LMFDB - Embedded newform 3042.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1,3042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3042, 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("3042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.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:	$24.2904922949$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} -3.00000 q^{7} -1.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} -3.00000 q^{7} -1.00000 q^{8} -3.00000 q^{10} +3.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -6.00000 q^{19} +3.00000 q^{20} -6.00000 q^{23} +4.00000 q^{25} -3.00000 q^{28} -1.00000 q^{32} -3.00000 q^{34} -9.00000 q^{35} -3.00000 q^{37} +6.00000 q^{38} -3.00000 q^{40} +1.00000 q^{43} +6.00000 q^{46} +3.00000 q^{47} +2.00000 q^{49} -4.00000 q^{50} +6.00000 q^{53} +3.00000 q^{56} -6.00000 q^{59} -8.00000 q^{61} +1.00000 q^{64} -12.0000 q^{67} +3.00000 q^{68} +9.00000 q^{70} -15.0000 q^{71} -6.00000 q^{73} +3.00000 q^{74} -6.00000 q^{76} +10.0000 q^{79} +3.00000 q^{80} -6.00000 q^{83} +9.00000 q^{85} -1.00000 q^{86} -6.00000 q^{89} -6.00000 q^{92} -3.00000 q^{94} -18.0000 q^{95} -12.0000 q^{97} -2.00000 q^{98} +O(q^{100})$

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.00000	−0.948683
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	3.00000	0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	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	$0.715123\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	0	0
$28$	−3.00000	−0.566947
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.00000	−0.514496
$35$	−9.00000	−1.52128
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	−3.00000	−0.474342
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	0	0
$46$	6.00000	0.884652
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	−4.00000	−0.565685
$51$	0	0
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	3.00000	0.400892
$57$	0	0
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	3.00000	0.363803
$69$	0	0
$70$	9.00000	1.07571
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	3.00000	0.348743
$75$	0	0
$76$	−6.00000	−0.688247
$77$	0	0
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	9.00000	0.976187
$86$	−1.00000	−0.107833
$87$	0	0
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	−6.00000	−0.625543
$93$	0	0
$94$	−3.00000	−0.309426
$95$	−18.0000	−1.84676
$96$	0	0
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	−2.00000	−0.202031
$99$	0	0
$100$	4.00000	0.400000
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	0	0
$106$	−6.00000	−0.582772
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	9.00000	0.862044	0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000	0.862044	0.431022	+	0.902342i	$0.358153\pi$
$110$	0	0
$111$	0	0
$112$	−3.00000	−0.283473
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−18.0000	−1.67851
$116$	0	0
$117$	0	0
$118$	6.00000	0.552345
$119$	−9.00000	−0.825029
$120$	0	0
$121$	−11.0000	−1.00000
$122$	8.00000	0.724286
$123$	0	0
$124$	0	0
$125$	−3.00000	−0.268328
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	0	0
$133$	18.0000	1.56080
$134$	12.0000	1.03664
$135$	0	0
$136$	−3.00000	−0.257248
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	−9.00000	−0.760639
$141$	0	0
$142$	15.0000	1.25877
$143$	0	0
$144$	0	0
$145$	0	0
$146$	6.00000	0.496564
$147$	0	0
$148$	−3.00000	−0.246598
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	15.0000	1.22068	0.610341	−	0.792139i	$-0.291032\pi$
