LMFDB - Embedded newform 45.5.g.b.28.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,5,Mod(28,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

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

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

chi := DirichletCharacter("45.28");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	45.g (of order $4$, degree $2$, 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:	$4.65164833877$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 5)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			28.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	45.28
Dual form			45.5.g.b.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 - 1.00000i) q^{2} +14.0000i q^{4} +(-20.0000 + 15.0000i) q^{5} +(-26.0000 + 26.0000i) q^{7} +(30.0000 + 30.0000i) q^{8} +(-5.00000 + 35.0000i) q^{10} +8.00000 q^{11} +(139.000 + 139.000i) q^{13} +52.0000i q^{14} -164.000 q^{16} +(1.00000 - 1.00000i) q^{17} -180.000i q^{19} +(-210.000 - 280.000i) q^{20} +(8.00000 - 8.00000i) q^{22} +(166.000 + 166.000i) q^{23} +(175.000 - 600.000i) q^{25} +278.000 q^{26} +(-364.000 - 364.000i) q^{28} -480.000i q^{29} +572.000 q^{31} +(-644.000 + 644.000i) q^{32} -2.00000i q^{34} +(130.000 - 910.000i) q^{35} +(-251.000 + 251.000i) q^{37} +(-180.000 - 180.000i) q^{38} +(-1050.00 - 150.000i) q^{40} +1688.00 q^{41} +(1474.00 + 1474.00i) q^{43} +112.000i q^{44} +332.000 q^{46} +(-2474.00 + 2474.00i) q^{47} +1049.00i q^{49} +(-425.000 - 775.000i) q^{50} +(-1946.00 + 1946.00i) q^{52} +(3331.00 + 3331.00i) q^{53} +(-160.000 + 120.000i) q^{55} -1560.00 q^{56} +(-480.000 - 480.000i) q^{58} -3660.00i q^{59} +1592.00 q^{61} +(572.000 - 572.000i) q^{62} -1336.00i q^{64} +(-4865.00 - 695.000i) q^{65} +(874.000 - 874.000i) q^{67} +(14.0000 + 14.0000i) q^{68} +(-780.000 - 1040.00i) q^{70} +6068.00 q^{71} +(-791.000 - 791.000i) q^{73} +502.000i q^{74} +2520.00 q^{76} +(-208.000 + 208.000i) q^{77} -9120.00i q^{79} +(3280.00 - 2460.00i) q^{80} +(1688.00 - 1688.00i) q^{82} +(-5654.00 - 5654.00i) q^{83} +(-5.00000 + 35.0000i) q^{85} +2948.00 q^{86} +(240.000 + 240.000i) q^{88} +2160.00i q^{89} -7228.00 q^{91} +(-2324.00 + 2324.00i) q^{92} +4948.00i q^{94} +(2700.00 + 3600.00i) q^{95} +(-6551.00 + 6551.00i) q^{97} +(1049.00 + 1049.00i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 40 q^{5} - 52 q^{7} + 60 q^{8} - 10 q^{10} + 16 q^{11} + 278 q^{13} - 328 q^{16} + 2 q^{17} - 420 q^{20} + 16 q^{22} + 332 q^{23} + 350 q^{25} + 556 q^{26} - 728 q^{28} + 1144 q^{31} - 1288 q^{32}+ \cdots + 2098 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 40 * q^5 - 52 * q^7 + 60 * q^8 - 10 * q^10 + 16 * q^11 + 278 * q^13 - 328 * q^16 + 2 * q^17 - 420 * q^20 + 16 * q^22 + 332 * q^23 + 350 * q^25 + 556 * q^26 - 728 * q^28 + 1144 * q^31 - 1288 * q^32 + 260 * q^35 - 502 * q^37 - 360 * q^38 - 2100 * q^40 + 3376 * q^41 + 2948 * q^43 + 664 * q^46 - 4948 * q^47 - 850 * q^50 - 3892 * q^52 + 6662 * q^53 - 320 * q^55 - 3120 * q^56 - 960 * q^58 + 3184 * q^61 + 1144 * q^62 - 9730 * q^65 + 1748 * q^67 + 28 * q^68 - 1560 * q^70 + 12136 * q^71 - 1582 * q^73 + 5040 * q^76 - 416 * q^77 + 6560 * q^80 + 3376 * q^82 - 11308 * q^83 - 10 * q^85 + 5896 * q^86 + 480 * q^88 - 14456 * q^91 - 4648 * q^92 + 5400 * q^95 - 13102 * q^97 + 2098 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	1.00000	−	1.00000i	0.250000	−	0.250000i	−0.570971	−	0.820971i	$-0.693433\pi$
$2$	1.00000	−	1.00000i	0.250000	−	0.250000i	0.820971	+	0.570971i	$0.193433\pi$
$3$		0			0
$3$		0			0
$4$			14.0000i			0.875000i
$5$	−20.0000	+	15.0000i	−0.800000	+	0.600000i
$5$	−20.0000	+	15.0000i	−0.800000	+	0.600000i
$6$		0			0
$7$	−26.0000	+	26.0000i	−0.530612	+	0.530612i	−0.920755	−	0.390142i	$-0.872426\pi$
$7$	−26.0000	+	26.0000i	−0.530612	+	0.530612i	0.390142	+	0.920755i	$0.372426\pi$
$8$	30.0000	+	30.0000i	0.468750	+	0.468750i
$9$		0			0
$10$	−5.00000	+	35.0000i	−0.0500000	+	0.350000i
$11$	8.00000			0.0661157			0.0330579	−	0.999453i	$-0.489475\pi$
$11$	8.00000			0.0661157			0.0330579	+	0.999453i	$0.489475\pi$
$12$		0			0
$13$	139.000	+	139.000i	0.822485	+	0.822485i	0.986464	−	0.163979i	$-0.0524328\pi$
$13$	139.000	+	139.000i	0.822485	+	0.822485i	−0.163979	+	0.986464i	$0.552433\pi$
$14$			52.0000i			0.265306i
$15$		0			0
$16$	−164.000			−0.640625
$17$	1.00000	−	1.00000i	0.00346021	−	0.00346021i	−0.705375	−	0.708835i	$-0.749221\pi$
$17$	1.00000	−	1.00000i	0.00346021	−	0.00346021i	0.708835	+	0.705375i	$0.249221\pi$
$18$		0			0
$19$		−	180.000i		−	0.498615i	−0.968424	−	0.249307i	$-0.919797\pi$
$19$		−	180.000i		−	0.498615i	0.968424	−	0.249307i	$-0.0802030\pi$
$20$	−210.000	−	280.000i	−0.525000	−	0.700000i
$21$		0			0
$22$	8.00000	−	8.00000i	0.0165289	−	0.0165289i
$23$	166.000	+	166.000i	0.313800	+	0.313800i	0.846380	−	0.532580i	$-0.178777\pi$
$23$	166.000	+	166.000i	0.313800	+	0.313800i	−0.532580	+	0.846380i	$0.678777\pi$
$24$		0			0
$25$	175.000	−	600.000i	0.280000	−	0.960000i
$26$	278.000			0.411243
$27$		0			0
$28$	−364.000	−	364.000i	−0.464286	−	0.464286i
$29$		−	480.000i		−	0.570749i	−0.958416	−	0.285375i	$-0.907882\pi$
$29$		−	480.000i		−	0.570749i	0.958416	−	0.285375i	$-0.0921180\pi$
$30$		0			0
$31$	572.000			0.595213			0.297607	−	0.954689i	$-0.403812\pi$
$31$	572.000			0.595213			0.297607	+	0.954689i	$0.403812\pi$
$32$	−644.000	+	644.000i	−0.628906	+	0.628906i
$33$		0			0
$34$		−	2.00000i		−	0.00173010i
$35$	130.000	−	910.000i	0.106122	−	0.742857i
$36$		0			0
$37$	−251.000	+	251.000i	−0.183346	+	0.183346i	−0.792812	−	0.609466i	$-0.791384\pi$
$37$	−251.000	+	251.000i	−0.183346	+	0.183346i	0.609466	+	0.792812i	$0.291384\pi$
$38$	−180.000	−	180.000i	−0.124654	−	0.124654i
$39$		0			0
$40$	−1050.00	−	150.000i	−0.656250	−	0.0937500i
$41$	1688.00			1.00416			0.502082	−	0.864820i	$-0.332568\pi$
$41$	1688.00			1.00416			0.502082	+	0.864820i	$0.332568\pi$
$42$		0			0
$43$	1474.00	+	1474.00i	0.797188	+	0.797188i	0.982651	−	0.185463i	$-0.0593786\pi$
$43$	1474.00	+	1474.00i	0.797188	+	0.797188i	−0.185463	+	0.982651i	$0.559379\pi$
$44$			112.000i			0.0578512i
$45$		0			0
$46$	332.000			0.156900
$47$	−2474.00	+	2474.00i	−1.11996	+	1.11996i	−0.128218	+	0.991746i	$0.540926\pi$
$47$	−2474.00	+	2474.00i	−1.11996	+	1.11996i	−0.991746	+	0.128218i	$0.959074\pi$
$48$		0			0
$49$			1049.00i			0.436901i
$50$	−425.000	−	775.000i	−0.170000	−	0.310000i
$51$		0			0
$52$	−1946.00	+	1946.00i	−0.719675	+	0.719675i
$53$	3331.00	+	3331.00i	1.18583	+	1.18583i	0.978209	+	0.207622i	$0.0665724\pi$
$53$	3331.00	+	3331.00i	1.18583	+	1.18583i	0.207622	+	0.978209i	$0.433428\pi$
$54$		0			0
$55$	−160.000	+	120.000i	−0.0528926	+	0.0396694i
$56$	−1560.00			−0.497449
$57$		0			0
$58$	−480.000	−	480.000i	−0.142687	−	0.142687i
$59$		−	3660.00i		−	1.05142i	−0.850663	−	0.525711i	$-0.823799\pi$
$59$		−	3660.00i		−	1.05142i	0.850663	−	0.525711i	$-0.176201\pi$
$60$		0			0
$61$	1592.00			0.427842			0.213921	−	0.976851i	$-0.431376\pi$
$61$	1592.00			0.427842			0.213921	+	0.976851i	$0.431376\pi$
$62$	572.000	−	572.000i	0.148803	−	0.148803i
$63$		0			0
$64$		−	1336.00i		−	0.326172i
$65$	−4865.00	−	695.000i	−1.15148	−	0.164497i
