LMFDB - Embedded newform 2366.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2366,2,Mod(1,2366)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2366, 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("2366.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2366 = 2 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2366.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:	$18.8926051182$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 182)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2366.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +3.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +3.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} +5.00000 q^{11} +3.00000 q^{12} -1.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} +6.00000 q^{18} -2.00000 q^{19} -3.00000 q^{21} +5.00000 q^{22} +5.00000 q^{23} +3.00000 q^{24} -5.00000 q^{25} +9.00000 q^{27} -1.00000 q^{28} +4.00000 q^{29} -1.00000 q^{31} +1.00000 q^{32} +15.0000 q^{33} -4.00000 q^{34} +6.00000 q^{36} -7.00000 q^{37} -2.00000 q^{38} +9.00000 q^{41} -3.00000 q^{42} -12.0000 q^{43} +5.00000 q^{44} +5.00000 q^{46} +7.00000 q^{47} +3.00000 q^{48} +1.00000 q^{49} -5.00000 q^{50} -12.0000 q^{51} -4.00000 q^{53} +9.00000 q^{54} -1.00000 q^{56} -6.00000 q^{57} +4.00000 q^{58} +6.00000 q^{59} +13.0000 q^{61} -1.00000 q^{62} -6.00000 q^{63} +1.00000 q^{64} +15.0000 q^{66} -11.0000 q^{67} -4.00000 q^{68} +15.0000 q^{69} +6.00000 q^{72} -7.00000 q^{73} -7.00000 q^{74} -15.0000 q^{75} -2.00000 q^{76} -5.00000 q^{77} -17.0000 q^{79} +9.00000 q^{81} +9.00000 q^{82} -4.00000 q^{83} -3.00000 q^{84} -12.0000 q^{86} +12.0000 q^{87} +5.00000 q^{88} -14.0000 q^{89} +5.00000 q^{92} -3.00000 q^{93} +7.00000 q^{94} +3.00000 q^{96} -5.00000 q^{97} +1.00000 q^{98} +30.0000 q^{99} +O(q^{100})\)

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
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	3.00000	1.22474
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	6.00000	2.00000
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	3.00000	0.866025
$13$	0	0
$13$	0	0
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	6.00000	1.41421
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	5.00000	1.06600
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	3.00000	0.612372
$25$	−5.00000	−1.00000
$26$	0	0
$27$	9.00000	1.73205
$28$	−1.00000	−0.188982
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	1.00000	0.176777
$33$	15.0000	2.61116
$34$	−4.00000	−0.685994
$35$	0	0
$36$	6.00000	1.00000
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	−3.00000	−0.462910
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	5.00000	0.753778
$45$	0	0
$46$	5.00000	0.737210
$47$	7.00000	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$48$	3.00000	0.433013
$49$	1.00000	0.142857
$50$	−5.00000	−0.707107
$51$	−12.0000	−1.68034
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	9.00000	1.22474
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−6.00000	−0.794719
$58$	4.00000	0.525226
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	13.0000	1.66448	0.832240	−	0.554416i	$-0.187058\pi$
$61$	13.0000	1.66448	0.832240	+	0.554416i	$0.187058\pi$
$62$	−1.00000	−0.127000
$63$	−6.00000	−0.755929
$64$	1.00000	0.125000
$65$	0	0
$66$	15.0000	1.84637
$67$	−11.0000	−1.34386	−0.671932	−	0.740613i	$-0.734535\pi$
$67$	−11.0000	−1.34386	−0.671932	+	0.740613i	$0.734535\pi$
$68$	−4.00000	−0.485071
$69$	15.0000	1.80579
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	6.00000	0.707107
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	−7.00000	−0.813733
$75$	−15.0000	−1.73205
$76$	−2.00000	−0.229416
$77$	−5.00000	−0.569803
$78$	0	0
$79$	−17.0000	−1.91265	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−17.0000	−1.91265	−0.956325	+	0.292306i	$0.905577\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	9.00000	0.993884
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	−3.00000	−0.327327
$85$	0	0
$86$	−12.0000	−1.29399
$87$	12.0000	1.28654
$88$	5.00000	0.533002
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	0	0