$151$	15.0000	1.22068	0.610341	+	0.792139i	$0.291032\pi$
$152$	6.00000	0.486664
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	−10.0000	−0.795557
$159$	0	0
$160$	−3.00000	−0.237171
$161$	18.0000	1.41860
$162$	0	0
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	0	0
$165$	0	0
$166$	6.00000	0.465690
$167$	−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
$169$	0	0
$170$	−9.00000	−0.690268
$171$	0	0
$172$	1.00000	0.0762493
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	−12.0000	−0.907115
$176$	0	0
$177$	0	0
$178$	6.00000	0.449719
$179$	15.0000	1.12115	0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000	1.12115	0.560576	+	0.828103i	$0.310580\pi$
$180$	0	0
$181$	−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
$183$	0	0
$184$	6.00000	0.442326
$185$	−9.00000	−0.661693
$186$	0	0
$187$	0	0
$188$	3.00000	0.218797
$189$	0	0
$190$	18.0000	1.30586
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	12.0000	0.861550
$195$	0	0
$196$	2.00000	0.142857
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	−4.00000	−0.282843
$201$	0	0
$202$	−12.0000	−0.844317
$203$	0	0
$204$	0	0
$205$	0	0
$206$	14.0000	0.975426
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−23.0000	−1.58339	−0.791693	−	0.610920i	$-0.790800\pi$
$211$	−23.0000	−1.58339	−0.791693	+	0.610920i	$0.790800\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−12.0000	−0.820303
$215$	3.00000	0.204598
$216$	0	0
$217$	0	0
$218$	−9.00000	−0.609557
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−9.00000	−0.602685	−0.301342	−	0.953516i	$-0.597435\pi$
$223$	−9.00000	−0.602685	−0.301342	+	0.953516i	$0.597435\pi$
$224$	3.00000	0.200446
$225$	0	0
$226$	−6.00000	−0.399114
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	−9.00000	−0.594737	−0.297368	−	0.954763i	$-0.596109\pi$
$229$	−9.00000	−0.594737	−0.297368	+	0.954763i	$0.596109\pi$
$230$	18.0000	1.18688
$231$	0	0
$232$	0	0
$233$	−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
$235$	9.00000	0.587095
$236$	−6.00000	−0.390567
$237$	0	0
$238$	9.00000	0.583383
$239$	−9.00000	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$240$	0	0
$241$	30.0000	1.93247	0.966235	−	0.257663i	$-0.0829523\pi$
$241$	30.0000	1.93247	0.966235	+	0.257663i	$0.0829523\pi$
$242$	11.0000	0.707107
$243$	0	0
$244$	−8.00000	−0.512148
$245$	6.00000	0.383326
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	3.00000	0.189737
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	0	0
$254$	−2.00000	−0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	3.00000	0.187135	0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000	0.187135	0.0935674	+	0.995613i	$0.470173\pi$
$258$	0	0
$259$	9.00000	0.559233
$260$	0	0
$261$	0	0
$262$	−3.00000	−0.185341
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	18.0000	1.10573
$266$	−18.0000	−1.10365
$267$	0	0
$268$	−12.0000	−0.733017
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−15.0000	−0.911185	−0.455593	−	0.890188i	$-0.650573\pi$
$271$	−15.0000	−0.911185	−0.455593	+	0.890188i	$0.650573\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	−18.0000	−1.08742
$275$	0	0
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	−5.00000	−0.299880
$279$	0	0
$280$	9.00000	0.537853
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−15.0000	−0.890086
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	0	0
$292$	−6.00000	−0.351123
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	0	0
$295$	−18.0000	−1.04800
