LMFDB - Embedded newform 8880.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8880,2,Mod(1,8880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8880, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 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("8880.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8880.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:	$70.9071569949$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4440)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8880.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +3.00000 q^{13} -1.00000 q^{15} +3.00000 q^{17} +7.00000 q^{19} +3.00000 q^{21} -3.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +10.0000 q^{29} -10.0000 q^{31} -1.00000 q^{33} -3.00000 q^{35} -1.00000 q^{37} -3.00000 q^{39} -12.0000 q^{41} +12.0000 q^{43} +1.00000 q^{45} +4.00000 q^{47} +2.00000 q^{49} -3.00000 q^{51} +9.00000 q^{53} +1.00000 q^{55} -7.00000 q^{57} +6.00000 q^{59} +2.00000 q^{61} -3.00000 q^{63} +3.00000 q^{65} -10.0000 q^{67} +3.00000 q^{69} -2.00000 q^{71} +15.0000 q^{73} -1.00000 q^{75} -3.00000 q^{77} -4.00000 q^{79} +1.00000 q^{81} -9.00000 q^{83} +3.00000 q^{85} -10.0000 q^{87} -11.0000 q^{89} -9.00000 q^{91} +10.0000 q^{93} +7.00000 q^{95} +8.00000 q^{97} +1.00000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	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
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−3.00000	−0.507093
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	−3.00000	−0.480384
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	−3.00000	−0.420084
$52$	0	0
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−7.00000	−0.927173
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	−3.00000	−0.377964
$64$	0	0
$65$	3.00000	0.372104
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	0	0
$69$	3.00000	0.361158
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	15.0000	1.75562	0.877809	−	0.479012i	$-0.159005\pi$
$73$	15.0000	1.75562	0.877809	+	0.479012i	$0.159005\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−3.00000	−0.341882
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	0	0
$87$	−10.0000	−1.07211
$88$	0	0
$89$	−11.0000	−1.16600	−0.582999	−	0.812473i	$-0.698121\pi$
$89$	−11.0000	−1.16600	−0.582999	+	0.812473i	$0.698121\pi$
$90$	0	0
$91$	−9.00000	−0.943456
$92$	0	0
$93$	10.0000	1.03695
$94$	0	0
$95$	7.00000	0.718185
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0	0
$107$	13.0000	1.25676	0.628379	−	0.777908i	$-0.283719\pi$
$107$	13.0000	1.25676	0.628379	+	0.777908i	$0.283719\pi$
$108$	0	0
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	0	0
$113$	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
$115$	−3.00000	−0.279751
$116$	0	0
$117$	3.00000	0.277350
$118$	0	0
$119$	−9.00000	−0.825029
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	12.0000	1.08200
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	0	0
$129$	−12.0000	−1.05654
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	−21.0000	−1.82093
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	3.00000	0.250873
$144$	0	0
$145$	10.0000	0.830455
$146$	0	0
$147$	−2.00000	−0.164957
$148$	0	0