$92$	5.00000	0.521286
$93$	−3.00000	−0.311086
$94$	7.00000	0.721995
$95$	0	0
$96$	3.00000	0.306186
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	1.00000	0.101015
$99$	30.0000	3.01511
$100$	−5.00000	−0.500000
$101$	15.0000	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$102$	−12.0000	−1.18818
$103$	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
$105$	0	0
$106$	−4.00000	−0.388514
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	9.00000	0.866025
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	−21.0000	−1.99323
$112$	−1.00000	−0.0944911
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	−6.00000	−0.561951
$115$	0	0
$116$	4.00000	0.371391
$117$	0	0
$118$	6.00000	0.552345
$119$	4.00000	0.366679
$120$	0	0
$121$	14.0000	1.27273
$122$	13.0000	1.17696
$123$	27.0000	2.43451
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	−6.00000	−0.534522
$127$	9.00000	0.798621	0.399310	−	0.916816i	$-0.369250\pi$
$127$	9.00000	0.798621	0.399310	+	0.916816i	$0.369250\pi$
$128$	1.00000	0.0883883
$129$	−36.0000	−3.16962
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	15.0000	1.30558
$133$	2.00000	0.173422
$134$	−11.0000	−0.950255
$135$	0	0
$136$	−4.00000	−0.342997
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	15.0000	1.27688
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	21.0000	1.76852
$142$	0	0
$143$	0	0
$144$	6.00000	0.500000
$145$	0	0
$146$	−7.00000	−0.579324
$147$	3.00000	0.247436
$148$	−7.00000	−0.575396
$149$	−7.00000	−0.573462	−0.286731	−	0.958011i	$-0.592569\pi$
$149$	−7.00000	−0.573462	−0.286731	+	0.958011i	$0.592569\pi$
$150$	−15.0000	−1.22474
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	−2.00000	−0.162221
$153$	−24.0000	−1.94029
$154$	−5.00000	−0.402911
$155$	0	0
$156$	0	0
$157$	1.00000	0.0798087	0.0399043	−	0.999204i	$-0.487295\pi$
$157$	1.00000	0.0798087	0.0399043	+	0.999204i	$0.487295\pi$
$158$	−17.0000	−1.35245
$159$	−12.0000	−0.951662
$160$	0	0
$161$	−5.00000	−0.394055
$162$	9.00000	0.707107
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	−3.00000	−0.231455
$169$	0	0
$170$	0	0
$171$	−12.0000	−0.917663
$172$	−12.0000	−0.914991
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	12.0000	0.909718
$175$	5.00000	0.377964
$176$	5.00000	0.376889
$177$	18.0000	1.35296
$178$	−14.0000	−1.04934
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	0	0
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	0	0
$183$	39.0000	2.88296
$184$	5.00000	0.368605
$185$	0	0
$186$	−3.00000	−0.219971
$187$	−20.0000	−1.46254
$188$	7.00000	0.510527
$189$	−9.00000	−0.654654
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	3.00000	0.216506
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−5.00000	−0.358979
$195$	0	0
$196$	1.00000	0.0714286
$197$	−27.0000	−1.92367	−0.961835	−	0.273629i	$-0.911776\pi$
$197$	−27.0000	−1.92367	−0.961835	+	0.273629i	$0.911776\pi$
$198$	30.0000	2.13201
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	−5.00000	−0.353553
$201$	−33.0000	−2.32764
$202$	15.0000	1.05540
$203$	−4.00000	−0.280745
$204$	−12.0000	−0.840168
$205$	0	0
$206$	6.00000	0.418040
$207$	30.0000	2.08514
$208$	0	0
$209$	−10.0000	−0.691714
$210$	0	0
$211$	−14.0000	−0.963800	−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000	−0.963800	−0.481900	+	0.876226i	$0.660053\pi$
$212$	−4.00000	−0.274721
$213$	0	0
$214$	−8.00000	−0.546869
$215$	0	0
$216$	9.00000	0.612372
$217$	1.00000	0.0678844
$218$	18.0000	1.21911
$219$	−21.0000	−1.41905
$220$	0	0
$221$	0	0
$222$	−21.0000	−1.40943
$223$	−3.00000	−0.200895	−0.100447	−	0.994942i	$-0.532027\pi$
$223$	−3.00000	−0.200895	−0.100447	+	0.994942i	$0.532027\pi$
$224$	−1.00000	−0.0668153
$225$	−30.0000	−2.00000
$226$	1.00000	0.0665190
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−6.00000	−0.397360
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	0	0
$231$	−15.0000	−0.986928
$232$	4.00000	0.262613
$233$	−21.0000	−1.37576	−0.687878	−	0.725826i	$-0.741458\pi$
$233$	−21.0000	−1.37576	−0.687878	+	0.725826i	$0.741458\pi$
$234$	0	0
$235$	0	0
$236$	6.00000	0.390567
$237$	−51.0000	−3.31281
