LMFDB - Embedded newform 624.4.bc.d.463.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(31,624)]

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([2, 0, 0, 3]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("624.31");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.bc (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:	$36.8171918436$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$14$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			463.3
Character	$\chi$	$=$	624.463
Dual form			624.4.bc.d.31.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.00000i q^{3} +(-8.48307 - 8.48307i) q^{5} +(-5.05973 - 5.05973i) q^{7} -9.00000 q^{9} +(9.98793 + 9.98793i) q^{11} +(-4.24079 + 46.6799i) q^{13} +(-25.4492 + 25.4492i) q^{15} +61.7223i q^{17} +(-51.7737 + 51.7737i) q^{19} +(-15.1792 + 15.1792i) q^{21} +81.7373 q^{23} +18.9248i q^{25} +27.0000i q^{27} -24.7234 q^{29} +(-55.5963 + 55.5963i) q^{31} +(29.9638 - 29.9638i) q^{33} +85.8440i q^{35} +(275.273 - 275.273i) q^{37} +(140.040 + 12.7224i) q^{39} +(-29.8795 - 29.8795i) q^{41} +16.9411 q^{43} +(76.3476 + 76.3476i) q^{45} +(188.989 + 188.989i) q^{47} -291.798i q^{49} +185.167 q^{51} +115.819 q^{53} -169.457i q^{55} +(155.321 + 155.321i) q^{57} +(368.361 + 368.361i) q^{59} +569.709 q^{61} +(45.5376 + 45.5376i) q^{63} +(431.964 - 360.014i) q^{65} +(-500.494 + 500.494i) q^{67} -245.212i q^{69} +(52.4392 - 52.4392i) q^{71} +(138.074 - 138.074i) q^{73} +56.7744 q^{75} -101.072i q^{77} -417.808i q^{79} +81.0000 q^{81} +(-781.441 + 781.441i) q^{83} +(523.595 - 523.595i) q^{85} +74.1701i q^{87} +(555.705 - 555.705i) q^{89} +(257.645 - 214.731i) q^{91} +(166.789 + 166.789i) q^{93} +878.400 q^{95} +(98.2402 + 98.2402i) q^{97} +(-89.8914 - 89.8914i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 4 q^{5} + 8 q^{7} - 252 q^{9} - 64 q^{11} - 32 q^{13} + 12 q^{15} - 56 q^{19} + 24 q^{21} - 384 q^{23} - 32 q^{29} + 168 q^{31} - 192 q^{33} + 412 q^{37} - 252 q^{39} + 1340 q^{41} - 624 q^{43} - 36 q^{45}+ \cdots + 576 q^{99}+O(q^{100}) $ 28 * q + 4 * q^5 + 8 * q^7 - 252 * q^9 - 64 * q^11 - 32 * q^13 + 12 * q^15 - 56 * q^19 + 24 * q^21 - 384 * q^23 - 32 * q^29 + 168 * q^31 - 192 * q^33 + 412 * q^37 - 252 * q^39 + 1340 * q^41 - 624 * q^43 - 36 * q^45 + 688 * q^47 - 312 * q^51 + 2152 * q^53 + 168 * q^57 - 624 * q^59 - 216 * q^61 - 72 * q^63 + 60 * q^65 + 2024 * q^67 - 448 * q^71 - 724 * q^73 + 1716 * q^75 + 2268 * q^81 - 2288 * q^83 - 1528 * q^85 - 1876 * q^89 + 3448 * q^91 - 504 * q^93 - 192 * q^95 + 76 * q^97 + 576 * 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
$3$		−	3.00000i		−	0.577350i
$3$		−	3.00000i		−	0.577350i
$4$		0			0
$5$	−8.48307	−	8.48307i	−0.758748	−	0.758748i	0.217346	−	0.976095i	$-0.430260\pi$
$5$	−8.48307	−	8.48307i	−0.758748	−	0.758748i	−0.976095	+	0.217346i	$0.930260\pi$
$6$		0			0
$7$	−5.05973	−	5.05973i	−0.273200	−	0.273200i	0.557187	−	0.830387i	$-0.311881\pi$
$7$	−5.05973	−	5.05973i	−0.273200	−	0.273200i	−0.830387	+	0.557187i	$0.811881\pi$
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	9.98793	+	9.98793i	0.273770	+	0.273770i	0.830616	−	0.556846i	$-0.187988\pi$
$11$	9.98793	+	9.98793i	0.273770	+	0.273770i	−0.556846	+	0.830616i	$0.687988\pi$
$12$		0			0
$13$	−4.24079	+	46.6799i	−0.0904756	+	0.995899i
$13$	−4.24079	+	46.6799i	−0.0904756	+	0.995899i
$14$		0			0
$15$	−25.4492	+	25.4492i	−0.438064	+	0.438064i
$16$		0			0
$17$			61.7223i			0.880580i	0.897856	+	0.440290i	$0.145124\pi$
$17$			61.7223i			0.880580i	−0.897856	+	0.440290i	$0.854876\pi$
$18$		0			0
$19$	−51.7737	+	51.7737i	−0.625143	+	0.625143i	−0.946842	−	0.321699i	$-0.895746\pi$
$19$	−51.7737	+	51.7737i	−0.625143	+	0.625143i	0.321699	+	0.946842i	$0.395746\pi$
$20$		0			0
$21$	−15.1792	+	15.1792i	−0.157732	+	0.157732i
$22$		0			0
$23$	81.7373			0.741017			0.370509	−	0.928829i	$-0.379183\pi$
$23$	81.7373			0.741017			0.370509	+	0.928829i	$0.379183\pi$
$24$		0			0
$25$			18.9248i			0.151398i
$26$		0			0
$27$			27.0000i			0.192450i
$28$		0			0
$29$	−24.7234			−0.158311			−0.0791554	−	0.996862i	$-0.525222\pi$
$29$	−24.7234			−0.158311			−0.0791554	+	0.996862i	$0.525222\pi$
$30$		0			0
$31$	−55.5963	+	55.5963i	−0.322110	+	0.322110i	−0.849576	−	0.527466i	$-0.823142\pi$
$31$	−55.5963	+	55.5963i	−0.322110	+	0.322110i	0.527466	+	0.849576i	$0.323142\pi$
$32$		0			0
$33$	29.9638	−	29.9638i	0.158061	−	0.158061i
$34$		0			0
$35$			85.8440i			0.414580i
$36$		0			0
$37$	275.273	−	275.273i	1.22310	−	1.22310i	0.256575	−	0.966524i	$-0.417406\pi$
$37$	275.273	−	275.273i	1.22310	−	1.22310i	0.966524	−	0.256575i	$-0.0825939\pi$
$38$		0			0
$39$	140.040	+	12.7224i	0.574982	+	0.0522361i
$40$		0			0
$41$	−29.8795	−	29.8795i	−0.113814	−	0.113814i	0.647906	−	0.761720i	$-0.275645\pi$
$41$	−29.8795	−	29.8795i	−0.113814	−	0.113814i	−0.761720	+	0.647906i	$0.775645\pi$
$42$		0			0
$43$	16.9411			0.0600813			0.0300406	−	0.999549i	$-0.490436\pi$
$43$	16.9411			0.0600813			0.0300406	+	0.999549i	$0.490436\pi$
$44$		0			0
$45$	76.3476	+	76.3476i	0.252916	+	0.252916i
$46$		0			0
$47$	188.989	+	188.989i	0.586529	+	0.586529i	0.936690	−	0.350160i	$-0.113873\pi$
$47$	188.989	+	188.989i	0.586529	+	0.586529i	−0.350160	+	0.936690i	$0.613873\pi$
$48$		0			0
$49$		−	291.798i		−	0.850724i
$50$		0			0
$51$	185.167			0.508403
$52$		0			0
$53$	115.819			0.300169			0.150085	−	0.988673i	$-0.452045\pi$
$53$	115.819			0.300169			0.150085	+	0.988673i	$0.452045\pi$
$54$		0			0
$55$		−	169.457i		−	0.415446i
$56$		0			0
$57$	155.321	+	155.321i	0.360926	+	0.360926i
$58$		0			0
$59$	368.361	+	368.361i	0.812822	+	0.812822i	0.985056	−	0.172234i	$-0.0550987\pi$
$59$	368.361	+	368.361i	0.812822	+	0.812822i	−0.172234	+	0.985056i	$0.555099\pi$
$60$		0			0
$61$	569.709			1.19580			0.597899	−	0.801571i	$-0.296002\pi$
$61$	569.709			1.19580			0.597899	+	0.801571i	$0.296002\pi$
$62$		0			0
$63$	45.5376	+	45.5376i	0.0910666	+	0.0910666i
$64$		0			0
$65$	431.964	−	360.014i	0.824285	−	0.686988i
$66$		0			0
$67$	−500.494	+	500.494i	−0.912613	+	0.912613i	−0.996477	−	0.0838641i	$-0.973274\pi$
$67$	−500.494	+	500.494i	−0.912613	+	0.912613i	0.0838641	+	0.996477i	$0.473274\pi$
$68$		0			0
$69$		−	245.212i		−	0.427826i
$70$		0			0
