LMFDB - Embedded newform 624.4.bc.c.31.8

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			31.8
Character	$\chi$	$=$	624.31
Dual form			624.4.bc.c.463.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.00000i q^{3} +(-0.273393 + 0.273393i) q^{5} +(-21.0333 + 21.0333i) q^{7} -9.00000 q^{9} +(25.8675 - 25.8675i) q^{11} +(-15.7390 + 44.1507i) q^{13} +(0.820180 + 0.820180i) q^{15} -111.650i q^{17} +(-64.0398 - 64.0398i) q^{19} +(63.0999 + 63.0999i) q^{21} -92.7291 q^{23} +124.851i q^{25} +27.0000i q^{27} +254.637 q^{29} +(173.320 + 173.320i) q^{31} +(-77.6026 - 77.6026i) q^{33} -11.5007i q^{35} +(90.5331 + 90.5331i) q^{37} +(132.452 + 47.2169i) q^{39} +(235.240 - 235.240i) q^{41} +291.487 q^{43} +(2.46054 - 2.46054i) q^{45} +(35.6910 - 35.6910i) q^{47} -541.801i q^{49} -334.951 q^{51} +655.515 q^{53} +14.1440i q^{55} +(-192.119 + 192.119i) q^{57} +(-64.7380 + 64.7380i) q^{59} +243.569 q^{61} +(189.300 - 189.300i) q^{63} +(-7.76758 - 16.3734i) q^{65} +(-286.961 - 286.961i) q^{67} +278.187i q^{69} +(-707.169 - 707.169i) q^{71} +(530.346 + 530.346i) q^{73} +374.552 q^{75} +1088.16i q^{77} -1264.84i q^{79} +81.0000 q^{81} +(358.933 + 358.933i) q^{83} +(30.5244 + 30.5244i) q^{85} -763.910i q^{87} +(-374.949 - 374.949i) q^{89} +(-597.593 - 1259.68i) q^{91} +(519.959 - 519.959i) q^{93} +35.0161 q^{95} +(759.324 - 759.324i) q^{97} +(-232.808 + 232.808i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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
$5$	−0.273393	+	0.273393i	−0.0244530	+	0.0244530i	−0.719228	−	0.694775i	$-0.755504\pi$
$5$	−0.273393	+	0.273393i	−0.0244530	+	0.0244530i	0.694775	+	0.719228i	$0.255504\pi$
$6$		0			0
$7$	−21.0333	+	21.0333i	−1.13569	+	1.13569i	−0.146478	+	0.989214i	$0.546794\pi$
$7$	−21.0333	+	21.0333i	−1.13569	+	1.13569i	−0.989214	+	0.146478i	$0.953206\pi$
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	25.8675	−	25.8675i	0.709032	−	0.709032i	−0.257300	−	0.966332i	$-0.582833\pi$
$11$	25.8675	−	25.8675i	0.709032	−	0.709032i	0.966332	+	0.257300i	$0.0828327\pi$
$12$		0			0
$13$	−15.7390	+	44.1507i	−0.335785	+	0.941939i
$13$	−15.7390	+	44.1507i	−0.335785	+	0.941939i
$14$		0			0
$15$	0.820180	+	0.820180i	0.0141180	+	0.0141180i
$16$		0			0
$17$		−	111.650i		−	1.59289i	−0.604710	−	0.796446i	$-0.706711\pi$
$17$		−	111.650i		−	1.59289i	0.604710	−	0.796446i	$-0.293289\pi$
$18$		0			0
$19$	−64.0398	−	64.0398i	−0.773250	−	0.773250i	0.205424	−	0.978673i	$-0.434143\pi$
$19$	−64.0398	−	64.0398i	−0.773250	−	0.773250i	−0.978673	+	0.205424i	$0.934143\pi$
$20$		0			0
$21$	63.0999	+	63.0999i	0.655692	+	0.655692i
$22$		0			0
$23$	−92.7291			−0.840668			−0.420334	−	0.907369i	$-0.638087\pi$
$23$	−92.7291			−0.840668			−0.420334	+	0.907369i	$0.638087\pi$
$24$		0			0
$25$			124.851i			0.998804i
$26$		0			0
$27$			27.0000i			0.192450i
$28$		0			0
$29$	254.637			1.63051			0.815256	−	0.579101i	$-0.196596\pi$
$29$	254.637			1.63051			0.815256	+	0.579101i	$0.196596\pi$
$30$		0			0
$31$	173.320	+	173.320i	1.00417	+	1.00417i	0.999991	+	0.00417411i	$0.00132866\pi$
$31$	173.320	+	173.320i	1.00417	+	1.00417i	0.00417411	+	0.999991i	$0.498671\pi$
$32$		0			0
$33$	−77.6026	−	77.6026i	−0.409360	−	0.409360i
$34$		0			0
$35$		−	11.5007i		−	0.0555422i
$36$		0			0
$37$	90.5331	+	90.5331i	0.402258	+	0.402258i	0.879028	−	0.476770i	$-0.158193\pi$
$37$	90.5331	+	90.5331i	0.402258	+	0.402258i	−0.476770	+	0.879028i	$0.658193\pi$
$38$		0			0
$39$	132.452	+	47.2169i	0.543829	+	0.193865i
$40$		0			0
$41$	235.240	−	235.240i	0.896057	−	0.896057i	−0.0990278	−	0.995085i	$-0.531573\pi$
$41$	235.240	−	235.240i	0.896057	−	0.896057i	0.995085	+	0.0990278i	$0.0315733\pi$
$42$		0			0
$43$	291.487			1.03375			0.516876	−	0.856060i	$-0.327095\pi$
$43$	291.487			1.03375			0.516876	+	0.856060i	$0.327095\pi$
$44$		0			0
$45$	2.46054	−	2.46054i	0.00815101	−	0.00815101i
$46$		0			0
$47$	35.6910	−	35.6910i	0.110767	−	0.110767i	−0.649551	−	0.760318i	$-0.725043\pi$
$47$	35.6910	−	35.6910i	0.110767	−	0.110767i	0.760318	+	0.649551i	$0.225043\pi$
$48$		0			0
$49$		−	541.801i		−	1.57959i
$50$		0			0
$51$	−334.951			−0.919657
$52$		0			0
$53$	655.515			1.69891			0.849453	−	0.527665i	$-0.176932\pi$
$53$	655.515			1.69891			0.849453	+	0.527665i	$0.176932\pi$
$54$		0			0
$55$			14.1440i			0.0346760i
$56$		0			0
$57$	−192.119	+	192.119i	−0.446436	+	0.446436i
$58$		0			0
$59$	−64.7380	+	64.7380i	−0.142850	+	0.142850i	−0.774915	−	0.632065i	$-0.782208\pi$
$59$	−64.7380	+	64.7380i	−0.142850	+	0.142850i	0.632065	+	0.774915i	$0.282208\pi$
$60$		0			0
$61$	243.569			0.511244			0.255622	−	0.966777i	$-0.417720\pi$
$61$	243.569			0.511244			0.255622	+	0.966777i	$0.417720\pi$
$62$		0			0
$63$	189.300	−	189.300i	0.378564	−	0.378564i
$64$		0			0
$65$	−7.76758	−	16.3734i	−0.0148223	−	0.0312442i
$66$		0			0
$67$	−286.961	−	286.961i	−0.523251	−	0.523251i	0.395300	−	0.918552i	$-0.370640\pi$
$67$	−286.961	−	286.961i	−0.523251	−	0.523251i	−0.918552	+	0.395300i	$0.870640\pi$
$68$		0			0
$69$			278.187i			0.485360i
$70$		0			0
$71$	−707.169	−	707.169i	−1.18205	−	1.18205i	−0.979212	−	0.202838i	$-0.934983\pi$
$71$	−707.169	−	707.169i	−1.18205	−	1.18205i	−0.202838	−	0.979212i	$-0.565017\pi$
$72$		0			0
$73$	530.346	+	530.346i	0.850305	+	0.850305i	0.990171	−	0.139865i	$-0.0446670\pi$
$73$	530.346	+	530.346i	0.850305	+	0.850305i	−0.139865	+	0.990171i	$0.544667\pi$
$74$		0			0
$75$	374.552			0.576660
$76$		0			0
$77$			1088.16i			1.61048i
$78$		0			0
$79$		−	1264.84i		−	1.80134i	−0.434507	−	0.900668i	$-0.643077\pi$
$79$		−	1264.84i		−	1.80134i	0.434507	−	0.900668i	$-0.356923\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	358.933	+	358.933i	0.474674	+	0.474674i	0.903424	−	0.428749i	$-0.141046\pi$
$83$	358.933	+	358.933i	0.474674	+	0.474674i	−0.428749	+	0.903424i	$0.641046\pi$
$84$		0			0
$85$	30.5244	+	30.5244i	0.0389510	+	0.0389510i
$86$		0			0
$87$		−	763.910i		−	0.941377i
$88$		0			0
$89$	−374.949	−	374.949i	−0.446567	−	0.446567i	0.447645	−	0.894212i	$-0.352263\pi$