$296$	3.00000	0.174371
$297$	0	0
$298$	−6.00000	−0.347571
$299$	0	0
$300$	0	0
$301$	−3.00000	−0.172917
$302$	−15.0000	−0.863153
$303$	0	0
$304$	−6.00000	−0.344124
$305$	−24.0000	−1.37424
$306$	0	0
$307$	−18.0000	−1.02731	−0.513657	−	0.857996i	$-0.671710\pi$
$307$	−18.0000	−1.02731	−0.513657	+	0.857996i	$0.671710\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	19.0000	1.07394	0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000	1.07394	0.536972	+	0.843600i	$0.319568\pi$
$314$	22.0000	1.24153
$315$	0	0
$316$	10.0000	0.562544
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	0	0
$320$	3.00000	0.167705
$321$	0	0
$322$	−18.0000	−1.00310
$323$	−18.0000	−1.00155
$324$	0	0
$325$	0	0
$326$	6.00000	0.332309
$327$	0	0
$328$	0	0
$329$	−9.00000	−0.496186
$330$	0	0
$331$	30.0000	1.64895	0.824475	−	0.565899i	$-0.191471\pi$
$331$	30.0000	1.64895	0.824475	+	0.565899i	$0.191471\pi$
$332$	−6.00000	−0.329293
$333$	0	0
$334$	12.0000	0.656611
$335$	−36.0000	−1.96689
$336$	0	0
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	0	0
$339$	0	0
$340$	9.00000	0.488094
$341$	0	0
$342$	0	0
$343$	15.0000	0.809924
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	6.00000	0.322562
$347$	−33.0000	−1.77153	−0.885766	−	0.464131i	$-0.846367\pi$
$347$	−33.0000	−1.77153	−0.885766	+	0.464131i	$0.846367\pi$
$348$	0	0
$349$	21.0000	1.12410	0.562052	−	0.827102i	$-0.310012\pi$
$349$	21.0000	1.12410	0.562052	+	0.827102i	$0.310012\pi$
$350$	12.0000	0.641427
$351$	0	0
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	−45.0000	−2.38835
$356$	−6.00000	−0.317999
$357$	0	0
$358$	−15.0000	−0.792775
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	2.00000	0.105118
$363$	0	0
$364$	0	0
$365$	−18.0000	−0.942163
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	9.00000	0.467888
$371$	−18.0000	−0.934513
$372$	0	0
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	0	0
$375$	0	0
$376$	−3.00000	−0.154713
$377$	0	0
$378$	0	0
$379$	−6.00000	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$380$	−18.0000	−0.923381
$381$	0	0
$382$	12.0000	0.613973
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	0	0
$385$	0	0
$386$	6.00000	0.305392
$387$	0	0
$388$	−12.0000	−0.609208
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	−2.00000	−0.101015
$393$	0	0
$394$	3.00000	0.151138
$395$	30.0000	1.50946
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	−20.0000	−1.00251
$399$	0	0
$400$	4.00000	0.200000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	0	0
$404$	12.0000	0.597022
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	0	0
$412$	−14.0000	−0.689730
$413$	18.0000	0.885722
$414$	0	0
$415$	−18.0000	−0.883585
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−15.0000	−0.732798	−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000	−0.732798	−0.366399	+	0.930458i	$0.619409\pi$
$420$	0	0
$421$	−15.0000	−0.731055	−0.365528	−	0.930800i	$-0.619111\pi$
$421$	−15.0000	−0.731055	−0.365528	+	0.930800i	$0.619111\pi$
$422$	23.0000	1.11962
$423$	0	0
$424$	−6.00000	−0.291386
$425$	12.0000	0.582086
$426$	0	0
$427$	24.0000	1.16144
$428$	12.0000	0.580042
$429$	0	0
$430$	−3.00000	−0.144673
$431$	15.0000	0.722525	0.361262	−	0.932464i	$-0.382346\pi$
$431$	15.0000	0.722525	0.361262	+	0.932464i	$0.382346\pi$
$432$	0	0
$433$	11.0000	0.528626	0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000	0.528626	0.264313	+	0.964437i	$0.414855\pi$
$434$	0	0
$435$	0	0
$436$	9.00000	0.431022
$437$	36.0000	1.72211
$438$	0	0