$71$	52.4392	−	52.4392i	0.0876533	−	0.0876533i	−0.661921	−	0.749574i	$-0.730258\pi$
$71$	52.4392	−	52.4392i	0.0876533	−	0.0876533i	0.749574	+	0.661921i	$0.230258\pi$
$72$		0			0
$73$	138.074	−	138.074i	0.221374	−	0.221374i	−0.587703	−	0.809077i	$-0.699968\pi$
$73$	138.074	−	138.074i	0.221374	−	0.221374i	0.809077	+	0.587703i	$0.199968\pi$
$74$		0			0
$75$	56.7744			0.0874099
$76$		0			0
$77$		−	101.072i		−	0.149588i
$78$		0			0
$79$		−	417.808i		−	0.595027i	−0.954718	−	0.297513i	$-0.903843\pi$
$79$		−	417.808i		−	0.595027i	0.954718	−	0.297513i	$-0.0961573\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	−781.441	+	781.441i	−1.03343	+	1.03343i	−0.0340035	+	0.999422i	$0.510826\pi$
$83$	−781.441	+	781.441i	−1.03343	+	1.03343i	−0.999422	+	0.0340035i	$0.989174\pi$
$84$		0			0
$85$	523.595	−	523.595i	0.668139	−	0.668139i
$86$		0			0
$87$			74.1701i			0.0914008i
$88$		0			0
$89$	555.705	−	555.705i	0.661849	−	0.661849i	−0.293966	−	0.955816i	$-0.594975\pi$
$89$	555.705	−	555.705i	0.661849	−	0.661849i	0.955816	+	0.293966i	$0.0949754\pi$
$90$		0			0
$91$	257.645	−	214.731i	0.296797	−	0.247361i
$92$		0			0
$93$	166.789	+	166.789i	0.185970	+	0.185970i
$94$		0			0
$95$	878.400			0.948652
$96$		0			0
$97$	98.2402	+	98.2402i	0.102833	+	0.102833i	0.756651	−	0.653819i	$-0.226834\pi$
$97$	98.2402	+	98.2402i	0.102833	+	0.102833i	−0.653819	+	0.756651i	$0.726834\pi$
$98$		0			0
$99$	−89.8914	−	89.8914i	−0.0912568	−	0.0912568i
$100$		0			0
$101$			698.072i			0.687730i	0.939019	+	0.343865i	$0.111736\pi$
$101$			698.072i			0.687730i	−0.939019	+	0.343865i	$0.888264\pi$
$102$		0			0
$103$	362.337			0.346623			0.173311	−	0.984867i	$-0.444553\pi$
$103$	362.337			0.346623			0.173311	+	0.984867i	$0.444553\pi$
$104$		0			0
$105$	257.532			0.239358
$106$		0			0
$107$			1293.65i			1.16880i	0.811464	+	0.584402i	$0.198671\pi$
$107$			1293.65i			1.16880i	−0.811464	+	0.584402i	$0.801329\pi$
$108$		0			0
$109$	896.432	+	896.432i	0.787731	+	0.787731i	0.981122	−	0.193391i	$-0.0619487\pi$
$109$	896.432	+	896.432i	0.787731	+	0.787731i	−0.193391	+	0.981122i	$0.561949\pi$
$110$		0			0
$111$	−825.820	−	825.820i	−0.706157	−	0.706157i
$112$		0			0
$113$	1317.62			1.09691			0.548456	−	0.836180i	$-0.315216\pi$
$113$	1317.62			1.09691			0.548456	+	0.836180i	$0.315216\pi$
$114$		0			0
$115$	−693.382	−	693.382i	−0.562246	−	0.562246i
$116$		0			0
$117$	38.1671	−	420.119i	0.0301585	−	0.331966i
$118$		0			0
$119$	312.298	−	312.298i	0.240574	−	0.240574i
$120$		0			0
$121$		−	1131.48i		−	0.850099i
$122$		0			0
$123$	−89.6384	+	89.6384i	−0.0657107	+	0.0657107i
$124$		0			0
$125$	−899.843	+	899.843i	−0.643875	+	0.643875i
$126$		0			0
$127$	593.408			0.414618			0.207309	−	0.978276i	$-0.433530\pi$
$127$	593.408			0.414618			0.207309	+	0.978276i	$0.433530\pi$
$128$		0			0
$129$		−	50.8233i		−	0.0346879i
$130$		0			0
$131$			55.8307i			0.0372362i	0.999827	+	0.0186181i	$0.00592667\pi$
$131$			55.8307i			0.0372362i	−0.999827	+	0.0186181i	$0.994073\pi$
$132$		0			0
$133$	523.922			0.341578
$134$		0			0
$135$	229.043	−	229.043i	0.146021	−	0.146021i
$136$		0			0
$137$	−109.442	+	109.442i	−0.0682499	+	0.0682499i	−0.740408	−	0.672158i	$-0.765368\pi$
$137$	−109.442	+	109.442i	−0.0682499	+	0.0682499i	0.672158	+	0.740408i	$0.265368\pi$
$138$		0			0
$139$			1268.74i			0.774195i	0.922039	+	0.387098i	$0.126522\pi$
$139$			1268.74i			0.774195i	−0.922039	+	0.387098i	$0.873478\pi$
$140$		0			0
$141$	566.967	−	566.967i	0.338633	−	0.338633i
$142$		0			0
$143$	−508.593	+	423.879i	−0.297417	+	0.247878i
$144$		0			0
$145$	209.730	+	209.730i	0.120118	+	0.120118i
$146$		0			0
$147$	−875.395			−0.491166
$148$		0			0
$149$	818.726	+	818.726i	0.450152	+	0.450152i	0.895405	−	0.445253i	$-0.146886\pi$
$149$	818.726	+	818.726i	0.450152	+	0.450152i	−0.445253	+	0.895405i	$0.646886\pi$
$150$		0			0
$151$	432.964	+	432.964i	0.233338	+	0.233338i	0.814085	−	0.580746i	$-0.197239\pi$
$151$	432.964	+	432.964i	0.233338	+	0.233338i	−0.580746	+	0.814085i	$0.697239\pi$
$152$		0			0
$153$		−	555.501i		−	0.293527i
$154$		0			0
$155$	943.255			0.488800
$156$		0			0
$157$	1061.33			0.539510			0.269755	−	0.962929i	$-0.413057\pi$
$157$	1061.33			0.539510			0.269755	+	0.962929i	$0.413057\pi$
$158$		0			0
$159$		−	347.457i		−	0.173303i
$160$		0			0
$161$	−413.568	−	413.568i	−0.202446	−	0.202446i
$162$		0			0
$163$	327.180	+	327.180i	0.157219	+	0.157219i	0.781333	−	0.624114i	$-0.214540\pi$
$163$	327.180	+	327.180i	0.157219	+	0.157219i	−0.624114	+	0.781333i	$0.714540\pi$
$164$		0			0
$165$	−508.370			−0.239858
$166$		0			0
$167$	2717.01	+	2717.01i	1.25897	+	1.25897i	0.951584	+	0.307389i	$0.0994552\pi$
$167$	2717.01	+	2717.01i	1.25897	+	1.25897i	0.307389	+	0.951584i	$0.400545\pi$
$168$		0			0
$169$	−2161.03	−	395.919i	−0.983628	−	0.180209i
$170$		0			0
$171$	465.964	−	465.964i	0.208381	−	0.208381i
$172$		0			0
$173$		−	2635.72i		−	1.15832i	−0.815213	−	0.579161i	$-0.803380\pi$
$173$		−	2635.72i		−	1.15832i	0.815213	−	0.579161i	$-0.196620\pi$
$174$		0			0
$175$	95.7544	−	95.7544i	0.0413620	−	0.0413620i
$176$		0			0
$177$	1105.08	−	1105.08i	0.469283	−	0.469283i
$178$		0			0
$179$	−1912.17			−0.798447			−0.399224	−	0.916854i	$-0.630720\pi$
$179$	−1912.17			−0.798447			−0.399224	+	0.916854i	$0.630720\pi$
$180$		0			0
$181$			1505.59i			0.618285i	0.951016	+	0.309143i	$0.100042\pi$
$181$			1505.59i			0.618285i	−0.951016	+	0.309143i	$0.899958\pi$
$182$		0			0
$183$		−	1709.13i		−	0.690395i
$184$		0			0
$185$	−4670.32			−1.85605
$186$		0			0
$187$	−616.479	+	616.479i	−0.241077	+	0.241077i
$188$		0			0
$189$	136.613	−	136.613i	0.0525773	−	0.0525773i
$190$		0			0
$191$			4878.43i			1.84812i	0.382249	+	0.924060i	$0.375150\pi$
$191$			4878.43i			1.84812i	−0.382249	+	0.924060i	$0.624850\pi$
$192$		0			0
$193$	1630.98	−	1630.98i	0.608294	−	0.608294i	−0.334206	−	0.942500i	$-0.608468\pi$
$193$	1630.98	−	1630.98i	0.608294	−	0.608294i	0.942500	+	0.334206i	$0.108468\pi$
$194$		0			0
$195$	−1080.04	−	1295.89i	−0.396633	−	0.475901i
$196$		0			0
$197$	−12.2682	−	12.2682i	−0.00443693	−	0.00443693i	0.704885	−	0.709322i	$-0.250999\pi$
$197$	−12.2682	−	12.2682i	−0.00443693	−	0.00443693i	−0.709322	+	0.704885i	$0.750999\pi$
$198$		0			0