$89$	−374.949	−	374.949i	−0.446567	−	0.446567i	−0.894212	+	0.447645i	$0.852263\pi$
$90$		0			0
$91$	−597.593	−	1259.68i	−0.688405	−	1.45110i
$92$		0			0
$93$	519.959	−	519.959i	0.579755	−	0.579755i
$94$		0			0
$95$	35.0161			0.0378166
$96$		0			0
$97$	759.324	−	759.324i	0.794822	−	0.794822i	−0.187452	−	0.982274i	$-0.560023\pi$
$97$	759.324	−	759.324i	0.794822	−	0.794822i	0.982274	+	0.187452i	$0.0600229\pi$
$98$		0			0
$99$	−232.808	+	232.808i	−0.236344	+	0.236344i
$100$		0			0
$101$		−	778.258i		−	0.766728i	−0.923597	−	0.383364i	$-0.874765\pi$
$101$		−	778.258i		−	0.766728i	0.923597	−	0.383364i	$-0.125235\pi$
$102$		0			0
$103$	−1400.37			−1.33963			−0.669817	−	0.742526i	$-0.733627\pi$
$103$	−1400.37			−1.33963			−0.669817	+	0.742526i	$0.733627\pi$
$104$		0			0
$105$	−34.5022			−0.0320673
$106$		0			0
$107$		−	67.6349i		−	0.0611076i	−0.999533	−	0.0305538i	$-0.990273\pi$
$107$		−	67.6349i		−	0.0611076i	0.999533	−	0.0305538i	$-0.00972709\pi$
$108$		0			0
$109$	−89.4906	+	89.4906i	−0.0786389	+	0.0786389i	−0.745332	−	0.666693i	$-0.767709\pi$
$109$	−89.4906	+	89.4906i	−0.0786389	+	0.0786389i	0.666693	+	0.745332i	$0.267709\pi$
$110$		0			0
$111$	271.599	−	271.599i	0.232244	−	0.232244i
$112$		0			0
$113$	1831.00			1.52430			0.762152	−	0.647398i	$-0.224143\pi$
$113$	1831.00			1.52430			0.762152	+	0.647398i	$0.224143\pi$
$114$		0			0
$115$	25.3515	−	25.3515i	0.0205569	−	0.0205569i
$116$		0			0
$117$	141.651	−	397.356i	0.111928	−	0.313980i
$118$		0			0
$119$	2348.38	+	2348.38i	1.80904	+	1.80904i
$120$		0			0
$121$		−	7.25759i		−	0.00545274i
$122$		0			0
$123$	−705.720	−	705.720i	−0.517339	−	0.517339i
$124$		0			0
$125$	−68.3074	−	68.3074i	−0.0488768	−	0.0488768i
$126$		0			0
$127$	2749.79			1.92129			0.960647	−	0.277773i	$-0.0895963\pi$
$127$	2749.79			1.92129			0.960647	+	0.277773i	$0.0895963\pi$
$128$		0			0
$129$		−	874.461i		−	0.596837i
$130$		0			0
$131$		−	2159.32i		−	1.44016i	−0.693892	−	0.720079i	$-0.744105\pi$
$131$		−	2159.32i		−	1.44016i	0.693892	−	0.720079i	$-0.255895\pi$
$132$		0			0
$133$	2693.94			1.75635
$134$		0			0
$135$	−7.38162	−	7.38162i	−0.00470599	−	0.00470599i
$136$		0			0
$137$	1905.40	+	1905.40i	1.18824	+	1.18824i	0.977553	+	0.210692i	$0.0675716\pi$
$137$	1905.40	+	1905.40i	1.18824	+	1.18824i	0.210692	+	0.977553i	$0.432428\pi$
$138$		0			0
$139$			1429.73i			0.872431i	0.899842	+	0.436215i	$0.143681\pi$
$139$			1429.73i			0.872431i	−0.899842	+	0.436215i	$0.856319\pi$
$140$		0			0
$141$	−107.073	−	107.073i	−0.0639515	−	0.0639515i
$142$		0			0
$143$	734.942	+	1549.20i	0.429783	+	0.905947i
$144$		0			0
$145$	−69.6159	+	69.6159i	−0.0398710	+	0.0398710i
$146$		0			0
$147$	−1625.40			−0.911979
$148$		0			0
$149$	−1750.08	+	1750.08i	−0.962226	+	0.962226i	−0.999312	−	0.0370857i	$-0.988193\pi$
$149$	−1750.08	+	1750.08i	−0.962226	+	0.962226i	0.0370857	+	0.999312i	$0.488193\pi$
$150$		0			0
$151$	421.983	−	421.983i	0.227421	−	0.227421i	−0.584194	−	0.811614i	$-0.698589\pi$
$151$	421.983	−	421.983i	0.227421	−	0.227421i	0.811614	+	0.584194i	$0.198589\pi$
$152$		0			0
$153$			1004.85i			0.530964i
$154$		0			0
$155$	−94.7688			−0.0491098
$156$		0			0
$157$	−3172.08			−1.61248			−0.806239	−	0.591589i	$-0.798501\pi$
$157$	−3172.08			−1.61248			−0.806239	+	0.591589i	$0.798501\pi$
$158$		0			0
$159$		−	1966.55i		−	0.980863i
$160$		0			0
$161$	1950.40	−	1950.40i	0.954740	−	0.954740i
$162$		0			0
$163$	62.0039	−	62.0039i	0.0297946	−	0.0297946i	−0.692053	−	0.721847i	$-0.743293\pi$
$163$	62.0039	−	62.0039i	0.0297946	−	0.0297946i	0.721847	+	0.692053i	$0.243293\pi$
$164$		0			0
$165$	42.4320			0.0200202
$166$		0			0
$167$	−2608.31	+	2608.31i	−1.20861	+	1.20861i	−0.237128	+	0.971478i	$0.576206\pi$
$167$	−2608.31	+	2608.31i	−1.20861	+	1.20861i	−0.971478	+	0.237128i	$0.923794\pi$
$168$		0			0
$169$	−1701.57	−	1389.77i	−0.774497	−	0.632577i
$170$		0			0
$171$	576.358	+	576.358i	0.257750	+	0.257750i
$172$		0			0
$173$		−	538.180i		−	0.236515i	−0.992983	−	0.118258i	$-0.962269\pi$
$173$		−	538.180i		−	0.236515i	0.992983	−	0.118258i	$-0.0377308\pi$
$174$		0			0
$175$	−2626.02	−	2626.02i	−1.13433	−	1.13433i
$176$		0			0
$177$	194.214	+	194.214i	0.0824746	+	0.0824746i
$178$		0			0
$179$	755.184			0.315336			0.157668	−	0.987492i	$-0.449602\pi$
$179$	755.184			0.315336			0.157668	+	0.987492i	$0.449602\pi$
$180$		0			0
$181$			26.3232i			0.0108099i	0.999985	+	0.00540494i	$0.00172045\pi$
$181$			26.3232i			0.0108099i	−0.999985	+	0.00540494i	$0.998280\pi$
$182$		0			0
$183$		−	730.708i		−	0.295167i
$184$		0			0
$185$	−49.5022			−0.0196728
$186$		0			0
$187$	−2888.12	−	2888.12i	−1.12941	−	1.12941i
$188$		0			0
$189$	−567.900	−	567.900i	−0.218564	−	0.218564i
$190$		0			0
$191$			959.409i			0.363457i	0.983349	+	0.181729i	$0.0581693\pi$
$191$			959.409i			0.363457i	−0.983349	+	0.181729i	$0.941831\pi$
$192$		0			0
$193$	2734.09	+	2734.09i	1.01971	+	1.01971i	0.999802	+	0.0199077i	$0.00633724\pi$
$193$	2734.09	+	2734.09i	1.01971	+	1.01971i	0.0199077	+	0.999802i	$0.493663\pi$
$194$		0			0
$195$	−49.1203	+	23.3027i	−0.0180389	+	0.00855766i
$196$		0			0
$197$	2713.88	−	2713.88i	0.981502	−	0.981502i	−0.0183301	−	0.999832i	$-0.505835\pi$
$197$	2713.88	−	2713.88i	0.981502	−	0.981502i	0.999832	+	0.0183301i	$0.00583497\pi$
$198$		0			0
$199$	4469.99			1.59231			0.796154	−	0.605095i	$-0.206865\pi$
$199$	4469.99			1.59231			0.796154	+	0.605095i	$0.206865\pi$
$200$		0			0
$201$	−860.883	+	860.883i	−0.302099	+	0.302099i
$202$		0			0
$203$	−5355.85	+	5355.85i	−1.85176	+	1.85176i
$204$		0			0
$205$			128.626i			0.0438226i
$206$		0			0
$207$	834.562			0.280223
$208$		0			0
$209$	−3313.10			−1.09652
$210$		0			0
$211$			3871.83i			1.26326i	0.775271	+	0.631629i	$0.217613\pi$
$211$			3871.83i			1.26326i	−0.775271	+	0.631629i	$0.782387\pi$
$212$		0			0
$213$	−2121.51	+	2121.51i	−0.682457	+	0.682457i
$214$		0			0
$215$	−79.6906	+	79.6906i	−0.0252784	+	0.0252784i
$216$		0			0
$217$	−7290.97			−2.28085
$218$		0			0
$219$	1591.04	−	1591.04i	0.490924	−	0.490924i
$220$		0			0