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	21.0000	0.997740	0.498870	−	0.866677i	$-0.333748\pi$
$443$	21.0000	0.997740	0.498870	+	0.866677i	$0.333748\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	9.00000	0.426162
$447$	0	0
$448$	−3.00000	−0.141737
$449$	24.0000	1.13263	0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000	1.13263	0.566315	+	0.824189i	$0.308369\pi$
$450$	0	0
$451$	0	0
$452$	6.00000	0.282216
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	9.00000	0.420542
$459$	0	0
$460$	−18.0000	−0.839254
$461$	15.0000	0.698620	0.349310	−	0.937007i	$-0.386416\pi$
$461$	15.0000	0.698620	0.349310	+	0.937007i	$0.386416\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	0	0
$465$	0	0
$466$	21.0000	0.972806
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	36.0000	1.66233
$470$	−9.00000	−0.415139
$471$	0	0
$472$	6.00000	0.276172
$473$	0	0
$474$	0	0
$475$	−24.0000	−1.10120
$476$	−9.00000	−0.412514
$477$	0	0
$478$	9.00000	0.411650
$479$	39.0000	1.78196	0.890978	−	0.454047i	$-0.150020\pi$
$479$	39.0000	1.78196	0.890978	+	0.454047i	$0.150020\pi$
$480$	0	0
$481$	0	0
$482$	−30.0000	−1.36646
$483$	0	0
$484$	−11.0000	−0.500000
$485$	−36.0000	−1.63468
$486$	0	0
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	8.00000	0.362143
$489$	0	0
$490$	−6.00000	−0.271052
$491$	27.0000	1.21849	0.609246	−	0.792981i	$-0.291472\pi$
$491$	27.0000	1.21849	0.609246	+	0.792981i	$0.291472\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	45.0000	2.01853
$498$	0	0
$499$	−36.0000	−1.61158	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	−3.00000	−0.134164
$501$	0	0
$502$	−12.0000	−0.535586
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	36.0000	1.60198
$506$	0	0
$507$	0	0
$508$	2.00000	0.0887357
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	18.0000	0.796273
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−3.00000	−0.132324
$515$	−42.0000	−1.85074
$516$	0	0
$517$	0	0
$518$	−9.00000	−0.395437
$519$	0	0
$520$	0	0
$521$	−27.0000	−1.18289	−0.591446	−	0.806345i	$-0.701443\pi$
$521$	−27.0000	−1.18289	−0.591446	+	0.806345i	$0.701443\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	3.00000	0.131056
$525$	0	0
$526$	24.0000	1.04645
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	−18.0000	−0.781870
$531$	0	0
$532$	18.0000	0.780399
$533$	0	0
$534$	0	0
$535$	36.0000	1.55642
$536$	12.0000	0.518321
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	15.0000	0.644900	0.322450	−	0.946586i	$-0.395494\pi$
$541$	15.0000	0.644900	0.322450	+	0.946586i	$0.395494\pi$
$542$	15.0000	0.644305
$543$	0	0
$544$	−3.00000	−0.128624
$545$	27.0000	1.15655
$546$	0	0
$547$	−37.0000	−1.58201	−0.791003	−	0.611812i	$-0.790441\pi$
$547$	−37.0000	−1.58201	−0.791003	+	0.611812i	$0.790441\pi$
$548$	18.0000	0.768922
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−30.0000	−1.27573
$554$	8.00000	0.339887
$555$	0	0
$556$	5.00000	0.212047
$557$	−27.0000	−1.14403	−0.572013	−	0.820244i	$-0.693837\pi$
$557$	−27.0000	−1.14403	−0.572013	+	0.820244i	$0.693837\pi$
$558$	0	0
$559$	0	0
$560$	−9.00000	−0.380319
$561$	0	0
$562$	30.0000	1.26547
$563$	39.0000	1.64365	0.821827	−	0.569737i	$-0.192955\pi$
$563$	39.0000	1.64365	0.821827	+	0.569737i	$0.192955\pi$
$564$	0	0
$565$	18.0000	0.757266
$566$	4.00000	0.168133
$567$	0	0
$568$	15.0000	0.629386
$569$	45.0000	1.88650	0.943249	−	0.332086i	$-0.107752\pi$
$569$	45.0000	1.88650	0.943249	+	0.332086i	$0.107752\pi$
$570$	0	0
$571$	23.0000	0.962520	0.481260	−	0.876578i	$-0.340179\pi$