$199$	3705.30			1.31991			0.659955	−	0.751305i	$-0.270575\pi$
$199$	3705.30			1.31991			0.659955	+	0.751305i	$0.270575\pi$
$200$		0			0
$201$	1501.48	+	1501.48i	0.526897	+	0.526897i
$202$		0			0
$203$	125.094	+	125.094i	0.0432505	+	0.0432505i
$204$		0			0
$205$			506.939i			0.172713i
$206$		0			0
$207$	−735.635			−0.247006
$208$		0			0
$209$	−1034.23			−0.342291
$210$		0			0
$211$		−	3304.42i		−	1.07813i	−0.842264	−	0.539066i	$-0.818777\pi$
$211$		−	3304.42i		−	1.07813i	0.842264	−	0.539066i	$-0.181223\pi$
$212$		0			0
$213$	−157.317	−	157.317i	−0.0506067	−	0.0506067i
$214$		0			0
$215$	−143.713	−	143.713i	−0.0455866	−	0.0455866i
$216$		0			0
$217$	562.605			0.176000
$218$		0			0
$219$	−414.221	−	414.221i	−0.127810	−	0.127810i
$220$		0			0
$221$	−2881.19	−	261.751i	−0.876969	−	0.0796710i
$222$		0			0
$223$	409.577	−	409.577i	0.122992	−	0.122992i	−0.642931	−	0.765924i	$-0.722282\pi$
$223$	409.577	−	409.577i	0.122992	−	0.122992i	0.765924	+	0.642931i	$0.222282\pi$
$224$		0			0
$225$		−	170.323i		−	0.0504661i
$226$		0			0
$227$	−1804.89	+	1804.89i	−0.527729	+	0.527729i	−0.919895	−	0.392165i	$-0.871726\pi$
$227$	−1804.89	+	1804.89i	−0.527729	+	0.527729i	0.392165	+	0.919895i	$0.371726\pi$
$228$		0			0
$229$	−3016.37	+	3016.37i	−0.870426	+	0.870426i	−0.992519	−	0.122092i	$-0.961040\pi$
$229$	−3016.37	+	3016.37i	−0.870426	+	0.870426i	0.122092	+	0.992519i	$0.461040\pi$
$230$		0			0
$231$	−303.217			−0.0863647
$232$		0			0
$233$		−	709.245i		−	0.199417i	−0.995017	−	0.0997086i	$-0.968209\pi$
$233$		−	709.245i		−	0.199417i	0.995017	−	0.0997086i	$-0.0317911\pi$
$234$		0			0
$235$		−	3206.41i		−	0.890056i
$236$		0			0
$237$	−1253.43			−0.343539
$238$		0			0
$239$	977.682	−	977.682i	0.264607	−	0.264607i	−0.562316	−	0.826923i	$-0.690089\pi$
$239$	977.682	−	977.682i	0.264607	−	0.264607i	0.826923	+	0.562316i	$0.190089\pi$
$240$		0			0
$241$	−3040.89	+	3040.89i	−0.812785	+	0.812785i	−0.985051	−	0.172266i	$-0.944891\pi$
$241$	−3040.89	+	3040.89i	−0.812785	+	0.812785i	0.172266	+	0.985051i	$0.444891\pi$
$242$		0			0
$243$		−	243.000i		−	0.0641500i
$244$		0			0
$245$	−2475.34	+	2475.34i	−0.645485	+	0.645485i
$246$		0			0
$247$	−2197.23	−	2636.36i	−0.566019	−	0.679139i
$248$		0			0
$249$	2344.32	+	2344.32i	0.596648	+	0.596648i
$250$		0			0
$251$	1550.70			0.389958			0.194979	−	0.980807i	$-0.437536\pi$
$251$	1550.70			0.389958			0.194979	+	0.980807i	$0.437536\pi$
$252$		0			0
$253$	816.386	+	816.386i	0.202869	+	0.202869i
$254$		0			0
$255$	−1570.78	−	1570.78i	−0.385750	−	0.385750i
$256$		0			0
$257$			6530.26i			1.58501i	0.609868	+	0.792503i	$0.291222\pi$
$257$			6530.26i			1.58501i	−0.609868	+	0.792503i	$0.708778\pi$
$258$		0			0
$259$	−2785.62			−0.668301
$260$		0			0
$261$	222.510			0.0527703
$262$		0			0
$263$		−	1775.56i		−	0.416295i	−0.978097	−	0.208148i	$-0.933257\pi$
$263$		−	1775.56i		−	0.416295i	0.978097	−	0.208148i	$-0.0667434\pi$
$264$		0			0
$265$	−982.501	−	982.501i	−0.227753	−	0.227753i
$266$		0			0
$267$	−1667.11	−	1667.11i	−0.382119	−	0.382119i
$268$		0			0
$269$	−4308.52			−0.976562			−0.488281	−	0.872687i	$-0.662376\pi$
$269$	−4308.52			−0.976562			−0.488281	+	0.872687i	$0.662376\pi$
$270$		0			0
$271$	−4453.18	−	4453.18i	−0.998196	−	0.998196i	0.00180207	−	0.999998i	$-0.499426\pi$
$271$	−4453.18	−	4453.18i	−0.998196	−	0.998196i	−0.999998	+	0.00180207i	$0.999426\pi$
$272$		0			0
$273$	−644.192	−	772.935i	−0.142814	−	0.171356i
$274$		0			0
$275$	−189.020	+	189.020i	−0.0414484	+	0.0414484i
$276$		0			0
$277$		−	5207.50i		−	1.12956i	−0.825242	−	0.564780i	$-0.808961\pi$
$277$		−	5207.50i		−	1.12956i	0.825242	−	0.564780i	$-0.191039\pi$
$278$		0			0
$279$	500.367	−	500.367i	0.107370	−	0.107370i
$280$		0			0
$281$	−4994.91	+	4994.91i	−1.06040	+	1.06040i	−0.0623403	+	0.998055i	$0.519856\pi$
$281$	−4994.91	+	4994.91i	−1.06040	+	1.06040i	−0.998055	+	0.0623403i	$0.980144\pi$
$282$		0			0
$283$	−3035.85			−0.637676			−0.318838	−	0.947809i	$-0.603293\pi$
$283$	−3035.85			−0.637676			−0.318838	+	0.947809i	$0.603293\pi$
$284$		0			0
$285$		−	2635.20i		−	0.547705i
$286$		0			0
$287$			302.364i			0.0621881i
$288$		0			0
$289$	1103.35			0.224578
$290$		0			0
$291$	294.721	−	294.721i	0.0593705	−	0.0593705i
$292$		0			0
$293$	−3724.32	+	3724.32i	−0.742584	+	0.742584i	−0.973075	−	0.230490i	$-0.925967\pi$
$293$	−3724.32	+	3724.32i	−0.742584	+	0.742584i	0.230490	+	0.973075i	$0.425967\pi$
$294$		0			0
$295$		−	6249.66i		−	1.23345i
$296$		0			0
$297$	−269.674	+	269.674i	−0.0526872	+	0.0526872i
$298$		0			0
$299$	−346.630	+	3815.49i	−0.0670440	+	0.737978i
$300$		0			0
$301$	−85.7174	−	85.7174i	−0.0164142	−	0.0164142i
$302$		0			0
$303$	2094.21			0.397061
$304$		0			0
$305$	−4832.88	−	4832.88i	−0.907311	−	0.907311i
$306$		0			0
$307$	−3862.19	−	3862.19i	−0.718002	−	0.718002i	0.250194	−	0.968196i	$-0.419506\pi$
$307$	−3862.19	−	3862.19i	−0.718002	−	0.718002i	−0.968196	+	0.250194i	$0.919506\pi$
$308$		0			0
$309$		−	1087.01i		−	0.200123i
$310$		0			0
$311$	8058.08			1.46923			0.734617	−	0.678482i	$-0.237362\pi$
$311$	8058.08			1.46923			0.734617	+	0.678482i	$0.237362\pi$
$312$		0			0
$313$	−2817.68			−0.508833			−0.254417	−	0.967095i	$-0.581883\pi$
$313$	−2817.68			−0.508833			−0.254417	+	0.967095i	$0.581883\pi$
$314$		0			0
$315$		−	772.596i		−	0.138193i
$316$		0			0
$317$	3472.70	+	3472.70i	0.615288	+	0.615288i	0.944319	−	0.329031i	$-0.106722\pi$
$317$	3472.70	+	3472.70i	0.615288	+	0.615288i	−0.329031	+	0.944319i	$0.606722\pi$
$318$		0			0
$319$	−246.935	−	246.935i	−0.0433408	−	0.0433408i
$320$		0			0
$321$	3880.96			0.674810
$322$		0			0
$323$	−3195.60	−	3195.60i	−0.550488	−	0.550488i
$324$		0			0
$325$	−883.408	−	80.2561i	−0.150777	−	0.0136979i
$326$		0			0
$327$	2689.30	−	2689.30i	0.454796	−	0.454796i
$328$		0			0
$329$		−	1912.47i		−	0.320479i
$330$		0			0
$331$	−2748.24	+	2748.24i	−0.456365	+	0.456365i	−0.897460	−	0.441095i	$-0.854590\pi$
$331$	−2748.24	+	2748.24i	−0.456365	+	0.456365i	0.441095	+	0.897460i	$0.354590\pi$
$332$		0			0
$333$	−2477.46	+	2477.46i	−0.407700	+	0.407700i
$334$		0			0
$335$	8491.45			1.38489
$336$		0			0
$337$			764.136i			0.123517i	0.998091	+	0.0617583i	$0.0196708\pi$
$337$			764.136i			0.123517i	−0.998091	+	0.0617583i	$0.980329\pi$