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	−7.00000	−0.569652	−0.284826	−	0.958579i	$-0.591936\pi$
$151$	−7.00000	−0.569652	−0.284826	+	0.958579i	$0.591936\pi$
$152$	0	0
$153$	3.00000	0.242536
$154$	0	0
$155$	−10.0000	−0.803219
$156$	0	0
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	0	0
$159$	−9.00000	−0.713746
$160$	0	0
$161$	9.00000	0.709299
$162$	0	0
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	0	0
$165$	−1.00000	−0.0778499
$166$	0	0
$167$	11.0000	0.851206	0.425603	−	0.904910i	$-0.360062\pi$
$167$	11.0000	0.851206	0.425603	+	0.904910i	$0.360062\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	7.00000	0.535303
$172$	0	0
$173$	−1.00000	−0.0760286	−0.0380143	−	0.999277i	$-0.512103\pi$
$173$	−1.00000	−0.0760286	−0.0380143	+	0.999277i	$0.512103\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	0	0
$177$	−6.00000	−0.450988
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	3.00000	0.219382
$188$	0	0
$189$	3.00000	0.218218
$190$	0	0
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	0	0
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	0	0
$195$	−3.00000	−0.214834
$196$	0	0
$197$	11.0000	0.783718	0.391859	−	0.920025i	$-0.371832\pi$
$197$	11.0000	0.783718	0.391859	+	0.920025i	$0.371832\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	10.0000	0.705346
$202$	0	0
$203$	−30.0000	−2.10559
$204$	0	0
$205$	−12.0000	−0.838116
$206$	0	0
$207$	−3.00000	−0.208514
$208$	0	0
$209$	7.00000	0.484200
$210$	0	0
$211$	−18.0000	−1.23917	−0.619586	−	0.784929i	$-0.712699\pi$
$211$	−18.0000	−1.23917	−0.619586	+	0.784929i	$0.712699\pi$
$212$	0	0
$213$	2.00000	0.137038
$214$	0	0
$215$	12.0000	0.818393
$216$	0	0
$217$	30.0000	2.03653
$218$	0	0
$219$	−15.0000	−1.01361
$220$	0	0
$221$	9.00000	0.605406
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	16.0000	1.06196	0.530979	−	0.847385i	$-0.321824\pi$
$227$	16.0000	1.06196	0.530979	+	0.847385i	$0.321824\pi$
$228$	0	0
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	0	0
$237$	4.00000	0.259828
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.00000	0.127775
$246$	0	0
$247$	21.0000	1.33620
$248$	0	0
$249$	9.00000	0.570352
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	0	0
$253$	−3.00000	−0.188608
$254$	0	0
$255$	−3.00000	−0.187867
$256$	0	0
$257$	7.00000	0.436648	0.218324	−	0.975876i	$-0.429941\pi$
$257$	7.00000	0.436648	0.218324	+	0.975876i	$0.429941\pi$
$258$	0	0
$259$	3.00000	0.186411
$260$	0	0
$261$	10.0000	0.618984
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	9.00000	0.552866
$266$	0	0
$267$	11.0000	0.673189
$268$	0	0
$269$	21.0000	1.28039	0.640196	−	0.768211i	$-0.278853\pi$
$269$	21.0000	1.28039	0.640196	+	0.768211i	$0.278853\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	0	0
$273$	9.00000	0.544705
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	−11.0000	−0.660926	−0.330463	−	0.943819i	$-0.607205\pi$
$277$	−11.0000	−0.660926	−0.330463	+	0.943819i	$0.607205\pi$
$278$	0	0
$279$	−10.0000	−0.598684
$280$	0	0
$281$	17.0000	1.01413	0.507067	−	0.861906i	$-0.330729\pi$
$281$	17.0000	1.01413	0.507067	+	0.861906i	$0.330729\pi$
$282$	0	0
$283$	−17.0000	−1.01055	−0.505273	−	0.862960i	$-0.668608\pi$