$66$		0			0
$67$	874.000	−	874.000i	0.194698	−	0.194698i	−0.603025	−	0.797723i	$-0.706038\pi$
$67$	874.000	−	874.000i	0.194698	−	0.194698i	0.797723	+	0.603025i	$0.206038\pi$
$68$	14.0000	+	14.0000i	0.00302768	+	0.00302768i
$69$		0			0
$70$	−780.000	−	1040.00i	−0.159184	−	0.212245i
$71$	6068.00			1.20373			0.601865	−	0.798598i	$-0.294425\pi$
$71$	6068.00			1.20373			0.601865	+	0.798598i	$0.294425\pi$
$72$		0			0
$73$	−791.000	−	791.000i	−0.148433	−	0.148433i	0.628985	−	0.777418i	$-0.283471\pi$
$73$	−791.000	−	791.000i	−0.148433	−	0.148433i	−0.777418	+	0.628985i	$0.783471\pi$
$74$			502.000i			0.0916728i
$75$		0			0
$76$	2520.00			0.436288
$77$	−208.000	+	208.000i	−0.0350818	+	0.0350818i
$78$		0			0
$79$		−	9120.00i		−	1.46130i	−0.682750	−	0.730652i	$-0.739216\pi$
$79$		−	9120.00i		−	1.46130i	0.682750	−	0.730652i	$-0.260784\pi$
$80$	3280.00	−	2460.00i	0.512500	−	0.384375i
$81$		0			0
$82$	1688.00	−	1688.00i	0.251041	−	0.251041i
$83$	−5654.00	−	5654.00i	−0.820729	−	0.820729i	0.165484	−	0.986213i	$-0.447081\pi$
$83$	−5654.00	−	5654.00i	−0.820729	−	0.820729i	−0.986213	+	0.165484i	$0.947081\pi$
$84$		0			0
$85$	−5.00000	+	35.0000i	−0.000692042	+	0.00484429i
$86$	2948.00			0.398594
$87$		0			0
$88$	240.000	+	240.000i	0.0309917	+	0.0309917i
$89$			2160.00i			0.272693i	0.990661	+	0.136346i	$0.0435360\pi$
$89$			2160.00i			0.272693i	−0.990661	+	0.136346i	$0.956464\pi$
$90$		0			0
$91$	−7228.00			−0.872841
$92$	−2324.00	+	2324.00i	−0.274575	+	0.274575i
$93$		0			0
$94$			4948.00i			0.559982i
$95$	2700.00	+	3600.00i	0.299169	+	0.398892i
$96$		0			0
$97$	−6551.00	+	6551.00i	−0.696248	+	0.696248i	−0.963599	−	0.267351i	$-0.913852\pi$
$97$	−6551.00	+	6551.00i	−0.696248	+	0.696248i	0.267351	+	0.963599i	$0.413852\pi$
$98$	1049.00	+	1049.00i	0.109225	+	0.109225i
$99$		0			0
$100$	8400.00	+	2450.00i	0.840000	+	0.245000i
$101$	−16102.0			−1.57847			−0.789236	−	0.614090i	$-0.789523\pi$
$101$	−16102.0			−1.57847			−0.789236	+	0.614090i	$0.789523\pi$
$102$		0			0
$103$	994.000	+	994.000i	0.0936940	+	0.0936940i	0.752400	−	0.658706i	$-0.228896\pi$
$103$	994.000	+	994.000i	0.0936940	+	0.0936940i	−0.658706	+	0.752400i	$0.728896\pi$
$104$			8340.00i			0.771080i
$105$		0			0
$106$	6662.00			0.592916
$107$	8326.00	−	8326.00i	0.727225	−	0.727225i	−0.242841	−	0.970066i	$-0.578079\pi$
$107$	8326.00	−	8326.00i	0.727225	−	0.727225i	0.970066	+	0.242841i	$0.0780793\pi$
$108$		0			0
$109$			17010.0i			1.43170i	0.698255	+	0.715849i	$0.253960\pi$
$109$			17010.0i			1.43170i	−0.698255	+	0.715849i	$0.746040\pi$
$110$	−40.0000	+	280.000i	−0.00330579	+	0.0231405i
$111$		0			0
$112$	4264.00	−	4264.00i	0.339923	−	0.339923i
$113$	5161.00	+	5161.00i	0.404182	+	0.404182i	0.879704	−	0.475522i	$-0.157741\pi$
$113$	5161.00	+	5161.00i	0.404182	+	0.404182i	−0.475522	+	0.879704i	$0.657741\pi$
$114$		0			0
$115$	−5810.00	−	830.000i	−0.439319	−	0.0627599i
$116$	6720.00			0.499405
$117$		0			0
$118$	−3660.00	−	3660.00i	−0.262856	−	0.262856i
$119$			52.0000i			0.00367206i
$120$		0			0
$121$	−14577.0			−0.995629
$122$	1592.00	−	1592.00i	0.106960	−	0.106960i
$123$		0			0
$124$			8008.00i			0.520812i
$125$	5500.00	+	14625.0i	0.352000	+	0.936000i
$126$		0			0
$127$	12574.0	−	12574.0i	0.779590	−	0.779590i	−0.200171	−	0.979761i	$-0.564150\pi$
$127$	12574.0	−	12574.0i	0.779590	−	0.779590i	0.979761	+	0.200171i	$0.0641499\pi$
$128$	−11640.0	−	11640.0i	−0.710449	−	0.710449i
$129$		0			0
$130$	−5560.00	+	4170.00i	−0.328994	+	0.246746i
$131$	−20272.0			−1.18128			−0.590642	−	0.806934i	$-0.701125\pi$
$131$	−20272.0			−1.18128			−0.590642	+	0.806934i	$0.701125\pi$
$132$		0			0
$133$	4680.00	+	4680.00i	0.264571	+	0.264571i
$134$		−	1748.00i		−	0.0973491i
$135$		0			0
$136$	60.0000			0.00324394
$137$	10351.0	−	10351.0i	0.551494	−	0.551494i	−0.375378	−	0.926872i	$-0.622487\pi$
$137$	10351.0	−	10351.0i	0.551494	−	0.551494i	0.926872	+	0.375378i	$0.122487\pi$
$138$		0			0
$139$		−	27060.0i		−	1.40055i	−0.713874	−	0.700274i	$-0.753061\pi$
$139$		−	27060.0i		−	1.40055i	0.713874	−	0.700274i	$-0.246939\pi$
$140$	12740.0	+	1820.00i	0.650000	+	0.0928571i
$141$		0			0
$142$	6068.00	−	6068.00i	0.300932	−	0.300932i
$143$	1112.00	+	1112.00i	0.0543792	+	0.0543792i
$144$		0			0
$145$	7200.00	+	9600.00i	0.342449	+	0.456599i
$146$	−1582.00			−0.0742166
$147$		0			0
$148$	−3514.00	−	3514.00i	−0.160427	−	0.160427i
$149$		−	16350.0i		−	0.736453i	−0.929736	−	0.368227i	$-0.879965\pi$
$149$		−	16350.0i		−	0.736453i	0.929736	−	0.368227i	$-0.120035\pi$
$150$		0			0
$151$	1052.00			0.0461383			0.0230692	−	0.999734i	$-0.492656\pi$
$151$	1052.00			0.0461383			0.0230692	+	0.999734i	$0.492656\pi$
$152$	5400.00	−	5400.00i	0.233726	−	0.233726i
$153$		0			0
$154$			416.000i			0.0175409i
$155$	−11440.0	+	8580.00i	−0.476171	+	0.357128i
$156$		0			0
$157$	6499.00	−	6499.00i	0.263662	−	0.263662i	−0.562878	−	0.826540i	$-0.690306\pi$
$157$	6499.00	−	6499.00i	0.263662	−	0.263662i	0.826540	+	0.562878i	$0.190306\pi$
$158$	−9120.00	−	9120.00i	−0.365326	−	0.365326i
$159$		0			0
$160$	3220.00	−	22540.0i	0.125781	−	0.880469i
$161$	−8632.00			−0.333012
$162$		0			0
$163$	−19286.0	−	19286.0i	−0.725884	−	0.725884i	0.243913	−	0.969797i	$-0.421569\pi$
$163$	−19286.0	−	19286.0i	−0.725884	−	0.725884i	−0.969797	+	0.243913i	$0.921569\pi$
$164$			23632.0i			0.878644i
$165$		0			0
$166$	−11308.0			−0.410364
$167$	15526.0	−	15526.0i	0.556707	−	0.556707i	−0.371661	−	0.928368i	$-0.621212\pi$
$167$	15526.0	−	15526.0i	0.556707	−	0.556707i	0.928368	+	0.371661i	$0.121212\pi$
$168$		0			0
$169$			10081.0i			0.352964i
$170$	30.0000	+	40.0000i	0.00103806	+	0.00138408i
$171$		0			0
$172$	−20636.0	+	20636.0i	−0.697539	+	0.697539i
$173$	16891.0	+	16891.0i	0.564369	+	0.564369i	0.930545	−	0.366176i	$-0.119333\pi$
$173$	16891.0	+	16891.0i	0.564369	+	0.564369i	−0.366176	+	0.930545i	$0.619333\pi$
$174$		0			0
$175$	11050.0	+	20150.0i	0.360816	+	0.657959i
$176$	−1312.00			−0.0423554
$177$		0			0
$178$	2160.00	+	2160.00i	0.0681732	+	0.0681732i
$179$			10620.0i			0.331450i	0.986172	+	0.165725i	$0.0529965\pi$
$179$			10620.0i			0.331450i	−0.986172	+	0.165725i	$0.947004\pi$
$180$		0			0
$181$	24122.0			0.736302			0.368151	−	0.929766i	$-0.379991\pi$
$181$	24122.0			0.736302			0.368151	+	0.929766i	$0.379991\pi$
$182$	−7228.00	+	7228.00i	−0.218210	+	0.218210i
$183$		0			0
$184$			9960.00i			0.294187i
$185$	1255.00	−	8785.00i	0.0366691	−	0.256684i
$186$		0			0
$187$	8.00000	−	8.00000i	0.000228774	−	0.000228774i
$188$	−34636.0	−	34636.0i	−0.979968	−	0.979968i
$189$		0			0
$190$	6300.00	+	900.000i	0.174515	+	0.0249307i
$191$	45188.0			1.23867			0.619336	−	0.785126i	$-0.287402\pi$
$191$	45188.0			1.23867			0.619336	+	0.785126i	$0.287402\pi$
$192$		0			0
$193$	39199.0	+	39199.0i	1.05235	+	1.05235i	0.998552	+	0.0537986i	$0.0171329\pi$
$193$	39199.0	+	39199.0i	1.05235	+	1.05235i	0.0537986	+	0.998552i	$0.482867\pi$
$194$			13102.0i			0.348124i
$195$		0			0
$196$	−14686.0			−0.382289
$197$	−28349.0	+	28349.0i	−0.730475	+	0.730475i	−0.970714	−	0.240239i	$-0.922774\pi$
$197$	−28349.0	+	28349.0i	−0.730475	+	0.730475i	0.240239	+	0.970714i	$0.422774\pi$
$198$		0			0
