LMFDB - Embedded newform 539.2.b.c.538.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [539,2,Mod(538,539)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(539, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("539.538");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 539 = 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	539.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$4.30393666895$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 32 x^{14} - 16 x^{13} + 372 x^{12} - 176 x^{11} + 2096 x^{10} - 1120 x^{9} + 7010 x^{8} + \cdots + 8449 $ x^16 + 32x^14 - 16x^13 + 372x^12 - 176x^11 + 2096x^10 - 1120x^9 + 7010x^8 - 3808x^7 + 13568x^6 - 8816x^5 + 15628x^4 - 11216x^3 + 8272x^2 - 11968x + 8449
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			538.1
Root			$-0.707107 - 0.806331i$ of defining polynomial
Character	$\chi$	$=$	539.538
Dual form			539.2.b.c.538.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.60728i q^{2} -2.82079i q^{3} -4.79793 q^{4} -1.84776i q^{5} -7.35461 q^{6} +7.29501i q^{8} -4.95687 q^{9} +O(q^{10})$	\(q-2.60728i q^{2} -2.82079i q^{3} -4.79793 q^{4} -1.84776i q^{5} -7.35461 q^{6} +7.29501i q^{8} -4.95687 q^{9} -4.81763 q^{10} +(-0.112389 - 3.31472i) q^{11} +13.5340i q^{12} +4.27619 q^{13} -5.21215 q^{15} +9.42429 q^{16} +3.30270 q^{17} +12.9240i q^{18} +3.58783 q^{19} +8.86542i q^{20} +(-8.64242 + 0.293029i) q^{22} -4.98737 q^{23} +20.5777 q^{24} +1.58579 q^{25} -11.1492i q^{26} +5.51994i q^{27} -3.27285i q^{29} +13.5895i q^{30} +1.58375i q^{31} -9.98180i q^{32} +(-9.35014 + 0.317025i) q^{33} -8.61108i q^{34} +23.7827 q^{36} +0.946795 q^{37} -9.35449i q^{38} -12.0622i q^{39} +13.4794 q^{40} +1.77125 q^{41} +12.0622i q^{43} +(0.539234 + 15.9038i) q^{44} +9.15911i q^{45} +13.0035i q^{46} -8.15307i q^{47} -26.5840i q^{48} -4.13460i q^{50} -9.31623i q^{51} -20.5169 q^{52} -0.871553 q^{53} +14.3920 q^{54} +(-6.12480 + 0.207667i) q^{55} -10.1205i q^{57} -8.53325 q^{58} +6.68835i q^{59} +25.0075 q^{60} +4.67942 q^{61} +4.12927 q^{62} -7.17680 q^{64} -7.90136i q^{65} +(0.826575 + 24.3785i) q^{66} -8.66949 q^{67} -15.8461 q^{68} +14.0683i q^{69} +7.09378 q^{71} -36.1604i q^{72} +9.35014 q^{73} -2.46857i q^{74} -4.47318i q^{75} -17.2142 q^{76} -31.4497 q^{78} +6.48649i q^{79} -17.4138i q^{80} +0.699980 q^{81} -4.61816i q^{82} -8.55237 q^{83} -6.10259i q^{85} +31.4497 q^{86} -9.23204 q^{87} +(24.1809 - 0.819876i) q^{88} -7.58170i q^{89} +23.8804 q^{90} +23.9291 q^{92} +4.46742 q^{93} -21.2574 q^{94} -6.62944i q^{95} -28.1566 q^{96} -3.12677i q^{97} +(0.557097 + 16.4307i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4} - 16 q^{9}+O(q^{10}) $ 16 * q - 32 * q^4 - 16 * q^9	$ 16 q - 32 q^{4} - 16 q^{9} - 16 q^{15} + 16 q^{16} - 32 q^{22} - 16 q^{23} + 48 q^{25} + 96 q^{36} + 64 q^{37} + 48 q^{44} - 32 q^{53} - 80 q^{58} + 112 q^{60} - 32 q^{64} - 112 q^{67} - 48 q^{71} - 16 q^{78} - 16 q^{81} + 16 q^{86} + 96 q^{88} + 16 q^{92} - 16 q^{99}+O(q^{100}) $ 16 * q - 32 * q^4 - 16 * q^9 - 16 * q^15 + 16 * q^16 - 32 * q^22 - 16 * q^23 + 48 * q^25 + 96 * q^36 + 64 * q^37 + 48 * q^44 - 32 * q^53 - 80 * q^58 + 112 * q^60 - 32 * q^64 - 112 * q^67 - 48 * q^71 - 16 * q^78 - 16 * q^81 + 16 * q^86 + 96 * q^88 + 16 * q^92 - 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/539\mathbb{Z}\right)^\times$.

$n$	$199$	$442$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	2.60728i		−	1.84363i	−0.387632	−	0.921814i	$-0.626707\pi$
$2$		−	2.60728i		−	1.84363i	0.387632	−	0.921814i	$-0.373293\pi$
$3$		−	2.82079i		−	1.62859i	−0.580454	−	0.814293i	$-0.697125\pi$
$3$		−	2.82079i		−	1.62859i	0.580454	−	0.814293i	$-0.302875\pi$
$4$	−4.79793			−2.39897
$5$		−	1.84776i		−	0.826343i	−0.910653	−	0.413171i	$-0.864421\pi$
$5$		−	1.84776i		−	0.826343i	0.910653	−	0.413171i	$-0.135579\pi$
$6$	−7.35461			−3.00251
$7$		0			0
$7$		0			0
$8$			7.29501i			2.57917i
$9$	−4.95687			−1.65229
$10$	−4.81763			−1.52347
$11$	−0.112389	−	3.31472i	−0.0338865	−	0.999426i
$11$	−0.112389	−	3.31472i	−0.0338865	−	0.999426i
$12$			13.5340i			3.90692i
$13$	4.27619			1.18600			0.593000	−	0.805202i	$-0.297943\pi$
$13$	4.27619			1.18600			0.593000	+	0.805202i	$0.297943\pi$
$14$		0			0
$15$	−5.21215			−1.34577
$16$	9.42429			2.35607
$17$	3.30270			0.801022			0.400511	−	0.916292i	$-0.368833\pi$
$17$	3.30270			0.801022			0.400511	+	0.916292i	$0.368833\pi$
$18$			12.9240i			3.04621i
$19$	3.58783			0.823104			0.411552	−	0.911386i	$-0.364987\pi$
$19$	3.58783			0.823104			0.411552	+	0.911386i	$0.364987\pi$
$20$			8.86542i			1.98237i
$21$		0			0
$22$	−8.64242	+	0.293029i	−1.84257	+	0.0624741i
$23$	−4.98737			−1.03994			−0.519969	−	0.854185i	$-0.674057\pi$
$23$	−4.98737			−1.03994			−0.519969	+	0.854185i	$0.674057\pi$
$24$	20.5777			4.20041
$25$	1.58579			0.317157
$26$		−	11.1492i		−	2.18654i
$27$			5.51994i			1.06231i
$28$		0			0
$29$		−	3.27285i		−	0.607753i	−0.952711	−	0.303877i	$-0.901719\pi$
$29$		−	3.27285i		−	0.607753i	0.952711	−	0.303877i	$-0.0982811\pi$
$30$			13.5895i			2.48110i
$31$			1.58375i			0.284449i	0.989834	+	0.142225i	$0.0454255\pi$
$31$			1.58375i			0.284449i	−0.989834	+	0.142225i	$0.954574\pi$
$32$		−	9.98180i		−	1.76455i
$33$	−9.35014	+	0.317025i	−1.62765	+	0.0551870i
$34$		−	8.61108i		−	1.47679i
$35$		0			0
$36$	23.7827			3.96379
$37$	0.946795			0.155652			0.0778261	−	0.996967i	$-0.475202\pi$
$37$	0.946795			0.155652			0.0778261	+	0.996967i	$0.475202\pi$
$38$		−	9.35449i		−	1.51750i
$39$		−	12.0622i		−	1.93150i
$40$	13.4794			2.13128
$41$	1.77125			0.276623			0.138312	−	0.990389i	$-0.455832\pi$
$41$	1.77125			0.276623			0.138312	+	0.990389i	$0.455832\pi$
$42$		0			0
$43$			12.0622i			1.83947i	0.392535	+	0.919737i	$0.371598\pi$
$43$			12.0622i			1.83947i	−0.392535	+	0.919737i	$0.628402\pi$
$44$	0.539234	+	15.9038i	0.0812925	+	2.39759i
$45$			9.15911i			1.36536i
$46$			13.0035i			1.91726i
$47$		−	8.15307i		−	1.18925i	−0.804004	−	0.594624i	$-0.797301\pi$
$47$		−	8.15307i		−	1.18925i	0.804004	−	0.594624i	$-0.202699\pi$
$48$		−	26.5840i		−	3.83707i
$49$		0			0
$50$		−	4.13460i		−	0.584720i
$51$		−	9.31623i		−	1.30453i
$52$	−20.5169			−2.84518
$53$	−0.871553			−0.119717			−0.0598585	−	0.998207i	$-0.519065\pi$
$53$	−0.871553			−0.119717			−0.0598585	+	0.998207i	$0.519065\pi$
$54$	14.3920			1.95851
$55$	−6.12480	+	0.207667i	−0.825868	+	0.0280018i
$56$		0			0
$57$		−	10.1205i		−	1.34050i
