LMFDB - Embedded newform 960.2.h.g.191.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,2,Mod(191,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("960.191");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$960 = 2^{6} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	960.h (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$7.66563859404$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.342102016.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16$ x^8 + x^6 + 4x^4 + 4x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{4}$
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.8
Root			$1.17915 + 0.780776i$ of defining polynomial
Character	$\chi$	$=$	960.191
Dual form			960.2.h.g.191.7

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.51022 + 0.848071i) q^{3} -1.00000i q^{5} +3.02045i q^{7} +(1.56155 + 2.56155i) q^{9} -1.32431 q^{11} +5.12311 q^{13} +(0.848071 - 1.51022i) q^{15} -2.00000i q^{17} +1.32431i q^{19} +(-2.56155 + 4.56155i) q^{21} +0.371834 q^{23} -1.00000 q^{25} +(0.185917 + 5.19283i) q^{27} +3.12311i q^{29} +4.71659i q^{31} +(-2.00000 - 1.12311i) q^{33} +3.02045 q^{35} -5.12311 q^{37} +(7.73704 + 4.34475i) q^{39} +1.12311i q^{41} +7.73704i q^{43} +(2.56155 - 1.56155i) q^{45} +3.02045 q^{47} -2.12311 q^{49} +(1.69614 - 3.02045i) q^{51} -12.2462i q^{53} +1.32431i q^{55} +(-1.12311 + 2.00000i) q^{57} +14.1498 q^{59} -3.12311 q^{61} +(-7.73704 + 4.71659i) q^{63} -5.12311i q^{65} -4.34475i q^{67} +(0.561553 + 0.315342i) q^{69} +3.39228 q^{71} +8.24621 q^{73} +(-1.51022 - 0.848071i) q^{75} -4.00000i q^{77} -8.10887i q^{79} +(-4.12311 + 8.00000i) q^{81} -15.1022 q^{83} -2.00000 q^{85} +(-2.64861 + 4.71659i) q^{87} -10.2462i q^{89} +15.4741i q^{91} +(-4.00000 + 7.12311i) q^{93} +1.32431 q^{95} -6.00000 q^{97} +(-2.06798 - 3.39228i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 4 q^{9} + 8 q^{13} - 4 q^{21} - 8 q^{25} - 16 q^{33} - 8 q^{37} + 4 q^{45} + 16 q^{49} + 24 q^{57} + 8 q^{61} - 12 q^{69} - 16 q^{85} - 32 q^{93} - 48 q^{97}+O(q^{100})$ 8 * q - 4 * q^9 + 8 * q^13 - 4 * q^21 - 8 * q^25 - 16 * q^33 - 8 * q^37 + 4 * q^45 + 16 * q^49 + 24 * q^57 + 8 * q^61 - 12 * q^69 - 16 * q^85 - 32 * q^93 - 48 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$ .

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$-1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$		0			0
$2$		0			0
$3$	1.51022	+	0.848071i	0.871928	+	0.489634i
$3$	1.51022	+	0.848071i	0.871928	+	0.489634i
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$			3.02045i			1.14162i	0.821081	+	0.570811i	$0.193371\pi$
$7$			3.02045i			1.14162i	−0.821081	+	0.570811i	$0.806629\pi$
$8$		0			0
$9$	1.56155	+	2.56155i	0.520518	+	0.853851i
$10$		0			0
$11$	−1.32431			−0.399294			−0.199647	−	0.979868i	$-0.563979\pi$
$11$	−1.32431			−0.399294			−0.199647	+	0.979868i	$0.563979\pi$
$12$		0			0
$13$	5.12311			1.42089			0.710447	−	0.703751i	$-0.248493\pi$
$13$	5.12311			1.42089			0.710447	+	0.703751i	$0.248493\pi$
$14$		0			0
$15$	0.848071	−	1.51022i	0.218971	−	0.389938i
$16$		0			0
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$		0			0
$19$			1.32431i			0.303817i	0.988395	+	0.151908i	$0.0485419\pi$
$19$			1.32431i			0.303817i	−0.988395	+	0.151908i	$0.951458\pi$
$20$		0			0
$21$	−2.56155	+	4.56155i	−0.558977	+	0.995412i
$22$		0			0
$23$	0.371834			0.0775328			0.0387664	−	0.999248i	$-0.487657\pi$
$23$	0.371834			0.0775328			0.0387664	+	0.999248i	$0.487657\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	0.185917	+	5.19283i	0.0357798	+	0.999360i
$28$		0			0
$29$			3.12311i			0.579946i	0.957035	+	0.289973i	$0.0936464\pi$
$29$			3.12311i			0.579946i	−0.957035	+	0.289973i	$0.906354\pi$
$30$		0			0
$31$			4.71659i			0.847124i	0.905867	+	0.423562i	$0.139220\pi$
$31$			4.71659i			0.847124i	−0.905867	+	0.423562i	$0.860780\pi$
$32$		0			0
$33$	−2.00000	−	1.12311i	−0.348155	−	0.195508i
$34$		0			0
$35$	3.02045			0.510549
$36$		0			0
$37$	−5.12311			−0.842233			−0.421117	−	0.907006i	$-0.638362\pi$
$37$	−5.12311			−0.842233			−0.421117	+	0.907006i	$0.638362\pi$
$38$		0			0
$39$	7.73704	+	4.34475i	1.23892	+	0.695718i
$40$		0			0
$41$			1.12311i			0.175400i	0.996147	+	0.0876998i	$0.0279516\pi$
$41$			1.12311i			0.175400i	−0.996147	+	0.0876998i	$0.972048\pi$
$42$		0			0
$43$			7.73704i			1.17989i	0.807445	+	0.589944i	$0.200850\pi$
$43$			7.73704i			1.17989i	−0.807445	+	0.589944i	$0.799150\pi$
$44$		0			0
$45$	2.56155	−	1.56155i	0.381854	−	0.232783i
$46$		0			0
$47$	3.02045			0.440578			0.220289	−	0.975435i	$-0.429300\pi$
$47$	3.02045			0.440578			0.220289	+	0.975435i	$0.429300\pi$
$48$		0			0
$49$	−2.12311			−0.303301
$50$		0			0
$51$	1.69614	−	3.02045i	0.237507	−	0.422947i
$52$		0			0
$53$		−	12.2462i		−	1.68215i	−0.540921	−	0.841073i	$-0.681924\pi$
$53$		−	12.2462i		−	1.68215i	0.540921	−	0.841073i	$-0.318076\pi$
$54$		0			0
$55$			1.32431i			0.178570i
$56$		0			0
$57$	−1.12311	+	2.00000i	−0.148759	+	0.264906i
$58$		0			0
$59$	14.1498			1.84214			0.921071	−	0.389394i	$-0.127315\pi$
$59$	14.1498			1.84214			0.921071	+	0.389394i	$0.127315\pi$
$60$		0			0
$61$	−3.12311			−0.399873			−0.199936	−	0.979809i	$-0.564074\pi$
$61$	−3.12311			−0.399873			−0.199936	+	0.979809i	$0.564074\pi$
$62$		0			0
$63$	−7.73704	+	4.71659i	−0.974775	+	0.594234i
$64$		0			0
$65$		−	5.12311i		−	0.635443i
$66$		0			0
$67$		−	4.34475i		−	0.530796i	−0.964139	−	0.265398i	$-0.914497\pi$
$67$		−	4.34475i		−	0.530796i	0.964139	−	0.265398i	$-0.0855034\pi$
$68$		0			0
$69$	0.561553	+	0.315342i	0.0676030	+	0.0379627i
$70$		0			0
$71$	3.39228			0.402590			0.201295	−	0.979531i	$-0.435485\pi$