$571$	23.0000	0.962520	0.481260	+	0.876578i	$0.340179\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−24.0000	−1.00087
$576$	0	0
$577$	−42.0000	−1.74848	−0.874241	−	0.485491i	$-0.838641\pi$
$577$	−42.0000	−1.74848	−0.874241	+	0.485491i	$0.838641\pi$
$578$	8.00000	0.332756
$579$	0	0
$580$	0	0
$581$	18.0000	0.746766
$582$	0	0
$583$	0	0
$584$	6.00000	0.248282
$585$	0	0
$586$	9.00000	0.371787
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	0	0
$589$	0	0
$590$	18.0000	0.741048
$591$	0	0
$592$	−3.00000	−0.123299
$593$	36.0000	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$594$	0	0
$595$	−27.0000	−1.10689
$596$	6.00000	0.245770
$597$	0	0
$598$	0	0
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	0	0
$601$	37.0000	1.50926	0.754631	−	0.656150i	$-0.227816\pi$
$601$	37.0000	1.50926	0.754631	+	0.656150i	$0.227816\pi$
$602$	3.00000	0.122271
$603$	0	0
$604$	15.0000	0.610341
$605$	−33.0000	−1.34164
$606$	0	0
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	6.00000	0.243332
$609$	0	0
$610$	24.0000	0.971732
$611$	0	0
$612$	0	0
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	18.0000	0.726421
$615$	0	0
$616$	0	0
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	0	0
$619$	−24.0000	−0.964641	−0.482321	−	0.875995i	$-0.660206\pi$
$619$	−24.0000	−0.964641	−0.482321	+	0.875995i	$0.660206\pi$
$620$	0	0
$621$	0	0
$622$	18.0000	0.721734
$623$	18.0000	0.721155
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−19.0000	−0.759393
$627$	0	0
$628$	−22.0000	−0.877896
$629$	−9.00000	−0.358854
$630$	0	0
$631$	15.0000	0.597141	0.298570	−	0.954388i	$-0.403490\pi$
$631$	15.0000	0.597141	0.298570	+	0.954388i	$0.403490\pi$
$632$	−10.0000	−0.397779
$633$	0	0
$634$	18.0000	0.714871
$635$	6.00000	0.238103
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	−3.00000	−0.118585
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	36.0000	1.41970	0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000	1.41970	0.709851	+	0.704352i	$0.248762\pi$
$644$	18.0000	0.709299
$645$	0	0
$646$	18.0000	0.708201
$647$	−42.0000	−1.65119	−0.825595	−	0.564263i	$-0.809160\pi$
$647$	−42.0000	−1.65119	−0.825595	+	0.564263i	$0.809160\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−6.00000	−0.234978
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	0	0
$655$	9.00000	0.351659
$656$	0	0
$657$	0	0
$658$	9.00000	0.350857
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	−30.0000	−1.16598
$663$	0	0
$664$	6.00000	0.232845
$665$	54.0000	2.09403
$666$	0	0
$667$	0	0
$668$	−12.0000	−0.464294
$669$	0	0
$670$	36.0000	1.39080
$671$	0	0
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	13.0000	0.500741
$675$	0	0
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	36.0000	1.38155
$680$	−9.00000	−0.345134
$681$	0	0
$682$	0	0
$683$	6.00000	0.229584	0.114792	−	0.993390i	$-0.463380\pi$
$683$	6.00000	0.229584	0.114792	+	0.993390i	$0.463380\pi$
$684$	0	0
$685$	54.0000	2.06323
$686$	−15.0000	−0.572703
$687$	0	0
$688$	1.00000	0.0381246
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	33.0000	1.25266
$695$	15.0000	0.568982
$696$	0	0
$697$	0	0
$698$	−21.0000	−0.794862
$699$	0	0
$700$	−12.0000	−0.453557
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	18.0000	0.678883
$704$	0	0
$705$	0	0
$706$	6.00000	0.225813
$707$	−36.0000	−1.35392
$708$	0	0
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	45.0000	1.68882
$711$	0	0
$712$	6.00000	0.224860
$713$	0	0
$714$	0	0
$715$	0	0
$716$	15.0000	0.560576
