LMFDB - Embedded newform 7742.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7742,2,Mod(1,7742)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7742, 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("7742.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7742 = 2 \cdot 7^{2} \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7742.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:	$61.8201812449$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 158)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7742.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} -2.00000 q^{9} +3.00000 q^{10} -1.00000 q^{12} -5.00000 q^{13} +3.00000 q^{15} +1.00000 q^{16} +2.00000 q^{18} -2.00000 q^{19} -3.00000 q^{20} -6.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} +5.00000 q^{26} +5.00000 q^{27} -3.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} -2.00000 q^{36} +2.00000 q^{37} +2.00000 q^{38} +5.00000 q^{39} +3.00000 q^{40} +12.0000 q^{41} +8.00000 q^{43} +6.00000 q^{45} +6.00000 q^{46} +9.00000 q^{47} -1.00000 q^{48} -4.00000 q^{50} -5.00000 q^{52} +6.00000 q^{53} -5.00000 q^{54} +2.00000 q^{57} +9.00000 q^{59} +3.00000 q^{60} -8.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +15.0000 q^{65} -4.00000 q^{67} +6.00000 q^{69} -9.00000 q^{71} +2.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} -4.00000 q^{75} -2.00000 q^{76} -5.00000 q^{78} +1.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -12.0000 q^{82} -18.0000 q^{83} -8.00000 q^{86} -9.00000 q^{89} -6.00000 q^{90} -6.00000 q^{92} -4.00000 q^{93} -9.00000 q^{94} +6.00000 q^{95} +1.00000 q^{96} -17.0000 q^{97} +O(q^{100})\)

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	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	1.00000	0.500000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	−2.00000	−0.666667
$10$	3.00000	0.948683
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−1.00000	−0.288675
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	2.00000	0.471405
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−3.00000	−0.670820
$21$	0	0
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	1.00000	0.204124
$25$	4.00000	0.800000
$26$	5.00000	0.980581
$27$	5.00000	0.962250
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−3.00000	−0.547723
$31$	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
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−2.00000	−0.333333
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	2.00000	0.324443
$39$	5.00000	0.800641
$40$	3.00000	0.474342
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	6.00000	0.894427
$46$	6.00000	0.884652
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−4.00000	−0.565685
$51$	0	0
$52$	−5.00000	−0.693375
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−5.00000	−0.680414
$55$	0	0
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	3.00000	0.387298
$61$	−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
$63$	0	0
$64$	1.00000	0.125000
$65$	15.0000	1.86052
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	2.00000	0.235702
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−2.00000	−0.232495
$75$	−4.00000	−0.461880
$76$	−2.00000	−0.229416
$77$	0	0
$78$	−5.00000	−0.566139
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	−12.0000	−1.32518
$83$	−18.0000	−1.97576	−0.987878	−	0.155230i	$-0.950388\pi$
$83$	−18.0000	−1.97576	−0.987878	+	0.155230i	$0.950388\pi$
$84$	0	0
$85$	0	0
$86$	−8.00000	−0.862662
$87$	0	0
$88$	0	0
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	−6.00000	−0.632456
$91$	0	0
$92$	−6.00000	−0.625543
$93$	−4.00000	−0.414781