$283$	−17.0000	−1.01055	−0.505273	+	0.862960i	$0.668608\pi$
$284$	0	0
$285$	−7.00000	−0.414644
$286$	0	0
$287$	36.0000	2.12501
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−8.00000	−0.468968
$292$	0	0
$293$	1.00000	0.0584206	0.0292103	−	0.999573i	$-0.490701\pi$
$293$	1.00000	0.0584206	0.0292103	+	0.999573i	$0.490701\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−9.00000	−0.520483
$300$	0	0
$301$	−36.0000	−2.07501
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	−14.0000	−0.799022	−0.399511	−	0.916728i	$-0.630820\pi$
$307$	−14.0000	−0.799022	−0.399511	+	0.916728i	$0.630820\pi$
$308$	0	0
$309$	12.0000	0.682656
$310$	0	0
$311$	−32.0000	−1.81455	−0.907277	−	0.420534i	$-0.861843\pi$
$311$	−32.0000	−1.81455	−0.907277	+	0.420534i	$0.861843\pi$
$312$	0	0
$313$	4.00000	0.226093	0.113047	−	0.993590i	$-0.463939\pi$
$313$	4.00000	0.226093	0.113047	+	0.993590i	$0.463939\pi$
$314$	0	0
$315$	−3.00000	−0.169031
$316$	0	0
$317$	14.0000	0.786318	0.393159	−	0.919470i	$-0.371382\pi$
$317$	14.0000	0.786318	0.393159	+	0.919470i	$0.371382\pi$
$318$	0	0
$319$	10.0000	0.559893
$320$	0	0
$321$	−13.0000	−0.725589
$322$	0	0
$323$	21.0000	1.16847
$324$	0	0
$325$	3.00000	0.166410
$326$	0	0
$327$	9.00000	0.497701
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	0	0
$333$	−1.00000	−0.0547997
$334$	0	0
$335$	−10.0000	−0.546358
$336$	0	0
$337$	−1.00000	−0.0544735	−0.0272367	−	0.999629i	$-0.508671\pi$
$337$	−1.00000	−0.0544735	−0.0272367	+	0.999629i	$0.508671\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−10.0000	−0.541530
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	3.00000	0.161515
$346$	0	0
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	0	0
$349$	−28.0000	−1.49881	−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000	−1.49881	−0.749403	+	0.662114i	$0.769659\pi$
$350$	0	0
$351$	−3.00000	−0.160128
$352$	0	0
$353$	−34.0000	−1.80964	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−34.0000	−1.80964	−0.904819	+	0.425797i	$0.859994\pi$
$354$	0	0
$355$	−2.00000	−0.106149
$356$	0	0
$357$	9.00000	0.476331
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	0	0
$363$	10.0000	0.524864
$364$	0	0
$365$	15.0000	0.785136
$366$	0	0
$367$	5.00000	0.260998	0.130499	−	0.991448i	$-0.458342\pi$
$367$	5.00000	0.260998	0.130499	+	0.991448i	$0.458342\pi$
$368$	0	0
$369$	−12.0000	−0.624695
$370$	0	0
$371$	−27.0000	−1.40177
$372$	0	0
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	30.0000	1.54508
$378$	0	0
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	0	0
$381$	13.0000	0.666010
$382$	0	0
$383$	−33.0000	−1.68622	−0.843111	−	0.537740i	$-0.819278\pi$
$383$	−33.0000	−1.68622	−0.843111	+	0.537740i	$0.819278\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	0	0
$387$	12.0000	0.609994
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	−9.00000	−0.455150
$392$	0	0
$393$	−8.00000	−0.403547
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	20.0000	1.00377	0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000	1.00377	0.501886	+	0.864934i	$0.332640\pi$
$398$	0	0
$399$	21.0000	1.05131
$400$	0	0
$401$	13.0000	0.649189	0.324595	−	0.945853i	$-0.394772\pi$
$401$	13.0000	0.649189	0.324595	+	0.945853i	$0.394772\pi$