$221$	4929.44	+	1757.26i	1.50041	+	0.534869i
$222$		0			0
$223$	622.771	+	622.771i	0.187013	+	0.187013i	0.794403	−	0.607391i	$-0.207784\pi$
$223$	622.771	+	622.771i	0.187013	+	0.187013i	−0.607391	+	0.794403i	$0.707784\pi$
$224$		0			0
$225$		−	1123.65i		−	0.332935i
$226$		0			0
$227$	2515.85	+	2515.85i	0.735607	+	0.735607i	0.971725	−	0.236117i	$-0.0758750\pi$
$227$	2515.85	+	2515.85i	0.735607	+	0.735607i	−0.236117	+	0.971725i	$0.575875\pi$
$228$		0			0
$229$	−153.691	−	153.691i	−0.0443502	−	0.0443502i	0.684584	−	0.728934i	$-0.259984\pi$
$229$	−153.691	−	153.691i	−0.0443502	−	0.0443502i	−0.728934	+	0.684584i	$0.759984\pi$
$230$		0			0
$231$	3264.48			0.929814
$232$		0			0
$233$			1246.98i			0.350612i	0.984514	+	0.175306i	$0.0560914\pi$
$233$			1246.98i			0.350612i	−0.984514	+	0.175306i	$0.943909\pi$
$234$		0			0
$235$			19.5153i			0.00541719i
$236$		0			0
$237$	−3794.52			−1.04000
$238$		0			0
$239$	1522.92	+	1522.92i	0.412173	+	0.412173i	0.882495	−	0.470322i	$-0.155862\pi$
$239$	1522.92	+	1522.92i	0.412173	+	0.412173i	−0.470322	+	0.882495i	$0.655862\pi$
$240$		0			0
$241$	1140.78	+	1140.78i	0.304912	+	0.304912i	0.842932	−	0.538020i	$-0.180828\pi$
$241$	1140.78	+	1140.78i	0.304912	+	0.304912i	−0.538020	+	0.842932i	$0.680828\pi$
$242$		0			0
$243$		−	243.000i		−	0.0641500i
$244$		0			0
$245$	148.125	+	148.125i	0.0386259	+	0.0386259i
$246$		0			0
$247$	3835.32	−	1819.48i	0.987999	−	0.468708i
$248$		0			0
$249$	1076.80	−	1076.80i	0.274053	−	0.274053i
$250$		0			0
$251$	−87.5945			−0.0220275			−0.0110138	−	0.999939i	$-0.503506\pi$
$251$	−87.5945			−0.0220275			−0.0110138	+	0.999939i	$0.503506\pi$
$252$		0			0
$253$	−2398.67	+	2398.67i	−0.596060	+	0.596060i
$254$		0			0
$255$	91.5733	−	91.5733i	0.0224884	−	0.0224884i
$256$		0			0
$257$		−	6685.04i		−	1.62257i	−0.584649	−	0.811286i	$-0.698768\pi$
$257$		−	6685.04i		−	1.62257i	0.584649	−	0.811286i	$-0.301232\pi$
$258$		0			0
$259$	−3808.42			−0.913682
$260$		0			0
$261$	−2291.73			−0.543504
$262$		0			0
$263$		−	401.346i		−	0.0940990i	−0.998893	−	0.0470495i	$-0.985018\pi$
$263$		−	401.346i		−	0.0940990i	0.998893	−	0.0470495i	$-0.0149819\pi$
$264$		0			0
$265$	−179.213	+	179.213i	−0.0415434	+	0.0415434i
$266$		0			0
$267$	−1124.85	+	1124.85i	−0.257826	+	0.257826i
$268$		0			0
$269$	−682.208			−0.154628			−0.0773141	−	0.997007i	$-0.524634\pi$
$269$	−682.208			−0.154628			−0.0773141	+	0.997007i	$0.524634\pi$
$270$		0			0
$271$	−2700.91	+	2700.91i	−0.605419	+	0.605419i	−0.941745	−	0.336327i	$-0.890815\pi$
$271$	−2700.91	+	2700.91i	−0.605419	+	0.605419i	0.336327	+	0.941745i	$0.390815\pi$
$272$		0			0
$273$	−3779.04	+	1792.78i	−0.837793	+	0.397451i
$274$		0			0
$275$	3229.57	+	3229.57i	0.708184	+	0.708184i
$276$		0			0
$277$		−	8547.11i		−	1.85396i	−0.375115	−	0.926978i	$-0.622397\pi$
$277$		−	8547.11i		−	1.85396i	0.375115	−	0.926978i	$-0.377603\pi$
$278$		0			0
$279$	−1559.88	−	1559.88i	−0.334722	−	0.334722i
$280$		0			0
$281$	−1468.43	−	1468.43i	−0.311741	−	0.311741i	0.533842	−	0.845584i	$-0.320748\pi$
$281$	−1468.43	−	1468.43i	−0.311741	−	0.311741i	−0.845584	+	0.533842i	$0.820748\pi$
$282$		0			0
$283$	−3121.63			−0.655696			−0.327848	−	0.944731i	$-0.606323\pi$
$283$	−3121.63			−0.655696			−0.327848	+	0.944731i	$0.606323\pi$
$284$		0			0
$285$		−	105.048i		−	0.0218334i
$286$		0			0
$287$			9895.76i			2.03529i
$288$		0			0
$289$	−7552.78			−1.53731
$290$		0			0
$291$	−2277.97	−	2277.97i	−0.458891	−	0.458891i
$292$		0			0
$293$	−970.105	−	970.105i	−0.193427	−	0.193427i	0.603748	−	0.797175i	$-0.293673\pi$
$293$	−970.105	−	970.105i	−0.193427	−	0.193427i	−0.797175	+	0.603748i	$0.793673\pi$
$294$		0			0
$295$		−	35.3978i		−	0.00698624i
$296$		0			0
$297$	698.423	+	698.423i	0.136453	+	0.136453i
$298$		0			0
$299$	1459.46	−	4094.06i	0.282283	−	0.791858i
$300$		0			0
$301$	−6130.94	+	6130.94i	−1.17402	+	1.17402i
$302$		0			0
$303$	−2334.77			−0.442671
$304$		0			0
$305$	−66.5902	+	66.5902i	−0.0125015	+	0.0125015i
$306$		0			0
$307$	1890.33	−	1890.33i	0.351423	−	0.351423i	−0.509216	−	0.860639i	$-0.670065\pi$
$307$	1890.33	−	1890.33i	0.351423	−	0.351423i	0.860639	+	0.509216i	$0.170065\pi$
$308$		0			0
$309$			4201.10i			0.773438i
$310$		0			0
$311$	3592.16			0.654960			0.327480	−	0.944858i	$-0.393801\pi$
$311$	3592.16			0.654960			0.327480	+	0.944858i	$0.393801\pi$
$312$		0			0
$313$	963.010			0.173906			0.0869529	−	0.996212i	$-0.472287\pi$
$313$	963.010			0.173906			0.0869529	+	0.996212i	$0.472287\pi$
$314$		0			0
$315$			103.507i			0.0185141i
$316$		0			0
$317$	−6422.68	+	6422.68i	−1.13796	+	1.13796i	−0.149146	+	0.988815i	$0.547652\pi$
$317$	−6422.68	+	6422.68i	−1.13796	+	1.13796i	−0.988815	+	0.149146i	$0.952348\pi$
$318$		0			0
$319$	6586.82	−	6586.82i	1.15609	−	1.15609i
$320$		0			0
$321$	−202.905			−0.0352805
$322$		0			0
$323$	−7150.06	+	7150.06i	−1.23170	+	1.23170i
$324$		0			0
$325$	−5512.24	−	1965.02i	−0.940812	−	0.335383i
$326$		0			0
$327$	268.472	+	268.472i	0.0454022	+	0.0454022i
$328$		0			0
$329$			1501.40i			0.251595i
$330$		0			0
$331$	−4569.55	−	4569.55i	−0.758807	−	0.758807i	0.217298	−	0.976105i	$-0.430276\pi$
$331$	−4569.55	−	4569.55i	−0.758807	−	0.758807i	−0.976105	+	0.217298i	$0.930276\pi$
$332$		0			0
$333$	−814.797	−	814.797i	−0.134086	−	0.134086i
$334$		0			0
$335$	156.906			0.0255902
$336$		0			0
$337$		−	88.7401i		−	0.0143442i	−0.999974	−	0.00717208i	$-0.997717\pi$
$337$		−	88.7401i		−	0.0143442i	0.999974	−	0.00717208i	$-0.00228296\pi$
$338$		0			0
$339$		−	5493.01i		−	0.880058i
$340$		0			0
$341$	8966.70			1.42397
$342$		0			0
$343$	4181.44	+	4181.44i	0.658241	+	0.658241i
$344$		0			0
$345$	−76.0545	−	76.0545i	−0.0118685	−	0.0118685i
$346$		0			0
$347$		−	2219.46i		−	0.343363i	−0.985152	−	0.171682i	$-0.945080\pi$
$347$		−	2219.46i		−	0.343363i	0.985152	−	0.171682i	$-0.0549200\pi$
$348$		0			0
$349$	348.999	+	348.999i	0.0535287	+	0.0535287i	0.733364	−	0.679836i	$-0.237949\pi$
$349$	348.999	+	348.999i	0.0535287	+	0.0535287i	−0.679836	+	0.733364i	$0.737949\pi$
