LMFDB - Embedded newform 930.2.a.m.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [930,2,Mod(1,930)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("930.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(930, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 930 = 2 \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	930.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,-1,1,-1,-1,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$7.42608738798$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	930.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +5.00000 q^{11} -1.00000 q^{12} -6.00000 q^{13} +3.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} +5.00000 q^{19} -1.00000 q^{20} -3.00000 q^{21} +5.00000 q^{22} +5.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{27} +3.00000 q^{28} +8.00000 q^{29} +1.00000 q^{30} -1.00000 q^{31} +1.00000 q^{32} -5.00000 q^{33} -4.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} +4.00000 q^{37} +5.00000 q^{38} +6.00000 q^{39} -1.00000 q^{40} -3.00000 q^{42} -7.00000 q^{43} +5.00000 q^{44} -1.00000 q^{45} +5.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} -6.00000 q^{52} +11.0000 q^{53} -1.00000 q^{54} -5.00000 q^{55} +3.00000 q^{56} -5.00000 q^{57} +8.00000 q^{58} +14.0000 q^{59} +1.00000 q^{60} -6.00000 q^{61} -1.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} +6.00000 q^{65} -5.00000 q^{66} +10.0000 q^{67} -4.00000 q^{68} -5.00000 q^{69} -3.00000 q^{70} -9.00000 q^{71} +1.00000 q^{72} -3.00000 q^{73} +4.00000 q^{74} -1.00000 q^{75} +5.00000 q^{76} +15.0000 q^{77} +6.00000 q^{78} -7.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -12.0000 q^{83} -3.00000 q^{84} +4.00000 q^{85} -7.00000 q^{86} -8.00000 q^{87} +5.00000 q^{88} -9.00000 q^{89} -1.00000 q^{90} -18.0000 q^{91} +5.00000 q^{92} +1.00000 q^{93} +6.00000 q^{94} -5.00000 q^{95} -1.00000 q^{96} +10.0000 q^{97} +2.00000 q^{98} +5.00000 q^{99} +O(q^{100})\)

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
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	−1.00000	−0.288675
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	3.00000	0.801784
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	1.00000	0.235702
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	−1.00000	−0.223607
$21$	−3.00000	−0.654654
$22$	5.00000	1.06600
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	−6.00000	−1.17670
$27$	−1.00000	−0.192450
$28$	3.00000	0.566947
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	1.00000	0.182574
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000	0.176777
$33$	−5.00000	−0.870388
$34$	−4.00000	−0.685994
$35$	−3.00000	−0.507093
$36$	1.00000	0.166667
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	5.00000	0.811107
$39$	6.00000	0.960769
$40$	−1.00000	−0.158114
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−3.00000	−0.462910
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	5.00000	0.753778
$45$	−1.00000	−0.149071
$46$	5.00000	0.737210
$47$	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
$49$	2.00000	0.285714
$50$	1.00000	0.141421
$51$	4.00000	0.560112
$52$	−6.00000	−0.832050
$53$	11.0000	1.51097	0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000	1.51097	0.755483	+	0.655168i	$0.227402\pi$
$54$	−1.00000	−0.136083
$55$	−5.00000	−0.674200
$56$	3.00000	0.400892
$57$	−5.00000	−0.662266
$58$	8.00000	1.05045
$59$	14.0000	1.82264	0.911322	−	0.411693i	$-0.135063\pi$
$59$	14.0000	1.82264	0.911322	+	0.411693i	$0.135063\pi$
$60$	1.00000	0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−1.00000	−0.127000
