			0
$640$	−356583.	−	91436.5i	−0.870565	−	0.223234i
$641$	−480663.	+	277511.i	−1.16984	+	0.675405i	−0.953641	−	0.300945i	$-0.902698\pi$
$641$	−480663.	+	277511.i	−1.16984	+	0.675405i	−0.216195	+	0.976350i	$0.569365\pi$
$642$		0			0
$643$	−560077.	−	323360.i	−1.35465	−	0.782105i	−0.365749	−	0.930713i	$-0.619187\pi$
$643$	−560077.	−	323360.i	−1.35465	−	0.782105i	−0.988896	+	0.148609i	$0.952521\pi$
$644$	−116215.	−	67096.5i	−0.280213	−	0.161781i
$645$		0			0
$646$	2544.55	+	4407.28i	0.00609741	+	0.0105610i
$647$	−347688.			−0.830579			−0.415289	−	0.909689i	$-0.636320\pi$
$647$	−347688.			−0.830579			−0.415289	+	0.909689i	$0.636320\pi$
$648$		0			0
$649$	−379906.			−0.901959
$650$	−125350.	+	2703.93i	−0.296686	+	0.00639983i
$651$		0			0
$652$	−197267.	−	113892.i	−0.464043	−	0.267915i
$653$	−304539.	+	527478.i	−0.714196	+	1.23702i	0.249073	+	0.968485i	$0.419874\pi$
$653$	−304539.	+	527478.i	−0.714196	+	1.23702i	−0.963269	+	0.268538i	$0.913459\pi$
$654$		0			0
$655$	685699.	−	191681.i	1.59827	−	0.446783i
$656$		−	275952.i		−	0.641247i
$657$		0			0
$658$		−	25197.0i		−	0.0581965i
$659$	−540048.	+	311797.i	−1.24354	+	0.717961i	−0.969814	−	0.243845i	$-0.921591\pi$
$659$	−540048.	+	311797.i	−1.24354	+	0.717961i	−0.273731	+	0.961806i	$0.588258\pi$
$660$		0			0
$661$	−184642.	+	319810.i	−0.422598	+	0.731962i	−0.996193	−	0.0871775i	$-0.972215\pi$
$661$	−184642.	+	319810.i	−0.422598	+	0.731962i	0.573594	+	0.819140i	$0.305549\pi$
$662$	45286.2	−	78438.0i	0.103336	−	0.178983i
$663$		0			0
$664$	−56303.1	−	97519.8i	−0.127701	−	0.221185i
$665$	6336.84	−	24712.3i	0.0143294	−	0.0558818i
$666$		0			0
$667$			672630.i			1.51191i
$668$	−126172.	−	218536.i	−0.282755	−	0.489746i
$669$		0			0
$670$	163881.	+	160384.i	0.365073	+	0.357283i
$671$	186635.	+	107754.i	0.414523	+	0.239325i
$672$		0			0
$673$	130097.	−	75111.4i	0.287234	−	0.165835i	−0.349460	−	0.936951i	$-0.613635\pi$
$673$	130097.	−	75111.4i	0.287234	−	0.165835i	0.636694	+	0.771117i	$0.280301\pi$
$674$			91063.7i			0.200459i
$675$		0			0
$676$	74481.8			0.162988
$677$	290147.	+	502550.i	0.633055	+	1.09648i	0.986924	+	0.161188i	$0.0515326\pi$
$677$	290147.	+	502550.i	0.633055	+	1.09648i	−0.353869	+	0.935295i	$0.615134\pi$
$678$		0			0
$679$	133879.	−	231886.i	0.290385	−	0.502961i
$680$	−52480.1	+	53624.4i	−0.113495	+	0.115970i
$681$		0			0
$682$	168786.	−	97448.7i	0.362884	−	0.209511i
$683$	−11404.6			−0.0244477			−0.0122238	−	0.999925i	$-0.503891\pi$
$683$	−11404.6			−0.0244477			−0.0122238	+	0.999925i	$0.503891\pi$
$684$		0			0
$685$	635356.	+	162921.i	1.35405	+	0.347212i
$686$	81862.7	−	47263.4i	0.173955	−	0.100433i
$687$		0			0
$688$	355099.	+	205017.i	0.750192	+	0.433124i