$350$		0			0
$351$	−1192.07	−	424.952i	−0.181276	−	0.0646218i
$352$		0			0
$353$	−3197.63	+	3197.63i	−0.482132	+	0.482132i	−0.905812	−	0.423680i	$-0.860738\pi$
$353$	−3197.63	+	3197.63i	−0.482132	+	0.482132i	0.423680	+	0.905812i	$0.360738\pi$
$354$		0			0
$355$	386.671			0.0578094
$356$		0			0
$357$	7045.13	−	7045.13i	1.04445	−	1.04445i
$358$		0			0
$359$	3873.97	−	3873.97i	0.569528	−	0.569528i	−0.362468	−	0.931996i	$-0.618066\pi$
$359$	3873.97	−	3873.97i	0.569528	−	0.569528i	0.931996	+	0.362468i	$0.118066\pi$
$360$		0			0
$361$			1343.20i			0.195830i
$362$		0			0
$363$	−21.7728			−0.00314814
$364$		0			0
$365$	−289.986			−0.0415851
$366$		0			0
$367$		−	10811.6i		−	1.53777i	−0.639385	−	0.768887i	$-0.720811\pi$
$367$		−	10811.6i		−	1.53777i	0.639385	−	0.768887i	$-0.279189\pi$
$368$		0			0
$369$	−2117.16	+	2117.16i	−0.298686	+	0.298686i
$370$		0			0
$371$	−13787.7	+	13787.7i	−1.92943	+	1.92943i
$372$		0			0
$373$	−952.323			−0.132197			−0.0660984	−	0.997813i	$-0.521055\pi$
$373$	−952.323			−0.132197			−0.0660984	+	0.997813i	$0.521055\pi$
$374$		0			0
$375$	−204.922	+	204.922i	−0.0282190	+	0.0282190i
$376$		0			0
$377$	−4007.72	+	11242.4i	−0.547501	+	1.53584i
$378$		0			0
$379$	1274.63	+	1274.63i	0.172754	+	0.172754i	0.788188	−	0.615434i	$-0.211019\pi$
$379$	1274.63	+	1274.63i	0.172754	+	0.172754i	−0.615434	+	0.788188i	$0.711019\pi$
$380$		0			0
$381$		−	8249.37i		−	1.10926i
$382$		0			0
$383$	8644.16	+	8644.16i	1.15325	+	1.15325i	0.985896	+	0.167356i	$0.0535230\pi$
$383$	8644.16	+	8644.16i	1.15325	+	1.15325i	0.167356	+	0.985896i	$0.446477\pi$
$384$		0			0
$385$	−297.495	−	297.495i	−0.0393812	−	0.0393812i
$386$		0			0
$387$	−2623.38			−0.344584
$388$		0			0
$389$		−	4083.99i		−	0.532305i	−0.963931	−	0.266152i	$-0.914248\pi$
$389$		−	4083.99i		−	0.532305i	0.963931	−	0.266152i	$-0.0857524\pi$
$390$		0			0
$391$			10353.2i			1.33909i
$392$		0			0
$393$	−6477.96			−0.831476
$394$		0			0
$395$	345.799	+	345.799i	0.0440481	+	0.0440481i
$396$		0			0
$397$	2127.11	+	2127.11i	0.268908	+	0.268908i	0.828660	−	0.559752i	$-0.189104\pi$
$397$	2127.11	+	2127.11i	0.268908	+	0.268908i	−0.559752	+	0.828660i	$0.689104\pi$
$398$		0			0
$399$		−	8081.82i		−	1.01403i
$400$		0			0
$401$	−7622.18	−	7622.18i	−0.949211	−	0.949211i	0.0495604	−	0.998771i	$-0.484218\pi$
$401$	−7622.18	−	7622.18i	−0.949211	−	0.949211i	−0.998771	+	0.0495604i	$0.984218\pi$
$402$		0			0
$403$	−10380.1	+	4924.31i	−1.28305	+	0.608679i
$404$		0			0
$405$	−22.1448	+	22.1448i	−0.00271700	+	0.00271700i
$406$		0			0
$407$	4683.73			0.570427
$408$		0			0
$409$	9151.38	−	9151.38i	1.10637	−	1.10637i	0.112750	−	0.993623i	$-0.464034\pi$
$409$	9151.38	−	9151.38i	1.10637	−	1.10637i	0.993623	−	0.112750i	$-0.0359659\pi$
$410$		0			0
$411$	5716.21	−	5716.21i	0.686033	−	0.686033i
$412$		0			0
$413$		−	2723.31i		−	0.324468i
$414$		0			0
$415$	−196.260			−0.0232145
$416$		0			0
$417$	4289.18			0.503698
$418$		0			0
$419$		−	4782.66i		−	0.557633i	−0.960344	−	0.278817i	$-0.910058\pi$
$419$		−	4782.66i		−	0.557633i	0.960344	−	0.278817i	$-0.0899422\pi$
$420$		0			0
$421$	−793.867	+	793.867i	−0.0919019	+	0.0919019i	−0.751563	−	0.659661i	$-0.770700\pi$
$421$	−793.867	+	793.867i	−0.0919019	+	0.0919019i	0.659661	+	0.751563i	$0.270700\pi$
$422$		0			0
$423$	−321.219	+	321.219i	−0.0369224	+	0.0369224i
$424$		0			0
$425$	13939.6			1.59099
$426$		0			0
$427$	−5123.07	+	5123.07i	−0.580616	+	0.580616i
$428$		0			0
$429$	4647.59	−	2204.83i	0.523049	−	0.248135i
$430$		0			0
$431$	2361.03	+	2361.03i	0.263867	+	0.263867i	0.826623	−	0.562756i	$-0.190259\pi$
$431$	2361.03	+	2361.03i	0.263867	+	0.263867i	−0.562756	+	0.826623i	$0.690259\pi$
$432$		0			0
$433$			14581.4i			1.61833i	0.587580	+	0.809166i	$0.300081\pi$
$433$			14581.4i			1.61833i	−0.587580	+	0.809166i	$0.699919\pi$
$434$		0			0
$435$	208.848	+	208.848i	0.0230195	+	0.0230195i
$436$		0			0
$437$	5938.36	+	5938.36i	0.650046	+	0.650046i
$438$		0			0
$439$	2833.46			0.308049			0.154025	−	0.988067i	$-0.450777\pi$
$439$	2833.46			0.308049			0.154025	+	0.988067i	$0.450777\pi$
$440$		0			0
$441$			4876.21i			0.526531i
$442$		0			0
$443$			16412.0i			1.76018i	0.474808	+	0.880089i	$0.342517\pi$
$443$			16412.0i			1.76018i	−0.474808	+	0.880089i	$0.657483\pi$
$444$		0			0
$445$	205.017			0.0218398
$446$		0			0
$447$	5250.23	+	5250.23i	0.555542	+	0.555542i
$448$		0			0
$449$	−1800.87	−	1800.87i	−0.189284	−	0.189284i	0.606103	−	0.795386i	$-0.292732\pi$
$449$	−1800.87	−	1800.87i	−0.189284	−	0.189284i	−0.795386	+	0.606103i	$0.792732\pi$
$450$		0			0
$451$		−	12170.2i		−	1.27067i
$452$		0			0
$453$	−1265.95	−	1265.95i	−0.131301	−	0.131301i
$454$		0			0
$455$	507.765	+	181.010i	0.0523174	+	0.0186502i
$456$		0			0
$457$	6711.08	−	6711.08i	0.686939	−	0.686939i	−0.274615	−	0.961554i	$-0.588550\pi$
$457$	6711.08	−	6711.08i	0.686939	−	0.686939i	0.961554	+	0.274615i	$0.0885504\pi$
$458$		0			0
$459$	3014.56			0.306552
$460$		0			0
$461$	−4702.09	+	4702.09i	−0.475050	+	0.475050i	−0.903544	−	0.428495i	$-0.859044\pi$
$461$	−4702.09	+	4702.09i	−0.475050	+	0.475050i	0.428495	+	0.903544i	$0.359044\pi$
$462$		0			0
$463$	7769.17	−	7769.17i	0.779835	−	0.779835i	−0.199967	−	0.979803i	$-0.564084\pi$
$463$	7769.17	−	7769.17i	0.779835	−	0.779835i	0.979803	+	0.199967i	$0.0640835\pi$
$464$		0			0
$465$			284.306i			0.0283535i
$466$		0			0
$467$	6414.73			0.635628			0.317814	−	0.948153i	$-0.397051\pi$
$467$	6414.73			0.635628			0.317814	+	0.948153i	$0.397051\pi$
$468$		0			0
$469$	12071.5			1.18851
$470$		0			0
$471$			9516.23i			0.930965i
$472$		0			0
$473$	7540.05	−	7540.05i	0.732964	−	0.732964i
$474$		0			0
$475$	7995.40	−	7995.40i	0.772325	−	0.772325i
$476$		0			0
$477$	−5899.64			−0.566302
$478$		0			0
$479$	1920.28	−	1920.28i	0.183173	−	0.183173i	−0.609564	−	0.792737i	$-0.708655\pi$
$479$	1920.28	−	1920.28i	0.183173	−	0.183173i	0.792737	+	0.609564i	$0.208655\pi$
$480$		0			0
$481$	−5421.99	+	2572.20i	−0.513974	+	0.243830i
$482$		0			0
$483$	−5851.20	−	5851.20i	−0.551219	−	0.551219i
