LMFDB - Embedded newform 1078.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1078,2,Mod(1,1078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, 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("1078.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1078 = 2 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1078.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:	$8.60787333789$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1078.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} +5.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} -2.00000 q^{18} +2.00000 q^{19} -1.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} +5.00000 q^{26} -5.00000 q^{27} +3.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +6.00000 q^{34} -2.00000 q^{36} +2.00000 q^{37} +2.00000 q^{38} +5.00000 q^{39} -6.00000 q^{41} -4.00000 q^{43} -1.00000 q^{44} +6.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} -5.00000 q^{50} +6.00000 q^{51} +5.00000 q^{52} -12.0000 q^{53} -5.00000 q^{54} +2.00000 q^{57} +3.00000 q^{58} -3.00000 q^{59} -7.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -1.00000 q^{66} -13.0000 q^{67} +6.00000 q^{68} +6.00000 q^{69} -12.0000 q^{71} -2.00000 q^{72} -10.0000 q^{73} +2.00000 q^{74} -5.00000 q^{75} +2.00000 q^{76} +5.00000 q^{78} -1.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +6.00000 q^{83} -4.00000 q^{86} +3.00000 q^{87} -1.00000 q^{88} +6.00000 q^{89} +6.00000 q^{92} +8.00000 q^{93} +6.00000 q^{94} +1.00000 q^{96} -13.0000 q^{97} +2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	−2.00000	−0.471405
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	−1.00000	−0.213201
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	1.00000	0.204124
$25$	−5.00000	−1.00000
$26$	5.00000	0.980581
$27$	−5.00000	−0.962250
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	6.00000	1.02899
$35$	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
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	6.00000	0.884652
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−5.00000	−0.707107
$51$	6.00000	0.840168
$52$	5.00000	0.693375
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	−5.00000	−0.680414
$55$	0	0
$56$	0	0
$57$	2.00000	0.264906
$58$	3.00000	0.393919
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	−13.0000	−1.58820	−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−13.0000	−1.58820	−0.794101	+	0.607785i	$0.792058\pi$
$68$	6.00000	0.727607
$69$	6.00000	0.722315
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	−2.00000	−0.235702
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	2.00000	0.232495
$75$	−5.00000	−0.577350
$76$	2.00000	0.229416
$77$	0	0
$78$	5.00000	0.566139
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	3.00000	0.321634
$88$	−1.00000	−0.106600
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	6.00000	0.625543
$93$	8.00000	0.829561
$94$	6.00000	0.618853
$95$	0	0
$96$	1.00000	0.102062
$97$	−13.0000	−1.31995	−0.659975	−	0.751288i	$-0.729433\pi$
$97$	−13.0000	−1.31995	−0.659975	+	0.751288i	$0.729433\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	−5.00000	−0.500000
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	6.00000	0.594089
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	5.00000	0.490290
$105$	0	0
$106$	−12.0000	−1.16554
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	−5.00000	−0.481125
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	2.00000	0.187317
$115$	0	0
$116$	3.00000	0.278543
$117$	−10.0000	−0.924500
$118$	−3.00000	−0.276172
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−7.00000	−0.633750
$123$	−6.00000	−0.541002
$124$	8.00000	0.718421