$71$	3.39228			0.402590			0.201295	+	0.979531i	$0.435485\pi$
$72$		0			0
$73$	8.24621			0.965146			0.482573	−	0.875856i	$-0.339702\pi$
$73$	8.24621			0.965146			0.482573	+	0.875856i	$0.339702\pi$
$74$		0			0
$75$	−1.51022	−	0.848071i	−0.174386	−	0.0979267i
$76$		0			0
$77$		−	4.00000i		−	0.455842i
$78$		0			0
$79$		−	8.10887i		−	0.912319i	−0.889898	−	0.456160i	$-0.849225\pi$
$79$		−	8.10887i		−	0.912319i	0.889898	−	0.456160i	$-0.150775\pi$
$80$		0			0
$81$	−4.12311	+	8.00000i	−0.458123	+	0.888889i
$82$		0			0
$83$	−15.1022			−1.65769			−0.828843	−	0.559481i	$-0.811000\pi$
$83$	−15.1022			−1.65769			−0.828843	+	0.559481i	$0.811000\pi$
$84$		0			0
$85$	−2.00000			−0.216930
$86$		0			0
$87$	−2.64861	+	4.71659i	−0.283961	+	0.505671i
$88$		0			0
$89$		−	10.2462i		−	1.08610i	−0.839702	−	0.543048i	$-0.817270\pi$
$89$		−	10.2462i		−	1.08610i	0.839702	−	0.543048i	$-0.182730\pi$
$90$		0			0
$91$			15.4741i			1.62212i
$92$		0			0
$93$	−4.00000	+	7.12311i	−0.414781	+	0.738632i
$94$		0			0
$95$	1.32431			0.135871
$96$		0			0
$97$	−6.00000			−0.609208			−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000			−0.609208			−0.304604	+	0.952479i	$0.598524\pi$
$98$		0			0
$99$	−2.06798	−	3.39228i	−0.207839	−	0.340937i
$100$		0			0
$101$		−	0.876894i		−	0.0872543i	−0.999048	−	0.0436271i	$-0.986109\pi$
$101$		−	0.876894i		−	0.0872543i	0.999048	−	0.0436271i	$-0.0138914\pi$
$102$		0			0
$103$			9.80501i			0.966117i	0.875588	+	0.483058i	$0.160474\pi$
$103$			9.80501i			0.966117i	−0.875588	+	0.483058i	$0.839526\pi$
$104$		0			0
$105$	4.56155	+	2.56155i	0.445162	+	0.249982i
$106$		0			0
$107$	3.02045			0.291998			0.145999	−	0.989285i	$-0.453360\pi$
$107$	3.02045			0.291998			0.145999	+	0.989285i	$0.453360\pi$
$108$		0			0
$109$	0.876894			0.0839912			0.0419956	−	0.999118i	$-0.486628\pi$
$109$	0.876894			0.0839912			0.0419956	+	0.999118i	$0.486628\pi$
$110$		0			0
$111$	−7.73704	−	4.34475i	−0.734367	−	0.412386i
$112$		0			0
$113$		−	14.0000i		−	1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$		−	14.0000i		−	1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$		0			0
$115$		−	0.371834i		−	0.0346737i
$116$		0			0
$117$	8.00000	+	13.1231i	0.739600	+	1.21323i
$118$		0			0
$119$	6.04090			0.553768
$120$		0			0
$121$	−9.24621			−0.840565
$122$		0			0
$123$	−0.952473	+	1.69614i	−0.0858816	+	0.152936i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$		−	15.1022i		−	1.34011i	−0.742313	−	0.670054i	$-0.766271\pi$
$127$		−	15.1022i		−	1.34011i	0.742313	−	0.670054i	$-0.233729\pi$
$128$		0			0
$129$	−6.56155	+	11.6847i	−0.577713	+	1.02878i
$130$		0			0
$131$	−5.46026			−0.477065			−0.238532	−	0.971135i	$-0.576666\pi$
$131$	−5.46026			−0.477065			−0.238532	+	0.971135i	$0.576666\pi$
$132$		0			0
$133$	−4.00000			−0.346844
$134$		0			0
$135$	5.19283	−	0.185917i	0.446927	−	0.0160012i
$136$		0			0
$137$			8.24621i			0.704521i	0.935902	+	0.352261i	$0.114587\pi$
$137$			8.24621i			0.704521i	−0.935902	+	0.352261i	$0.885413\pi$
$138$		0			0
$139$		−	17.5420i		−	1.48790i	−0.668237	−	0.743949i	$-0.732951\pi$
$139$		−	17.5420i		−	1.48790i	0.668237	−	0.743949i	$-0.267049\pi$
$140$		0			0
$141$	4.56155	+	2.56155i	0.384152	+	0.215722i
$142$		0			0
$143$	−6.78456			−0.567354
$144$		0			0
$145$	3.12311			0.259360
$146$		0			0
$147$	−3.20636	−	1.80054i	−0.264457	−	0.148506i
$148$		0			0
$149$		−	14.0000i		−	1.14692i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$		−	14.0000i		−	1.14692i	0.819232	−	0.573462i	$-0.194400\pi$
$150$		0			0
$151$			7.36520i			0.599372i	0.954038	+	0.299686i	$0.0968819\pi$
$151$			7.36520i			0.599372i	−0.954038	+	0.299686i	$0.903118\pi$
$152$		0			0
$153$	5.12311	−	3.12311i	0.414179	−	0.252488i
$154$		0			0
$155$	4.71659			0.378846
$156$		0			0
$157$	3.36932			0.268901			0.134450	−	0.990920i	$-0.457073\pi$
$157$	3.36932			0.268901			0.134450	+	0.990920i	$0.457073\pi$
$158$		0			0
$159$	10.3857	−	18.4945i	0.823636	−	1.46671i
$160$		0			0
$161$			1.12311i			0.0885131i
$162$		0			0
$163$		−	15.6829i		−	1.22838i	−0.789159	−	0.614189i	$-0.789483\pi$
$163$		−	15.6829i		−	1.22838i	0.789159	−	0.614189i	$-0.210517\pi$
$164$		0			0
$165$	−1.12311	+	2.00000i	−0.0874337	+	0.155700i
$166$		0			0
$167$	−9.06134			−0.701188			−0.350594	−	0.936528i	$-0.614020\pi$
$167$	−9.06134			−0.701188			−0.350594	+	0.936528i	$0.614020\pi$
$168$		0			0
$169$	13.2462			1.01894
$170$		0			0
$171$	−3.39228	+	2.06798i	−0.259414	+	0.158142i
$172$		0			0
$173$			2.00000i			0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$			2.00000i			0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$		0			0
$175$		−	3.02045i		−	0.228324i
$176$		0			0
$177$	21.3693	+	12.0000i	1.60622	+	0.901975i
$178$		0			0
$179$	10.0138			0.748468			0.374234	−	0.927334i	$-0.377906\pi$
$179$	10.0138			0.748468			0.374234	+	0.927334i	$0.377906\pi$
$180$		0			0
$181$	12.2462			0.910254			0.455127	−	0.890427i	$-0.349594\pi$
$181$	12.2462			0.910254			0.455127	+	0.890427i	$0.349594\pi$
$182$		0			0
$183$	−4.71659	−	2.64861i	−0.348660	−	0.195791i
$184$		0			0
$185$			5.12311i			0.376658i
$186$		0			0
$187$			2.64861i			0.193686i
$188$		0			0
$189$	−15.6847	+	0.561553i	−1.14089	+	0.0408470i
$190$		0			0
$191$	−24.9073			−1.80223			−0.901113	−	0.433585i	$-0.857248\pi$
$191$	−24.9073			−1.80223			−0.901113	+	0.433585i	$0.857248\pi$
$192$		0			0
$193$	−0.246211			−0.0177227			−0.00886134	−	0.999961i	$-0.502821\pi$
$193$	−0.246211			−0.0177227			−0.00886134	+	0.999961i	$0.502821\pi$
$194$		0			0
$195$	4.34475	−	7.73704i	0.311134	−	0.554061i
$196$		0			0
$197$			4.24621i			0.302530i	0.988493	+	0.151265i	$0.0483347\pi$