$338$		0			0
$339$		−	3952.85i		−	0.633302i
$340$		0			0
$341$	−1110.59			−0.176368
$342$		0			0
$343$	−3211.91	+	3211.91i	−0.505617	+	0.505617i
$344$		0			0
$345$	−2080.15	+	2080.15i	−0.324613	+	0.324613i
$346$		0			0
$347$			11556.8i			1.78791i	0.448159	+	0.893954i	$0.352080\pi$
$347$			11556.8i			1.78791i	−0.448159	+	0.893954i	$0.647920\pi$
$348$		0			0
$349$	8368.94	−	8368.94i	1.28361	−	1.28361i	0.345009	−	0.938599i	$-0.387876\pi$
$349$	8368.94	−	8368.94i	1.28361	−	1.28361i	0.938599	−	0.345009i	$-0.112124\pi$
$350$		0			0
$351$	−1260.36	−	114.501i	−0.191661	−	0.0174120i
$352$		0			0
$353$	900.605	+	900.605i	0.135791	+	0.135791i	0.771735	−	0.635944i	$-0.219389\pi$
$353$	900.605	+	900.605i	0.135791	+	0.135791i	−0.635944	+	0.771735i	$0.719389\pi$
$354$		0			0
$355$	−889.690			−0.133014
$356$		0			0
$357$	−936.895	−	936.895i	−0.138896	−	0.138896i
$358$		0			0
$359$	1122.48	+	1122.48i	0.165020	+	0.165020i	0.784786	−	0.619767i	$-0.212773\pi$
$359$	1122.48	+	1122.48i	0.165020	+	0.165020i	−0.619767	+	0.784786i	$0.712773\pi$
$360$		0			0
$361$			1497.96i			0.218393i
$362$		0			0
$363$	−3394.45			−0.490805
$364$		0			0
$365$	−2342.58			−0.335935
$366$		0			0
$367$		−	13608.0i		−	1.93551i	−0.251899	−	0.967754i	$-0.581055\pi$
$367$		−	13608.0i		−	1.93551i	0.251899	−	0.967754i	$-0.418945\pi$
$368$		0			0
$369$	268.915	+	268.915i	0.0379381	+	0.0379381i
$370$		0			0
$371$	−586.014	−	586.014i	−0.0820062	−	0.0820062i
$372$		0			0
$373$	−854.063			−0.118557			−0.0592785	−	0.998241i	$-0.518880\pi$
$373$	−854.063			−0.118557			−0.0592785	+	0.998241i	$0.518880\pi$
$374$		0			0
$375$	2699.53	+	2699.53i	0.371741	+	0.371741i
$376$		0			0
$377$	104.847	−	1154.08i	0.0143233	−	0.157662i
$378$		0			0
$379$	−2772.08	+	2772.08i	−0.375705	+	0.375705i	−0.869550	−	0.493845i	$-0.835591\pi$
$379$	−2772.08	+	2772.08i	−0.375705	+	0.375705i	0.493845	+	0.869550i	$0.335591\pi$
$380$		0			0
$381$		−	1780.22i		−	0.239380i
$382$		0			0
$383$	−633.997	+	633.997i	−0.0845842	+	0.0845842i	−0.748133	−	0.663549i	$-0.769050\pi$
$383$	−633.997	+	633.997i	−0.0845842	+	0.0845842i	0.663549	+	0.748133i	$0.269050\pi$
$384$		0			0
$385$	−857.404	+	857.404i	−0.113500	+	0.113500i
$386$		0			0
$387$	−152.470			−0.0200271
$388$		0			0
$389$		−	1280.86i		−	0.166946i	−0.996510	−	0.0834731i	$-0.973399\pi$
$389$		−	1280.86i		−	0.166946i	0.996510	−	0.0834731i	$-0.0266013\pi$
$390$		0			0
$391$			5045.01i			0.652525i
$392$		0			0
$393$	167.492			0.0214984
$394$		0			0
$395$	−3544.30	+	3544.30i	−0.451476	+	0.451476i
$396$		0			0
$397$	208.309	−	208.309i	0.0263343	−	0.0263343i	−0.693817	−	0.720151i	$-0.744072\pi$
$397$	208.309	−	208.309i	0.0263343	−	0.0263343i	0.720151	+	0.693817i	$0.244072\pi$
$398$		0			0
$399$		−	1571.77i		−	0.197210i
$400$		0			0
$401$	976.355	−	976.355i	0.121588	−	0.121588i	−0.643694	−	0.765283i	$-0.722599\pi$
$401$	976.355	−	976.355i	0.121588	−	0.121588i	0.765283	+	0.643694i	$0.222599\pi$
$402$		0			0
$403$	−2359.46	−	2831.01i	−0.291645	−	0.349932i
$404$		0			0
$405$	−687.128	−	687.128i	−0.0843054	−	0.0843054i
$406$		0			0
$407$	5498.82			0.669697
$408$		0			0
$409$	−1212.43	−	1212.43i	−0.146579	−	0.146579i	0.630009	−	0.776588i	$-0.283051\pi$
$409$	−1212.43	−	1212.43i	−0.146579	−	0.146579i	−0.776588	+	0.630009i	$0.783051\pi$
$410$		0			0
$411$	328.325	+	328.325i	0.0394041	+	0.0394041i
$412$		0			0
$413$		−	3727.61i		−	0.444125i
$414$		0			0
$415$	13258.0			1.56822
$416$		0			0
$417$	3806.22			0.446982
$418$		0			0
$419$			13160.9i			1.53450i	0.641350	+	0.767248i	$0.278375\pi$
$419$			13160.9i			1.53450i	−0.641350	+	0.767248i	$0.721625\pi$
$420$		0			0
$421$	1057.97	+	1057.97i	0.122475	+	0.122475i	0.765688	−	0.643212i	$-0.222399\pi$
$421$	1057.97	+	1057.97i	0.122475	+	0.122475i	−0.643212	+	0.765688i	$0.722399\pi$
$422$		0			0
$423$	−1700.90	−	1700.90i	−0.195510	−	0.195510i
$424$		0			0
$425$	−1168.08			−0.133318
$426$		0			0
$427$	−2882.57	−	2882.57i	−0.326692	−	0.326692i
$428$		0			0
$429$	1271.64	+	1525.78i	0.143112	+	0.171714i
$430$		0			0
$431$	−863.251	+	863.251i	−0.0964764	+	0.0964764i	−0.753698	−	0.657221i	$-0.771732\pi$
$431$	−863.251	+	863.251i	−0.0964764	+	0.0964764i	0.657221	+	0.753698i	$0.271732\pi$
$432$		0			0
$433$		−	2491.00i		−	0.276466i	−0.990400	−	0.138233i	$-0.955858\pi$
$433$		−	2491.00i		−	0.276466i	0.990400	−	0.138233i	$-0.0441423\pi$
$434$		0			0
$435$	629.190	−	629.190i	0.0693502	−	0.0693502i
$436$		0			0
$437$	−4231.84	+	4231.84i	−0.463241	+	0.463241i
$438$		0			0
$439$	15318.0			1.66535			0.832675	−	0.553762i	$-0.186808\pi$
$439$	15318.0			1.66535			0.832675	+	0.553762i	$0.186808\pi$
$440$		0			0
$441$			2626.18i			0.283575i
$442$		0			0
$443$			3524.90i			0.378043i	0.981973	+	0.189022i	$0.0605316\pi$
$443$			3524.90i			0.378043i	−0.981973	+	0.189022i	$0.939468\pi$
$444$		0			0
$445$	−9428.16			−1.00435
$446$		0			0
$447$	2456.18	−	2456.18i	0.259895	−	0.259895i
$448$		0			0
$449$	−1065.75	+	1065.75i	−0.112018	+	0.112018i	−0.760894	−	0.648876i	$-0.775239\pi$
$449$	−1065.75	+	1065.75i	−0.112018	+	0.112018i	0.648876	+	0.760894i	$0.275239\pi$
$450$		0			0
$451$		−	596.868i		−	0.0623180i
$452$		0			0
$453$	1298.89	−	1298.89i	0.134718	−	0.134718i
$454$		0			0
$455$	−4007.19	−	364.046i	−0.412879	−	0.0375094i
$456$		0			0
$457$	5244.01	+	5244.01i	0.536772	+	0.536772i	0.922579	−	0.385808i	$-0.126077\pi$
$457$	5244.01	+	5244.01i	0.536772	+	0.536772i	−0.385808	+	0.922579i	$0.626077\pi$
$458$		0			0
$459$	−1666.50			−0.169468
$460$		0			0
$461$	797.343	+	797.343i	0.0805553	+	0.0805553i	0.746236	−	0.665681i	$-0.231859\pi$
$461$	797.343	+	797.343i	0.0805553	+	0.0805553i	−0.665681	+	0.746236i	$0.731859\pi$
$462$		0			0
$463$	5719.24	+	5719.24i	0.574073	+	0.574073i	0.933264	−	0.359191i	$-0.116947\pi$
$463$	5719.24	+	5719.24i	0.574073	+	0.574073i	−0.359191	+	0.933264i	$0.616947\pi$
$464$		0			0
$465$		−	2829.76i		−	0.282209i
$466$		0			0
$467$	−14196.0			−1.40667			−0.703335	−	0.710859i	$-0.748306\pi$
$467$	−14196.0			−1.40667			−0.703335	+	0.710859i	$0.748306\pi$
$468$		0			0
$469$	5064.73			0.498651
$470$		0			0
$471$		−	3183.98i		−	0.311486i
$472$		0			0
$473$	169.207	+	169.207i	0.0164485	+	0.0164485i
$474$		0			0