$125$	0	0
$126$	0	0
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	−13.0000	−1.12303
$135$	0	0
$136$	6.00000	0.514496
$137$	15.0000	1.28154	0.640768	−	0.767734i	$-0.278616\pi$
$137$	15.0000	1.28154	0.640768	+	0.767734i	$0.278616\pi$
$138$	6.00000	0.510754
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	−12.0000	−1.00702
$143$	−5.00000	−0.418121
$144$	−2.00000	−0.166667
$145$	0	0
$146$	−10.0000	−0.827606
$147$	0	0
$148$	2.00000	0.164399
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	−5.00000	−0.408248
$151$	−1.00000	−0.0813788	−0.0406894	−	0.999172i	$-0.512955\pi$
$151$	−1.00000	−0.0813788	−0.0406894	+	0.999172i	$0.512955\pi$
$152$	2.00000	0.162221
$153$	−12.0000	−0.970143
$154$	0	0
$155$	0	0
$156$	5.00000	0.400320
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	−1.00000	−0.0795557
$159$	−12.0000	−0.951662
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	17.0000	1.33154	0.665771	−	0.746156i	$-0.268103\pi$
$163$	17.0000	1.33154	0.665771	+	0.746156i	$0.268103\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	6.00000	0.465690
$167$	9.00000	0.696441	0.348220	−	0.937413i	$-0.386786\pi$
$167$	9.00000	0.696441	0.348220	+	0.937413i	$0.386786\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	−4.00000	−0.305888
$172$	−4.00000	−0.304997
$173$	−15.0000	−1.14043	−0.570214	−	0.821496i	$-0.693140\pi$
$173$	−15.0000	−1.14043	−0.570214	+	0.821496i	$0.693140\pi$
$174$	3.00000	0.227429
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−3.00000	−0.225494
$178$	6.00000	0.449719
$179$	9.00000	0.672692	0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000	0.672692	0.336346	+	0.941739i	$0.390809\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	−7.00000	−0.517455
$184$	6.00000	0.442326
$185$	0	0
$186$	8.00000	0.586588
$187$	−6.00000	−0.438763
$188$	6.00000	0.437595
$189$	0	0
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	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	$0.790854\pi$
$194$	−13.0000	−0.933346
$195$	0	0
$196$	0	0
$197$	−9.00000	−0.641223	−0.320612	−	0.947211i	$-0.603888\pi$
$197$	−9.00000	−0.641223	−0.320612	+	0.947211i	$0.603888\pi$
$198$	2.00000	0.142134
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	−5.00000	−0.353553
$201$	−13.0000	−0.916949
$202$	3.00000	0.211079
$203$	0	0
$204$	6.00000	0.420084
$205$	0	0
$206$	−4.00000	−0.278693
$207$	−12.0000	−0.834058
$208$	5.00000	0.346688
$209$	−2.00000	−0.138343
$210$	0	0
$211$	26.0000	1.78991	0.894957	−	0.446153i	$-0.147206\pi$
$211$	26.0000	1.78991	0.894957	+	0.446153i	$0.147206\pi$
$212$	−12.0000	−0.824163
$213$	−12.0000	−0.822226
$214$	0	0
$215$	0	0
$216$	−5.00000	−0.340207
$217$	0	0
$218$	14.0000	0.948200
$219$	−10.0000	−0.675737
$220$	0	0
$221$	30.0000	2.01802
$222$	2.00000	0.134231
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	0	0
$225$	10.0000	0.666667
$226$	9.00000	0.598671
$227$	−6.00000	−0.398234	−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000	−0.398234	−0.199117	+	0.979976i	$0.563807\pi$
$228$	2.00000	0.132453
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	0	0
$231$	0	0
$232$	3.00000	0.196960
$233$	−30.0000	−1.96537	−0.982683	−	0.185296i	$-0.940675\pi$
$233$	−30.0000	−1.96537	−0.982683	+	0.185296i	$0.940675\pi$
$234$	−10.0000	−0.653720
$235$	0	0
$236$	−3.00000	−0.195283
$237$	−1.00000	−0.0649570
$238$	0	0
$239$	21.0000	1.35838	0.679189	−	0.733964i	$-0.262332\pi$
$239$	21.0000	1.35838	0.679189	+	0.733964i	$0.262332\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	1.00000	0.0642824
$243$	16.0000	1.02640
$244$	−7.00000	−0.448129
$245$	0	0
$246$	−6.00000	−0.382546
$247$	10.0000	0.636285
$248$	8.00000	0.508001
$249$	6.00000	0.380235
$250$	0	0