$197$			4.24621i			0.302530i	−0.988493	+	0.151265i	$0.951665\pi$
$198$		0			0
$199$		−	5.46026i		−	0.387067i	−0.981094	−	0.193534i	$-0.938005\pi$
$199$		−	5.46026i		−	0.387067i	0.981094	−	0.193534i	$-0.0619949\pi$
$200$		0			0
$201$	3.68466	−	6.56155i	0.259896	−	0.462816i
$202$		0			0
$203$	−9.43318			−0.662079
$204$		0			0
$205$	1.12311			0.0784411
$206$		0			0
$207$	0.580639	+	0.952473i	0.0403572	+	0.0662014i
$208$		0			0
$209$		−	1.75379i		−	0.121312i
$210$		0			0
$211$		−	16.7984i		−	1.15645i	−0.815878	−	0.578224i	$-0.803746\pi$
$211$		−	16.7984i		−	1.15645i	0.815878	−	0.578224i	$-0.196254\pi$
$212$		0			0
$213$	5.12311	+	2.87689i	0.351029	+	0.197122i
$214$		0			0
$215$	7.73704			0.527662
$216$		0			0
$217$	−14.2462			−0.967096
$218$		0			0
$219$	12.4536	+	6.99337i	0.841538	+	0.472568i
$220$		0			0
$221$		−	10.2462i		−	0.689235i
$222$		0			0
$223$		−	8.31768i		−	0.556993i	−0.960437	−	0.278496i	$-0.910164\pi$
$223$		−	8.31768i		−	0.556993i	0.960437	−	0.278496i	$-0.0898360\pi$
$224$		0			0
$225$	−1.56155	−	2.56155i	−0.104104	−	0.170770i
$226$		0			0
$227$	−21.8868			−1.45268			−0.726339	−	0.687337i	$-0.758780\pi$
$227$	−21.8868			−1.45268			−0.726339	+	0.687337i	$0.758780\pi$
$228$		0			0
$229$	16.2462			1.07358			0.536790	−	0.843716i	$-0.319637\pi$
$229$	16.2462			1.07358			0.536790	+	0.843716i	$0.319637\pi$
$230$		0			0
$231$	3.39228	−	6.04090i	0.223196	−	0.397462i
$232$		0			0
$233$			10.0000i			0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$			10.0000i			0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$		0			0
$235$		−	3.02045i		−	0.197032i
$236$		0			0
$237$	6.87689	−	12.2462i	0.446702	−	0.795477i
$238$		0			0
$239$	17.3790			1.12416			0.562078	−	0.827084i	$-0.310002\pi$
$239$	17.3790			1.12416			0.562078	+	0.827084i	$0.310002\pi$
$240$		0			0
$241$	13.3693			0.861193			0.430597	−	0.902544i	$-0.358303\pi$
$241$	13.3693			0.861193			0.430597	+	0.902544i	$0.358303\pi$
$242$		0			0
$243$	−13.0114	+	8.58511i	−0.834680	+	0.550735i
$244$		0			0
$245$			2.12311i			0.135640i
$246$		0			0
$247$			6.78456i			0.431691i
$248$		0			0
$249$	−22.8078	−	12.8078i	−1.44538	−	0.811659i
$250$		0			0
$251$	−18.7033			−1.18054			−0.590272	−	0.807205i	$-0.700979\pi$
$251$	−18.7033			−1.18054			−0.590272	+	0.807205i	$0.700979\pi$
$252$		0			0
$253$	−0.492423			−0.0309583
$254$		0			0
$255$	−3.02045	−	1.69614i	−0.189148	−	0.106216i
$256$		0			0
$257$		−	30.4924i		−	1.90207i	−0.309091	−	0.951033i	$-0.600025\pi$
$257$		−	30.4924i		−	1.90207i	0.309091	−	0.951033i	$-0.399975\pi$
$258$		0			0
$259$		−	15.4741i		−	0.961512i
$260$		0			0
$261$	−8.00000	+	4.87689i	−0.495188	+	0.301872i
$262$		0			0
$263$	−23.7917			−1.46706			−0.733531	−	0.679656i	$-0.762129\pi$
$263$	−23.7917			−1.46706			−0.733531	+	0.679656i	$0.762129\pi$
$264$		0			0
$265$	−12.2462			−0.752279
$266$		0			0
$267$	8.68951	−	15.4741i	0.531789	−	0.946998i
$268$		0			0
$269$			14.0000i			0.853595i	0.904347	+	0.426798i	$0.140358\pi$
$269$			14.0000i			0.853595i	−0.904347	+	0.426798i	$0.859642\pi$
$270$		0			0
$271$		−	15.3110i		−	0.930080i	−0.885290	−	0.465040i	$-0.846040\pi$
$271$		−	15.3110i		−	0.930080i	0.885290	−	0.465040i	$-0.153960\pi$
$272$		0			0
$273$	−13.1231	+	23.3693i	−0.794246	+	1.41438i
$274$		0			0
$275$	1.32431			0.0798587
$276$		0			0
$277$	−23.3693			−1.40413			−0.702063	−	0.712115i	$-0.747738\pi$
$277$	−23.3693			−1.40413			−0.702063	+	0.712115i	$0.747738\pi$
$278$		0			0
$279$	−12.0818	+	7.36520i	−0.723318	+	0.440943i
$280$		0			0
$281$		−	13.6155i		−	0.812234i	−0.913821	−	0.406117i	$-0.866882\pi$
$281$		−	13.6155i		−	0.812234i	0.913821	−	0.406117i	$-0.133118\pi$
$282$		0			0
$283$			23.2111i			1.37976i	0.723925	+	0.689879i	$0.242336\pi$
$283$			23.2111i			1.37976i	−0.723925	+	0.689879i	$0.757664\pi$
$284$		0			0
$285$	2.00000	+	1.12311i	0.118470	+	0.0665270i
$286$		0			0
$287$	−3.39228			−0.200240
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$	−9.06134	−	5.08842i	−0.531185	−	0.298289i
$292$		0			0
$293$			2.49242i			0.145609i	0.997346	+	0.0728044i	$0.0231949\pi$
$293$			2.49242i			0.145609i	−0.997346	+	0.0728044i	$0.976805\pi$
$294$		0			0
$295$		−	14.1498i		−	0.823831i
$296$		0			0
$297$	−0.246211	−	6.87689i	−0.0142866	−	0.399038i
$298$		0			0
$299$	1.90495			0.110166
$300$		0			0
$301$	−23.3693			−1.34699
$302$		0			0
$303$	0.743668	−	1.32431i	0.0427226	−	0.0760794i
$304$		0			0
$305$			3.12311i			0.178829i
$306$		0			0
$307$			11.1293i			0.635184i	0.948227	+	0.317592i	$0.102874\pi$
$307$			11.1293i			0.635184i	−0.948227	+	0.317592i	$0.897126\pi$
$308$		0			0
$309$	−8.31534	+	14.8078i	−0.473043	+	0.842384i
$310$		0			0
$311$	20.7713			1.17783			0.588916	−	0.808194i	$-0.299555\pi$
$311$	20.7713			1.17783			0.588916	+	0.808194i	$0.299555\pi$
$312$		0			0
$313$	−22.4924			−1.27135			−0.635673	−	0.771958i	$-0.719278\pi$
$313$	−22.4924			−1.27135			−0.635673	+	0.771958i	$0.719278\pi$
$314$		0			0
$315$	4.71659	+	7.73704i	0.265750	+	0.435933i
$316$		0			0
$317$		−	16.7386i		−	0.940135i	−0.882630	−	0.470068i	$-0.844230\pi$
$317$		−	16.7386i		−	0.940135i	0.882630	−	0.470068i	$-0.155770\pi$
$318$		0			0
$319$		−	4.13595i		−	0.231569i
$320$		0			0
$321$	4.56155	+	2.56155i	0.254601	+	0.142972i
$322$		0			0
$323$	2.64861			0.147373
$324$		0			0
$325$	−5.12311			−0.284179
$326$		0			0
$327$	1.32431	+	0.743668i	0.0732343	+	0.0411249i
$328$		0			0
$329$			9.12311i			0.502973i
$330$		0			0
$331$		−	3.22925i		−	0.177496i	−0.996054	−	0.0887479i	$-0.971713\pi$
$331$		−	3.22925i		−	0.177496i	0.996054	−	0.0887479i	$-0.0282865\pi$
$332$		0			0