$475$	−979.808	−	979.808i	−0.0946456	−	0.0946456i
$476$		0			0
$477$	−1042.37			−0.100056
$478$		0			0
$479$	5205.23	+	5205.23i	0.496520	+	0.496520i	0.910353	−	0.413833i	$-0.135810\pi$
$479$	5205.23	+	5205.23i	0.496520	+	0.496520i	−0.413833	+	0.910353i	$0.635810\pi$
$480$		0			0
$481$	11682.4	+	14017.1i	1.10742	+	1.32874i
$482$		0			0
$483$	−1240.71	+	1240.71i	−0.116882	+	0.116882i
$484$		0			0
$485$		−	1666.76i		−	0.156048i
$486$		0			0
$487$	−4613.50	+	4613.50i	−0.429276	+	0.429276i	−0.888382	−	0.459105i	$-0.848170\pi$
$487$	−4613.50	+	4613.50i	−0.429276	+	0.429276i	0.459105	+	0.888382i	$0.348170\pi$
$488$		0			0
$489$	981.540	−	981.540i	0.0907705	−	0.0907705i
$490$		0			0
$491$	6586.64			0.605399			0.302700	−	0.953086i	$-0.402112\pi$
$491$	6586.64			0.605399			0.302700	+	0.953086i	$0.402112\pi$
$492$		0			0
$493$		−	1525.98i		−	0.139405i
$494$		0			0
$495$			1525.11i			0.138482i
$496$		0			0
$497$	−530.656			−0.0478937
$498$		0			0
$499$	−13544.8	+	13544.8i	−1.21513	+	1.21513i	−0.245811	+	0.969318i	$0.579054\pi$
$499$	−13544.8	+	13544.8i	−1.21513	+	1.21513i	−0.969318	+	0.245811i	$0.920946\pi$
$500$		0			0
$501$	8151.03	−	8151.03i	0.726868	−	0.726868i
$502$		0			0
$503$		−	6839.28i		−	0.606259i	−0.952949	−	0.303130i	$-0.901968\pi$
$503$		−	6839.28i		−	0.606259i	0.952949	−	0.303130i	$-0.0980315\pi$
$504$		0			0
$505$	5921.79	−	5921.79i	0.521814	−	0.521814i
$506$		0			0
$507$	−1187.76	+	6483.09i	−0.104044	+	0.567898i
$508$		0			0
$509$	13300.7	+	13300.7i	1.15824	+	1.15824i	0.984854	+	0.173384i	$0.0554701\pi$
$509$	13300.7	+	13300.7i	1.15824	+	1.15824i	0.173384	+	0.984854i	$0.444530\pi$
$510$		0			0
$511$	−1397.23			−0.120959
$512$		0			0
$513$	−1397.89	−	1397.89i	−0.120309	−	0.120309i
$514$		0			0
$515$	−3073.73	−	3073.73i	−0.263000	−	0.263000i
$516$		0			0
$517$			3775.22i			0.321149i
$518$		0			0
$519$	−7907.15			−0.668758
$520$		0			0
$521$	16665.8			1.40142			0.700712	−	0.713444i	$-0.252866\pi$
$521$	16665.8			1.40142			0.700712	+	0.713444i	$0.252866\pi$
$522$		0			0
$523$			9810.51i			0.820236i	0.912032	+	0.410118i	$0.134513\pi$
$523$			9810.51i			0.820236i	−0.912032	+	0.410118i	$0.865487\pi$
$524$		0			0
$525$	−287.263	−	287.263i	−0.0238804	−	0.0238804i
$526$		0			0
$527$	−3431.54	−	3431.54i	−0.283643	−	0.283643i
$528$		0			0
$529$	−5486.02			−0.450894
$530$		0			0
$531$	−3315.25	−	3315.25i	−0.270941	−	0.270941i
$532$		0			0
$533$	1521.48	−	1268.06i	0.123645	−	0.103050i
$534$		0			0
$535$	10974.1	−	10974.1i	0.886829	−	0.886829i
$536$		0			0
$537$			5736.50i			0.460984i
$538$		0			0
$539$	2914.46	−	2914.46i	0.232903	−	0.232903i
$540$		0			0
$541$	8525.03	−	8525.03i	0.677486	−	0.677486i	−0.281945	−	0.959431i	$-0.590980\pi$
$541$	8525.03	−	8525.03i	0.677486	−	0.677486i	0.959431	+	0.281945i	$0.0909795\pi$
$542$		0			0
$543$	4516.77			0.356967
$544$		0			0
$545$		−	15209.0i		−	1.19538i
$546$		0			0
$547$			13678.6i			1.06921i	0.845103	+	0.534604i	$0.179539\pi$
$547$			13678.6i			1.06921i	−0.845103	+	0.534604i	$0.820461\pi$
$548$		0			0
$549$	−5127.38			−0.398600
$550$		0			0
$551$	1280.02	−	1280.02i	0.0989669	−	0.0989669i
$552$		0			0
$553$	−2114.00	+	2114.00i	−0.162561	+	0.162561i
$554$		0			0
$555$			14011.0i			1.07159i
$556$		0			0
$557$	4698.17	−	4698.17i	0.357393	−	0.357393i	−0.505458	−	0.862851i	$-0.668677\pi$
$557$	4698.17	−	4698.17i	0.357393	−	0.357393i	0.862851	+	0.505458i	$0.168677\pi$
$558$		0			0
$559$	−71.8437	+	790.810i	−0.00543589	+	0.0598349i
$560$		0			0
$561$	1849.44	+	1849.44i	0.139186	+	0.139186i
$562$		0			0
$563$	20088.1			1.50376			0.751878	−	0.659302i	$-0.229148\pi$
$563$	20088.1			1.50376			0.751878	+	0.659302i	$0.229148\pi$
$564$		0			0
$565$	−11177.4	−	11177.4i	−0.832280	−	0.832280i
$566$		0			0
$567$	−409.838	−	409.838i	−0.0303555	−	0.0303555i
$568$		0			0
$569$		−	11263.5i		−	0.829859i	−0.909853	−	0.414930i	$-0.863806\pi$
$569$		−	11263.5i		−	0.829859i	0.909853	−	0.414930i	$-0.136194\pi$
$570$		0			0
$571$	22058.1			1.61664			0.808321	−	0.588743i	$-0.200377\pi$
$571$	22058.1			1.61664			0.808321	+	0.588743i	$0.200377\pi$
$572$		0			0
$573$	14635.3			1.06701
$574$		0			0
$575$			1546.86i			0.112189i
$576$		0			0
$577$	−1203.51	−	1203.51i	−0.0868334	−	0.0868334i	0.662356	−	0.749189i	$-0.269557\pi$
$577$	−1203.51	−	1203.51i	−0.0868334	−	0.0868334i	−0.749189	+	0.662356i	$0.769557\pi$
$578$		0			0
$579$	−4892.95	−	4892.95i	−0.351199	−	0.351199i
$580$		0			0
$581$	7907.76			0.564663
$582$		0			0
$583$	1156.79	+	1156.79i	0.0821775	+	0.0821775i
$584$		0			0
$585$	−3887.67	+	3240.13i	−0.274762	+	0.228996i
$586$		0			0
$587$	877.916	−	877.916i	0.0617300	−	0.0617300i	−0.675568	−	0.737298i	$-0.736101\pi$
$587$	877.916	−	877.916i	0.0617300	−	0.0617300i	0.737298	+	0.675568i	$0.236101\pi$
$588$		0			0
$589$		−	5756.86i		−	0.402729i
$590$		0			0
$591$	−36.8047	+	36.8047i	−0.00256166	+	0.00256166i
$592$		0			0
$593$	19122.8	−	19122.8i	1.32425	−	1.32425i	0.413945	−	0.910302i	$-0.364151\pi$
$593$	19122.8	−	19122.8i	1.32425	−	1.32425i	0.910302	−	0.413945i	$-0.135849\pi$
$594$		0			0
$595$	−5298.49			−0.365071
$596$		0			0
$597$		−	11115.9i		−	0.762050i
$598$		0			0
$599$		−	5662.39i		−	0.386242i	−0.981175	−	0.193121i	$-0.938139\pi$
$599$		−	5662.39i		−	0.386242i	0.981175	−	0.193121i	$-0.0618610\pi$
$600$		0			0
$601$	19396.2			1.31645			0.658226	−	0.752820i	$-0.271307\pi$
$601$	19396.2			1.31645			0.658226	+	0.752820i	$0.271307\pi$
$602$		0			0
$603$	4504.45	−	4504.45i	0.304204	−	0.304204i
$604$		0			0
$605$	−9598.44	+	9598.44i	−0.645012	+	0.645012i
$606$		0			0
$607$			14814.1i			0.990588i	0.868725	+	0.495294i	$0.164940\pi$
$607$			14814.1i			0.990588i	−0.868725	+	0.495294i	$0.835060\pi$
$608$		0			0
$609$	375.281	−	375.281i	0.0249707	−	0.0249707i
$610$		0			0
$611$	−9623.45	+	8020.53i	−0.637190	+	0.531057i
$612$		0			0
$613$	−2280.88	−	2280.88i	−0.150284	−	0.150284i	0.627961	−	0.778245i	$-0.283890\pi$
$613$	−2280.88	−	2280.88i	−0.150284	−	0.150284i	−0.778245	+	0.627961i	$0.783890\pi$
$614$		0			0
$615$	1520.82			0.0997159
$616$		0			0
$617$	−19276.8	−	19276.8i	−1.25779	−	1.25779i	−0.952148	−	0.305639i	$-0.901130\pi$
$617$	−19276.8	−	19276.8i	−1.25779	−	1.25779i	−0.305639	−	0.952148i	$-0.598870\pi$
$618$		0			0
