LMFDB - Embedded newform 624.3.ba.b.385.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,3,Mod(385,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("624.385");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	624.ba (of order $4$, 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:	$17.0027684961$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.1579585536.10
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 4x^{6} + 28x^{5} - 38x^{4} + 8x^{3} + 200x^{2} - 352x + 484 $ x^8 - 2x^7 - 4x^6 + 28x^5 - 38x^4 + 8x^3 + 200x^2 - 352*x + 484
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			385.2
Root			$0.252411 + 1.79004i$ of defining polynomial
Character	$\chi$	$=$	624.385
Dual form			624.3.ba.b.577.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.73205 q^{3} +(1.58008 + 1.58008i) q^{5} +(-3.96400 + 3.96400i) q^{7} +3.00000 q^{9} +(10.0345 - 10.0345i) q^{11} +(-5.41846 + 11.8170i) q^{13} +(-2.73677 - 2.73677i) q^{15} +6.40175i q^{17} +(-7.73551 - 7.73551i) q^{19} +(6.86585 - 6.86585i) q^{21} +34.8297i q^{23} -20.0067i q^{25} -5.19615 q^{27} +8.25124 q^{29} +(-13.8633 - 13.8633i) q^{31} +(-17.3803 + 17.3803i) q^{33} -12.5269 q^{35} +(-37.2522 + 37.2522i) q^{37} +(9.38504 - 20.4676i) q^{39} +(-28.7020 - 28.7020i) q^{41} +42.7195i q^{43} +(4.74023 + 4.74023i) q^{45} +(-37.5964 + 37.5964i) q^{47} +17.5734i q^{49} -11.0882i q^{51} -96.7934 q^{53} +31.7107 q^{55} +(13.3983 + 13.3983i) q^{57} +(-37.8935 + 37.8935i) q^{59} +91.2864 q^{61} +(-11.8920 + 11.8920i) q^{63} +(-27.2333 + 10.1101i) q^{65} +(42.4500 + 42.4500i) q^{67} -60.3268i q^{69} +(-29.8579 - 29.8579i) q^{71} +(-10.9949 + 10.9949i) q^{73} +34.6526i q^{75} +79.5538i q^{77} -102.031 q^{79} +9.00000 q^{81} +(85.2758 + 85.2758i) q^{83} +(-10.1153 + 10.1153i) q^{85} -14.2916 q^{87} +(-63.8510 + 63.8510i) q^{89} +(-25.3636 - 68.3212i) q^{91} +(24.0119 + 24.0119i) q^{93} -24.4454i q^{95} +(-58.7023 - 58.7023i) q^{97} +(30.1036 - 30.1036i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 20 q^{5} - 8 q^{7} + 24 q^{9} + 20 q^{11} + 8 q^{13} - 12 q^{15} + 40 q^{19} - 48 q^{21} + 56 q^{29} - 32 q^{31} - 36 q^{33} - 104 q^{35} - 16 q^{37} - 24 q^{39} - 116 q^{41} - 60 q^{45} - 100 q^{47}+ \cdots + 60 q^{99}+O(q^{100}) $ 8 * q - 20 * q^5 - 8 * q^7 + 24 * q^9 + 20 * q^11 + 8 * q^13 - 12 * q^15 + 40 * q^19 - 48 * q^21 + 56 * q^29 - 32 * q^31 - 36 * q^33 - 104 * q^35 - 16 * q^37 - 24 * q^39 - 116 * q^41 - 60 * q^45 - 100 * q^47 - 232 * q^53 + 176 * q^55 + 120 * q^57 - 316 * q^59 + 160 * q^61 - 24 * q^63 - 92 * q^65 - 176 * q^67 - 4 * q^71 - 64 * q^73 - 208 * q^79 + 72 * q^81 + 212 * q^83 - 168 * q^85 + 24 * q^87 - 452 * q^89 + 400 * q^91 - 24 * q^93 + 128 * q^97 + 60 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$1$	$1$

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			0
$2$		0			0
$3$	−1.73205			−0.577350
$3$	−1.73205			−0.577350
$4$		0			0
$5$	1.58008	+	1.58008i	0.316015	+	0.316015i	0.847234	−	0.531219i	$-0.178266\pi$
$5$	1.58008	+	1.58008i	0.316015	+	0.316015i	−0.531219	+	0.847234i	$0.678266\pi$
$6$		0			0
$7$	−3.96400	+	3.96400i	−0.566286	+	0.566286i	−0.931086	−	0.364800i	$-0.881137\pi$
$7$	−3.96400	+	3.96400i	−0.566286	+	0.566286i	0.364800	+	0.931086i	$0.381137\pi$
$8$		0			0
$9$	3.00000			0.333333
$10$		0			0
$11$	10.0345	−	10.0345i	0.912230	−	0.912230i	−0.0842172	−	0.996447i	$-0.526839\pi$
$11$	10.0345	−	10.0345i	0.912230	−	0.912230i	0.996447	+	0.0842172i	$0.0268389\pi$
$12$		0			0
$13$	−5.41846	+	11.8170i	−0.416804	+	0.908996i
$13$	−5.41846	+	11.8170i	−0.416804	+	0.908996i
$14$		0			0
$15$	−2.73677	−	2.73677i	−0.182452	−	0.182452i
$16$		0			0
$17$			6.40175i			0.376574i	0.982114	+	0.188287i	$0.0602934\pi$
$17$			6.40175i			0.376574i	−0.982114	+	0.188287i	$0.939707\pi$
$18$		0			0
$19$	−7.73551	−	7.73551i	−0.407132	−	0.407132i	0.473605	−	0.880737i	$-0.342952\pi$
$19$	−7.73551	−	7.73551i	−0.407132	−	0.407132i	−0.880737	+	0.473605i	$0.842952\pi$
$20$		0			0
$21$	6.86585	−	6.86585i	0.326945	−	0.326945i
$22$		0			0
$23$			34.8297i			1.51434i	0.653221	+	0.757168i	$0.273417\pi$
$23$			34.8297i			1.51434i	−0.653221	+	0.757168i	$0.726583\pi$
$24$		0			0
$25$		−	20.0067i		−	0.800269i
$26$		0			0
$27$	−5.19615			−0.192450
$28$		0			0
$29$	8.25124			0.284526			0.142263	−	0.989829i	$-0.454562\pi$
$29$	8.25124			0.284526			0.142263	+	0.989829i	$0.454562\pi$
$30$		0			0
$31$	−13.8633	−	13.8633i	−0.447202	−	0.447202i	0.447221	−	0.894423i	$-0.352414\pi$
$31$	−13.8633	−	13.8633i	−0.447202	−	0.447202i	−0.894423	+	0.447221i	$0.852414\pi$
$32$		0			0
$33$	−17.3803	+	17.3803i	−0.526676	+	0.526676i
$34$		0			0
$35$	−12.5269			−0.357910
$36$		0			0
$37$	−37.2522	+	37.2522i	−1.00682	+	1.00682i	−0.00683875	+	0.999977i	$0.502177\pi$
$37$	−37.2522	+	37.2522i	−1.00682	+	1.00682i	−0.999977	+	0.00683875i	$0.997823\pi$
$38$		0			0
$39$	9.38504	−	20.4676i	0.240642	−	0.524809i
$40$		0			0
$41$	−28.7020	−	28.7020i	−0.700050	−	0.700050i	0.264371	−	0.964421i	$-0.414836\pi$
$41$	−28.7020	−	28.7020i	−0.700050	−	0.700050i	−0.964421	+	0.264371i	$0.914836\pi$
$42$		0			0
$43$			42.7195i			0.993477i	0.867900	+	0.496739i	$0.165469\pi$
$43$			42.7195i			0.993477i	−0.867900	+	0.496739i	$0.834531\pi$
$44$		0			0
$45$	4.74023	+	4.74023i	0.105338	+	0.105338i
$46$		0			0
$47$	−37.5964	+	37.5964i	−0.799924	+	0.799924i	−0.983083	−	0.183159i	$-0.941368\pi$
$47$	−37.5964	+	37.5964i	−0.799924	+	0.799924i	0.183159	+	0.983083i	$0.441368\pi$
$48$		0			0
$49$			17.5734i			0.358641i
$50$		0			0
$51$		−	11.0882i		−	0.217415i
$52$		0			0
$53$	−96.7934			−1.82629			−0.913145	−	0.407635i	$-0.866354\pi$
$53$	−96.7934			−1.82629			−0.913145	+	0.407635i	$0.866354\pi$
$54$		0			0
$55$	31.7107			0.576558
$56$		0			0
$57$	13.3983	+	13.3983i	0.235058	+	0.235058i
$58$		0			0
$59$	−37.8935	+	37.8935i	−0.642262	+	0.642262i	−0.951111	−	0.308849i	$-0.900056\pi$
$59$	−37.8935	+	37.8935i	−0.642262	+	0.642262i	0.308849	+	0.951111i	$0.400056\pi$
$60$		0			0
$61$	91.2864			1.49650			0.748249	−	0.663418i	$-0.230895\pi$
$61$	91.2864			1.49650			0.748249	+	0.663418i	$0.230895\pi$
$62$		0			0
$63$	−11.8920	+	11.8920i	−0.188762	+	0.188762i
$64$		0			0
$65$	−27.2333	+	10.1101i	−0.418973	+	0.155540i
$66$		0			0