$333$	−8.00000	−	13.1231i	−0.438397	−	0.719142i
$334$		0			0
$335$	−4.34475			−0.237379
$336$		0			0
$337$	−1.50758			−0.0821230			−0.0410615	−	0.999157i	$-0.513074\pi$
$337$	−1.50758			−0.0821230			−0.0410615	+	0.999157i	$0.513074\pi$
$338$		0			0
$339$	11.8730	−	21.1431i	0.644852	−	1.14834i
$340$		0			0
$341$		−	6.24621i		−	0.338251i
$342$		0			0
$343$			14.7304i			0.795367i
$344$		0			0
$345$	0.315342	−	0.561553i	0.0169774	−	0.0302330i
$346$		0			0
$347$	22.6305			1.21487			0.607434	−	0.794370i	$-0.292199\pi$
$347$	22.6305			1.21487			0.607434	+	0.794370i	$0.292199\pi$
$348$		0			0
$349$	14.0000			0.749403			0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000			0.749403			0.374701	+	0.927146i	$0.377745\pi$
$350$		0			0
$351$	0.952473	+	26.6034i	0.0508392	+	1.41998i
$352$		0			0
$353$			20.2462i			1.07760i	0.842435	+	0.538799i	$0.181122\pi$
$353$			20.2462i			1.07760i	−0.842435	+	0.538799i	$0.818878\pi$
$354$		0			0
$355$		−	3.39228i		−	0.180044i
$356$		0			0
$357$	9.12311	+	5.12311i	0.482846	+	0.271144i
$358$		0			0
$359$	21.5150			1.13552			0.567758	−	0.823195i	$-0.307811\pi$
$359$	21.5150			1.13552			0.567758	+	0.823195i	$0.307811\pi$
$360$		0			0
$361$	17.2462			0.907695
$362$		0			0
$363$	−13.9638	−	7.84144i	−0.732912	−	0.411569i
$364$		0			0
$365$		−	8.24621i		−	0.431626i
$366$		0			0
$367$			10.9663i			0.572436i	0.958165	+	0.286218i	$0.0923981\pi$
$367$			10.9663i			0.572436i	−0.958165	+	0.286218i	$0.907602\pi$
$368$		0			0
$369$	−2.87689	+	1.75379i	−0.149765	+	0.0912986i
$370$		0			0
$371$	36.9890			1.92038
$372$		0			0
$373$	9.12311			0.472377			0.236188	−	0.971707i	$-0.424102\pi$
$373$	9.12311			0.472377			0.236188	+	0.971707i	$0.424102\pi$
$374$		0			0
$375$	−0.848071	+	1.51022i	−0.0437942	+	0.0779876i
$376$		0			0
$377$			16.0000i			0.824042i
$378$		0			0
$379$			18.7033i			0.960725i	0.877070	+	0.480363i	$0.159495\pi$
$379$			18.7033i			0.960725i	−0.877070	+	0.480363i	$0.840505\pi$
$380$		0			0
$381$	12.8078	−	22.8078i	0.656162	−	1.16848i
$382$		0			0
$383$	−15.1022			−0.771688			−0.385844	−	0.922564i	$-0.626090\pi$
$383$	−15.1022			−0.771688			−0.385844	+	0.922564i	$0.626090\pi$
$384$		0			0
$385$	−4.00000			−0.203859
$386$		0			0
$387$	−19.8188	+	12.0818i	−1.00745	+	0.614152i
$388$		0			0
$389$		−	20.7386i		−	1.05149i	−0.850642	−	0.525745i	$-0.823787\pi$
$389$		−	20.7386i		−	1.05149i	0.850642	−	0.525745i	$-0.176213\pi$
$390$		0			0
$391$		−	0.743668i		−	0.0376089i
$392$		0			0
$393$	−8.24621	−	4.63068i	−0.415966	−	0.233587i
$394$		0			0
$395$	−8.10887			−0.408002
$396$		0			0
$397$	−14.8769			−0.746650			−0.373325	−	0.927701i	$-0.621782\pi$
$397$	−14.8769			−0.746650			−0.373325	+	0.927701i	$0.621782\pi$
$398$		0			0
$399$	−6.04090	−	3.39228i	−0.302423	−	0.169827i
$400$		0			0
$401$		−	24.0000i		−	1.19850i	−0.800561	−	0.599251i	$-0.795465\pi$
$401$		−	24.0000i		−	1.19850i	0.800561	−	0.599251i	$-0.204535\pi$
$402$		0			0
$403$			24.1636i			1.20367i
$404$		0			0
$405$	8.00000	+	4.12311i	0.397523	+	0.204879i
$406$		0			0
$407$	6.78456			0.336298
$408$		0			0
$409$	25.3693			1.25443			0.627216	−	0.778845i	$-0.284194\pi$
$409$	25.3693			1.25443			0.627216	+	0.778845i	$0.284194\pi$
$410$		0			0
$411$	−6.99337	+	12.4536i	−0.344957	+	0.614292i
$412$		0			0
$413$			42.7386i			2.10303i
$414$		0			0
$415$			15.1022i			0.741340i
$416$		0			0
$417$	14.8769	−	26.4924i	0.728525	−	1.29734i
$418$		0			0
$419$	−7.36520			−0.359814			−0.179907	−	0.983684i	$-0.557580\pi$
$419$	−7.36520			−0.359814			−0.179907	+	0.983684i	$0.557580\pi$
$420$		0			0
$421$	25.3693			1.23642			0.618212	−	0.786011i	$-0.287857\pi$
$421$	25.3693			1.23642			0.618212	+	0.786011i	$0.287857\pi$
$422$		0			0
$423$	4.71659	+	7.73704i	0.229328	+	0.376188i
$424$		0			0
$425$			2.00000i			0.0970143i
$426$		0			0
$427$		−	9.43318i		−	0.456503i
$428$		0			0
$429$	−10.2462	−	5.75379i	−0.494692	−	0.277796i
$430$		0			0
$431$	−16.6354			−0.801297			−0.400648	−	0.916232i	$-0.631215\pi$
$431$	−16.6354			−0.801297			−0.400648	+	0.916232i	$0.631215\pi$
$432$		0			0
$433$	18.0000			0.865025			0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000			0.865025			0.432512	+	0.901628i	$0.357627\pi$
$434$		0			0
$435$	4.71659	+	2.64861i	0.226143	+	0.126991i
$436$		0			0
$437$			0.492423i			0.0235558i
$438$		0			0
$439$		−	9.27015i		−	0.442440i	−0.975224	−	0.221220i	$-0.928996\pi$
$439$		−	9.27015i		−	0.442440i	0.975224	−	0.221220i	$-0.0710039\pi$
$440$		0			0
$441$	−3.31534	−	5.43845i	−0.157873	−	0.258974i
$442$		0			0
$443$	16.5896			0.788195			0.394097	−	0.919069i	$-0.371057\pi$
$443$	16.5896			0.788195			0.394097	+	0.919069i	$0.371057\pi$
$444$		0			0
$445$	−10.2462			−0.485717
$446$		0			0
$447$	11.8730	−	21.1431i	0.561573	−	1.00004i
$448$		0			0
$449$			27.3693i			1.29164i	0.763491	+	0.645819i	$0.223484\pi$
$449$			27.3693i			1.29164i	−0.763491	+	0.645819i	$0.776516\pi$
$450$		0			0
$451$		−	1.48734i		−	0.0700359i
$452$		0			0
$453$	−6.24621	+	11.1231i	−0.293473	+	0.522609i
$454$		0			0
$455$	15.4741			0.725436
$456$		0			0
$457$	−10.0000			−0.467780			−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000			−0.467780			−0.233890	+	0.972263i	$0.575146\pi$
$458$		0			0
$459$	10.3857	−	0.371834i	0.484761	−	0.0173557i
$460$		0			0
$461$			41.8617i			1.94970i	0.222872	+	0.974848i	$0.428457\pi$
$461$			41.8617i			1.94970i	−0.222872	+	0.974848i	$0.571543\pi$
$462$		0			0
$463$		−	3.02045i		−	0.140372i	−0.997534	−	0.0701861i	$-0.977641\pi$
$463$		−	3.02045i		−	0.140372i	0.997534	−	0.0701861i	$-0.0223593\pi$
$464$		0			0
$465$	7.12311	+	4.00000i	0.330326	+	0.185496i
$466$		0			0