$484$		0			0
$485$			415.188i			0.0388716i
$486$		0			0
$487$	3787.66	+	3787.66i	0.352433	+	0.352433i	0.861014	−	0.508581i	$-0.169830\pi$
$487$	3787.66	+	3787.66i	0.352433	+	0.352433i	−0.508581	+	0.861014i	$0.669830\pi$
$488$		0			0
$489$	−186.012	−	186.012i	−0.0172019	−	0.0172019i
$490$		0			0
$491$	−10052.6			−0.923964			−0.461982	−	0.886889i	$-0.652862\pi$
$491$	−10052.6			−0.923964			−0.461982	+	0.886889i	$0.652862\pi$
$492$		0			0
$493$		−	28430.3i		−	2.59723i
$494$		0			0
$495$		−	127.296i		−	0.0115587i
$496$		0			0
$497$	29748.2			2.68489
$498$		0			0
$499$	−2487.89	−	2487.89i	−0.223193	−	0.223193i	0.586649	−	0.809841i	$-0.300447\pi$
$499$	−2487.89	−	2487.89i	−0.223193	−	0.223193i	−0.809841	+	0.586649i	$0.800447\pi$
$500$		0			0
$501$	7824.94	+	7824.94i	0.697789	+	0.697789i
$502$		0			0
$503$		−	7122.94i		−	0.631404i	−0.948858	−	0.315702i	$-0.897760\pi$
$503$		−	7122.94i		−	0.631404i	0.948858	−	0.315702i	$-0.102240\pi$
$504$		0			0
$505$	212.770	+	212.770i	0.0187488	+	0.0187488i
$506$		0			0
$507$	−4169.32	+	5104.71i	−0.365219	+	0.447156i
$508$		0			0
$509$	−21.1389	+	21.1389i	−0.00184080	+	0.00184080i	−0.708027	−	0.706186i	$-0.750414\pi$
$509$	−21.1389	+	21.1389i	−0.00184080	+	0.00184080i	0.706186	+	0.708027i	$0.250414\pi$
$510$		0			0
$511$	−22309.9			−1.93137
$512$		0			0
$513$	1729.08	−	1729.08i	0.148812	−	0.148812i
$514$		0			0
$515$	382.851	−	382.851i	0.0327581	−	0.0327581i
$516$		0			0
$517$		−	1846.47i		−	0.157075i
$518$		0			0
$519$	−1614.54			−0.136552
$520$		0			0
$521$	1146.31			0.0963926			0.0481963	−	0.998838i	$-0.484653\pi$
$521$	1146.31			0.0963926			0.0481963	+	0.998838i	$0.484653\pi$
$522$		0			0
$523$		−	2668.61i		−	0.223117i	−0.993758	−	0.111559i	$-0.964416\pi$
$523$		−	2668.61i		−	0.223117i	0.993758	−	0.111559i	$-0.0355843\pi$
$524$		0			0
$525$	−7878.06	+	7878.06i	−0.654908	+	0.654908i
$526$		0			0
$527$	19351.2	−	19351.2i	1.59953	−	1.59953i
$528$		0			0
$529$	−3568.31			−0.293277
$530$		0			0
$531$	582.642	−	582.642i	0.0476167	−	0.0476167i
$532$		0			0
$533$	6683.58	+	14088.5i	0.543149	+	1.14491i
$534$		0			0
$535$	18.4909	+	18.4909i	0.00149427	+	0.00149427i
$536$		0			0
$537$		−	2265.55i		−	0.182059i
$538$		0			0
$539$	−14015.0	−	14015.0i	−1.11998	−	1.11998i
$540$		0			0
$541$	6831.12	+	6831.12i	0.542870	+	0.542870i	0.924369	−	0.381499i	$-0.124592\pi$
$541$	6831.12	+	6831.12i	0.542870	+	0.542870i	−0.381499	+	0.924369i	$0.624592\pi$
$542$		0			0
$543$	78.9695			0.00624108
$544$		0			0
$545$		−	48.9322i		−	0.00384592i
$546$		0			0
$547$			2808.78i			0.219551i	0.993956	+	0.109776i	$0.0350133\pi$
$547$			2808.78i			0.219551i	−0.993956	+	0.109776i	$0.964987\pi$
$548$		0			0
$549$	−2192.12			−0.170415
$550$		0			0
$551$	−16306.9	−	16306.9i	−1.26079	−	1.26079i
$552$		0			0
$553$	26603.8	+	26603.8i	2.04576	+	2.04576i
$554$		0			0
$555$			148.507i			0.0113581i
$556$		0			0
$557$	−708.506	−	708.506i	−0.0538965	−	0.0538965i	0.679645	−	0.733541i	$-0.262134\pi$
$557$	−708.506	−	708.506i	−0.0538965	−	0.0538965i	−0.733541	+	0.679645i	$0.762134\pi$
$558$		0			0
$559$	−4587.70	+	12869.4i	−0.347118	+	0.973731i
$560$		0			0
$561$	−8664.35	+	8664.35i	−0.652066	+	0.652066i
$562$		0			0
$563$	−5784.89			−0.433044			−0.216522	−	0.976278i	$-0.569471\pi$
$563$	−5784.89			−0.433044			−0.216522	+	0.976278i	$0.569471\pi$
$564$		0			0
$565$	−500.584	+	500.584i	−0.0372739	+	0.0372739i
$566$		0			0
$567$	−1703.70	+	1703.70i	−0.126188	+	0.126188i
$568$		0			0
$569$		−	22693.4i		−	1.67198i	−0.548744	−	0.835991i	$-0.684894\pi$
$569$		−	22693.4i		−	1.67198i	0.548744	−	0.835991i	$-0.315106\pi$
$570$		0			0
$571$	11986.8			0.878517			0.439258	−	0.898361i	$-0.355241\pi$
$571$	11986.8			0.878517			0.439258	+	0.898361i	$0.355241\pi$
$572$		0			0
$573$	2878.23			0.209842
$574$		0			0
$575$		−	11577.3i		−	0.839663i
$576$		0			0
$577$	−10669.9	+	10669.9i	−0.769836	+	0.769836i	−0.978078	−	0.208241i	$-0.933226\pi$
$577$	−10669.9	+	10669.9i	−0.769836	+	0.769836i	0.208241	+	0.978078i	$0.433226\pi$
$578$		0			0
$579$	8202.26	−	8202.26i	0.588730	−	0.588730i
$580$		0			0
$581$	−15099.1			−1.07817
$582$		0			0
$583$	16956.6	−	16956.6i	1.20458	−	1.20458i
$584$		0			0
$585$	69.9082	+	147.361i	0.00494077	+	0.0104147i
$586$		0			0
$587$	−18497.4	−	18497.4i	−1.30063	−	1.30063i	−0.927966	−	0.372664i	$-0.878444\pi$
$587$	−18497.4	−	18497.4i	−1.30063	−	1.30063i	−0.372664	−	0.927966i	$-0.621556\pi$
$588$		0			0
$589$		−	22198.7i		−	1.55294i
$590$		0			0
$591$	−8141.64	−	8141.64i	−0.566670	−	0.566670i
$592$		0			0
$593$	−9906.75	−	9906.75i	−0.686039	−	0.686039i	0.275315	−	0.961354i	$-0.411218\pi$
$593$	−9906.75	−	9906.75i	−0.686039	−	0.686039i	−0.961354	+	0.275315i	$0.911218\pi$
$594$		0			0
$595$	−1284.06			−0.0884728
$596$		0			0
$597$		−	13410.0i		−	0.919319i
$598$		0			0
$599$			9307.98i			0.634914i	0.948273	+	0.317457i	$0.102829\pi$
$599$			9307.98i			0.634914i	−0.948273	+	0.317457i	$0.897171\pi$
$600$		0			0
$601$	5679.52			0.385478			0.192739	−	0.981250i	$-0.438263\pi$
$601$	5679.52			0.385478			0.192739	+	0.981250i	$0.438263\pi$
$602$		0			0
$603$	2582.65	+	2582.65i	0.174417	+	0.174417i
$604$		0			0
$605$	1.98418	+	1.98418i	0.000133336	+	0.000133336i
$606$		0			0
$607$		−	2238.02i		−	0.149652i	−0.997197	−	0.0748259i	$-0.976160\pi$
$607$		−	2238.02i		−	0.149652i	0.997197	−	0.0748259i	$-0.0238401\pi$
$608$		0			0
$609$	16067.6	+	16067.6i	1.06911	+	1.06911i
$610$		0			0
$611$	1014.04	+	2137.52i	0.0671420	+	0.141530i
$612$		0			0
$613$	−11118.2	+	11118.2i	−0.732562	+	0.732562i	−0.971127	−	0.238564i	$-0.923323\pi$
$613$	−11118.2	+	11118.2i	−0.732562	+	0.732562i	0.238564	+	0.971127i	$0.423323\pi$
$614$		0			0
$615$	385.878			0.0253010
$616$		0			0
$617$	11356.7	−	11356.7i	0.741009	−	0.741009i	−0.231764	−	0.972772i	$-0.574450\pi$
$617$	11356.7	−	11356.7i	0.741009	−	0.741009i	0.972772	+	0.231764i	$0.0744496\pi$
$618$		0			0
$619$	10331.5	−	10331.5i	0.670854	−	0.670854i	−0.287059	−	0.957913i	$-0.592678\pi$
