LMFDB - Embedded newform 762.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [762,2,Mod(1,762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(762, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("762.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 762 = 2 \cdot 3 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	762.a (trivial)

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:	yes
Analytic conductor:	$6.08460063402$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	762.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +5.00000 q^{11} +1.00000 q^{12} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} +5.00000 q^{22} +3.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} +1.00000 q^{27} +1.00000 q^{28} +2.00000 q^{29} -1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +5.00000 q^{33} -3.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} -4.00000 q^{37} -1.00000 q^{38} -1.00000 q^{40} +7.00000 q^{41} +1.00000 q^{42} +2.00000 q^{43} +5.00000 q^{44} -1.00000 q^{45} +3.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} -4.00000 q^{50} -3.00000 q^{51} +5.00000 q^{53} +1.00000 q^{54} -5.00000 q^{55} +1.00000 q^{56} -1.00000 q^{57} +2.00000 q^{58} -10.0000 q^{59} -1.00000 q^{60} -8.00000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +5.00000 q^{66} -2.00000 q^{67} -3.00000 q^{68} +3.00000 q^{69} -1.00000 q^{70} -12.0000 q^{71} +1.00000 q^{72} -10.0000 q^{73} -4.00000 q^{74} -4.00000 q^{75} -1.00000 q^{76} +5.00000 q^{77} -4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +7.00000 q^{82} +1.00000 q^{84} +3.00000 q^{85} +2.00000 q^{86} +2.00000 q^{87} +5.00000 q^{88} -8.00000 q^{89} -1.00000 q^{90} +3.00000 q^{92} +4.00000 q^{93} +6.00000 q^{94} +1.00000 q^{95} +1.00000 q^{96} -14.0000 q^{97} -6.00000 q^{98} +5.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	1.00000	0.408248
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	1.00000	0.288675
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	1.00000	0.267261
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	−1.00000	−0.223607
$21$	1.00000	0.218218
$22$	5.00000	1.06600
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	1.00000	0.204124
$25$	−4.00000	−0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	1.00000	0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	−1.00000	−0.182574
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	5.00000	0.870388
$34$	−3.00000	−0.514496
$35$	−1.00000	−0.169031
$36$	1.00000	0.166667
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−1.00000	−0.158114
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	1.00000	0.154303
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	5.00000	0.753778
$45$	−1.00000	−0.149071
$46$	3.00000	0.442326
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	−4.00000	−0.565685
$51$	−3.00000	−0.420084
$52$	0	0
$53$	5.00000	0.686803	0.343401	−	0.939189i	$-0.388421\pi$
$53$	5.00000	0.686803	0.343401	+	0.939189i	$0.388421\pi$
$54$	1.00000	0.136083
$55$	−5.00000	−0.674200
$56$	1.00000	0.133631
$57$	−1.00000	−0.132453
$58$	2.00000	0.262613
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	−1.00000	−0.129099
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	4.00000	0.508001
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	5.00000	0.615457
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	−3.00000	−0.363803
$69$	3.00000	0.361158
$70$	−1.00000	−0.119523
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	1.00000	0.117851
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−4.00000	−0.464991
$75$	−4.00000	−0.461880
$76$	−1.00000	−0.114708
$77$	5.00000	0.569803
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	7.00000	0.773021
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	1.00000	0.109109
$85$	3.00000	0.325396
$86$	2.00000	0.215666
$87$	2.00000	0.214423
$88$	5.00000	0.533002
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	3.00000	0.312772
$93$	4.00000	0.414781
$94$	6.00000	0.618853
$95$	1.00000	0.102598
$96$	1.00000	0.102062
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−6.00000	−0.606092
$99$	5.00000	0.502519
$100$	−4.00000	−0.400000
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	−3.00000	−0.297044