$94$	−9.00000	−0.928279
$95$	6.00000	0.615587
$96$	1.00000	0.102062
$97$	−17.0000	−1.72609	−0.863044	−	0.505128i	$-0.831445\pi$
$97$	−17.0000	−1.72609	−0.863044	+	0.505128i	$0.831445\pi$
$98$	0	0
$99$	0	0
$100$	4.00000	0.400000
$101$	15.0000	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$102$	0	0
$103$	13.0000	1.28093	0.640464	−	0.767988i	$-0.278742\pi$
$103$	13.0000	1.28093	0.640464	+	0.767988i	$0.278742\pi$
$104$	5.00000	0.490290
$105$	0	0
$106$	−6.00000	−0.582772
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	5.00000	0.481125
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	−2.00000	−0.187317
$115$	18.0000	1.67851
$116$	0	0
$117$	10.0000	0.924500
$118$	−9.00000	−0.828517
$119$	0	0
$120$	−3.00000	−0.273861
$121$	−11.0000	−1.00000
$122$	8.00000	0.724286
$123$	−12.0000	−1.08200
$124$	4.00000	0.359211
$125$	3.00000	0.268328
$126$	0	0
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	−1.00000	−0.0883883
$129$	−8.00000	−0.704361
$130$	−15.0000	−1.31559
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	0	0
$134$	4.00000	0.345547
$135$	−15.0000	−1.29099
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	−6.00000	−0.510754
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	−9.00000	−0.757937
$142$	9.00000	0.755263
$143$	0	0
$144$	−2.00000	−0.166667
$145$	0	0
$146$	2.00000	0.165521
$147$	0	0
$148$	2.00000	0.164399
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	4.00000	0.326599
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	0	0
$155$	−12.0000	−0.963863
$156$	5.00000	0.400320
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	−1.00000	−0.0795557
$159$	−6.00000	−0.475831
$160$	3.00000	0.237171
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	18.0000	1.39707
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	4.00000	0.305888
$172$	8.00000	0.609994
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−9.00000	−0.676481
$178$	9.00000	0.674579
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	6.00000	0.447214
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	8.00000	0.591377
$184$	6.00000	0.442326
$185$	−6.00000	−0.441129
$186$	4.00000	0.293294
$187$	0	0
$188$	9.00000	0.656392
$189$	0	0
$190$	−6.00000	−0.435286
$191$	−15.0000	−1.08536	−0.542681	−	0.839939i	$-0.682591\pi$
$191$	−15.0000	−1.08536	−0.542681	+	0.839939i	$0.682591\pi$
$192$	−1.00000	−0.0721688
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	17.0000	1.22053
$195$	−15.0000	−1.07417
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	−4.00000	−0.282843
$201$	4.00000	0.282138
$202$	−15.0000	−1.05540
$203$	0	0
$204$	0	0
$205$	−36.0000	−2.51435
$206$	−13.0000	−0.905753
$207$	12.0000	0.834058
$208$	−5.00000	−0.346688
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	6.00000	0.412082
$213$	9.00000	0.616670
$214$	−3.00000	−0.205076
$215$	−24.0000	−1.63679
$216$	−5.00000	−0.340207
$217$	0	0
$218$	−2.00000	−0.135457
$219$	2.00000	0.135147
$220$	0	0
$221$	0	0
$222$	2.00000	0.134231
$223$	−26.0000	−1.74109	−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000	−1.74109	−0.870544	+	0.492090i	$0.836233\pi$
$224$	0	0
$225$	−8.00000	−0.533333
$226$	−18.0000	−1.19734
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	2.00000	0.132453
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	−18.0000	−1.18688
$231$	0	0
$232$	0	0
$233$	−12.0000	−0.786146	−0.393073	−	0.919507i	$-0.628588\pi$
$233$	−12.0000	−0.786146	−0.393073	+	0.919507i	$0.628588\pi$
$234$	−10.0000	−0.653720
$235$	−27.0000	−1.76129
$236$	9.00000	0.585850
$237$	−1.00000	−0.0649570
$238$	0	0
$239$	18.0000	1.16432	0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000	1.16432	0.582162	+	0.813073i	$0.302207\pi$