$251$	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
$253$	−6.00000	−0.377217
$254$	−13.0000	−0.815693
$255$	0	0
$256$	1.00000	0.0625000
$257$	27.0000	1.68421	0.842107	−	0.539311i	$-0.181315\pi$
$257$	27.0000	1.68421	0.842107	+	0.539311i	$0.181315\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	6.00000	0.370681
$263$	−3.00000	−0.184988	−0.0924940	−	0.995713i	$-0.529484\pi$
$263$	−3.00000	−0.184988	−0.0924940	+	0.995713i	$0.529484\pi$
$264$	−1.00000	−0.0615457
$265$	0	0
$266$	0	0
$267$	6.00000	0.367194
$268$	−13.0000	−0.794101
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	−25.0000	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	15.0000	0.906183
$275$	5.00000	0.301511
$276$	6.00000	0.361158
$277$	−1.00000	−0.0600842	−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000	−0.0600842	−0.0300421	+	0.999549i	$0.509564\pi$
$278$	−4.00000	−0.239904
$279$	−16.0000	−0.957895
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	6.00000	0.357295
$283$	−22.0000	−1.30776	−0.653882	−	0.756596i	$-0.726861\pi$
$283$	−22.0000	−1.30776	−0.653882	+	0.756596i	$0.726861\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	−5.00000	−0.295656
$287$	0	0
$288$	−2.00000	−0.117851
$289$	19.0000	1.11765
$290$	0	0
$291$	−13.0000	−0.762073
$292$	−10.0000	−0.585206
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	0	0
$296$	2.00000	0.116248
$297$	5.00000	0.290129
$298$	−6.00000	−0.347571
$299$	30.0000	1.73494
$300$	−5.00000	−0.288675
$301$	0	0
$302$	−1.00000	−0.0575435
$303$	3.00000	0.172345
$304$	2.00000	0.114708
$305$	0	0
$306$	−12.0000	−0.685994
$307$	32.0000	1.82634	0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000	1.82634	0.913168	+	0.407583i	$0.133628\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	5.00000	0.283069
$313$	−7.00000	−0.395663	−0.197832	−	0.980236i	$-0.563390\pi$
$313$	−7.00000	−0.395663	−0.197832	+	0.980236i	$0.563390\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	−24.0000	−1.34797	−0.673987	−	0.738743i	$-0.735420\pi$
$317$	−24.0000	−1.34797	−0.673987	+	0.738743i	$0.735420\pi$
$318$	−12.0000	−0.672927
$319$	−3.00000	−0.167968
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.0000	0.667698
$324$	1.00000	0.0555556
$325$	−25.0000	−1.38675
$326$	17.0000	0.941543
$327$	14.0000	0.774202
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	−13.0000	−0.714545	−0.357272	−	0.934000i	$-0.616293\pi$
$331$	−13.0000	−0.714545	−0.357272	+	0.934000i	$0.616293\pi$
$332$	6.00000	0.329293
$333$	−4.00000	−0.219199
$334$	9.00000	0.492458
$335$	0	0
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	12.0000	0.652714
$339$	9.00000	0.488813
$340$	0	0
$341$	−8.00000	−0.433224
$342$	−4.00000	−0.216295
$343$	0	0
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−15.0000	−0.806405
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	3.00000	0.160817
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	−25.0000	−1.33440
$352$	−1.00000	−0.0533002
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	−3.00000	−0.159448
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	9.00000	0.475665
$359$	−9.00000	−0.475002	−0.237501	−	0.971387i	$-0.576328\pi$
$359$	−9.00000	−0.475002	−0.237501	+	0.971387i	$0.576328\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−10.0000	−0.525588
$363$	1.00000	0.0524864
$364$	0	0
$365$	0	0
$366$	−7.00000	−0.365896
$367$	38.0000	1.98358	0.991792	−	0.127862i	$-0.0408116\pi$
$367$	38.0000	1.98358	0.991792	+	0.127862i	$0.0408116\pi$
$368$	6.00000	0.312772
$369$	12.0000	0.624695
$370$	0	0
$371$	0	0
$372$	8.00000	0.414781
$373$	11.0000	0.569558	0.284779	−	0.958593i	$-0.408080\pi$
$373$	11.0000	0.569558	0.284779	+	0.958593i	$0.408080\pi$