$402$	0	0
$403$	−30.0000	−1.49441
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−1.00000	−0.0495682
$408$	0	0
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	0	0
$413$	−18.0000	−0.885722
$414$	0	0
$415$	−9.00000	−0.441793
$416$	0	0
$417$	−14.0000	−0.685583
$418$	0	0
$419$	39.0000	1.90527	0.952637	−	0.304109i	$-0.0983586\pi$
$419$	39.0000	1.90527	0.952637	+	0.304109i	$0.0983586\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	3.00000	0.145521
$426$	0	0
$427$	−6.00000	−0.290360
$428$	0	0
$429$	−3.00000	−0.144841
$430$	0	0
$431$	−23.0000	−1.10787	−0.553936	−	0.832560i	$-0.686875\pi$
$431$	−23.0000	−1.10787	−0.553936	+	0.832560i	$0.686875\pi$
$432$	0	0
$433$	21.0000	1.00920	0.504598	−	0.863355i	$-0.331641\pi$
$433$	21.0000	1.00920	0.504598	+	0.863355i	$0.331641\pi$
$434$	0	0
$435$	−10.0000	−0.479463
$436$	0	0
$437$	−21.0000	−1.00457
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	−11.0000	−0.521450
$446$	0	0
$447$	22.0000	1.04056
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	0	0
$453$	7.00000	0.328889
$454$	0	0
$455$	−9.00000	−0.421927
$456$	0	0
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	0	0
$459$	−3.00000	−0.140028
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	10.0000	0.463739
$466$	0	0
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	0	0
$469$	30.0000	1.38527
$470$	0	0
$471$	−12.0000	−0.552931
$472$	0	0
$473$	12.0000	0.551761
$474$	0	0
$475$	7.00000	0.321182
$476$	0	0
$477$	9.00000	0.412082
$478$	0	0
$479$	29.0000	1.32504	0.662522	−	0.749043i	$-0.269486\pi$
$479$	29.0000	1.32504	0.662522	+	0.749043i	$0.269486\pi$
$480$	0	0
$481$	−3.00000	−0.136788
$482$	0	0
$483$	−9.00000	−0.409514
$484$	0	0
$485$	8.00000	0.363261
$486$	0	0
$487$	38.0000	1.72194	0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000	1.72194	0.860972	+	0.508652i	$0.169856\pi$
$488$	0	0
$489$	−11.0000	−0.497437
$490$	0	0
$491$	−3.00000	−0.135388	−0.0676941	−	0.997706i	$-0.521564\pi$
$491$	−3.00000	−0.135388	−0.0676941	+	0.997706i	$0.521564\pi$
$492$	0	0
$493$	30.0000	1.35113
$494$	0	0
$495$	1.00000	0.0449467
$496$	0	0
$497$	6.00000	0.269137
$498$	0	0
$499$	17.0000	0.761025	0.380512	−	0.924776i	$-0.375748\pi$
$499$	17.0000	0.761025	0.380512	+	0.924776i	$0.375748\pi$
$500$	0	0
$501$	−11.0000	−0.491444
$502$	0	0
$503$	4.00000	0.178351	0.0891756	−	0.996016i	$-0.471577\pi$
$503$	4.00000	0.178351	0.0891756	+	0.996016i	$0.471577\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	4.00000	0.177646
$508$	0	0
$509$	−29.0000	−1.28540	−0.642701	−	0.766117i	$-0.722186\pi$
$509$	−29.0000	−1.28540	−0.642701	+	0.766117i	$0.722186\pi$
$510$	0	0
$511$	−45.0000	−1.99068
$512$	0	0
$513$	−7.00000	−0.309058
$514$	0	0
$515$	−12.0000	−0.528783
$516$	0	0
$517$	4.00000	0.175920
$518$	0	0
$519$	1.00000	0.0438951
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	0	0
$525$	3.00000	0.130931
$526$	0	0
$527$	−30.0000	−1.30682
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	−36.0000	−1.55933
$534$	0	0
$535$	13.0000	0.562039
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	33.0000	1.41878	0.709390	−	0.704816i	$-0.248970\pi$
$541$	33.0000	1.41878	0.709390	+	0.704816i	$0.248970\pi$
$542$	0	0