$240$	3.00000	0.193649
$241$	−17.0000	−1.09507	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	−17.0000	−1.09507	−0.547533	+	0.836784i	$0.684433\pi$
$242$	11.0000	0.707107
$243$	−16.0000	−1.02640
$244$	−8.00000	−0.512148
$245$	0	0
$246$	12.0000	0.765092
$247$	10.0000	0.636285
$248$	−4.00000	−0.254000
$249$	18.0000	1.14070
$250$	−3.00000	−0.189737
$251$	27.0000	1.70422	0.852112	−	0.523359i	$-0.175321\pi$
$251$	27.0000	1.70422	0.852112	+	0.523359i	$0.175321\pi$
$252$	0	0
$253$	0	0
$254$	7.00000	0.439219
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	8.00000	0.498058
$259$	0	0
$260$	15.0000	0.930261
$261$	0	0
$262$	6.00000	0.370681
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	0	0
$265$	−18.0000	−1.10573
$266$	0	0
$267$	9.00000	0.550791
$268$	−4.00000	−0.244339
$269$	21.0000	1.28039	0.640196	−	0.768211i	$-0.278853\pi$
$269$	21.0000	1.28039	0.640196	+	0.768211i	$0.278853\pi$
$270$	15.0000	0.912871
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	0	0
$274$	−12.0000	−0.724947
$275$	0	0
$276$	6.00000	0.361158
$277$	17.0000	1.02143	0.510716	−	0.859750i	$-0.329381\pi$
$277$	17.0000	1.02143	0.510716	+	0.859750i	$0.329381\pi$
$278$	5.00000	0.299880
$279$	−8.00000	−0.478947
$280$	0	0
$281$	15.0000	0.894825	0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000	0.894825	0.447412	+	0.894328i	$0.352346\pi$
$282$	9.00000	0.535942
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	−9.00000	−0.534052
$285$	−6.00000	−0.355409
$286$	0	0
$287$	0	0
$288$	2.00000	0.117851
$289$	−17.0000	−1.00000
$290$	0	0
$291$	17.0000	0.996558
$292$	−2.00000	−0.117041
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	−27.0000	−1.57200
$296$	−2.00000	−0.116248
$297$	0	0
$298$	−12.0000	−0.695141
$299$	30.0000	1.73494
$300$	−4.00000	−0.230940
$301$	0	0
$302$	−8.00000	−0.460348
$303$	−15.0000	−0.861727
$304$	−2.00000	−0.114708
$305$	24.0000	1.37424
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	−13.0000	−0.739544
$310$	12.0000	0.681554
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	−5.00000	−0.283069
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	1.00000	0.0562544
$317$	−15.0000	−0.842484	−0.421242	−	0.906948i	$-0.638406\pi$
$317$	−15.0000	−0.842484	−0.421242	+	0.906948i	$0.638406\pi$
$318$	6.00000	0.336463
$319$	0	0
$320$	−3.00000	−0.167705
$321$	−3.00000	−0.167444
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	−20.0000	−1.10940
$326$	−20.0000	−1.10770
$327$	−2.00000	−0.110600
$328$	−12.0000	−0.662589
$329$	0	0
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−18.0000	−0.987878
$333$	−4.00000	−0.219199
$334$	12.0000	0.656611
$335$	12.0000	0.655630
$336$	0	0
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	−12.0000	−0.652714
$339$	−18.0000	−0.977626
$340$	0	0
$341$	0	0
$342$	−4.00000	−0.216295
$343$	0	0
$344$	−8.00000	−0.431331
$345$	−18.0000	−0.969087
$346$	−6.00000	−0.322562
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	0	0
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	0	0
$351$	−25.0000	−1.33440
$352$	0	0
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	9.00000	0.478345
$355$	27.0000	1.43301
$356$	−9.00000	−0.476999
$357$	0	0
$358$	−18.0000	−0.951330
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	−6.00000	−0.316228
$361$	−15.0000	−0.789474
$362$	2.00000	0.105118
$363$	11.0000	0.577350
$364$	0	0
$365$	6.00000	0.314054
$366$	−8.00000	−0.418167
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	−6.00000	−0.312772
$369$	−24.0000	−1.24939
$370$	6.00000	0.311925
$371$	0	0
$372$	−4.00000	−0.207390
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	−9.00000	−0.464140