$619$	10331.5	−	10331.5i	0.670854	−	0.670854i	0.957913	+	0.287059i	$0.0926776\pi$
$620$		0			0
$621$		−	2503.69i		−	0.161787i
$622$		0			0
$623$	15772.8			1.01433
$624$		0			0
$625$	−15569.0			−0.996414
$626$		0			0
$627$			9939.31i			0.633075i
$628$		0			0
$629$	10108.0	−	10108.0i	0.640753	−	0.640753i
$630$		0			0
$631$	−5854.45	+	5854.45i	−0.369353	+	0.369353i	−0.867241	−	0.497888i	$-0.834109\pi$
$631$	−5854.45	+	5854.45i	−0.369353	+	0.369353i	0.497888	+	0.867241i	$0.334109\pi$
$632$		0			0
$633$	11615.5			0.729342
$634$		0			0
$635$	−751.773	+	751.773i	−0.0469814	+	0.0469814i
$636$		0			0
$637$	23920.9	+	8527.38i	1.48788	+	0.530404i
$638$		0			0
$639$	6364.53	+	6364.53i	0.394017	+	0.394017i
$640$		0			0
$641$		−	10683.0i		−	0.658275i	−0.944282	−	0.329137i	$-0.893242\pi$
$641$		−	10683.0i		−	0.658275i	0.944282	−	0.329137i	$-0.106758\pi$
$642$		0			0
$643$	−4982.94	−	4982.94i	−0.305611	−	0.305611i	0.537593	−	0.843204i	$-0.319334\pi$
$643$	−4982.94	−	4982.94i	−0.305611	−	0.305611i	−0.843204	+	0.537593i	$0.819334\pi$
$644$		0			0
$645$	239.072	+	239.072i	0.0145945	+	0.0145945i
$646$		0			0
$647$	2784.86			0.169218			0.0846091	−	0.996414i	$-0.473036\pi$
$647$	2784.86			0.169218			0.0846091	+	0.996414i	$0.473036\pi$
$648$		0			0
$649$			3349.22i			0.202571i
$650$		0			0
$651$			21872.9i			1.31685i
$652$		0			0
$653$	26879.1			1.61081			0.805405	−	0.592724i	$-0.201948\pi$
$653$	26879.1			1.61081			0.805405	+	0.592724i	$0.201948\pi$
$654$		0			0
$655$	590.344	+	590.344i	0.0352162	+	0.0352162i
$656$		0			0
$657$	−4773.11	−	4773.11i	−0.283435	−	0.283435i
$658$		0			0
$659$			6679.58i			0.394840i	0.980319	+	0.197420i	$0.0632563\pi$
$659$			6679.58i			0.394840i	−0.980319	+	0.197420i	$0.936744\pi$
$660$		0			0
$661$	8985.44	+	8985.44i	0.528734	+	0.528734i	0.920195	−	0.391461i	$-0.128030\pi$
$661$	8985.44	+	8985.44i	0.528734	+	0.528734i	−0.391461	+	0.920195i	$0.628030\pi$
$662$		0			0
$663$	5271.78	−	14788.3i	0.308807	−	0.866260i
$664$		0			0
$665$	−736.505	+	736.505i	−0.0429480	+	0.0429480i
$666$		0			0
$667$	−23612.2			−1.37072
$668$		0			0
$669$	1868.31	−	1868.31i	0.107972	−	0.107972i
$670$		0			0
$671$	6300.54	−	6300.54i	0.362488	−	0.362488i
$672$		0			0
$673$			6040.00i			0.345951i	0.984926	+	0.172975i	$0.0553381\pi$
$673$			6040.00i			0.345951i	−0.984926	+	0.172975i	$0.944662\pi$
$674$		0			0
$675$	−3370.96			−0.192220
$676$		0			0
$677$	−17765.9			−1.00857			−0.504284	−	0.863538i	$-0.668243\pi$
$677$	−17765.9			−1.00857			−0.504284	+	0.863538i	$0.668243\pi$
$678$		0			0
$679$			31942.2i			1.80535i
$680$		0			0
$681$	7547.55	−	7547.55i	0.424703	−	0.424703i
$682$		0			0
$683$	17350.8	−	17350.8i	0.972048	−	0.972048i	−0.0275722	−	0.999620i	$-0.508778\pi$
$683$	17350.8	−	17350.8i	0.972048	−	0.972048i	0.999620	+	0.0275722i	$0.00877763\pi$
$684$		0			0
$685$	−1041.85			−0.0581123
$686$		0			0
$687$	−461.073	+	461.073i	−0.0256056	+	0.0256056i
$688$		0			0
$689$	−10317.1	+	28941.5i	−0.570466	+	1.60026i
$690$		0			0
$691$	23320.9	+	23320.9i	1.28389	+	1.28389i	0.938435	+	0.345455i	$0.112275\pi$
$691$	23320.9	+	23320.9i	1.28389	+	1.28389i	0.345455	+	0.938435i	$0.387725\pi$
$692$		0			0
$693$		−	9793.44i		−	0.536828i
$694$		0			0
$695$	−390.878	−	390.878i	−0.0213336	−	0.0213336i
$696$		0			0
$697$	−26264.6	−	26264.6i	−1.42732	−	1.42732i
$698$		0			0
$699$	3740.95			0.202426
$700$		0			0
$701$			19117.6i			1.03005i	0.857176	+	0.515024i	$0.172217\pi$
$701$			19117.6i			1.03005i	−0.857176	+	0.515024i	$0.827783\pi$
$702$		0			0
$703$		−	11595.4i		−	0.622092i
$704$		0			0
$705$	58.5460			0.00312762
$706$		0			0
$707$	16369.3	+	16369.3i	0.870768	+	0.870768i
$708$		0			0
$709$	7633.77	+	7633.77i	0.404361	+	0.404361i	0.879767	−	0.475406i	$-0.157699\pi$
$709$	7633.77	+	7633.77i	0.404361	+	0.404361i	−0.475406	+	0.879767i	$0.657699\pi$
$710$		0			0
$711$			11383.6i			0.600446i
$712$		0			0
$713$	−16071.8	−	16071.8i	−0.844170	−	0.844170i
$714$		0			0
$715$	−624.468	−	222.612i	−0.0326626	−	0.0116437i
$716$		0			0
$717$	4568.75	−	4568.75i	0.237968	−	0.237968i
$718$		0			0
$719$	−736.056			−0.0381784			−0.0190892	−	0.999818i	$-0.506077\pi$
$719$	−736.056			−0.0381784			−0.0190892	+	0.999818i	$0.506077\pi$
$720$		0			0
$721$	29454.4	−	29454.4i	1.52141	−	1.52141i
$722$		0			0
$723$	3422.33	−	3422.33i	0.176041	−	0.176041i
$724$		0			0
$725$			31791.5i			1.62856i
$726$		0			0
$727$	−7583.38			−0.386867			−0.193433	−	0.981113i	$-0.561962\pi$
$727$	−7583.38			−0.386867			−0.193433	+	0.981113i	$0.561962\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$		−	32544.6i		−	1.64666i
$732$		0			0
$733$	−556.909	+	556.909i	−0.0280626	+	0.0280626i	−0.720999	−	0.692936i	$-0.756317\pi$
$733$	−556.909	+	556.909i	−0.0280626	+	0.0280626i	0.692936	+	0.720999i	$0.256317\pi$
$734$		0			0
$735$	444.374	−	444.374i	0.0223007	−	0.0223007i
$736$		0			0
$737$	−14845.9			−0.742004
$738$		0			0
$739$	186.504	−	186.504i	0.00928373	−	0.00928373i	−0.702450	−	0.711733i	$-0.747910\pi$
$739$	186.504	−	186.504i	0.00928373	−	0.00928373i	0.711733	+	0.702450i	$0.247910\pi$
$740$		0			0
$741$	−5458.45	−	11506.0i	−0.270609	−	0.570422i
$742$		0			0
$743$	−20707.9	−	20707.9i	−1.02248	−	1.02248i	−0.999742	−	0.0227341i	$-0.992763\pi$
$743$	−20707.9	−	20707.9i	−1.02248	−	1.02248i	−0.0227341	−	0.999742i	$-0.507237\pi$
$744$		0			0
$745$		−	956.917i		−	0.0470587i
$746$		0			0
$747$	−3230.39	−	3230.39i	−0.158225	−	0.158225i
$748$		0			0
$749$	1422.59	+	1422.59i	0.0693995	+	0.0693995i
$750$		0			0
$751$	5271.99			0.256162			0.128081	−	0.991764i	$-0.459118\pi$
$751$	5271.99			0.256162			0.128081	+	0.991764i	$0.459118\pi$
$752$		0			0
$753$			262.783i			0.0127176i
$754$		0			0
$755$			230.735i			0.0111223i
$756$		0			0
$757$	4172.85			0.200350			0.100175	−	0.994970i	$-0.468060\pi$
$757$	4172.85			0.200350			0.100175	+	0.994970i	$0.468060\pi$
$758$		0			0
$759$	7196.02	+	7196.02i	0.344136	+	0.344136i
$760$		0			0
$761$	−1713.92	−	1713.92i	−0.0816422	−	0.0816422i	0.665106	−	0.746749i	$-0.268386\pi$