$67$	42.4500	+	42.4500i	0.633581	+	0.633581i	0.948964	−	0.315383i	$-0.102133\pi$
$67$	42.4500	+	42.4500i	0.633581	+	0.633581i	−0.315383	+	0.948964i	$0.602133\pi$
$68$		0			0
$69$		−	60.3268i		−	0.874302i
$70$		0			0
$71$	−29.8579	−	29.8579i	−0.420534	−	0.420534i	0.464853	−	0.885388i	$-0.346107\pi$
$71$	−29.8579	−	29.8579i	−0.420534	−	0.420534i	−0.885388	+	0.464853i	$0.846107\pi$
$72$		0			0
$73$	−10.9949	+	10.9949i	−0.150615	+	0.150615i	−0.778393	−	0.627777i	$-0.783965\pi$
$73$	−10.9949	+	10.9949i	−0.150615	+	0.150615i	0.627777	+	0.778393i	$0.283965\pi$
$74$		0			0
$75$			34.6526i			0.462035i
$76$		0			0
$77$			79.5538i			1.03317i
$78$		0			0
$79$	−102.031			−1.29153			−0.645764	−	0.763537i	$-0.723461\pi$
$79$	−102.031			−1.29153			−0.645764	+	0.763537i	$0.723461\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$	85.2758	+	85.2758i	1.02742	+	1.02742i	0.999613	+	0.0278064i	$0.00885219\pi$
$83$	85.2758	+	85.2758i	1.02742	+	1.02742i	0.0278064	+	0.999613i	$0.491148\pi$
$84$		0			0
$85$	−10.1153	+	10.1153i	−0.119003	+	0.119003i
$86$		0			0
$87$	−14.2916			−0.164271
$88$		0			0
$89$	−63.8510	+	63.8510i	−0.717427	+	0.717427i	−0.968078	−	0.250651i	$-0.919355\pi$
$89$	−63.8510	+	63.8510i	−0.717427	+	0.717427i	0.250651	+	0.968078i	$0.419355\pi$
$90$		0			0
$91$	−25.3636	−	68.3212i	−0.278721	−	0.750782i
$92$		0			0
$93$	24.0119	+	24.0119i	0.258192	+	0.258192i
$94$		0			0
$95$		−	24.4454i		−	0.257320i
$96$		0			0
$97$	−58.7023	−	58.7023i	−0.605178	−	0.605178i	0.336504	−	0.941682i	$-0.390755\pi$
$97$	−58.7023	−	58.7023i	−0.605178	−	0.605178i	−0.941682	+	0.336504i	$0.890755\pi$
$98$		0			0
$99$	30.1036	−	30.1036i	0.304077	−	0.304077i
$100$		0			0
$101$			130.126i			1.28837i	0.764869	+	0.644186i	$0.222804\pi$
$101$			130.126i			1.28837i	−0.764869	+	0.644186i	$0.777196\pi$
$102$		0			0
$103$		−	27.9217i		−	0.271085i	−0.990772	−	0.135542i	$-0.956722\pi$
$103$		−	27.9217i		−	0.271085i	0.990772	−	0.135542i	$-0.0432777\pi$
$104$		0			0
$105$	21.6971			0.206640
$106$		0			0
$107$	−166.168			−1.55297			−0.776486	−	0.630134i	$-0.783000\pi$
$107$	−166.168			−1.55297			−0.776486	+	0.630134i	$0.783000\pi$
$108$		0			0
$109$	−4.56393	−	4.56393i	−0.0418709	−	0.0418709i	0.685861	−	0.727732i	$-0.259426\pi$
$109$	−4.56393	−	4.56393i	−0.0418709	−	0.0418709i	−0.727732	+	0.685861i	$0.759426\pi$
$110$		0			0
$111$	64.5226	−	64.5226i	0.581285	−	0.581285i
$112$		0			0
$113$	−60.7525			−0.537633			−0.268817	−	0.963191i	$-0.586633\pi$
$113$	−60.7525			−0.537633			−0.268817	+	0.963191i	$0.586633\pi$
$114$		0			0
$115$	−55.0336	+	55.0336i	−0.478553	+	0.478553i
$116$		0			0
$117$	−16.2554	+	35.4509i	−0.138935	+	0.302999i
$118$		0			0
$119$	−25.3765	−	25.3765i	−0.213248	−	0.213248i
$120$		0			0
$121$		−	80.3837i		−	0.664328i
$122$		0			0
$123$	49.7134	+	49.7134i	0.404174	+	0.404174i
$124$		0			0
$125$	71.1141	−	71.1141i	0.568913	−	0.568913i
$126$		0			0
$127$		−	130.207i		−	1.02525i	−0.858613	−	0.512624i	$-0.828673\pi$
$127$		−	130.207i		−	1.02525i	0.858613	−	0.512624i	$-0.171327\pi$
$128$		0			0
$129$		−	73.9924i		−	0.573584i
$130$		0			0
$131$	−135.531			−1.03459			−0.517293	−	0.855809i	$-0.673060\pi$
$131$	−135.531			−1.03459			−0.517293	+	0.855809i	$0.673060\pi$
$132$		0			0
$133$	61.3271			0.461106
$134$		0			0
$135$	−8.21032	−	8.21032i	−0.0608172	−	0.0608172i
$136$		0			0
$137$	135.519	−	135.519i	0.989190	−	0.989190i	−0.0107526	−	0.999942i	$-0.503423\pi$
$137$	135.519	−	135.519i	0.989190	−	0.989190i	0.999942	+	0.0107526i	$0.00342272\pi$
$138$		0			0
$139$	244.006			1.75544			0.877720	−	0.479174i	$-0.159064\pi$
$139$	244.006			1.75544			0.877720	+	0.479174i	$0.159064\pi$
$140$		0			0
$141$	65.1189	−	65.1189i	0.461836	−	0.461836i
$142$		0			0
$143$	64.2059	+	172.949i	0.448992	+	1.20944i
$144$		0			0
$145$	13.0376	+	13.0376i	0.0899145	+	0.0899145i
$146$		0			0
$147$		−	30.4380i		−	0.207061i
$148$		0			0
$149$	174.887	+	174.887i	1.17374	+	1.17374i	0.981312	+	0.192425i	$0.0616351\pi$
$149$	174.887	+	174.887i	1.17374	+	1.17374i	0.192425	+	0.981312i	$0.438365\pi$
$150$		0			0
$151$	38.1095	−	38.1095i	0.252381	−	0.252381i	−0.569565	−	0.821946i	$-0.692888\pi$
$151$	38.1095	−	38.1095i	0.252381	−	0.252381i	0.821946	+	0.569565i	$0.192888\pi$
$152$		0			0
$153$			19.2053i			0.125525i
$154$		0			0
$155$		−	43.8101i		−	0.282646i
$156$		0			0
$157$	150.359			0.957699			0.478849	−	0.877897i	$-0.341054\pi$
$157$	150.359			0.957699			0.478849	+	0.877897i	$0.341054\pi$
$158$		0			0
$159$	167.651			1.05441
$160$		0			0
$161$	−138.065	−	138.065i	−0.857547	−	0.857547i
$162$		0			0
$163$	167.544	−	167.544i	1.02788	−	1.02788i	0.0282759	−	0.999600i	$-0.490998\pi$
$163$	167.544	−	167.544i	1.02788	−	1.02788i	0.999600	−	0.0282759i	$-0.00900169\pi$
$164$		0			0
$165$	−54.9245			−0.332876
$166$		0			0
$167$	33.7608	−	33.7608i	0.202160	−	0.202160i	−0.598765	−	0.800925i	$-0.704342\pi$
$167$	33.7608	−	33.7608i	0.202160	−	0.202160i	0.800925	+	0.598765i	$0.204342\pi$
$168$		0			0
$169$	−110.281	−	128.059i	−0.652548	−	0.757747i
$170$		0			0
$171$	−23.2065	−	23.2065i	−0.135711	−	0.135711i
$172$		0			0
$173$			150.230i			0.868383i	0.900821	+	0.434191i	$0.142966\pi$
$173$			150.230i			0.868383i	−0.900821	+	0.434191i	$0.857034\pi$
$174$		0			0
$175$	79.3066	+	79.3066i	0.453181	+	0.453181i
$176$		0			0
$177$	65.6334	−	65.6334i	0.370810	−	0.370810i
$178$		0			0
$179$		−	247.106i		−	1.38048i	−0.723582	−	0.690239i	$-0.757505\pi$
$179$		−	247.106i		−	1.38048i	0.723582	−	0.690239i	$-0.242495\pi$
$180$		0			0
$181$		−	111.674i		−	0.616986i	−0.951227	−	0.308493i	$-0.900175\pi$
$181$		−	111.674i		−	0.616986i	0.951227	−	0.308493i	$-0.0998247\pi$
$182$		0			0
$183$	−158.113			−0.864004
$184$		0			0
$185$	−117.723			−0.636338
$186$		0			0
$187$	64.2386	+	64.2386i	0.343522	+	0.343522i
$188$		0			0
$189$	20.5976	−	20.5976i	0.108982	−	0.108982i
$190$		0			0
$191$	64.6991			0.338739			0.169369	−	0.985553i	$-0.445827\pi$
$191$	64.6991			0.338739			0.169369	+	0.985553i	$0.445827\pi$
$192$		0			0
$193$	97.0986	−	97.0986i	0.503102	−	0.503102i	−0.409299	−	0.912400i	$-0.634227\pi$
$193$	97.0986	−	97.0986i	0.503102	−	0.503102i	0.912400	+	0.409299i	$0.134227\pi$
$194$		0			0
$195$	47.1694	−	17.5112i	0.241894	−	0.0898012i
$196$		0			0