$199$			1800.00i			0.0454534i	0.999742	+	0.0227267i	$0.00723476\pi$
$199$			1800.00i			0.0454534i	−0.999742	+	0.0227267i	$0.992765\pi$
$200$	23250.0	−	12750.0i	0.581250	−	0.318750i
$201$		0			0
$202$	−16102.0	+	16102.0i	−0.394618	+	0.394618i
$203$	12480.0	+	12480.0i	0.302846	+	0.302846i
$204$		0			0
$205$	−33760.0	+	25320.0i	−0.803331	+	0.602499i
$206$	1988.00			0.0468470
$207$		0			0
$208$	−22796.0	−	22796.0i	−0.526905	−	0.526905i
$209$		−	1440.00i		−	0.0329663i
$210$		0			0
$211$	18392.0			0.413108			0.206554	−	0.978435i	$-0.433775\pi$
$211$	18392.0			0.413108			0.206554	+	0.978435i	$0.433775\pi$
$212$	−46634.0	+	46634.0i	−1.03760	+	1.03760i
$213$		0			0
$214$		−	16652.0i		−	0.363613i
$215$	−51590.0	−	7370.00i	−1.11606	−	0.159438i
$216$		0			0
$217$	−14872.0	+	14872.0i	−0.315827	+	0.315827i
$218$	17010.0	+	17010.0i	0.357924	+	0.357924i
$219$		0			0
$220$	−1680.00	−	2240.00i	−0.0347107	−	0.0462810i
$221$	278.000			0.00569194
$222$		0			0
$223$	−866.000	−	866.000i	−0.0174144	−	0.0174144i	0.698346	−	0.715760i	$-0.253920\pi$
$223$	−866.000	−	866.000i	−0.0174144	−	0.0174144i	−0.715760	+	0.698346i	$0.753920\pi$
$224$		−	33488.0i		−	0.667411i
$225$		0			0
$226$	10322.0			0.202091
$227$	45226.0	−	45226.0i	0.877681	−	0.877681i	−0.115614	−	0.993294i	$-0.536883\pi$
$227$	45226.0	−	45226.0i	0.877681	−	0.877681i	0.993294	+	0.115614i	$0.0368835\pi$
$228$		0			0
$229$		−	39120.0i		−	0.745981i	−0.927835	−	0.372991i	$-0.878332\pi$
$229$		−	39120.0i		−	0.745981i	0.927835	−	0.372991i	$-0.121668\pi$
$230$	−6640.00	+	4980.00i	−0.125520	+	0.0941399i
$231$		0			0
$232$	14400.0	−	14400.0i	0.267539	−	0.267539i
$233$	33121.0	+	33121.0i	0.610087	+	0.610087i	0.942969	−	0.332882i	$-0.108021\pi$
$233$	33121.0	+	33121.0i	0.610087	+	0.610087i	−0.332882	+	0.942969i	$0.608021\pi$
$234$		0			0
$235$	12370.0	−	86590.0i	0.223993	−	1.56795i
$236$	51240.0			0.919994
$237$		0			0
$238$	52.0000	+	52.0000i	0.000918014	+	0.000918014i
$239$		−	88440.0i		−	1.54829i	−0.633007	−	0.774146i	$-0.718180\pi$
$239$		−	88440.0i		−	1.54829i	0.633007	−	0.774146i	$-0.281820\pi$
$240$		0			0
$241$	20312.0			0.349718			0.174859	−	0.984593i	$-0.444053\pi$
$241$	20312.0			0.349718			0.174859	+	0.984593i	$0.444053\pi$
$242$	−14577.0	+	14577.0i	−0.248907	+	0.248907i
$243$		0			0
$244$			22288.0i			0.374362i
$245$	−15735.0	−	20980.0i	−0.262141	−	0.349521i
$246$		0			0
$247$	25020.0	−	25020.0i	0.410103	−	0.410103i
$248$	17160.0	+	17160.0i	0.279006	+	0.279006i
$249$		0			0
$250$	20125.0	+	9125.00i	0.322000	+	0.146000i
$251$	−74752.0			−1.18652			−0.593260	−	0.805011i	$-0.702160\pi$
$251$	−74752.0			−1.18652			−0.593260	+	0.805011i	$0.702160\pi$
$252$		0			0
$253$	1328.00	+	1328.00i	0.0207471	+	0.0207471i
$254$		−	25148.0i		−	0.389795i
$255$		0			0
$256$	−1904.00			−0.0290527
$257$	−37799.0	+	37799.0i	−0.572287	+	0.572287i	−0.932767	−	0.360480i	$-0.882613\pi$
$257$	−37799.0	+	37799.0i	−0.572287	+	0.572287i	0.360480	+	0.932767i	$0.382613\pi$
$258$		0			0
$259$		−	13052.0i		−	0.194571i
$260$	9730.00	−	68110.0i	0.143935	−	1.00754i
$261$		0			0
$262$	−20272.0	+	20272.0i	−0.295321	+	0.295321i
$263$	33586.0	+	33586.0i	0.485564	+	0.485564i	0.906903	−	0.421339i	$-0.138440\pi$
$263$	33586.0	+	33586.0i	0.485564	+	0.485564i	−0.421339	+	0.906903i	$0.638440\pi$
$264$		0			0
$265$	−116585.	−	16655.0i	−1.66016	−	0.237166i
$266$	9360.00			0.132286
$267$		0			0
$268$	12236.0	+	12236.0i	0.170361	+	0.170361i
$269$			28530.0i			0.394273i	0.980376	+	0.197137i	$0.0631642\pi$
$269$			28530.0i			0.394273i	−0.980376	+	0.197137i	$0.936836\pi$
$270$		0			0
$271$	−7468.00			−0.101687			−0.0508435	−	0.998707i	$-0.516191\pi$
$271$	−7468.00			−0.101687			−0.0508435	+	0.998707i	$0.516191\pi$
$272$	−164.000	+	164.000i	−0.00221670	+	0.00221670i
$273$		0			0
$274$		−	20702.0i		−	0.275747i
$275$	1400.00	−	4800.00i	0.0185124	−	0.0634711i
$276$		0			0
$277$	6499.00	−	6499.00i	0.0847007	−	0.0847007i	−0.663487	−	0.748188i	$-0.730924\pi$
$277$	6499.00	−	6499.00i	0.0847007	−	0.0847007i	0.748188	+	0.663487i	$0.230924\pi$
$278$	−27060.0	−	27060.0i	−0.350137	−	0.350137i
$279$		0			0
$280$	31200.0	−	23400.0i	0.397959	−	0.298469i
$281$	97928.0			1.24021			0.620104	−	0.784520i	$-0.287091\pi$
$281$	97928.0			1.24021			0.620104	+	0.784520i	$0.287091\pi$
$282$		0			0
$283$	59854.0	+	59854.0i	0.747344	+	0.747344i	0.973980	−	0.226636i	$-0.0727728\pi$
$283$	59854.0	+	59854.0i	0.747344	+	0.747344i	−0.226636	+	0.973980i	$0.572773\pi$
$284$			84952.0i			1.05326i
$285$		0			0
$286$	2224.00			0.0271896
$287$	−43888.0	+	43888.0i	−0.532822	+	0.532822i
$288$		0			0
$289$			83519.0i			0.999976i
$290$	16800.0	+	2400.00i	0.199762	+	0.0285375i
$291$		0			0
$292$	11074.0	−	11074.0i	0.129879	−	0.129879i
$293$	−28499.0	−	28499.0i	−0.331967	−	0.331967i	0.521366	−	0.853333i	$-0.325423\pi$
$293$	−28499.0	−	28499.0i	−0.331967	−	0.331967i	−0.853333	+	0.521366i	$0.825423\pi$
$294$		0			0
$295$	54900.0	+	73200.0i	0.630853	+	0.841138i
$296$	−15060.0			−0.171886
$297$		0			0
$298$	−16350.0	−	16350.0i	−0.184113	−	0.184113i
$299$			46148.0i			0.516191i
$300$		0			0
$301$	−76648.0			−0.845995
$302$	1052.00	−	1052.00i	0.0115346	−	0.0115346i
$303$		0			0
$304$			29520.0i			0.319425i
$305$	−31840.0	+	23880.0i	−0.342274	+	0.256705i
$306$		0			0
$307$	−117926.	+	117926.i	−1.25122	+	1.25122i	−0.296043	+	0.955175i	$0.595667\pi$
$307$	−117926.	+	117926.i	−1.25122	+	1.25122i	−0.955175	+	0.296043i	$0.904333\pi$
$308$	−2912.00	−	2912.00i	−0.0306966	−	0.0306966i
$309$		0			0
$310$	−2860.00	+	20020.0i	−0.0297607	+	0.208325i
$311$	−3892.00			−0.0402395			−0.0201197	−	0.999798i	$-0.506405\pi$
$311$	−3892.00			−0.0402395			−0.0201197	+	0.999798i	$0.506405\pi$
$312$		0			0
$313$	−49961.0	−	49961.0i	−0.509967	−	0.509967i	0.404549	−	0.914516i	$-0.367429\pi$
$313$	−49961.0	−	49961.0i	−0.509967	−	0.509967i	−0.914516	+	0.404549i	$0.867429\pi$
$314$		−	12998.0i		−	0.131831i
$315$		0			0
$316$	127680.			1.27864
$317$	−17099.0	+	17099.0i	−0.170158	+	0.170158i	−0.787049	−	0.616891i	$-0.788392\pi$
$317$	−17099.0	+	17099.0i	−0.170158	+	0.170158i	0.616891	+	0.787049i	$0.288392\pi$
$318$		0			0
$319$		−	3840.00i		−	0.0377355i
$320$	20040.0	+	26720.0i	0.195703	+	0.260937i
$321$		0			0
$322$	−8632.00	+	8632.00i	−0.0832530	+	0.0832530i
$323$	−180.000	−	180.000i	−0.00172531	−	0.00172531i
$324$		0			0
$325$	107725.	−	59075.0i	1.01988	−	0.559290i
$326$	−38572.0			−0.362942
$327$		0			0
$328$	50640.0	+	50640.0i	0.470702	+	0.470702i
$329$		−	128648.i		−	1.18853i
$330$		0			0
$331$	−143128.			−1.30638			−0.653189	−	0.757195i	$-0.726569\pi$
$331$	−143128.			−1.30638			−0.653189	+	0.757195i	$0.726569\pi$
$332$	79156.0	−	79156.0i	0.718138	−	0.718138i
$333$		0			0
$334$		−	31052.0i		−	0.278353i
$335$	−4370.00	+	30590.0i	−0.0389396	+	0.272577i
$336$		0			0
$337$	103249.	−	103249.i	0.909130	−	0.909130i	−0.0870719	−	0.996202i	$-0.527751\pi$
$337$	103249.	−	103249.i	0.909130	−	0.909130i	0.996202	+	0.0870719i	$0.0277510\pi$
$338$	10081.0	+	10081.0i	0.0882410	+	0.0882410i
$339$		0			0
$340$	−490.000	−	70.0000i	−0.00423875	−	0.000605536i
$341$	4576.00			0.0393529
$342$		0			0
$343$	−89700.0	−	89700.0i	−0.762437	−	0.762437i