$467$	−2.27678			−0.105357			−0.0526784	−	0.998612i	$-0.516776\pi$
$467$	−2.27678			−0.105357			−0.0526784	+	0.998612i	$0.516776\pi$
$468$		0			0
$469$	13.1231			0.605969
$470$		0			0
$471$	5.08842	+	2.85742i	0.234462	+	0.131663i
$472$		0			0
$473$		−	10.2462i		−	0.471121i
$474$		0			0
$475$		−	1.32431i		−	0.0607634i
$476$		0			0
$477$	31.3693	−	19.1231i	1.43630	−	0.875587i
$478$		0			0
$479$	25.6509			1.17202			0.586010	−	0.810304i	$-0.300698\pi$
$479$	25.6509			1.17202			0.586010	+	0.810304i	$0.300698\pi$
$480$		0			0
$481$	−26.2462			−1.19672
$482$		0			0
$483$	−0.952473	+	1.69614i	−0.0433390	+	0.0771771i
$484$		0			0
$485$			6.00000i			0.272446i
$486$		0			0
$487$			25.2791i			1.14550i	0.819728	+	0.572752i	$0.194124\pi$
$487$			25.2791i			1.14550i	−0.819728	+	0.572752i	$0.805876\pi$
$488$		0			0
$489$	13.3002	−	23.6847i	0.601455	−	1.07106i
$490$		0			0
$491$	−26.9752			−1.21737			−0.608687	−	0.793410i	$-0.708304\pi$
$491$	−26.9752			−1.21737			−0.608687	+	0.793410i	$0.708304\pi$
$492$		0			0
$493$	6.24621			0.281315
$494$		0			0
$495$	−3.39228	+	2.06798i	−0.152472	+	0.0929486i
$496$		0			0
$497$			10.2462i			0.459605i
$498$		0			0
$499$			32.2725i			1.44471i	0.691521	+	0.722357i	$0.256941\pi$
$499$			32.2725i			1.44471i	−0.691521	+	0.722357i	$0.743059\pi$
$500$		0			0
$501$	−13.6847	−	7.68466i	−0.611385	−	0.343325i
$502$		0			0
$503$	14.3586			0.640217			0.320109	−	0.947381i	$-0.396281\pi$
$503$	14.3586			0.640217			0.320109	+	0.947381i	$0.396281\pi$
$504$		0			0
$505$	−0.876894			−0.0390213
$506$		0			0
$507$	20.0047	+	11.2337i	0.888442	+	0.498907i
$508$		0			0
$509$			11.1231i			0.493023i	0.969140	+	0.246511i	$0.0792843\pi$
$509$			11.1231i			0.493023i	−0.969140	+	0.246511i	$0.920716\pi$
$510$		0			0
$511$			24.9073i			1.10183i
$512$		0			0
$513$	−6.87689	+	0.246211i	−0.303622	+	0.0108705i
$514$		0			0
$515$	9.80501			0.432060
$516$		0			0
$517$	−4.00000			−0.175920
$518$		0			0
$519$	−1.69614	+	3.02045i	−0.0744523	+	0.132583i
$520$		0			0
$521$			38.2462i			1.67560i	0.545980	+	0.837798i	$0.316158\pi$
$521$			38.2462i			1.67560i	−0.545980	+	0.837798i	$0.683842\pi$
$522$		0			0
$523$		−	35.2929i		−	1.54325i	−0.636077	−	0.771625i	$-0.719444\pi$
$523$		−	35.2929i		−	1.54325i	0.636077	−	0.771625i	$-0.280556\pi$
$524$		0			0
$525$	2.56155	−	4.56155i	0.111795	−	0.199082i
$526$		0			0
$527$	9.43318			0.410916
$528$		0			0
$529$	−22.8617			−0.993989
$530$		0			0
$531$	22.0956	+	36.2454i	0.958868	+	1.57292i
$532$		0			0
$533$			5.75379i			0.249224i
$534$		0			0
$535$		−	3.02045i		−	0.130585i
$536$		0			0
$537$	15.1231	+	8.49242i	0.652610	+	0.366475i
$538$		0			0
$539$	2.81164			0.121106
$540$		0			0
$541$	26.9848			1.16017			0.580085	−	0.814556i	$-0.303020\pi$
$541$	26.9848			1.16017			0.580085	+	0.814556i	$0.303020\pi$
$542$		0			0
$543$	18.4945	+	10.3857i	0.793676	+	0.445691i
$544$		0			0
$545$		−	0.876894i		−	0.0375620i
$546$		0			0
$547$			5.83209i			0.249362i	0.992197	+	0.124681i	$0.0397908\pi$
$547$			5.83209i			0.249362i	−0.992197	+	0.124681i	$0.960209\pi$
$548$		0			0
$549$	−4.87689	−	8.00000i	−0.208141	−	0.341432i
$550$		0			0
$551$	−4.13595			−0.176197
$552$		0			0
$553$	24.4924			1.04152
$554$		0			0
$555$	−4.34475	+	7.73704i	−0.184425	+	0.328419i
$556$		0			0
$557$			36.2462i			1.53580i	0.640569	+	0.767901i	$0.278699\pi$
$557$			36.2462i			1.53580i	−0.640569	+	0.767901i	$0.721301\pi$
$558$		0			0
$559$			39.6377i			1.67649i
$560$		0			0
$561$	−2.24621	+	4.00000i	−0.0948351	+	0.168880i
$562$		0			0
$563$	7.90007			0.332948			0.166474	−	0.986046i	$-0.446762\pi$
$563$	7.90007			0.332948			0.166474	+	0.986046i	$0.446762\pi$
$564$		0			0
$565$	−14.0000			−0.588984
$566$		0			0
$567$	−24.1636	−	12.4536i	−1.01478	−	0.523003i
$568$		0			0
$569$			13.1231i			0.550149i	0.961423	+	0.275075i	$0.0887026\pi$
$569$			13.1231i			0.550149i	−0.961423	+	0.275075i	$0.911297\pi$
$570$		0			0
$571$			33.0161i			1.38168i	0.723007	+	0.690841i	$0.242759\pi$
$571$			33.0161i			1.38168i	−0.723007	+	0.690841i	$0.757241\pi$
$572$		0			0
$573$	−37.6155	−	21.1231i	−1.57141	−	0.882430i
$574$		0			0
$575$	−0.371834			−0.0155066
$576$		0			0
$577$	−32.2462			−1.34243			−0.671214	−	0.741264i	$-0.734227\pi$
$577$	−32.2462			−1.34243			−0.671214	+	0.741264i	$0.734227\pi$
$578$		0			0
$579$	−0.371834	−	0.208805i	−0.0154529	−	0.00867762i
$580$		0			0
$581$		−	45.6155i		−	1.89245i
$582$		0			0
$583$			16.2177i			0.671670i
$584$		0			0
$585$	13.1231	−	8.00000i	0.542574	−	0.330759i
$586$		0			0
$587$	−1.85917			−0.0767362			−0.0383681	−	0.999264i	$-0.512216\pi$
$587$	−1.85917			−0.0767362			−0.0383681	+	0.999264i	$0.512216\pi$
$588$		0			0
$589$	−6.24621			−0.257371
$590$		0			0
$591$	−3.60109	+	6.41273i	−0.148129	+	0.263784i
$592$		0			0
$593$		−	8.24621i		−	0.338631i	−0.985562	−	0.169316i	$-0.945844\pi$
$593$		−	8.24621i		−	0.338631i	0.985562	−	0.169316i	$-0.0541557\pi$
$594$		0			0
$595$		−	6.04090i		−	0.247653i
$596$		0			0
$597$	4.63068	−	8.24621i	0.189521	−	0.337495i
$598$		0			0
$599$	−44.1912			−1.80560			−0.902802	−	0.430056i	$-0.858494\pi$
$599$	−44.1912			−1.80560			−0.902802	+	0.430056i	$0.858494\pi$
$600$		0			0
$601$	23.1231			0.943211			0.471606	−	0.881810i	$-0.343675\pi$
$601$	23.1231			0.943211			0.471606	+	0.881810i	$0.343675\pi$
$602$		0			0
$603$	11.1293	−	6.78456i	0.453221	−	0.276289i
$604$		0			0
$605$			9.24621i			0.375912i
$606$		0			0
$607$		−	4.50778i		−	0.182965i	−0.995807	−	0.0914827i	$-0.970839\pi$
$607$		−	4.50778i		−	0.182965i	0.995807	−	0.0914827i	$-0.0291606\pi$
$608$		0			0
$609$	−14.2462	−	8.00000i	−0.577286	−	0.324176i
$610$		0			0
$611$	15.4741			0.626014