$197$	48.8725	+	48.8725i	0.248084	+	0.248084i	0.820184	−	0.572100i	$-0.193871\pi$
$197$	48.8725	+	48.8725i	0.248084	+	0.248084i	−0.572100	+	0.820184i	$0.693871\pi$
$198$		0			0
$199$			139.570i			0.701358i	0.936496	+	0.350679i	$0.114049\pi$
$199$			139.570i			0.701358i	−0.936496	+	0.350679i	$0.885951\pi$
$200$		0			0
$201$	−73.5255	−	73.5255i	−0.365798	−	0.365798i
$202$		0			0
$203$	−32.7079	+	32.7079i	−0.161123	+	0.161123i
$204$		0			0
$205$		−	90.7029i		−	0.442453i
$206$		0			0
$207$			104.489i			0.504778i
$208$		0			0
$209$	−155.244			−0.742796
$210$		0			0
$211$	−52.9556			−0.250975			−0.125487	−	0.992095i	$-0.540049\pi$
$211$	−52.9556			−0.250975			−0.125487	+	0.992095i	$0.540049\pi$
$212$		0			0
$213$	51.7154	+	51.7154i	0.242796	+	0.242796i
$214$		0			0
$215$	−67.5001	+	67.5001i	−0.313954	+	0.313954i
$216$		0			0
$217$	109.908			0.506489
$218$		0			0
$219$	19.0438	−	19.0438i	0.0869578	−	0.0869578i
$220$		0			0
$221$	−75.6492	−	34.6876i	−0.342304	−	0.156958i
$222$		0			0
$223$	−70.4483	−	70.4483i	−0.315912	−	0.315912i	0.531283	−	0.847195i	$-0.321710\pi$
$223$	−70.4483	−	70.4483i	−0.315912	−	0.315912i	−0.847195	+	0.531283i	$0.821710\pi$
$224$		0			0
$225$		−	60.0201i		−	0.266756i
$226$		0			0
$227$	95.0195	+	95.0195i	0.418588	+	0.418588i	0.884717	−	0.466129i	$-0.154352\pi$
$227$	95.0195	+	95.0195i	0.418588	+	0.418588i	−0.466129	+	0.884717i	$0.654352\pi$
$228$		0			0
$229$	66.9011	−	66.9011i	0.292144	−	0.292144i	−0.545782	−	0.837927i	$-0.683767\pi$
$229$	66.9011	−	66.9011i	0.292144	−	0.292144i	0.837927	+	0.545782i	$0.183767\pi$
$230$		0			0
$231$		−	137.791i		−	0.596499i
$232$		0			0
$233$			81.9200i			0.351588i	0.984427	+	0.175794i	$0.0562493\pi$
$233$			81.9200i			0.351588i	−0.984427	+	0.175794i	$0.943751\pi$
$234$		0			0
$235$	−118.811			−0.505577
$236$		0			0
$237$	176.722			0.745664
$238$		0			0
$239$	−5.12005	−	5.12005i	−0.0214228	−	0.0214228i	0.696314	−	0.717737i	$-0.254822\pi$
$239$	−5.12005	−	5.12005i	−0.0214228	−	0.0214228i	−0.717737	+	0.696314i	$0.754822\pi$
$240$		0			0
$241$	−152.203	+	152.203i	−0.631546	+	0.631546i	−0.948456	−	0.316910i	$-0.897355\pi$
$241$	−152.203	+	152.203i	−0.631546	+	0.631546i	0.316910	+	0.948456i	$0.397355\pi$
$242$		0			0
$243$	−15.5885			−0.0641500
$244$		0			0
$245$	−27.7673	+	27.7673i	−0.113336	+	0.113336i
$246$		0			0
$247$	133.325	−	49.4956i	0.539776	−	0.200387i
$248$		0			0
$249$	−147.702	−	147.702i	−0.593181	−	0.593181i
$250$		0			0
$251$			221.877i			0.883974i	0.897022	+	0.441987i	$0.145726\pi$
$251$			221.877i			0.883974i	−0.897022	+	0.441987i	$0.854274\pi$
$252$		0			0
$253$	349.500	+	349.500i	1.38142	+	1.38142i
$254$		0			0
$255$	17.5201	−	17.5201i	0.0687064	−	0.0687064i
$256$		0			0
$257$		−	15.0363i		−	0.0585068i	−0.999572	−	0.0292534i	$-0.990687\pi$
$257$		−	15.0363i		−	0.0585068i	0.999572	−	0.0292534i	$-0.00931298\pi$
$258$		0			0
$259$		−	295.335i		−	1.14029i
$260$		0			0
$261$	24.7537			0.0948419
$262$		0			0
$263$	105.723			0.401990			0.200995	−	0.979592i	$-0.435582\pi$
$263$	105.723			0.401990			0.200995	+	0.979592i	$0.435582\pi$
$264$		0			0
$265$	−152.941	−	152.941i	−0.577136	−	0.577136i
$266$		0			0
$267$	110.593	−	110.593i	0.414207	−	0.414207i
$268$		0			0
$269$	168.740			0.627286			0.313643	−	0.949541i	$-0.398451\pi$
$269$	168.740			0.627286			0.313643	+	0.949541i	$0.398451\pi$
$270$		0			0
$271$	52.4787	−	52.4787i	0.193648	−	0.193648i	−0.603622	−	0.797271i	$-0.706276\pi$
$271$	52.4787	−	52.4787i	0.193648	−	0.193648i	0.797271	+	0.603622i	$0.206276\pi$
$272$		0			0
$273$	43.9311	+	118.336i	0.160920	+	0.433464i
$274$		0			0
$275$	−200.758	−	200.758i	−0.730029	−	0.730029i
$276$		0			0
$277$			153.018i			0.552412i	0.961098	+	0.276206i	$0.0890772\pi$
$277$			153.018i			0.552412i	−0.961098	+	0.276206i	$0.910923\pi$
$278$		0			0
$279$	−41.5898	−	41.5898i	−0.149067	−	0.149067i
$280$		0			0
$281$	37.8887	−	37.8887i	0.134835	−	0.134835i	−0.636468	−	0.771303i	$-0.719605\pi$
$281$	37.8887	−	37.8887i	0.134835	−	0.134835i	0.771303	+	0.636468i	$0.219605\pi$
$282$		0			0
$283$			34.7222i			0.122693i	0.998117	+	0.0613467i	$0.0195395\pi$
$283$			34.7222i			0.122693i	−0.998117	+	0.0613467i	$0.980460\pi$
$284$		0			0
$285$			42.3407i			0.148564i
$286$		0			0
$287$	227.550			0.792857
$288$		0			0
$289$	248.018			0.858192
$290$		0			0
$291$	101.675	+	101.675i	0.349400	+	0.349400i
$292$		0			0
$293$	54.4668	−	54.4668i	0.185894	−	0.185894i	−0.608025	−	0.793918i	$-0.708038\pi$
$293$	54.4668	−	54.4668i	0.185894	−	0.185894i	0.793918	+	0.608025i	$0.208038\pi$
$294$		0			0
$295$	−119.749			−0.405929
$296$		0			0
$297$	−52.1410	+	52.1410i	−0.175559	+	0.175559i
$298$		0			0
$299$	−411.581	−	188.723i	−1.37652	−	0.631182i
$300$		0			0
$301$	−169.340	−	169.340i	−0.562592	−	0.562592i
$302$		0			0
$303$		−	225.384i		−	0.743842i
$304$		0			0
$305$	144.240	+	144.240i	0.472916	+	0.472916i
$306$		0			0
$307$	−223.385	+	223.385i	−0.727639	+	0.727639i	−0.970149	−	0.242510i	$-0.922029\pi$
$307$	−223.385	+	223.385i	−0.727639	+	0.727639i	0.242510	+	0.970149i	$0.422029\pi$
$308$		0			0
$309$			48.3619i			0.156511i
$310$		0			0
$311$			516.479i			1.66070i	0.557240	+	0.830352i	$0.311860\pi$
$311$			516.479i			1.66070i	−0.557240	+	0.830352i	$0.688140\pi$
$312$		0			0
$313$	−149.235			−0.476788			−0.238394	−	0.971169i	$-0.576621\pi$
$313$	−149.235			−0.476788			−0.238394	+	0.971169i	$0.576621\pi$
$314$		0			0
$315$	−37.5806			−0.119303
$316$		0			0
$317$	−188.163	−	188.163i	−0.593574	−	0.593574i	0.345021	−	0.938595i	$-0.387872\pi$
$317$	−188.163	−	188.163i	−0.593574	−	0.593574i	−0.938595	+	0.345021i	$0.887872\pi$
$318$		0			0
$319$	82.7974	−	82.7974i	0.259553	−	0.259553i
$320$		0			0
$321$	287.811			0.896609
$322$		0			0
$323$	49.5208	−	49.5208i	0.153315	−	0.153315i
$324$		0			0
$325$	236.418	+	108.406i	0.727441	+	0.333555i
$326$		0			0
$327$	7.90495	+	7.90495i	0.0241742	+	0.0241742i
$328$		0			0
$329$		−	298.065i		−	0.905972i
$330$		0			0
$331$	133.749	+	133.749i	0.404074	+	0.404074i	0.879666	−	0.475592i	$-0.157766\pi$
$331$	133.749	+	133.749i	0.404074	+	0.404074i	−0.475592	+	0.879666i	$0.657766\pi$
$332$		0			0
$333$	−111.757	+	111.757i	−0.335605	+	0.335605i
$334$		0			0
$335$			134.148i			0.400443i
$336$		0			0