$377$	0	0
$378$	0	0
$379$	11.0000	0.565032	0.282516	−	0.959263i	$-0.408831\pi$
$379$	11.0000	0.565032	0.282516	+	0.959263i	$0.408831\pi$
$380$	6.00000	0.307794
$381$	7.00000	0.358621
$382$	15.0000	0.767467
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	22.0000	1.11977
$387$	−16.0000	−0.813326
$388$	−17.0000	−0.863044
$389$	9.00000	0.456318	0.228159	−	0.973624i	$-0.426729\pi$
$389$	9.00000	0.456318	0.228159	+	0.973624i	$0.426729\pi$
$390$	15.0000	0.759555
$391$	0	0
$392$	0	0
$393$	6.00000	0.302660
$394$	−6.00000	−0.302276
$395$	−3.00000	−0.150946
$396$	0	0
$397$	7.00000	0.351320	0.175660	−	0.984451i	$-0.443794\pi$
$397$	7.00000	0.351320	0.175660	+	0.984451i	$0.443794\pi$
$398$	11.0000	0.551380
$399$	0	0
$400$	4.00000	0.200000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	−4.00000	−0.199502
$403$	−20.0000	−0.996271
$404$	15.0000	0.746278
$405$	−3.00000	−0.149071
$406$	0	0
$407$	0	0
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	36.0000	1.77791
$411$	−12.0000	−0.591916
$412$	13.0000	0.640464
$413$	0	0
$414$	−12.0000	−0.589768
$415$	54.0000	2.65076
$416$	5.00000	0.245145
$417$	5.00000	0.244851
$418$	0	0
$419$	−27.0000	−1.31904	−0.659518	−	0.751689i	$-0.729240\pi$
$419$	−27.0000	−1.31904	−0.659518	+	0.751689i	$0.729240\pi$
$420$	0	0
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	4.00000	0.194717
$423$	−18.0000	−0.875190
$424$	−6.00000	−0.291386
$425$	0	0
$426$	−9.00000	−0.436051
$427$	0	0
$428$	3.00000	0.145010
$429$	0	0
$430$	24.0000	1.15738
$431$	−6.00000	−0.289010	−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000	−0.289010	−0.144505	+	0.989504i	$0.546159\pi$
$432$	5.00000	0.240563
$433$	−29.0000	−1.39365	−0.696826	−	0.717241i	$-0.745405\pi$
$433$	−29.0000	−1.39365	−0.696826	+	0.717241i	$0.745405\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	12.0000	0.574038
$438$	−2.00000	−0.0955637
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	−2.00000	−0.0949158
$445$	27.0000	1.27992
$446$	26.0000	1.23114
$447$	−12.0000	−0.567581
$448$	0	0
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	8.00000	0.377124
$451$	0	0
$452$	18.0000	0.846649
$453$	−8.00000	−0.375873
$454$	12.0000	0.563188
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	−37.0000	−1.73079	−0.865393	−	0.501093i	$-0.832931\pi$
$457$	−37.0000	−1.73079	−0.865393	+	0.501093i	$0.832931\pi$
$458$	−22.0000	−1.02799
$459$	0	0
$460$	18.0000	0.839254
$461$	−42.0000	−1.95614	−0.978068	−	0.208288i	$-0.933211\pi$
$461$	−42.0000	−1.95614	−0.978068	+	0.208288i	$0.933211\pi$
$462$	0	0
$463$	41.0000	1.90543	0.952716	−	0.303863i	$-0.0982765\pi$
$463$	41.0000	1.90543	0.952716	+	0.303863i	$0.0982765\pi$
$464$	0	0
$465$	12.0000	0.556487
$466$	12.0000	0.555889
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	10.0000	0.462250
$469$	0	0
$470$	27.0000	1.24542
$471$	−4.00000	−0.184310
$472$	−9.00000	−0.414259
$473$	0	0
$474$	1.00000	0.0459315
$475$	−8.00000	−0.367065
$476$	0	0
$477$	−12.0000	−0.549442
$478$	−18.0000	−0.823301
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	−3.00000	−0.136931
$481$	−10.0000	−0.455961
$482$	17.0000	0.774329
$483$	0	0
$484$	−11.0000	−0.500000
$485$	51.0000	2.31579
$486$	16.0000	0.725775
$487$	−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
$489$	−20.0000	−0.904431
$490$	0	0
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	−12.0000	−0.541002
$493$	0	0
$494$	−10.0000	−0.449921
$495$	0	0
$496$	4.00000	0.179605
$497$	0	0
$498$	−18.0000	−0.806599
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	3.00000	0.134164
$501$	12.0000	0.536120
$502$	−27.0000	−1.20507
$503$	21.0000	0.936344	0.468172	−	0.883637i	$-0.344913\pi$