$63$	3.00000	0.377964
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	−5.00000	−0.615457
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	−4.00000	−0.485071
$69$	−5.00000	−0.601929
$70$	−3.00000	−0.358569
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	1.00000	0.117851
$73$	−3.00000	−0.351123	−0.175562	−	0.984468i	$-0.556174\pi$
$73$	−3.00000	−0.351123	−0.175562	+	0.984468i	$0.556174\pi$
$74$	4.00000	0.464991
$75$	−1.00000	−0.115470
$76$	5.00000	0.573539
$77$	15.0000	1.70941
$78$	6.00000	0.679366
$79$	−7.00000	−0.787562	−0.393781	−	0.919204i	$-0.628833\pi$
$79$	−7.00000	−0.787562	−0.393781	+	0.919204i	$0.628833\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	−3.00000	−0.327327
$85$	4.00000	0.433861
$86$	−7.00000	−0.754829
$87$	−8.00000	−0.857690
$88$	5.00000	0.533002
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	−1.00000	−0.105409
$91$	−18.0000	−1.88691
$92$	5.00000	0.521286
$93$	1.00000	0.103695
$94$	6.00000	0.618853
$95$	−5.00000	−0.512989
$96$	−1.00000	−0.102062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	2.00000	0.202031
$99$	5.00000	0.502519
$100$	1.00000	0.100000
$101$	−1.00000	−0.0995037	−0.0497519	−	0.998762i	$-0.515843\pi$
$101$	−1.00000	−0.0995037	−0.0497519	+	0.998762i	$0.515843\pi$
$102$	4.00000	0.396059
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	−6.00000	−0.588348
$105$	3.00000	0.292770
$106$	11.0000	1.06841
$107$	−17.0000	−1.64345	−0.821726	−	0.569883i	$-0.806989\pi$
$107$	−17.0000	−1.64345	−0.821726	+	0.569883i	$0.806989\pi$
$108$	−1.00000	−0.0962250
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	−5.00000	−0.476731
$111$	−4.00000	−0.379663
$112$	3.00000	0.283473
$113$	−1.00000	−0.0940721	−0.0470360	−	0.998893i	$-0.514978\pi$
$113$	−1.00000	−0.0940721	−0.0470360	+	0.998893i	$0.514978\pi$
$114$	−5.00000	−0.468293
$115$	−5.00000	−0.466252
$116$	8.00000	0.742781
$117$	−6.00000	−0.554700
$118$	14.0000	1.28880
$119$	−12.0000	−1.10004
$120$	1.00000	0.0912871
$121$	14.0000	1.27273
$122$	−6.00000	−0.543214
$123$	0	0
$124$	−1.00000	−0.0898027
$125$	−1.00000	−0.0894427
$126$	3.00000	0.267261
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	1.00000	0.0883883
$129$	7.00000	0.616316
$130$	6.00000	0.526235
$131$	2.00000	0.174741	0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000	0.174741	0.0873704	+	0.996176i	$0.472154\pi$
$132$	−5.00000	−0.435194
$133$	15.0000	1.30066
$134$	10.0000	0.863868
$135$	1.00000	0.0860663
$136$	−4.00000	−0.342997
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	−5.00000	−0.425628
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	−3.00000	−0.253546
$141$	−6.00000	−0.505291
$142$	−9.00000	−0.755263
$143$	−30.0000	−2.50873
$144$	1.00000	0.0833333
$145$	−8.00000	−0.664364
$146$	−3.00000	−0.248282
$147$	−2.00000	−0.164957
$148$	4.00000	0.328798
$149$	13.0000	1.06500	0.532501	−	0.846430i	$-0.321252\pi$
$149$	13.0000	1.06500	0.532501	+	0.846430i	$0.321252\pi$
$150$	−1.00000	−0.0816497
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	5.00000	0.405554
$153$	−4.00000	−0.323381
$154$	15.0000	1.20873
$155$	1.00000	0.0803219
$156$	6.00000	0.480384
$157$	−3.00000	−0.239426	−0.119713	−	0.992809i	$-0.538197\pi$
$157$	−3.00000	−0.239426	−0.119713	+	0.992809i	$0.538197\pi$
$158$	−7.00000	−0.556890
$159$	−11.0000	−0.872357
$160$	−1.00000	−0.0790569
$161$	15.0000	1.18217
$162$	1.00000	0.0785674
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	0	0
$165$	5.00000	0.389249
$166$	−12.0000	−0.931381
$167$	−3.00000	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$168$	−3.00000	−0.231455