$58$	−8.53325			−1.12047
$59$			6.68835i			0.870749i	0.900249	+	0.435374i	$0.143384\pi$
$59$			6.68835i			0.870749i	−0.900249	+	0.435374i	$0.856616\pi$
$60$	25.0075			3.22846
$61$	4.67942			0.599138			0.299569	−	0.954075i	$-0.403157\pi$
$61$	4.67942			0.599138			0.299569	+	0.954075i	$0.403157\pi$
$62$	4.12927			0.524418
$63$		0			0
$64$	−7.17680			−0.897101
$65$		−	7.90136i		−	0.980043i
$66$	0.826575	+	24.3785i	0.101744	+	3.00078i
$67$	−8.66949			−1.05915			−0.529573	−	0.848264i	$-0.677648\pi$
$67$	−8.66949			−1.05915			−0.529573	+	0.848264i	$0.677648\pi$
$68$	−15.8461			−1.92163
$69$			14.0683i			1.69363i
$70$		0			0
$71$	7.09378			0.841877			0.420938	−	0.907089i	$-0.361701\pi$
$71$	7.09378			0.841877			0.420938	+	0.907089i	$0.361701\pi$
$72$		−	36.1604i		−	4.26155i
$73$	9.35014			1.09435			0.547176	−	0.837018i	$-0.315703\pi$
$73$	9.35014			1.09435			0.547176	+	0.837018i	$0.315703\pi$
$74$		−	2.46857i		−	0.286965i
$75$		−	4.47318i		−	0.516518i
$76$	−17.2142			−1.97460
$77$		0			0
$78$	−31.4497			−3.56098
$79$			6.48649i			0.729787i	0.931049	+	0.364894i	$0.118895\pi$
$79$			6.48649i			0.729787i	−0.931049	+	0.364894i	$0.881105\pi$
$80$		−	17.4138i		−	1.94692i
$81$	0.699980			0.0777755
$82$		−	4.61816i		−	0.509991i
$83$	−8.55237			−0.938745			−0.469372	−	0.883000i	$-0.655520\pi$
$83$	−8.55237			−0.938745			−0.469372	+	0.883000i	$0.655520\pi$
$84$		0			0
$85$		−	6.10259i		−	0.661919i
$86$	31.4497			3.39131
$87$	−9.23204			−0.989778
$88$	24.1809	−	0.819876i	2.57769	−	0.0873991i
$89$		−	7.58170i		−	0.803658i	−0.915715	−	0.401829i	$-0.868375\pi$
$89$		−	7.58170i		−	0.803658i	0.915715	−	0.401829i	$-0.131625\pi$
$90$	23.8804			2.51722
$91$		0			0
$92$	23.9291			2.49478
$93$	4.46742			0.463250
$94$	−21.2574			−2.19253
$95$		−	6.62944i		−	0.680166i
$96$	−28.1566			−2.87372
$97$		−	3.12677i		−	0.317475i	−0.987321	−	0.158738i	$-0.949258\pi$
$97$		−	3.12677i		−	0.317475i	0.987321	−	0.158738i	$-0.0507424\pi$
$98$		0			0
$99$	0.557097	+	16.4307i	0.0559903	+	1.65134i
$100$	−7.60850			−0.760850
$101$	−3.70593			−0.368754			−0.184377	−	0.982856i	$-0.559027\pi$
$101$	−3.70593			−0.368754			−0.184377	+	0.982856i	$0.559027\pi$
$102$	−24.2901			−2.40508
$103$			17.1498i			1.68982i	0.534908	+	0.844910i	$0.320346\pi$
$103$			17.1498i			1.68982i	−0.534908	+	0.844910i	$0.679654\pi$
$104$			31.1948i			3.05890i
$105$		0			0
$106$			2.27239i			0.220714i
$107$			12.0030i			1.16038i	0.814483	+	0.580188i	$0.197021\pi$
$107$			12.0030i			1.16038i	−0.814483	+	0.580188i	$0.802979\pi$
$108$		−	26.4843i		−	2.54845i
$109$			13.7020i			1.31241i	0.754582	+	0.656206i	$0.227840\pi$
$109$			13.7020i			1.31241i	−0.754582	+	0.656206i	$0.772160\pi$
$110$	0.541448	+	15.9691i	0.0516250	+	1.52259i
$111$		−	2.67071i		−	0.253493i
$112$		0			0
$113$	9.40320			0.884579			0.442290	−	0.896872i	$-0.354166\pi$
$113$	9.40320			0.884579			0.442290	+	0.896872i	$0.354166\pi$
$114$	−26.3871			−2.47138
$115$			9.21546i			0.859346i
$116$			15.7029i			1.45798i
$117$	−21.1965			−1.95962
$118$	17.4384			1.60534
$119$		0			0
$120$		−	38.0226i		−	3.47098i
$121$	−10.9747	+	0.745074i	−0.997703	+	0.0677340i
$122$		−	12.2006i		−	1.10459i
$123$		−	4.99634i		−	0.450505i
$124$		−	7.59870i		−	0.682384i
$125$		−	12.1689i		−	1.08842i
$126$		0			0
$127$		−	17.0586i		−	1.51370i	−0.653586	−	0.756852i	$-0.726736\pi$
$127$		−	17.0586i		−	1.51370i	0.653586	−	0.756852i	$-0.273264\pi$
$128$		−	1.25163i		−	0.110629i
$129$	34.0251			2.99574
$130$	−20.6011			−1.80684
$131$	−15.8374			−1.38372			−0.691862	−	0.722030i	$-0.743209\pi$
$131$	−15.8374			−1.38372			−0.691862	+	0.722030i	$0.743209\pi$
$132$	44.8613	−	1.52107i	3.90468	−	0.132392i
$133$		0			0
$134$			22.6038i			1.95267i
$135$	10.1995			0.877834
$136$			24.0932i			2.06598i
$137$	−12.1133			−1.03491			−0.517453	−	0.855712i	$-0.673120\pi$
$137$	−12.1133			−1.03491			−0.517453	+	0.855712i	$0.673120\pi$
$138$	36.6802			3.12242
$139$	−0.679664			−0.0576484			−0.0288242	−	0.999584i	$-0.509176\pi$
$139$	−0.679664			−0.0576484			−0.0288242	+	0.999584i	$0.509176\pi$
$140$		0			0
$141$	−22.9981			−1.93679
$142$		−	18.4955i		−	1.55211i
$143$	−0.480595	−	14.1744i	−0.0401894	−	1.18532i
$144$	−46.7150			−3.89292
$145$	−6.04744			−0.502213
$146$		−	24.3785i		−	2.01758i
$147$		0			0
$148$	−4.54266			−0.373404
$149$			8.63037i			0.707027i	0.935429	+	0.353514i	$0.115013\pi$
$149$			8.63037i			0.707027i	−0.935429	+	0.353514i	$0.884987\pi$
$150$	−11.6628			−0.952267
$151$		−	9.43466i		−	0.767781i	−0.923379	−	0.383891i	$-0.874584\pi$
$151$		−	9.43466i		−	0.767781i	0.923379	−	0.383891i	$-0.125416\pi$
$152$			26.1732i			2.12293i
$153$	−16.3711			−1.32352
$154$		0			0
$155$	2.92638			0.235052
$156$			57.8738i			4.63361i
$157$		−	17.4138i		−	1.38977i	−0.719119	−	0.694887i	$-0.755454\pi$
$157$		−	17.4138i		−	1.38977i	0.719119	−	0.694887i	$-0.244546\pi$
$158$	16.9121			1.34546
$159$			2.45847i			0.194969i
$160$	−18.4440			−1.45812
$161$		0			0
$162$		−	1.82505i		−	0.143389i
$163$	−15.8158			−1.23879			−0.619394	−	0.785080i	$-0.712622\pi$
$163$	−15.8158			−1.23879			−0.619394	+	0.785080i	$0.712622\pi$
$164$	−8.49836			−0.663610
$165$	0.585786	+	17.2768i	0.0456034	+	1.34500i
$166$			22.2985i			1.73070i
$167$	2.86284			0.221534			0.110767	−	0.993846i	$-0.464669\pi$
$167$	2.86284			0.221534			0.110767	+	0.993846i	$0.464669\pi$
$168$		0			0
$169$	5.28577			0.406597
$170$	−15.9112			−1.22033
$171$	−17.7844			−1.36001
$172$		−	57.8738i		−	4.41284i
$173$	20.6386			1.56912			0.784560	−	0.620053i	$-0.212889\pi$
$173$	20.6386			1.56912			0.784560	+	0.620053i	$0.212889\pi$
$174$			24.0705i			1.82478i
$175$		0			0
$176$	−1.05918	−	31.2389i	−0.0798390	−	2.35472i
$177$	18.8664			1.41809
$178$	−19.7676			−1.48165
$179$	24.0880			1.80042			0.900211	−	0.435454i	$-0.143412\pi$
$179$	24.0880			1.80042			0.900211	+	0.435454i	$0.143412\pi$
$180$		−	43.9448i		−	3.27545i
$181$			3.10219i			0.230584i	0.993332	+	0.115292i	$0.0367804\pi$
$181$			3.10219i			0.230584i	−0.993332	+	0.115292i	$0.963220\pi$
$182$		0			0
$183$		−	13.1997i		−	0.975748i
$184$		−	36.3829i		−	2.68218i
$185$		−	1.74945i		−	0.128622i
$186$		−	11.6478i		−	0.854060i
$187$	−0.371186	−	10.9475i	−0.0271438	−	0.800562i
$188$			39.1179i			2.85296i
$189$		0			0
$190$	−17.2848			−1.25397
$191$	6.82843			0.494088			0.247044	−	0.969004i	$-0.420541\pi$
$191$	6.82843			0.494088			0.247044	+	0.969004i	$0.420541\pi$