$717$	0	0
$718$	−24.0000	−0.895672
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	42.0000	1.56416
$722$	−17.0000	−0.632674
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	0	0
$727$	−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
$729$	0	0
$730$	18.0000	0.666210
$731$	3.00000	0.110959
$732$	0	0
$733$	−9.00000	−0.332423	−0.166211	−	0.986090i	$-0.553153\pi$
$733$	−9.00000	−0.332423	−0.166211	+	0.986090i	$0.553153\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	6.00000	0.221163
$737$	0	0
$738$	0	0
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	−9.00000	−0.330847
$741$	0	0
$742$	18.0000	0.660801
$743$	39.0000	1.43077	0.715386	−	0.698730i	$-0.246251\pi$
$743$	39.0000	1.43077	0.715386	+	0.698730i	$0.246251\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	−4.00000	−0.146450
$747$	0	0
$748$	0	0
$749$	−36.0000	−1.31541
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	3.00000	0.109399
$753$	0	0
$754$	0	0
$755$	45.0000	1.63772
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	6.00000	0.217930
$759$	0	0
$760$	18.0000	0.652929
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	−27.0000	−0.977466
$764$	−12.0000	−0.434145
$765$	0	0
$766$	−9.00000	−0.325183
$767$	0	0
$768$	0	0
$769$	24.0000	0.865462	0.432731	−	0.901523i	$-0.357550\pi$
$769$	24.0000	0.865462	0.432731	+	0.901523i	$0.357550\pi$
$770$	0	0
$771$	0	0
$772$	−6.00000	−0.215945
$773$	−21.0000	−0.755318	−0.377659	−	0.925945i	$-0.623271\pi$
$773$	−21.0000	−0.755318	−0.377659	+	0.925945i	$0.623271\pi$
$774$	0	0
$775$	0	0
$776$	12.0000	0.430775
$777$	0	0
$778$	30.0000	1.07555
$779$	0	0
$780$	0	0
$781$	0	0
$782$	18.0000	0.643679
$783$	0	0
$784$	2.00000	0.0714286
$785$	−66.0000	−2.35564
$786$	0	0
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	−3.00000	−0.106871
$789$	0	0
$790$	−30.0000	−1.06735
$791$	−18.0000	−0.640006
$792$	0	0
$793$	0	0
$794$	18.0000	0.638796
$795$	0	0
$796$	20.0000	0.708881
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	9.00000	0.318397
$800$	−4.00000	−0.141421
$801$	0	0
$802$	−30.0000	−1.05934
$803$	0	0
$804$	0	0
$805$	54.0000	1.90325
$806$	0	0
$807$	0	0
$808$	−12.0000	−0.422159
$809$	−15.0000	−0.527372	−0.263686	−	0.964609i	$-0.584938\pi$
$809$	−15.0000	−0.527372	−0.263686	+	0.964609i	$0.584938\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−18.0000	−0.630512
$816$	0	0
$817$	−6.00000	−0.209913
$818$	6.00000	0.209785
$819$	0	0
$820$	0	0
$821$	−45.0000	−1.57051	−0.785255	−	0.619172i	$-0.787468\pi$
$821$	−45.0000	−1.57051	−0.785255	+	0.619172i	$0.787468\pi$
$822$	0	0
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	14.0000	0.487713
$825$	0	0
$826$	−18.0000	−0.626300
$827$	18.0000	0.625921	0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000	0.625921	0.312961	+	0.949766i	$0.398679\pi$
$828$	0	0
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	18.0000	0.624789
$831$	0	0
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	−36.0000	−1.24583
$836$	0	0
$837$	0	0
$838$	15.0000	0.518166
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	15.0000	0.516934
$843$	0	0
$844$	−23.0000	−0.791693
$845$	0	0
$846$	0	0
$847$	33.0000	1.13389
$848$	6.00000	0.206041
$849$	0	0
$850$	−12.0000	−0.411597
$851$	18.0000	0.617032
$852$	0	0
$853$	39.0000	1.33533	0.667667	−	0.744460i	$-0.267293\pi$
$853$	39.0000	1.33533	0.667667	+	0.744460i	$0.267293\pi$
$854$	−24.0000	−0.821263
$855$	0	0
$856$	−12.0000	−0.410152
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	3.00000	0.102299