$374$	−6.00000	−0.310253
$375$	0	0
$376$	6.00000	0.309426
$377$	15.0000	0.772539
$378$	0	0
$379$	−25.0000	−1.28416	−0.642082	−	0.766636i	$-0.721929\pi$
$379$	−25.0000	−1.28416	−0.642082	+	0.766636i	$0.721929\pi$
$380$	0	0
$381$	−13.0000	−0.666010
$382$	−18.0000	−0.920960
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−22.0000	−1.11977
$387$	8.00000	0.406663
$388$	−13.0000	−0.659975
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	36.0000	1.82060
$392$	0	0
$393$	6.00000	0.302660
$394$	−9.00000	−0.453413
$395$	0	0
$396$	2.00000	0.100504
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	14.0000	0.701757
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−9.00000	−0.449439	−0.224719	−	0.974424i	$-0.572147\pi$
$401$	−9.00000	−0.449439	−0.224719	+	0.974424i	$0.572147\pi$
$402$	−13.0000	−0.648381
$403$	40.0000	1.99254
$404$	3.00000	0.149256
$405$	0	0
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	6.00000	0.297044
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	0	0
$411$	15.0000	0.739895
$412$	−4.00000	−0.197066
$413$	0	0
$414$	−12.0000	−0.589768
$415$	0	0
$416$	5.00000	0.245145
$417$	−4.00000	−0.195881
$418$	−2.00000	−0.0978232
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−16.0000	−0.779792	−0.389896	−	0.920859i	$-0.627489\pi$
$421$	−16.0000	−0.779792	−0.389896	+	0.920859i	$0.627489\pi$
$422$	26.0000	1.26566
$423$	−12.0000	−0.583460
$424$	−12.0000	−0.582772
$425$	−30.0000	−1.45521
$426$	−12.0000	−0.581402
$427$	0	0
$428$	0	0
$429$	−5.00000	−0.241402
$430$	0	0
$431$	15.0000	0.722525	0.361262	−	0.932464i	$-0.382346\pi$
$431$	15.0000	0.722525	0.361262	+	0.932464i	$0.382346\pi$
$432$	−5.00000	−0.240563
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	0	0
$436$	14.0000	0.670478
$437$	12.0000	0.574038
$438$	−10.0000	−0.477818
$439$	−19.0000	−0.906821	−0.453410	−	0.891302i	$-0.649793\pi$
$439$	−19.0000	−0.906821	−0.453410	+	0.891302i	$0.649793\pi$
$440$	0	0
$441$	0	0
$442$	30.0000	1.42695
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	−10.0000	−0.473514
$447$	−6.00000	−0.283790
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	10.0000	0.471405
$451$	6.00000	0.282529
$452$	9.00000	0.423324
$453$	−1.00000	−0.0469841
$454$	−6.00000	−0.281594
$455$	0	0
$456$	2.00000	0.0936586
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	20.0000	0.934539
$459$	−30.0000	−1.40028
$460$	0	0
$461$	39.0000	1.81641	0.908206	−	0.418524i	$-0.137453\pi$
$461$	39.0000	1.81641	0.908206	+	0.418524i	$0.137453\pi$
$462$	0	0
$463$	26.0000	1.20832	0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000	1.20832	0.604161	+	0.796862i	$0.293508\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−30.0000	−1.38972
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−10.0000	−0.462250
$469$	0	0
$470$	0	0
$471$	−4.00000	−0.184310
$472$	−3.00000	−0.138086
$473$	4.00000	0.183920
$474$	−1.00000	−0.0459315
$475$	−10.0000	−0.458831
$476$	0	0
$477$	24.0000	1.09888
$478$	21.0000	0.960518
$479$	−3.00000	−0.137073	−0.0685367	−	0.997649i	$-0.521833\pi$
$479$	−3.00000	−0.137073	−0.0685367	+	0.997649i	$0.521833\pi$
$480$	0	0
$481$	10.0000	0.455961
$482$	−10.0000	−0.455488
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	16.0000	0.725775
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	−7.00000	−0.316875
$489$	17.0000	0.768767
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	−6.00000	−0.270501
$493$	18.0000	0.810679
$494$	10.0000	0.449921
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	6.00000	0.268866
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	0	0
$501$	9.00000	0.402090
$502$	12.0000	0.535586
$503$	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