$344$			88440.0i			0.747363i
$345$		0			0
$346$	33782.0			0.282185
$347$	104626.	−	104626.i	0.868922	−	0.868922i	−0.123431	−	0.992353i	$-0.539390\pi$
$347$	104626.	−	104626.i	0.868922	−	0.868922i	0.992353	+	0.123431i	$0.0393899\pi$
$348$		0			0
$349$		−	94800.0i		−	0.778319i	−0.921170	−	0.389159i	$-0.872766\pi$
$349$		−	94800.0i		−	0.778319i	0.921170	−	0.389159i	$-0.127234\pi$
$350$	31200.0	+	9100.00i	0.254694	+	0.0742857i
$351$		0			0
$352$	−5152.00	+	5152.00i	−0.0415806	+	0.0415806i
$353$	−63569.0	−	63569.0i	−0.510148	−	0.510148i	0.404424	−	0.914572i	$-0.367472\pi$
$353$	−63569.0	−	63569.0i	−0.510148	−	0.510148i	−0.914572	+	0.404424i	$0.867472\pi$
$354$		0			0
$355$	−121360.	+	91020.0i	−0.962984	+	0.722238i
$356$	−30240.0			−0.238606
$357$		0			0
$358$	10620.0	+	10620.0i	0.0828626	+	0.0828626i
$359$			141840.i			1.10055i	0.834983	+	0.550275i	$0.185477\pi$
$359$			141840.i			1.10055i	−0.834983	+	0.550275i	$0.814523\pi$
$360$		0			0
$361$	97921.0			0.751383
$362$	24122.0	−	24122.0i	0.184076	−	0.184076i
$363$		0			0
$364$		−	101192.i		−	0.763736i
$365$	27685.0	+	3955.00i	0.207806	+	0.0296866i
$366$		0			0
$367$	142174.	−	142174.i	1.05557	−	1.05557i	0.0572103	−	0.998362i	$-0.481779\pi$
$367$	142174.	−	142174.i	1.05557	−	1.05557i	0.998362	−	0.0572103i	$-0.0182206\pi$
$368$	−27224.0	−	27224.0i	−0.201028	−	0.201028i
$369$		0			0
$370$	−7530.00	−	10040.0i	−0.0550037	−	0.0733382i
$371$	−173212.			−1.25843
$372$		0			0
$373$	10309.0	+	10309.0i	0.0740967	+	0.0740967i	0.743184	−	0.669087i	$-0.233315\pi$
$373$	10309.0	+	10309.0i	0.0740967	+	0.0740967i	−0.669087	+	0.743184i	$0.733315\pi$
$374$		−	16.0000i		−	0.000114387i
$375$		0			0
$376$	−148440.			−1.04997
$377$	66720.0	−	66720.0i	0.469433	−	0.469433i
$378$		0			0
$379$			115380.i			0.803253i	0.915804	+	0.401626i	$0.131555\pi$
$379$			115380.i			0.803253i	−0.915804	+	0.401626i	$0.868445\pi$
$380$	−50400.0	+	37800.0i	−0.349030	+	0.261773i
$381$		0			0
$382$	45188.0	−	45188.0i	0.309668	−	0.309668i
$383$	−62654.0	−	62654.0i	−0.427121	−	0.427121i	0.460525	−	0.887647i	$-0.347661\pi$
$383$	−62654.0	−	62654.0i	−0.427121	−	0.427121i	−0.887647	+	0.460525i	$0.847661\pi$
$384$		0			0
$385$	1040.00	−	7280.00i	0.00701636	−	0.0491145i
$386$	78398.0			0.526175
$387$		0			0
$388$	−91714.0	−	91714.0i	−0.609217	−	0.609217i
$389$		−	132690.i		−	0.876878i	−0.898761	−	0.438439i	$-0.855532\pi$
$389$		−	132690.i		−	0.876878i	0.898761	−	0.438439i	$-0.144468\pi$
$390$		0			0
$391$	332.000			0.00217162
$392$	−31470.0	+	31470.0i	−0.204797	+	0.204797i
$393$		0			0
$394$			56698.0i			0.365237i
$395$	136800.	+	182400.i	0.876783	+	1.16904i
$396$		0			0
$397$	−88451.0	+	88451.0i	−0.561205	+	0.561205i	−0.929650	−	0.368444i	$-0.879890\pi$
$397$	−88451.0	+	88451.0i	−0.561205	+	0.561205i	0.368444	+	0.929650i	$0.379890\pi$
$398$	1800.00	+	1800.00i	0.0113633	+	0.0113633i
$399$		0			0
$400$	−28700.0	+	98400.0i	−0.179375	+	0.615000i
$401$	−63202.0			−0.393045			−0.196522	−	0.980499i	$-0.562965\pi$
$401$	−63202.0			−0.393045			−0.196522	+	0.980499i	$0.562965\pi$
$402$		0			0
$403$	79508.0	+	79508.0i	0.489554	+	0.489554i
$404$		−	225428.i		−	1.38116i
$405$		0			0
$406$	24960.0			0.151423
$407$	−2008.00	+	2008.00i	−0.0121220	+	0.0121220i
$408$		0			0
$409$		−	52890.0i		−	0.316175i	−0.987425	−	0.158087i	$-0.949467\pi$
$409$		−	52890.0i		−	0.316175i	0.987425	−	0.158087i	$-0.0505327\pi$
$410$	−8440.00	+	59080.0i	−0.0502082	+	0.351457i
$411$		0			0
$412$	−13916.0	+	13916.0i	−0.0819823	+	0.0819823i
$413$	95160.0	+	95160.0i	0.557897	+	0.557897i
$414$		0			0
$415$	197890.	+	28270.0i	1.14902	+	0.164146i
$416$	−179032.			−1.03453
$417$		0			0
$418$	−1440.00	−	1440.00i	−0.00824157	−	0.00824157i
$419$			256980.i			1.46376i	0.681431	+	0.731882i	$0.261358\pi$
$419$			256980.i			1.46376i	−0.681431	+	0.731882i	$0.738642\pi$
$420$		0			0
$421$	186632.			1.05298			0.526492	−	0.850180i	$-0.323507\pi$
$421$	186632.			1.05298			0.526492	+	0.850180i	$0.323507\pi$
$422$	18392.0	−	18392.0i	0.103277	−	0.103277i
$423$		0			0
$424$			199860.i			1.11172i
$425$	−425.000	−	775.000i	−0.00235294	−	0.00429066i
$426$		0			0
$427$	−41392.0	+	41392.0i	−0.227018	+	0.227018i
$428$	116564.	+	116564.i	0.636322	+	0.636322i
$429$		0			0
$430$	−58960.0	+	44220.0i	−0.318875	+	0.239156i
$431$	208028.			1.11987			0.559935	−	0.828537i	$-0.310826\pi$
$431$	208028.			1.11987			0.559935	+	0.828537i	$0.310826\pi$
$432$		0			0
$433$	−151271.	−	151271.i	−0.806826	−	0.806826i	0.177326	−	0.984152i	$-0.443255\pi$
$433$	−151271.	−	151271.i	−0.806826	−	0.806826i	−0.984152	+	0.177326i	$0.943255\pi$
$434$			29744.0i			0.157914i
$435$		0			0
$436$	−238140.			−1.25274
$437$	29880.0	−	29880.0i	0.156465	−	0.156465i
$438$		0			0
$439$			158640.i			0.823159i	0.911374	+	0.411579i	$0.135023\pi$
$439$			158640.i			0.823159i	−0.911374	+	0.411579i	$0.864977\pi$
$440$	−8400.00	−	1200.00i	−0.0433884	−	0.00619835i
$441$		0			0
$442$	278.000	−	278.000i	0.00142298	−	0.00142298i
$443$	−252974.	−	252974.i	−1.28905	−	1.28905i	−0.935367	−	0.353679i	$-0.884930\pi$
$443$	−252974.	−	252974.i	−1.28905	−	1.28905i	−0.353679	−	0.935367i	$-0.615070\pi$
$444$		0			0
$445$	−32400.0	−	43200.0i	−0.163616	−	0.218154i
$446$	−1732.00			−0.00870719
$447$		0			0
$448$	34736.0	+	34736.0i	0.173071	+	0.173071i
$449$			123750.i			0.613836i	0.951736	+	0.306918i	$0.0992978\pi$
$449$			123750.i			0.613836i	−0.951736	+	0.306918i	$0.900702\pi$
$450$		0			0
$451$	13504.0			0.0663910
$452$	−72254.0	+	72254.0i	−0.353659	+	0.353659i
$453$		0			0
$454$		−	90452.0i		−	0.438840i
$455$	144560.	−	108420.i	0.698273	−	0.523705i
$456$		0			0
$457$	−32201.0	+	32201.0i	−0.154183	+	0.154183i	−0.779983	−	0.625800i	$-0.784773\pi$
$457$	−32201.0	+	32201.0i	−0.154183	+	0.154183i	0.625800	+	0.779983i	$0.284773\pi$
$458$	−39120.0	−	39120.0i	−0.186495	−	0.186495i
$459$		0			0
$460$	11620.0	−	81340.0i	0.0549149	−	0.384405i
$461$	−75142.0			−0.353574			−0.176787	−	0.984249i	$-0.556570\pi$
$461$	−75142.0			−0.353574			−0.176787	+	0.984249i	$0.556570\pi$
$462$		0			0
$463$	235714.	+	235714.i	1.09957	+	1.09957i	0.994461	+	0.105111i	$0.0335197\pi$
$463$	235714.	+	235714.i	1.09957	+	1.09957i	0.105111	+	0.994461i	$0.466480\pi$
$464$			78720.0i			0.365636i
$465$		0			0
$466$	66242.0			0.305043
$467$	−28574.0	+	28574.0i	−0.131020	+	0.131020i	−0.769576	−	0.638556i	$-0.779532\pi$
$467$	−28574.0	+	28574.0i	−0.131020	+	0.131020i	0.638556	+	0.769576i	$0.279532\pi$
$468$		0			0
$469$			45448.0i			0.206618i
$470$	−74220.0	−	98960.0i	−0.335989	−	0.447986i
$471$		0			0
$472$	109800.	−	109800.i	0.492854	−	0.492854i
$473$	11792.0	+	11792.0i	0.0527066	+	0.0527066i
$474$		0			0
$475$	−108000.	−	31500.0i	−0.478670	−	0.139612i
$476$	−728.000			−0.00321305
$477$		0			0
$478$	−88440.0	−	88440.0i	−0.387073	−	0.387073i
$479$			26520.0i			0.115585i	0.998329	+	0.0577926i	$0.0184062\pi$
$479$			26520.0i			0.115585i	−0.998329	+	0.0577926i	$0.981594\pi$
$480$		0			0
$481$	−69778.0			−0.301598
$482$	20312.0	−	20312.0i	0.0874296	−	0.0874296i
$483$		0			0
$484$		−	204078.i		−	0.871175i
$485$	32755.0	−	229285.i	0.139250	−	0.974748i
$486$		0			0