$612$		0			0
$613$	−9.12311			−0.368479			−0.184239	−	0.982881i	$-0.558982\pi$
$613$	−9.12311			−0.368479			−0.184239	+	0.982881i	$0.558982\pi$
$614$		0			0
$615$	1.69614	+	0.952473i	0.0683950	+	0.0384074i
$616$		0			0
$617$			14.0000i			0.563619i	0.959470	+	0.281809i	$0.0909346\pi$
$617$			14.0000i			0.563619i	−0.959470	+	0.281809i	$0.909065\pi$
$618$		0			0
$619$		−	28.1365i		−	1.13090i	−0.824782	−	0.565451i	$-0.808702\pi$
$619$		−	28.1365i		−	1.13090i	0.824782	−	0.565451i	$-0.191298\pi$
$620$		0			0
$621$	0.0691303	+	1.93087i	0.00277410	+	0.0774831i
$622$		0			0
$623$	30.9481			1.23991
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	1.48734	−	2.64861i	0.0593985	−	0.105775i
$628$		0			0
$629$			10.2462i			0.408543i
$630$		0			0
$631$		−	39.8007i		−	1.58444i	−0.610235	−	0.792220i	$-0.708925\pi$
$631$		−	39.8007i		−	1.58444i	0.610235	−	0.792220i	$-0.291075\pi$
$632$		0			0
$633$	14.2462	−	25.3693i	0.566236	−	1.00834i
$634$		0			0
$635$	−15.1022			−0.599314
$636$		0			0
$637$	−10.8769			−0.430958
$638$		0			0
$639$	5.29723	+	8.68951i	0.209555	+	0.343752i
$640$		0			0
$641$			6.38447i			0.252171i	0.992019	+	0.126086i	$0.0402414\pi$
$641$			6.38447i			0.252171i	−0.992019	+	0.126086i	$0.959759\pi$
$642$		0			0
$643$			3.60109i			0.142013i	0.997476	+	0.0710065i	$0.0226211\pi$
$643$			3.60109i			0.142013i	−0.997476	+	0.0710065i	$0.977379\pi$
$644$		0			0
$645$	11.6847	+	6.56155i	0.460083	+	0.258361i
$646$		0			0
$647$	−36.6172			−1.43957			−0.719786	−	0.694197i	$-0.755760\pi$
$647$	−36.6172			−1.43957			−0.719786	+	0.694197i	$0.755760\pi$
$648$		0			0
$649$	−18.7386			−0.735556
$650$		0			0
$651$	−21.5150	−	12.0818i	−0.843238	−	0.473523i
$652$		0			0
$653$		−	38.9848i		−	1.52559i	−0.646638	−	0.762797i	$-0.723825\pi$
$653$		−	38.9848i		−	1.52559i	0.646638	−	0.762797i	$-0.276175\pi$
$654$		0			0
$655$			5.46026i			0.213350i
$656$		0			0
$657$	12.8769	+	21.1231i	0.502375	+	0.824091i
$658$		0			0
$659$	−24.7442			−0.963898			−0.481949	−	0.876199i	$-0.660071\pi$
$659$	−24.7442			−0.963898			−0.481949	+	0.876199i	$0.660071\pi$
$660$		0			0
$661$	−28.1080			−1.09327			−0.546636	−	0.837370i	$-0.684092\pi$
$661$	−28.1080			−1.09327			−0.546636	+	0.837370i	$0.684092\pi$
$662$		0			0
$663$	8.68951	−	15.4741i	0.337473	−	0.600963i
$664$		0			0
$665$			4.00000i			0.155113i
$666$		0			0
$667$			1.16128i			0.0449648i
$668$		0			0
$669$	7.05398	−	12.5616i	0.272722	−	0.485658i
$670$		0			0
$671$	4.13595			0.159667
$672$		0			0
$673$	22.4924			0.867019			0.433510	−	0.901149i	$-0.357275\pi$
$673$	22.4924			0.867019			0.433510	+	0.901149i	$0.357275\pi$
$674$		0			0
$675$	−0.185917	−	5.19283i	−0.00715595	−	0.199872i
$676$		0			0
$677$			1.50758i			0.0579409i	0.999580	+	0.0289705i	$0.00922287\pi$
$677$			1.50758i			0.0579409i	−0.999580	+	0.0289705i	$0.990777\pi$
$678$		0			0
$679$		−	18.1227i		−	0.695485i
$680$		0			0
$681$	−33.0540	−	18.5616i	−1.26663	−	0.711280i
$682$		0			0
$683$	−7.90007			−0.302288			−0.151144	−	0.988512i	$-0.548296\pi$
$683$	−7.90007			−0.302288			−0.151144	+	0.988512i	$0.548296\pi$
$684$		0			0
$685$	8.24621			0.315072
$686$		0			0
$687$	24.5354	+	13.7779i	0.936085	+	0.525661i
$688$		0			0
$689$		−	62.7386i		−	2.39015i
$690$		0			0
$691$		−	18.2857i		−	0.695621i	−0.937565	−	0.347811i	$-0.886925\pi$
$691$		−	18.2857i		−	0.695621i	0.937565	−	0.347811i	$-0.113075\pi$
$692$		0			0
$693$	10.2462	−	6.24621i	0.389221	−	0.237274i
$694$		0			0
$695$	−17.5420			−0.665408
$696$		0			0
$697$	2.24621			0.0850813
$698$		0			0
$699$	−8.48071	+	15.1022i	−0.320770	+	0.571219i
$700$		0			0
$701$			17.5076i			0.661252i	0.943762	+	0.330626i	$0.107260\pi$
$701$			17.5076i			0.661252i	−0.943762	+	0.330626i	$0.892740\pi$
$702$		0			0
$703$		−	6.78456i		−	0.255885i
$704$		0			0
$705$	2.56155	−	4.56155i	0.0964737	−	0.171798i
$706$		0			0
$707$	2.64861			0.0996114
$708$		0			0
$709$	−6.49242			−0.243828			−0.121914	−	0.992541i	$-0.538903\pi$
$709$	−6.49242			−0.243828			−0.121914	+	0.992541i	$0.538903\pi$
$710$		0			0
$711$	20.7713	−	12.6624i	0.778985	−	0.474878i
$712$		0			0
$713$			1.75379i			0.0656799i
$714$		0			0
$715$			6.78456i			0.253728i
$716$		0			0
$717$	26.2462	+	14.7386i	0.980183	+	0.550424i
$718$		0			0
$719$	−30.9481			−1.15417			−0.577086	−	0.816684i	$-0.695810\pi$
$719$	−30.9481			−1.15417			−0.577086	+	0.816684i	$0.695810\pi$
$720$		0			0
$721$	−29.6155			−1.10294
$722$		0			0
$723$	20.1907	+	11.3381i	0.750899	+	0.421669i
$724$		0			0
$725$		−	3.12311i		−	0.115989i
$726$		0			0
$727$		−	10.9663i		−	0.406717i	−0.979104	−	0.203359i	$-0.934814\pi$
$727$		−	10.9663i		−	0.406717i	0.979104	−	0.203359i	$-0.0651857\pi$
$728$		0			0
$729$	−26.9309	+	1.93087i	−0.997440	+	0.0715137i
$730$		0			0
$731$	15.4741			0.572329
$732$		0			0
$733$	26.8769			0.992721			0.496360	−	0.868117i	$-0.334669\pi$
$733$	26.8769			0.992721			0.496360	+	0.868117i	$0.334669\pi$
$734$		0			0
$735$	−1.80054	+	3.20636i	−0.0664140	+	0.118269i
$736$		0			0
$737$			5.75379i			0.211944i
$738$		0			0
$739$			26.9752i			0.992300i	0.868237	+	0.496150i	$0.165253\pi$
$739$			26.9752i			0.992300i	−0.868237	+	0.496150i	$0.834747\pi$
$740$		0			0
$741$	−5.75379	+	10.2462i	−0.211371	+	0.376404i
$742$		0			0
$743$	−9.80501			−0.359711			−0.179856	−	0.983693i	$-0.557563\pi$
$743$	−9.80501			−0.359711			−0.179856	+	0.983693i	$0.557563\pi$
$744$		0			0
$745$	−14.0000			−0.512920
$746$		0			0
$747$	−23.5829	−	38.6852i	−0.862855	−	1.41542i
$748$		0			0
$749$			9.12311i			0.333351i
$750$		0			0
$751$			11.5012i			0.419683i	0.977735	+	0.209842i	$0.0672948\pi$
$751$			11.5012i			0.419683i	−0.977735	+	0.209842i	$0.932705\pi$