$503$	21.0000	0.936344	0.468172	+	0.883637i	$0.344913\pi$
$504$	0	0
$505$	−45.0000	−2.00247
$506$	0	0
$507$	−12.0000	−0.532939
$508$	−7.00000	−0.310575
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−10.0000	−0.441511
$514$	−18.0000	−0.793946
$515$	−39.0000	−1.71855
$516$	−8.00000	−0.352180
$517$	0	0
$518$	0	0
$519$	−6.00000	−0.263371
$520$	−15.0000	−0.657794
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	0	0
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	18.0000	0.784837
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	18.0000	0.781870
$531$	−18.0000	−0.781133
$532$	0	0
$533$	−60.0000	−2.59889
$534$	−9.00000	−0.389468
$535$	−9.00000	−0.389104
$536$	4.00000	0.172774
$537$	−18.0000	−0.776757
$538$	−21.0000	−0.905374
$539$	0	0
$540$	−15.0000	−0.645497
$541$	29.0000	1.24681	0.623404	−	0.781900i	$-0.285749\pi$
$541$	29.0000	1.24681	0.623404	+	0.781900i	$0.285749\pi$
$542$	−16.0000	−0.687259
$543$	2.00000	0.0858282
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	26.0000	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	12.0000	0.512615
$549$	16.0000	0.682863
$550$	0	0
$551$	0	0
$552$	−6.00000	−0.255377
$553$	0	0
$554$	−17.0000	−0.722261
$555$	6.00000	0.254686
$556$	−5.00000	−0.212047
$557$	15.0000	0.635570	0.317785	−	0.948163i	$-0.397061\pi$
$557$	15.0000	0.635570	0.317785	+	0.948163i	$0.397061\pi$
$558$	8.00000	0.338667
$559$	−40.0000	−1.69182
$560$	0	0
$561$	0	0
$562$	−15.0000	−0.632737
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	−9.00000	−0.378968
$565$	−54.0000	−2.27180
$566$	14.0000	0.588464
$567$	0	0
$568$	9.00000	0.377632
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	6.00000	0.251312
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	0	0
$573$	15.0000	0.626634
$574$	0	0
$575$	−24.0000	−1.00087
$576$	−2.00000	−0.0833333
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	17.0000	0.707107
$579$	22.0000	0.914289
$580$	0	0
$581$	0	0
$582$	−17.0000	−0.704673
$583$	0	0
$584$	2.00000	0.0827606
$585$	−30.0000	−1.24035
$586$	−6.00000	−0.247858
$587$	21.0000	0.866763	0.433381	−	0.901211i	$-0.357320\pi$
$587$	21.0000	0.866763	0.433381	+	0.901211i	$0.357320\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	27.0000	1.11157
$591$	−6.00000	−0.246807
$592$	2.00000	0.0821995
$593$	21.0000	0.862367	0.431183	−	0.902264i	$-0.358096\pi$
$593$	21.0000	0.862367	0.431183	+	0.902264i	$0.358096\pi$
$594$	0	0
$595$	0	0
$596$	12.0000	0.491539
$597$	11.0000	0.450200
$598$	−30.0000	−1.22679
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	4.00000	0.163299
$601$	28.0000	1.14214	0.571072	−	0.820900i	$-0.306528\pi$
$601$	28.0000	1.14214	0.571072	+	0.820900i	$0.306528\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	8.00000	0.325515
$605$	33.0000	1.34164
$606$	15.0000	0.609333
$607$	−5.00000	−0.202944	−0.101472	−	0.994838i	$-0.532355\pi$
$607$	−5.00000	−0.202944	−0.101472	+	0.994838i	$0.532355\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	−24.0000	−0.971732
$611$	−45.0000	−1.82051
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	20.0000	0.807134
$615$	36.0000	1.45166
$616$	0	0
$617$	−39.0000	−1.57008	−0.785040	−	0.619445i	$-0.787358\pi$
$617$	−39.0000	−1.57008	−0.785040	+	0.619445i	$0.787358\pi$
$618$	13.0000	0.522937
$619$	1.00000	0.0401934	0.0200967	−	0.999798i	$-0.493603\pi$
$619$	1.00000	0.0401934	0.0200967	+	0.999798i	$0.493603\pi$
$620$	−12.0000	−0.481932
$621$	−30.0000	−1.20386
$622$	12.0000	0.481156
$623$	0	0
$624$	5.00000	0.200160
$625$	−29.0000	−1.16000
$626$	−10.0000	−0.399680
$627$	0	0
$628$	4.00000	0.159617
$629$	0	0
$630$	0	0
$631$	−43.0000	−1.71180	−0.855901	−	0.517139i	$-0.826997\pi$