$337$		−	202.419i		−	0.600651i	−0.953837	−	0.300325i	$-0.902905\pi$
$337$		−	202.419i		−	0.600651i	0.953837	−	0.300325i	$-0.0970952\pi$
$338$		0			0
$339$	105.226			0.310403
$340$		0			0
$341$	−278.223			−0.815903
$342$		0			0
$343$	−263.897	−	263.897i	−0.769379	−	0.769379i
$344$		0			0
$345$	95.3210	−	95.3210i	0.276293	−	0.276293i
$346$		0			0
$347$	−190.870			−0.550059			−0.275029	−	0.961436i	$-0.588688\pi$
$347$	−190.870			−0.550059			−0.275029	+	0.961436i	$0.588688\pi$
$348$		0			0
$349$	−472.034	+	472.034i	−1.35253	+	1.35253i	−0.469713	+	0.882819i	$0.655643\pi$
$349$	−472.034	+	472.034i	−1.35253	+	1.35253i	−0.882819	+	0.469713i	$0.844357\pi$
$350$		0			0
$351$	28.1551	−	61.4027i	0.0802140	−	0.174936i
$352$		0			0
$353$	−298.499	−	298.499i	−0.845606	−	0.845606i	0.143975	−	0.989581i	$-0.454012\pi$
$353$	−298.499	−	298.499i	−0.845606	−	0.845606i	−0.989581	+	0.143975i	$0.954012\pi$
$354$		0			0
$355$		−	94.3556i		−	0.265791i
$356$		0			0
$357$	43.9535	+	43.9535i	0.123119	+	0.123119i
$358$		0			0
$359$	86.3614	−	86.3614i	0.240561	−	0.240561i	−0.576521	−	0.817082i	$-0.695590\pi$
$359$	86.3614	−	86.3614i	0.240561	−	0.240561i	0.817082	+	0.576521i	$0.195590\pi$
$360$		0			0
$361$		−	241.324i		−	0.668487i
$362$		0			0
$363$			139.229i			0.383550i
$364$		0			0
$365$	−34.7456			−0.0951935
$366$		0			0
$367$	54.1950			0.147670			0.0738352	−	0.997270i	$-0.476476\pi$
$367$	54.1950			0.147670			0.0738352	+	0.997270i	$0.476476\pi$
$368$		0			0
$369$	−86.1061	−	86.1061i	−0.233350	−	0.233350i
$370$		0			0
$371$	383.689	−	383.689i	1.03420	−	1.03420i
$372$		0			0
$373$	−17.9362			−0.0480862			−0.0240431	−	0.999711i	$-0.507654\pi$
$373$	−17.9362			−0.0480862			−0.0240431	+	0.999711i	$0.507654\pi$
$374$		0			0
$375$	−123.173	+	123.173i	−0.328462	+	0.328462i
$376$		0			0
$377$	−44.7090	+	97.5045i	−0.118592	+	0.258633i
$378$		0			0
$379$	281.565	+	281.565i	0.742916	+	0.742916i	0.973138	−	0.230222i	$-0.0739453\pi$
$379$	281.565	+	281.565i	0.742916	+	0.742916i	−0.230222	+	0.973138i	$0.573945\pi$
$380$		0			0
$381$			225.524i			0.591927i
$382$		0			0
$383$	244.827	+	244.827i	0.639236	+	0.639236i	0.950367	−	0.311131i	$-0.100708\pi$
$383$	244.827	+	244.827i	0.639236	+	0.639236i	−0.311131	+	0.950367i	$0.600708\pi$
$384$		0			0
$385$	−125.701	+	125.701i	−0.326496	+	0.326496i
$386$		0			0
$387$			128.159i			0.331159i
$388$		0			0
$389$			582.261i			1.49681i	0.663240	+	0.748407i	$0.269181\pi$
$389$			582.261i			1.49681i	−0.663240	+	0.748407i	$0.730819\pi$
$390$		0			0
$391$	−222.971			−0.570259
$392$		0			0
$393$	234.746			0.597318
$394$		0			0
$395$	−161.216	−	161.216i	−0.408143	−	0.408143i
$396$		0			0
$397$	372.060	−	372.060i	0.937179	−	0.937179i	−0.0609613	−	0.998140i	$-0.519417\pi$
$397$	372.060	−	372.060i	0.937179	−	0.937179i	0.998140	+	0.0609613i	$0.0194166\pi$
$398$		0			0
$399$	−106.222			−0.266220
$400$		0			0
$401$	−380.251	+	380.251i	−0.948258	+	0.948258i	−0.998726	−	0.0504677i	$-0.983929\pi$
$401$	−380.251	+	380.251i	−0.948258	+	0.948258i	0.0504677	+	0.998726i	$0.483929\pi$
$402$		0			0
$403$	238.939	−	88.7040i	0.592901	−	0.220109i
$404$		0			0
$405$	14.2207	+	14.2207i	0.0351128	+	0.0351128i
$406$		0			0
$407$			747.616i			1.83689i
$408$		0			0
$409$	58.5825	+	58.5825i	0.143233	+	0.143233i	0.775087	−	0.631854i	$-0.217706\pi$
$409$	58.5825	+	58.5825i	0.143233	+	0.143233i	−0.631854	+	0.775087i	$0.717706\pi$
$410$		0			0
$411$	−234.726	+	234.726i	−0.571109	+	0.571109i
$412$		0			0
$413$		−	300.420i		−	0.727408i
$414$		0			0
$415$			269.485i			0.649361i
$416$		0			0
$417$	−422.631			−1.01350
$418$		0			0
$419$	451.990			1.07874			0.539368	−	0.842070i	$-0.318663\pi$
$419$	451.990			1.07874			0.539368	+	0.842070i	$0.318663\pi$
$420$		0			0
$421$	54.5548	+	54.5548i	0.129584	+	0.129584i	0.768924	−	0.639340i	$-0.220792\pi$
$421$	54.5548	+	54.5548i	0.129584	+	0.129584i	−0.639340	+	0.768924i	$0.720792\pi$
$422$		0			0
$423$	−112.789	+	112.789i	−0.266641	+	0.266641i
$424$		0			0
$425$	128.078			0.301360
$426$		0			0
$427$	−361.859	+	361.859i	−0.847446	+	0.847446i
$428$		0			0
$429$	−111.208	−	299.557i	−0.259226	−	0.698268i
$430$		0			0
$431$	−422.858	−	422.858i	−0.981110	−	0.981110i	0.0187153	−	0.999825i	$-0.494042\pi$
$431$	−422.858	−	422.858i	−0.981110	−	0.981110i	−0.999825	+	0.0187153i	$0.994042\pi$
$432$		0			0
$433$			621.635i			1.43565i	0.696225	+	0.717824i	$0.254862\pi$
$433$			621.635i			1.43565i	−0.696225	+	0.717824i	$0.745138\pi$
$434$		0			0
$435$	−22.5818	−	22.5818i	−0.0519121	−	0.0519121i
$436$		0			0
$437$	269.425	−	269.425i	0.616534	−	0.616534i
$438$		0			0
$439$			416.992i			0.949868i	0.880021	+	0.474934i	$0.157528\pi$
$439$			416.992i			0.949868i	−0.880021	+	0.474934i	$0.842472\pi$
$440$		0			0
$441$			52.7202i			0.119547i
$442$		0			0
$443$	−581.904			−1.31355			−0.656776	−	0.754086i	$-0.728080\pi$
$443$	−581.904			−1.31355			−0.656776	+	0.754086i	$0.728080\pi$
$444$		0			0
$445$	−201.779			−0.453436
$446$		0			0
$447$	−302.913	−	302.913i	−0.677657	−	0.677657i
$448$		0			0
$449$	−66.7587	+	66.7587i	−0.148683	+	0.148683i	−0.777530	−	0.628846i	$-0.783527\pi$
$449$	−66.7587	+	66.7587i	−0.148683	+	0.148683i	0.628846	+	0.777530i	$0.283527\pi$
$450$		0			0
$451$	−576.023			−1.27721
$452$		0			0
$453$	−66.0077	+	66.0077i	−0.145712	+	0.145712i
$454$		0			0
$455$	67.8762	−	148.029i	0.149179	−	0.325339i
$456$		0			0
$457$	−103.815	−	103.815i	−0.227167	−	0.227167i	0.584341	−	0.811508i	$-0.301353\pi$
$457$	−103.815	−	103.815i	−0.227167	−	0.227167i	−0.811508	+	0.584341i	$0.801353\pi$
$458$		0			0
$459$		−	33.2645i		−	0.0724716i
$460$		0			0
$461$	−203.938	−	203.938i	−0.442381	−	0.442381i	0.450430	−	0.892812i	$-0.351271\pi$
$461$	−203.938	−	203.938i	−0.442381	−	0.442381i	−0.892812	+	0.450430i	$0.851271\pi$
$462$		0			0
$463$	481.370	−	481.370i	1.03968	−	1.03968i	0.0404953	−	0.999180i	$-0.487106\pi$
$463$	481.370	−	481.370i	1.03968	−	1.03968i	0.999180	−	0.0404953i	$-0.0128936\pi$
$464$		0			0
$465$			75.8812i			0.163185i
$466$		0			0
$467$		−	7.71730i		−	0.0165253i	−0.999966	−	0.00826263i	$-0.997370\pi$
$467$		−	7.71730i		−	0.0165253i	0.999966	−	0.00826263i	$-0.00263011\pi$
$468$		0			0
$469$	−336.543			−0.717576
$470$		0			0
$471$	−260.429			−0.552928
$472$		0			0