$689$	−10226.3	−	5904.18i	−0.0215418	−	0.0124372i
$690$		0			0
$691$	50403.6	+	87301.6i	0.105562	+	0.182838i	0.913967	−	0.405787i	$-0.133003\pi$
$691$	50403.6	+	87301.6i	0.105562	+	0.182838i	−0.808406	+	0.588625i	$0.799669\pi$
$692$	287289.			0.599938
$693$		0			0
$694$	113423.			0.235494
$695$	−224181.	−	801959.i	−0.464118	−	1.66028i
$696$		0			0
$697$	−106410.	−	61435.9i	−0.219037	−	0.126461i
$698$	82478.7	−	142857.i	0.169290	−	0.293219i
$699$		0			0
$700$	180712.	−	3898.15i	0.368800	−	0.00795540i
$701$			220659.i			0.449040i	0.974469	+	0.224520i	$0.0720814\pi$
$701$			220659.i			0.449040i	−0.974469	+	0.224520i	$0.927919\pi$
$702$		0			0
$703$			33503.2i			0.0677916i
$704$	−237862.	+	137330.i	−0.479933	+	0.277089i
$705$		0			0
$706$	−12115.0	+	20983.8i	−0.0243060	+	0.0420993i
$707$	69350.1	−	120118.i	0.138742	−	0.240308i
$708$		0			0
$709$	64731.1	+	112118.i	0.128772	+	0.223039i	0.923201	−	0.384318i	$-0.125563\pi$
$709$	64731.1	+	112118.i	0.128772	+	0.223039i	−0.794429	+	0.607357i	$0.792230\pi$
$710$	−31024.5	+	120989.i	−0.0615444	+	0.240010i
$711$		0			0
$712$			342806.i			0.676221i
$713$	356328.	+	617179.i	0.700924	+	1.21404i
$714$		0			0
$715$	−371570.	+	379671.i	−0.726823	+	0.742670i
$716$	85365.8	+	49286.0i	0.166517	+	0.0961385i
$717$		0			0
$718$	−86187.4	+	49760.3i	−0.167184	+	0.0965238i
$719$			354459.i			0.685659i	0.939398	+	0.342829i	$0.111385\pi$
$719$			354459.i			0.685659i	−0.939398	+	0.342829i	$0.888615\pi$
$720$		0			0
$721$	−91564.3			−0.176139
$722$	−69826.2	−	120943.i	−0.133950	−	0.232009i
$723$		0			0
$724$	−116490.	+	201767.i	−0.222235	+	0.384923i
$725$	−469823.	−	774679.i	−0.893836	−	1.47382i
$726$		0			0
$727$	173161.	−	99974.6i	0.327628	−	0.189156i	−0.327159	−	0.944969i	$-0.606091\pi$
$727$	173161.	−	99974.6i	0.327628	−	0.189156i	0.654788	+	0.755813i	$0.272758\pi$
$728$	120729.			0.227797
$729$		0			0
$730$	54901.3	−	214103.i	0.103024	−	0.401770i
$731$	158113.	−	91286.7i	0.295892	−	0.170833i
$732$		0			0
$733$	−746774.	−	431150.i	−1.38989	−	0.802455i	−0.396590	−	0.917996i	$-0.629807\pi$
$733$	−746774.	−	431150.i	−1.38989	−	0.802455i	−0.993303	+	0.115541i	$0.963140\pi$
$734$	18720.2	+	10808.1i	0.0347471	+	0.0200612i
$735$		0			0
$736$	175914.	+	304691.i	0.324746	+	0.562477i
$737$	971573.			1.78871
$738$		0			0
$739$	684605.			1.25358			0.626789	−	0.779189i	$-0.284369\pi$
$739$	684605.			1.25358			0.626789	+	0.779189i	$0.284369\pi$
$740$	−228609.	+	63905.8i	−0.417475	+	0.116702i
$741$		0			0
$742$	−1193.18	−	688.881i	−0.00216719	−	0.00125123i
$743$	−400928.	+	694427.i	−0.726254	+	1.25791i	0.232202	+	0.972668i	$0.425407\pi$
$743$	−400928.	+	694427.i	−0.726254	+	1.25791i	−0.958456	+	0.285241i	$0.907926\pi$