$752$		0			0
$753$	−28.2462	−	15.8617i	−1.02935	−	0.578034i
$754$		0			0
$755$	7.36520			0.268047
$756$		0			0
$757$	−10.8769			−0.395327			−0.197664	−	0.980270i	$-0.563335\pi$
$757$	−10.8769			−0.395327			−0.197664	+	0.980270i	$0.563335\pi$
$758$		0			0
$759$	−0.743668	−	0.417609i	−0.0269934	−	0.0151582i
$760$		0			0
$761$			31.2311i			1.13212i	0.824362	+	0.566062i	$0.191534\pi$
$761$			31.2311i			1.13212i	−0.824362	+	0.566062i	$0.808466\pi$
$762$		0			0
$763$			2.64861i			0.0958863i
$764$		0			0
$765$	−3.12311	−	5.12311i	−0.112916	−	0.185226i
$766$		0			0
$767$	72.4908			2.61749
$768$		0			0
$769$	38.9848			1.40583			0.702915	−	0.711274i	$-0.251882\pi$
$769$	38.9848			1.40583			0.702915	+	0.711274i	$0.251882\pi$
$770$		0			0
$771$	25.8597	−	46.0504i	0.931315	−	1.65846i
$772$		0			0
$773$		−	0.246211i		−	0.00885560i	−0.999990	−	0.00442780i	$-0.998591\pi$
$773$		−	0.246211i		−	0.00885560i	0.999990	−	0.00442780i	$-0.00140942\pi$
$774$		0			0
$775$		−	4.71659i		−	0.169425i
$776$		0			0
$777$	13.1231	−	23.3693i	0.470789	−	0.838370i
$778$		0			0
$779$	−1.48734			−0.0532894
$780$		0			0
$781$	−4.49242			−0.160752
$782$		0			0
$783$	−16.2177	+	0.580639i	−0.579575	+	0.0207503i
$784$		0			0
$785$		−	3.36932i		−	0.120256i
$786$		0			0
$787$			42.0775i			1.49990i	0.661495	+	0.749950i	$0.269922\pi$
$787$			42.0775i			1.49990i	−0.661495	+	0.749950i	$0.730078\pi$
$788$		0			0
$789$	−35.9309	−	20.1771i	−1.27917	−	0.718323i
$790$		0			0
$791$	42.2863			1.50353
$792$		0			0
$793$	−16.0000			−0.568177
$794$		0			0
$795$	−18.4945	−	10.3857i	−0.655933	−	0.368341i
$796$		0			0
$797$			12.7386i			0.451226i	0.974217	+	0.225613i	$0.0724384\pi$
$797$			12.7386i			0.451226i	−0.974217	+	0.225613i	$0.927562\pi$
$798$		0			0
$799$		−	6.04090i		−	0.213712i
$800$		0			0
$801$	26.2462	−	16.0000i	0.927364	−	0.565332i
$802$		0			0
$803$	−10.9205			−0.385377
$804$		0			0
$805$	1.12311			0.0395843
$806$		0			0
$807$	−11.8730	+	21.1431i	−0.417949	+	0.744274i
$808$		0			0
$809$			29.7538i			1.04609i	0.852306	+	0.523044i	$0.175204\pi$
$809$			29.7538i			1.04609i	−0.852306	+	0.523044i	$0.824796\pi$
$810$		0			0
$811$		−	46.2592i		−	1.62438i	−0.583393	−	0.812190i	$-0.698275\pi$
$811$		−	46.2592i		−	1.62438i	0.583393	−	0.812190i	$-0.301725\pi$
$812$		0			0
$813$	12.9848	−	23.1231i	0.455398	−	0.810963i
$814$		0			0
$815$	−15.6829			−0.549347
$816$		0			0
$817$	−10.2462			−0.358470
$818$		0			0
$819$	−39.6377	+	24.1636i	−1.38505	+	0.844344i
$820$		0			0
$821$			53.2311i			1.85778i	0.370360	+	0.928888i	$0.379234\pi$
$821$			53.2311i			1.85778i	−0.370360	+	0.928888i	$0.620766\pi$
$822$		0			0
$823$		−	48.2814i		−	1.68298i	−0.540270	−	0.841492i	$-0.681678\pi$
$823$		−	48.2814i		−	1.68298i	0.540270	−	0.841492i	$-0.318322\pi$
$824$		0			0
$825$	2.00000	+	1.12311i	0.0696311	+	0.0391015i
$826$		0			0
$827$	17.7509			0.617258			0.308629	−	0.951183i	$-0.400130\pi$
$827$	17.7509			0.617258			0.308629	+	0.951183i	$0.400130\pi$
$828$		0			0
$829$	8.87689			0.308307			0.154154	−	0.988047i	$-0.450735\pi$
$829$	8.87689			0.308307			0.154154	+	0.988047i	$0.450735\pi$
$830$		0			0
$831$	−35.2929	−	19.8188i	−1.22430	−	0.687508i
$832$		0			0
$833$			4.24621i			0.147122i
$834$		0			0
$835$			9.06134i			0.313581i
$836$		0			0
$837$	−24.4924	+	0.876894i	−0.846582	+	0.0303099i
$838$		0			0
$839$	−17.7051			−0.611247			−0.305624	−	0.952152i	$-0.598865\pi$
$839$	−17.7051			−0.611247			−0.305624	+	0.952152i	$0.598865\pi$
$840$		0			0
$841$	19.2462			0.663662
$842$		0			0
$843$	11.5469	−	20.5625i	0.397697	−	0.708210i
$844$		0			0
$845$		−	13.2462i		−	0.455684i
$846$		0			0
$847$		−	27.9277i		−	0.959607i
$848$		0			0
$849$	−19.6847	+	35.0540i	−0.675576	+	1.20305i
$850$		0			0
$851$	−1.90495			−0.0653007
$852$		0			0
$853$	7.86174			0.269181			0.134590	−	0.990901i	$-0.457028\pi$
$853$	7.86174			0.269181			0.134590	+	0.990901i	$0.457028\pi$
$854$		0			0
$855$	2.06798	+	3.39228i	0.0707233	+	0.116014i
$856$		0			0
$857$			20.7386i			0.708418i	0.935166	+	0.354209i	$0.115250\pi$
$857$			20.7386i			0.708418i	−0.935166	+	0.354209i	$0.884750\pi$
$858$		0			0
$859$		−	33.4337i		−	1.14074i	−0.821386	−	0.570372i	$-0.806799\pi$
$859$		−	33.4337i		−	1.14074i	0.821386	−	0.570372i	$-0.193201\pi$
$860$		0			0
$861$	−5.12311	−	2.87689i	−0.174595	−	0.0980443i
$862$		0			0
$863$	10.5487			0.359081			0.179541	−	0.983751i	$-0.442539\pi$
$863$	10.5487			0.359081			0.179541	+	0.983751i	$0.442539\pi$
$864$		0			0
$865$	2.00000			0.0680020
$866$		0			0
$867$	19.6329	+	11.0249i	0.666769	+	0.374426i
$868$		0			0
$869$			10.7386i			0.364283i
$870$		0			0
$871$		−	22.2586i		−	0.754205i
$872$		0			0
$873$	−9.36932	−	15.3693i	−0.317103	−	0.520173i
$874$		0			0
$875$	−3.02045			−0.102110
$876$		0			0
$877$	−37.6155			−1.27019			−0.635093	−	0.772436i	$-0.719038\pi$
$877$	−37.6155			−1.27019			−0.635093	+	0.772436i	$0.719038\pi$
$878$		0			0
$879$	−2.11375	+	3.76412i	−0.0712950	+	0.126960i
$880$		0			0
$881$		−	0.630683i		−	0.0212483i	−0.999944	−	0.0106241i	$-0.996618\pi$
$881$		−	0.630683i		−	0.0212483i	0.999944	−	0.0106241i	$-0.00338183\pi$
$882$		0			0
$883$		−	22.4674i		−	0.756090i	−0.925787	−	0.378045i	$-0.876597\pi$
$883$		−	22.4674i		−	0.756090i	0.925787	−	0.378045i	$-0.123403\pi$
$884$		0			0
$885$	12.0000	−	21.3693i	0.403376	−	0.718322i
$886$		0			0
$887$	−51.6737			−1.73503			−0.867516	−	0.497409i	$-0.834285\pi$
$887$	−51.6737			−1.73503			−0.867516	+	0.497409i	$0.834285\pi$
$888$		0			0
$889$	45.6155			1.52990
$890$		0			0
$891$	5.46026	−	10.5945i	0.182925	−	0.354928i
$892$		0			0
$893$			4.00000i			0.133855i