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	5.00000	0.485643
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	1.00000	0.0962250
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−5.00000	−0.476731
$111$	−4.00000	−0.379663
$112$	1.00000	0.0944911
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	−1.00000	−0.0936586
$115$	−3.00000	−0.279751
$116$	2.00000	0.185695
$117$	0	0
$118$	−10.0000	−0.920575
$119$	−3.00000	−0.275010
$120$	−1.00000	−0.0912871
$121$	14.0000	1.27273
$122$	−8.00000	−0.724286
$123$	7.00000	0.631169
$124$	4.00000	0.359211
$125$	9.00000	0.804984
$126$	1.00000	0.0890871
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	1.00000	0.0883883
$129$	2.00000	0.176090
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	5.00000	0.435194
$133$	−1.00000	−0.0867110
$134$	−2.00000	−0.172774
$135$	−1.00000	−0.0860663
$136$	−3.00000	−0.257248
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	3.00000	0.255377
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	−1.00000	−0.0845154
$141$	6.00000	0.505291
$142$	−12.0000	−1.00702
$143$	0	0
$144$	1.00000	0.0833333
$145$	−2.00000	−0.166091
$146$	−10.0000	−0.827606
$147$	−6.00000	−0.494872
$148$	−4.00000	−0.328798
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	−4.00000	−0.326599
$151$	−9.00000	−0.732410	−0.366205	−	0.930534i	$-0.619343\pi$
$151$	−9.00000	−0.732410	−0.366205	+	0.930534i	$0.619343\pi$
$152$	−1.00000	−0.0811107
$153$	−3.00000	−0.242536
$154$	5.00000	0.402911
$155$	−4.00000	−0.321288
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	−4.00000	−0.318223
$159$	5.00000	0.396526
$160$	−1.00000	−0.0790569
$161$	3.00000	0.236433
$162$	1.00000	0.0785674
$163$	17.0000	1.33154	0.665771	−	0.746156i	$-0.268103\pi$
$163$	17.0000	1.33154	0.665771	+	0.746156i	$0.268103\pi$
$164$	7.00000	0.546608
$165$	−5.00000	−0.389249
$166$	0	0
$167$	7.00000	0.541676	0.270838	−	0.962625i	$-0.412699\pi$
$167$	7.00000	0.541676	0.270838	+	0.962625i	$0.412699\pi$
$168$	1.00000	0.0771517
$169$	−13.0000	−1.00000
$170$	3.00000	0.230089
$171$	−1.00000	−0.0764719
$172$	2.00000	0.152499
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	2.00000	0.151620
$175$	−4.00000	−0.302372
$176$	5.00000	0.376889
$177$	−10.0000	−0.751646
$178$	−8.00000	−0.599625
$179$	19.0000	1.42013	0.710063	−	0.704138i	$-0.248666\pi$
$179$	19.0000	1.42013	0.710063	+	0.704138i	$0.248666\pi$
$180$	−1.00000	−0.0745356
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	0	0
$183$	−8.00000	−0.591377
$184$	3.00000	0.221163
$185$	4.00000	0.294086
$186$	4.00000	0.293294
$187$	−15.0000	−1.09691
$188$	6.00000	0.437595
$189$	1.00000	0.0727393
$190$	1.00000	0.0725476
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	1.00000	0.0721688
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	−6.00000	−0.428571
$197$	16.0000	1.13995	0.569976	−	0.821661i	$-0.306952\pi$
$197$	16.0000	1.13995	0.569976	+	0.821661i	$0.306952\pi$
$198$	5.00000	0.355335
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	−4.00000	−0.282843
$201$	−2.00000	−0.141069
$202$	−3.00000	−0.211079
$203$	2.00000	0.140372
$204$	−3.00000	−0.210042
$205$	−7.00000	−0.488901
$206$	6.00000	0.418040
$207$	3.00000	0.208514
$208$	0	0
$209$	−5.00000	−0.345857
$210$	−1.00000	−0.0690066
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	5.00000	0.343401
$213$	−12.0000	−0.822226
$214$	−4.00000	−0.273434
$215$	−2.00000	−0.136399
$216$	1.00000	0.0680414
$217$	4.00000	0.271538
$218$	−2.00000	−0.135457
$219$	−10.0000	−0.675737
$220$	−5.00000	−0.337100
$221$	0	0
$222$	−4.00000	−0.268462
$223$	−21.0000	−1.40626	−0.703132	−	0.711059i	$-0.748216\pi$
$223$	−21.0000	−1.40626	−0.703132	+	0.711059i	$0.748216\pi$
$224$	1.00000	0.0668153
$225$	−4.00000	−0.266667
$226$	9.00000	0.598671
$227$	−17.0000	−1.12833	−0.564165	−	0.825662i	$-0.690802\pi$
$227$	−17.0000	−1.12833	−0.564165	+	0.825662i	$0.690802\pi$
$228$	−1.00000	−0.0662266
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	−3.00000	−0.197814
$231$	5.00000	0.328976
$232$	2.00000	0.131306
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	−10.0000	−0.650945
$237$	−4.00000	−0.259828
$238$	−3.00000	−0.194461
$239$	9.00000	0.582162	0.291081	−	0.956698i	$-0.405985\pi$