$192$			20.2443i			1.46101i
$193$		−	7.29078i		−	0.524802i	−0.964959	−	0.262401i	$-0.915486\pi$
$193$		−	7.29078i		−	0.524802i	0.964959	−	0.262401i	$-0.0845142\pi$
$194$	−8.15238			−0.585307
$195$	−22.2881			−1.59608
$196$		0			0
$197$			4.62851i			0.329768i	0.986313	+	0.164884i	$0.0527249\pi$
$197$			4.62851i			0.329768i	−0.986313	+	0.164884i	$0.947275\pi$
$198$	42.8394	−	1.45251i	3.04446	−	0.103225i
$199$		−	8.81837i		−	0.625118i	−0.949898	−	0.312559i	$-0.898814\pi$
$199$		−	8.81837i		−	0.625118i	0.949898	−	0.312559i	$-0.101186\pi$
$200$			11.5683i			0.818004i
$201$			24.4548i			1.72491i
$202$			9.66242i			0.679845i
$203$		0			0
$204$			44.6987i			3.12953i
$205$		−	3.27285i		−	0.228586i
$206$	44.7144			3.11540
$207$	24.7218			1.71828
$208$	40.3000			2.79430
$209$	−0.403231	−	11.8926i	−0.0278921	−	0.822631i
$210$		0			0
$211$		−	9.25702i		−	0.637280i	−0.947876	−	0.318640i	$-0.896774\pi$
$211$		−	9.25702i		−	0.637280i	0.947876	−	0.318640i	$-0.103226\pi$
$212$	4.18165			0.287197
$213$		−	20.0101i		−	1.37107i
$214$	31.2953			2.13930
$215$	22.2881			1.52004
$216$	−40.2680			−2.73989
$217$		0			0
$218$	35.7250			2.41960
$219$		−	26.3748i		−	1.78224i
$220$	29.3864	−	0.996374i	1.98123	−	0.0671755i
$221$	14.1230			0.950013
$222$	−6.96331			−0.467347
$223$		−	17.3218i		−	1.15996i	−0.814632	−	0.579978i	$-0.803061\pi$
$223$		−	17.3218i		−	1.15996i	0.814632	−	0.579978i	$-0.196939\pi$
$224$		0			0
$225$	−7.86054			−0.524036
$226$		−	24.5168i		−	1.63084i
$227$	−25.7665			−1.71018			−0.855092	−	0.518476i	$-0.826500\pi$
$227$	−25.7665			−1.71018			−0.855092	+	0.518476i	$0.826500\pi$
$228$			48.5576i			3.21580i
$229$			7.78775i			0.514629i	0.966328	+	0.257314i	$0.0828376\pi$
$229$			7.78775i			0.514629i	−0.966328	+	0.257314i	$0.917162\pi$
$230$	24.0273			1.58431
$231$		0			0
$232$	23.8755			1.56750
$233$			11.9032i			0.779806i	0.920856	+	0.389903i	$0.127491\pi$
$233$			11.9032i			0.779806i	−0.920856	+	0.389903i	$0.872509\pi$
$234$			55.2653i			3.61281i
$235$	−15.0649			−0.982726
$236$		−	32.0902i		−	2.08890i
$237$	18.2970			1.18852
$238$		0			0
$239$		−	6.54570i		−	0.423406i	−0.977334	−	0.211703i	$-0.932099\pi$
$239$		−	6.54570i		−	0.423406i	0.977334	−	0.211703i	$-0.0679010\pi$
$240$	−49.1208			−3.17073
$241$	23.7655			1.53087			0.765436	−	0.643511i	$-0.222523\pi$
$241$	23.7655			1.53087			0.765436	+	0.643511i	$0.222523\pi$
$242$	1.94262	+	28.6143i	0.124876	+	1.83939i
$243$			14.5853i			0.935648i
$244$	−22.4515			−1.43731
$245$		0			0
$246$	−13.0269			−0.830564
$247$	15.3422			0.976202
$248$	−11.5534			−0.733644
$249$			24.1245i			1.52883i
$250$	−31.7279			−2.00665
$251$		−	12.6830i		−	0.800542i	−0.916397	−	0.400271i	$-0.868916\pi$
$251$		−	12.6830i		−	0.800542i	0.916397	−	0.400271i	$-0.131084\pi$
$252$		0			0
$253$	0.560524	+	16.5317i	0.0352398	+	1.03934i
$254$	−44.4766			−2.79071
$255$	−17.2142			−1.07799
$256$	−17.6170			−1.10106
$257$		−	19.7973i		−	1.23492i	−0.786601	−	0.617462i	$-0.788161\pi$
$257$		−	19.7973i		−	1.23492i	0.786601	−	0.617462i	$-0.211839\pi$
$258$		−	88.7130i		−	5.52303i
$259$		0			0
$260$			37.9102i			2.35109i
$261$			16.2231i			1.00419i
$262$			41.2927i			2.55107i
$263$		−	11.9193i		−	0.734975i	−0.930029	−	0.367487i	$-0.880218\pi$
$263$		−	11.9193i		−	0.734975i	0.930029	−	0.367487i	$-0.119782\pi$
$264$	−2.31270	−	68.2093i	−0.142337	−	4.19799i
$265$			1.61042i			0.0989273i
$266$		0			0
$267$	−21.3864			−1.30883
$268$	41.5956			2.54086
$269$			27.4317i			1.67254i	0.548320	+	0.836269i	$0.315268\pi$
$269$			27.4317i			1.67254i	−0.548320	+	0.836269i	$0.684732\pi$
$270$		−	26.5930i		−	1.61840i
$271$	5.22868			0.317620			0.158810	−	0.987309i	$-0.449234\pi$
$271$	5.22868			0.317620			0.158810	+	0.987309i	$0.449234\pi$
$272$	31.1256			1.88727
$273$		0			0
$274$			31.5827i			1.90798i
$275$	−0.178225	−	5.25644i	−0.0107473	−	0.316975i
$276$		−	67.4989i		−	4.06296i
$277$		−	19.0595i		−	1.14517i	−0.819844	−	0.572587i	$-0.805940\pi$
$277$		−	19.0595i		−	1.14517i	0.819844	−	0.572587i	$-0.194060\pi$
$278$			1.77208i			0.106282i
$279$		−	7.85043i		−	0.469993i
$280$		0			0
$281$		−	7.90136i		−	0.471356i	−0.971831	−	0.235678i	$-0.924269\pi$
$281$		−	7.90136i		−	0.471356i	0.971831	−	0.235678i	$-0.0757310\pi$
$282$			59.9626i			3.57072i
$283$	3.43310			0.204077			0.102038	−	0.994780i	$-0.467464\pi$
$283$	3.43310			0.204077			0.102038	+	0.994780i	$0.467464\pi$
$284$	−34.0355			−2.01963
$285$	−18.7003			−1.10771
$286$	−36.9566	+	1.25305i	−2.18529	+	0.0740943i
$287$		0			0
$288$			49.4785i			2.91555i
$289$	−6.09218			−0.358363
$290$			15.7674i			0.925893i
$291$	−8.81997			−0.517036
$292$	−44.8613			−2.62531
$293$	21.1965			1.23831			0.619157	−	0.785267i	$-0.287475\pi$
$293$	21.1965			1.23831			0.619157	+	0.785267i	$0.287475\pi$
$294$		0			0
$295$	12.3585			0.719537
$296$			6.90688i			0.401454i
$297$	18.2970	−	0.620379i	1.06170	−	0.0359980i
$298$	22.5018			1.30350
$299$	−21.3269			−1.23337
$300$			21.4620i			1.23911i
$301$		0			0
$302$	−24.5988			−1.41550
$303$			10.4537i			0.600547i
$304$	33.8127			1.93929
$305$		−	8.64644i		−	0.495093i
$306$			42.6840i			2.44008i
$307$	−14.6969			−0.838798			−0.419399	−	0.907802i	$-0.637759\pi$
$307$	−14.6969			−0.838798			−0.419399	+	0.907802i	$0.637759\pi$
$308$		0			0
$309$	48.3761			2.75202
$310$		−	7.62990i		−	0.433349i
$311$			8.31863i			0.471706i	0.971789	+	0.235853i	$0.0757885\pi$
$311$			8.31863i			0.471706i	−0.971789	+	0.235853i	$0.924212\pi$
$312$	87.9941			4.98168
$313$			16.0713i			0.908402i	0.890899	+	0.454201i	$0.150075\pi$
$313$			16.0713i			0.908402i	−0.890899	+	0.454201i	$0.849925\pi$
$314$	−45.4028			−2.56223
$315$		0			0
$316$		−	31.1217i		−	1.75073i
$317$	16.7422			0.940334			0.470167	−	0.882577i	$-0.344194\pi$
$317$	16.7422			0.940334			0.470167	+	0.882577i	$0.344194\pi$
$318$	6.40993			0.359451
$319$	−10.8486	+	0.367832i	−0.607404	+	0.0205946i
$320$			13.2610i			0.741313i
$321$	33.8581			1.88977
$322$		0			0
$323$	11.8495			0.659325
$324$	−3.35846			−0.186581
$325$	6.78112			0.376149
$326$			41.2363i			2.28387i
$327$	38.6505			2.13738
$328$			12.9213i			0.713460i
$329$		0			0
$330$	45.0455	−	1.52731i	2.47968	−	0.0840758i
$331$	22.6440			1.24462			0.622312	−	0.782769i	$-0.286193\pi$
$331$	22.6440			1.24462			0.622312	+	0.782769i	$0.286193\pi$
$332$	41.0337			2.25202
$333$	−4.69315			−0.257183
$334$		−	7.46425i		−	0.408426i
$335$			16.0191i			0.875218i
$336$		0			0