$238$	4.00000	0.259281
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	30.0000	1.93247	0.966235	−	0.257663i	$-0.0829523\pi$
$241$	30.0000	1.93247	0.966235	+	0.257663i	$0.0829523\pi$
$242$	14.0000	0.899954
$243$	0	0
$244$	13.0000	0.832240
$245$	0	0
$246$	27.0000	1.72146
$247$	0	0
$248$	−1.00000	−0.0635001
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−13.0000	−0.820553	−0.410276	−	0.911961i	$-0.634568\pi$
$251$	−13.0000	−0.820553	−0.410276	+	0.911961i	$0.634568\pi$
$252$	−6.00000	−0.377964
$253$	25.0000	1.57174
$254$	9.00000	0.564710
$255$	0	0
$256$	1.00000	0.0625000
$257$	28.0000	1.74659	0.873296	−	0.487190i	$-0.161978\pi$
$257$	28.0000	1.74659	0.873296	+	0.487190i	$0.161978\pi$
$258$	−36.0000	−2.24126
$259$	7.00000	0.434959
$260$	0	0
$261$	24.0000	1.48556
$262$	8.00000	0.494242
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	15.0000	0.923186
$265$	0	0
$266$	2.00000	0.122628
$267$	−42.0000	−2.57036
$268$	−11.0000	−0.671932
$269$	13.0000	0.792624	0.396312	−	0.918116i	$-0.370290\pi$
$269$	13.0000	0.792624	0.396312	+	0.918116i	$0.370290\pi$
$270$	0	0
$271$	1.00000	0.0607457	0.0303728	−	0.999539i	$-0.490331\pi$
$271$	1.00000	0.0607457	0.0303728	+	0.999539i	$0.490331\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	−18.0000	−1.08742
$275$	−25.0000	−1.50756
$276$	15.0000	0.902894
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	4.00000	0.239904
$279$	−6.00000	−0.359211
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	21.0000	1.25053
$283$	3.00000	0.178331	0.0891657	−	0.996017i	$-0.471580\pi$
$283$	3.00000	0.178331	0.0891657	+	0.996017i	$0.471580\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.00000	−0.531253
$288$	6.00000	0.353553
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−15.0000	−0.879316
$292$	−7.00000	−0.409644
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	−7.00000	−0.406867
$297$	45.0000	2.61116
$298$	−7.00000	−0.405499
$299$	0	0
$300$	−15.0000	−0.866025
$301$	12.0000	0.691669
$302$	12.0000	0.690522
$303$	45.0000	2.58518
$304$	−2.00000	−0.114708
$305$	0	0
$306$	−24.0000	−1.37199
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	−5.00000	−0.284901
$309$	18.0000	1.02398
$310$	0	0
$311$	26.0000	1.47432	0.737162	−	0.675716i	$-0.236165\pi$
$311$	26.0000	1.47432	0.737162	+	0.675716i	$0.236165\pi$
$312$	0	0
$313$	−30.0000	−1.69570	−0.847850	−	0.530236i	$-0.822103\pi$
$313$	−30.0000	−1.69570	−0.847850	+	0.530236i	$0.822103\pi$
$314$	1.00000	0.0564333
$315$	0	0
$316$	−17.0000	−0.956325
$317$	−11.0000	−0.617822	−0.308911	−	0.951091i	$-0.599964\pi$
$317$	−11.0000	−0.617822	−0.308911	+	0.951091i	$0.599964\pi$
$318$	−12.0000	−0.672927
$319$	20.0000	1.11979
$320$	0	0
$321$	−24.0000	−1.33955
$322$	−5.00000	−0.278639
$323$	8.00000	0.445132
$324$	9.00000	0.500000
$325$	0	0
$326$	−4.00000	−0.221540
$327$	54.0000	2.98621
$328$	9.00000	0.496942
$329$	−7.00000	−0.385922
$330$	0	0
$331$	−3.00000	−0.164895	−0.0824475	−	0.996595i	$-0.526274\pi$
$331$	−3.00000	−0.164895	−0.0824475	+	0.996595i	$0.526274\pi$
$332$	−4.00000	−0.219529
$333$	−42.0000	−2.30159
$334$	−8.00000	−0.437741
$335$	0	0
$336$	−3.00000	−0.163663
$337$	9.00000	0.490261	0.245131	−	0.969490i	$-0.421169\pi$
$337$	9.00000	0.490261	0.245131	+	0.969490i	$0.421169\pi$
$338$	0	0
$339$	3.00000	0.162938
$340$	0	0
$341$	−5.00000	−0.270765
$342$	−12.0000	−0.648886
$343$	−1.00000	−0.0539949
$344$	−12.0000	−0.646997
$345$	0	0
$346$	18.0000	0.967686
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	12.0000	0.643268
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	5.00000	0.267261
$351$	0	0
$352$	5.00000	0.266501
$353$	−3.00000	−0.159674	−0.0798369	−	0.996808i	$-0.525440\pi$
$353$	−3.00000	−0.159674	−0.0798369	+	0.996808i	$0.525440\pi$
$354$	18.0000	0.956689
$355$	0	0
$356$	−14.0000	−0.741999
$357$	12.0000	0.635107
$358$	−2.00000	−0.105703
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−5.00000	−0.262794
$363$	42.0000	2.20443
$364$	0	0
$365$	0	0
$366$	39.0000	2.03856
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	5.00000	0.260643
$369$	54.0000	2.81113
$370$	0	0