$473$	428.670	+	428.670i	0.906280	+	0.906280i
$474$		0			0
$475$	−154.762	+	154.762i	−0.325815	+	0.325815i
$476$		0			0
$477$	−290.380			−0.608763
$478$		0			0
$479$	−24.7867	+	24.7867i	−0.0517467	+	0.0517467i	−0.732507	−	0.680760i	$-0.761650\pi$
$479$	−24.7867	+	24.7867i	−0.0517467	+	0.0517467i	0.680760	+	0.732507i	$0.261650\pi$
$480$		0			0
$481$	−238.358	−	642.056i	−0.495546	−	1.33484i
$482$		0			0
$483$	239.136	+	239.136i	0.495105	+	0.495105i
$484$		0			0
$485$		−	185.508i		−	0.382491i
$486$		0			0
$487$	−245.469	−	245.469i	−0.504044	−	0.504044i	0.408648	−	0.912692i	$-0.366000\pi$
$487$	−245.469	−	245.469i	−0.504044	−	0.504044i	−0.912692	+	0.408648i	$0.866000\pi$
$488$		0			0
$489$	−290.194	+	290.194i	−0.593444	+	0.593444i
$490$		0			0
$491$			25.3996i			0.0517304i	0.999665	+	0.0258652i	$0.00823406\pi$
$491$			25.3996i			0.0517304i	−0.999665	+	0.0258652i	$0.991766\pi$
$492$		0			0
$493$			52.8224i			0.107145i
$494$		0			0
$495$	95.1320			0.192186
$496$		0			0
$497$	236.714			0.476285
$498$		0			0
$499$	328.094	+	328.094i	0.657502	+	0.657502i	0.954788	−	0.297286i	$-0.0960816\pi$
$499$	328.094	+	328.094i	0.657502	+	0.657502i	−0.297286	+	0.954788i	$0.596082\pi$
$500$		0			0
$501$	−58.4753	+	58.4753i	−0.116717	+	0.116717i
$502$		0			0
$503$	399.152			0.793543			0.396771	−	0.917918i	$-0.370131\pi$
$503$	399.152			0.793543			0.396771	+	0.917918i	$0.370131\pi$
$504$		0			0
$505$	−205.608	+	205.608i	−0.407145	+	0.407145i
$506$		0			0
$507$	191.012	+	221.805i	0.376749	+	0.437486i
$508$		0			0
$509$	655.020	+	655.020i	1.28688	+	1.28688i	0.936674	+	0.350203i	$0.113887\pi$
$509$	655.020	+	655.020i	1.28688	+	1.28688i	0.350203	+	0.936674i	$0.386113\pi$
$510$		0			0
$511$		−	87.1678i		−	0.170583i
$512$		0			0
$513$	40.1949	+	40.1949i	0.0783526	+	0.0783526i
$514$		0			0
$515$	44.1185	−	44.1185i	0.0856670	−	0.0856670i
$516$		0			0
$517$			754.525i			1.45943i
$518$		0			0
$519$		−	260.206i		−	0.501361i
$520$		0			0
$521$	287.758			0.552318			0.276159	−	0.961112i	$-0.410938\pi$
$521$	287.758			0.552318			0.276159	+	0.961112i	$0.410938\pi$
$522$		0			0
$523$	148.325			0.283604			0.141802	−	0.989895i	$-0.454710\pi$
$523$	148.325			0.283604			0.141802	+	0.989895i	$0.454710\pi$
$524$		0			0
$525$	−137.363	−	137.363i	−0.261644	−	0.261644i
$526$		0			0
$527$	88.7492	−	88.7492i	0.168405	−	0.168405i
$528$		0			0
$529$	−684.108			−1.29321
$530$		0			0
$531$	−113.680	+	113.680i	−0.214087	+	0.214087i
$532$		0			0
$533$	494.692	−	183.650i	0.928127	−	0.344559i
$534$		0			0
$535$	−262.558	−	262.558i	−0.490763	−	0.490763i
$536$		0			0
$537$			427.999i			0.797019i
$538$		0			0
$539$	176.341	+	176.341i	0.327163	+	0.327163i
$540$		0			0
$541$	−378.819	+	378.819i	−0.700221	+	0.700221i	−0.964458	−	0.264237i	$-0.914880\pi$
$541$	−378.819	+	378.819i	−0.700221	+	0.700221i	0.264237	+	0.964458i	$0.414880\pi$
$542$		0			0
$543$			193.426i			0.356217i
$544$		0			0
$545$		−	14.4227i		−	0.0264637i
$546$		0			0
$547$	−63.2197			−0.115575			−0.0577876	−	0.998329i	$-0.518405\pi$
$547$	−63.2197			−0.115575			−0.0577876	+	0.998329i	$0.518405\pi$
$548$		0			0
$549$	273.859			0.498833
$550$		0			0
$551$	−63.8276	−	63.8276i	−0.115839	−	0.115839i
$552$		0			0
$553$	404.450	−	404.450i	0.731374	−	0.731374i
$554$		0			0
$555$	203.901			0.367390
$556$		0			0
$557$	599.495	−	599.495i	1.07629	−	1.07629i	0.0794543	−	0.996839i	$-0.474682\pi$
$557$	599.495	−	599.495i	1.07629	−	1.07629i	0.996839	−	0.0794543i	$-0.0253178\pi$
$558$		0			0
$559$	−504.815	−	231.474i	−0.903067	−	0.414086i
$560$		0			0
$561$	−111.264	−	111.264i	−0.198332	−	0.198332i
$562$		0			0
$563$		−	79.3967i		−	0.141024i	−0.997511	−	0.0705122i	$-0.977537\pi$
$563$		−	79.3967i		−	0.141024i	0.997511	−	0.0705122i	$-0.0224634\pi$
$564$		0			0
$565$	−95.9937	−	95.9937i	−0.169900	−	0.169900i
$566$		0			0
$567$	−35.6760	+	35.6760i	−0.0629207	+	0.0629207i
$568$		0			0
$569$		−	27.2834i		−	0.0479497i	−0.999713	−	0.0239748i	$-0.992368\pi$
$569$		−	27.2834i		−	0.0479497i	0.999713	−	0.0239748i	$-0.00763216\pi$
$570$		0			0
$571$		−	73.0650i		−	0.127960i	−0.997951	−	0.0639799i	$-0.979621\pi$
$571$		−	73.0650i		−	0.127960i	0.997951	−	0.0639799i	$-0.0203793\pi$
$572$		0			0
$573$	−112.062			−0.195571
$574$		0			0
$575$	696.828			1.21187
$576$		0			0
$577$	172.432	+	172.432i	0.298841	+	0.298841i	0.840560	−	0.541719i	$-0.182226\pi$
$577$	172.432	+	172.432i	0.298841	+	0.298841i	−0.541719	+	0.840560i	$0.682226\pi$
$578$		0			0
$579$	−168.180	+	168.180i	−0.290466	+	0.290466i
$580$		0			0
$581$	−676.067			−1.16363
$582$		0			0
$583$	−971.276	+	971.276i	−1.66600	+	1.66600i
$584$		0			0
$585$	−81.6998	+	30.3303i	−0.139658	+	0.0518467i
$586$		0			0
$587$	−766.322	−	766.322i	−1.30549	−	1.30549i	−0.924635	−	0.380854i	$-0.875630\pi$
$587$	−766.322	−	766.322i	−1.30549	−	1.30549i	−0.380854	−	0.924635i	$-0.624370\pi$
$588$		0			0
$589$			214.479i			0.364141i
$590$		0			0
$591$	−84.6496	−	84.6496i	−0.143231	−	0.143231i
$592$		0			0
$593$	120.360	−	120.360i	0.202969	−	0.202969i	−0.598302	−	0.801271i	$-0.704158\pi$
$593$	120.360	−	120.360i	0.202969	−	0.202969i	0.801271	+	0.598302i	$0.204158\pi$
$594$		0			0
$595$		−	80.1938i		−	0.134779i
$596$		0			0
$597$		−	241.743i		−	0.404929i
$598$		0			0
$599$	−323.635			−0.540292			−0.270146	−	0.962819i	$-0.587072\pi$
$599$	−323.635			−0.540292			−0.270146	+	0.962819i	$0.587072\pi$
$600$		0			0
$601$	1108.97			1.84521			0.922607	−	0.385742i	$-0.126054\pi$
$601$	1108.97			1.84521			0.922607	+	0.385742i	$0.126054\pi$
$602$		0			0
$603$	127.350	+	127.350i	0.211194	+	0.211194i
$604$		0			0
$605$	127.012	−	127.012i	0.209938	−	0.209938i
$606$		0			0
$607$	196.769			0.324167			0.162084	−	0.986777i	$-0.448179\pi$
$607$	196.769			0.324167			0.162084	+	0.986777i	$0.448179\pi$
$608$		0			0
$609$	56.6518	−	56.6518i	0.0930243	−	0.0930243i
$610$		0			0
$611$	−240.561	−	647.990i	−0.393716	−	1.06054i
$612$		0			0
$613$	−702.697	−	702.697i	−1.14633	−	1.14633i	−0.987270	−	0.159055i	$-0.949155\pi$
$613$	−702.697	−	702.697i	−1.14633	−	1.14633i	−0.159055	−	0.987270i	$-0.550845\pi$
$614$		0			0
$615$			157.102i			0.255450i
$616$		0			0
$617$	−305.273	−	305.273i	−0.494771	−	0.494771i	0.415035	−	0.909805i	$-0.363769\pi$
$617$	−305.273	−	305.273i	−0.494771	−	0.494771i	−0.909805	+	0.415035i	$0.863769\pi$