$861$	0	0
$862$	−15.0000	−0.510902
$863$	39.0000	1.32758	0.663788	−	0.747921i	$-0.268948\pi$
$863$	39.0000	1.32758	0.663788	+	0.747921i	$0.268948\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	−11.0000	−0.373795
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−9.00000	−0.304778
$873$	0	0
$874$	−36.0000	−1.21772
$875$	9.00000	0.304256
$876$	0	0
$877$	3.00000	0.101303	0.0506514	−	0.998716i	$-0.483870\pi$
$877$	3.00000	0.101303	0.0506514	+	0.998716i	$0.483870\pi$
$878$	−10.0000	−0.337484
$879$	0	0
$880$	0	0
$881$	−33.0000	−1.11180	−0.555899	−	0.831250i	$-0.687626\pi$
$881$	−33.0000	−1.11180	−0.555899	+	0.831250i	$0.687626\pi$
$882$	0	0
$883$	11.0000	0.370179	0.185090	−	0.982722i	$-0.440742\pi$
$883$	11.0000	0.370179	0.185090	+	0.982722i	$0.440742\pi$
$884$	0	0
$885$	0	0
$886$	−21.0000	−0.705509
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	−6.00000	−0.201234
$890$	18.0000	0.603361
$891$	0	0
$892$	−9.00000	−0.301342
$893$	−18.0000	−0.602347
$894$	0	0
$895$	45.0000	1.50418
$896$	3.00000	0.100223
$897$	0	0
$898$	−24.0000	−0.800890
$899$	0	0
$900$	0	0
$901$	18.0000	0.599667
$902$	0	0
$903$	0	0
$904$	−6.00000	−0.199557
$905$	−6.00000	−0.199447
$906$	0	0
$907$	17.0000	0.564476	0.282238	−	0.959344i	$-0.408923\pi$
$907$	17.0000	0.564476	0.282238	+	0.959344i	$0.408923\pi$
$908$	12.0000	0.398234
$909$	0	0
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	0	0
$914$	−18.0000	−0.595387
$915$	0	0
$916$	−9.00000	−0.297368
$917$	−9.00000	−0.297206
$918$	0	0
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	18.0000	0.593442
$921$	0	0
$922$	−15.0000	−0.493999
$923$	0	0
$924$	0	0
$925$	−12.0000	−0.394558
$926$	−24.0000	−0.788689
$927$	0	0
$928$	0	0
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	−21.0000	−0.687878
$933$	0	0
$934$	12.0000	0.392652
$935$	0	0
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	−36.0000	−1.17544
$939$	0	0
$940$	9.00000	0.293548
$941$	45.0000	1.46696	0.733479	−	0.679712i	$-0.237895\pi$
$941$	45.0000	1.46696	0.733479	+	0.679712i	$0.237895\pi$
$942$	0	0
$943$	0	0
$944$	−6.00000	−0.195283
$945$	0	0
$946$	0	0
$947$	48.0000	1.55979	0.779895	−	0.625910i	$-0.215272\pi$
$947$	48.0000	1.55979	0.779895	+	0.625910i	$0.215272\pi$
$948$	0	0
$949$	0	0
$950$	24.0000	0.778663
$951$	0	0
$952$	9.00000	0.291692
$953$	9.00000	0.291539	0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000	0.291539	0.145769	+	0.989319i	$0.453434\pi$
$954$	0	0
$955$	−36.0000	−1.16493
$956$	−9.00000	−0.291081
$957$	0	0
$958$	−39.0000	−1.26003
$959$	−54.0000	−1.74375
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	30.0000	0.966235
$965$	−18.0000	−0.579441
$966$	0	0
$967$	3.00000	0.0964735	0.0482367	−	0.998836i	$-0.484640\pi$
$967$	3.00000	0.0964735	0.0482367	+	0.998836i	$0.484640\pi$
$968$	11.0000	0.353553
$969$	0	0
$970$	36.0000	1.15589
$971$	−27.0000	−0.866471	−0.433236	−	0.901281i	$-0.642628\pi$
$971$	−27.0000	−0.866471	−0.433236	+	0.901281i	$0.642628\pi$
$972$	0	0
$973$	−15.0000	−0.480878
$974$	12.0000	0.384505
$975$	0	0
$976$	−8.00000	−0.256074
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	0	0
$979$	0	0
$980$	6.00000	0.191663
$981$	0	0
$982$	−27.0000	−0.861605
$983$	−9.00000	−0.287055	−0.143528	−	0.989646i	$-0.545845\pi$
$983$	−9.00000	−0.287055	−0.143528	+	0.989646i	$0.545845\pi$
$984$	0	0
$985$	−9.00000	−0.286764
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.00000	−0.190789
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	0	0
$993$	0	0