$239$	9.00000	0.582162	0.291081	+	0.956698i	$0.405985\pi$
$240$	−1.00000	−0.0645497
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	14.0000	0.899954
$243$	1.00000	0.0641500
$244$	−8.00000	−0.512148
$245$	6.00000	0.383326
$246$	7.00000	0.446304
$247$	0	0
$248$	4.00000	0.254000
$249$	0	0
$250$	9.00000	0.569210
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	1.00000	0.0629941
$253$	15.0000	0.943042
$254$	1.00000	0.0627456
$255$	3.00000	0.187867
$256$	1.00000	0.0625000
$257$	20.0000	1.24757	0.623783	−	0.781598i	$-0.285595\pi$
$257$	20.0000	1.24757	0.623783	+	0.781598i	$0.285595\pi$
$258$	2.00000	0.124515
$259$	−4.00000	−0.248548
$260$	0	0
$261$	2.00000	0.123797
$262$	−15.0000	−0.926703
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	5.00000	0.307729
$265$	−5.00000	−0.307148
$266$	−1.00000	−0.0613139
$267$	−8.00000	−0.489592
$268$	−2.00000	−0.122169
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	−1.00000	−0.0608581
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	−2.00000	−0.120824
$275$	−20.0000	−1.20605
$276$	3.00000	0.180579
$277$	5.00000	0.300421	0.150210	−	0.988654i	$-0.452005\pi$
$277$	5.00000	0.300421	0.150210	+	0.988654i	$0.452005\pi$
$278$	−14.0000	−0.839664
$279$	4.00000	0.239474
$280$	−1.00000	−0.0597614
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	6.00000	0.357295
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	−12.0000	−0.712069
$285$	1.00000	0.0592349
$286$	0	0
$287$	7.00000	0.413197
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	−2.00000	−0.117444
$291$	−14.0000	−0.820695
$292$	−10.0000	−0.585206
$293$	−21.0000	−1.22683	−0.613417	−	0.789760i	$-0.710205\pi$
$293$	−21.0000	−1.22683	−0.613417	+	0.789760i	$0.710205\pi$
$294$	−6.00000	−0.349927
$295$	10.0000	0.582223
$296$	−4.00000	−0.232495
$297$	5.00000	0.290129
$298$	10.0000	0.579284
$299$	0	0
$300$	−4.00000	−0.230940
$301$	2.00000	0.115278
$302$	−9.00000	−0.517892
$303$	−3.00000	−0.172345
$304$	−1.00000	−0.0573539
$305$	8.00000	0.458079
$306$	−3.00000	−0.171499
$307$	14.0000	0.799022	0.399511	−	0.916728i	$-0.369180\pi$
$307$	14.0000	0.799022	0.399511	+	0.916728i	$0.369180\pi$
$308$	5.00000	0.284901
$309$	6.00000	0.341328
$310$	−4.00000	−0.227185
$311$	25.0000	1.41762	0.708810	−	0.705399i	$-0.249232\pi$
$311$	25.0000	1.41762	0.708810	+	0.705399i	$0.249232\pi$
$312$	0	0
$313$	34.0000	1.92179	0.960897	−	0.276907i	$-0.0893093\pi$
$313$	34.0000	1.92179	0.960897	+	0.276907i	$0.0893093\pi$
$314$	4.00000	0.225733
$315$	−1.00000	−0.0563436
$316$	−4.00000	−0.225018
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	5.00000	0.280386
$319$	10.0000	0.559893
$320$	−1.00000	−0.0559017
$321$	−4.00000	−0.223258
$322$	3.00000	0.167183
$323$	3.00000	0.166924
$324$	1.00000	0.0555556
$325$	0	0
$326$	17.0000	0.941543
$327$	−2.00000	−0.110600
$328$	7.00000	0.386510
$329$	6.00000	0.330791
$330$	−5.00000	−0.275241
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	7.00000	0.383023
$335$	2.00000	0.109272
$336$	1.00000	0.0545545
$337$	16.0000	0.871576	0.435788	−	0.900049i	$-0.356470\pi$
$337$	16.0000	0.871576	0.435788	+	0.900049i	$0.356470\pi$
$338$	−13.0000	−0.707107
$339$	9.00000	0.488813
$340$	3.00000	0.162698
$341$	20.0000	1.08306
$342$	−1.00000	−0.0540738
$343$	−13.0000	−0.701934
$344$	2.00000	0.107833
$345$	−3.00000	−0.161515
$346$	6.00000	0.322562
$347$	−30.0000	−1.61048	−0.805242	−	0.592946i	$-0.797965\pi$
$347$	−30.0000	−1.61048	−0.805242	+	0.592946i	$0.797965\pi$
$348$	2.00000	0.107211
$349$	7.00000	0.374701	0.187351	−	0.982293i	$-0.440010\pi$
$349$	7.00000	0.374701	0.187351	+	0.982293i	$0.440010\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	5.00000	0.266501
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	−10.0000	−0.531494
$355$	12.0000	0.636894
$356$	−8.00000	−0.423999
$357$	−3.00000	−0.158777
$358$	19.0000	1.00418
$359$	31.0000	1.63612	0.818059	−	0.575135i	$-0.195050\pi$
$359$	31.0000	1.63612	0.818059	+	0.575135i	$0.195050\pi$
$360$	−1.00000	−0.0527046
$361$	−18.0000	−0.947368
$362$	7.00000	0.367912
$363$	14.0000	0.734809
$364$	0	0
$365$	10.0000	0.523424
$366$	−8.00000	−0.418167
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	3.00000	0.156386
$369$	7.00000	0.364405