$619$	2569.30	+	2569.30i	0.166832	+	0.166832i	0.785585	−	0.618753i	$-0.212362\pi$
$619$	2569.30	+	2569.30i	0.166832	+	0.166832i	−0.618753	+	0.785585i	$0.712362\pi$
$620$		0			0
$621$			2206.91i			0.142609i
$622$		0			0
$623$	−5623.43			−0.361634
$624$		0			0
$625$	17632.5			1.12848
$626$		0			0
$627$			3102.68i			0.197622i
$628$		0			0
$629$	16990.5	+	16990.5i	1.07704	+	1.07704i
$630$		0			0
$631$	−4105.02	−	4105.02i	−0.258983	−	0.258983i	0.565657	−	0.824640i	$-0.308623\pi$
$631$	−4105.02	−	4105.02i	−0.258983	−	0.258983i	−0.824640	+	0.565657i	$0.808623\pi$
$632$		0			0
$633$	−9913.27			−0.622460
$634$		0			0
$635$	−5033.92	−	5033.92i	−0.314590	−	0.314590i
$636$		0			0
$637$	13621.1	+	1237.45i	0.847235	+	0.0769698i
$638$		0			0
$639$	−471.952	+	471.952i	−0.0292178	+	0.0292178i
$640$		0			0
$641$		−	9085.38i		−	0.559830i	−0.960025	−	0.279915i	$-0.909694\pi$
$641$		−	9085.38i		−	0.559830i	0.960025	−	0.279915i	$-0.0903062\pi$
$642$		0			0
$643$	14860.8	−	14860.8i	0.911436	−	0.911436i	−0.0849491	−	0.996385i	$-0.527073\pi$
$643$	14860.8	−	14860.8i	0.911436	−	0.911436i	0.996385	+	0.0849491i	$0.0270728\pi$
$644$		0			0
$645$	−431.138	+	431.138i	−0.0263194	+	0.0263194i
$646$		0			0
$647$	−11180.1			−0.679342			−0.339671	−	0.940544i	$-0.610316\pi$
$647$	−11180.1			−0.679342			−0.339671	+	0.940544i	$0.610316\pi$
$648$		0			0
$649$			7358.32i			0.445053i
$650$		0			0
$651$		−	1687.81i		−	0.101614i
$652$		0			0
$653$	11294.7			0.676868			0.338434	−	0.940990i	$-0.390103\pi$
$653$	11294.7			0.676868			0.338434	+	0.940990i	$0.390103\pi$
$654$		0			0
$655$	473.615	−	473.615i	0.0282529	−	0.0282529i
$656$		0			0
$657$	−1242.66	+	1242.66i	−0.0737914	+	0.0737914i
$658$		0			0
$659$		−	21547.7i		−	1.27372i	−0.770980	−	0.636860i	$-0.780233\pi$
$659$		−	21547.7i		−	1.27372i	0.770980	−	0.636860i	$-0.219767\pi$
$660$		0			0
$661$	−3005.91	+	3005.91i	−0.176878	+	0.176878i	−0.789993	−	0.613115i	$-0.789916\pi$
$661$	−3005.91	+	3005.91i	−0.176878	+	0.176878i	0.613115	+	0.789993i	$0.289916\pi$
$662$		0			0
$663$	−785.254	+	8643.58i	−0.0459981	+	0.506318i
$664$		0			0
$665$	−4444.47	−	4444.47i	−0.259171	−	0.259171i
$666$		0			0
$667$	−2020.82			−0.117311
$668$		0			0
$669$	−1228.73	−	1228.73i	−0.0710097	−	0.0710097i
$670$		0			0
$671$	5690.21	+	5690.21i	0.327374	+	0.327374i
$672$		0			0
$673$			2465.87i			0.141236i	0.997503	+	0.0706182i	$0.0224972\pi$
$673$			2465.87i			0.141236i	−0.997503	+	0.0706182i	$0.977503\pi$
$674$		0			0
$675$	−510.970			−0.0291366
$676$		0			0
$677$	28362.0			1.61010			0.805052	−	0.593204i	$-0.202137\pi$
$677$	28362.0			1.61010			0.805052	+	0.593204i	$0.202137\pi$
$678$		0			0
$679$		−	994.138i		−	0.0561878i
$680$		0			0
$681$	5414.66	+	5414.66i	0.304685	+	0.304685i
$682$		0			0
$683$	−19029.3	−	19029.3i	−1.06608	−	1.06608i	−0.997656	−	0.0684262i	$-0.978202\pi$
$683$	−19029.3	−	19029.3i	−1.06608	−	1.06608i	−0.0684262	−	0.997656i	$-0.521798\pi$
$684$		0			0
$685$	1856.80			0.103569
$686$		0			0
$687$	9049.12	+	9049.12i	0.502541	+	0.502541i
$688$		0			0
$689$	−491.164	+	5406.43i	−0.0271580	+	0.298938i
$690$		0			0
$691$	−6025.48	+	6025.48i	−0.331722	+	0.331722i	−0.853240	−	0.521518i	$-0.825366\pi$
$691$	−6025.48	+	6025.48i	−0.331722	+	0.331722i	0.521518	+	0.853240i	$0.325366\pi$
$692$		0			0
$693$			909.652i			0.0498627i
$694$		0			0
$695$	10762.8	−	10762.8i	0.587419	−	0.587419i
$696$		0			0
$697$	1844.23	−	1844.23i	0.100223	−	0.100223i
$698$		0			0
$699$	−2127.74			−0.115134
$700$		0			0
$701$			7225.59i			0.389311i	0.980872	+	0.194655i	$0.0623589\pi$
$701$			7225.59i			0.389311i	−0.980872	+	0.194655i	$0.937641\pi$
$702$		0			0
$703$			28503.9i			1.52922i
$704$		0			0
$705$	−9619.24			−0.513874
$706$		0			0
$707$	3532.05	−	3532.05i	0.187888	−	0.187888i
$708$		0			0
$709$	−21938.5	+	21938.5i	−1.16209	+	1.16209i	−0.178067	+	0.984018i	$0.556984\pi$
$709$	−21938.5	+	21938.5i	−1.16209	+	1.16209i	−0.984018	+	0.178067i	$0.943016\pi$
$710$		0			0
$711$			3760.28i			0.198342i
$712$		0			0
$713$	−4544.29	+	4544.29i	−0.238689	+	0.238689i
$714$		0			0
$715$	7910.22	+	718.629i	0.413742	+	0.0375877i
$716$		0			0
$717$	−2933.05	−	2933.05i	−0.152771	−	0.152771i
$718$		0			0
$719$	−6101.99			−0.316503			−0.158252	−	0.987399i	$-0.550586\pi$
$719$	−6101.99			−0.316503			−0.158252	+	0.987399i	$0.550586\pi$
$720$		0			0
$721$	−1833.33	−	1833.33i	−0.0946973	−	0.0946973i
$722$		0			0
$723$	9122.68	+	9122.68i	0.469262	+	0.469262i
$724$		0			0
$725$		−	467.885i		−	0.0239680i
$726$		0			0
$727$	−7298.20			−0.372318			−0.186159	−	0.982520i	$-0.559604\pi$
$727$	−7298.20			−0.372318			−0.186159	+	0.982520i	$0.559604\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$			1045.64i			0.0529064i
$732$		0			0
$733$	−8713.09	−	8713.09i	−0.439052	−	0.439052i	0.452641	−	0.891693i	$-0.350482\pi$
$733$	−8713.09	−	8713.09i	−0.439052	−	0.439052i	−0.891693	+	0.452641i	$0.850482\pi$
$734$		0			0
$735$	7426.03	+	7426.03i	0.372671	+	0.372671i
$736$		0			0
$737$	−9997.80			−0.499693
$738$		0			0
$739$	−23479.8	−	23479.8i	−1.16877	−	1.16877i	−0.982499	−	0.186268i	$-0.940361\pi$
$739$	−23479.8	−	23479.8i	−1.16877	−	1.16877i	−0.186268	−	0.982499i	$-0.559639\pi$
$740$		0			0
$741$	−7909.07	+	6591.70i	−0.392101	+	0.326791i
$742$		0			0
$743$	−1232.15	+	1232.15i	−0.0608387	+	0.0608387i	−0.736872	−	0.676033i	$-0.763698\pi$
$743$	−1232.15	+	1232.15i	−0.0608387	+	0.0608387i	0.676033	+	0.736872i	$0.263698\pi$
$744$		0			0
$745$		−	13890.6i		−	0.683104i
$746$		0			0
$747$	7032.97	−	7032.97i	0.344475	−	0.344475i
$748$		0			0
$749$	6545.53	−	6545.53i	0.319317	−	0.319317i
$750$		0			0
$751$	12349.0			0.600028			0.300014	−	0.953935i	$-0.403009\pi$
$751$	12349.0			0.600028			0.300014	+	0.953935i	$0.403009\pi$
$752$		0			0
$753$		−	4652.11i		−	0.225142i
$754$		0			0
$755$		−	7345.72i		−	0.354090i
$756$		0			0
$757$	−40932.2			−1.96527			−0.982633	−	0.185561i	$-0.940590\pi$
$757$	−40932.2			−1.96527			−0.982633	+	0.185561i	$0.940590\pi$
$758$		0			0
$759$	2449.16	−	2449.16i	0.117126	−	0.117126i
$760$		0			0
$761$	−21725.1	+	21725.1i	−1.03487	+	1.03487i	−0.0354955	+	0.999370i	$0.511301\pi$
$761$	−21725.1	+	21725.1i	−1.03487	+	1.03487i	−0.999370	+	0.0354955i	$0.988699\pi$
$762$		0			0