$505$	0	0
$506$	−6.00000	−0.266733
$507$	12.0000	0.532939
$508$	−13.0000	−0.576782
$509$	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
$511$	0	0
$512$	1.00000	0.0441942
$513$	−10.0000	−0.441511
$514$	27.0000	1.19092
$515$	0	0
$516$	−4.00000	−0.176090
$517$	−6.00000	−0.263880
$518$	0	0
$519$	−15.0000	−0.658427
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	−6.00000	−0.262613
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	−3.00000	−0.130806
$527$	48.0000	2.09091
$528$	−1.00000	−0.0435194
$529$	13.0000	0.565217
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	−30.0000	−1.29944
$534$	6.00000	0.259645
$535$	0	0
$536$	−13.0000	−0.561514
$537$	9.00000	0.388379
$538$	−24.0000	−1.03471
$539$	0	0
$540$	0	0
$541$	5.00000	0.214967	0.107483	−	0.994207i	$-0.465721\pi$
$541$	5.00000	0.214967	0.107483	+	0.994207i	$0.465721\pi$
$542$	−25.0000	−1.07384
$543$	−10.0000	−0.429141
$544$	6.00000	0.257248
$545$	0	0
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	15.0000	0.640768
$549$	14.0000	0.597505
$550$	5.00000	0.213201
$551$	6.00000	0.255609
$552$	6.00000	0.255377
$553$	0	0
$554$	−1.00000	−0.0424859
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	−16.0000	−0.677334
$559$	−20.0000	−0.845910
$560$	0	0
$561$	−6.00000	−0.253320
$562$	−6.00000	−0.253095
$563$	18.0000	0.758610	0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000	0.758610	0.379305	+	0.925272i	$0.376163\pi$
$564$	6.00000	0.252646
$565$	0	0
$566$	−22.0000	−0.924729
$567$	0	0
$568$	−12.0000	−0.503509
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	−5.00000	−0.209061
$573$	−18.0000	−0.751961
$574$	0	0
$575$	−30.0000	−1.25109
$576$	−2.00000	−0.0833333
$577$	−7.00000	−0.291414	−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000	−0.291414	−0.145707	+	0.989328i	$0.546546\pi$
$578$	19.0000	0.790296
$579$	−22.0000	−0.914289
$580$	0	0
$581$	0	0
$582$	−13.0000	−0.538867
$583$	12.0000	0.496989
$584$	−10.0000	−0.413803
$585$	0	0
$586$	−18.0000	−0.743573
$587$	9.00000	0.371470	0.185735	−	0.982600i	$-0.440533\pi$
$587$	9.00000	0.371470	0.185735	+	0.982600i	$0.440533\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	0	0
$591$	−9.00000	−0.370211
$592$	2.00000	0.0821995
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	5.00000	0.205152
$595$	0	0
$596$	−6.00000	−0.245770
$597$	14.0000	0.572982
$598$	30.0000	1.22679
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	−5.00000	−0.204124
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	26.0000	1.05880
$604$	−1.00000	−0.0406894
$605$	0	0
$606$	3.00000	0.121867
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	0	0
$611$	30.0000	1.21367
$612$	−12.0000	−0.485071
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	32.0000	1.29141
$615$	0	0
$616$	0	0
$617$	21.0000	0.845428	0.422714	−	0.906263i	$-0.361077\pi$
$617$	21.0000	0.845428	0.422714	+	0.906263i	$0.361077\pi$
$618$	−4.00000	−0.160904
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	0	0
$621$	−30.0000	−1.20386
$622$	24.0000	0.962312
$623$	0	0
$624$	5.00000	0.200160
$625$	25.0000	1.00000
$626$	−7.00000	−0.279776
$627$	−2.00000	−0.0798723
$628$	−4.00000	−0.159617
$629$	12.0000	0.478471
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	−1.00000	−0.0397779
$633$	26.0000	1.03341
$634$	−24.0000	−0.953162
$635$	0	0
$636$	−12.0000	−0.475831
$637$	0	0
$638$	−3.00000	−0.118771
$639$	24.0000	0.949425
$640$	0	0
$641$	9.00000	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$642$	0	0
$643$	−19.0000	−0.749287	−0.374643	−	0.927169i	$-0.622235\pi$
$643$	−19.0000	−0.749287	−0.374643	+	0.927169i	$0.622235\pi$
$644$	0	0
$645$	0	0
$646$	12.0000	0.472134
$647$	36.0000	1.41531	0.707653	−	0.706560i	$-0.249754\pi$