$618$		0			0
$619$	−97.1707	+	97.1707i	−0.156980	+	0.156980i	−0.781227	−	0.624247i	$-0.785406\pi$
$619$	−97.1707	+	97.1707i	−0.156980	+	0.156980i	0.624247	+	0.781227i	$0.285406\pi$
$620$		0			0
$621$		−	180.980i		−	0.291434i
$622$		0			0
$623$		−	506.211i		−	0.812538i
$624$		0			0
$625$	−275.436			−0.440698
$626$		0			0
$627$	268.891			0.428854
$628$		0			0
$629$	−238.479	−	238.479i	−0.379140	−	0.379140i
$630$		0			0
$631$	368.719	−	368.719i	0.584341	−	0.584341i	−0.351752	−	0.936093i	$-0.614414\pi$
$631$	368.719	−	368.719i	0.584341	−	0.584341i	0.936093	+	0.351752i	$0.114414\pi$
$632$		0			0
$633$	91.7219			0.144900
$634$		0			0
$635$	205.736	−	205.736i	0.323994	−	0.323994i
$636$		0			0
$637$	−207.664	−	95.2206i	−0.326003	−	0.149483i
$638$		0			0
$639$	−89.5738	−	89.5738i	−0.140178	−	0.140178i
$640$		0			0
$641$			779.737i			1.21644i	0.793769	+	0.608219i	$0.208116\pi$
$641$			779.737i			1.21644i	−0.793769	+	0.608219i	$0.791884\pi$
$642$		0			0
$643$	719.483	+	719.483i	1.11895	+	1.11895i	0.991896	+	0.127051i	$0.0405512\pi$
$643$	719.483	+	719.483i	1.11895	+	1.11895i	0.127051	+	0.991896i	$0.459449\pi$
$644$		0			0
$645$	116.914	−	116.914i	0.181262	−	0.181262i
$646$		0			0
$647$			313.980i			0.485286i	0.970116	+	0.242643i	$0.0780144\pi$
$647$			313.980i			0.485286i	−0.970116	+	0.242643i	$0.921986\pi$
$648$		0			0
$649$			760.487i			1.17178i
$650$		0			0
$651$	−190.366			−0.292421
$652$		0			0
$653$	−280.037			−0.428847			−0.214424	−	0.976741i	$-0.568787\pi$
$653$	−280.037			−0.428847			−0.214424	+	0.976741i	$0.568787\pi$
$654$		0			0
$655$	−214.149	−	214.149i	−0.326945	−	0.326945i
$656$		0			0
$657$	−32.9848	+	32.9848i	−0.0502051	+	0.0502051i
$658$		0			0
$659$	1021.28			1.54974			0.774871	−	0.632119i	$-0.217815\pi$
$659$	1021.28			1.54974			0.774871	+	0.632119i	$0.217815\pi$
$660$		0			0
$661$	638.747	−	638.747i	0.966334	−	0.966334i	−0.0331178	−	0.999451i	$-0.510544\pi$
$661$	638.747	−	638.747i	0.966334	−	0.966334i	0.999451	+	0.0331178i	$0.0105436\pi$
$662$		0			0
$663$	131.028	+	60.0807i	0.197629	+	0.0906195i
$664$		0			0
$665$	96.9016	+	96.9016i	0.145717	+	0.145717i
$666$		0			0
$667$			287.388i			0.430867i
$668$		0			0
$669$	122.020	+	122.020i	0.182392	+	0.182392i
$670$		0			0
$671$	916.016	−	916.016i	1.36515	−	1.36515i
$672$		0			0
$673$			236.547i			0.351482i	0.984436	+	0.175741i	$0.0562320\pi$
$673$			236.547i			0.351482i	−0.984436	+	0.175741i	$0.943768\pi$
$674$		0			0
$675$			103.958i			0.154012i
$676$		0			0
$677$	−788.423			−1.16458			−0.582292	−	0.812980i	$-0.697844\pi$
$677$	−788.423			−1.16458			−0.582292	+	0.812980i	$0.697844\pi$
$678$		0			0
$679$	465.392			0.685408
$680$		0			0
$681$	−164.579	−	164.579i	−0.241672	−	0.241672i
$682$		0			0
$683$	278.551	−	278.551i	0.407834	−	0.407834i	−0.473149	−	0.880983i	$-0.656883\pi$
$683$	278.551	−	278.551i	0.407834	−	0.407834i	0.880983	+	0.473149i	$0.156883\pi$
$684$		0			0
$685$	428.261			0.625198
$686$		0			0
$687$	−115.876	+	115.876i	−0.168670	+	0.168670i
$688$		0			0
$689$	524.471	−	1143.80i	0.761206	−	1.66009i
$690$		0			0
$691$	64.4993	+	64.4993i	0.0933420	+	0.0933420i	0.752236	−	0.658894i	$-0.228975\pi$
$691$	64.4993	+	64.4993i	0.0933420	+	0.0933420i	−0.658894	+	0.752236i	$0.728975\pi$
$692$		0			0
$693$			238.661i			0.344389i
$694$		0			0
$695$	385.549	+	385.549i	0.554746	+	0.554746i
$696$		0			0
$697$	183.743	−	183.743i	0.263620	−	0.263620i
$698$		0			0
$699$		−	141.890i		−	0.202989i
$700$		0			0
$701$			136.339i			0.194493i	0.995260	+	0.0972463i	$0.0310035\pi$
$701$			136.339i			0.194493i	−0.995260	+	0.0972463i	$0.968997\pi$
$702$		0			0
$703$	576.329			0.819814
$704$		0			0
$705$	205.786			0.291895
$706$		0			0
$707$	−515.818	−	515.818i	−0.729587	−	0.729587i
$708$		0			0
$709$	789.276	−	789.276i	1.11322	−	1.11322i	0.120513	−	0.992712i	$-0.461546\pi$
$709$	789.276	−	789.276i	1.11322	−	1.11322i	0.992712	−	0.120513i	$-0.0384540\pi$
$710$		0			0
$711$	−306.092			−0.430509
$712$		0			0
$713$	482.854	−	482.854i	0.677214	−	0.677214i
$714$		0			0
$715$	−171.823	+	374.723i	−0.240312	+	0.524089i
$716$		0			0
$717$	8.86818	+	8.86818i	0.0123685	+	0.0123685i
$718$		0			0
$719$			949.863i			1.32109i	0.750787	+	0.660544i	$0.229674\pi$
$719$			949.863i			1.32109i	−0.750787	+	0.660544i	$0.770326\pi$
$720$		0			0
$721$	110.682	+	110.682i	0.153512	+	0.153512i
$722$		0			0
$723$	263.623	−	263.623i	0.364623	−	0.364623i
$724$		0			0
$725$		−	165.080i		−	0.227697i
$726$		0			0
$727$			195.690i			0.269175i	0.990902	+	0.134588i	$0.0429710\pi$
$727$			195.690i			0.269175i	−0.990902	+	0.134588i	$0.957029\pi$
$728$		0			0
$729$	27.0000			0.0370370
$730$		0			0
$731$	−273.480			−0.374117
$732$		0			0
$733$	−373.535	−	373.535i	−0.509597	−	0.509597i	0.404806	−	0.914403i	$-0.367339\pi$
$733$	−373.535	−	373.535i	−0.509597	−	0.509597i	−0.914403	+	0.404806i	$0.867339\pi$
$734$		0			0
$735$	48.0944	−	48.0944i	0.0654345	−	0.0654345i
$736$		0			0
$737$	851.931			1.15594
$738$		0			0
$739$	868.098	−	868.098i	1.17469	−	1.17469i	0.193615	−	0.981078i	$-0.437979\pi$
$739$	868.098	−	868.098i	1.17469	−	1.17469i	0.981078	−	0.193615i	$-0.0620211\pi$
$740$		0			0
$741$	−230.925	+	85.7289i	−0.311640	+	0.115694i
$742$		0			0
$743$	744.696	+	744.696i	1.00228	+	1.00228i	0.999997	+	0.00228459i	$0.000727208\pi$
$743$	744.696	+	744.696i	1.00228	+	1.00228i	0.00228459	+	0.999997i	$0.499273\pi$
$744$		0			0
$745$			552.669i			0.741837i
$746$		0			0
$747$	255.828	+	255.828i	0.342473	+	0.342473i
$748$		0			0
$749$	658.690	−	658.690i	0.879426	−	0.879426i
$750$		0			0
$751$		−	267.175i		−	0.355760i	−0.984052	−	0.177880i	$-0.943076\pi$
$751$		−	267.175i		−	0.355760i	0.984052	−	0.177880i	$-0.0569238\pi$
$752$		0			0
$753$		−	384.303i		−	0.510362i
$754$		0			0
$755$	120.432			0.159513
$756$		0			0
$757$	−421.716			−0.557088			−0.278544	−	0.960423i	$-0.589852\pi$
$757$	−421.716			−0.557088			−0.278544	+	0.960423i	$0.589852\pi$
$758$		0			0
$759$	−605.351	−	605.351i	−0.797565	−	0.797565i
$760$		0			0
$761$	937.965	−	937.965i	1.23254	−	1.23254i	0.269558	−	0.962984i	$-0.413122\pi$
$761$	937.965	−	937.965i	1.23254	−	1.23254i	0.962984	−	0.269558i	$-0.0868776\pi$
$762$		0			0
$763$	36.1828			0.0474218
$764$		0			0