$744$		0			0
$745$	126738.	−	35428.6i	0.228347	−	0.0638324i
$746$			119679.i			0.215050i
$747$		0			0
$748$			152775.i			0.273053i
$749$	−342391.	+	197679.i	−0.610321	+	0.352369i
$750$		0			0
$751$	−254539.	+	440875.i	−0.451310	+	0.781692i	−0.998468	−	0.0553377i	$-0.982376\pi$
$751$	−254539.	+	440875.i	−0.451310	+	0.781692i	0.547158	+	0.837030i	$0.315710\pi$
$752$	117786.	−	204012.i	0.208286	−	0.360761i
$753$		0			0
$754$	−145402.	−	251843.i	−0.255757	−	0.442983i
$755$	−273012.	−	70006.9i	−0.478948	−	0.122814i
$756$		0			0
$757$			766769.i			1.33805i	0.743239	+	0.669026i	$0.233288\pi$
$757$			766769.i			1.33805i	−0.743239	+	0.669026i	$0.766712\pi$
$758$	41350.8	+	71621.6i	0.0719689	+	0.124654i
$759$		0			0
$760$	−30789.8	+	31461.2i	−0.0533065	+	0.0544688i
$761$	−283879.	−	163898.i	−0.490190	−	0.283011i	0.234463	−	0.972125i	$-0.424667\pi$
$761$	−283879.	−	163898.i	−0.490190	−	0.283011i	−0.724653	+	0.689114i	$0.758000\pi$
$762$		0			0
$763$	−38319.3	+	22123.6i	−0.0658216	+	0.0380021i
$764$			432990.i			0.741808i
$765$		0			0
$766$	−236688.			−0.403384
$767$	300292.	+	520120.i	0.510449	+	0.884124i
$768$		0			0
$769$	−197438.	+	341972.i	−0.333870	+	0.578280i	−0.983267	−	0.182170i	$-0.941688\pi$
$769$	−197438.	+	341972.i	−0.333870	+	0.578280i	0.649397	+	0.760449i	$0.275021\pi$
$770$	−43353.6	+	44298.8i	−0.0731212	+	0.0747155i
$771$		0			0
$772$	−551109.	+	318183.i	−0.924705	+	0.533878i
$773$	−121076.			−0.202628			−0.101314	−	0.994855i	$-0.532305\pi$
$773$	−121076.			−0.202628			−0.101314	+	0.994855i	$0.532305\pi$
$774$		0			0
$775$	−841479.	−	461924.i	−1.40101	−	0.769072i
$776$	−400118.	+	231008.i	−0.664454	+	0.383623i
$777$		0			0
$778$	−53606.5	−	30949.7i	−0.0885642	−	0.0511326i
$779$	−62430.3	−	36044.1i	−0.102878	−	0.0593964i
$780$		0			0
$781$	264612.	+	458321.i	0.433818	+	0.751395i
$782$	45211.3			0.0739321
$783$		0			0
$784$	403716.			0.656816
$785$	484888.	−	135546.i	0.786868	−	0.219962i
$786$		0			0
$787$	523274.	+	302112.i	0.844849	+	0.487774i	0.858910	−	0.512127i	$-0.171142\pi$
$787$	523274.	+	302112.i	0.844849	+	0.487774i	−0.0140603	+	0.999901i	$0.504476\pi$
$788$	357638.	−	619446.i	0.575958	−	0.997588i
$789$		0			0
$790$	63775.9	+	228145.i	0.102189	+	0.365558i
$791$			195588.i			0.312601i
$792$		0			0
$793$		−	340691.i		−	0.541769i
$794$	241217.	−	139267.i	0.382619	−	0.220905i
$795$		0			0
$796$	75833.5	−	131347.i	0.119684	−	0.207298i
$797$	−444851.	+	770505.i	−0.700322	+	1.21299i	0.268031	+	0.963410i	$0.413627\pi$
$797$	−444851.	+	770505.i	−0.700322	+	1.21299i	−0.968353	+	0.249584i	$0.919706\pi$
$798$		0			0
$799$	−52446.2	−	90839.5i	−0.0821524	−	0.142292i