$647$	36.0000	1.41531	0.707653	+	0.706560i	$0.249754\pi$
$648$	1.00000	0.0392837
$649$	3.00000	0.117760
$650$	−25.0000	−0.980581
$651$	0	0
$652$	17.0000	0.665771
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	14.0000	0.547443
$655$	0	0
$656$	−6.00000	−0.234261
$657$	20.0000	0.780274
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−34.0000	−1.32245	−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000	−1.32245	−0.661223	+	0.750189i	$0.729962\pi$
$662$	−13.0000	−0.505259
$663$	30.0000	1.16510
$664$	6.00000	0.232845
$665$	0	0
$666$	−4.00000	−0.154997
$667$	18.0000	0.696963
$668$	9.00000	0.348220
$669$	−10.0000	−0.386622
$670$	0	0
$671$	7.00000	0.270232
$672$	0	0
$673$	−4.00000	−0.154189	−0.0770943	−	0.997024i	$-0.524564\pi$
$673$	−4.00000	−0.154189	−0.0770943	+	0.997024i	$0.524564\pi$
$674$	−22.0000	−0.847408
$675$	25.0000	0.962250
$676$	12.0000	0.461538
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	9.00000	0.345643
$679$	0	0
$680$	0	0
$681$	−6.00000	−0.229920
$682$	−8.00000	−0.306336
$683$	−45.0000	−1.72188	−0.860939	−	0.508709i	$-0.830123\pi$
$683$	−45.0000	−1.72188	−0.860939	+	0.508709i	$0.830123\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	0	0
$687$	20.0000	0.763048
$688$	−4.00000	−0.152499
$689$	−60.0000	−2.28582
$690$	0	0
$691$	35.0000	1.33146	0.665731	−	0.746191i	$-0.268120\pi$
$691$	35.0000	1.33146	0.665731	+	0.746191i	$0.268120\pi$
$692$	−15.0000	−0.570214
$693$	0	0
$694$	12.0000	0.455514
$695$	0	0
$696$	3.00000	0.113715
$697$	−36.0000	−1.36360
$698$	−34.0000	−1.28692
$699$	−30.0000	−1.13470
$700$	0	0
$701$	−3.00000	−0.113308	−0.0566542	−	0.998394i	$-0.518043\pi$
$701$	−3.00000	−0.113308	−0.0566542	+	0.998394i	$0.518043\pi$
$702$	−25.0000	−0.943564
$703$	4.00000	0.150863
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	30.0000	1.12906
$707$	0	0
$708$	−3.00000	−0.112747
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	2.00000	0.0750059
$712$	6.00000	0.224860
$713$	48.0000	1.79761
$714$	0	0
$715$	0	0
$716$	9.00000	0.336346
$717$	21.0000	0.784259
$718$	−9.00000	−0.335877
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	0	0
$721$	0	0
$722$	−15.0000	−0.558242
$723$	−10.0000	−0.371904
$724$	−10.0000	−0.371647
$725$	−15.0000	−0.557086
$726$	1.00000	0.0371135
$727$	50.0000	1.85440	0.927199	−	0.374570i	$-0.122210\pi$
$727$	50.0000	1.85440	0.927199	+	0.374570i	$0.122210\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−24.0000	−0.887672
$732$	−7.00000	−0.258727
$733$	−43.0000	−1.58824	−0.794121	−	0.607760i	$-0.792068\pi$
$733$	−43.0000	−1.58824	−0.794121	+	0.607760i	$0.792068\pi$
$734$	38.0000	1.40261
$735$	0	0
$736$	6.00000	0.221163
$737$	13.0000	0.478861
$738$	12.0000	0.441726
$739$	2.00000	0.0735712	0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000	0.0735712	0.0367856	+	0.999323i	$0.488288\pi$
$740$	0	0
$741$	10.0000	0.367359
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	11.0000	0.402739
$747$	−12.0000	−0.439057
$748$	−6.00000	−0.219382
$749$	0	0
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	6.00000	0.218797
$753$	12.0000	0.437304
$754$	15.0000	0.546268
$755$	0	0
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	−25.0000	−0.908041
$759$	−6.00000	−0.217786
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	−13.0000	−0.470940
$763$	0	0
$764$	−18.0000	−0.651217
$765$	0	0
$766$	12.0000	0.433578
$767$	−15.0000	−0.541619
$768$	1.00000	0.0360844
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	27.0000	0.972381
$772$	−22.0000	−0.791797
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	8.00000	0.287554
$775$	−40.0000	−1.43684
$776$	−13.0000	−0.466673
$777$	0	0
$778$	12.0000	0.430221