$994$	−45.0000	−1.42731
$995$	60.0000	1.90213
$996$	0	0
$997$	8.00000	0.253363	0.126681	−	0.991943i	$-0.459567\pi$
$997$	8.00000	0.253363	0.126681	+	0.991943i	$0.459567\pi$
$998$	36.0000	1.13956
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.a.g.1.1		1
3.2	odd	2		338.2.a.d.1.1		1
12.11	even	2		2704.2.a.j.1.1		1
13.5	odd	4		234.2.b.b.181.2		2
13.8	odd	4		234.2.b.b.181.1		2
13.12	even	2		3042.2.a.j.1.1		1
15.14	odd	2		8450.2.a.h.1.1		1
39.2	even	12		338.2.e.c.147.2		4
39.5	even	4		26.2.b.a.25.1	✓	2
39.8	even	4		26.2.b.a.25.2	yes	2
39.11	even	12		338.2.e.c.147.1		4
39.17	odd	6		338.2.c.f.315.1		2
39.20	even	12		338.2.e.c.23.2		4
39.23	odd	6		338.2.c.f.191.1		2
39.29	odd	6		338.2.c.b.191.1		2
39.32	even	12		338.2.e.c.23.1		4
39.35	odd	6		338.2.c.b.315.1		2
39.38	odd	2		338.2.a.b.1.1		1
52.31	even	4		1872.2.c.f.1585.1		2
52.47	even	4		1872.2.c.f.1585.2		2
156.47	odd	4		208.2.f.a.129.1		2
156.83	odd	4		208.2.f.a.129.2		2
156.155	even	2		2704.2.a.k.1.1		1
195.8	odd	4		650.2.c.d.649.1		2
195.44	even	4		650.2.d.b.51.2		2
195.47	odd	4		650.2.c.a.649.2		2
195.83	odd	4		650.2.c.a.649.1		2
195.122	odd	4		650.2.c.d.649.2		2
195.164	even	4		650.2.d.b.51.1		2
195.194	odd	2		8450.2.a.u.1.1		1
273.5	odd	12		1274.2.n.c.753.1		4
273.44	even	12		1274.2.n.d.753.1		4
273.47	odd	12		1274.2.n.c.753.2		4
273.83	odd	4		1274.2.d.c.883.1		2
273.86	even	12		1274.2.n.d.753.2		4
273.122	odd	12		1274.2.n.c.961.2		4
273.125	odd	4		1274.2.d.c.883.2		2
273.164	odd	12		1274.2.n.c.961.1		4
273.200	even	12		1274.2.n.d.961.2		4
273.242	even	12		1274.2.n.d.961.1		4
312.5	even	4		832.2.f.d.129.1		2
312.83	odd	4		832.2.f.b.129.1		2
312.125	even	4		832.2.f.d.129.2		2
312.203	odd	4		832.2.f.b.129.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.b.a.25.1	✓	2	39.5	even	4
26.2.b.a.25.2	yes	2	39.8	even	4
208.2.f.a.129.1		2	156.47	odd	4
208.2.f.a.129.2		2	156.83	odd	4
234.2.b.b.181.1		2	13.8	odd	4
234.2.b.b.181.2		2	13.5	odd	4
338.2.a.b.1.1		1	39.38	odd	2
338.2.a.d.1.1		1	3.2	odd	2
338.2.c.b.191.1		2	39.29	odd	6
338.2.c.b.315.1		2	39.35	odd	6
338.2.c.f.191.1		2	39.23	odd	6
338.2.c.f.315.1		2	39.17	odd	6
338.2.e.c.23.1		4	39.32	even	12
338.2.e.c.23.2		4	39.20	even	12
338.2.e.c.147.1		4	39.11	even	12
338.2.e.c.147.2		4	39.2	even	12
650.2.c.a.649.1		2	195.83	odd	4
650.2.c.a.649.2		2	195.47	odd	4
650.2.c.d.649.1		2	195.8	odd	4
650.2.c.d.649.2		2	195.122	odd	4
650.2.d.b.51.1		2	195.164	even	4
650.2.d.b.51.2		2	195.44	even	4
832.2.f.b.129.1		2	312.83	odd	4
832.2.f.b.129.2		2	312.203	odd	4
832.2.f.d.129.1		2	312.5	even	4
832.2.f.d.129.2		2	312.125	even	4
1274.2.d.c.883.1		2	273.83	odd	4
1274.2.d.c.883.2		2	273.125	odd	4
1274.2.n.c.753.1		4	273.5	odd	12
1274.2.n.c.753.2		4	273.47	odd	12
1274.2.n.c.961.1		4	273.164	odd	12
1274.2.n.c.961.2		4	273.122	odd	12
1274.2.n.d.753.1		4	273.44	even	12
1274.2.n.d.753.2		4	273.86	even	12
1274.2.n.d.961.1		4	273.242	even	12
1274.2.n.d.961.2		4	273.200	even	12
1872.2.c.f.1585.1		2	52.31	even	4
1872.2.c.f.1585.2		2	52.47	even	4
2704.2.a.j.1.1		1	12.11	even	2
2704.2.a.k.1.1		1	156.155	even	2
3042.2.a.g.1.1		1	1.1	even	1	trivial
3042.2.a.j.1.1		1	13.12	even	2
8450.2.a.h.1.1		1	15.14	odd	2
8450.2.a.u.1.1		1	195.194	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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.a (trivial)

Introduction

L-functions

Modular forms

Varieties

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Representations

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