$761$	−1713.92	−	1713.92i	−0.0816422	−	0.0816422i	−0.746749	+	0.665106i	$0.768386\pi$
$762$		0			0
$763$		−	3764.57i		−	0.178619i
$764$		0			0
$765$	−274.720	−	274.720i	−0.0129837	−	0.0129837i
$766$		0			0
$767$	−1839.32	−	3877.13i	−0.0865893	−	0.182523i
$768$		0			0
$769$	11330.4	−	11330.4i	0.531321	−	0.531321i	−0.389645	−	0.920965i	$-0.627402\pi$
$769$	11330.4	−	11330.4i	0.531321	−	0.531321i	0.920965	+	0.389645i	$0.127402\pi$
$770$		0			0
$771$	−20055.1			−0.936793
$772$		0			0
$773$	−24535.7	+	24535.7i	−1.14164	+	1.14164i	−0.153492	+	0.988150i	$0.549052\pi$
$773$	−24535.7	+	24535.7i	−1.14164	+	1.14164i	−0.988150	+	0.153492i	$0.950948\pi$
$774$		0			0
$775$	−21639.0	+	21639.0i	−1.00296	+	1.00296i
$776$		0			0
$777$			11425.3i			0.527515i
$778$		0			0
$779$	−30129.5			−1.38575
$780$		0			0
$781$	−36585.4			−1.67622
$782$		0			0
$783$			6875.19i			0.313792i
$784$		0			0
$785$	867.224	−	867.224i	0.0394300	−	0.0394300i
$786$		0			0
$787$	−8238.29	+	8238.29i	−0.373143	+	0.373143i	−0.868621	−	0.495478i	$-0.834993\pi$
$787$	−8238.29	+	8238.29i	−0.373143	+	0.373143i	0.495478	+	0.868621i	$0.334993\pi$
$788$		0			0
$789$	−1204.04			−0.0543281
$790$		0			0
$791$	−38512.1	+	38512.1i	−1.73114	+	1.73114i
$792$		0			0
$793$	−3833.53	+	10753.8i	−0.171668	+	0.481560i
$794$		0			0
$795$	537.640	+	537.640i	0.0239851	+	0.0239851i
$796$		0			0
$797$			7079.26i			0.314630i	0.987548	+	0.157315i	$0.0502838\pi$
$797$			7079.26i			0.314630i	−0.987548	+	0.157315i	$0.949716\pi$
$798$		0			0
$799$	−3984.91	−	3984.91i	−0.176440	−	0.176440i
$800$		0			0
$801$	3374.54	+	3374.54i	0.148856	+	0.148856i
$802$		0			0
$803$	27437.5			1.20579
$804$		0			0
$805$			1066.45i			0.0466926i
$806$		0			0
$807$			2046.62i			0.0892746i
$808$		0			0
$809$	−35129.6			−1.52669			−0.763344	−	0.645993i	$-0.776444\pi$
$809$	−35129.6			−1.52669			−0.763344	+	0.645993i	$0.776444\pi$
$810$		0			0
$811$	2803.23	+	2803.23i	0.121375	+	0.121375i	0.765185	−	0.643810i	$-0.222647\pi$
$811$	2803.23	+	2803.23i	0.121375	+	0.121375i	−0.643810	+	0.765185i	$0.722647\pi$
$812$		0			0
$813$	8102.72	+	8102.72i	0.349539	+	0.349539i
$814$		0			0
$815$			33.9029i			0.00145714i
$816$		0			0
$817$	−18666.8	−	18666.8i	−0.799349	−	0.799349i
$818$		0			0
$819$	5378.34	+	11337.1i	0.229468	+	0.483700i
$820$		0			0
$821$	−24188.6	+	24188.6i	−1.02825	+	1.02825i	−0.0286563	+	0.999589i	$0.509123\pi$
$821$	−24188.6	+	24188.6i	−1.02825	+	1.02825i	−0.999589	+	0.0286563i	$0.990877\pi$
$822$		0			0
$823$	21771.5			0.922122			0.461061	−	0.887368i	$-0.347469\pi$
$823$	21771.5			0.922122			0.461061	+	0.887368i	$0.347469\pi$
$824$		0			0
$825$	9688.72	−	9688.72i	0.408870	−	0.408870i
$826$		0			0
$827$	10484.4	−	10484.4i	0.440844	−	0.440844i	−0.451451	−	0.892296i	$-0.649094\pi$
$827$	10484.4	−	10484.4i	0.440844	−	0.440844i	0.892296	+	0.451451i	$0.149094\pi$
$828$		0			0
$829$		−	10416.0i		−	0.436386i	−0.975906	−	0.218193i	$-0.929984\pi$
$829$		−	10416.0i		−	0.436386i	0.975906	−	0.218193i	$-0.0700163\pi$
$830$		0			0
$831$	−25641.3			−1.07038
$832$		0			0
$833$	−60492.2			−2.51612
$834$		0			0
$835$		−	1426.19i		−	0.0591082i
$836$		0			0
$837$	−4679.63	+	4679.63i	−0.193252	+	0.193252i
$838$		0			0
$839$	12549.6	−	12549.6i	0.516402	−	0.516402i	−0.400078	−	0.916481i	$-0.631017\pi$
$839$	12549.6	−	12549.6i	0.516402	−	0.516402i	0.916481	+	0.400078i	$0.131017\pi$
$840$		0			0
$841$	40450.8			1.65857
$842$		0			0
$843$	−4405.30	+	4405.30i	−0.179984	+	0.179984i
$844$		0			0
$845$	845.152	−	85.2435i	0.0344072	−	0.00347037i
$846$		0			0
$847$	152.651	+	152.651i	0.00619263	+	0.00619263i
$848$		0			0
$849$			9364.90i			0.378566i
$850$		0			0
$851$	−8395.05	−	8395.05i	−0.338165	−	0.338165i
$852$		0			0
$853$	30365.1	+	30365.1i	1.21885	+	1.21885i	0.968034	+	0.250818i	$0.0806996\pi$
$853$	30365.1	+	30365.1i	1.21885	+	1.21885i	0.250818	+	0.968034i	$0.419300\pi$
$854$		0			0
$855$	−315.145			−0.0126055
$856$		0			0
$857$		−	13468.8i		−	0.536855i	−0.963300	−	0.268427i	$-0.913496\pi$
$857$		−	13468.8i		−	0.536855i	0.963300	−	0.268427i	$-0.0865039\pi$
$858$		0			0
$859$			44650.0i			1.77350i	0.462248	+	0.886751i	$0.347043\pi$
$859$			44650.0i			1.77350i	−0.462248	+	0.886751i	$0.652957\pi$
$860$		0			0
$861$	29687.3			1.17508
$862$		0			0
$863$	18961.1	+	18961.1i	0.747906	+	0.747906i	0.974086	−	0.226179i	$-0.0726235\pi$
$863$	18961.1	+	18961.1i	0.747906	+	0.747906i	−0.226179	+	0.974086i	$0.572624\pi$
$864$		0			0
$865$	147.135	+	147.135i	0.00578351	+	0.00578351i
$866$		0			0
$867$			22658.3i			0.887564i
$868$		0			0
$869$	−32718.3	−	32718.3i	−1.27721	−	1.27721i
$870$		0			0
$871$	17186.0	−	8153.06i	0.668571	−	0.317171i
$872$		0			0
$873$	−6833.92	+	6833.92i	−0.264941	+	0.264941i
$874$		0			0
$875$	2873.46			0.111018
$876$		0			0
$877$	10734.2	−	10734.2i	0.413306	−	0.413306i	−0.469582	−	0.882889i	$-0.655595\pi$
$877$	10734.2	−	10734.2i	0.413306	−	0.413306i	0.882889	+	0.469582i	$0.155595\pi$
$878$		0			0
$879$	−2910.32	+	2910.32i	−0.111675	+	0.111675i
$880$		0			0
$881$		−	40328.3i		−	1.54222i	−0.636702	−	0.771110i	$-0.719702\pi$
$881$		−	40328.3i		−	1.54222i	0.636702	−	0.771110i	$-0.280298\pi$
$882$		0			0
$883$	8935.90			0.340563			0.170281	−	0.985395i	$-0.445532\pi$
$883$	8935.90			0.340563			0.170281	+	0.985395i	$0.445532\pi$
$884$		0			0
$885$	−106.194			−0.00403351
$886$		0			0
$887$		−	7581.44i		−	0.286990i	−0.989651	−	0.143495i	$-0.954166\pi$
$887$		−	7581.44i		−	0.286990i	0.989651	−	0.143495i	$-0.0458341\pi$
$888$		0			0
$889$	−57837.2	+	57837.2i	−2.18200	+	2.18200i
$890$		0			0
$891$	2095.27	−	2095.27i	0.0787813	−	0.0787813i
$892$		0			0
$893$	−4571.29			−0.171301
$894$		0			0
$895$	−206.462	+	206.462i	−0.00771092	+	0.00771092i
$896$		0			0
$897$	−12282.2	−	4378.38i	−0.457179	−	0.162976i
$898$		0			0
$899$	44133.5	+	44133.5i	1.63730	+	1.63730i
$900$		0			0
$901$		−	73188.5i		−	2.70617i
$902$		0			0
$903$	18392.8	+	18392.8i	0.677823	+	0.677823i
$904$		0			0
$905$	−7.19658	−	7.19658i	−0.000264334	−	0.000264334i
$906$		0			0