$337$			15.0416i			0.819367i	0.912228	+	0.409684i	$0.134361\pi$
$337$			15.0416i			0.819367i	−0.912228	+	0.409684i	$0.865639\pi$
$338$		−	13.7815i		−	0.749615i
$339$		−	26.5245i		−	1.44061i
$340$			29.2798i			1.58792i
$341$	5.24967	−	0.177995i	0.284286	−	0.00963898i
$342$			46.3690i			2.50735i
$343$		0			0
$344$	−87.9941			−4.74432
$345$	25.9949			1.39952
$346$		−	53.8106i		−	2.89288i
$347$			21.2355i			1.13998i	0.821651	+	0.569991i	$0.193053\pi$
$347$			21.2355i			1.13998i	−0.821651	+	0.569991i	$0.806947\pi$
$348$	44.2947			2.37444
$349$	27.2440			1.45834			0.729168	−	0.684335i	$-0.239907\pi$
$349$	27.2440			1.45834			0.729168	+	0.684335i	$0.239907\pi$
$350$		0			0
$351$			23.6043i			1.25990i
$352$	−33.0869	+	1.12184i	−1.76354	+	0.0597944i
$353$			24.5201i			1.30507i	0.757757	+	0.652537i	$0.226295\pi$
$353$			24.5201i			1.30507i	−0.757757	+	0.652537i	$0.773705\pi$
$354$		−	49.1902i		−	2.61443i
$355$		−	13.1076i		−	0.695679i
$356$			36.3765i			1.92795i
$357$		0			0
$358$		−	62.8043i		−	3.31931i
$359$		−	16.3821i		−	0.864616i	−0.901726	−	0.432308i	$-0.857699\pi$
$359$		−	16.3821i		−	0.864616i	0.901726	−	0.432308i	$-0.142301\pi$
$360$	−66.8158			−3.52150
$361$	−6.12750			−0.322500
$362$	8.08829			0.425111
$363$	2.10170	+	30.9575i	0.110311	+	1.62485i
$364$		0			0
$365$		−	17.2768i		−	0.904309i
$366$	−34.4153			−1.79892
$367$			0.541030i			0.0282415i	0.999900	+	0.0141208i	$0.00449493\pi$
$367$			0.541030i			0.0282415i	−0.999900	+	0.0141208i	$0.995505\pi$
$368$	−47.0024			−2.45017
$369$	−8.77988			−0.457063
$370$	−4.56131			−0.237131
$371$		0			0
$372$	−21.4344			−1.11132
$373$			29.0616i			1.50475i	0.658733	+	0.752376i	$0.271092\pi$
$373$			29.0616i			1.50475i	−0.658733	+	0.752376i	$0.728908\pi$
$374$	−28.5433	+	0.967788i	−1.47594	+	0.0500431i
$375$	−34.3261			−1.77259
$376$	59.4767			3.06728
$377$		−	13.9953i		−	0.720796i
$378$		0			0
$379$	−0.941949			−0.0483846			−0.0241923	−	0.999707i	$-0.507701\pi$
$379$	−0.941949			−0.0483846			−0.0241923	+	0.999707i	$0.507701\pi$
$380$			31.8076i			1.63170i
$381$	−48.1187			−2.46520
$382$		−	17.8037i		−	0.910914i
$383$		−	1.45839i		−	0.0745201i	−0.999306	−	0.0372600i	$-0.988137\pi$
$383$		−	1.45839i		−	0.0745201i	0.999306	−	0.0372600i	$-0.0118630\pi$
$384$	−3.53058			−0.180169
$385$		0			0
$386$	−19.0091			−0.967539
$387$		−	59.7910i		−	3.03935i
$388$			15.0020i			0.761613i
$389$	6.83944			0.346773			0.173387	−	0.984854i	$-0.444529\pi$
$389$	6.83944			0.346773			0.173387	+	0.984854i	$0.444529\pi$
$390$			58.1114i			2.94259i
$391$	−16.4718			−0.833014
$392$		0			0
$393$			44.6741i			2.25351i
$394$	12.0678			0.607969
$395$	11.9855			0.603055
$396$	−2.67291	−	78.8332i	−0.134319	−	3.96152i
$397$		−	12.3733i		−	0.620997i	−0.950574	−	0.310499i	$-0.899504\pi$
$397$		−	12.3733i		−	0.620997i	0.950574	−	0.310499i	$-0.100496\pi$
$398$	−22.9920			−1.15248
$399$		0			0
$400$	14.9449			0.747246
$401$	−19.6018			−0.978866			−0.489433	−	0.872041i	$-0.662796\pi$
$401$	−19.6018			−0.978866			−0.489433	+	0.872041i	$0.662796\pi$
$402$	63.7607			3.18009
$403$			6.77239i			0.337357i
$404$	17.7808			0.884628
$405$		−	1.29339i		−	0.0642693i
$406$		0			0
$407$	−0.106409	−	3.13836i	−0.00527450	−	0.155563i
$408$	67.9620			3.36462
$409$	−14.8396			−0.733772			−0.366886	−	0.930266i	$-0.619576\pi$
$409$	−14.8396			−0.733772			−0.366886	+	0.930266i	$0.619576\pi$
$410$	−8.53325			−0.421427
$411$			34.1690i			1.68543i
$412$		−	82.2836i		−	4.05382i
$413$		0			0
$414$		−	64.4567i		−	3.16787i
$415$			15.8027i			0.775725i
$416$		−	42.6840i		−	2.09276i
$417$			1.91719i			0.0938853i
$418$	−31.0075	+	1.05134i	−1.51663	+	0.0514227i
$419$		−	10.7644i		−	0.525877i	−0.964813	−	0.262939i	$-0.915308\pi$
$419$		−	10.7644i		−	0.525877i	0.964813	−	0.262939i	$-0.0846917\pi$
$420$		0			0
$421$	−22.4013			−1.09177			−0.545887	−	0.837859i	$-0.683807\pi$
$421$	−22.4013			−1.09177			−0.545887	+	0.837859i	$0.683807\pi$
$422$	−24.1357			−1.17491
$423$			40.4137i			1.96498i
$424$		−	6.35798i		−	0.308771i
$425$	5.23738			0.254050
$426$	−52.1720			−2.52774
$427$		0			0
$428$		−	57.5897i		−	2.78370i
$429$	−39.9829	+	1.35566i	−1.93039	+	0.0654518i
$430$		−	58.1114i		−	2.80238i
$431$			9.39997i			0.452781i	0.974037	+	0.226390i	$0.0726925\pi$
$431$			9.39997i			0.452781i	−0.974037	+	0.226390i	$0.927308\pi$
$432$			52.0215i			2.50289i
$433$			12.0749i			0.580282i	0.956984	+	0.290141i	$0.0937021\pi$
$433$			12.0749i			0.580282i	−0.956984	+	0.290141i	$0.906298\pi$
$434$		0			0
$435$			17.0586i			0.817896i
$436$		−	65.7412i		−	3.14843i
$437$	−17.8938			−0.855977
$438$	−68.7666			−3.28580
$439$	38.6505			1.84469			0.922343	−	0.386371i	$-0.126272\pi$
$439$	38.6505			1.84469			0.922343	+	0.386371i	$0.126272\pi$
$440$	−1.51493	−	44.6805i	−0.0722217	−	2.13006i
$441$		0			0
$442$		−	36.8226i		−	1.75147i
$443$	13.4963			0.641229			0.320615	−	0.947210i	$-0.396111\pi$
$443$	13.4963			0.641229			0.320615	+	0.947210i	$0.396111\pi$
$444$			12.8139i			0.608121i
$445$	−14.0091			−0.664097
$446$	−45.1630			−2.13853
$447$	24.3445			1.15145
$448$		0			0
$449$	−23.6949			−1.11823			−0.559115	−	0.829090i	$-0.688859\pi$
$449$	−23.6949			−1.11823			−0.559115	+	0.829090i	$0.688859\pi$
$450$			20.4947i			0.966128i
$451$	−0.199069	−	5.87121i	−0.00937379	−	0.276465i
$452$	−45.1159			−2.12208
$453$	−26.6132			−1.25040
$454$			67.1807i			3.15294i
$455$		0			0
$456$	73.8292			3.45737
$457$		−	14.3126i		−	0.669514i	−0.942304	−	0.334757i	$-0.891346\pi$
$457$		−	14.3126i		−	0.669514i	0.942304	−	0.334757i	$-0.108654\pi$
$458$	20.3049			0.948784
$459$			18.2307i			0.850936i
$460$		−	44.2151i		−	2.06154i
$461$	25.7002			1.19698			0.598489	−	0.801131i	$-0.295768\pi$
$461$	25.7002			1.19698			0.598489	+	0.801131i	$0.295768\pi$
$462$		0			0
$463$	−30.0222			−1.39525			−0.697624	−	0.716464i	$-0.745759\pi$
$463$	−30.0222			−1.39525			−0.697624	+	0.716464i	$0.745759\pi$
$464$		−	30.8443i		−	1.43191i
$465$		−	8.25471i		−	0.382803i
$466$	31.0351			1.43767
$467$		−	30.2018i		−	1.39757i	−0.715331	−	0.698786i	$-0.753724\pi$
$467$		−	30.2018i		−	1.39757i	0.715331	−	0.698786i	$-0.246276\pi$
$468$	101.699			4.70106
$469$		0			0
$470$			39.2785i			1.81178i
$471$	−49.1208			−2.26337
$472$	−48.7915			−2.24581
$473$	39.9829	−	1.35566i	1.83842	−	0.0623333i
$474$		−	47.7056i		−	2.19119i
$475$	5.68953			0.261053
$476$		0			0
$477$	4.32018			0.197807
$478$	−17.0665			−0.780604