$779$	−12.0000	−0.429945
$780$	0	0
$781$	12.0000	0.429394
$782$	36.0000	1.28736
$783$	−15.0000	−0.536056
$784$	0	0
$785$	0	0
$786$	6.00000	0.214013
$787$	14.0000	0.499046	0.249523	−	0.968369i	$-0.419726\pi$
$787$	14.0000	0.499046	0.249523	+	0.968369i	$0.419726\pi$
$788$	−9.00000	−0.320612
$789$	−3.00000	−0.106803
$790$	0	0
$791$	0	0
$792$	2.00000	0.0710669
$793$	−35.0000	−1.24289
$794$	2.00000	0.0709773
$795$	0	0
$796$	14.0000	0.496217
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	0	0
$799$	36.0000	1.27359
$800$	−5.00000	−0.176777
$801$	−12.0000	−0.423999
$802$	−9.00000	−0.317801
$803$	10.0000	0.352892
$804$	−13.0000	−0.458475
$805$	0	0
$806$	40.0000	1.40894
$807$	−24.0000	−0.844840
$808$	3.00000	0.105540
$809$	36.0000	1.26569	0.632846	−	0.774277i	$-0.281886\pi$
$809$	36.0000	1.26569	0.632846	+	0.774277i	$0.281886\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	0	0
$813$	−25.0000	−0.876788
$814$	−2.00000	−0.0701000
$815$	0	0
$816$	6.00000	0.210042
$817$	−8.00000	−0.279885
$818$	−4.00000	−0.139857
$819$	0	0
$820$	0	0
$821$	27.0000	0.942306	0.471153	−	0.882051i	$-0.343838\pi$
$821$	27.0000	0.942306	0.471153	+	0.882051i	$0.343838\pi$
$822$	15.0000	0.523185
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	−4.00000	−0.139347
$825$	5.00000	0.174078
$826$	0	0
$827$	42.0000	1.46048	0.730242	−	0.683189i	$-0.239408\pi$
$827$	42.0000	1.46048	0.730242	+	0.683189i	$0.239408\pi$
$828$	−12.0000	−0.417029
$829$	−16.0000	−0.555703	−0.277851	−	0.960624i	$-0.589622\pi$
$829$	−16.0000	−0.555703	−0.277851	+	0.960624i	$0.589622\pi$
$830$	0	0
$831$	−1.00000	−0.0346896
$832$	5.00000	0.173344
$833$	0	0
$834$	−4.00000	−0.138509
$835$	0	0
$836$	−2.00000	−0.0691714
$837$	−40.0000	−1.38260
$838$	12.0000	0.414533
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−16.0000	−0.551396
$843$	−6.00000	−0.206651
$844$	26.0000	0.894957
$845$	0	0
$846$	−12.0000	−0.412568
$847$	0	0
$848$	−12.0000	−0.412082
$849$	−22.0000	−0.755038
$850$	−30.0000	−1.02899
$851$	12.0000	0.411355
$852$	−12.0000	−0.411113
$853$	−58.0000	−1.98588	−0.992941	−	0.118609i	$-0.962157\pi$
$853$	−58.0000	−1.98588	−0.992941	+	0.118609i	$0.962157\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	48.0000	1.63965	0.819824	−	0.572615i	$-0.194071\pi$
$857$	48.0000	1.63965	0.819824	+	0.572615i	$0.194071\pi$
$858$	−5.00000	−0.170697
$859$	35.0000	1.19418	0.597092	−	0.802173i	$-0.296323\pi$
$859$	35.0000	1.19418	0.597092	+	0.802173i	$0.296323\pi$
$860$	0	0
$861$	0	0
$862$	15.0000	0.510902
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	14.0000	0.475739
$867$	19.0000	0.645274
$868$	0	0
$869$	1.00000	0.0339227
$870$	0	0
$871$	−65.0000	−2.20244
$872$	14.0000	0.474100
$873$	26.0000	0.879967
$874$	12.0000	0.405906
$875$	0	0
$876$	−10.0000	−0.337869
$877$	−37.0000	−1.24940	−0.624701	−	0.780864i	$-0.714779\pi$
$877$	−37.0000	−1.24940	−0.624701	+	0.780864i	$0.714779\pi$
$878$	−19.0000	−0.641219
$879$	−18.0000	−0.607125
$880$	0	0
$881$	−15.0000	−0.505363	−0.252681	−	0.967550i	$-0.581312\pi$
$881$	−15.0000	−0.505363	−0.252681	+	0.967550i	$0.581312\pi$
$882$	0	0
$883$	−19.0000	−0.639401	−0.319700	−	0.947519i	$-0.603582\pi$
$883$	−19.0000	−0.639401	−0.319700	+	0.947519i	$0.603582\pi$
$884$	30.0000	1.00901
$885$	0	0
$886$	12.0000	0.403148
$887$	57.0000	1.91387	0.956936	−	0.290298i	$-0.0937544\pi$
$887$	57.0000	1.91387	0.956936	+	0.290298i	$0.0937544\pi$
$888$	2.00000	0.0671156
$889$	0	0
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	−10.0000	−0.334825
$893$	12.0000	0.401565
$894$	−6.00000	−0.200670
$895$	0	0
$896$	0	0
$897$	30.0000	1.00167
$898$	30.0000	1.00111
$899$	24.0000	0.800445
$900$	10.0000	0.333333
$901$	−72.0000	−2.39867