$487$	41374.0	−	41374.0i	0.174449	−	0.174449i	−0.614482	−	0.788931i	$-0.710635\pi$
$487$	41374.0	−	41374.0i	0.174449	−	0.174449i	0.788931	+	0.614482i	$0.210635\pi$
$488$	47760.0	+	47760.0i	0.200551	+	0.200551i
$489$		0			0
$490$	−36715.0	−	5245.00i	−0.152915	−	0.0218451i
$491$	149288.			0.619244			0.309622	−	0.950860i	$-0.399797\pi$
$491$	149288.			0.619244			0.309622	+	0.950860i	$0.399797\pi$
$492$		0			0
$493$	−480.000	−	480.000i	−0.00197491	−	0.00197491i
$494$		−	50040.0i		−	0.205052i
$495$		0			0
$496$	−93808.0			−0.381309
$497$	−157768.	+	157768.i	−0.638714	+	0.638714i
$498$		0			0
$499$		−	284100.i		−	1.14096i	−0.821312	−	0.570480i	$-0.806757\pi$
$499$		−	284100.i		−	1.14096i	0.821312	−	0.570480i	$-0.193243\pi$
$500$	−204750.	+	77000.0i	−0.819000	+	0.308000i
$501$		0			0
$502$	−74752.0	+	74752.0i	−0.296630	+	0.296630i
$503$	117406.	+	117406.i	0.464039	+	0.464039i	0.899977	−	0.435938i	$-0.143583\pi$
$503$	117406.	+	117406.i	0.464039	+	0.464039i	−0.435938	+	0.899977i	$0.643583\pi$
$504$		0			0
$505$	322040.	−	241530.i	1.26278	−	0.947084i
$506$	2656.00			0.0103735
$507$		0			0
$508$	176036.	+	176036.i	0.682141	+	0.682141i
$509$		−	234960.i		−	0.906898i	−0.891282	−	0.453449i	$-0.850193\pi$
$509$		−	234960.i		−	0.906898i	0.891282	−	0.453449i	$-0.149807\pi$
$510$		0			0
$511$	41132.0			0.157521
$512$	184336.	−	184336.i	0.703186	−	0.703186i
$513$		0			0
$514$			75598.0i			0.286144i
$515$	−34790.0	−	4970.00i	−0.131172	−	0.0187388i
$516$		0			0
$517$	−19792.0	+	19792.0i	−0.0740472	+	0.0740472i
$518$	−13052.0	−	13052.0i	−0.0486427	−	0.0486427i
$519$		0			0
$520$	−125100.	−	166800.i	−0.462648	−	0.616864i
$521$	171218.			0.630774			0.315387	−	0.948963i	$-0.397866\pi$
$521$	171218.			0.630774			0.315387	+	0.948963i	$0.397866\pi$
$522$		0			0
$523$	−332666.	−	332666.i	−1.21620	−	1.21620i	−0.968952	−	0.247248i	$-0.920474\pi$
$523$	−332666.	−	332666.i	−1.21620	−	1.21620i	−0.247248	−	0.968952i	$-0.579526\pi$
$524$		−	283808.i		−	1.03362i
$525$		0			0
$526$	67172.0			0.242782
$527$	572.000	−	572.000i	0.00205956	−	0.00205956i
$528$		0			0
$529$		−	224729.i		−	0.803060i
$530$	−133240.	+	99930.0i	−0.474333	+	0.355749i
$531$		0			0
$532$	−65520.0	+	65520.0i	−0.231500	+	0.231500i
$533$	234632.	+	234632.i	0.825910	+	0.825910i
$534$		0			0
$535$	−41630.0	+	291410.i	−0.145445	+	1.01812i
$536$	52440.0			0.182530
$537$		0			0
$538$	28530.0	+	28530.0i	0.0985683	+	0.0985683i
$539$			8392.00i			0.0288860i
$540$		0			0
$541$	13862.0			0.0473621			0.0236811	−	0.999720i	$-0.492461\pi$
$541$	13862.0			0.0473621			0.0236811	+	0.999720i	$0.492461\pi$
$542$	−7468.00	+	7468.00i	−0.0254218	+	0.0254218i
$543$		0			0
$544$			1288.00i			0.00435229i
$545$	−255150.	−	340200.i	−0.859019	−	1.14536i
$546$		0			0
$547$	−180026.	+	180026.i	−0.601673	+	0.601673i	−0.940756	−	0.339083i	$-0.889883\pi$
$547$	−180026.	+	180026.i	−0.601673	+	0.601673i	0.339083	+	0.940756i	$0.389883\pi$
$548$	144914.	+	144914.i	0.482558	+	0.482558i
$549$		0			0
$550$	−3400.00	−	6200.00i	−0.0112397	−	0.0204959i
$551$	−86400.0			−0.284584
$552$		0			0
$553$	237120.	+	237120.i	0.775386	+	0.775386i
$554$		−	12998.0i		−	0.0423503i
$555$		0			0
$556$	378840.			1.22548
$557$	−422549.	+	422549.i	−1.36197	+	1.36197i	−0.490560	+	0.871408i	$0.663208\pi$
$557$	−422549.	+	422549.i	−1.36197	+	1.36197i	−0.871408	+	0.490560i	$0.836792\pi$
$558$		0			0
$559$			409772.i			1.31135i
$560$	−21320.0	+	149240.i	−0.0679847	+	0.475893i
$561$		0			0
$562$	97928.0	−	97928.0i	0.310052	−	0.310052i
$563$	−105314.	−	105314.i	−0.332253	−	0.332253i	0.521188	−	0.853442i	$-0.325489\pi$
$563$	−105314.	−	105314.i	−0.332253	−	0.332253i	−0.853442	+	0.521188i	$0.825489\pi$
$564$		0			0
$565$	−180635.	−	25805.0i	−0.565855	−	0.0808364i
$566$	119708.			0.373672
$567$		0			0
$568$	182040.	+	182040.i	0.564248	+	0.564248i
$569$			85830.0i			0.265103i	0.991176	+	0.132551i	$0.0423170\pi$
$569$			85830.0i			0.265103i	−0.991176	+	0.132551i	$0.957683\pi$
$570$		0			0
$571$	−142168.			−0.436043			−0.218022	−	0.975944i	$-0.569960\pi$
$571$	−142168.			−0.436043			−0.218022	+	0.975944i	$0.569960\pi$
$572$	−15568.0	+	15568.0i	−0.0475818	+	0.0475818i
$573$		0			0
$574$			87776.0i			0.266411i
$575$	128650.	−	70550.0i	0.389112	−	0.213384i
$576$		0			0
$577$	−154601.	+	154601.i	−0.464366	+	0.464366i	−0.900084	−	0.435717i	$-0.856495\pi$
$577$	−154601.	+	154601.i	−0.464366	+	0.464366i	0.435717	+	0.900084i	$0.356495\pi$
$578$	83519.0	+	83519.0i	0.249994	+	0.249994i
$579$		0			0
$580$	−134400.	+	100800.i	−0.399524	+	0.299643i
$581$	294008.			0.870977
$582$		0			0
$583$	26648.0	+	26648.0i	0.0784021	+	0.0784021i
$584$		−	47460.0i		−	0.139156i
$585$		0			0
$586$	−56998.0			−0.165983
$587$	−222974.	+	222974.i	−0.647110	+	0.647110i	−0.952294	−	0.305184i	$-0.901282\pi$
$587$	−222974.	+	222974.i	−0.647110	+	0.647110i	0.305184	+	0.952294i	$0.401282\pi$
$588$		0			0
$589$		−	102960.i		−	0.296782i
$590$	128100.	+	18300.0i	0.367998	+	0.0525711i
$591$		0			0
$592$	41164.0	−	41164.0i	0.117456	−	0.117456i
$593$	−148049.	−	148049.i	−0.421014	−	0.421014i	0.464539	−	0.885553i	$-0.346220\pi$
$593$	−148049.	−	148049.i	−0.421014	−	0.421014i	−0.885553	+	0.464539i	$0.846220\pi$
$594$		0			0
$595$	−780.000	−	1040.00i	−0.00220323	−	0.00293765i
$596$	228900.			0.644397
$597$		0			0
$598$	46148.0	+	46148.0i	0.129048	+	0.129048i
$599$			31800.0i			0.0886285i	0.999018	+	0.0443143i	$0.0141103\pi$
$599$			31800.0i			0.0886285i	−0.999018	+	0.0443143i	$0.985890\pi$
$600$		0			0
$601$	−71848.0			−0.198914			−0.0994571	−	0.995042i	$-0.531711\pi$
$601$	−71848.0			−0.198914			−0.0994571	+	0.995042i	$0.531711\pi$
$602$	−76648.0	+	76648.0i	−0.211499	+	0.211499i
$603$		0			0
$604$			14728.0i			0.0403710i
$605$	291540.	−	218655.i	0.796503	−	0.597377i
$606$		0			0
$607$	−13526.0	+	13526.0i	−0.0367106	+	0.0367106i	−0.725224	−	0.688513i	$-0.758264\pi$
$607$	−13526.0	+	13526.0i	−0.0367106	+	0.0367106i	0.688513	+	0.725224i	$0.258264\pi$
$608$	115920.	+	115920.i	0.313582	+	0.313582i
$609$		0			0
$610$	−7960.00	+	55720.0i	−0.0213921	+	0.149745i
$611$	−687772.			−1.84231
$612$		0			0
$613$	−303461.	−	303461.i	−0.807573	−	0.807573i	0.176693	−	0.984266i	$-0.443460\pi$
$613$	−303461.	−	303461.i	−0.807573	−	0.807573i	−0.984266	+	0.176693i	$0.943460\pi$
$614$			235852.i			0.625609i
$615$		0			0
$616$	−12480.0			−0.0328892
$617$	−122399.	+	122399.i	−0.321520	+	0.321520i	−0.849350	−	0.527830i	$-0.823006\pi$
$617$	−122399.	+	122399.i	−0.321520	+	0.321520i	0.527830	+	0.849350i	$0.323006\pi$
$618$		0			0
$619$			110220.i			0.287660i	0.989602	+	0.143830i	$0.0459418\pi$
$619$			110220.i			0.287660i	−0.989602	+	0.143830i	$0.954058\pi$
$620$	−120120.	−	160160.i	−0.312487	−	0.416649i
$621$		0			0
$622$	−3892.00	+	3892.00i	−0.0100599	+	0.0100599i
$623$	−56160.0	−	56160.0i	−0.144694	−	0.144694i
$624$		0			0
$625$	−329375.	−	210000.i	−0.843200	−	0.537600i
$626$	−99922.0			−0.254984
$627$		0			0
$628$	90986.0	+	90986.0i	0.230704	+	0.230704i
$629$			502.000i			0.00126883i
$630$		0			0
$631$	620372.			1.55809			0.779047	−	0.626966i	$-0.215703\pi$
$631$	620372.			1.55809			0.779047	+	0.626966i	$0.215703\pi$