$631$	−43.0000	−1.71180	−0.855901	+	0.517139i	$0.826997\pi$
$632$	−1.00000	−0.0397779
$633$	4.00000	0.158986
$634$	15.0000	0.595726
$635$	21.0000	0.833360
$636$	−6.00000	−0.237915
$637$	0	0
$638$	0	0
$639$	18.0000	0.712069
$640$	3.00000	0.118585
$641$	27.0000	1.06644	0.533218	−	0.845978i	$-0.320983\pi$
$641$	27.0000	1.06644	0.533218	+	0.845978i	$0.320983\pi$
$642$	3.00000	0.118401
$643$	−14.0000	−0.552106	−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000	−0.552106	−0.276053	+	0.961142i	$0.589027\pi$
$644$	0	0
$645$	24.0000	0.944999
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	20.0000	0.784465
$651$	0	0
$652$	20.0000	0.783260
$653$	−21.0000	−0.821794	−0.410897	−	0.911682i	$-0.634784\pi$
$653$	−21.0000	−0.821794	−0.410897	+	0.911682i	$0.634784\pi$
$654$	2.00000	0.0782062
$655$	18.0000	0.703318
$656$	12.0000	0.468521
$657$	4.00000	0.156055
$658$	0	0
$659$	21.0000	0.818044	0.409022	−	0.912525i	$-0.365870\pi$
$659$	21.0000	0.818044	0.409022	+	0.912525i	$0.365870\pi$
$660$	0	0
$661$	4.00000	0.155582	0.0777910	−	0.996970i	$-0.475213\pi$
$661$	4.00000	0.155582	0.0777910	+	0.996970i	$0.475213\pi$
$662$	−8.00000	−0.310929
$663$	0	0
$664$	18.0000	0.698535
$665$	0	0
$666$	4.00000	0.154997
$667$	0	0
$668$	−12.0000	−0.464294
$669$	26.0000	1.00522
$670$	−12.0000	−0.463600
$671$	0	0
$672$	0	0
$673$	−28.0000	−1.07932	−0.539660	−	0.841883i	$-0.681447\pi$
$673$	−28.0000	−1.07932	−0.539660	+	0.841883i	$0.681447\pi$
$674$	13.0000	0.500741
$675$	20.0000	0.769800
$676$	12.0000	0.461538
$677$	−15.0000	−0.576497	−0.288248	−	0.957556i	$-0.593073\pi$
$677$	−15.0000	−0.576497	−0.288248	+	0.957556i	$0.593073\pi$
$678$	18.0000	0.691286
$679$	0	0
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	30.0000	1.14792	0.573959	−	0.818884i	$-0.305407\pi$
$683$	30.0000	1.14792	0.573959	+	0.818884i	$0.305407\pi$
$684$	4.00000	0.152944
$685$	−36.0000	−1.37549
$686$	0	0
$687$	−22.0000	−0.839352
$688$	8.00000	0.304997
$689$	−30.0000	−1.14291
$690$	18.0000	0.685248
$691$	−35.0000	−1.33146	−0.665731	−	0.746191i	$-0.731880\pi$
$691$	−35.0000	−1.33146	−0.665731	+	0.746191i	$0.731880\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	6.00000	0.227757
$695$	15.0000	0.568982
$696$	0	0
$697$	0	0
$698$	−28.0000	−1.05982
$699$	12.0000	0.453882
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	25.0000	0.943564
$703$	−4.00000	−0.150863
$704$	0	0
$705$	27.0000	1.01688
$706$	24.0000	0.903252
$707$	0	0
$708$	−9.00000	−0.338241
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	−27.0000	−1.01329
$711$	−2.00000	−0.0750059
$712$	9.00000	0.337289
$713$	−24.0000	−0.898807
$714$	0	0
$715$	0	0
$716$	18.0000	0.672692
$717$	−18.0000	−0.672222
$718$	15.0000	0.559795
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	6.00000	0.223607
$721$	0	0
$722$	15.0000	0.558242
$723$	17.0000	0.632237
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	−11.0000	−0.408248
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	−6.00000	−0.222070
$731$	0	0
$732$	8.00000	0.295689
$733$	−50.0000	−1.84679	−0.923396	−	0.383849i	$-0.874598\pi$
$733$	−50.0000	−1.84679	−0.923396	+	0.383849i	$0.874598\pi$
$734$	−28.0000	−1.03350
$735$	0	0
$736$	6.00000	0.221163
$737$	0	0
$738$	24.0000	0.883452
$739$	−43.0000	−1.58178	−0.790890	−	0.611958i	$-0.790382\pi$
$739$	−43.0000	−1.58178	−0.790890	+	0.611958i	$0.790382\pi$
$740$	−6.00000	−0.220564
$741$	−10.0000	−0.367359
$742$	0	0
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	4.00000	0.146647
$745$	−36.0000	−1.31894
$746$	22.0000	0.805477
$747$	36.0000	1.31717
$748$	0	0
$749$	0	0
$750$	3.00000	0.109545
$751$	50.0000	1.82453	0.912263	−	0.409605i	$-0.134333\pi$