$763$		−	9071.41i		−	0.430415i
$764$		0			0
$765$	−4712.35	+	4712.35i	−0.222713	+	0.222713i
$766$		0			0
$767$	−18757.2	+	15632.9i	−0.883028	+	0.735947i
$768$		0			0
$769$	2654.34	+	2654.34i	0.124471	+	0.124471i	0.766598	−	0.642127i	$-0.221948\pi$
$769$	2654.34	+	2654.34i	0.124471	+	0.124471i	−0.642127	+	0.766598i	$0.721948\pi$
$770$		0			0
$771$	19590.8			0.915103
$772$		0			0
$773$	−9788.90	−	9788.90i	−0.455475	−	0.455475i	0.441692	−	0.897167i	$-0.354378\pi$
$773$	−9788.90	−	9788.90i	−0.455475	−	0.455475i	−0.897167	+	0.441692i	$0.854378\pi$
$774$		0			0
$775$	−1052.15	−	1052.15i	−0.0487669	−	0.0487669i
$776$		0			0
$777$			8356.85i			0.385843i
$778$		0			0
$779$	3093.94			0.142300
$780$		0			0
$781$	1047.52			0.0479938
$782$		0			0
$783$		−	667.531i		−	0.0304669i
$784$		0			0
$785$	−9003.30	−	9003.30i	−0.409352	−	0.409352i
$786$		0			0
$787$	6923.36	+	6923.36i	0.313585	+	0.313585i	0.846297	−	0.532712i	$-0.178827\pi$
$787$	6923.36	+	6923.36i	0.313585	+	0.313585i	−0.532712	+	0.846297i	$0.678827\pi$
$788$		0			0
$789$	−5326.68			−0.240348
$790$		0			0
$791$	−6666.79	−	6666.79i	−0.299676	−	0.299676i
$792$		0			0
$793$	−2416.01	+	26594.0i	−0.108191	+	1.19089i
$794$		0			0
$795$	−2947.50	+	2947.50i	−0.131493	+	0.131493i
$796$		0			0
$797$			25520.7i			1.13424i	0.823636	+	0.567119i	$0.191942\pi$
$797$			25520.7i			1.13424i	−0.823636	+	0.567119i	$0.808058\pi$
$798$		0			0
$799$	−11664.8	+	11664.8i	−0.516486	+	0.516486i
$800$		0			0
$801$	−5001.34	+	5001.34i	−0.220616	+	0.220616i
$802$		0			0
$803$	2758.14			0.121211
$804$		0			0
$805$			7016.65i			0.307211i
$806$		0			0
$807$			12925.6i			0.563818i
$808$		0			0
$809$	−12674.7			−0.550825			−0.275413	−	0.961326i	$-0.588815\pi$
$809$	−12674.7			−0.550825			−0.275413	+	0.961326i	$0.588815\pi$
$810$		0			0
$811$	−20413.5	+	20413.5i	−0.883867	+	0.883867i	−0.993925	−	0.110058i	$-0.964896\pi$
$811$	−20413.5	+	20413.5i	−0.883867	+	0.883867i	0.110058	+	0.993925i	$0.464896\pi$
$812$		0			0
$813$	−13359.5	+	13359.5i	−0.576309	+	0.576309i
$814$		0			0
$815$		−	5550.98i		−	0.238580i
$816$		0			0
$817$	−877.105	+	877.105i	−0.0375594	+	0.0375594i
$818$		0			0
$819$	−2318.81	+	1932.58i	−0.0989324	+	0.0824538i
$820$		0			0
$821$	12215.4	+	12215.4i	0.519272	+	0.519272i	0.917351	−	0.398079i	$-0.130323\pi$
$821$	12215.4	+	12215.4i	0.519272	+	0.519272i	−0.398079	+	0.917351i	$0.630323\pi$
$822$		0			0
$823$	−25616.6			−1.08498			−0.542489	−	0.840063i	$-0.682518\pi$
$823$	−25616.6			−1.08498			−0.542489	+	0.840063i	$0.682518\pi$
$824$		0			0
$825$	567.059	+	567.059i	0.0239303	+	0.0239303i
$826$		0			0
$827$	−14802.1	−	14802.1i	−0.622395	−	0.622395i	0.323748	−	0.946143i	$-0.395057\pi$
$827$	−14802.1	−	14802.1i	−0.622395	−	0.622395i	−0.946143	+	0.323748i	$0.895057\pi$
$828$		0			0
$829$		−	41460.8i		−	1.73702i	−0.495668	−	0.868512i	$-0.665077\pi$
$829$		−	41460.8i		−	1.73702i	0.495668	−	0.868512i	$-0.334923\pi$
$830$		0			0
$831$	−15622.5			−0.652152
$832$		0			0
$833$	18010.5			0.749131
$834$		0			0
$835$		−	46097.1i		−	1.91049i
$836$		0			0
$837$	−1501.10	−	1501.10i	−0.0619900	−	0.0619900i
$838$		0			0
$839$	−13209.4	−	13209.4i	−0.543551	−	0.543551i	0.381017	−	0.924568i	$-0.375574\pi$
$839$	−13209.4	−	13209.4i	−0.543551	−	0.543551i	−0.924568	+	0.381017i	$0.875574\pi$
$840$		0			0
$841$	−23777.8			−0.974938
$842$		0			0
$843$	14984.7	+	14984.7i	0.612219	+	0.612219i
$844$		0			0
$845$	14973.6	+	21690.8i	0.609593	+	0.883060i
$846$		0			0
$847$	−5724.99	+	5724.99i	−0.232247	+	0.232247i
$848$		0			0
$849$			9107.54i			0.368162i
$850$		0			0
$851$	22500.1	−	22500.1i	0.906337	−	0.906337i
$852$		0			0
$853$	−9629.74	+	9629.74i	−0.386537	+	0.386537i	−0.873450	−	0.486913i	$-0.838123\pi$
$853$	−9629.74	+	9629.74i	−0.386537	+	0.386537i	0.486913	+	0.873450i	$0.338123\pi$
$854$		0			0
$855$	−7905.60			−0.316217
$856$		0			0
$857$		−	1749.69i		−	0.0697414i	−0.999392	−	0.0348707i	$-0.988898\pi$
$857$		−	1749.69i		−	0.0697414i	0.999392	−	0.0348707i	$-0.0111019\pi$
$858$		0			0
$859$			45205.4i			1.79556i	0.440442	+	0.897781i	$0.354822\pi$
$859$			45205.4i			1.79556i	−0.440442	+	0.897781i	$0.645178\pi$
$860$		0			0
$861$	907.092			0.0359043
$862$		0			0
$863$	20072.5	−	20072.5i	0.791745	−	0.791745i	−0.190033	−	0.981778i	$-0.560859\pi$
$863$	20072.5	−	20072.5i	0.791745	−	0.791745i	0.981778	+	0.190033i	$0.0608595\pi$
$864$		0			0
$865$	−22359.0	+	22359.0i	−0.878875	+	0.878875i
$866$		0			0
$867$		−	3310.06i		−	0.129660i
$868$		0			0
$869$	4173.04	−	4173.04i	0.162901	−	0.162901i
$870$		0			0
$871$	−21240.5	−	25485.5i	−0.826301	−	0.991439i
$872$		0			0
$873$	−884.162	−	884.162i	−0.0342776	−	0.0342776i
$874$		0			0
$875$	9105.92			0.351813
$876$		0			0
$877$	−33660.3	−	33660.3i	−1.29604	−	1.29604i	−0.930986	−	0.365055i	$-0.881050\pi$
$877$	−33660.3	−	33660.3i	−1.29604	−	1.29604i	−0.365055	−	0.930986i	$-0.618950\pi$
$878$		0			0
$879$	11173.0	+	11173.0i	0.428731	+	0.428731i
$880$		0			0
$881$		−	10925.0i		−	0.417791i	−0.977938	−	0.208896i	$-0.933013\pi$
$881$		−	10925.0i		−	0.417791i	0.977938	−	0.208896i	$-0.0669869\pi$
$882$		0			0
$883$	38958.2			1.48477			0.742384	−	0.669975i	$-0.233695\pi$
$883$	38958.2			1.48477			0.742384	+	0.669975i	$0.233695\pi$
$884$		0			0
$885$	−18749.0			−0.712135
$886$		0			0
$887$		−	35783.0i		−	1.35454i	−0.735734	−	0.677270i	$-0.763163\pi$
$887$		−	35783.0i		−	1.35454i	0.735734	−	0.677270i	$-0.236837\pi$
$888$		0			0
$889$	−3002.48	−	3002.48i	−0.113273	−	0.113273i
$890$		0			0
$891$	809.023	+	809.023i	0.0304189	+	0.0304189i
$892$		0			0
$893$	−19569.3			−0.733329
$894$		0			0
$895$	16221.0	+	16221.0i	0.605820	+	0.605820i
$896$		0			0
$897$	11446.5	+	1039.89i	0.426072	+	0.0387079i
$898$		0			0
$899$	1374.53	−	1374.53i	0.0509934	−	0.0509934i
$900$		0			0
$901$			7148.63i			0.264323i
$902$		0			0
$903$	−257.152	+	257.152i	−0.00947674	+	0.00947674i
$904$		0			0
$905$	12772.0	−	12772.0i	0.469123	−	0.469123i
$906$		0			0
$907$	−6527.32			−0.238959			−0.119480	−	0.992837i	$-0.538123\pi$
$907$	−6527.32			−0.238959			−0.119480	+	0.992837i	$0.538123\pi$
$908$		0			0
$909$		−	6282.64i		−	0.229243i
$910$		0			0
$911$		−	9359.62i		−	0.340393i	−0.985410	−	0.170197i	$-0.945560\pi$