$800$	−415425.	−	228045.i	−0.649102	−	0.356320i
$801$		0			0
$802$		−	61546.2i		−	0.0956868i
$803$	−468260.	−	811049.i	−0.726199	−	1.25781i
$804$		0			0
$805$	−161982.	−	158525.i	−0.249962	−	0.244629i
$806$	−266830.	−	154054.i	−0.410737	−	0.237139i
$807$		0			0
$808$	−207263.	+	119663.i	−0.317468	+	0.183290i
$809$		−	351052.i		−	0.536382i	−0.963366	−	0.268191i	$-0.913574\pi$
$809$		−	351052.i		−	0.536382i	0.963366	−	0.268191i	$-0.0864258\pi$
$810$		0			0
$811$	−316054.			−0.480529			−0.240264	−	0.970707i	$-0.577234\pi$
$811$	−316054.			−0.480529			−0.240264	+	0.970707i	$0.577234\pi$
$812$	209620.	+	363072.i	0.317922	+	0.550657i
$813$		0			0
$814$	40699.4	−	70493.5i	0.0614242	−	0.106390i
$815$	−274954.	−	269087.i	−0.413947	−	0.405114i
$816$		0			0
$817$	92764.3	−	53557.5i	0.138975	−	0.0802373i
$818$	−111985.			−0.167360
$819$		0			0
$820$	126865.	−	494746.i	0.188674	−	0.735791i
$821$	853848.	−	492970.i	1.26676	−	0.731364i	0.292387	−	0.956300i	$-0.405551\pi$
$821$	853848.	−	492970.i	1.26676	−	0.731364i	0.974374	+	0.224936i	$0.0722172\pi$
$822$		0			0
$823$	547461.	+	316077.i	0.808264	+	0.466651i	0.846353	−	0.532623i	$-0.178794\pi$
$823$	547461.	+	316077.i	0.808264	+	0.466651i	−0.0380887	+	0.999274i	$0.512127\pi$
$824$	136827.	+	78997.0i	0.201519	+	0.116347i
$825$		0			0
$826$	35037.1	+	60686.0i	0.0513532	+	0.0889464i
$827$	309027.			0.451840			0.225920	−	0.974146i	$-0.427461\pi$
$827$	309027.			0.451840			0.225920	+	0.974146i	$0.427461\pi$
$828$		0			0
$829$	−832364.			−1.21117			−0.605584	−	0.795782i	$-0.707060\pi$
$829$	−832364.			−1.21117			−0.605584	+	0.795782i	$0.707060\pi$
$830$	−24605.4	−	88020.6i	−0.0357169	−	0.127770i
$831$		0			0
$832$	376030.	+	217101.i	0.543221	+	0.313629i
$833$	89880.3	−	155677.i	0.129531	−	0.224355i
$834$		0			0
$835$	−114741.	−	410462.i	−0.164568	−	0.588709i
$836$			89632.1i			0.128248i
$837$		0			0
$838$			63933.0i			0.0910410i
$839$	−718506.	+	414830.i	−1.02072	+	0.589313i	−0.914312	−	0.405010i	$-0.867268\pi$
$839$	−718506.	+	414830.i	−1.02072	+	0.589313i	−0.106407	+	0.994323i	$0.533935\pi$
$840$		0			0
$841$	697061.	−	1.20734e6i	0.985550	−	1.70702i
$842$	−128626.	+	222786.i	−0.181427	+	0.314241i
$843$		0			0
$844$	82402.8	+	142726.i	0.115680	+	0.200363i
$845$	121854.	+	31246.3i	0.170658	+	0.0437608i
$846$		0			0
$847$		−	23433.4i		−	0.0326639i
$848$	−6440.51	−	11155.3i	−0.00895630	−	0.0155128i
$849$		0			0
$850$	−52070.5	+	31579.4i	−0.0720699	+	0.0437085i
$851$	257765.	+	148820.i	0.355930	+	0.205496i
$852$		0			0
$853$	607611.	−	350804.i	0.835079	−	0.482133i	−0.0205095	−	0.999790i	$-0.506529\pi$
$853$	607611.	−	350804.i	0.835079	−	0.482133i	0.855588	+	0.517657i	$0.173195\pi$