$371$	4.00000	0.207670
$372$	−3.00000	−0.155543
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	−20.0000	−1.03418
$375$	0	0
$376$	7.00000	0.360997
$377$	0	0
$378$	−9.00000	−0.462910
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	27.0000	1.38325
$382$	−16.0000	−0.818631
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	−4.00000	−0.203595
$387$	−72.0000	−3.65997
$388$	−5.00000	−0.253837
$389$	8.00000	0.405616	0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000	0.405616	0.202808	+	0.979219i	$0.434993\pi$
$390$	0	0
$391$	−20.0000	−1.01144
$392$	1.00000	0.0505076
$393$	24.0000	1.21064
$394$	−27.0000	−1.36024
$395$	0	0
$396$	30.0000	1.50756
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	−20.0000	−1.00251
$399$	6.00000	0.300376
$400$	−5.00000	−0.250000
$401$	28.0000	1.39825	0.699127	−	0.714998i	$-0.253572\pi$
$401$	28.0000	1.39825	0.699127	+	0.714998i	$0.253572\pi$
$402$	−33.0000	−1.64589
$403$	0	0
$404$	15.0000	0.746278
$405$	0	0
$406$	−4.00000	−0.198517
$407$	−35.0000	−1.73489
$408$	−12.0000	−0.594089
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	−54.0000	−2.66362
$412$	6.00000	0.295599
$413$	−6.00000	−0.295241
$414$	30.0000	1.47442
$415$	0	0
$416$	0	0
$417$	12.0000	0.587643
$418$	−10.0000	−0.489116
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	0	0
$421$	35.0000	1.70580	0.852898	−	0.522078i	$-0.174843\pi$
$421$	35.0000	1.70580	0.852898	+	0.522078i	$0.174843\pi$
$422$	−14.0000	−0.681509
$423$	42.0000	2.04211
$424$	−4.00000	−0.194257
$425$	20.0000	0.970143
$426$	0	0
$427$	−13.0000	−0.629114
$428$	−8.00000	−0.386695
$429$	0	0
$430$	0	0
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	9.00000	0.433013
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	1.00000	0.0480015
$435$	0	0
$436$	18.0000	0.862044
$437$	−10.0000	−0.478365
$438$	−21.0000	−1.00342
$439$	−2.00000	−0.0954548	−0.0477274	−	0.998860i	$-0.515198\pi$
$439$	−2.00000	−0.0954548	−0.0477274	+	0.998860i	$0.515198\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	−21.0000	−0.996616
$445$	0	0
$446$	−3.00000	−0.142054
$447$	−21.0000	−0.993266
$448$	−1.00000	−0.0472456
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	−30.0000	−1.41421
$451$	45.0000	2.11897
$452$	1.00000	0.0470360
$453$	36.0000	1.69143
$454$	12.0000	0.563188
$455$	0	0
$456$	−6.00000	−0.280976
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	−16.0000	−0.747631
$459$	−36.0000	−1.68034
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	−15.0000	−0.697863
$463$	−12.0000	−0.557687	−0.278844	−	0.960337i	$-0.589951\pi$
$463$	−12.0000	−0.557687	−0.278844	+	0.960337i	$0.589951\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	−21.0000	−0.972806
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	0	0
$469$	11.0000	0.507933
$470$	0	0
$471$	3.00000	0.138233
$472$	6.00000	0.276172
$473$	−60.0000	−2.75880
$474$	−51.0000	−2.34251
$475$	10.0000	0.458831
$476$	4.00000	0.183340
$477$	−24.0000	−1.09888
$478$	6.00000	0.274434
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	0	0
$482$	30.0000	1.36646
$483$	−15.0000	−0.682524
$484$	14.0000	0.636364
$485$	0	0
$486$	0	0
$487$	26.0000	1.17817	0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000	1.17817	0.589086	+	0.808070i	$0.299488\pi$
$488$	13.0000	0.588482
$489$	−12.0000	−0.542659
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	27.0000	1.21725
$493$	−16.0000	−0.720604
$494$	0	0
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	0	0
$498$	−12.0000	−0.537733
$499$	−1.00000	−0.0447661	−0.0223831	−	0.999749i	$-0.507125\pi$
$499$	−1.00000	−0.0447661	−0.0223831	+	0.999749i	$0.507125\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	−13.0000	−0.580218
$503$	−40.0000	−1.78351	−0.891756	−	0.452517i	$-0.850526\pi$
$503$	−40.0000	−1.78351	−0.891756	+	0.452517i	$0.850526\pi$
$504$	−6.00000	−0.267261
$505$	0	0
$506$	25.0000	1.11139
$507$	0	0
$508$	9.00000	0.399310
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	1.00000	0.0441942
$513$	−18.0000	−0.794719
$514$	28.0000	1.23503
$515$	0	0
$516$	−36.0000	−1.58481
$517$	35.0000	1.53930
$518$	7.00000	0.307562