$765$	−30.3458	+	30.3458i	−0.0396677	+	0.0396677i
$766$		0			0
$767$	−242.461	−	653.109i	−0.316116	−	0.851512i
$768$		0			0
$769$	749.082	+	749.082i	0.974098	+	0.974098i	0.999673	−	0.0255747i	$-0.00814156\pi$
$769$	749.082	+	749.082i	0.974098	+	0.974098i	−0.0255747	+	0.999673i	$0.508142\pi$
$770$		0			0
$771$			26.0436i			0.0337789i
$772$		0			0
$773$	292.773	+	292.773i	0.378749	+	0.378749i	0.870651	−	0.491902i	$-0.163698\pi$
$773$	292.773	+	292.773i	0.378749	+	0.378749i	−0.491902	+	0.870651i	$0.663698\pi$
$774$		0			0
$775$	−277.358	+	277.358i	−0.357882	+	0.357882i
$776$		0			0
$777$			511.536i			0.658347i
$778$		0			0
$779$			444.050i			0.570025i
$780$		0			0
$781$	−599.221			−0.767248
$782$		0			0
$783$	−42.8747			−0.0547570
$784$		0			0
$785$	237.578	+	237.578i	0.302648	+	0.302648i
$786$		0			0
$787$	992.205	−	992.205i	1.26074	−	1.26074i	0.310010	−	0.950733i	$-0.399668\pi$
$787$	992.205	−	992.205i	1.26074	−	1.26074i	0.950733	−	0.310010i	$-0.100332\pi$
$788$		0			0
$789$	−183.118			−0.232089
$790$		0			0
$791$	240.823	−	240.823i	0.304454	−	0.304454i
$792$		0			0
$793$	−494.631	+	1078.73i	−0.623747	+	1.36031i
$794$		0			0
$795$	264.902	+	264.902i	0.333209	+	0.333209i
$796$		0			0
$797$		−	96.6901i		−	0.121318i	−0.998159	−	0.0606588i	$-0.980680\pi$
$797$		−	96.6901i		−	0.121318i	0.998159	−	0.0606588i	$-0.0193202\pi$
$798$		0			0
$799$	−240.683	−	240.683i	−0.301230	−	0.301230i
$800$		0			0
$801$	−191.553	+	191.553i	−0.239142	+	0.239142i
$802$		0			0
$803$			220.658i			0.274792i
$804$		0			0
$805$		−	436.307i		−	0.541996i
$806$		0			0
$807$	−292.266			−0.362163
$808$		0			0
$809$	−800.289			−0.989233			−0.494616	−	0.869111i	$-0.664691\pi$
$809$	−800.289			−0.989233			−0.494616	+	0.869111i	$0.664691\pi$
$810$		0			0
$811$	−1015.91	−	1015.91i	−1.25267	−	1.25267i	−0.954520	−	0.298147i	$-0.903631\pi$
$811$	−1015.91	−	1015.91i	−1.25267	−	1.25267i	−0.298147	−	0.954520i	$-0.596369\pi$
$812$		0			0
$813$	−90.8958	+	90.8958i	−0.111803	+	0.111803i
$814$		0			0
$815$	529.464			0.649649
$816$		0			0
$817$	330.457	−	330.457i	0.404476	−	0.404476i
$818$		0			0
$819$	−76.0909	−	204.964i	−0.0929071	−	0.250261i
$820$		0			0
$821$	−515.901	−	515.901i	−0.628381	−	0.628381i	0.319280	−	0.947661i	$-0.396559\pi$
$821$	−515.901	−	515.901i	−0.628381	−	0.628381i	−0.947661	+	0.319280i	$0.896559\pi$
$822$		0			0
$823$		−	563.080i		−	0.684179i	−0.939667	−	0.342090i	$-0.888865\pi$
$823$		−	563.080i		−	0.684179i	0.939667	−	0.342090i	$-0.111135\pi$
$824$		0			0
$825$	347.723	+	347.723i	0.421483	+	0.421483i
$826$		0			0
$827$	129.875	−	129.875i	0.157044	−	0.157044i	−0.624212	−	0.781255i	$-0.714580\pi$
$827$	129.875	−	129.875i	0.157044	−	0.157044i	0.781255	+	0.624212i	$0.214580\pi$
$828$		0			0
$829$		−	1140.27i		−	1.37548i	−0.725957	−	0.687740i	$-0.758603\pi$
$829$		−	1140.27i		−	1.37548i	0.725957	−	0.687740i	$-0.241397\pi$
$830$		0			0
$831$		−	265.035i		−	0.318935i
$832$		0			0
$833$	−112.500			−0.135055
$834$		0			0
$835$	106.689			0.127771
$836$		0			0
$837$	72.0356	+	72.0356i	0.0860641	+	0.0860641i
$838$		0			0
$839$	−373.307	+	373.307i	−0.444942	+	0.444942i	−0.893669	−	0.448727i	$-0.851878\pi$
$839$	−373.307	+	373.307i	−0.444942	+	0.444942i	0.448727	+	0.893669i	$0.351878\pi$
$840$		0			0
$841$	−772.917			−0.919045
$842$		0			0
$843$	−65.6252	+	65.6252i	−0.0778473	+	0.0778473i
$844$		0			0
$845$	28.0916	−	376.595i	0.0332445	−	0.445675i
$846$		0			0
$847$	318.641	+	318.641i	0.376200	+	0.376200i
$848$		0			0
$849$		−	60.1406i		−	0.0708370i
$850$		0			0
$851$	−1297.48	−	1297.48i	−1.52466	−	1.52466i
$852$		0			0
$853$	−65.1164	+	65.1164i	−0.0763381	+	0.0763381i	−0.744245	−	0.667907i	$-0.767190\pi$
$853$	−65.1164	+	65.1164i	−0.0763381	+	0.0763381i	0.667907	+	0.744245i	$0.267190\pi$
$854$		0			0
$855$		−	73.3362i		−	0.0857733i
$856$		0			0
$857$		−	98.7570i		−	0.115236i	−0.998339	−	0.0576178i	$-0.981650\pi$
$857$		−	98.7570i		−	0.115236i	0.998339	−	0.0576178i	$-0.0183505\pi$
$858$		0			0
$859$	−959.271			−1.11673			−0.558365	−	0.829595i	$-0.688571\pi$
$859$	−959.271			−1.11673			−0.558365	+	0.829595i	$0.688571\pi$
$860$		0			0
$861$	−394.128			−0.457756
$862$		0			0
$863$	30.4807	+	30.4807i	0.0353194	+	0.0353194i	0.724546	−	0.689227i	$-0.242050\pi$
$863$	30.4807	+	30.4807i	0.0353194	+	0.0353194i	−0.689227	+	0.724546i	$0.742050\pi$
$864$		0			0
$865$	−237.375	+	237.375i	−0.274422	+	0.274422i
$866$		0			0
$867$	−429.579			−0.495478
$868$		0			0
$869$	−1023.83	+	1023.83i	−1.17817	+	1.17817i
$870$		0			0
$871$	−731.642	+	271.616i	−0.840003	+	0.311844i
$872$		0			0
$873$	−176.107	−	176.107i	−0.201726	−	0.201726i
$874$		0			0
$875$			563.793i			0.644334i
$876$		0			0
$877$	66.1101	+	66.1101i	0.0753821	+	0.0753821i	0.743793	−	0.668410i	$-0.233025\pi$
$877$	66.1101	+	66.1101i	0.0753821	+	0.0753821i	−0.668410	+	0.743793i	$0.733025\pi$
$878$		0			0
$879$	−94.3393	+	94.3393i	−0.107326	+	0.107326i
$880$		0			0
$881$			825.315i			0.936794i	0.883518	+	0.468397i	$0.155168\pi$
$881$			825.315i			0.936794i	−0.883518	+	0.468397i	$0.844832\pi$
$882$		0			0
$883$			1501.16i			1.70007i	0.526724	+	0.850037i	$0.323420\pi$
$883$			1501.16i			1.70007i	−0.526724	+	0.850037i	$0.676580\pi$
$884$		0			0
$885$	207.412			0.234363
$886$		0			0
$887$	122.414			0.138009			0.0690045	−	0.997616i	$-0.478018\pi$
$887$	122.414			0.138009			0.0690045	+	0.997616i	$0.478018\pi$
$888$		0			0
$889$	516.139	+	516.139i	0.580584	+	0.580584i
$890$		0			0
$891$	90.3108	−	90.3108i	0.101359	−	0.101359i
$892$		0			0
$893$	581.655			0.651349
$894$		0			0
$895$	390.446	−	390.446i	0.436252	−	0.436252i
$896$		0			0
$897$	712.879	+	326.878i	0.794737	+	0.364413i
$898$		0			0
$899$	−114.389	−	114.389i	−0.127240	−	0.127240i
$900$		0			0
$901$		−	619.647i		−	0.687732i
$902$		0			0
$903$	293.306	+	293.306i	0.324813	+	0.324813i
$904$		0			0
$905$	176.454	−	176.454i	0.194977	−	0.194977i
$906$		0			0
$907$			976.517i			1.07665i	0.842739	+	0.538323i	$0.180942\pi$
$907$			976.517i			1.07665i	−0.842739	+	0.538323i	$0.819058\pi$
$908$		0			0
$909$			390.377i			0.429457i
$910$		0			0
$911$	1291.48			1.41765			0.708825	−	0.705384i	$-0.249225\pi$
$911$	1291.48			1.41765			0.708825	+	0.705384i	$0.249225\pi$
$912$		0			0
$913$	1711.41			1.87449