$370$	4.00000	0.207950
$371$	5.00000	0.259587
$372$	4.00000	0.207390
$373$	17.0000	0.880227	0.440113	−	0.897942i	$-0.354938\pi$
$373$	17.0000	0.880227	0.440113	+	0.897942i	$0.354938\pi$
$374$	−15.0000	−0.775632
$375$	9.00000	0.464758
$376$	6.00000	0.309426
$377$	0	0
$378$	1.00000	0.0514344
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	1.00000	0.0512989
$381$	1.00000	0.0512316
$382$	10.0000	0.511645
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	1.00000	0.0510310
$385$	−5.00000	−0.254824
$386$	−2.00000	−0.101797
$387$	2.00000	0.101666
$388$	−14.0000	−0.710742
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	−9.00000	−0.455150
$392$	−6.00000	−0.303046
$393$	−15.0000	−0.756650
$394$	16.0000	0.806068
$395$	4.00000	0.201262
$396$	5.00000	0.251259
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	−10.0000	−0.501255
$399$	−1.00000	−0.0500626
$400$	−4.00000	−0.200000
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	−2.00000	−0.0997509
$403$	0	0
$404$	−3.00000	−0.149256
$405$	−1.00000	−0.0496904
$406$	2.00000	0.0992583
$407$	−20.0000	−0.991363
$408$	−3.00000	−0.148522
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	−7.00000	−0.345705
$411$	−2.00000	−0.0986527
$412$	6.00000	0.295599
$413$	−10.0000	−0.492068
$414$	3.00000	0.147442
$415$	0	0
$416$	0	0
$417$	−14.0000	−0.685583
$418$	−5.00000	−0.244558
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	−1.00000	−0.0487950
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	−12.0000	−0.584151
$423$	6.00000	0.291730
$424$	5.00000	0.242821
$425$	12.0000	0.582086
$426$	−12.0000	−0.581402
$427$	−8.00000	−0.387147
$428$	−4.00000	−0.193347
$429$	0	0
$430$	−2.00000	−0.0964486
$431$	−2.00000	−0.0963366	−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000	−0.0963366	−0.0481683	+	0.998839i	$0.515338\pi$
$432$	1.00000	0.0481125
$433$	35.0000	1.68199	0.840996	−	0.541041i	$-0.181970\pi$
$433$	35.0000	1.68199	0.840996	+	0.541041i	$0.181970\pi$
$434$	4.00000	0.192006
$435$	−2.00000	−0.0958927
$436$	−2.00000	−0.0957826
$437$	−3.00000	−0.143509
$438$	−10.0000	−0.477818
$439$	13.0000	0.620456	0.310228	−	0.950662i	$-0.399595\pi$
$439$	13.0000	0.620456	0.310228	+	0.950662i	$0.399595\pi$
$440$	−5.00000	−0.238366
$441$	−6.00000	−0.285714
$442$	0	0
$443$	3.00000	0.142534	0.0712672	−	0.997457i	$-0.477296\pi$
$443$	3.00000	0.142534	0.0712672	+	0.997457i	$0.477296\pi$
$444$	−4.00000	−0.189832
$445$	8.00000	0.379236
$446$	−21.0000	−0.994379
$447$	10.0000	0.472984
$448$	1.00000	0.0472456
$449$	9.00000	0.424736	0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000	0.424736	0.212368	+	0.977190i	$0.431882\pi$
$450$	−4.00000	−0.188562
$451$	35.0000	1.64809
$452$	9.00000	0.423324
$453$	−9.00000	−0.422857
$454$	−17.0000	−0.797850
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	3.00000	0.140334	0.0701670	−	0.997535i	$-0.477647\pi$
$457$	3.00000	0.140334	0.0701670	+	0.997535i	$0.477647\pi$
$458$	6.00000	0.280362
$459$	−3.00000	−0.140028
$460$	−3.00000	−0.139876
$461$	21.0000	0.978068	0.489034	−	0.872265i	$-0.337349\pi$
$461$	21.0000	0.978068	0.489034	+	0.872265i	$0.337349\pi$
$462$	5.00000	0.232621
$463$	2.00000	0.0929479	0.0464739	−	0.998920i	$-0.485202\pi$
$463$	2.00000	0.0929479	0.0464739	+	0.998920i	$0.485202\pi$
$464$	2.00000	0.0928477
$465$	−4.00000	−0.185496
$466$	10.0000	0.463241
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	0	0
$469$	−2.00000	−0.0923514
$470$	−6.00000	−0.276759
$471$	4.00000	0.184310
$472$	−10.0000	−0.460287
$473$	10.0000	0.459800
$474$	−4.00000	−0.183726
$475$	4.00000	0.183533
$476$	−3.00000	−0.137505
$477$	5.00000	0.228934
$478$	9.00000	0.411650
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	4.00000	0.182195
$483$	3.00000	0.136505
$484$	14.0000	0.636364
$485$	14.0000	0.635707
$486$	1.00000	0.0453609
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	−8.00000	−0.362143
$489$	17.0000	0.768767
$490$	6.00000	0.271052
$491$	2.00000	0.0902587	0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000	0.0902587	0.0451294	+	0.998981i	$0.485630\pi$
$492$	7.00000	0.315584
$493$	−6.00000	−0.270226
$494$	0	0
$495$	−5.00000	−0.224733
$496$	4.00000	0.179605
$497$	−12.0000	−0.538274
$498$	0	0
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	9.00000	0.402492