$751$	50.0000	1.82453	0.912263	+	0.409605i	$0.134333\pi$
$752$	9.00000	0.328196
$753$	−27.0000	−0.983935
$754$	0	0
$755$	−24.0000	−0.873449
$756$	0	0
$757$	29.0000	1.05402	0.527011	−	0.849858i	$-0.323312\pi$
$757$	29.0000	1.05402	0.527011	+	0.849858i	$0.323312\pi$
$758$	−11.0000	−0.399538
$759$	0	0
$760$	−6.00000	−0.217643
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	−7.00000	−0.253583
$763$	0	0
$764$	−15.0000	−0.542681
$765$	0	0
$766$	6.00000	0.216789
$767$	−45.0000	−1.62486
$768$	−1.00000	−0.0360844
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	−22.0000	−0.791797
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	16.0000	0.575108
$775$	16.0000	0.574737
$776$	17.0000	0.610264
$777$	0	0
$778$	−9.00000	−0.322666
$779$	−24.0000	−0.859889
$780$	−15.0000	−0.537086
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−12.0000	−0.428298
$786$	−6.00000	−0.214013
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	6.00000	0.213741
$789$	18.0000	0.640817
$790$	3.00000	0.106735
$791$	0	0
$792$	0	0
$793$	40.0000	1.42044
$794$	−7.00000	−0.248421
$795$	18.0000	0.638394
$796$	−11.0000	−0.389885
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	0	0
$800$	−4.00000	−0.141421
$801$	18.0000	0.635999
$802$	6.00000	0.211867
$803$	0	0
$804$	4.00000	0.141069
$805$	0	0
$806$	20.0000	0.704470
$807$	−21.0000	−0.739235
$808$	−15.0000	−0.527698
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	3.00000	0.105409
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	−16.0000	−0.561144
$814$	0	0
$815$	−60.0000	−2.10171
$816$	0	0
$817$	−16.0000	−0.559769
$818$	−22.0000	−0.769212
$819$	0	0
$820$	−36.0000	−1.25717
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	12.0000	0.418548
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	−13.0000	−0.452876
$825$	0	0
$826$	0	0
$827$	21.0000	0.730242	0.365121	−	0.930960i	$-0.381028\pi$
$827$	21.0000	0.730242	0.365121	+	0.930960i	$0.381028\pi$
$828$	12.0000	0.417029
$829$	16.0000	0.555703	0.277851	−	0.960624i	$-0.410378\pi$
$829$	16.0000	0.555703	0.277851	+	0.960624i	$0.410378\pi$
$830$	−54.0000	−1.87437
$831$	−17.0000	−0.589723
$832$	−5.00000	−0.173344
$833$	0	0
$834$	−5.00000	−0.173136
$835$	36.0000	1.24583
$836$	0	0
$837$	20.0000	0.691301
$838$	27.0000	0.932700
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−17.0000	−0.585859
$843$	−15.0000	−0.516627
$844$	−4.00000	−0.137686
$845$	−36.0000	−1.23844
$846$	18.0000	0.618853
$847$	0	0
$848$	6.00000	0.206041
$849$	14.0000	0.480479
$850$	0	0
$851$	−12.0000	−0.411355
$852$	9.00000	0.308335
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	−12.0000	−0.410391
$856$	−3.00000	−0.102538
$857$	−21.0000	−0.717346	−0.358673	−	0.933463i	$-0.616771\pi$
$857$	−21.0000	−0.717346	−0.358673	+	0.933463i	$0.616771\pi$
$858$	0	0
$859$	−23.0000	−0.784750	−0.392375	−	0.919805i	$-0.628346\pi$
$859$	−23.0000	−0.784750	−0.392375	+	0.919805i	$0.628346\pi$
$860$	−24.0000	−0.818393
$861$	0	0
$862$	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	$0.366057\pi$
$864$	−5.00000	−0.170103
$865$	−18.0000	−0.612018
$866$	29.0000	0.985460
$867$	17.0000	0.577350
$868$	0	0
$869$	0	0
$870$	0	0
$871$	20.0000	0.677674
$872$	−2.00000	−0.0677285
$873$	34.0000	1.15073
$874$	−12.0000	−0.405906
$875$	0	0
$876$	2.00000	0.0675737
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	26.0000	0.877457
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	29.0000	0.975928	0.487964	−	0.872864i	$-0.337740\pi$
$883$	29.0000	0.975928	0.487964	+	0.872864i	$0.337740\pi$
$884$	0	0
$885$	27.0000	0.907595
$886$	36.0000	1.20944
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	2.00000	0.0671156