$907$	6949.46			0.254414			0.127207	−	0.991876i	$-0.459399\pi$
$907$	6949.46			0.254414			0.127207	+	0.991876i	$0.459399\pi$
$908$		0			0
$909$			7004.32i			0.255576i
$910$		0			0
$911$		−	30432.9i		−	1.10679i	−0.832918	−	0.553396i	$-0.813332\pi$
$911$		−	30432.9i		−	1.10679i	0.832918	−	0.553396i	$-0.186668\pi$
$912$		0			0
$913$	18569.4			0.673119
$914$		0			0
$915$	199.771	+	199.771i	0.00721772	+	0.00721772i
$916$		0			0
$917$	45417.7	+	45417.7i	1.63558	+	1.63558i
$918$		0			0
$919$			18081.4i			0.649020i	0.945882	+	0.324510i	$0.105199\pi$
$919$			18081.4i			0.649020i	−0.945882	+	0.324510i	$0.894801\pi$
$920$		0			0
$921$	−5670.99	−	5670.99i	−0.202894	−	0.202894i
$922$		0			0
$923$	42352.1	−	20091.9i	1.51033	−	0.716505i
$924$		0			0
$925$	−11303.1	+	11303.1i	−0.401777	+	0.401777i
$926$		0			0
$927$	12603.3			0.446545
$928$		0			0
$929$	12783.7	−	12783.7i	0.451476	−	0.451476i	−0.444368	−	0.895844i	$-0.646572\pi$
$929$	12783.7	−	12783.7i	0.451476	−	0.451476i	0.895844	+	0.444368i	$0.146572\pi$
$930$		0			0
$931$	−34696.8	+	34696.8i	−1.22142	+	1.22142i
$932$		0			0
$933$		−	10776.5i		−	0.378142i
$934$		0			0
$935$	1579.18			0.0552351
$936$		0			0
$937$	−2344.53			−0.0817421			−0.0408710	−	0.999164i	$-0.513013\pi$
$937$	−2344.53			−0.0817421			−0.0408710	+	0.999164i	$0.513013\pi$
$938$		0			0
$939$		−	2889.03i		−	0.100405i
$940$		0			0
$941$	−18313.4	+	18313.4i	−0.634432	+	0.634432i	−0.949176	−	0.314745i	$-0.898081\pi$
$941$	−18313.4	+	18313.4i	−0.634432	+	0.634432i	0.314745	+	0.949176i	$0.398081\pi$
$942$		0			0
$943$	−21813.6	+	21813.6i	−0.753286	+	0.753286i
$944$		0			0
$945$	310.520			0.0106891
$946$		0			0
$947$	30759.0	−	30759.0i	1.05547	−	1.05547i	0.0571038	−	0.998368i	$-0.481813\pi$
$947$	30759.0	−	30759.0i	1.05547	−	1.05547i	0.998368	−	0.0571038i	$-0.0181866\pi$
$948$		0			0
$949$	−31762.2	+	15068.1i	−1.08645	+	0.515416i
$950$		0			0
$951$	19268.0	+	19268.0i	0.657002	+	0.657002i
$952$		0			0
$953$		−	37108.6i		−	1.26135i	−0.776048	−	0.630674i	$-0.782778\pi$
$953$		−	37108.6i		−	1.26135i	0.776048	−	0.630674i	$-0.217222\pi$
$954$		0			0
$955$	−262.296	−	262.296i	−0.00888764	−	0.00888764i
$956$		0			0
$957$	−19760.5	−	19760.5i	−0.667466	−	0.667466i
$958$		0			0
$959$	−80153.8			−2.69896
$960$		0			0
$961$			30288.4i			1.01670i
$962$		0			0
$963$			608.714i			0.0203692i
$964$		0			0
$965$	−1494.96			−0.0498700
$966$		0			0
$967$	19976.2	+	19976.2i	0.664312	+	0.664312i	0.956394	−	0.292081i	$-0.0943477\pi$
$967$	19976.2	+	19976.2i	0.664312	+	0.664312i	−0.292081	+	0.956394i	$0.594348\pi$
$968$		0			0
$969$	21450.2	+	21450.2i	0.711124	+	0.711124i
$970$		0			0
$971$		−	30509.9i		−	1.00835i	−0.863601	−	0.504176i	$-0.831796\pi$
$971$		−	30509.9i		−	1.00835i	0.863601	−	0.504176i	$-0.168204\pi$
$972$		0			0
$973$	−30071.9	−	30071.9i	−0.990813	−	0.990813i
$974$		0			0
$975$	−5895.05	+	16536.7i	−0.193634	+	0.543178i
$976$		0			0
$977$	4511.07	−	4511.07i	0.147719	−	0.147719i	−0.629379	−	0.777098i	$-0.716691\pi$
$977$	4511.07	−	4511.07i	0.147719	−	0.147719i	0.777098	+	0.629379i	$0.216691\pi$
$978$		0			0
$979$	−19398.0			−0.633261
$980$		0			0
$981$	805.415	−	805.415i	0.0262130	−	0.0262130i
$982$		0			0
$983$	−34195.2	+	34195.2i	−1.10952	+	1.10952i	−0.116306	+	0.993213i	$0.537105\pi$
$983$	−34195.2	+	34195.2i	−1.10952	+	1.10952i	−0.993213	+	0.116306i	$0.962895\pi$
$984$		0			0
$985$			1483.91i			0.0480014i
$986$		0			0
$987$	4504.20			0.145258
$988$		0			0
$989$	−27029.3			−0.869042
$990$		0			0
$991$		−	48201.6i		−	1.54508i	−0.634966	−	0.772540i	$-0.718986\pi$
$991$		−	48201.6i		−	1.54508i	0.634966	−	0.772540i	$-0.281014\pi$
$992$		0			0
$993$	−13708.6	+	13708.6i	−0.438097	+	0.438097i
$994$		0			0
$995$	−1222.06	+	1222.06i	−0.0389367	+	0.0389367i
$996$		0			0
$997$	−35777.2			−1.13648			−0.568242	−	0.822861i	$-0.692376\pi$
$997$	−35777.2			−1.13648			−0.568242	+	0.822861i	$0.692376\pi$
$998$		0			0
$999$	−2444.39	+	2444.39i	−0.0774146	+	0.0774146i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.bc.c.31.8	✓	28
4.3	odd	2		624.4.bc.d.31.8	yes	28
13.8	odd	4		624.4.bc.d.463.8	yes	28
52.47	even	4	inner	624.4.bc.c.463.8	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
624.4.bc.c.31.8	✓	28	1.1	even	1	trivial
624.4.bc.c.463.8	yes	28	52.47	even	4	inner
624.4.bc.d.31.8	yes	28	4.3	odd	2
624.4.bc.d.463.8	yes	28	13.8	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.bc (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +(-0.273393 + 0.273393i) q^{5} +(-21.0333 + 21.0333i) q^{7} -9.00000 q^{9} +(25.8675 - 25.8675i) q^{11} +(-15.7390 + 44.1507i) q^{13} +(0.820180 + 0.820180i) q^{15} -111.650i q^{17} +(-64.0398 - 64.0398i) q^{19} +(63.0999 + 63.0999i) q^{21} -92.7291 q^{23} +124.851i q^{25} +27.0000i q^{27} +254.637 q^{29} +(173.320 + 173.320i) q^{31} +(-77.6026 - 77.6026i) q^{33} -11.5007i q^{35} +(90.5331 + 90.5331i) q^{37} +(132.452 + 47.2169i) q^{39} +(235.240 - 235.240i) q^{41} +291.487 q^{43} +(2.46054 - 2.46054i) q^{45} +(35.6910 - 35.6910i) q^{47} -541.801i q^{49} -334.951 q^{51} +655.515 q^{53} +14.1440i q^{55} +(-192.119 + 192.119i) q^{57} +(-64.7380 + 64.7380i) q^{59} +243.569 q^{61} +(189.300 - 189.300i) q^{63} +(-7.76758 - 16.3734i) q^{65} +(-286.961 - 286.961i) q^{67} +278.187i q^{69} +(-707.169 - 707.169i) q^{71} +(530.346 + 530.346i) q^{73} +374.552 q^{75} +1088.16i q^{77} -1264.84i q^{79} +81.0000 q^{81} +(358.933 + 358.933i) q^{83} +(30.5244 + 30.5244i) q^{85} -763.910i q^{87} +(-374.949 - 374.949i) q^{89} +(-597.593 - 1259.68i) q^{91} +(519.959 - 519.959i) q^{93} +35.0161 q^{95} +(759.324 - 759.324i) q^{97} +(-232.808 + 232.808i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 4 q^{5} - 8 q^{7} - 252 q^{9} + 64 q^{11} - 32 q^{13} - 12 q^{15} + 56 q^{19} + 24 q^{21} + 384 q^{23} - 32 q^{29} - 168 q^{31} - 192 q^{33} + 412 q^{37} + 252 q^{39} + 1340 q^{41} + 624 q^{43} - 36 q^{45}+ \cdots - 576 q^{99}+O(q^{100}) \) 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

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