$479$	30.4495			1.39128			0.695638	−	0.718393i	$-0.255122\pi$
$479$	30.4495			1.39128			0.695638	+	0.718393i	$0.255122\pi$
$480$			52.0266i			2.37468i
$481$	4.04867			0.184604
$482$		−	61.9635i		−	2.82236i
$483$		0			0
$484$	52.6561	−	3.57482i	2.39346	−	0.162492i
$485$	−5.77752			−0.262344
$486$	38.0281			1.72499
$487$	−1.28644			−0.0582941			−0.0291470	−	0.999575i	$-0.509279\pi$
$487$	−1.28644			−0.0582941			−0.0291470	+	0.999575i	$0.509279\pi$
$488$			34.1364i			1.54528i
$489$			44.6131i			2.01747i
$490$		0			0
$491$		−	4.06107i		−	0.183274i	−0.995792	−	0.0916368i	$-0.970790\pi$
$491$		−	4.06107i		−	0.183274i	0.995792	−	0.0916368i	$-0.0292099\pi$
$492$			23.9721i			1.08075i
$493$		−	10.8092i		−	0.486824i
$494$		−	40.0015i		−	1.79975i
$495$	30.3599	−	1.02938i	1.36458	−	0.0462672i
$496$			14.9257i			0.670183i
$497$		0			0
$498$	62.8994			2.81859
$499$	−19.0360			−0.852170			−0.426085	−	0.904683i	$-0.640108\pi$
$499$	−19.0360			−0.852170			−0.426085	+	0.904683i	$0.640108\pi$
$500$			58.3858i			2.61109i
$501$		−	8.07549i		−	0.360786i
$502$	−33.0681			−1.47590
$503$	−42.2557			−1.88409			−0.942044	−	0.335489i	$-0.891098\pi$
$503$	−42.2557			−1.88409			−0.942044	+	0.335489i	$0.891098\pi$
$504$		0			0
$505$			6.84767i			0.304717i
$506$	43.1029	−	1.46145i	1.91616	−	0.0649692i
$507$		−	14.9101i		−	0.662179i
$508$			81.8459i			3.63133i
$509$			37.7416i			1.67287i	0.548067	+	0.836434i	$0.315364\pi$
$509$			37.7416i			1.67287i	−0.548067	+	0.836434i	$0.684636\pi$
$510$			44.8822i			1.98742i
$511$		0			0
$512$			43.4292i			1.91932i
$513$			19.8046i			0.874394i
$514$	−51.6173			−2.27674
$515$	31.6887			1.39637
$516$	−163.250			−7.18668
$517$	−27.0251	+	0.916313i	−1.18856	+	0.0402994i
$518$		0			0
$519$		−	58.2171i		−	2.55545i
$520$	57.6405			2.52770
$521$		−	9.29390i		−	0.407173i	−0.979057	−	0.203587i	$-0.934740\pi$
$521$		−	9.29390i		−	0.407173i	0.979057	−	0.203587i	$-0.0652598\pi$
$522$	42.2983			1.85134
$523$	−3.32369			−0.145335			−0.0726674	−	0.997356i	$-0.523151\pi$
$523$	−3.32369			−0.145335			−0.0726674	+	0.997356i	$0.523151\pi$
$524$	75.9869			3.31951
$525$		0			0
$526$	−31.0770			−1.35502
$527$			5.23063i			0.227850i
$528$	−88.1185	+	2.98774i	−3.83486	+	0.130025i
$529$	1.87385			0.0814716
$530$	4.19882			0.182385
$531$		−	33.1533i		−	1.43873i
$532$		0			0
$533$	7.57421			0.328076
$534$			55.7604i			2.41299i
$535$	22.1787			0.958869
$536$		−	63.2440i		−	2.73172i
$537$		−	67.9473i		−	2.93214i
$538$	71.5221			3.08354
$539$		0			0
$540$	−48.9366			−2.10590
$541$			2.82033i			0.121255i	0.998160	+	0.0606277i	$0.0193103\pi$
$541$			2.82033i			0.121255i	−0.998160	+	0.0606277i	$0.980690\pi$
$542$		−	13.6327i		−	0.585573i
$543$	8.75063			0.375526
$544$		−	32.9669i		−	1.41344i
$545$	25.3180			1.08450
$546$		0			0
$547$			30.6702i			1.31136i	0.755038	+	0.655681i	$0.227618\pi$
$547$			30.6702i			1.31136i	−0.755038	+	0.655681i	$0.772382\pi$
$548$	58.1186			2.48270
$549$	−23.1953			−0.989951
$550$	−13.7050	+	0.464682i	−0.584384	+	0.0198141i
$551$		−	11.7424i		−	0.500244i
$552$	−102.629			−4.36816
$553$		0			0
$554$	−49.6936			−2.11128
$555$	−4.93484			−0.209472
$556$	3.26098			0.138296
$557$		−	30.2337i		−	1.28104i	−0.767940	−	0.640522i	$-0.778718\pi$
$557$		−	30.2337i		−	1.28104i	0.767940	−	0.640522i	$-0.221282\pi$
$558$	−20.4683			−0.866492
$559$			51.5804i			2.18162i
$560$		0			0
$561$	−30.8807	+	1.04704i	−1.30378	+	0.0442060i
$562$	−20.6011			−0.869005
$563$	−27.1259			−1.14322			−0.571609	−	0.820526i	$-0.693681\pi$
$563$	−27.1259			−1.14322			−0.571609	+	0.820526i	$0.693681\pi$
$564$	110.343			4.64630
$565$		−	17.3749i		−	0.730966i
$566$		−	8.95107i		−	0.376241i
$567$		0			0
$568$			51.7492i			2.17135i
$569$			16.5317i			0.693046i	0.938041	+	0.346523i	$0.112638\pi$
$569$			16.5317i			0.693046i	−0.938041	+	0.346523i	$0.887362\pi$
$570$			48.7569i			2.04220i
$571$			33.2978i			1.39347i	0.717330	+	0.696734i	$0.245364\pi$
$571$			33.2978i			1.39347i	−0.717330	+	0.696734i	$0.754636\pi$
$572$	2.30586	+	68.0076i	0.0964130	+	2.84354i
$573$		−	19.2616i		−	0.804664i
$574$		0			0
$575$	−7.90890			−0.329824
$576$	35.5745			1.48227
$577$		−	30.3404i		−	1.26309i	−0.775340	−	0.631544i	$-0.782421\pi$
$577$		−	30.3404i		−	1.26309i	0.775340	−	0.631544i	$-0.217579\pi$
$578$			15.8840i			0.660689i
$579$	−20.5658			−0.854684
$580$	29.0152			1.20479
$581$		0			0
$582$			22.9962i			0.953222i
$583$	0.0979527	+	2.88895i	0.00405679	+	0.119648i
$584$			68.2093i			2.82252i
$585$			39.1661i			1.61932i
$586$		−	55.2653i		−	2.28299i
$587$			4.45108i			0.183716i	0.995772	+	0.0918578i	$0.0292805\pi$
$587$			4.45108i			0.183716i	−0.995772	+	0.0918578i	$0.970719\pi$
$588$		0			0
$589$			5.68220i			0.234131i
$590$		−	32.2220i		−	1.32656i
$591$	13.0561			0.537055
$592$	8.92288			0.366728
$593$	27.9689			1.14855			0.574273	−	0.818664i	$-0.305285\pi$
$593$	27.9689			1.14855			0.574273	+	0.818664i	$0.305285\pi$
$594$	−1.61750	−	47.7056i	−0.0663670	−	1.95738i
$595$		0			0
$596$		−	41.4079i		−	1.69614i
$597$	−24.8748			−1.01806
$598$			55.6053i			2.27387i
$599$	2.72326			0.111269			0.0556346	−	0.998451i	$-0.482282\pi$
$599$	2.72326			0.111269			0.0556346	+	0.998451i	$0.482282\pi$
$600$	32.6318			1.33219
$601$	−11.6830			−0.476558			−0.238279	−	0.971197i	$-0.576583\pi$
$601$	−11.6830			−0.476558			−0.238279	+	0.971197i	$0.576583\pi$
$602$		0			0
$603$	42.9735			1.75002
$604$			45.2668i			1.84188i
$605$	1.37672	+	20.2787i	0.0559715	+	0.824445i
$606$	27.2557			1.10719
$607$	38.2523			1.55261			0.776307	−	0.630355i	$-0.217091\pi$
$607$	38.2523			1.55261			0.776307	+	0.630355i	$0.217091\pi$
$608$		−	35.8130i		−	1.45241i
$609$		0			0
$610$	−22.5437			−0.912768
$611$		−	34.8640i		−	1.41045i
$612$	78.5473			3.17509
$613$		−	0.183542i		−	0.00741318i	−0.999993	−	0.00370659i	$-0.998820\pi$
$613$		−	0.183542i		−	0.00741318i	0.999993	−	0.00370659i	$-0.00117985\pi$
$614$			38.3191i			1.54643i
$615$	−9.23204			−0.372272
$616$		0			0
$617$	9.73115			0.391761			0.195881	−	0.980628i	$-0.437243\pi$
$617$	9.73115			0.391761			0.195881	+	0.980628i	$0.437243\pi$
$618$		−	126.130i		−	5.07370i
$619$			11.5017i			0.462294i	0.972919	+	0.231147i	$0.0742479\pi$
$619$			11.5017i			0.462294i	−0.972919	+	0.231147i	$0.925752\pi$
$620$	−14.0406			−0.563883
$621$		−	27.5300i		−	1.10474i
$622$	21.6890			0.869651
$623$		0			0
$624$		−	113.678i		−	4.55076i
$625$	−14.5563			−0.582254