$501$	7.00000	0.312737
$502$	−21.0000	−0.937276
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	1.00000	0.0445435
$505$	3.00000	0.133498
$506$	15.0000	0.666831
$507$	−13.0000	−0.577350
$508$	1.00000	0.0443678
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	3.00000	0.132842
$511$	−10.0000	−0.442374
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	20.0000	0.882162
$515$	−6.00000	−0.264392
$516$	2.00000	0.0880451
$517$	30.0000	1.31940
$518$	−4.00000	−0.175750
$519$	6.00000	0.263371
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	2.00000	0.0875376
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−15.0000	−0.655278
$525$	−4.00000	−0.174574
$526$	12.0000	0.523225
$527$	−12.0000	−0.522728
$528$	5.00000	0.217597
$529$	−14.0000	−0.608696
$530$	−5.00000	−0.217186
$531$	−10.0000	−0.433963
$532$	−1.00000	−0.0433555
$533$	0	0
$534$	−8.00000	−0.346194
$535$	4.00000	0.172935
$536$	−2.00000	−0.0863868
$537$	19.0000	0.819911
$538$	−10.0000	−0.431131
$539$	−30.0000	−1.29219
$540$	−1.00000	−0.0430331
$541$	31.0000	1.33279	0.666397	−	0.745597i	$-0.267836\pi$
$541$	31.0000	1.33279	0.666397	+	0.745597i	$0.267836\pi$
$542$	20.0000	0.859074
$543$	7.00000	0.300399
$544$	−3.00000	−0.128624
$545$	2.00000	0.0856706
$546$	0	0
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	−2.00000	−0.0854358
$549$	−8.00000	−0.341432
$550$	−20.0000	−0.852803
$551$	−2.00000	−0.0852029
$552$	3.00000	0.127688
$553$	−4.00000	−0.170097
$554$	5.00000	0.212430
$555$	4.00000	0.169791
$556$	−14.0000	−0.593732
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	4.00000	0.169334
$559$	0	0
$560$	−1.00000	−0.0422577
$561$	−15.0000	−0.633300
$562$	16.0000	0.674919
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	6.00000	0.252646
$565$	−9.00000	−0.378633
$566$	4.00000	0.168133
$567$	1.00000	0.0419961
$568$	−12.0000	−0.503509
$569$	17.0000	0.712677	0.356339	−	0.934357i	$-0.384025\pi$
$569$	17.0000	0.712677	0.356339	+	0.934357i	$0.384025\pi$
$570$	1.00000	0.0418854
$571$	−30.0000	−1.25546	−0.627730	−	0.778431i	$-0.716016\pi$
$571$	−30.0000	−1.25546	−0.627730	+	0.778431i	$0.716016\pi$
$572$	0	0
$573$	10.0000	0.417756
$574$	7.00000	0.292174
$575$	−12.0000	−0.500435
$576$	1.00000	0.0416667
$577$	25.0000	1.04076	0.520382	−	0.853934i	$-0.325790\pi$
$577$	25.0000	1.04076	0.520382	+	0.853934i	$0.325790\pi$
$578$	−8.00000	−0.332756
$579$	−2.00000	−0.0831172
$580$	−2.00000	−0.0830455
$581$	0	0
$582$	−14.0000	−0.580319
$583$	25.0000	1.03539
$584$	−10.0000	−0.413803
$585$	0	0
$586$	−21.0000	−0.867502
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	−6.00000	−0.247436
$589$	−4.00000	−0.164817
$590$	10.0000	0.411693
$591$	16.0000	0.658152
$592$	−4.00000	−0.164399
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	5.00000	0.205152
$595$	3.00000	0.122988
$596$	10.0000	0.409616
$597$	−10.0000	−0.409273
$598$	0	0
$599$	−23.0000	−0.939755	−0.469877	−	0.882732i	$-0.655702\pi$
$599$	−23.0000	−0.939755	−0.469877	+	0.882732i	$0.655702\pi$
$600$	−4.00000	−0.163299
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	2.00000	0.0815139
$603$	−2.00000	−0.0814463
$604$	−9.00000	−0.366205
$605$	−14.0000	−0.569181
$606$	−3.00000	−0.121867
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	−1.00000	−0.0405554
$609$	2.00000	0.0810441
$610$	8.00000	0.323911
$611$	0	0
$612$	−3.00000	−0.121268
$613$	−23.0000	−0.928961	−0.464481	−	0.885583i	$-0.653759\pi$
$613$	−23.0000	−0.928961	−0.464481	+	0.885583i	$0.653759\pi$
$614$	14.0000	0.564994
$615$	−7.00000	−0.282267
$616$	5.00000	0.201456
$617$	−40.0000	−1.61034	−0.805170	−	0.593045i	$-0.797926\pi$
$617$	−40.0000	−1.61034	−0.805170	+	0.593045i	$0.797926\pi$
$618$	6.00000	0.241355
$619$	18.0000	0.723481	0.361741	−	0.932279i	$-0.382183\pi$
$619$	18.0000	0.723481	0.361741	+	0.932279i	$0.382183\pi$
$620$	−4.00000	−0.160644
$621$	3.00000	0.120386
$622$	25.0000	1.00241
$623$	−8.00000	−0.320513
$624$	0	0
$625$	11.0000	0.440000
$626$	34.0000	1.35891
$627$	−5.00000	−0.199681
$628$	4.00000	0.159617
$629$	12.0000	0.478471
$630$	−1.00000	−0.0398410
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	−4.00000	−0.159111
$633$	−12.0000	−0.476957
$634$	3.00000	0.119145
$635$	−1.00000	−0.0396838