$169$	23.0000	1.76923
$170$	4.00000	0.306786
$171$	5.00000	0.382360
$172$	−7.00000	−0.533745
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	−8.00000	−0.606478
$175$	3.00000	0.226779
$176$	5.00000	0.376889
$177$	−14.0000	−1.05230
$178$	−9.00000	−0.674579
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	−1.00000	−0.0745356
$181$	−21.0000	−1.56092	−0.780459	−	0.625207i	$-0.785014\pi$
$181$	−21.0000	−1.56092	−0.780459	+	0.625207i	$0.785014\pi$
$182$	−18.0000	−1.33425
$183$	6.00000	0.443533
$184$	5.00000	0.368605
$185$	−4.00000	−0.294086
$186$	1.00000	0.0733236
$187$	−20.0000	−1.46254
$188$	6.00000	0.437595
$189$	−3.00000	−0.218218
$190$	−5.00000	−0.362738
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	−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	$0.790854\pi$
$194$	10.0000	0.717958
$195$	−6.00000	−0.429669
$196$	2.00000	0.142857
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	5.00000	0.355335
$199$	1.00000	0.0708881	0.0354441	−	0.999372i	$-0.488715\pi$
$199$	1.00000	0.0708881	0.0354441	+	0.999372i	$0.488715\pi$
$200$	1.00000	0.0707107
$201$	−10.0000	−0.705346
$202$	−1.00000	−0.0703598
$203$	24.0000	1.68447
$204$	4.00000	0.280056
$205$	0	0
$206$	−16.0000	−1.11477
$207$	5.00000	0.347524
$208$	−6.00000	−0.416025
$209$	25.0000	1.72929
$210$	3.00000	0.207020
$211$	−19.0000	−1.30801	−0.654007	−	0.756489i	$-0.726913\pi$
$211$	−19.0000	−1.30801	−0.654007	+	0.756489i	$0.726913\pi$
$212$	11.0000	0.755483
$213$	9.00000	0.616670
$214$	−17.0000	−1.16210
$215$	7.00000	0.477396
$216$	−1.00000	−0.0680414
$217$	−3.00000	−0.203653
$218$	−12.0000	−0.812743
$219$	3.00000	0.202721
$220$	−5.00000	−0.337100
$221$	24.0000	1.61441
$222$	−4.00000	−0.268462
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	3.00000	0.200446
$225$	1.00000	0.0666667
$226$	−1.00000	−0.0665190
$227$	−13.0000	−0.862840	−0.431420	−	0.902151i	$-0.641987\pi$
$227$	−13.0000	−0.862840	−0.431420	+	0.902151i	$0.641987\pi$
$228$	−5.00000	−0.331133
$229$	−5.00000	−0.330409	−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000	−0.330409	−0.165205	+	0.986259i	$0.552828\pi$
$230$	−5.00000	−0.329690
$231$	−15.0000	−0.986928
$232$	8.00000	0.525226
$233$	1.00000	0.0655122	0.0327561	−	0.999463i	$-0.489572\pi$
$233$	1.00000	0.0655122	0.0327561	+	0.999463i	$0.489572\pi$
$234$	−6.00000	−0.392232
$235$	−6.00000	−0.391397
$236$	14.0000	0.911322
$237$	7.00000	0.454699
$238$	−12.0000	−0.777844
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	1.00000	0.0645497
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	14.0000	0.899954
$243$	−1.00000	−0.0641500
$244$	−6.00000	−0.384111
$245$	−2.00000	−0.127775
$246$	0	0
$247$	−30.0000	−1.90885
$248$	−1.00000	−0.0635001
$249$	12.0000	0.760469
$250$	−1.00000	−0.0632456
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	3.00000	0.188982
$253$	25.0000	1.57174
$254$	−20.0000	−1.25491
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	7.00000	0.436648	0.218324	−	0.975876i	$-0.429941\pi$
$257$	7.00000	0.436648	0.218324	+	0.975876i	$0.429941\pi$
$258$	7.00000	0.435801
$259$	12.0000	0.745644
$260$	6.00000	0.372104
$261$	8.00000	0.495188
$262$	2.00000	0.123560
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	−5.00000	−0.307729
$265$	−11.0000	−0.675725
$266$	15.0000	0.919709
$267$	9.00000	0.550791
$268$	10.0000	0.610847
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	1.00000	0.0608581
$271$	19.0000	1.15417	0.577084	−	0.816685i	$-0.304191\pi$
$271$	19.0000	1.15417	0.577084	+	0.816685i	$0.304191\pi$
$272$	−4.00000	−0.242536
$273$	18.0000	1.08941
$274$	−6.00000	−0.362473