$632$	273600.	−	273600.i	0.684986	−	0.684986i
$633$		0			0
$634$			34198.0i			0.0850790i
$635$	−62870.0	+	440090.i	−0.155918	+	1.09143i
$636$		0			0
$637$	−145811.	+	145811.i	−0.359345	+	0.359345i
$638$	−3840.00	−	3840.00i	−0.00943387	−	0.00943387i
$639$		0			0
$640$	407400.	+	58200.0i	0.994629	+	0.142090i
$641$	722888.			1.75936			0.879680	−	0.475565i	$-0.157756\pi$
$641$	722888.			1.75936			0.879680	+	0.475565i	$0.157756\pi$
$642$		0			0
$643$	−393026.	−	393026.i	−0.950603	−	0.950603i	0.0482328	−	0.998836i	$-0.484641\pi$
$643$	−393026.	−	393026.i	−0.950603	−	0.950603i	−0.998836	+	0.0482328i	$0.984641\pi$
$644$		−	120848.i		−	0.291385i
$645$		0			0
$646$	−360.000			−0.000862656
$647$	338626.	−	338626.i	0.808931	−	0.808931i	−0.175541	−	0.984472i	$-0.556167\pi$
$647$	338626.	−	338626.i	0.808931	−	0.808931i	0.984472	+	0.175541i	$0.0561674\pi$
$648$		0			0
$649$		−	29280.0i		−	0.0695155i
$650$	48650.0	−	166800.i	0.115148	−	0.394793i
$651$		0			0
$652$	270004.	−	270004.i	0.635148	−	0.635148i
$653$	−254669.	−	254669.i	−0.597241	−	0.597241i	0.342336	−	0.939577i	$-0.388782\pi$
$653$	−254669.	−	254669.i	−0.597241	−	0.597241i	−0.939577	+	0.342336i	$0.888782\pi$
$654$		0			0
$655$	405440.	−	304080.i	0.945027	−	0.708770i
$656$	−276832.			−0.643293
$657$		0			0
$658$	−128648.	−	128648.i	−0.297133	−	0.297133i
$659$		−	603660.i		−	1.39002i	−0.718999	−	0.695011i	$-0.755400\pi$
$659$		−	603660.i		−	1.39002i	0.718999	−	0.695011i	$-0.244600\pi$
$660$		0			0
$661$	14792.0			0.0338551			0.0169275	−	0.999857i	$-0.494612\pi$
$661$	14792.0			0.0338551			0.0169275	+	0.999857i	$0.494612\pi$
$662$	−143128.	+	143128.i	−0.326594	+	0.326594i
$663$		0			0
$664$		−	339240.i		−	0.769433i
$665$	−163800.	−	23400.0i	−0.370400	−	0.0529142i
$666$		0			0
$667$	79680.0	−	79680.0i	0.179101	−	0.179101i
$668$	217364.	+	217364.i	0.487119	+	0.487119i
$669$		0			0
$670$	26220.0	+	34960.0i	0.0584094	+	0.0778793i
$671$	12736.0			0.0282871
$672$		0			0
$673$	300409.	+	300409.i	0.663258	+	0.663258i	0.956147	−	0.292888i	$-0.0946164\pi$
$673$	300409.	+	300409.i	0.663258	+	0.663258i	−0.292888	+	0.956147i	$0.594616\pi$
$674$		−	206498.i		−	0.454565i
$675$		0			0
$676$	−141134.			−0.308843
$677$	256051.	−	256051.i	0.558662	−	0.558662i	−0.370264	−	0.928926i	$-0.620733\pi$
$677$	256051.	−	256051.i	0.558662	−	0.558662i	0.928926	+	0.370264i	$0.120733\pi$
$678$		0			0
$679$		−	340652.i		−	0.738876i
$680$	−1200.00	+	900.000i	−0.00259516	+	0.00194637i
$681$		0			0
$682$	4576.00	−	4576.00i	0.00983824	−	0.00983824i
$683$	−341954.	−	341954.i	−0.733038	−	0.733038i	0.238183	−	0.971220i	$-0.423448\pi$
$683$	−341954.	−	341954.i	−0.733038	−	0.733038i	−0.971220	+	0.238183i	$0.923448\pi$
$684$		0			0
$685$	−51755.0	+	362285.i	−0.110299	+	0.772092i
$686$	−179400.			−0.381219
$687$		0			0
$688$	−241736.	−	241736.i	−0.510698	−	0.510698i
$689$			926018.i			1.95066i
$690$		0			0
$691$	−717688.			−1.50307			−0.751536	−	0.659692i	$-0.770687\pi$
$691$	−717688.			−1.50307			−0.751536	+	0.659692i	$0.770687\pi$
$692$	−236474.	+	236474.i	−0.493823	+	0.493823i
$693$		0			0
$694$		−	209252.i		−	0.434461i
$695$	405900.	+	541200.i	0.840329	+	1.12044i
$696$		0			0
$697$	1688.00	−	1688.00i	0.00347462	−	0.00347462i
$698$	−94800.0	−	94800.0i	−0.194580	−	0.194580i
$699$		0			0
$700$	−282100.	+	154700.i	−0.575714	+	0.315714i
$701$	−267352.			−0.544061			−0.272030	−	0.962289i	$-0.587695\pi$
$701$	−267352.			−0.544061			−0.272030	+	0.962289i	$0.587695\pi$
$702$		0			0
$703$	45180.0	+	45180.0i	0.0914188	+	0.0914188i
$704$		−	10688.0i		−	0.0215651i
$705$		0			0
$706$	−127138.			−0.255074
$707$	418652.	−	418652.i	0.837557	−	0.837557i
$708$		0			0
$709$			159360.i			0.317020i	0.987357	+	0.158510i	$0.0506690\pi$
$709$			159360.i			0.317020i	−0.987357	+	0.158510i	$0.949331\pi$
$710$	−30340.0	+	212380.i	−0.0601865	+	0.421305i
$711$		0			0
$712$	−64800.0	+	64800.0i	−0.127825	+	0.127825i
$713$	94952.0	+	94952.0i	0.186778	+	0.186778i
$714$		0			0
$715$	−38920.0	−	5560.00i	−0.0761309	−	0.0108758i
$716$	−148680.			−0.290019
$717$		0			0
$718$	141840.	+	141840.i	0.275138	+	0.275138i
$719$			364680.i			0.705430i	0.935731	+	0.352715i	$0.114742\pi$
$719$			364680.i			0.705430i	−0.935731	+	0.352715i	$0.885258\pi$
$720$		0			0
$721$	−51688.0			−0.0994304
$722$	97921.0	−	97921.0i	0.187846	−	0.187846i
$723$		0			0
$724$			337708.i			0.644265i
$725$	−288000.	−	84000.0i	−0.547919	−	0.159810i
$726$		0			0
$727$	716374.	−	716374.i	1.35541	−	1.35541i	0.475925	−	0.879486i	$-0.342113\pi$
$727$	716374.	−	716374.i	1.35541	−	1.35541i	0.879486	−	0.475925i	$-0.157887\pi$
$728$	−216840.	−	216840.i	−0.409144	−	0.409144i
$729$		0			0
$730$	31640.0	−	23730.0i	0.0593732	−	0.0445299i
$731$	2948.00			0.00551687
$732$		0			0
$733$	641029.	+	641029.i	1.19308	+	1.19308i	0.976198	+	0.216883i	$0.0695889\pi$
$733$	641029.	+	641029.i	1.19308	+	1.19308i	0.216883	+	0.976198i	$0.430411\pi$
$734$		−	284348.i		−	0.527786i
$735$		0			0
$736$	−213808.			−0.394701
$737$	6992.00	−	6992.00i	0.0128726	−	0.0128726i
$738$		0			0
$739$			607140.i			1.11173i	0.831272	+	0.555866i	$0.187613\pi$
$739$			607140.i			1.11173i	−0.831272	+	0.555866i	$0.812387\pi$
$740$	122990.	+	17570.0i	0.224598	+	0.0320855i
$741$		0			0
$742$	−173212.	+	173212.i	−0.314608	+	0.314608i
$743$	248026.	+	248026.i	0.449283	+	0.449283i	0.895116	−	0.445833i	$-0.147093\pi$
$743$	248026.	+	248026.i	0.449283	+	0.449283i	−0.445833	+	0.895116i	$0.647093\pi$
$744$		0			0
$745$	245250.	+	327000.i	0.441872	+	0.589163i
$746$	20618.0			0.0370484
$747$		0			0
$748$	112.000	+	112.000i	0.000200177	+	0.000200177i
$749$			432952.i			0.771749i
$750$		0			0
$751$	169052.			0.299737			0.149869	−	0.988706i	$-0.452115\pi$
$751$	169052.			0.299737			0.149869	+	0.988706i	$0.452115\pi$
$752$	405736.	−	405736.i	0.717477	−	0.717477i
$753$		0			0
$754$		−	133440.i		−	0.234716i
$755$	−21040.0	+	15780.0i	−0.0369107	+	0.0276830i
$756$		0			0
$757$	−166301.	+	166301.i	−0.290204	+	0.290204i	−0.837161	−	0.546957i	$-0.815786\pi$
$757$	−166301.	+	166301.i	−0.290204	+	0.290204i	0.546957	+	0.837161i	$0.315786\pi$
$758$	115380.	+	115380.i	0.200813	+	0.200813i
$759$		0			0
$760$	−27000.0	+	189000.i	−0.0467452	+	0.327216i
$761$	−557842.			−0.963256			−0.481628	−	0.876376i	$-0.659954\pi$
$761$	−557842.			−0.963256			−0.481628	+	0.876376i	$0.659954\pi$
$762$		0			0
$763$	−442260.	−	442260.i	−0.759676	−	0.759676i
$764$			632632.i			1.08384i
$765$		0			0
$766$	−125308.			−0.213561
$767$	508740.	−	508740.i	0.864779	−	0.864779i
$768$		0			0
$769$			678720.i			1.14773i	0.818952	+	0.573863i	$0.194556\pi$
$769$			678720.i			1.14773i	−0.818952	+	0.573863i	$0.805444\pi$
$770$	−6240.00	−	8320.00i	−0.0105245	−	0.0140327i
$771$		0			0
$772$	−548786.	+	548786.i	−0.920807	+	0.920807i
$773$	164341.	+	164341.i	0.275034	+	0.275034i	0.831123	−	0.556089i	$-0.187699\pi$
$773$	164341.	+	164341.i	0.275034	+	0.275034i	−0.556089	+	0.831123i	$0.687699\pi$
$774$		0			0
$775$	100100.	−	343200.i	0.166660	−	0.571405i
$776$	−393060.			−0.652733
$777$		0			0
$778$	−132690.	−	132690.i	−0.219219	−	0.219219i
$779$		−	303840.i		−	0.500691i
$780$		0			0
$781$	48544.0			0.0795854