$543$	−2.00000	−0.0858282
$544$	0	0
$545$	−9.00000	−0.385518
$546$	0	0
$547$	−9.00000	−0.384812	−0.192406	−	0.981315i	$-0.561629\pi$
$547$	−9.00000	−0.384812	−0.192406	+	0.981315i	$0.561629\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	70.0000	2.98210
$552$	0	0
$553$	12.0000	0.510292
$554$	0	0
$555$	1.00000	0.0424476
$556$	0	0
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	0	0
$559$	36.0000	1.52264
$560$	0	0
$561$	−3.00000	−0.126660
$562$	0	0
$563$	14.0000	0.590030	0.295015	−	0.955493i	$-0.404675\pi$
$563$	14.0000	0.590030	0.295015	+	0.955493i	$0.404675\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	0	0
$567$	−3.00000	−0.125988
$568$	0	0
$569$	−45.0000	−1.88650	−0.943249	−	0.332086i	$-0.892248\pi$
$569$	−45.0000	−1.88650	−0.943249	+	0.332086i	$0.892248\pi$
$570$	0	0
$571$	30.0000	1.25546	0.627730	−	0.778431i	$-0.283984\pi$
$571$	30.0000	1.25546	0.627730	+	0.778431i	$0.283984\pi$
$572$	0	0
$573$	−15.0000	−0.626634
$574$	0	0
$575$	−3.00000	−0.125109
$576$	0	0
$577$	26.0000	1.08239	0.541197	−	0.840896i	$-0.317971\pi$
$577$	26.0000	1.08239	0.541197	+	0.840896i	$0.317971\pi$
$578$	0	0
$579$	−22.0000	−0.914289
$580$	0	0
$581$	27.0000	1.12015
$582$	0	0
$583$	9.00000	0.372742
$584$	0	0
$585$	3.00000	0.124035
$586$	0	0
$587$	16.0000	0.660391	0.330195	−	0.943913i	$-0.392885\pi$
$587$	16.0000	0.660391	0.330195	+	0.943913i	$0.392885\pi$
$588$	0	0
$589$	−70.0000	−2.88430
$590$	0	0
$591$	−11.0000	−0.452480
$592$	0	0
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	0	0
$595$	−9.00000	−0.368964
$596$	0	0
$597$	0	0
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	0	0
$603$	−10.0000	−0.407231
$604$	0	0
$605$	−10.0000	−0.406558
$606$	0	0
$607$	26.0000	1.05531	0.527654	−	0.849460i	$-0.323072\pi$
$607$	26.0000	1.05531	0.527654	+	0.849460i	$0.323072\pi$
$608$	0	0
$609$	30.0000	1.21566
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	34.0000	1.37325	0.686624	−	0.727013i	$-0.259092\pi$
$613$	34.0000	1.37325	0.686624	+	0.727013i	$0.259092\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	0	0
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	0	0
$623$	33.0000	1.32212
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−7.00000	−0.279553
$628$	0	0
$629$	−3.00000	−0.119618
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	0	0
$633$	18.0000	0.715436
$634$	0	0
$635$	−13.0000	−0.515889
$636$	0	0
$637$	6.00000	0.237729
$638$	0	0
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	48.0000	1.89589	0.947943	−	0.318440i	$-0.103159\pi$
$641$	48.0000	1.89589	0.947943	+	0.318440i	$0.103159\pi$
$642$	0	0
$643$	5.00000	0.197181	0.0985904	−	0.995128i	$-0.468567\pi$
$643$	5.00000	0.197181	0.0985904	+	0.995128i	$0.468567\pi$
$644$	0	0
$645$	−12.0000	−0.472500
$646$	0	0
$647$	−27.0000	−1.06148	−0.530740	−	0.847535i	$-0.678086\pi$
$647$	−27.0000	−1.06148	−0.530740	+	0.847535i	$0.678086\pi$
$648$	0	0
$649$	6.00000	0.235521
$650$	0	0
$651$	−30.0000	−1.17579
$652$	0	0
$653$	−16.0000	−0.626128	−0.313064	−	0.949732i	$-0.601356\pi$
$653$	−16.0000	−0.626128	−0.313064	+	0.949732i	$0.601356\pi$
$654$	0	0
$655$	8.00000	0.312586
$656$	0	0
$657$	15.0000	0.585206
$658$	0	0
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	0	0