$519$	54.0000	2.37034
$520$	0	0
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	24.0000	1.05045
$523$	−13.0000	−0.568450	−0.284225	−	0.958758i	$-0.591736\pi$
$523$	−13.0000	−0.568450	−0.284225	+	0.958758i	$0.591736\pi$
$524$	8.00000	0.349482
$525$	15.0000	0.654654
$526$	0	0
$527$	4.00000	0.174243
$528$	15.0000	0.652791
$529$	2.00000	0.0869565
$530$	0	0
$531$	36.0000	1.56227
$532$	2.00000	0.0867110
$533$	0	0
$534$	−42.0000	−1.81752
$535$	0	0
$536$	−11.0000	−0.475128
$537$	−6.00000	−0.258919
$538$	13.0000	0.560470
$539$	5.00000	0.215365
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	1.00000	0.0429537
$543$	−15.0000	−0.643712
$544$	−4.00000	−0.171499
$545$	0	0
$546$	0	0
$547$	−32.0000	−1.36822	−0.684111	−	0.729378i	$-0.739809\pi$
$547$	−32.0000	−1.36822	−0.684111	+	0.729378i	$0.739809\pi$
$548$	−18.0000	−0.768922
$549$	78.0000	3.32896
$550$	−25.0000	−1.06600
$551$	−8.00000	−0.340811
$552$	15.0000	0.638442
$553$	17.0000	0.722914
$554$	−10.0000	−0.424859
$555$	0	0
$556$	4.00000	0.169638
$557$	9.00000	0.381342	0.190671	−	0.981654i	$-0.438934\pi$
$557$	9.00000	0.381342	0.190671	+	0.981654i	$0.438934\pi$
$558$	−6.00000	−0.254000
$559$	0	0
$560$	0	0
$561$	−60.0000	−2.53320
$562$	16.0000	0.674919
$563$	35.0000	1.47507	0.737537	−	0.675307i	$-0.235989\pi$
$563$	35.0000	1.47507	0.737537	+	0.675307i	$0.235989\pi$
$564$	21.0000	0.884260
$565$	0	0
$566$	3.00000	0.126099
$567$	−9.00000	−0.377964
$568$	0	0
$569$	21.0000	0.880366	0.440183	−	0.897908i	$-0.354914\pi$
$569$	21.0000	0.880366	0.440183	+	0.897908i	$0.354914\pi$
$570$	0	0
$571$	22.0000	0.920671	0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000	0.920671	0.460336	+	0.887745i	$0.347729\pi$
$572$	0	0
$573$	−48.0000	−2.00523
$574$	−9.00000	−0.375653
$575$	−25.0000	−1.04257
$576$	6.00000	0.250000
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	−1.00000	−0.0415945
$579$	−12.0000	−0.498703
$580$	0	0
$581$	4.00000	0.165948
$582$	−15.0000	−0.621770
$583$	−20.0000	−0.828315
$584$	−7.00000	−0.289662
$585$	0	0
$586$	6.00000	0.247858
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	3.00000	0.123718
$589$	2.00000	0.0824086
$590$	0	0
$591$	−81.0000	−3.33189
$592$	−7.00000	−0.287698
$593$	46.0000	1.88899	0.944497	−	0.328521i	$-0.106550\pi$
$593$	46.0000	1.88899	0.944497	+	0.328521i	$0.106550\pi$
$594$	45.0000	1.84637
$595$	0	0
$596$	−7.00000	−0.286731
$597$	−60.0000	−2.45564
$598$	0	0
$599$	−9.00000	−0.367730	−0.183865	−	0.982952i	$-0.558861\pi$
$599$	−9.00000	−0.367730	−0.183865	+	0.982952i	$0.558861\pi$
$600$	−15.0000	−0.612372
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	12.0000	0.489083
$603$	−66.0000	−2.68773
$604$	12.0000	0.488273
$605$	0	0
$606$	45.0000	1.82800
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	−2.00000	−0.0811107
$609$	−12.0000	−0.486265
$610$	0	0
$611$	0	0
$612$	−24.0000	−0.970143
$613$	23.0000	0.928961	0.464481	−	0.885583i	$-0.346241\pi$
$613$	23.0000	0.928961	0.464481	+	0.885583i	$0.346241\pi$
$614$	−32.0000	−1.29141
$615$	0	0
$616$	−5.00000	−0.201456
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	18.0000	0.724066
$619$	−6.00000	−0.241160	−0.120580	−	0.992704i	$-0.538475\pi$
$619$	−6.00000	−0.241160	−0.120580	+	0.992704i	$0.538475\pi$
$620$	0	0
$621$	45.0000	1.80579
$622$	26.0000	1.04251
$623$	14.0000	0.560898
$624$	0	0
$625$	25.0000	1.00000
$626$	−30.0000	−1.19904
$627$	−30.0000	−1.19808
$628$	1.00000	0.0399043
$629$	28.0000	1.11643
$630$	0	0
$631$	26.0000	1.03504	0.517522	−	0.855670i	$-0.326855\pi$
$631$	26.0000	1.03504	0.517522	+	0.855670i	$0.326855\pi$
$632$	−17.0000	−0.676224
$633$	−42.0000	−1.66935
$634$	−11.0000	−0.436866
$635$	0	0
$636$	−12.0000	−0.475831
$637$	0	0
$638$	20.0000	0.791808
$639$	0	0
$640$	0	0
$641$	−33.0000	−1.30342	−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000	−1.30342	−0.651711	+	0.758468i	$0.725948\pi$
$642$	−24.0000	−0.947204
$643$	36.0000	1.41970	0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000	1.41970	0.709851	+	0.704352i	$0.248762\pi$
$644$	−5.00000	−0.197028
$645$	0	0
$646$	8.00000	0.314756