$782$	332.000	−	332.000i	0.000542906	−	0.000542906i
$783$		0			0
$784$		−	172036.i		−	0.279890i
$785$	−32495.0	+	227465.i	−0.0527324	+	0.369127i
$786$		0			0
$787$	323074.	−	323074.i	0.521618	−	0.521618i	−0.396442	−	0.918060i	$-0.629755\pi$
$787$	323074.	−	323074.i	0.521618	−	0.521618i	0.918060	+	0.396442i	$0.129755\pi$
$788$	−396886.	−	396886.i	−0.639166	−	0.639166i
$789$		0			0
$790$	319200.	+	45600.0i	0.511456	+	0.0730652i
$791$	−268372.			−0.428928
$792$		0			0
$793$	221288.	+	221288.i	0.351894	+	0.351894i
$794$			176902.i			0.280603i
$795$		0			0
$796$	−25200.0			−0.0397717
$797$	−510299.	+	510299.i	−0.803356	+	0.803356i	−0.983619	−	0.180262i	$-0.942305\pi$
$797$	−510299.	+	510299.i	−0.803356	+	0.803356i	0.180262	+	0.983619i	$0.442305\pi$
$798$		0			0
$799$			4948.00i			0.00775061i
$800$	273700.	+	499100.i	0.427656	+	0.779844i
$801$		0			0
$802$	−63202.0	+	63202.0i	−0.0982612	+	0.0982612i
$803$	−6328.00	−	6328.00i	−0.00981376	−	0.00981376i
$804$		0			0
$805$	172640.	−	129480.i	0.266409	−	0.199807i
$806$	159016.			0.244777
$807$		0			0
$808$	−483060.	−	483060.i	−0.739909	−	0.739909i
$809$		−	1.18656e6i		−	1.81298i	−0.422229	−	0.906489i	$-0.638752\pi$
$809$		−	1.18656e6i		−	1.81298i	0.422229	−	0.906489i	$-0.361248\pi$
$810$		0			0
$811$	−431008.			−0.655305			−0.327653	−	0.944798i	$-0.606258\pi$
$811$	−431008.			−0.655305			−0.327653	+	0.944798i	$0.606258\pi$
$812$	−174720.	+	174720.i	−0.264991	+	0.264991i
$813$		0			0
$814$			4016.00i			0.00606101i
$815$	675010.	+	96430.0i	1.01624	+	0.145177i
$816$		0			0
$817$	265320.	−	265320.i	0.397490	−	0.397490i
$818$	−52890.0	−	52890.0i	−0.0790436	−	0.0790436i
$819$		0			0
$820$	−354480.	−	472640.i	−0.527186	−	0.702915i
$821$	639368.			0.948560			0.474280	−	0.880374i	$-0.342709\pi$
$821$	639368.			0.948560			0.474280	+	0.880374i	$0.342709\pi$
$822$		0			0
$823$	−74966.0	−	74966.0i	−0.110679	−	0.110679i	0.649599	−	0.760277i	$-0.274937\pi$
$823$	−74966.0	−	74966.0i	−0.110679	−	0.110679i	−0.760277	+	0.649599i	$0.774937\pi$
$824$			59640.0i			0.0878382i
$825$		0			0
$826$	190320.			0.278949
$827$	144226.	−	144226.i	0.210879	−	0.210879i	−0.593762	−	0.804641i	$-0.702358\pi$
$827$	144226.	−	144226.i	0.210879	−	0.210879i	0.804641	+	0.593762i	$0.202358\pi$
$828$		0			0
$829$		−	685170.i		−	0.996987i	−0.866894	−	0.498493i	$-0.833887\pi$
$829$		−	685170.i		−	0.996987i	0.866894	−	0.498493i	$-0.166113\pi$
$830$	226160.	−	169620.i	0.328291	−	0.246219i
$831$		0			0
$832$	185704.	−	185704.i	0.268272	−	0.268272i
$833$	1049.00	+	1049.00i	0.00151177	+	0.00151177i
$834$		0			0
$835$	−77630.0	+	543410.i	−0.111341	+	0.779390i
$836$	20160.0			0.0288455
$837$		0			0
$838$	256980.	+	256980.i	0.365941	+	0.365941i
$839$			445560.i			0.632969i	0.948598	+	0.316484i	$0.102502\pi$
$839$			445560.i			0.632969i	−0.948598	+	0.316484i	$0.897498\pi$
$840$		0			0
$841$	476881.			0.674245
$842$	186632.	−	186632.i	0.263246	−	0.263246i
$843$		0			0
$844$			257488.i			0.361470i
$845$	−151215.	−	201620.i	−0.211778	−	0.282371i
$846$		0			0
$847$	379002.	−	379002.i	0.528293	−	0.528293i
$848$	−546284.	−	546284.i	−0.759673	−	0.759673i
$849$		0			0
$850$	−1200.00	−	350.000i	−0.00166090	−	0.000484429i
$851$	−83332.0			−0.115068
$852$		0			0
$853$	−319781.	−	319781.i	−0.439496	−	0.439496i	0.452347	−	0.891842i	$-0.350587\pi$
$853$	−319781.	−	319781.i	−0.439496	−	0.439496i	−0.891842	+	0.452347i	$0.850587\pi$
$854$			82784.0i			0.113509i
$855$		0			0
$856$	499560.			0.681774
$857$	−576449.	+	576449.i	−0.784873	+	0.784873i	−0.980649	−	0.195776i	$-0.937277\pi$
$857$	−576449.	+	576449.i	−0.784873	+	0.784873i	0.195776	+	0.980649i	$0.437277\pi$
$858$		0			0
$859$		−	807540.i		−	1.09440i	−0.837001	−	0.547202i	$-0.815693\pi$
$859$		−	807540.i		−	1.09440i	0.837001	−	0.547202i	$-0.184307\pi$
$860$	103180.	−	722260.i	0.139508	−	0.976555i
$861$		0			0
$862$	208028.	−	208028.i	0.279967	−	0.279967i
$863$	−507914.	−	507914.i	−0.681975	−	0.681975i	0.278470	−	0.960445i	$-0.410173\pi$
$863$	−507914.	−	507914.i	−0.681975	−	0.681975i	−0.960445	+	0.278470i	$0.910173\pi$
$864$		0			0
$865$	−591185.	−	84455.0i	−0.790117	−	0.112874i
$866$	−302542.			−0.403413
$867$		0			0
$868$	−208208.	−	208208.i	−0.276349	−	0.276349i
$869$		−	72960.0i		−	0.0966152i
$870$		0			0
$871$	242972.			0.320273
$872$	−510300.	+	510300.i	−0.671108	+	0.671108i
$873$		0			0
$874$		−	59760.0i		−	0.0782326i
$875$	−523250.	−	237250.i	−0.683429	−	0.309878i
$876$		0			0
$877$	−673901.	+	673901.i	−0.876187	+	0.876187i	−0.993138	−	0.116951i	$-0.962688\pi$
$877$	−673901.	+	673901.i	−0.876187	+	0.876187i	0.116951	+	0.993138i	$0.462688\pi$
$878$	158640.	+	158640.i	0.205790	+	0.205790i
$879$		0			0
$880$	26240.0	−	19680.0i	0.0338843	−	0.0254132i
$881$	1.33753e6			1.72326			0.861631	−	0.507536i	$-0.169444\pi$
$881$	1.33753e6			1.72326			0.861631	+	0.507536i	$0.169444\pi$
$882$		0			0
$883$	131554.	+	131554.i	0.168726	+	0.168726i	0.786419	−	0.617693i	$-0.211933\pi$
$883$	131554.	+	131554.i	0.168726	+	0.168726i	−0.617693	+	0.786419i	$0.711933\pi$
$884$			3892.00i			0.00498045i
$885$		0			0
$886$	−505948.			−0.644523
$887$	327826.	−	327826.i	0.416674	−	0.416674i	−0.467382	−	0.884056i	$-0.654803\pi$
$887$	327826.	−	327826.i	0.416674	−	0.416674i	0.884056	+	0.467382i	$0.154803\pi$
$888$		0			0
$889$			653848.i			0.827320i
$890$	−75600.0	−	10800.0i	−0.0954425	−	0.0136346i
$891$		0			0
$892$	12124.0	−	12124.0i	0.0152376	−	0.0152376i
$893$	445320.	+	445320.i	0.558431	+	0.558431i
$894$		0			0
$895$	−159300.	−	212400.i	−0.198870	−	0.265160i
$896$	605280.			0.753946
$897$		0			0
$898$	123750.	+	123750.i	0.153459	+	0.153459i
$899$		−	274560.i		−	0.339717i
$900$		0			0
$901$	6662.00			0.00820644
$902$	13504.0	−	13504.0i	0.0165978	−	0.0165978i
$903$		0			0
$904$			309660.i			0.378921i
$905$	−482440.	+	361830.i	−0.589042	+	0.441781i
$906$		0			0
$907$	105274.	−	105274.i	0.127970	−	0.127970i	−0.640221	−	0.768191i	$-0.721157\pi$
$907$	105274.	−	105274.i	0.127970	−	0.127970i	0.768191	+	0.640221i	$0.221157\pi$
$908$	633164.	+	633164.i	0.767970	+	0.767970i
$909$		0			0
$910$	36140.0	−	252980.i	0.0436421	−	0.305495i
$911$	−1.59209e6			−1.91837			−0.959183	−	0.282787i	$-0.908741\pi$
$911$	−1.59209e6			−1.91837			−0.959183	+	0.282787i	$0.908741\pi$
$912$		0			0
$913$	−45232.0	−	45232.0i	−0.0542631	−	0.0542631i
$914$			64402.0i			0.0770916i
$915$		0			0
$916$	547680.			0.652734
$917$	527072.	−	527072.i	0.626803	−	0.626803i
$918$		0			0
$919$		−	515880.i		−	0.610826i	−0.952220	−	0.305413i	$-0.901205\pi$
$919$		−	515880.i		−	0.610826i	0.952220	−	0.305413i	$-0.0987946\pi$
$920$	−149400.	−	199200.i	−0.176512	−	0.235350i
$921$		0			0
$922$	−75142.0	+	75142.0i	−0.0883936	+	0.0883936i
$923$	843452.	+	843452.i	0.990050	+	0.990050i
$924$		0			0
$925$	106675.	+	194525.i	0.124675	+	0.227348i
$926$	471428.			0.549786
$927$		0			0
$928$	309120.	+	309120.i	0.358948	+	0.358948i
$929$		−	1.60023e6i		−	1.85418i	−0.374844	−	0.927088i	$-0.622304\pi$
$929$		−	1.60023e6i		−	1.85418i	0.374844	−	0.927088i	$-0.377696\pi$
$930$		0			0
$931$	188820.			0.217846
$932$	−463694.	+	463694.i	−0.533826	+	0.533826i
$933$		0			0