$854$		−	39750.7i		−	0.0545041i
$855$		0			0
$856$	682191.			0.931019
$857$	−542612.	−	939831.i	−0.738801	−	1.27964i	−0.953035	−	0.302859i	$-0.902059\pi$
$857$	−542612.	−	939831.i	−0.738801	−	1.27964i	0.214234	−	0.976782i	$-0.431274\pi$
$858$		0			0
$859$	442391.	−	766243.i	0.599542	−	1.03844i	−0.393347	−	0.919390i	$-0.628683\pi$
$859$	442391.	−	766243.i	0.599542	−	1.03844i	0.992889	−	0.119047i	$-0.0379839\pi$
$860$	542393.	+	530819.i	0.733360	+	0.717711i
$861$		0			0
$862$	41328.9	−	23861.2i	0.0556210	−	0.0321128i
$863$	−848540.			−1.13933			−0.569667	−	0.821876i	$-0.692928\pi$
$863$	−848540.			−1.13933			−0.569667	+	0.821876i	$0.692928\pi$
$864$		0			0
$865$	470011.	+	120522.i	0.628169	+	0.161077i
$866$	97112.3	−	56067.8i	0.129491	−	0.0747615i
$867$		0			0
$868$	384678.	+	222094.i	0.510573	+	0.294779i
$869$	869248.	+	501861.i	1.15108	+	0.664575i
$870$		0			0
$871$	−767967.	−	1.33016e6i	−1.01229	−	1.75334i
$872$	76348.6			0.100408
$873$		0			0
$874$	26525.2			0.0347245
$875$	297285.	+	69434.2i	0.388290	+	0.0906895i
$876$		0			0
$877$	−445046.	−	256947.i	−0.578636	−	0.334076i	0.181955	−	0.983307i	$-0.441757\pi$
$877$	−445046.	−	256947.i	−0.578636	−	0.334076i	−0.760591	+	0.649231i	$0.775091\pi$
$878$	−56074.2	+	97123.4i	−0.0727402	+	0.125990i
$879$		0			0
$880$	−558100.	+	156012.i	−0.720687	+	0.201462i
$881$			222212.i			0.286297i	0.989701	+	0.143148i	$0.0457226\pi$
$881$			222212.i			0.286297i	−0.989701	+	0.143148i	$0.954277\pi$
$882$		0			0
$883$		−	248366.i		−	0.318545i	−0.987235	−	0.159272i	$-0.949085\pi$
$883$		−	248366.i		−	0.318545i	0.987235	−	0.159272i	$-0.0509148\pi$
$884$	209160.	−	120759.i	0.267654	−	0.154530i
$885$		0			0
$886$	−135517.	+	234722.i	−0.172634	+	0.299011i
$887$	432263.	−	748702.i	0.549415	−	0.951616i	−0.448899	−	0.893582i	$-0.648184\pi$
$887$	432263.	−	748702.i	0.549415	−	0.951616i	0.998315	−	0.0580331i	$-0.0184829\pi$
$888$		0			0
$889$	−191296.	−	331334.i	−0.242048	−	0.419240i
$890$	−69109.7	+	269514.i	−0.0872488	+	0.340252i
$891$		0			0
$892$		−	715528.i		−	0.899284i
$893$	−30769.9	−	53295.1i	−0.0385855	−	0.0668320i
$894$		0			0
$895$	118984.	+	116445.i	0.148540	+	0.145371i
$896$	249153.	+	143849.i	0.310349	+	0.179180i
$897$		0			0
$898$	−75154.8	+	43390.7i	−0.0931975	+	0.0538076i
$899$		−	2.22645e6i		−	2.75482i
$900$		0			0
$901$	−5735.47			−0.00706512
$902$	87572.3	+	151680.i	0.107635	+	0.186429i
$903$		0			0
$904$	168744.	−	292272.i	0.206486	−	0.357644i
$905$	−275226.	+	281227.i	−0.336041	+	0.343368i
$906$		0			0
$907$	−929719.	+	536773.i	−1.13015	+	0.652494i	−0.943973	−	0.330022i	$-0.892944\pi$
$907$	−929719.	+	536773.i	−1.13015	+	0.652494i	−0.186179	+	0.982516i	$0.559610\pi$