$894$		0			0
$895$		−	10.0138i		−	0.334725i
$896$		0			0
$897$	2.87689	+	1.61553i	0.0960567	+	0.0539409i
$898$		0			0
$899$	−14.7304			−0.491287
$900$		0			0
$901$	−24.4924			−0.815961
$902$		0			0
$903$	−35.2929	−	19.8188i	−1.17447	−	0.659529i
$904$		0			0
$905$		−	12.2462i		−	0.407078i
$906$		0			0
$907$		−	46.2134i		−	1.53449i	−0.641353	−	0.767246i	$-0.721627\pi$
$907$		−	46.2134i		−	1.53449i	0.641353	−	0.767246i	$-0.278373\pi$
$908$		0			0
$909$	2.24621	−	1.36932i	0.0745021	−	0.0454174i
$910$		0			0
$911$	−14.3128			−0.474204			−0.237102	−	0.971485i	$-0.576198\pi$
$911$	−14.3128			−0.474204			−0.237102	+	0.971485i	$0.576198\pi$
$912$		0			0
$913$	20.0000			0.661903
$914$		0			0
$915$	−2.64861	+	4.71659i	−0.0875605	+	0.155926i
$916$		0			0
$917$		−	16.4924i		−	0.544628i
$918$		0			0
$919$			49.9775i			1.64861i	0.566148	+	0.824303i	$0.308433\pi$
$919$			49.9775i			1.64861i	−0.566148	+	0.824303i	$0.691567\pi$
$920$		0			0
$921$	−9.43845	+	16.8078i	−0.311007	+	0.553835i
$922$		0			0
$923$	17.3790			0.572037
$924$		0			0
$925$	5.12311			0.168447
$926$		0			0
$927$	−25.1161	+	15.3110i	−0.824920	+	0.502881i
$928$		0			0
$929$			33.1231i			1.08673i	0.839495	+	0.543367i	$0.182851\pi$
$929$			33.1231i			1.08673i	−0.839495	+	0.543367i	$0.817149\pi$
$930$		0			0
$931$		−	2.81164i		−	0.0921479i
$932$		0			0
$933$	31.3693	+	17.6155i	1.02699	+	0.576707i
$934$		0			0
$935$	2.64861			0.0866189
$936$		0			0
$937$	−22.4924			−0.734795			−0.367398	−	0.930064i	$-0.619751\pi$
$937$	−22.4924			−0.734795			−0.367398	+	0.930064i	$0.619751\pi$
$938$		0			0
$939$	−33.9686	−	19.0752i	−1.10852	−	0.622494i
$940$		0			0
$941$			0.876894i			0.0285859i	0.999898	+	0.0142930i	$0.00454975\pi$
$941$			0.876894i			0.0285859i	−0.999898	+	0.0142930i	$0.995450\pi$
$942$		0			0
$943$			0.417609i			0.0135992i
$944$		0			0
$945$	0.561553	+	15.6847i	0.0182673	+	0.510222i
$946$		0			0
$947$	−32.4813			−1.05550			−0.527750	−	0.849400i	$-0.676964\pi$
$947$	−32.4813			−1.05550			−0.527750	+	0.849400i	$0.676964\pi$
$948$		0			0
$949$	42.2462			1.37137
$950$		0			0
$951$	14.1955	−	25.2791i	0.460322	−	0.819731i
$952$		0			0
$953$			22.4924i			0.728601i	0.931281	+	0.364301i	$0.118692\pi$
$953$			22.4924i			0.728601i	−0.931281	+	0.364301i	$0.881308\pi$
$954$		0			0
$955$			24.9073i			0.805980i
$956$		0			0
$957$	3.50758	−	6.24621i	0.113384	−	0.201911i
$958$		0			0
$959$	−24.9073			−0.804297
$960$		0			0
$961$	8.75379			0.282380
$962$		0			0
$963$	4.71659	+	7.73704i	0.151990	+	0.249323i
$964$		0			0
$965$			0.246211i			0.00792582i
$966$		0			0
$967$		−	26.4404i		−	0.850265i	−0.905131	−	0.425132i	$-0.860228\pi$
$967$		−	26.4404i		−	0.850265i	0.905131	−	0.425132i	$-0.139772\pi$
$968$		0			0
$969$	4.00000	+	2.24621i	0.128499	+	0.0721587i
$970$		0			0
$971$	−52.6261			−1.68885			−0.844427	−	0.535671i	$-0.820059\pi$
$971$	−52.6261			−1.68885			−0.844427	+	0.535671i	$0.820059\pi$
$972$		0			0
$973$	52.9848			1.69862
$974$		0			0
$975$	−7.73704	−	4.34475i	−0.247783	−	0.139144i
$976$		0			0
$977$		−	31.7538i		−	1.01589i	−0.861388	−	0.507947i	$-0.830405\pi$
$977$		−	31.7538i		−	1.01589i	0.861388	−	0.507947i	$-0.169595\pi$
$978$		0			0
$979$			13.5691i			0.433671i
$980$		0			0
$981$	1.36932	+	2.24621i	0.0437189	+	0.0717160i
$982$		0			0
$983$	40.0095			1.27610			0.638052	−	0.769993i	$-0.279740\pi$
$983$	40.0095			1.27610			0.638052	+	0.769993i	$0.279740\pi$
$984$		0			0
$985$	4.24621			0.135296
$986$		0			0
$987$	−7.73704	+	13.7779i	−0.246273	+	0.438556i
$988$		0			0
$989$			2.87689i			0.0914799i
$990$		0			0
$991$			33.0161i			1.04879i	0.851475	+	0.524396i	$0.175709\pi$
$991$			33.0161i			1.04879i	−0.851475	+	0.524396i	$0.824291\pi$
$992$		0			0
$993$	2.73863	−	4.87689i	0.0869079	−	0.154764i
$994$		0			0
$995$	−5.46026			−0.173102
$996$		0			0
$997$	33.6155			1.06461			0.532307	−	0.846551i	$-0.321325\pi$
$997$	33.6155			1.06461			0.532307	+	0.846551i	$0.321325\pi$
$998$		0			0
$999$	−0.952473	−	26.6034i	−0.0301349	−	0.841694i

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.2.h.g.191.8		8
3.2	odd	2	inner	960.2.h.g.191.2		8
4.3	odd	2	inner	960.2.h.g.191.1		8
8.3	odd	2		60.2.e.a.11.1	✓	8
8.5	even	2		60.2.e.a.11.7	yes	8
12.11	even	2	inner	960.2.h.g.191.7		8
24.5	odd	2		60.2.e.a.11.2	yes	8
24.11	even	2		60.2.e.a.11.8	yes	8
40.3	even	4		300.2.h.b.299.3		8
40.13	odd	4		300.2.h.b.299.2		8
40.19	odd	2		300.2.e.c.251.8		8
40.27	even	4		300.2.h.a.299.6		8
40.29	even	2		300.2.e.c.251.2		8
40.37	odd	4		300.2.h.a.299.7		8
120.29	odd	2		300.2.e.c.251.7		8
120.53	even	4		300.2.h.a.299.8		8
120.59	even	2		300.2.e.c.251.1		8
120.77	even	4		300.2.h.b.299.1		8
120.83	odd	4		300.2.h.a.299.5		8
120.107	odd	4		300.2.h.b.299.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.2.e.a.11.1	✓	8	8.3	odd	2
60.2.e.a.11.2	yes	8	24.5	odd	2
60.2.e.a.11.7	yes	8	8.5	even	2
60.2.e.a.11.8	yes	8	24.11	even	2
300.2.e.c.251.1		8	120.59	even	2
300.2.e.c.251.2		8	40.29	even	2
300.2.e.c.251.7		8	120.29	odd	2
300.2.e.c.251.8		8	40.19	odd	2
300.2.h.a.299.5		8	120.83	odd	4
300.2.h.a.299.6		8	40.27	even	4
300.2.h.a.299.7		8	40.37	odd	4
300.2.h.a.299.8		8	120.53	even	4
300.2.h.b.299.1		8	120.77	even	4
300.2.h.b.299.2		8	40.13	odd	4
300.2.h.b.299.3		8	40.3	even	4
300.2.h.b.299.4		8	120.107	odd	4
960.2.h.g.191.1		8	4.3	odd	2	inner
960.2.h.g.191.2		8	3.2	odd	2	inner
960.2.h.g.191.7		8	12.11	even	2	inner
960.2.h.g.191.8		8	1.1	even	1	trivial

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	960.2.h.g.191.8

Level	$960$
Weight	$2$
Character	960.191
Analytic conductor	$7.666$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$