$626$	41.9024			1.67476
$627$	−33.5467	+	1.13743i	−1.33973	+	0.0454247i
$628$			83.5503i			3.33402i
$629$	3.12698			0.124681
$630$		0			0
$631$	16.1028			0.641042			0.320521	−	0.947241i	$-0.396142\pi$
$631$	16.1028			0.641042			0.320521	+	0.947241i	$0.396142\pi$
$632$	−47.3190			−1.88225
$633$	−26.1121			−1.03786
$634$		−	43.6516i		−	1.73363i
$635$	−31.5201			−1.25084
$636$		−	11.7956i		−	0.467725i
$637$		0			0
$638$	0.959041	+	28.2853i	0.0379688	+	1.11983i
$639$	−35.1630			−1.39103
$640$	−2.31270			−0.0914176
$641$	−6.69315			−0.264363			−0.132182	−	0.991226i	$-0.542198\pi$
$641$	−6.69315			−0.264363			−0.132182	+	0.991226i	$0.542198\pi$
$642$		−	88.2776i		−	3.48404i
$643$		−	2.28381i		−	0.0900647i	−0.998986	−	0.0450323i	$-0.985661\pi$
$643$		−	2.28381i		−	0.0900647i	0.998986	−	0.0450323i	$-0.0143391\pi$
$644$		0			0
$645$		−	62.8701i		−	2.47551i
$646$		−	30.8951i		−	1.21555i
$647$		−	13.6100i		−	0.535062i	−0.963549	−	0.267531i	$-0.913792\pi$
$647$		−	13.6100i		−	0.535062i	0.963549	−	0.267531i	$-0.0862078\pi$
$648$			5.10636i			0.200597i
$649$	22.1700	−	0.751695i	0.870249	−	0.0295066i
$650$		−	17.6803i		−	0.693479i
$651$		0			0
$652$	75.8831			2.97181
$653$	13.0004			0.508744			0.254372	−	0.967106i	$-0.418131\pi$
$653$	13.0004			0.508744			0.254372	+	0.967106i	$0.418131\pi$
$654$		−	100.773i		−	3.94053i
$655$			29.2638i			1.14343i
$656$	16.6928			0.651745
$657$	−46.3475			−1.80819
$658$		0			0
$659$		−	12.0622i		−	0.469878i	−0.972010	−	0.234939i	$-0.924511\pi$
$659$		−	12.0622i		−	0.469878i	0.972010	−	0.234939i	$-0.0754890\pi$
$660$	−2.81056	−	82.8929i	−0.109401	−	3.22660i
$661$			43.7435i			1.70142i	0.525633	+	0.850712i	$0.323829\pi$
$661$			43.7435i			1.70142i	−0.525633	+	0.850712i	$0.676171\pi$
$662$		−	59.0393i		−	2.29463i
$663$		−	39.8379i		−	1.54718i
$664$		−	62.3896i		−	2.42119i
$665$		0			0
$666$			12.2364i			0.474150i
$667$			16.3229i			0.632026i
$668$	−13.7357			−0.531452
$669$	−48.8613			−1.88909
$670$	41.7664			1.61358
$671$	−0.525914	−	15.5110i	−0.0203027	−	0.598794i
$672$		0			0
$673$			24.5449i			0.946135i	0.881026	+	0.473067i	$0.156853\pi$
$673$			24.5449i			0.946135i	−0.881026	+	0.473067i	$0.843147\pi$
$674$	39.2177			1.51061
$675$			8.75344i			0.336920i
$676$	−25.3608			−0.975414
$677$	−34.0338			−1.30802			−0.654012	−	0.756484i	$-0.726915\pi$
$677$	−34.0338			−1.30802			−0.654012	+	0.756484i	$0.726915\pi$
$678$	−69.1569			−2.65596
$679$		0			0
$680$	44.5185			1.70720
$681$			72.6820i			2.78518i
$682$	−0.464084	−	13.6874i	−0.0177707	−	0.524117i
$683$	16.4568			0.629701			0.314850	−	0.949141i	$-0.398046\pi$
$683$	16.4568			0.629701			0.314850	+	0.949141i	$0.398046\pi$
$684$	85.3284			3.26261
$685$			22.3824i			0.855187i
$686$		0			0
$687$	21.9676			0.838117
$688$			113.678i			4.33393i
$689$	−3.72692			−0.141984
$690$		−	67.7761i		−	2.58019i
$691$			35.4680i			1.34927i	0.738153	+	0.674634i	$0.235698\pi$
$691$			35.4680i			1.34927i	−0.738153	+	0.674634i	$0.764302\pi$
$692$	−99.0224			−3.76427
$693$		0			0
$694$	55.3670			2.10170
$695$			1.25586i			0.0476373i
$696$		−	67.3478i		−	2.55281i
$697$	5.84992			0.221582
$698$		−	71.0327i		−	2.68863i
$699$	33.5765			1.26998
$700$		0			0
$701$			23.7041i			0.895291i	0.894211	+	0.447645i	$0.147737\pi$
$701$			23.7041i			0.895291i	−0.894211	+	0.447645i	$0.852263\pi$
$702$	61.5431			2.32279
$703$	3.39694			0.128118
$704$	0.806592	+	23.7891i	0.0303996	+	0.896585i
$705$			42.4950i			1.60045i
$706$	63.9310			2.40607
$707$		0			0
$708$	−90.5199			−3.40195
$709$	23.1114			0.867966			0.433983	−	0.900921i	$-0.357108\pi$
$709$	23.1114			0.867966			0.433983	+	0.900921i	$0.357108\pi$
$710$	−34.1752			−1.28257
$711$		−	32.1527i		−	1.20582i
$712$	55.3085			2.07277
$713$		−	7.89872i		−	0.295809i
$714$		0			0
$715$	−26.1908	+	0.888024i	−0.979480	+	0.0332102i
$716$	−115.573			−4.31915
$717$	−18.4641			−0.689553
$718$	−42.7129			−1.59403
$719$			29.8421i			1.11292i	0.830874	+	0.556461i	$0.187841\pi$
$719$			29.8421i			1.11292i	−0.830874	+	0.556461i	$0.812159\pi$
$720$			86.3181i			3.21689i
$721$		0			0
$722$			15.9761i			0.594570i
$723$		−	67.0377i		−	2.49316i
$724$		−	14.8841i		−	0.553163i
$725$		−	5.19004i		−	0.192753i
$726$	80.7149	−	5.47973i	2.99561	−	0.203372i
$727$			28.5107i			1.05740i	0.848808	+	0.528702i	$0.177321\pi$
$727$			28.5107i			1.05740i	−0.848808	+	0.528702i	$0.822679\pi$
$728$		0			0
$729$	43.2421			1.60156
$730$	−45.0455			−1.66721
$731$			39.8379i			1.47346i
$732$			63.3311i			2.34079i
$733$	−39.5750			−1.46174			−0.730869	−	0.682518i	$-0.760885\pi$
$733$	−39.5750			−1.46174			−0.730869	+	0.682518i	$0.760885\pi$
$734$	1.41062			0.0520669
$735$		0			0
$736$			49.7829i			1.83502i
$737$	0.974352	+	28.7369i	0.0358907	+	1.05854i
$738$			22.8917i			0.842654i
$739$			16.9748i			0.624430i	0.950012	+	0.312215i	$0.101071\pi$
$739$			16.9748i			0.624430i	−0.950012	+	0.312215i	$0.898929\pi$
$740$			8.39374i			0.308560i
$741$		−	43.2772i		−	1.58983i
$742$		0			0
$743$		−	39.9272i		−	1.46479i	−0.680882	−	0.732393i	$-0.738403\pi$
$743$		−	39.9272i		−	1.46479i	0.680882	−	0.732393i	$-0.261597\pi$
$744$			32.5898i			1.19480i
$745$	15.9468			0.584247
$746$	75.7719			2.77421
$747$	42.3930			1.55108
$748$	1.78093	+	52.5255i	0.0651171	+	1.92052i
$749$		0			0
$750$			89.4979i			3.26800i
$751$	19.4020			0.707988			0.353994	−	0.935248i	$-0.384823\pi$
$751$	19.4020			0.707988			0.353994	+	0.935248i	$0.384823\pi$
$752$		−	76.8369i		−	2.80195i
$753$	−35.7760			−1.30375
$754$	−36.4898			−1.32888
$755$	−17.4330			−0.634451
$756$		0			0
$757$	−12.9289			−0.469910			−0.234955	−	0.972006i	$-0.575494\pi$
$757$	−12.9289			−0.469910			−0.234955	+	0.972006i	$0.575494\pi$
$758$			2.45593i			0.0892033i
$759$	46.6326	−	1.58112i	1.69266	−	0.0573911i
$760$	48.3618			1.75427
$761$	9.68928			0.351236			0.175618	−	0.984458i	$-0.443808\pi$
$761$	9.68928			0.351236			0.175618	+	0.984458i	$0.443808\pi$
$762$			125.459i			4.54491i
$763$		0			0
$764$	−32.7623			−1.18530
$765$			30.2498i			1.09368i
$766$	−3.80243			−0.137387
$767$			28.6006i			1.03271i
$768$			49.6938i			1.79317i
$769$	7.36367			0.265541			0.132770	−	0.991147i	$-0.457613\pi$
$769$	7.36367			0.265541			0.132770	+	0.991147i	$0.457613\pi$
$770$		0			0
$771$	−55.8442			−2.01118
$772$			34.9807i			1.25898i
$773$		−	8.42604i		−	0.303064i	−0.988452	−	0.151532i	$-0.951579\pi$
$773$		−	8.42604i		−	0.303064i	0.988452	−	0.151532i	$-0.0484206\pi$