$661$	21.0000	0.816805	0.408403	−	0.912802i	$-0.366086\pi$
$661$	21.0000	0.816805	0.408403	+	0.912802i	$0.366086\pi$
$662$	0	0
$663$	−9.00000	−0.349531
$664$	0	0
$665$	−21.0000	−0.814345
$666$	0	0
$667$	−30.0000	−1.16160
$668$	0	0
$669$	−16.0000	−0.618596
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	−29.0000	−1.11787	−0.558934	−	0.829212i	$-0.688789\pi$
$673$	−29.0000	−1.11787	−0.558934	+	0.829212i	$0.688789\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−13.0000	−0.499631	−0.249815	−	0.968294i	$-0.580370\pi$
$677$	−13.0000	−0.499631	−0.249815	+	0.968294i	$0.580370\pi$
$678$	0	0
$679$	−24.0000	−0.921035
$680$	0	0
$681$	−16.0000	−0.613121
$682$	0	0
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	−26.0000	−0.991962
$688$	0	0
$689$	27.0000	1.02862
$690$	0	0
$691$	−22.0000	−0.836919	−0.418460	−	0.908235i	$-0.637430\pi$
$691$	−22.0000	−0.836919	−0.418460	+	0.908235i	$0.637430\pi$
$692$	0	0
$693$	−3.00000	−0.113961
$694$	0	0
$695$	14.0000	0.531050
$696$	0	0
$697$	−36.0000	−1.36360
$698$	0	0
$699$	10.0000	0.378235
$700$	0	0
$701$	−48.0000	−1.81293	−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000	−1.81293	−0.906467	+	0.422276i	$0.861231\pi$
$702$	0	0
$703$	−7.00000	−0.264010
$704$	0	0
$705$	−4.00000	−0.150649
$706$	0	0
$707$	18.0000	0.676960
$708$	0	0
$709$	17.0000	0.638448	0.319224	−	0.947679i	$-0.396578\pi$
$709$	17.0000	0.638448	0.319224	+	0.947679i	$0.396578\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	30.0000	1.12351
$714$	0	0
$715$	3.00000	0.112194
$716$	0	0
$717$	12.0000	0.448148
$718$	0	0
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	36.0000	1.34071
$722$	0	0
$723$	−10.0000	−0.371904
$724$	0	0
$725$	10.0000	0.371391
$726$	0	0
$727$	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
$729$	1.00000	0.0370370
$730$	0	0
$731$	36.0000	1.33151
$732$	0	0
$733$	46.0000	1.69905	0.849524	−	0.527549i	$-0.176889\pi$
$733$	46.0000	1.69905	0.849524	+	0.527549i	$0.176889\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	−10.0000	−0.368355
$738$	0	0
$739$	22.0000	0.809283	0.404642	−	0.914475i	$-0.367396\pi$
$739$	22.0000	0.809283	0.404642	+	0.914475i	$0.367396\pi$
$740$	0	0
$741$	−21.0000	−0.771454
$742$	0	0
$743$	−38.0000	−1.39408	−0.697042	−	0.717030i	$-0.745501\pi$
$743$	−38.0000	−1.39408	−0.697042	+	0.717030i	$0.745501\pi$
$744$	0	0
$745$	−22.0000	−0.806018
$746$	0	0
$747$	−9.00000	−0.329293
$748$	0	0
$749$	−39.0000	−1.42503
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	0	0
$753$	−2.00000	−0.0728841
$754$	0	0
$755$	−7.00000	−0.254756
$756$	0	0
$757$	−47.0000	−1.70824	−0.854122	−	0.520073i	$-0.825905\pi$
$757$	−47.0000	−1.70824	−0.854122	+	0.520073i	$0.825905\pi$
$758$	0	0
$759$	3.00000	0.108893
$760$	0	0
$761$	−10.0000	−0.362500	−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000	−0.362500	−0.181250	+	0.983437i	$0.558014\pi$
$762$	0	0
$763$	27.0000	0.977466
$764$	0	0
$765$	3.00000	0.108465
$766$	0	0
$767$	18.0000	0.649942
$768$	0	0
$769$	−42.0000	−1.51456	−0.757279	−	0.653091i	$-0.773472\pi$
$769$	−42.0000	−1.51456	−0.757279	+	0.653091i	$0.773472\pi$
$770$	0	0
$771$	−7.00000	−0.252099
$772$	0	0
$773$	−33.0000	−1.18693	−0.593464	−	0.804861i	$-0.702240\pi$