$636$	5.00000	0.198263
$637$	0	0
$638$	10.0000	0.395904
$639$	−12.0000	−0.474713
$640$	−1.00000	−0.0395285
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−4.00000	−0.157867
$643$	21.0000	0.828159	0.414080	−	0.910241i	$-0.364104\pi$
$643$	21.0000	0.828159	0.414080	+	0.910241i	$0.364104\pi$
$644$	3.00000	0.118217
$645$	−2.00000	−0.0787499
$646$	3.00000	0.118033
$647$	−3.00000	−0.117942	−0.0589711	−	0.998260i	$-0.518782\pi$
$647$	−3.00000	−0.117942	−0.0589711	+	0.998260i	$0.518782\pi$
$648$	1.00000	0.0392837
$649$	−50.0000	−1.96267
$650$	0	0
$651$	4.00000	0.156772
$652$	17.0000	0.665771
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	−2.00000	−0.0782062
$655$	15.0000	0.586098
$656$	7.00000	0.273304
$657$	−10.0000	−0.390137
$658$	6.00000	0.233904
$659$	−26.0000	−1.01282	−0.506408	−	0.862294i	$-0.669027\pi$
$659$	−26.0000	−1.01282	−0.506408	+	0.862294i	$0.669027\pi$
$660$	−5.00000	−0.194625
$661$	4.00000	0.155582	0.0777910	−	0.996970i	$-0.475213\pi$
$661$	4.00000	0.155582	0.0777910	+	0.996970i	$0.475213\pi$
$662$	−18.0000	−0.699590
$663$	0	0
$664$	0	0
$665$	1.00000	0.0387783
$666$	−4.00000	−0.154997
$667$	6.00000	0.232321
$668$	7.00000	0.270838
$669$	−21.0000	−0.811907
$670$	2.00000	0.0772667
$671$	−40.0000	−1.54418
$672$	1.00000	0.0385758
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	16.0000	0.616297
$675$	−4.00000	−0.153960
$676$	−13.0000	−0.500000
$677$	−50.0000	−1.92166	−0.960828	−	0.277145i	$-0.910612\pi$
$677$	−50.0000	−1.92166	−0.960828	+	0.277145i	$0.910612\pi$
$678$	9.00000	0.345643
$679$	−14.0000	−0.537271
$680$	3.00000	0.115045
$681$	−17.0000	−0.651441
$682$	20.0000	0.765840
$683$	40.0000	1.53056	0.765279	−	0.643699i	$-0.222601\pi$
$683$	40.0000	1.53056	0.765279	+	0.643699i	$0.222601\pi$
$684$	−1.00000	−0.0382360
$685$	2.00000	0.0764161
$686$	−13.0000	−0.496342
$687$	6.00000	0.228914
$688$	2.00000	0.0762493
$689$	0	0
$690$	−3.00000	−0.114208
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	6.00000	0.228086
$693$	5.00000	0.189934
$694$	−30.0000	−1.13878
$695$	14.0000	0.531050
$696$	2.00000	0.0758098
$697$	−21.0000	−0.795432
$698$	7.00000	0.264954
$699$	10.0000	0.378235
$700$	−4.00000	−0.151186
$701$	23.0000	0.868698	0.434349	−	0.900745i	$-0.356978\pi$
$701$	23.0000	0.868698	0.434349	+	0.900745i	$0.356978\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	5.00000	0.188445
$705$	−6.00000	−0.225973
$706$	−10.0000	−0.376355
$707$	−3.00000	−0.112827
$708$	−10.0000	−0.375823
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	12.0000	0.450352
$711$	−4.00000	−0.150012
$712$	−8.00000	−0.299813
$713$	12.0000	0.449404
$714$	−3.00000	−0.112272
$715$	0	0
$716$	19.0000	0.710063
$717$	9.00000	0.336111
$718$	31.0000	1.15691
$719$	−50.0000	−1.86469	−0.932343	−	0.361576i	$-0.882239\pi$
$719$	−50.0000	−1.86469	−0.932343	+	0.361576i	$0.882239\pi$
$720$	−1.00000	−0.0372678
$721$	6.00000	0.223452
$722$	−18.0000	−0.669891
$723$	4.00000	0.148762
$724$	7.00000	0.260153
$725$	−8.00000	−0.297113
$726$	14.0000	0.519589
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	10.0000	0.370117
$731$	−6.00000	−0.221918
$732$	−8.00000	−0.295689
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	4.00000	0.147643
$735$	6.00000	0.221313
$736$	3.00000	0.110581
$737$	−10.0000	−0.368355
$738$	7.00000	0.257674
$739$	−44.0000	−1.61857	−0.809283	−	0.587419i	$-0.800144\pi$
$739$	−44.0000	−1.61857	−0.809283	+	0.587419i	$0.800144\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	5.00000	0.183556
$743$	9.00000	0.330178	0.165089	−	0.986279i	$-0.447209\pi$
$743$	9.00000	0.330178	0.165089	+	0.986279i	$0.447209\pi$
$744$	4.00000	0.146647
$745$	−10.0000	−0.366372
$746$	17.0000	0.622414
$747$	0	0
$748$	−15.0000	−0.548454
$749$	−4.00000	−0.146157
$750$	9.00000	0.328634
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	6.00000	0.218797
$753$	−21.0000	−0.765283
$754$	0	0
$755$	9.00000	0.327544
$756$	1.00000	0.0363696
$757$	16.0000	0.581530	0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000	0.581530	0.290765	+	0.956795i	$0.406090\pi$
$758$	−12.0000	−0.435860
$759$	15.0000	0.544466
$760$	1.00000	0.0362738
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	1.00000	0.0362262
$763$	−2.00000	−0.0724049