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	9.00000	0.353553
$649$	30.0000	1.17760
$650$	0	0
$651$	3.00000	0.117579
$652$	−4.00000	−0.156652
$653$	−12.0000	−0.469596	−0.234798	−	0.972044i	$-0.575443\pi$
$653$	−12.0000	−0.469596	−0.234798	+	0.972044i	$0.575443\pi$
$654$	54.0000	2.11157
$655$	0	0
$656$	9.00000	0.351391
$657$	−42.0000	−1.63858
$658$	−7.00000	−0.272888
$659$	26.0000	1.01282	0.506408	−	0.862294i	$-0.330973\pi$
$659$	26.0000	1.01282	0.506408	+	0.862294i	$0.330973\pi$
$660$	0	0
$661$	20.0000	0.777910	0.388955	−	0.921257i	$-0.372836\pi$
$661$	20.0000	0.777910	0.388955	+	0.921257i	$0.372836\pi$
$662$	−3.00000	−0.116598
$663$	0	0
$664$	−4.00000	−0.155230
$665$	0	0
$666$	−42.0000	−1.62747
$667$	20.0000	0.774403
$668$	−8.00000	−0.309529
$669$	−9.00000	−0.347960
$670$	0	0
$671$	65.0000	2.50930
$672$	−3.00000	−0.115728
$673$	−9.00000	−0.346925	−0.173462	−	0.984841i	$-0.555495\pi$
$673$	−9.00000	−0.346925	−0.173462	+	0.984841i	$0.555495\pi$
$674$	9.00000	0.346667
$675$	−45.0000	−1.73205
$676$	0	0
$677$	3.00000	0.115299	0.0576497	−	0.998337i	$-0.481639\pi$
$677$	3.00000	0.115299	0.0576497	+	0.998337i	$0.481639\pi$
$678$	3.00000	0.115214
$679$	5.00000	0.191882
$680$	0	0
$681$	36.0000	1.37952
$682$	−5.00000	−0.191460
$683$	−3.00000	−0.114792	−0.0573959	−	0.998351i	$-0.518280\pi$
$683$	−3.00000	−0.114792	−0.0573959	+	0.998351i	$0.518280\pi$
$684$	−12.0000	−0.458831
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	−48.0000	−1.83131
$688$	−12.0000	−0.457496
$689$	0	0
$690$	0	0
$691$	36.0000	1.36950	0.684752	−	0.728776i	$-0.259910\pi$
$691$	36.0000	1.36950	0.684752	+	0.728776i	$0.259910\pi$
$692$	18.0000	0.684257
$693$	−30.0000	−1.13961
$694$	16.0000	0.607352
$695$	0	0
$696$	12.0000	0.454859
$697$	−36.0000	−1.36360
$698$	−2.00000	−0.0757011
$699$	−63.0000	−2.38288
$700$	5.00000	0.188982
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	0	0
$703$	14.0000	0.528020
$704$	5.00000	0.188445
$705$	0	0
$706$	−3.00000	−0.112906
$707$	−15.0000	−0.564133
$708$	18.0000	0.676481
$709$	−29.0000	−1.08912	−0.544559	−	0.838723i	$-0.683303\pi$
$709$	−29.0000	−1.08912	−0.544559	+	0.838723i	$0.683303\pi$
$710$	0	0
$711$	−102.000	−3.82530
$712$	−14.0000	−0.524672
$713$	−5.00000	−0.187251
$714$	12.0000	0.449089
$715$	0	0
$716$	−2.00000	−0.0747435
$717$	18.0000	0.672222
$718$	4.00000	0.149279
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	0	0
$721$	−6.00000	−0.223452
$722$	−15.0000	−0.558242
$723$	90.0000	3.34714
$724$	−5.00000	−0.185824
$725$	−20.0000	−0.742781
$726$	42.0000	1.55877
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	48.0000	1.77534
$732$	39.0000	1.44148
$733$	−20.0000	−0.738717	−0.369358	−	0.929287i	$-0.620423\pi$
$733$	−20.0000	−0.738717	−0.369358	+	0.929287i	$0.620423\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	5.00000	0.184302
$737$	−55.0000	−2.02595
$738$	54.0000	1.98777
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	0	0
$742$	4.00000	0.146845
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	−3.00000	−0.109985
$745$	0	0
$746$	4.00000	0.146450
$747$	−24.0000	−0.878114
$748$	−20.0000	−0.731272
$749$	8.00000	0.292314
$750$	0	0
$751$	25.0000	0.912263	0.456131	−	0.889912i	$-0.349235\pi$
$751$	25.0000	0.912263	0.456131	+	0.889912i	$0.349235\pi$
$752$	7.00000	0.255264
$753$	−39.0000	−1.42124
$754$	0	0
$755$	0	0
$756$	−9.00000	−0.327327
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	16.0000	0.581146
$759$	75.0000	2.72233
$760$	0	0
$761$	33.0000	1.19625	0.598125	−	0.801403i	$-0.295913\pi$
$761$	33.0000	1.19625	0.598125	+	0.801403i	$0.295913\pi$
$762$	27.0000	0.978107
$763$	−18.0000	−0.651644
$764$	−16.0000	−0.578860
$765$	0	0
$766$	9.00000	0.325183
$767$	0	0
$768$	3.00000	0.108253
$769$	1.00000	0.0360609	0.0180305	−	0.999837i	$-0.494260\pi$
$769$	1.00000	0.0360609	0.0180305	+	0.999837i	$0.494260\pi$
$770$	0	0
$771$	84.0000	3.02519
$772$	−4.00000	−0.143963
$773$	−34.0000	−1.22290	−0.611448	−	0.791285i	$-0.709412\pi$
$773$	−34.0000	−1.22290	−0.611448	+	0.791285i	$0.709412\pi$
$774$	−72.0000	−2.58799
$775$	5.00000	0.179605
$776$	−5.00000	−0.179490
$777$	21.0000	0.753371