$773$	−33.0000	−1.18693	−0.593464	+	0.804861i	$0.702240\pi$
$774$	0	0
$775$	−10.0000	−0.359211
$776$	0	0
$777$	−3.00000	−0.107624
$778$	0	0
$779$	−84.0000	−3.00961
$780$	0	0
$781$	−2.00000	−0.0715656
$782$	0	0
$783$	−10.0000	−0.357371
$784$	0	0
$785$	12.0000	0.428298
$786$	0	0
$787$	38.0000	1.35455	0.677277	−	0.735728i	$-0.263160\pi$
$787$	38.0000	1.35455	0.677277	+	0.735728i	$0.263160\pi$
$788$	0	0
$789$	−24.0000	−0.854423
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	6.00000	0.213066
$794$	0	0
$795$	−9.00000	−0.319197
$796$	0	0
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	0	0
$801$	−11.0000	−0.388666
$802$	0	0
$803$	15.0000	0.529339
$804$	0	0
$805$	9.00000	0.317208
$806$	0	0
$807$	−21.0000	−0.739235
$808$	0	0
$809$	−39.0000	−1.37117	−0.685583	−	0.727994i	$-0.740453\pi$
$809$	−39.0000	−1.37117	−0.685583	+	0.727994i	$0.740453\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	0	0
$813$	−12.0000	−0.420858
$814$	0	0
$815$	11.0000	0.385313
$816$	0	0
$817$	84.0000	2.93879
$818$	0	0
$819$	−9.00000	−0.314485
$820$	0	0
$821$	−13.0000	−0.453703	−0.226852	−	0.973929i	$-0.572843\pi$
$821$	−13.0000	−0.453703	−0.226852	+	0.973929i	$0.572843\pi$
$822$	0	0
$823$	51.0000	1.77775	0.888874	−	0.458151i	$-0.151488\pi$
$823$	51.0000	1.77775	0.888874	+	0.458151i	$0.151488\pi$
$824$	0	0
$825$	−1.00000	−0.0348155
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	−5.00000	−0.173657	−0.0868286	−	0.996223i	$-0.527673\pi$
$829$	−5.00000	−0.173657	−0.0868286	+	0.996223i	$0.527673\pi$
$830$	0	0
$831$	11.0000	0.381586
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	11.0000	0.380671
$836$	0	0
$837$	10.0000	0.345651
$838$	0	0
$839$	−10.0000	−0.345238	−0.172619	−	0.984989i	$-0.555223\pi$
$839$	−10.0000	−0.345238	−0.172619	+	0.984989i	$0.555223\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	0	0
$843$	−17.0000	−0.585511
$844$	0	0
$845$	−4.00000	−0.137604
$846$	0	0
$847$	30.0000	1.03081
$848$	0	0
$849$	17.0000	0.583438
$850$	0	0
$851$	3.00000	0.102839
$852$	0	0
$853$	23.0000	0.787505	0.393753	−	0.919216i	$-0.371177\pi$
$853$	23.0000	0.787505	0.393753	+	0.919216i	$0.371177\pi$
$854$	0	0
$855$	7.00000	0.239395
$856$	0	0
$857$	−7.00000	−0.239115	−0.119558	−	0.992827i	$-0.538148\pi$
$857$	−7.00000	−0.239115	−0.119558	+	0.992827i	$0.538148\pi$
$858$	0	0
$859$	5.00000	0.170598	0.0852989	−	0.996355i	$-0.472815\pi$
$859$	5.00000	0.170598	0.0852989	+	0.996355i	$0.472815\pi$
$860$	0	0
$861$	−36.0000	−1.22688
$862$	0	0
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	0	0
$865$	−1.00000	−0.0340010
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	−4.00000	−0.135691
$870$	0	0
$871$	−30.0000	−1.01651
$872$	0	0
$873$	8.00000	0.270759
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	40.0000	1.35070	0.675352	−	0.737496i	$-0.263992\pi$
$877$	40.0000	1.35070	0.675352	+	0.737496i	$0.263992\pi$
$878$	0	0
$879$	−1.00000	−0.0337292
$880$	0	0
$881$	−40.0000	−1.34763	−0.673817	−	0.738898i	$-0.735346\pi$
$881$	−40.0000	−1.34763	−0.673817	+	0.738898i	$0.735346\pi$
$882$	0	0
$883$	47.0000	1.58168	0.790838	−	0.612026i	$-0.209645\pi$
$883$	47.0000	1.58168	0.790838	+	0.612026i	$0.209645\pi$
$884$	0	0
$885$	−6.00000	−0.201688