$774$	−155.892			−5.60343
$775$			2.51148i			0.0902151i
$776$	22.8098			0.818824
$777$		0			0
$778$		−	17.8324i		−	0.639321i
$779$	6.35495			0.227690
$780$	106.937			3.82895
$781$	−0.797261	−	23.5139i	−0.0285282	−	0.841393i
$782$			42.9466i			1.53577i
$783$	18.0659			0.645624
$784$		0			0
$785$	−32.1765			−1.14843
$786$	116.478			4.15464
$787$	−3.47842			−0.123992			−0.0619961	−	0.998076i	$-0.519747\pi$
$787$	−3.47842			−0.123992			−0.0619961	+	0.998076i	$0.519747\pi$
$788$		−	22.2073i		−	0.791102i
$789$	−33.6218			−1.19697
$790$		−	31.2495i		−	1.11181i
$791$		0			0
$792$	−119.862	+	4.06402i	−4.25910	+	0.144409i
$793$	20.0101			0.710578
$794$	−32.2607			−1.14489
$795$	4.54266			0.161112
$796$			42.3099i			1.49964i
$797$			26.1007i			0.924535i	0.886741	+	0.462267i	$0.152964\pi$
$797$			26.1007i			0.924535i	−0.886741	+	0.462267i	$0.847036\pi$
$798$		0			0
$799$		−	26.9271i		−	0.952614i
$800$		−	15.8290i		−	0.559640i
$801$			37.5815i			1.32788i
$802$			51.1074i			1.80466i
$803$	−1.05085	−	30.9931i	−0.0370837	−	1.09372i
$804$		−	117.333i		−	4.13800i
$805$		0			0
$806$	17.6575			0.621961
$807$	77.3790			2.72387
$808$		−	27.0348i		−	0.951081i
$809$			45.7278i			1.60770i	0.594829	+	0.803852i	$0.297220\pi$
$809$			45.7278i			1.60770i	−0.594829	+	0.803852i	$0.702780\pi$
$810$	−3.37225			−0.118489
$811$	−4.50370			−0.158146			−0.0790731	−	0.996869i	$-0.525196\pi$
$811$	−4.50370			−0.158146			−0.0790731	+	0.996869i	$0.525196\pi$
$812$		0			0
$813$		−	14.7490i		−	0.517271i
$814$	−8.18260	+	0.277439i	−0.286800	+	0.00972423i
$815$			29.2238i			1.02366i
$816$		−	87.7989i		−	3.07358i
$817$			43.2772i			1.51408i
$818$			38.6911i			1.35280i
$819$		0			0
$820$			15.7029i			0.548370i
$821$			6.85432i			0.239217i	0.992821	+	0.119609i	$0.0381640\pi$
$821$			6.85432i			0.239217i	−0.992821	+	0.119609i	$0.961836\pi$
$822$	89.0883			3.10731
$823$	33.7311			1.17579			0.587897	−	0.808936i	$-0.299956\pi$
$823$	33.7311			1.17579			0.587897	+	0.808936i	$0.299956\pi$
$824$	−125.108			−4.35834
$825$	−14.8273	+	0.502734i	−0.516221	+	0.0175030i
$826$		0			0
$827$			4.06107i			0.141217i	0.997504	+	0.0706086i	$0.0224941\pi$
$827$			4.06107i			0.141217i	−0.997504	+	0.0706086i	$0.977506\pi$
$828$	−118.613			−4.12210
$829$		−	46.2049i		−	1.60476i	−0.596812	−	0.802381i	$-0.703566\pi$
$829$		−	46.2049i		−	1.60476i	0.596812	−	0.802381i	$-0.296434\pi$
$830$	41.2022			1.43015
$831$	−53.7629			−1.86502
$832$	−30.6894			−1.06396
$833$		0			0
$834$	4.99866			0.173090
$835$		−	5.28985i		−	0.183063i
$836$	1.93468	+	57.0601i	0.0669122	+	1.97346i
$837$	−8.74218			−0.302174
$838$	−28.0660			−0.969522
$839$			2.39354i			0.0826343i	0.999146	+	0.0413171i	$0.0131554\pi$
$839$			2.39354i			0.0826343i	−0.999146	+	0.0413171i	$0.986845\pi$
$840$		0			0
$841$	18.2884			0.630636
$842$			58.4066i			2.01283i
$843$	−22.2881			−0.767643
$844$			44.4146i			1.52881i
$845$		−	9.76682i		−	0.335989i
$846$	105.370			3.62270
$847$		0			0
$848$	−8.21377			−0.282062
$849$		−	9.68406i		−	0.332356i
$850$		−	13.6553i		−	0.468374i
$851$	−4.72202			−0.161869
$852$			96.0070i			3.28915i
$853$	−20.2353			−0.692844			−0.346422	−	0.938079i	$-0.612604\pi$
$853$	−20.2353			−0.692844			−0.346422	+	0.938079i	$0.612604\pi$
$854$		0			0
$855$			32.8613i			1.12383i
$856$	−87.5621			−2.99281
$857$	20.4074			0.697105			0.348552	−	0.937289i	$-0.386673\pi$
$857$	20.4074			0.697105			0.348552	+	0.937289i	$0.386673\pi$
$858$	3.53459	+	104.247i	0.120669	+	3.55893i
$859$			13.8282i			0.471812i	0.971776	+	0.235906i	$0.0758057\pi$
$859$			13.8282i			0.471812i	−0.971776	+	0.235906i	$0.924194\pi$
$860$	−106.937			−3.64652
$861$		0			0
$862$	24.5084			0.834759
$863$	−44.5210			−1.51551			−0.757756	−	0.652538i	$-0.773704\pi$
$863$	−44.5210			−1.51551			−0.757756	+	0.652538i	$0.773704\pi$
$864$	55.0989			1.87450
$865$		−	38.1351i		−	1.29663i
$866$	31.4826			1.06982
$867$			17.1848i			0.583625i
$868$		0			0
$869$	21.5009	−	0.729008i	0.729368	−	0.0247299i
$870$	44.4766			1.50790
$871$	−37.0723			−1.25615
$872$	−99.9561			−3.38494
$873$			15.4990i			0.524562i
$874$			46.6543i			1.57810i
$875$		0			0
$876$			126.545i			4.27554i
$877$		−	22.2139i		−	0.750111i	−0.927002	−	0.375056i	$-0.877624\pi$
$877$		−	22.2139i		−	0.750111i	0.927002	−	0.375056i	$-0.122376\pi$
$878$		−	100.773i		−	3.40092i
$879$		−	59.7910i		−	2.01670i
$880$	−57.7219	+	1.95712i	−1.94581	+	0.0659744i
$881$			44.2943i			1.49231i	0.665771	+	0.746157i	$0.268103\pi$
$881$			44.2943i			1.49231i	−0.665771	+	0.746157i	$0.731897\pi$
$882$		0			0
$883$	21.3173			0.717385			0.358692	−	0.933456i	$-0.383223\pi$
$883$	21.3173			0.717385			0.358692	+	0.933456i	$0.383223\pi$
$884$	−67.7610			−2.27905
$885$		−	34.8607i		−	1.17183i
$886$		−	35.1887i		−	1.18219i
$887$	36.3300			1.21984			0.609920	−	0.792463i	$-0.291201\pi$
$887$	36.3300			1.21984			0.609920	+	0.792463i	$0.291201\pi$
$888$	19.4829			0.653803
$889$		0			0
$890$			36.5258i			1.22435i
$891$	−0.0786698	−	2.32024i	−0.00263554	−	0.0777309i
$892$			83.1090i			2.78270i
$893$		−	29.2518i		−	0.978874i
$894$		−	63.4730i		−	2.12285i
$895$		−	44.5088i		−	1.48777i
$896$		0			0
$897$			60.1588i			2.00864i
$898$			61.7793i			2.06160i
$899$	5.18336			0.172875
$900$	37.7144			1.25715
$901$	−2.87848			−0.0958960
$902$	−15.3079	+	0.519030i	−0.509698	+	0.0172818i
$903$		0			0
$904$			68.5964i			2.28148i
$905$	5.73210			0.190541
$906$			69.3882i			2.30527i
$907$	−1.24612			−0.0413768			−0.0206884	−	0.999786i	$-0.506586\pi$
$907$	−1.24612			−0.0413768			−0.0206884	+	0.999786i	$0.506586\pi$
$908$	123.626			4.10267
$909$	18.3698			0.609289
$910$		0			0
$911$	12.1914			0.403920			0.201960	−	0.979394i	$-0.435269\pi$
$911$	12.1914			0.403920			0.201960	+	0.979394i	$0.435269\pi$
$912$		−	95.3787i		−	3.15831i
$913$	0.961190	+	28.3487i	0.0318108	+	0.938206i
$914$	−37.3170			−1.23434
$915$	−24.3898			−0.806302
$916$		−	37.3651i		−	1.23458i
$917$		0			0
$918$	47.5326			1.56881
$919$		−	29.5951i		−	0.976252i	−0.872773	−	0.488126i	$-0.837681\pi$
$919$		−	29.5951i		−	0.976252i	0.872773	−	0.488126i	$-0.162319\pi$
$920$	−67.2268			−2.21640
$921$			41.4570i			1.36605i
$922$		−	67.0078i		−	2.20678i
$923$	30.3343			0.998466
$924$		0			0
$925$	1.50142			0.0493662
$926$			78.2763i			2.57232i
$927$		−	85.0094i		−	2.79208i
$928$	−32.6689			−1.07241
$929$		−	10.4932i		−	0.344272i	−0.985073	−	0.172136i	$-0.944933\pi$