$778$	8.00000	0.286814
$779$	−18.0000	−0.644917
$780$	0	0
$781$	0	0
$782$	−20.0000	−0.715199
$783$	36.0000	1.28654
$784$	1.00000	0.0357143
$785$	0	0
$786$	24.0000	0.856052
$787$	−8.00000	−0.285169	−0.142585	−	0.989783i	$-0.545541\pi$
$787$	−8.00000	−0.285169	−0.142585	+	0.989783i	$0.545541\pi$
$788$	−27.0000	−0.961835
$789$	0	0
$790$	0	0
$791$	−1.00000	−0.0355559
$792$	30.0000	1.06600
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−20.0000	−0.708881
$797$	3.00000	0.106265	0.0531327	−	0.998587i	$-0.483079\pi$
$797$	3.00000	0.106265	0.0531327	+	0.998587i	$0.483079\pi$
$798$	6.00000	0.212398
$799$	−28.0000	−0.990569
$800$	−5.00000	−0.176777
$801$	−84.0000	−2.96799
$802$	28.0000	0.988714
$803$	−35.0000	−1.23512
$804$	−33.0000	−1.16382
$805$	0	0
$806$	0	0
$807$	39.0000	1.37287
$808$	15.0000	0.527698
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−52.0000	−1.82597	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000	−1.82597	−0.912983	+	0.407997i	$0.866228\pi$
$812$	−4.00000	−0.140372
$813$	3.00000	0.105215
$814$	−35.0000	−1.22675
$815$	0	0
$816$	−12.0000	−0.420084
$817$	24.0000	0.839654
$818$	6.00000	0.209785
$819$	0	0
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	−54.0000	−1.88347
$823$	−15.0000	−0.522867	−0.261434	−	0.965221i	$-0.584195\pi$
$823$	−15.0000	−0.522867	−0.261434	+	0.965221i	$0.584195\pi$
$824$	6.00000	0.209020
$825$	−75.0000	−2.61116
$826$	−6.00000	−0.208767
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	30.0000	1.04257
$829$	−54.0000	−1.87550	−0.937749	−	0.347314i	$-0.887094\pi$
$829$	−54.0000	−1.87550	−0.937749	+	0.347314i	$0.887094\pi$
$830$	0	0
$831$	−30.0000	−1.04069
$832$	0	0
$833$	−4.00000	−0.138592
$834$	12.0000	0.415526
$835$	0	0
$836$	−10.0000	−0.345857
$837$	−9.00000	−0.311086
$838$	−9.00000	−0.310900
$839$	−47.0000	−1.62262	−0.811310	−	0.584616i	$-0.801245\pi$
$839$	−47.0000	−1.62262	−0.811310	+	0.584616i	$0.801245\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	35.0000	1.20618
$843$	48.0000	1.65321
$844$	−14.0000	−0.481900
$845$	0	0
$846$	42.0000	1.44399
$847$	−14.0000	−0.481046
$848$	−4.00000	−0.137361
$849$	9.00000	0.308879
$850$	20.0000	0.685994
$851$	−35.0000	−1.19978
$852$	0	0
$853$	16.0000	0.547830	0.273915	−	0.961754i	$-0.411681\pi$
$853$	16.0000	0.547830	0.273915	+	0.961754i	$0.411681\pi$
$854$	−13.0000	−0.444851
$855$	0	0
$856$	−8.00000	−0.273434
$857$	2.00000	0.0683187	0.0341593	−	0.999416i	$-0.489125\pi$
$857$	2.00000	0.0683187	0.0341593	+	0.999416i	$0.489125\pi$
$858$	0	0
$859$	29.0000	0.989467	0.494734	−	0.869045i	$-0.335266\pi$
$859$	29.0000	0.989467	0.494734	+	0.869045i	$0.335266\pi$
$860$	0	0
$861$	−27.0000	−0.920158
$862$	6.00000	0.204361
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	9.00000	0.306186
$865$	0	0
$866$	26.0000	0.883516
$867$	−3.00000	−0.101885
$868$	1.00000	0.0339422
$869$	−85.0000	−2.88343
$870$	0	0
$871$	0	0
$872$	18.0000	0.609557
$873$	−30.0000	−1.01535
$874$	−10.0000	−0.338255
$875$	0	0
$876$	−21.0000	−0.709524
$877$	−13.0000	−0.438979	−0.219489	−	0.975615i	$-0.570439\pi$
$877$	−13.0000	−0.438979	−0.219489	+	0.975615i	$0.570439\pi$
$878$	−2.00000	−0.0674967
$879$	18.0000	0.607125
$880$	0	0
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	6.00000	0.202031
$883$	30.0000	1.00958	0.504790	−	0.863242i	$-0.331570\pi$
$883$	30.0000	1.00958	0.504790	+	0.863242i	$0.331570\pi$
$884$	0	0
$885$	0	0
$886$	−12.0000	−0.403148
$887$	54.0000	1.81314	0.906571	−	0.422053i	$-0.138690\pi$
$887$	54.0000	1.81314	0.906571	+	0.422053i	$0.138690\pi$
$888$	−21.0000	−0.704714
$889$	−9.00000	−0.301850
$890$	0	0
$891$	45.0000	1.50756
$892$	−3.00000	−0.100447
$893$	−14.0000	−0.468492
$894$	−21.0000	−0.702345
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−18.0000	−0.600668
$899$	−4.00000	−0.133407
$900$	−30.0000	−1.00000
$901$	16.0000	0.533037
$902$	45.0000	1.49834
$903$	36.0000	1.19800
$904$	1.00000	0.0332595
$905$	0	0
$906$	36.0000	1.19602
$907$	14.0000	0.464862	0.232431	−	0.972613i	$-0.425332\pi$
$907$	14.0000	0.464862	0.232431	+	0.972613i	$0.425332\pi$
$908$	12.0000	0.398234
$909$	90.0000	2.98511