$886$	0	0
$887$	30.0000	1.00730	0.503651	−	0.863907i	$-0.331990\pi$
$887$	30.0000	1.00730	0.503651	+	0.863907i	$0.331990\pi$
$888$	0	0
$889$	39.0000	1.30802
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	28.0000	0.936984
$894$	0	0
$895$	0	0
$896$	0	0
$897$	9.00000	0.300501
$898$	0	0
$899$	−100.000	−3.33519
$900$	0	0
$901$	27.0000	0.899500
$902$	0	0
$903$	36.0000	1.19800
$904$	0	0
$905$	2.00000	0.0664822
$906$	0	0
$907$	23.0000	0.763702	0.381851	−	0.924224i	$-0.375287\pi$
$907$	23.0000	0.763702	0.381851	+	0.924224i	$0.375287\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	−9.00000	−0.297857
$914$	0	0
$915$	−2.00000	−0.0661180
$916$	0	0
$917$	−24.0000	−0.792550
$918$	0	0
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	0	0
$921$	14.0000	0.461316
$922$	0	0
$923$	−6.00000	−0.197492
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	−12.0000	−0.394132
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	14.0000	0.458831
$932$	0	0
$933$	32.0000	1.04763
$934$	0	0
$935$	3.00000	0.0981105
$936$	0	0
$937$	−18.0000	−0.588034	−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000	−0.588034	−0.294017	+	0.955800i	$0.594992\pi$
$938$	0	0
$939$	−4.00000	−0.130535
$940$	0	0
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	0	0
$943$	36.0000	1.17232
$944$	0	0
$945$	3.00000	0.0975900
$946$	0	0
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	0	0
$949$	45.0000	1.46076
$950$	0	0
$951$	−14.0000	−0.453981
$952$	0	0
$953$	10.0000	0.323932	0.161966	−	0.986796i	$-0.448217\pi$
$953$	10.0000	0.323932	0.161966	+	0.986796i	$0.448217\pi$
$954$	0	0
$955$	15.0000	0.485389
$956$	0	0
$957$	−10.0000	−0.323254
$958$	0	0
$959$	−6.00000	−0.193750
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	13.0000	0.418919
$964$	0	0
$965$	22.0000	0.708205
$966$	0	0
$967$	−18.0000	−0.578841	−0.289420	−	0.957202i	$-0.593463\pi$
$967$	−18.0000	−0.578841	−0.289420	+	0.957202i	$0.593463\pi$
$968$	0	0
$969$	−21.0000	−0.674617
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	−42.0000	−1.34646
$974$	0	0
$975$	−3.00000	−0.0960769
$976$	0	0
$977$	7.00000	0.223950	0.111975	−	0.993711i	$-0.464282\pi$
$977$	7.00000	0.223950	0.111975	+	0.993711i	$0.464282\pi$
$978$	0	0
$979$	−11.0000	−0.351562
$980$	0	0
$981$	−9.00000	−0.287348
$982$	0	0
$983$	−38.0000	−1.21201	−0.606006	−	0.795460i	$-0.707229\pi$
$983$	−38.0000	−1.21201	−0.606006	+	0.795460i	$0.707229\pi$
$984$	0	0
$985$	11.0000	0.350489
$986$	0	0
$987$	12.0000	0.381964
$988$	0	0
$989$	−36.0000	−1.14473
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	0	0
$993$	−28.0000	−0.888553
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−3.00000	−0.0950110	−0.0475055	−	0.998871i	$-0.515127\pi$
$997$	−3.00000	−0.0950110	−0.0475055	+	0.998871i	$0.515127\pi$
$998$	0	0
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8880.2.a.j.1.1		1
4.3	odd	2		4440.2.a.i.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4440.2.a.i.1.1	✓	1	4.3	odd	2
8880.2.a.j.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:	\( N \)	\(=\)	\( 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8880.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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