$929$		−	10.4932i		−	0.344272i	0.985073	−	0.172136i	$-0.0550668\pi$
$930$	−21.5224			−0.705747
$931$		0			0
$932$		−	57.1109i		−	1.87073i
$933$	23.4651			0.768214
$934$	−78.7446			−2.57660
$935$	−20.2284	+	0.685863i	−0.661539	+	0.0224301i
$936$		−	154.629i		−	5.05420i
$937$	−22.1577			−0.723861			−0.361930	−	0.932205i	$-0.617882\pi$
$937$	−22.1577			−0.723861			−0.361930	+	0.932205i	$0.617882\pi$
$938$		0			0
$939$	45.3338			1.47941
$940$	72.2804			2.35753
$941$	10.3877			0.338630			0.169315	−	0.985562i	$-0.445844\pi$
$941$	10.3877			0.338630			0.169315	+	0.985562i	$0.445844\pi$
$942$			128.072i			4.17281i
$943$	−8.83390			−0.287671
$944$			63.0330i			2.05155i
$945$		0			0
$946$	−3.53459	−	104.247i	−0.114919	−	3.38936i
$947$	38.4477			1.24938			0.624691	−	0.780872i	$-0.285225\pi$
$947$	38.4477			1.24938			0.624691	+	0.780872i	$0.285225\pi$
$948$	−87.7880			−2.85122
$949$	39.9829			1.29790
$950$		−	14.8342i		−	0.481286i
$951$		−	47.2262i		−	1.53141i
$952$		0			0
$953$			1.18153i			0.0382735i	0.999817	+	0.0191367i	$0.00609178\pi$
$953$			1.18153i			0.0382735i	−0.999817	+	0.0191367i	$0.993908\pi$
$954$		−	11.2639i		−	0.364683i
$955$		−	12.6173i		−	0.408286i
$956$			31.4058i			1.01574i
$957$	1.03758	+	30.6016i	0.0335401	+	0.989210i
$958$		−	79.3906i		−	2.56499i
$959$		0			0
$960$	37.4066			1.20729
$961$	28.4918			0.919089
$962$		−	10.5560i		−	0.340340i
$963$		−	59.4975i		−	1.91728i
$964$	−114.025			−3.67251
$965$	−13.4716			−0.433666
$966$		0			0
$967$		−	38.1782i		−	1.22773i	−0.789411	−	0.613865i	$-0.789614\pi$
$967$		−	38.1782i		−	1.22773i	0.789411	−	0.613865i	$-0.210386\pi$
$968$	−5.43532	−	80.0608i	−0.174698	−	2.57325i
$969$		−	33.4250i		−	1.07377i
$970$			15.0636i			0.483664i
$971$		−	29.5171i		−	0.947250i	−0.880727	−	0.473625i	$-0.842945\pi$
$971$		−	29.5171i		−	0.947250i	0.880727	−	0.473625i	$-0.157055\pi$
$972$		−	69.9794i		−	2.24459i
$973$		0			0
$974$			3.35411i			0.107473i
$975$		−	19.1281i		−	0.612590i
$976$	44.1002			1.41161
$977$	−2.44499			−0.0782223			−0.0391112	−	0.999235i	$-0.512453\pi$
$977$	−2.44499			−0.0782223			−0.0391112	+	0.999235i	$0.512453\pi$
$978$	116.319			3.71947
$979$	−25.1312	+	0.852097i	−0.803197	+	0.0272331i
$980$		0			0
$981$		−	67.9190i		−	2.16849i
$982$	−10.5884			−0.337888
$983$		−	6.62529i		−	0.211314i	−0.994403	−	0.105657i	$-0.966305\pi$
$983$		−	6.62529i		−	0.211314i	0.994403	−	0.105657i	$-0.0336946\pi$
$984$	36.4483			1.16193
$985$	8.55237			0.272501
$986$	−28.1828			−0.897522
$987$		0			0
$988$	−73.6109			−2.34188
$989$		−	60.1588i		−	1.91294i
$990$	−2.68389	−	79.1569i	−0.0852996	−	2.51577i
$991$	8.97768			0.285185			0.142593	−	0.989781i	$-0.454456\pi$
$991$	8.97768			0.285185			0.142593	+	0.989781i	$0.454456\pi$
$992$	15.8086			0.501924
$993$		−	63.8739i		−	2.02698i
$994$		0			0
$995$	−16.2942			−0.516562
$996$		−	115.748i		−	3.66760i
$997$	37.0824			1.17441			0.587206	−	0.809438i	$-0.300228\pi$
$997$	37.0824			1.17441			0.587206	+	0.809438i	$0.300228\pi$
$998$			49.6324i			1.57108i
$999$			5.22625i			0.165351i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.b.c.538.1	✓	16
7.2	even	3		539.2.i.e.472.2		32
7.3	odd	6		539.2.i.e.362.16		32
7.4	even	3		539.2.i.e.362.15		32
7.5	odd	6		539.2.i.e.472.1		32
7.6	odd	2	inner	539.2.b.c.538.2	yes	16
11.10	odd	2	inner	539.2.b.c.538.15	yes	16
77.10	even	6		539.2.i.e.362.2		32
77.32	odd	6		539.2.i.e.362.1		32
77.54	even	6		539.2.i.e.472.15		32
77.65	odd	6		539.2.i.e.472.16		32
77.76	even	2	inner	539.2.b.c.538.16	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.b.c.538.1	✓	16	1.1	even	1	trivial
539.2.b.c.538.2	yes	16	7.6	odd	2	inner
539.2.b.c.538.15	yes	16	11.10	odd	2	inner
539.2.b.c.538.16	yes	16	77.76	even	2	inner
539.2.i.e.362.1		32	77.32	odd	6
539.2.i.e.362.2		32	77.10	even	6
539.2.i.e.362.15		32	7.4	even	3
539.2.i.e.362.16		32	7.3	odd	6
539.2.i.e.472.1		32	7.5	odd	6
539.2.i.e.472.2		32	7.2	even	3
539.2.i.e.472.15		32	77.54	even	6
539.2.i.e.472.16		32	77.65	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.60728i q^{2} -2.82079i q^{3} -4.79793 q^{4} -1.84776i q^{5} -7.35461 q^{6} +7.29501i q^{8} -4.95687 q^{9} +O(q^{10})\)	\(q-2.60728i q^{2} -2.82079i q^{3} -4.79793 q^{4} -1.84776i q^{5} -7.35461 q^{6} +7.29501i q^{8} -4.95687 q^{9} -4.81763 q^{10} +(-0.112389 - 3.31472i) q^{11} +13.5340i q^{12} +4.27619 q^{13} -5.21215 q^{15} +9.42429 q^{16} +3.30270 q^{17} +12.9240i q^{18} +3.58783 q^{19} +8.86542i q^{20} +(-8.64242 + 0.293029i) q^{22} -4.98737 q^{23} +20.5777 q^{24} +1.58579 q^{25} -11.1492i q^{26} +5.51994i q^{27} -3.27285i q^{29} +13.5895i q^{30} +1.58375i q^{31} -9.98180i q^{32} +(-9.35014 + 0.317025i) q^{33} -8.61108i q^{34} +23.7827 q^{36} +0.946795 q^{37} -9.35449i q^{38} -12.0622i q^{39} +13.4794 q^{40} +1.77125 q^{41} +12.0622i q^{43} +(0.539234 + 15.9038i) q^{44} +9.15911i q^{45} +13.0035i q^{46} -8.15307i q^{47} -26.5840i q^{48} -4.13460i q^{50} -9.31623i q^{51} -20.5169 q^{52} -0.871553 q^{53} +14.3920 q^{54} +(-6.12480 + 0.207667i) q^{55} -10.1205i q^{57} -8.53325 q^{58} +6.68835i q^{59} +25.0075 q^{60} +4.67942 q^{61} +4.12927 q^{62} -7.17680 q^{64} -7.90136i q^{65} +(0.826575 + 24.3785i) q^{66} -8.66949 q^{67} -15.8461 q^{68} +14.0683i q^{69} +7.09378 q^{71} -36.1604i q^{72} +9.35014 q^{73} -2.46857i q^{74} -4.47318i q^{75} -17.2142 q^{76} -31.4497 q^{78} +6.48649i q^{79} -17.4138i q^{80} +0.699980 q^{81} -4.61816i q^{82} -8.55237 q^{83} -6.10259i q^{85} +31.4497 q^{86} -9.23204 q^{87} +(24.1809 - 0.819876i) q^{88} -7.58170i q^{89} +23.8804 q^{90} +23.9291 q^{92} +4.46742 q^{93} -21.2574 q^{94} -6.62944i q^{95} -28.1566 q^{96} -3.12677i q^{97} +(0.557097 + 16.4307i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} - 16 q^{9}+O(q^{10}) \) 16 * q - 32 * q^4 - 16 * q^9	\( 16 q - 32 q^{4} - 16 q^{9} - 16 q^{15} + 16 q^{16} - 32 q^{22} - 16 q^{23} + 48 q^{25} + 96 q^{36} + 64 q^{37} + 48 q^{44} - 32 q^{53} - 80 q^{58} + 112 q^{60} - 32 q^{64} - 112 q^{67} - 48 q^{71} - 16 q^{78} - 16 q^{81} + 16 q^{86} + 96 q^{88} + 16 q^{92} - 16 q^{99}+O(q^{100}) \) 16 * q - 32 * q^4 - 16 * q^9 - 16 * q^15 + 16 * q^16 - 32 * q^22 - 16 * q^23 + 48 * q^25 + 96 * q^36 + 64 * q^37 + 48 * q^44 - 32 * q^53 - 80 * q^58 + 112 * q^60 - 32 * q^64 - 112 * q^67 - 48 * q^71 - 16 * q^78 - 16 * q^81 + 16 * q^86 + 96 * q^88 + 16 * q^92 - 16 * q^99

Introduction

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