$910$	0	0
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	−6.00000	−0.198680
$913$	−20.0000	−0.661903
$914$	10.0000	0.330771
$915$	0	0
$916$	−16.0000	−0.528655
$917$	−8.00000	−0.264183
$918$	−36.0000	−1.18818
$919$	31.0000	1.02260	0.511298	−	0.859404i	$-0.329165\pi$
$919$	31.0000	1.02260	0.511298	+	0.859404i	$0.329165\pi$
$920$	0	0
$921$	−96.0000	−3.16331
$922$	−12.0000	−0.395199
$923$	0	0
$924$	−15.0000	−0.493464
$925$	35.0000	1.15079
$926$	−12.0000	−0.394344
$927$	36.0000	1.18240
$928$	4.00000	0.131306
$929$	29.0000	0.951459	0.475730	−	0.879592i	$-0.342184\pi$
$929$	29.0000	0.951459	0.475730	+	0.879592i	$0.342184\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	−21.0000	−0.687878
$933$	78.0000	2.55361
$934$	−36.0000	−1.17796
$935$	0	0
$936$	0	0
$937$	−8.00000	−0.261349	−0.130674	−	0.991425i	$-0.541714\pi$
$937$	−8.00000	−0.261349	−0.130674	+	0.991425i	$0.541714\pi$
$938$	11.0000	0.359163
$939$	−90.0000	−2.93704
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	3.00000	0.0977453
$943$	45.0000	1.46540
$944$	6.00000	0.195283
$945$	0	0
$946$	−60.0000	−1.95077
$947$	16.0000	0.519930	0.259965	−	0.965618i	$-0.416289\pi$
$947$	16.0000	0.519930	0.259965	+	0.965618i	$0.416289\pi$
$948$	−51.0000	−1.65640
$949$	0	0
$950$	10.0000	0.324443
$951$	−33.0000	−1.07010
$952$	4.00000	0.129641
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	−24.0000	−0.777029
$955$	0	0
$956$	6.00000	0.194054
$957$	60.0000	1.93952
$958$	16.0000	0.516937
$959$	18.0000	0.581250
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	−48.0000	−1.54678
$964$	30.0000	0.966235
$965$	0	0
$966$	−15.0000	−0.482617
$967$	−2.00000	−0.0643157	−0.0321578	−	0.999483i	$-0.510238\pi$
$967$	−2.00000	−0.0643157	−0.0321578	+	0.999483i	$0.510238\pi$
$968$	14.0000	0.449977
$969$	24.0000	0.770991
$970$	0	0
$971$	9.00000	0.288824	0.144412	−	0.989518i	$-0.453871\pi$
$971$	9.00000	0.288824	0.144412	+	0.989518i	$0.453871\pi$
$972$	0	0
$973$	−4.00000	−0.128234
$974$	26.0000	0.833094
$975$	0	0
$976$	13.0000	0.416120
$977$	26.0000	0.831814	0.415907	−	0.909407i	$-0.363464\pi$
$977$	26.0000	0.831814	0.415907	+	0.909407i	$0.363464\pi$
$978$	−12.0000	−0.383718
$979$	−70.0000	−2.23721
$980$	0	0
$981$	108.000	3.44817
$982$	28.0000	0.893516
$983$	56.0000	1.78612	0.893061	−	0.449935i	$-0.148553\pi$
$983$	56.0000	1.78612	0.893061	+	0.449935i	$0.148553\pi$
$984$	27.0000	0.860729
$985$	0	0
$986$	−16.0000	−0.509544
$987$	−21.0000	−0.668437
$988$	0	0
$989$	−60.0000	−1.90789
$990$	0	0
$991$	59.0000	1.87420	0.937098	−	0.349065i	$-0.113501\pi$
$991$	59.0000	1.87420	0.937098	+	0.349065i	$0.113501\pi$
$992$	−1.00000	−0.0317500
$993$	−9.00000	−0.285606
$994$	0	0
$995$	0	0
$996$	−12.0000	−0.380235
$997$	33.0000	1.04512	0.522560	−	0.852602i	$-0.324977\pi$
$997$	33.0000	1.04512	0.522560	+	0.852602i	$0.324977\pi$
$998$	−1.00000	−0.0316544
$999$	−63.0000	−1.99323

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2366.2.a.o.1.1		1
13.5	odd	4		2366.2.d.i.337.1		2
13.8	odd	4		2366.2.d.i.337.2		2
13.12	even	2		182.2.a.b.1.1	✓	1
39.38	odd	2		1638.2.a.q.1.1		1
52.51	odd	2		1456.2.a.b.1.1		1
65.64	even	2		4550.2.a.o.1.1		1
91.12	odd	6		1274.2.f.u.1145.1		2
91.25	even	6		1274.2.f.m.79.1		2
91.38	odd	6		1274.2.f.u.79.1		2
91.51	even	6		1274.2.f.m.1145.1		2
91.90	odd	2		1274.2.a.a.1.1		1
104.51	odd	2		5824.2.a.be.1.1		1
104.77	even	2		5824.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.2.a.b.1.1	✓	1	13.12	even	2
1274.2.a.a.1.1		1	91.90	odd	2
1274.2.f.m.79.1		2	91.25	even	6
1274.2.f.m.1145.1		2	91.51	even	6
1274.2.f.u.79.1		2	91.38	odd	6
1274.2.f.u.1145.1		2	91.12	odd	6
1456.2.a.b.1.1		1	52.51	odd	2
1638.2.a.q.1.1		1	39.38	odd	2
2366.2.a.o.1.1		1	1.1	even	1	trivial
2366.2.d.i.337.1		2	13.5	odd	4
2366.2.d.i.337.2		2	13.8	odd	4
4550.2.a.o.1.1		1	65.64	even	2
5824.2.a.a.1.1		1	104.77	even	2
5824.2.a.be.1.1		1	104.51	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:	\( N \)	\(=\)	\( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2366.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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