$908$	291029.			0.352992
$909$		0			0
$910$	94916.8	+	24338.9i	0.114620	+	0.0293913i
$911$	872700.	−	503854.i	1.05155	−	0.607110i	0.128464	−	0.991714i	$-0.458995\pi$
$911$	872700.	−	503854.i	1.05155	−	0.607110i	0.923082	+	0.384604i	$0.125662\pi$
$912$		0			0
$913$	−335365.	−	193623.i	−0.402324	−	0.232282i
$914$	111384.	+	64307.7i	0.133331	+	0.0769787i
$915$		0			0
$916$	607741.	+	1.05264e6i	0.724315	+	1.25455i
$917$	−556440.			−0.661728
$918$		0			0
$919$	−576271.			−0.682332			−0.341166	−	0.940003i	$-0.610822\pi$
$919$	−576271.			−0.682332			−0.341166	+	0.940003i	$0.610822\pi$
$920$	105286.	+	376638.i	0.124393	+	0.444989i
$921$		0			0
$922$	−347955.	−	200892.i	−0.409318	−	0.236320i
$923$	418318.	−	724549.i	0.491025	−	0.850480i
$924$		0			0
$925$	−400820.	+	8646.11i	−0.468453	+	0.0101050i
$926$			101102.i			0.117907i
$927$		0			0
$928$		−	1.09916e6i		−	1.27634i
$929$	889734.	−	513688.i	1.03093	−	0.595207i	0.113679	−	0.993518i	$-0.463737\pi$
$929$	889734.	−	513688.i	1.03093	−	0.595207i	0.917251	+	0.398310i	$0.130403\pi$
$930$		0			0
$931$	52732.3	−	91335.1i	0.0608384	−	0.105375i
$932$	−273154.	+	473117.i	−0.314468	+	0.544674i
$933$		0			0
$934$	44915.0	+	77795.1i	0.0514870	+	0.0891782i
$935$	−64091.4	+	249943.i	−0.0733122	+	0.285902i
$936$		0			0
$937$			358312.i			0.408115i	0.978959	+	0.204057i	$0.0654129\pi$
$937$			358312.i			0.408115i	−0.978959	+	0.204057i	$0.934587\pi$
$938$	−89604.0	−	155199.i	−0.101841	−	0.176393i
$939$		0			0
$940$	304967.	−	311617.i	0.345142	−	0.352667i
$941$	−634781.	−	366491.i	−0.716877	−	0.413889i	0.0967250	−	0.995311i	$-0.469163\pi$
$941$	−634781.	−	366491.i	−0.716877	−	0.413889i	−0.813602	+	0.581422i	$0.802497\pi$
$942$		0			0
$943$	−554628.	+	320214.i	−0.623703	+	0.360095i
$944$			655140.i			0.735174i
$945$		0			0
$946$	−260245.			−0.290804
$947$	361257.	+	625716.i	0.402825	+	0.697713i	0.994066	−	0.108782i	$-0.0346950\pi$
$947$	361257.	+	625716.i	0.402825	+	0.697713i	−0.591241	+	0.806495i	$0.701362\pi$
$948$		0			0
$949$	−740260.	+	1.28217e6i	−0.821962	+	1.42368i
$950$	−30549.5	+	18527.5i	−0.0338499	+	0.0205291i
$951$		0			0
$952$	50783.3	−	29319.8i	0.0560335	−	0.0323509i
$953$	686812.			0.756227			0.378113	−	0.925759i	$-0.376573\pi$
$953$	686812.			0.756227			0.378113	+	0.925759i	$0.376573\pi$
$954$		0			0
$955$	−181646.	+	708383.i	−0.199168	+	0.776715i
$956$	−342519.	+	197754.i	−0.374774	+	0.216376i
$957$		0			0
$958$	85992.7	+	49647.9i	0.0936980	+	0.0540966i
$959$	−443938.	−	256308.i	−0.482709	−	0.278692i
$960$		0			0
$961$	−717709.	−	1.24311e6i	−0.777144	−	1.34605i
$962$	−128681.			−0.139048
$963$		0			0
$964$	−226111.			−0.243314
$965$	−1.03511e6	+	289356.i	−1.11156	+	0.310727i