$934$			57148.0i			0.0655100i
$935$	−40.0000	+	280.000i	−4.57548e−5	+	0.000320284i
$936$		0			0
$937$	−908801.	+	908801.i	−1.03512	+	1.03512i	−0.0357569	+	0.999361i	$0.511384\pi$
$937$	−908801.	+	908801.i	−1.03512	+	1.03512i	−0.999361	+	0.0357569i	$0.988616\pi$
$938$	45448.0	+	45448.0i	0.0516546	+	0.0516546i
$939$		0			0
$940$	1.21226e6	+	173180.i	1.37196	+	0.195994i
$941$	−455962.			−0.514931			−0.257466	−	0.966287i	$-0.582887\pi$
$941$	−455962.			−0.514931			−0.257466	+	0.966287i	$0.582887\pi$
$942$		0			0
$943$	280208.	+	280208.i	0.315106	+	0.315106i
$944$			600240.i			0.673567i
$945$		0			0
$946$	23584.0			0.0263533
$947$	−463274.	+	463274.i	−0.516580	+	0.516580i	−0.916535	−	0.399955i	$-0.869026\pi$
$947$	−463274.	+	463274.i	−0.516580	+	0.516580i	0.399955	+	0.916535i	$0.369026\pi$
$948$		0			0
$949$		−	219898.i		−	0.244168i
$950$	−139500.	+	76500.0i	−0.154571	+	0.0847645i
$951$		0			0
$952$	−1560.00	+	1560.00i	−0.00172128	+	0.00172128i
$953$	1.03428e6	+	1.03428e6i	1.13881	+	1.13881i	0.988663	+	0.150151i	$0.0479759\pi$
$953$	1.03428e6	+	1.03428e6i	1.13881	+	1.13881i	0.150151	+	0.988663i	$0.452024\pi$
$954$		0			0
$955$	−903760.	+	677820.i	−0.990938	+	0.743203i
$956$	1.23816e6			1.35476
$957$		0			0
$958$	26520.0	+	26520.0i	0.0288963	+	0.0288963i
$959$			538252.i			0.585259i
$960$		0			0
$961$	−596337.			−0.645721
$962$	−69778.0	+	69778.0i	−0.0753995	+	0.0753995i
$963$		0			0
$964$			284368.i			0.306004i
$965$	−1.37196e6	−	195995.i	−1.47329	−	0.210470i
$966$		0			0
$967$	−1.10253e6	+	1.10253e6i	−1.17906	+	1.17906i	−0.199076	+	0.979984i	$0.563794\pi$
$967$	−1.10253e6	+	1.10253e6i	−1.17906	+	1.17906i	−0.979984	+	0.199076i	$0.936206\pi$
$968$	−437310.	−	437310.i	−0.466701	−	0.466701i
$969$		0			0
$970$	−196530.	−	262040.i	−0.208874	−	0.278499i
$971$	264368.			0.280395			0.140198	−	0.990124i	$-0.455226\pi$
$971$	264368.			0.280395			0.140198	+	0.990124i	$0.455226\pi$
$972$		0			0
$973$	703560.	+	703560.i	0.743148	+	0.743148i
$974$		−	82748.0i		−	0.0872247i
$975$		0			0
$976$	−261088.			−0.274086
$977$	−866249.	+	866249.i	−0.907515	+	0.907515i	−0.996071	−	0.0885566i	$-0.971775\pi$
$977$	−866249.	+	866249.i	−0.907515	+	0.907515i	0.0885566	+	0.996071i	$0.471775\pi$
$978$		0			0
$979$			17280.0i			0.0180293i
$980$	293720.	−	220290.i	0.305831	−	0.229373i
$981$		0			0
$982$	149288.	−	149288.i	0.154811	−	0.154811i
$983$	−1.08205e6	−	1.08205e6i	−1.11980	−	1.11980i	−0.991770	−	0.128034i	$-0.959133\pi$
$983$	−1.08205e6	−	1.08205e6i	−1.11980	−	1.11980i	−0.128034	−	0.991770i	$-0.540867\pi$
$984$		0			0
$985$	141745.	−	992215.i	0.146095	−	1.02266i
$986$	−960.000			−0.000987455
$987$		0			0
$988$	350280.	+	350280.i	0.358840	+	0.358840i
$989$			489368.i			0.500314i
$990$		0			0
$991$	−54988.0			−0.0559913			−0.0279957	−	0.999608i	$-0.508912\pi$
$991$	−54988.0			−0.0559913			−0.0279957	+	0.999608i	$0.508912\pi$
$992$	−368368.	+	368368.i	−0.374333	+	0.374333i
$993$		0			0
$994$			315536.i			0.319357i
$995$	−27000.0	−	36000.0i	−0.0272720	−	0.0363627i
$996$		0			0
$997$	73099.0	−	73099.0i	0.0735396	−	0.0735396i	−0.669380	−	0.742920i	$-0.733440\pi$
$997$	73099.0	−	73099.0i	0.0735396	−	0.0735396i	0.742920	+	0.669380i	$0.233440\pi$
$998$	−284100.	−	284100.i	−0.285240	−	0.285240i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.5.g.b.28.1		2
3.2	odd	2		5.5.c.a.3.1	yes	2
5.2	odd	4	inner	45.5.g.b.37.1		2
5.3	odd	4		225.5.g.b.82.1		2
5.4	even	2		225.5.g.b.118.1		2
12.11	even	2		80.5.p.d.33.1		2
15.2	even	4		5.5.c.a.2.1	✓	2
15.8	even	4		25.5.c.a.7.1		2
15.14	odd	2		25.5.c.a.18.1		2
24.5	odd	2		320.5.p.h.193.1		2
24.11	even	2		320.5.p.c.193.1		2
60.23	odd	4		400.5.p.a.257.1		2
60.47	odd	4		80.5.p.d.17.1		2
60.59	even	2		400.5.p.a.193.1		2
120.77	even	4		320.5.p.h.257.1		2
120.107	odd	4		320.5.p.c.257.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.5.c.a.2.1	✓	2	15.2	even	4
5.5.c.a.3.1	yes	2	3.2	odd	2
25.5.c.a.7.1		2	15.8	even	4
25.5.c.a.18.1		2	15.14	odd	2
45.5.g.b.28.1		2	1.1	even	1	trivial
45.5.g.b.37.1		2	5.2	odd	4	inner
80.5.p.d.17.1		2	60.47	odd	4
80.5.p.d.33.1		2	12.11	even	2
225.5.g.b.82.1		2	5.3	odd	4
225.5.g.b.118.1		2	5.4	even	2
320.5.p.c.193.1		2	24.11	even	2
320.5.p.c.257.1		2	120.107	odd	4
320.5.p.h.193.1		2	24.5	odd	2
320.5.p.h.257.1		2	120.77	even	4
400.5.p.a.193.1		2	60.59	even	2
400.5.p.a.257.1		2	60.23	odd	4

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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	45.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{2} +14.0000i q^{4} +(-20.0000 + 15.0000i) q^{5} +(-26.0000 + 26.0000i) q^{7} +(30.0000 + 30.0000i) q^{8} +(-5.00000 + 35.0000i) q^{10} +8.00000 q^{11} +(139.000 + 139.000i) q^{13} +52.0000i q^{14} -164.000 q^{16} +(1.00000 - 1.00000i) q^{17} -180.000i q^{19} +(-210.000 - 280.000i) q^{20} +(8.00000 - 8.00000i) q^{22} +(166.000 + 166.000i) q^{23} +(175.000 - 600.000i) q^{25} +278.000 q^{26} +(-364.000 - 364.000i) q^{28} -480.000i q^{29} +572.000 q^{31} +(-644.000 + 644.000i) q^{32} -2.00000i q^{34} +(130.000 - 910.000i) q^{35} +(-251.000 + 251.000i) q^{37} +(-180.000 - 180.000i) q^{38} +(-1050.00 - 150.000i) q^{40} +1688.00 q^{41} +(1474.00 + 1474.00i) q^{43} +112.000i q^{44} +332.000 q^{46} +(-2474.00 + 2474.00i) q^{47} +1049.00i q^{49} +(-425.000 - 775.000i) q^{50} +(-1946.00 + 1946.00i) q^{52} +(3331.00 + 3331.00i) q^{53} +(-160.000 + 120.000i) q^{55} -1560.00 q^{56} +(-480.000 - 480.000i) q^{58} -3660.00i q^{59} +1592.00 q^{61} +(572.000 - 572.000i) q^{62} -1336.00i q^{64} +(-4865.00 - 695.000i) q^{65} +(874.000 - 874.000i) q^{67} +(14.0000 + 14.0000i) q^{68} +(-780.000 - 1040.00i) q^{70} +6068.00 q^{71} +(-791.000 - 791.000i) q^{73} +502.000i q^{74} +2520.00 q^{76} +(-208.000 + 208.000i) q^{77} -9120.00i q^{79} +(3280.00 - 2460.00i) q^{80} +(1688.00 - 1688.00i) q^{82} +(-5654.00 - 5654.00i) q^{83} +(-5.00000 + 35.0000i) q^{85} +2948.00 q^{86} +(240.000 + 240.000i) q^{88} +2160.00i q^{89} -7228.00 q^{91} +(-2324.00 + 2324.00i) q^{92} +4948.00i q^{94} +(2700.00 + 3600.00i) q^{95} +(-6551.00 + 6551.00i) q^{97} +(1049.00 + 1049.00i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 40 q^{5} - 52 q^{7} + 60 q^{8} - 10 q^{10} + 16 q^{11} + 278 q^{13} - 328 q^{16} + 2 q^{17} - 420 q^{20} + 16 q^{22} + 332 q^{23} + 350 q^{25} + 556 q^{26} - 728 q^{28} + 1144 q^{31} - 1288 q^{32}+ \cdots + 2098 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 40 * q^5 - 52 * q^7 + 60 * q^8 - 10 * q^10 + 16 * q^11 + 278 * q^13 - 328 * q^16 + 2 * q^17 - 420 * q^20 + 16 * q^22 + 332 * q^23 + 350 * q^25 + 556 * q^26 - 728 * q^28 + 1144 * q^31 - 1288 * q^32 + 260 * q^35 - 502 * q^37 - 360 * q^38 - 2100 * q^40 + 3376 * q^41 + 2948 * q^43 + 664 * q^46 - 4948 * q^47 - 850 * q^50 - 3892 * q^52 + 6662 * q^53 - 320 * q^55 - 3120 * q^56 - 960 * q^58 + 3184 * q^61 + 1144 * q^62 - 9730 * q^65 + 1748 * q^67 + 28 * q^68 - 1560 * q^70 + 12136 * q^71 - 1582 * q^73 + 5040 * q^76 - 416 * q^77 + 6560 * q^80 + 3376 * q^82 - 11308 * q^83 - 10 * q^85 + 5896 * q^86 + 480 * q^88 - 14456 * q^91 - 4648 * q^92 + 5400 * q^95 - 13102 * q^97 + 2098 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more