$764$	10.0000	0.361787
$765$	3.00000	0.108465
$766$	−8.00000	−0.289052
$767$	0	0
$768$	1.00000	0.0360844
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	−5.00000	−0.180187
$771$	20.0000	0.720282
$772$	−2.00000	−0.0719816
$773$	4.00000	0.143870	0.0719350	−	0.997409i	$-0.477083\pi$
$773$	4.00000	0.143870	0.0719350	+	0.997409i	$0.477083\pi$
$774$	2.00000	0.0718885
$775$	−16.0000	−0.574737
$776$	−14.0000	−0.502571
$777$	−4.00000	−0.143499
$778$	26.0000	0.932145
$779$	−7.00000	−0.250801
$780$	0	0
$781$	−60.0000	−2.14697
$782$	−9.00000	−0.321839
$783$	2.00000	0.0714742
$784$	−6.00000	−0.214286
$785$	−4.00000	−0.142766
$786$	−15.0000	−0.535032
$787$	−17.0000	−0.605985	−0.302992	−	0.952993i	$-0.597986\pi$
$787$	−17.0000	−0.605985	−0.302992	+	0.952993i	$0.597986\pi$
$788$	16.0000	0.569976
$789$	12.0000	0.427211
$790$	4.00000	0.142314
$791$	9.00000	0.320003
$792$	5.00000	0.177667
$793$	0	0
$794$	−22.0000	−0.780751
$795$	−5.00000	−0.177332
$796$	−10.0000	−0.354441
$797$	−14.0000	−0.495905	−0.247953	−	0.968772i	$-0.579758\pi$
$797$	−14.0000	−0.495905	−0.247953	+	0.968772i	$0.579758\pi$
$798$	−1.00000	−0.0353996
$799$	−18.0000	−0.636794
$800$	−4.00000	−0.141421
$801$	−8.00000	−0.282666
$802$	10.0000	0.353112
$803$	−50.0000	−1.76446
$804$	−2.00000	−0.0705346
$805$	−3.00000	−0.105736
$806$	0	0
$807$	−10.0000	−0.352017
$808$	−3.00000	−0.105540
$809$	−51.0000	−1.79306	−0.896532	−	0.442978i	$-0.853922\pi$
$809$	−51.0000	−1.79306	−0.896532	+	0.442978i	$0.853922\pi$
$810$	−1.00000	−0.0351364
$811$	−7.00000	−0.245803	−0.122902	−	0.992419i	$-0.539220\pi$
$811$	−7.00000	−0.245803	−0.122902	+	0.992419i	$0.539220\pi$
$812$	2.00000	0.0701862
$813$	20.0000	0.701431
$814$	−20.0000	−0.701000
$815$	−17.0000	−0.595484
$816$	−3.00000	−0.105021
$817$	−2.00000	−0.0699711
$818$	18.0000	0.629355
$819$	0	0
$820$	−7.00000	−0.244451
$821$	−11.0000	−0.383903	−0.191951	−	0.981404i	$-0.561482\pi$
$821$	−11.0000	−0.383903	−0.191951	+	0.981404i	$0.561482\pi$
$822$	−2.00000	−0.0697580
$823$	12.0000	0.418294	0.209147	−	0.977884i	$-0.432931\pi$
$823$	12.0000	0.418294	0.209147	+	0.977884i	$0.432931\pi$
$824$	6.00000	0.209020
$825$	−20.0000	−0.696311
$826$	−10.0000	−0.347945
$827$	44.0000	1.53003	0.765015	−	0.644013i	$-0.222732\pi$
$827$	44.0000	1.53003	0.765015	+	0.644013i	$0.222732\pi$
$828$	3.00000	0.104257
$829$	−31.0000	−1.07667	−0.538337	−	0.842729i	$-0.680947\pi$
$829$	−31.0000	−1.07667	−0.538337	+	0.842729i	$0.680947\pi$
$830$	0	0
$831$	5.00000	0.173448
$832$	0	0
$833$	18.0000	0.623663
$834$	−14.0000	−0.484780
$835$	−7.00000	−0.242245
$836$	−5.00000	−0.172929
$837$	4.00000	0.138260
$838$	−21.0000	−0.725433
$839$	21.0000	0.725001	0.362500	−	0.931984i	$-0.381923\pi$
$839$	21.0000	0.725001	0.362500	+	0.931984i	$0.381923\pi$
$840$	−1.00000	−0.0345033
$841$	−25.0000	−0.862069
$842$	2.00000	0.0689246
$843$	16.0000	0.551069
$844$	−12.0000	−0.413057
$845$	13.0000	0.447214
$846$	6.00000	0.206284
$847$	14.0000	0.481046
$848$	5.00000	0.171701
$849$	4.00000	0.137280
$850$	12.0000	0.411597
$851$	−12.0000	−0.411355
$852$	−12.0000	−0.411113
$853$	7.00000	0.239675	0.119838	−	0.992793i	$-0.461763\pi$
$853$	7.00000	0.239675	0.119838	+	0.992793i	$0.461763\pi$
$854$	−8.00000	−0.273754
$855$	1.00000	0.0341993
$856$	−4.00000	−0.136717
$857$	4.00000	0.136637	0.0683187	−	0.997664i	$-0.478237\pi$
$857$	4.00000	0.136637	0.0683187	+	0.997664i	$0.478237\pi$
$858$	0	0
$859$	−22.0000	−0.750630	−0.375315	−	0.926897i	$-0.622466\pi$
$859$	−22.0000	−0.750630	−0.375315	+	0.926897i	$0.622466\pi$
$860$	−2.00000	−0.0681994
$861$	7.00000	0.238559
$862$	−2.00000	−0.0681203
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	1.00000	0.0340207
$865$	−6.00000	−0.204006
$866$	35.0000	1.18935
$867$	−8.00000	−0.271694
$868$	4.00000	0.135769
$869$	−20.0000	−0.678454
$870$	−2.00000	−0.0678064
$871$	0	0
$872$	−2.00000	−0.0677285
$873$	−14.0000	−0.473828
$874$	−3.00000	−0.101477
$875$	9.00000	0.304256
$876$	−10.0000	−0.337869
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	13.0000	0.438729
$879$	−21.0000	−0.708312
$880$	−5.00000	−0.168550
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	−6.00000	−0.202031
$883$	−19.0000	−0.639401	−0.319700	−	0.947519i	$-0.603582\pi$