$966$		0			0
$967$	217686.	+	125681.i	0.232797	+	0.134406i	0.611862	−	0.790965i	$-0.290421\pi$
$967$	217686.	+	125681.i	0.232797	+	0.134406i	−0.379064	+	0.925370i	$0.623754\pi$
$968$	−20217.1	+	35017.1i	−0.0215759	+	0.0373705i
$969$		0			0
$970$	−361144.	+	100955.i	−0.383828	+	0.107296i
$971$			299319.i			0.317465i	0.987322	+	0.158733i	$0.0507408\pi$
$971$			299319.i			0.317465i	−0.987322	+	0.158733i	$0.949259\pi$
$972$		0			0
$973$			650784.i			0.687403i
$974$	227694.	−	131459.i	0.240012	−	0.138571i
$975$		0			0
$976$	185819.	−	321849.i	0.195070	−	0.337872i
$977$	352133.	−	609912.i	0.368907	−	0.638966i	−0.620488	−	0.784216i	$-0.713065\pi$
$977$	352133.	−	609912.i	0.368907	−	0.638966i	0.989395	+	0.145250i	$0.0463986\pi$
$978$		0			0
$979$	589446.	+	1.02095e6i	0.615004	+	1.06522i
$980$	723810.	+	185602.i	0.753654	+	0.193255i
$981$		0			0
$982$			250533.i			0.259802i
$983$	343643.	+	595207.i	0.355632	+	0.615972i	0.987226	−	0.159327i	$-0.0509324\pi$
$983$	343643.	+	595207.i	0.355632	+	0.615972i	−0.631594	+	0.775299i	$0.717599\pi$
$984$		0			0
$985$	844972.	−	863396.i	0.870903	−	0.889892i
$986$	−122323.	−	70623.5i	−0.125822	−	0.0726433i
$987$		0			0
$988$	122713.	−	70848.5i	0.125712	−	0.0725800i
$989$		−	951604.i		−	0.972890i
$990$		0			0
$991$	315848.			0.321611			0.160806	−	0.986986i	$-0.448591\pi$
$991$	315848.			0.321611			0.160806	+	0.986986i	$0.448591\pi$
$992$	−582286.	−	1.00855e6i	−0.591715	−	1.02488i
$993$		0			0
$994$	48808.0	−	84538.0i	0.0493990	−	0.0855617i
$995$	179168.	−	183074.i	0.180973	−	0.184919i
$996$		0			0
$997$	458423.	−	264671.i	0.461186	−	0.266266i	−0.251357	−	0.967895i	$-0.580877\pi$
$997$	458423.	−	264671.i	0.461186	−	0.266266i	0.712543	+	0.701629i	$0.247543\pi$
$998$	−11453.6			−0.0114995
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.5.h.a.44.13		44
3.2	odd	2		45.5.h.a.14.10	✓	44
5.4	even	2	inner	135.5.h.a.44.10		44
9.2	odd	6	inner	135.5.h.a.89.10		44
9.4	even	3		405.5.d.a.404.19		44
9.5	odd	6		405.5.d.a.404.25		44
9.7	even	3		45.5.h.a.29.13	yes	44
15.14	odd	2		45.5.h.a.14.13	yes	44
45.4	even	6		405.5.d.a.404.26		44
45.14	odd	6		405.5.d.a.404.20		44
45.29	odd	6	inner	135.5.h.a.89.13		44
45.34	even	6		45.5.h.a.29.10	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.5.h.a.14.10	✓	44	3.2	odd	2
45.5.h.a.14.13	yes	44	15.14	odd	2
45.5.h.a.29.10	yes	44	45.34	even	6
45.5.h.a.29.13	yes	44	9.7	even	3
135.5.h.a.44.10		44	5.4	even	2	inner
135.5.h.a.44.13		44	1.1	even	1	trivial
135.5.h.a.89.10		44	9.2	odd	6	inner
135.5.h.a.89.13		44	45.29	odd	6	inner
405.5.d.a.404.19		44	9.4	even	3
405.5.d.a.404.20		44	45.14	odd	6
405.5.d.a.404.25		44	9.5	odd	6