$914$		0			0
$915$	−249.830	−	249.830i	−0.273038	−	0.273038i
$916$		0			0
$917$	537.244	−	537.244i	0.585871	−	0.585871i
$918$		0			0
$919$	−1651.58			−1.79715			−0.898575	−	0.438819i	$-0.855397\pi$
$919$	−1651.58			−1.79715			−0.898575	+	0.438819i	$0.855397\pi$
$920$		0			0
$921$	386.914	−	386.914i	0.420103	−	0.420103i
$922$		0			0
$923$	514.614	−	191.046i	0.557544	−	0.206983i
$924$		0			0
$925$	745.293	+	745.293i	0.805723	+	0.805723i
$926$		0			0
$927$		−	83.7652i		−	0.0903616i
$928$		0			0
$929$	644.310	+	644.310i	0.693552	+	0.693552i	0.963012	−	0.269460i	$-0.0868451\pi$
$929$	644.310	+	644.310i	0.693552	+	0.693552i	−0.269460	+	0.963012i	$0.586845\pi$
$930$		0			0
$931$	135.939	−	135.939i	0.146014	−	0.146014i
$932$		0			0
$933$		−	894.567i		−	0.958807i
$934$		0			0
$935$			203.004i			0.217116i
$936$		0			0
$937$	−939.846			−1.00304			−0.501519	−	0.865147i	$-0.667225\pi$
$937$	−939.846			−1.00304			−0.501519	+	0.865147i	$0.667225\pi$
$938$		0			0
$939$	258.482			0.275273
$940$		0			0
$941$	−767.549	−	767.549i	−0.815674	−	0.815674i	0.169804	−	0.985478i	$-0.445687\pi$
$941$	−767.549	−	767.549i	−0.815674	−	0.815674i	−0.985478	+	0.169804i	$0.945687\pi$
$942$		0			0
$943$	999.684	−	999.684i	1.06011	−	1.06011i
$944$		0			0
$945$	65.0914			0.0688798
$946$		0			0
$947$	518.599	−	518.599i	0.547623	−	0.547623i	−0.378130	−	0.925753i	$-0.623433\pi$
$947$	518.599	−	518.599i	0.547623	−	0.547623i	0.925753	+	0.378130i	$0.123433\pi$
$948$		0			0
$949$	−70.3509	−	189.502i	−0.0741316	−	0.199686i
$950$		0			0
$951$	325.908	+	325.908i	0.342700	+	0.342700i
$952$		0			0
$953$			1429.04i			1.49952i	0.661709	+	0.749761i	$0.269831\pi$
$953$			1429.04i			1.49952i	−0.661709	+	0.749761i	$0.730169\pi$
$954$		0			0
$955$	102.230	+	102.230i	0.107047	+	0.107047i
$956$		0			0
$957$	−143.409	+	143.409i	−0.149853	+	0.149853i
$958$		0			0
$959$			1074.39i			1.12033i
$960$		0			0
$961$		−	576.620i		−	0.600020i
$962$		0			0
$963$	−498.504			−0.517657
$964$		0			0
$965$	306.847			0.317976
$966$		0			0
$967$	−507.433	−	507.433i	−0.524749	−	0.524749i	0.394253	−	0.919002i	$-0.371003\pi$
$967$	−507.433	−	507.433i	−0.524749	−	0.524749i	−0.919002	+	0.394253i	$0.871003\pi$
$968$		0			0
$969$	−85.7725	+	85.7725i	−0.0885165	+	0.0885165i
$970$		0			0
$971$	−832.781			−0.857652			−0.428826	−	0.903387i	$-0.641073\pi$
$971$	−832.781			−0.857652			−0.428826	+	0.903387i	$0.641073\pi$
$972$		0			0
$973$	−967.241	+	967.241i	−0.994081	+	0.994081i
$974$		0			0
$975$	−409.489	−	187.764i	−0.419988	−	0.192578i
$976$		0			0
$977$	−562.185	−	562.185i	−0.575419	−	0.575419i	0.358219	−	0.933638i	$-0.383384\pi$
$977$	−562.185	−	562.185i	−0.575419	−	0.575419i	−0.933638	+	0.358219i	$0.883384\pi$
$978$		0			0
$979$			1281.43i			1.30892i
$980$		0			0
$981$	−13.6918	−	13.6918i	−0.0139570	−	0.0139570i
$982$		0			0
$983$	−351.787	+	351.787i	−0.357870	+	0.357870i	−0.863027	−	0.505157i	$-0.831435\pi$
$983$	−351.787	+	351.787i	−0.357870	+	0.357870i	0.505157	+	0.863027i	$0.331435\pi$
$984$		0			0
$985$			154.445i			0.156797i
$986$		0			0
$987$			516.263i			0.523063i
$988$		0			0
$989$	−1487.91			−1.50446
$990$		0			0
$991$	−96.6030			−0.0974803			−0.0487401	−	0.998811i	$-0.515521\pi$
$991$	−96.6030			−0.0974803			−0.0487401	+	0.998811i	$0.515521\pi$
$992$		0			0
$993$	−231.659	−	231.659i	−0.233292	−	0.233292i
$994$		0			0
$995$	−220.532	+	220.532i	−0.221640	+	0.221640i
$996$		0			0
$997$	−1290.54			−1.29443			−0.647213	−	0.762309i	$-0.724065\pi$
$997$	−1290.54			−1.29443			−0.647213	+	0.762309i	$0.724065\pi$
$998$		0			0
$999$	193.568	−	193.568i	0.193762	−	0.193762i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.3.ba.b.385.2		8
4.3	odd	2		39.3.g.a.34.2	yes	8
12.11	even	2		117.3.j.b.73.3		8
13.5	odd	4	inner	624.3.ba.b.577.2		8
52.31	even	4		39.3.g.a.31.2	✓	8
156.83	odd	4		117.3.j.b.109.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.3.g.a.31.2	✓	8	52.31	even	4
39.3.g.a.34.2	yes	8	4.3	odd	2
117.3.j.b.73.3		8	12.11	even	2
117.3.j.b.109.3		8	156.83	odd	4
624.3.ba.b.385.2		8	1.1	even	1	trivial
624.3.ba.b.577.2		8	13.5	odd	4	inner

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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	624.ba (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +(1.58008 + 1.58008i) q^{5} +(-3.96400 + 3.96400i) q^{7} +3.00000 q^{9} +(10.0345 - 10.0345i) q^{11} +(-5.41846 + 11.8170i) q^{13} +(-2.73677 - 2.73677i) q^{15} +6.40175i q^{17} +(-7.73551 - 7.73551i) q^{19} +(6.86585 - 6.86585i) q^{21} +34.8297i q^{23} -20.0067i q^{25} -5.19615 q^{27} +8.25124 q^{29} +(-13.8633 - 13.8633i) q^{31} +(-17.3803 + 17.3803i) q^{33} -12.5269 q^{35} +(-37.2522 + 37.2522i) q^{37} +(9.38504 - 20.4676i) q^{39} +(-28.7020 - 28.7020i) q^{41} +42.7195i q^{43} +(4.74023 + 4.74023i) q^{45} +(-37.5964 + 37.5964i) q^{47} +17.5734i q^{49} -11.0882i q^{51} -96.7934 q^{53} +31.7107 q^{55} +(13.3983 + 13.3983i) q^{57} +(-37.8935 + 37.8935i) q^{59} +91.2864 q^{61} +(-11.8920 + 11.8920i) q^{63} +(-27.2333 + 10.1101i) q^{65} +(42.4500 + 42.4500i) q^{67} -60.3268i q^{69} +(-29.8579 - 29.8579i) q^{71} +(-10.9949 + 10.9949i) q^{73} +34.6526i q^{75} +79.5538i q^{77} -102.031 q^{79} +9.00000 q^{81} +(85.2758 + 85.2758i) q^{83} +(-10.1153 + 10.1153i) q^{85} -14.2916 q^{87} +(-63.8510 + 63.8510i) q^{89} +(-25.3636 - 68.3212i) q^{91} +(24.0119 + 24.0119i) q^{93} -24.4454i q^{95} +(-58.7023 - 58.7023i) q^{97} +(30.1036 - 30.1036i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 20 q^{5} - 8 q^{7} + 24 q^{9} + 20 q^{11} + 8 q^{13} - 12 q^{15} + 40 q^{19} - 48 q^{21} + 56 q^{29} - 32 q^{31} - 36 q^{33} - 104 q^{35} - 16 q^{37} - 24 q^{39} - 116 q^{41} - 60 q^{45} - 100 q^{47}+ \cdots + 60 q^{99}+O(q^{100}) \) 8 * q - 20 * q^5 - 8 * q^7 + 24 * q^9 + 20 * q^11 + 8 * q^13 - 12 * q^15 + 40 * q^19 - 48 * q^21 + 56 * q^29 - 32 * q^31 - 36 * q^33 - 104 * q^35 - 16 * q^37 - 24 * q^39 - 116 * q^41 - 60 * q^45 - 100 * q^47 - 232 * q^53 + 176 * q^55 + 120 * q^57 - 316 * q^59 + 160 * q^61 - 24 * q^63 - 92 * q^65 - 176 * q^67 - 4 * q^71 - 64 * q^73 - 208 * q^79 + 72 * q^81 + 212 * q^83 - 168 * q^85 + 24 * q^87 - 452 * q^89 + 400 * q^91 - 24 * q^93 + 128 * q^97 + 60 * q^99

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