$883$	−19.0000	−0.639401	−0.319700	+	0.947519i	$0.603582\pi$
$884$	0	0
$885$	10.0000	0.336146
$886$	3.00000	0.100787
$887$	−15.0000	−0.503651	−0.251825	−	0.967773i	$-0.581031\pi$
$887$	−15.0000	−0.503651	−0.251825	+	0.967773i	$0.581031\pi$
$888$	−4.00000	−0.134231
$889$	1.00000	0.0335389
$890$	8.00000	0.268161
$891$	5.00000	0.167506
$892$	−21.0000	−0.703132
$893$	−6.00000	−0.200782
$894$	10.0000	0.334450
$895$	−19.0000	−0.635100
$896$	1.00000	0.0334077
$897$	0	0
$898$	9.00000	0.300334
$899$	8.00000	0.266815
$900$	−4.00000	−0.133333
$901$	−15.0000	−0.499722
$902$	35.0000	1.16537
$903$	2.00000	0.0665558
$904$	9.00000	0.299336
$905$	−7.00000	−0.232688
$906$	−9.00000	−0.299005
$907$	19.0000	0.630885	0.315442	−	0.948945i	$-0.397847\pi$
$907$	19.0000	0.630885	0.315442	+	0.948945i	$0.397847\pi$
$908$	−17.0000	−0.564165
$909$	−3.00000	−0.0995037
$910$	0	0
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	−1.00000	−0.0331133
$913$	0	0
$914$	3.00000	0.0992312
$915$	8.00000	0.264472
$916$	6.00000	0.198246
$917$	−15.0000	−0.495344
$918$	−3.00000	−0.0990148
$919$	52.0000	1.71532	0.857661	−	0.514216i	$-0.171917\pi$
$919$	52.0000	1.71532	0.857661	+	0.514216i	$0.171917\pi$
$920$	−3.00000	−0.0989071
$921$	14.0000	0.461316
$922$	21.0000	0.691598
$923$	0	0
$924$	5.00000	0.164488
$925$	16.0000	0.526077
$926$	2.00000	0.0657241
$927$	6.00000	0.197066
$928$	2.00000	0.0656532
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	−4.00000	−0.131165
$931$	6.00000	0.196642
$932$	10.0000	0.327561
$933$	25.0000	0.818463
$934$	6.00000	0.196326
$935$	15.0000	0.490552
$936$	0	0
$937$	−14.0000	−0.457360	−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000	−0.457360	−0.228680	+	0.973502i	$0.573441\pi$
$938$	−2.00000	−0.0653023
$939$	34.0000	1.10955
$940$	−6.00000	−0.195698
$941$	32.0000	1.04317	0.521585	−	0.853199i	$-0.325341\pi$
$941$	32.0000	1.04317	0.521585	+	0.853199i	$0.325341\pi$
$942$	4.00000	0.130327
$943$	21.0000	0.683854
$944$	−10.0000	−0.325472
$945$	−1.00000	−0.0325300
$946$	10.0000	0.325128
$947$	−46.0000	−1.49480	−0.747400	−	0.664375i	$-0.768698\pi$
$947$	−46.0000	−1.49480	−0.747400	+	0.664375i	$0.768698\pi$
$948$	−4.00000	−0.129914
$949$	0	0
$950$	4.00000	0.129777
$951$	3.00000	0.0972817
$952$	−3.00000	−0.0972306
$953$	51.0000	1.65205	0.826026	−	0.563632i	$-0.190596\pi$
$953$	51.0000	1.65205	0.826026	+	0.563632i	$0.190596\pi$
$954$	5.00000	0.161881
$955$	−10.0000	−0.323592
$956$	9.00000	0.291081
$957$	10.0000	0.323254
$958$	−24.0000	−0.775405
$959$	−2.00000	−0.0645834
$960$	−1.00000	−0.0322749
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−4.00000	−0.128898
$964$	4.00000	0.128831
$965$	2.00000	0.0643823
$966$	3.00000	0.0965234
$967$	16.0000	0.514525	0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000	0.514525	0.257263	+	0.966342i	$0.417179\pi$
$968$	14.0000	0.449977
$969$	3.00000	0.0963739
$970$	14.0000	0.449513
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	1.00000	0.0320750
$973$	−14.0000	−0.448819
$974$	−16.0000	−0.512673
$975$	0	0
$976$	−8.00000	−0.256074
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	17.0000	0.543600
$979$	−40.0000	−1.27841
$980$	6.00000	0.191663
$981$	−2.00000	−0.0638551
$982$	2.00000	0.0638226
$983$	18.0000	0.574111	0.287055	−	0.957914i	$-0.407324\pi$
$983$	18.0000	0.574111	0.287055	+	0.957914i	$0.407324\pi$
$984$	7.00000	0.223152
$985$	−16.0000	−0.509802
$986$	−6.00000	−0.191079
$987$	6.00000	0.190982
$988$	0	0
$989$	6.00000	0.190789
$990$	−5.00000	−0.158910
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	4.00000	0.127000
$993$	−18.0000	−0.571213
$994$	−12.0000	−0.380617
$995$	10.0000	0.317021
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	24.0000	0.759707
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	762.2.a.f.1.1	✓	1
3.2	odd	2		2286.2.a.f.1.1		1
4.3	odd	2		6096.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
762.2.a.f.1.1	✓	1	1.1	even	1	trivial
2286.2.a.f.1.1		1	3.2	odd	2
6096.2.a.d.1.1		1	4.3	odd	2

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:	\( N \)	\(=\)	\( 762 = 2 \cdot 3 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	762.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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