405.5.d.a.404.26		44	45.4	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	135.h (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.547257 + 0.947877i) q^{2} +(7.40102 - 12.8189i) q^{4} +(17.4860 - 17.8673i) q^{5} +(-16.9207 + 9.76914i) q^{7} +33.7133 q^{8} +(26.5053 + 6.79660i) q^{10} +(100.405 - 57.9690i) q^{11} +(-158.728 - 91.6418i) q^{13} +(-18.5199 - 10.6925i) q^{14} +(-99.9665 - 173.147i) q^{16} -89.0232 q^{17} -52.2295 q^{19} +(-99.6252 - 356.388i) q^{20} +(109.895 + 63.4480i) q^{22} +(-232.002 + 401.839i) q^{23} +(-13.4788 - 624.855i) q^{25} -200.607i q^{26} +289.206i q^{28} +(1255.41 - 724.811i) q^{29} +(767.942 - 1330.11i) q^{31} +(379.121 - 656.657i) q^{32} +(-48.7186 - 84.3831i) q^{34} +(-121.327 + 473.149i) q^{35} -641.462i q^{37} +(-28.5830 - 49.5071i) q^{38} +(589.511 - 602.364i) q^{40} +(1195.31 + 690.111i) q^{41} +(-1776.09 + 1025.43i) q^{43} -1716.12i q^{44} -507.859 q^{46} +(589.129 + 1020.40i) q^{47} +(-1009.63 + 1748.73i) q^{49} +(584.909 - 354.732i) q^{50} +(-2349.50 + 1356.49i) q^{52} +64.4267 q^{53} +(719.940 - 2807.62i) q^{55} +(-570.451 + 329.350i) q^{56} +(1374.06 + 793.316i) q^{58} +(-2837.79 - 1638.40i) q^{59} +(929.409 + 1609.78i) q^{61} +1681.05 q^{62} -2369.02 q^{64} +(-4412.91 + 1233.59i) q^{65} +(7257.38 + 4190.05i) q^{67} +(-658.862 + 1141.18i) q^{68} +(-514.884 + 143.931i) q^{70} +4564.71i q^{71} -8077.75i q^{73} +(608.027 - 351.045i) q^{74} +(-386.551 + 669.527i) q^{76} +(-1132.62 + 1961.75i) q^{77} +(4328.70 + 7497.52i) q^{79} +(-4841.68 - 1241.52i) q^{80} +1510.67i q^{82} +(-1670.06 - 2892.62i) q^{83} +(-1556.66 + 1590.60i) q^{85} +(-1943.96 - 1122.34i) q^{86} +(3384.99 - 1954.33i) q^{88} +10168.3i q^{89} +3581.05 q^{91} +(3434.10 + 5948.04i) q^{92} +(-644.811 + 1116.84i) q^{94} +(-913.285 + 933.198i) q^{95} +(-11868.3 + 6852.15i) q^{97} -2210.10 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 162 q^{4} - 6 q^{5} + 28 q^{10} - 228 q^{11} - 282 q^{14} - 1058 q^{16} - 8 q^{19} + 2196 q^{20} - 148 q^{25} - 2370 q^{29} - 1112 q^{31} - 436 q^{34} - 850 q^{40} - 1830 q^{41} - 5668 q^{46} + 5396 q^{49}+ \cdots - 58746 q^{95}+O(q^{100}) \) 44 * q - 162 * q^4 - 6 * q^5 + 28 * q^10 - 228 * q^11 - 282 * q^14 - 1058 * q^16 - 8 * q^19 + 2196 * q^20 - 148 * q^25 - 2370 * q^29 - 1112 * q^31 - 436 * q^34 - 850 * q^40 - 1830 * q^41 - 5668 * q^46 + 5396 * q^49 - 13524 * q^50 - 3072 * q^55 + 8250 * q^56 + 27348 * q^59 + 2782 * q^61 + 8348 * q^64 + 24906 * q^65 + 4782 * q^70 - 64728 * q^74 - 1776 * q^76 + 7900 * q^79 + 6766 * q^85 - 9192 * q^86 + 1560 * q^91 - 18826 * q^94 - 58746 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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