$902$	6.00000	0.199778
$903$	0	0
$904$	9.00000	0.299336
$905$	0	0
$906$	−1.00000	−0.0332228
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	−6.00000	−0.199117
$909$	−6.00000	−0.199007
$910$	0	0
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	2.00000	0.0662266
$913$	−6.00000	−0.198571
$914$	−22.0000	−0.727695
$915$	0	0
$916$	20.0000	0.660819
$917$	0	0
$918$	−30.0000	−0.990148
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	32.0000	1.05444
$922$	39.0000	1.28440
$923$	−60.0000	−1.97492
$924$	0	0
$925$	−10.0000	−0.328798
$926$	26.0000	0.854413
$927$	8.00000	0.262754
$928$	3.00000	0.0984798
$929$	15.0000	0.492134	0.246067	−	0.969253i	$-0.420862\pi$
$929$	15.0000	0.492134	0.246067	+	0.969253i	$0.420862\pi$
$930$	0	0
$931$	0	0
$932$	−30.0000	−0.982683
$933$	24.0000	0.785725
$934$	−12.0000	−0.392652
$935$	0	0
$936$	−10.0000	−0.326860
$937$	−52.0000	−1.69877	−0.849383	−	0.527777i	$-0.823026\pi$
$937$	−52.0000	−1.69877	−0.849383	+	0.527777i	$0.823026\pi$
$938$	0	0
$939$	−7.00000	−0.228436
$940$	0	0
$941$	−27.0000	−0.880175	−0.440087	−	0.897955i	$-0.645053\pi$
$941$	−27.0000	−0.880175	−0.440087	+	0.897955i	$0.645053\pi$
$942$	−4.00000	−0.130327
$943$	−36.0000	−1.17232
$944$	−3.00000	−0.0976417
$945$	0	0
$946$	4.00000	0.130051
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	−1.00000	−0.0324785
$949$	−50.0000	−1.62307
$950$	−10.0000	−0.324443
$951$	−24.0000	−0.778253
$952$	0	0
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	24.0000	0.777029
$955$	0	0
$956$	21.0000	0.679189
$957$	−3.00000	−0.0969762
$958$	−3.00000	−0.0969256
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	10.0000	0.322413
$963$	0	0
$964$	−10.0000	−0.322078
$965$	0	0
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	1.00000	0.0321412
$969$	12.0000	0.385496
$970$	0	0
$971$	−33.0000	−1.05902	−0.529510	−	0.848304i	$-0.677624\pi$
$971$	−33.0000	−1.05902	−0.529510	+	0.848304i	$0.677624\pi$
$972$	16.0000	0.513200
$973$	0	0
$974$	−4.00000	−0.128168
$975$	−25.0000	−0.800641
$976$	−7.00000	−0.224065
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	17.0000	0.543600
$979$	−6.00000	−0.191761
$980$	0	0
$981$	−28.0000	−0.893971
$982$	−30.0000	−0.957338
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	18.0000	0.573237
$987$	0	0
$988$	10.0000	0.318142
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	8.00000	0.254000
$993$	−13.0000	−0.412543
$994$	0	0
$995$	0	0
$996$	6.00000	0.190117
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	8.00000	0.253236
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1078.2.a.k.1.1		1
3.2	odd	2		9702.2.a.o.1.1		1
4.3	odd	2		8624.2.a.l.1.1		1
7.2	even	3		154.2.e.b.67.1	yes	2
7.3	odd	6		1078.2.e.d.177.1		2
7.4	even	3		154.2.e.b.23.1	✓	2
7.5	odd	6		1078.2.e.d.67.1		2
7.6	odd	2		1078.2.a.i.1.1		1
21.2	odd	6		1386.2.k.n.991.1		2
21.11	odd	6		1386.2.k.n.793.1		2
21.20	even	2		9702.2.a.l.1.1		1
28.11	odd	6		1232.2.q.c.177.1		2
28.23	odd	6		1232.2.q.c.529.1		2
28.27	even	2		8624.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.e.b.23.1	✓	2	7.4	even	3
154.2.e.b.67.1	yes	2	7.2	even	3
1078.2.a.i.1.1		1	7.6	odd	2
1078.2.a.k.1.1		1	1.1	even	1	trivial
1078.2.e.d.67.1		2	7.5	odd	6
1078.2.e.d.177.1		2	7.3	odd	6
1232.2.q.c.177.1		2	28.11	odd	6
1232.2.q.c.529.1		2	28.23	odd	6
1386.2.k.n.793.1		2	21.11	odd	6
1386.2.k.n.991.1		2	21.2	odd	6
8624.2.a.l.1.1		1	4.3	odd	2
8624.2.a.t.1.1		1	28.27	even	2
9702.2.a.l.1.1		1	21.20	even	2
9702.2.a.o.1.1		1	3.2	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:	\( N \)	\(=\)	\( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1078.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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