$889$	0	0
$890$	−27.0000	−0.905042
$891$	0	0
$892$	−26.0000	−0.870544
$893$	−18.0000	−0.602347
$894$	12.0000	0.401340
$895$	−54.0000	−1.80502
$896$	0	0
$897$	−30.0000	−1.00167
$898$	−12.0000	−0.400445
$899$	0	0
$900$	−8.00000	−0.266667
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−18.0000	−0.598671
$905$	6.00000	0.199447
$906$	8.00000	0.265782
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	−12.0000	−0.398234
$909$	−30.0000	−0.995037
$910$	0	0
$911$	42.0000	1.39152	0.695761	−	0.718273i	$-0.255067\pi$
$911$	42.0000	1.39152	0.695761	+	0.718273i	$0.255067\pi$
$912$	2.00000	0.0662266
$913$	0	0
$914$	37.0000	1.22385
$915$	−24.0000	−0.793416
$916$	22.0000	0.726900
$917$	0	0
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−18.0000	−0.593442
$921$	20.0000	0.659022
$922$	42.0000	1.38320
$923$	45.0000	1.48119
$924$	0	0
$925$	8.00000	0.263038
$926$	−41.0000	−1.34734
$927$	−26.0000	−0.853952
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	−12.0000	−0.393496
$931$	0	0
$932$	−12.0000	−0.393073
$933$	12.0000	0.392862
$934$	6.00000	0.196326
$935$	0	0
$936$	−10.0000	−0.326860
$937$	52.0000	1.69877	0.849383	−	0.527777i	$-0.176974\pi$
$937$	52.0000	1.69877	0.849383	+	0.527777i	$0.176974\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	−27.0000	−0.880643
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	4.00000	0.130327
$943$	−72.0000	−2.34464
$944$	9.00000	0.292925
$945$	0	0
$946$	0	0
$947$	−60.0000	−1.94974	−0.974869	−	0.222779i	$-0.928487\pi$
$947$	−60.0000	−1.94974	−0.974869	+	0.222779i	$0.928487\pi$
$948$	−1.00000	−0.0324785
$949$	10.0000	0.324614
$950$	8.00000	0.259554
$951$	15.0000	0.486408
$952$	0	0
$953$	−3.00000	−0.0971795	−0.0485898	−	0.998819i	$-0.515473\pi$
$953$	−3.00000	−0.0971795	−0.0485898	+	0.998819i	$0.515473\pi$
$954$	12.0000	0.388514
$955$	45.0000	1.45617
$956$	18.0000	0.582162
$957$	0	0
$958$	−18.0000	−0.581554
$959$	0	0
$960$	3.00000	0.0968246
$961$	−15.0000	−0.483871
$962$	10.0000	0.322413
$963$	−6.00000	−0.193347
$964$	−17.0000	−0.547533
$965$	66.0000	2.12462
$966$	0	0
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	11.0000	0.353553
$969$	0	0
$970$	−51.0000	−1.63751
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	−16.0000	−0.513200
$973$	0	0
$974$	16.0000	0.512673
$975$	20.0000	0.640513
$976$	−8.00000	−0.256074
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	20.0000	0.639529
$979$	0	0
$980$	0	0
$981$	−4.00000	−0.127710
$982$	−15.0000	−0.478669
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	12.0000	0.382546
$985$	−18.0000	−0.573528
$986$	0	0
$987$	0	0
$988$	10.0000	0.318142
$989$	−48.0000	−1.52631
$990$	0	0
$991$	−25.0000	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$992$	−4.00000	−0.127000
$993$	−8.00000	−0.253872
$994$	0	0
$995$	33.0000	1.04617
$996$	18.0000	0.570352
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	4.00000	0.126618
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7742.2.a.b.1.1		1
7.6	odd	2		158.2.a.b.1.1	✓	1
21.20	even	2		1422.2.a.f.1.1		1
28.27	even	2		1264.2.a.c.1.1		1
35.34	odd	2		3950.2.a.g.1.1		1
56.13	odd	2		5056.2.a.d.1.1		1
56.27	even	2		5056.2.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
158.2.a.b.1.1	✓	1	7.6	odd	2
1264.2.a.c.1.1		1	28.27	even	2
1422.2.a.f.1.1		1	21.20	even	2
3950.2.a.g.1.1		1	35.34	odd	2
5056.2.a.d.1.1		1	56.13	odd	2
5056.2.a.l.1.1		1	56.27	even	2
7742.2.a.b.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7742 = 2 \cdot 7^{2} \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7742.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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