$911$		−	9359.62i		−	0.340393i	0.985410	−	0.170197i	$-0.0544403\pi$
$912$		0			0
$913$	−15610.0			−0.565843
$914$		0			0
$915$	−14498.6	+	14498.6i	−0.523836	+	0.523836i
$916$		0			0
$917$	282.488	−	282.488i	0.0101729	−	0.0101729i
$918$		0			0
$919$		−	25376.0i		−	0.910857i	−0.890272	−	0.455428i	$-0.849486\pi$
$919$		−	25376.0i		−	0.910857i	0.890272	−	0.455428i	$-0.150514\pi$
$920$		0			0
$921$	−11586.6	+	11586.6i	−0.414539	+	0.414539i
$922$		0			0
$923$	2225.47	+	2670.24i	0.0793633	+	0.0952243i
$924$		0			0
$925$	5209.49	+	5209.49i	0.185175	+	0.185175i
$926$		0			0
$927$	−3261.04			−0.115541
$928$		0			0
$929$	−14163.3	−	14163.3i	−0.500198	−	0.500198i	0.411302	−	0.911499i	$-0.365074\pi$
$929$	−14163.3	−	14163.3i	−0.500198	−	0.500198i	−0.911499	+	0.411302i	$0.865074\pi$
$930$		0			0
$931$	15107.5	+	15107.5i	0.531824	+	0.531824i
$932$		0			0
$933$		−	24174.2i		−	0.848263i
$934$		0			0
$935$	10459.3			0.365833
$936$		0			0
$937$	29274.6			1.02066			0.510331	−	0.859978i	$-0.329523\pi$
$937$	29274.6			1.02066			0.510331	+	0.859978i	$0.329523\pi$
$938$		0			0
$939$			8453.04i			0.293775i
$940$		0			0
$941$	39241.4	+	39241.4i	1.35944	+	1.35944i	0.874606	+	0.484835i	$0.161120\pi$
$941$	39241.4	+	39241.4i	1.35944	+	1.35944i	0.484835	+	0.874606i	$0.338880\pi$
$942$		0			0
$943$	−2442.27	−	2442.27i	−0.0843384	−	0.0843384i
$944$		0			0
$945$	−2317.79			−0.0797859
$946$		0			0
$947$	−12114.3	−	12114.3i	−0.415693	−	0.415693i	0.468023	−	0.883716i	$-0.344966\pi$
$947$	−12114.3	−	12114.3i	−0.415693	−	0.415693i	−0.883716	+	0.468023i	$0.844966\pi$
$948$		0			0
$949$	5859.73	+	7030.82i	0.200437	+	0.240495i
$950$		0			0
$951$	10418.1	−	10418.1i	0.355237	−	0.355237i
$952$		0			0
$953$		−	53542.6i		−	1.81995i	−0.414662	−	0.909976i	$-0.636100\pi$
$953$		−	53542.6i		−	1.81995i	0.414662	−	0.909976i	$-0.363900\pi$
$954$		0			0
$955$	41384.0	−	41384.0i	1.40226	−	1.40226i
$956$		0			0
$957$	−740.806	+	740.806i	−0.0250228	+	0.0250228i
$958$		0			0
$959$	1107.49			0.0372917
$960$		0			0
$961$			23609.1i			0.792491i
$962$		0			0
$963$		−	11642.9i		−	0.389602i
$964$		0			0
$965$	−27671.5			−0.923085
$966$		0			0
$967$	−24963.7	+	24963.7i	−0.830175	+	0.830175i	−0.987540	−	0.157365i	$-0.949700\pi$
$967$	−24963.7	+	24963.7i	−0.830175	+	0.830175i	0.157365	+	0.987540i	$0.449700\pi$
$968$		0			0
$969$	−9586.79	+	9586.79i	−0.317825	+	0.317825i
$970$		0			0
$971$		−	5776.30i		−	0.190907i	−0.995434	−	0.0954533i	$-0.969570\pi$
$971$		−	5776.30i		−	0.190907i	0.995434	−	0.0954533i	$-0.0304301\pi$
$972$		0			0
$973$	6419.48	−	6419.48i	0.211510	−	0.211510i
$974$		0			0
$975$	−240.768	+	2650.23i	−0.00790847	+	0.0870514i
$976$		0			0
$977$	−27576.9	−	27576.9i	−0.903032	−	0.903032i	0.0926652	−	0.995697i	$-0.470461\pi$
$977$	−27576.9	−	27576.9i	−0.903032	−	0.903032i	−0.995697	+	0.0926652i	$0.970461\pi$
$978$		0			0
$979$	11100.7			0.362390
$980$		0			0
$981$	−8067.89	−	8067.89i	−0.262577	−	0.262577i
$982$		0			0
$983$	−36938.2	−	36938.2i	−1.19852	−	1.19852i	−0.974610	−	0.223911i	$-0.928117\pi$
$983$	−36938.2	−	36938.2i	−1.19852	−	1.19852i	−0.223911	−	0.974610i	$-0.571883\pi$
$984$		0			0
$985$			208.145i			0.00673303i
$986$		0			0
$987$	−5737.40			−0.185029
$988$		0			0
$989$	1384.72			0.0445213
$990$		0			0
$991$		−	7160.60i		−	0.229530i	−0.993393	−	0.114765i	$-0.963389\pi$
$991$		−	7160.60i		−	0.229530i	0.993393	−	0.114765i	$-0.0366115\pi$
$992$		0			0
$993$	8244.72	+	8244.72i	0.263483	+	0.263483i
$994$		0			0
$995$	−31432.3	−	31432.3i	−1.00148	−	1.00148i
$996$		0			0
$997$	42655.2			1.35497			0.677484	−	0.735538i	$-0.263070\pi$
$997$	42655.2			1.35497			0.677484	+	0.735538i	$0.263070\pi$
$998$		0			0
$999$	7432.38	+	7432.38i	0.235386	+	0.235386i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.bc.d.463.3	yes	28
4.3	odd	2		624.4.bc.c.463.3	yes	28
13.5	odd	4		624.4.bc.c.31.3	✓	28
52.31	even	4	inner	624.4.bc.d.31.3	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
624.4.bc.c.31.3	✓	28	13.5	odd	4
624.4.bc.c.463.3	yes	28	4.3	odd	2
624.4.bc.d.31.3	yes	28	52.31	even	4	inner
624.4.bc.d.463.3	yes	28	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.bc (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +(-8.48307 - 8.48307i) q^{5} +(-5.05973 - 5.05973i) q^{7} -9.00000 q^{9} +(9.98793 + 9.98793i) q^{11} +(-4.24079 + 46.6799i) q^{13} +(-25.4492 + 25.4492i) q^{15} +61.7223i q^{17} +(-51.7737 + 51.7737i) q^{19} +(-15.1792 + 15.1792i) q^{21} +81.7373 q^{23} +18.9248i q^{25} +27.0000i q^{27} -24.7234 q^{29} +(-55.5963 + 55.5963i) q^{31} +(29.9638 - 29.9638i) q^{33} +85.8440i q^{35} +(275.273 - 275.273i) q^{37} +(140.040 + 12.7224i) q^{39} +(-29.8795 - 29.8795i) q^{41} +16.9411 q^{43} +(76.3476 + 76.3476i) q^{45} +(188.989 + 188.989i) q^{47} -291.798i q^{49} +185.167 q^{51} +115.819 q^{53} -169.457i q^{55} +(155.321 + 155.321i) q^{57} +(368.361 + 368.361i) q^{59} +569.709 q^{61} +(45.5376 + 45.5376i) q^{63} +(431.964 - 360.014i) q^{65} +(-500.494 + 500.494i) q^{67} -245.212i q^{69} +(52.4392 - 52.4392i) q^{71} +(138.074 - 138.074i) q^{73} +56.7744 q^{75} -101.072i q^{77} -417.808i q^{79} +81.0000 q^{81} +(-781.441 + 781.441i) q^{83} +(523.595 - 523.595i) q^{85} +74.1701i q^{87} +(555.705 - 555.705i) q^{89} +(257.645 - 214.731i) q^{91} +(166.789 + 166.789i) q^{93} +878.400 q^{95} +(98.2402 + 98.2402i) q^{97} +(-89.8914 - 89.8914i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 4 q^{5} + 8 q^{7} - 252 q^{9} - 64 q^{11} - 32 q^{13} + 12 q^{15} - 56 q^{19} + 24 q^{21} - 384 q^{23} - 32 q^{29} + 168 q^{31} - 192 q^{33} + 412 q^{37} - 252 q^{39} + 1340 q^{41} - 624 q^{43} - 36 q^{45}+ \cdots + 576 q^{99}+O(q^{100}) \) 28 * q + 4 * q^5 + 8 * q^7 - 252 * q^9 - 64 * q^11 - 32 * q^13 + 12 * q^15 - 56 * q^19 + 24 * q^21 - 384 * q^23 - 32 * q^29 + 168 * q^31 - 192 * q^33 + 412 * q^37 - 252 * q^39 + 1340 * q^41 - 624 * q^43 - 36 * q^45 + 688 * q^47 - 312 * q^51 + 2152 * q^53 + 168 * q^57 - 624 * q^59 - 216 * q^61 - 72 * q^63 + 60 * q^65 + 2024 * q^67 - 448 * q^71 - 724 * q^73 + 1716 * q^75 + 2268 * q^81 - 2288 * q^83 - 1528 * q^85 - 1876 * q^89 + 3448 * q^91 - 504 * q^93 - 192 * q^95 + 76 * q^97 + 576 * q^99

Introduction

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