$275$	5.00000	0.301511
$276$	−5.00000	−0.300965
$277$	−24.0000	−1.44202	−0.721010	−	0.692925i	$-0.756322\pi$
$277$	−24.0000	−1.44202	−0.721010	+	0.692925i	$0.756322\pi$
$278$	14.0000	0.839664
$279$	−1.00000	−0.0598684
$280$	−3.00000	−0.179284
$281$	−20.0000	−1.19310	−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000	−1.19310	−0.596550	+	0.802576i	$0.703462\pi$
$282$	−6.00000	−0.357295
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	−9.00000	−0.534052
$285$	5.00000	0.296174
$286$	−30.0000	−1.77394
$287$	0	0
$288$	1.00000	0.0589256
$289$	−1.00000	−0.0588235
$290$	−8.00000	−0.469776
$291$	−10.0000	−0.586210
$292$	−3.00000	−0.175562
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−2.00000	−0.116642
$295$	−14.0000	−0.815112
$296$	4.00000	0.232495
$297$	−5.00000	−0.290129
$298$	13.0000	0.753070
$299$	−30.0000	−1.73494
$300$	−1.00000	−0.0577350
$301$	−21.0000	−1.21042
$302$	−8.00000	−0.460348
$303$	1.00000	0.0574485
$304$	5.00000	0.286770
$305$	6.00000	0.343559
$306$	−4.00000	−0.228665
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	15.0000	0.854704
$309$	16.0000	0.910208
$310$	1.00000	0.0567962
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	6.00000	0.339683
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−3.00000	−0.169300
$315$	−3.00000	−0.169031
$316$	−7.00000	−0.393781
$317$	20.0000	1.12331	0.561656	−	0.827371i	$-0.310164\pi$
$317$	20.0000	1.12331	0.561656	+	0.827371i	$0.310164\pi$
$318$	−11.0000	−0.616849
$319$	40.0000	2.23957
$320$	−1.00000	−0.0559017
$321$	17.0000	0.948847
$322$	15.0000	0.835917
$323$	−20.0000	−1.11283
$324$	1.00000	0.0555556
$325$	−6.00000	−0.332820
$326$	6.00000	0.332309
$327$	12.0000	0.663602
$328$	0	0
$329$	18.0000	0.992372
$330$	5.00000	0.275241
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	−12.0000	−0.658586
$333$	4.00000	0.219199
$334$	−3.00000	−0.164153
$335$	−10.0000	−0.546358
$336$	−3.00000	−0.163663
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	23.0000	1.25104
$339$	1.00000	0.0543125
$340$	4.00000	0.216930
$341$	−5.00000	−0.270765
$342$	5.00000	0.270369
$343$	−15.0000	−0.809924
$344$	−7.00000	−0.377415
$345$	5.00000	0.269191
$346$	10.0000	0.537603
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	−8.00000	−0.428845
$349$	−8.00000	−0.428230	−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000	−0.428230	−0.214115	+	0.976808i	$0.568687\pi$
$350$	3.00000	0.160357
$351$	6.00000	0.320256
$352$	5.00000	0.266501
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	−14.0000	−0.744092
$355$	9.00000	0.477670
$356$	−9.00000	−0.476999
$357$	12.0000	0.635107
$358$	20.0000	1.05703
$359$	−17.0000	−0.897226	−0.448613	−	0.893726i	$-0.648082\pi$
$359$	−17.0000	−0.897226	−0.448613	+	0.893726i	$0.648082\pi$
$360$	−1.00000	−0.0527046
$361$	6.00000	0.315789
$362$	−21.0000	−1.10374
$363$	−14.0000	−0.734809
$364$	−18.0000	−0.943456
$365$	3.00000	0.157027
$366$	6.00000	0.313625
$367$	34.0000	1.77479	0.887393	−	0.461014i	$-0.152514\pi$
$367$	34.0000	1.77479	0.887393	+	0.461014i	$0.152514\pi$
$368$	5.00000	0.260643
$369$	0	0
$370$	−4.00000	−0.207950
$371$	33.0000	1.71327
$372$	1.00000	0.0518476
$373$	1.00000	0.0517780	0.0258890	−	0.999665i	$-0.491758\pi$
$373$	1.00000	0.0517780	0.0258890	+	0.999665i	$0.491758\pi$
$374$	−20.0000	−1.03418
$375$	1.00000	0.0516398
$376$	6.00000	0.309426
$377$	−48.0000	−2.47213
$378$	−3.00000	−0.154303
$379$	−17.0000	−0.873231	−0.436616	−	0.899648i	$-0.643823\pi$
$379$	−17.0000	−0.873231	−0.436616	+	0.899648i	$0.643823\pi$
$380$	−5.00000	−0.256495
$381$	20.0000	1.02463
$382$	16.0000	0.818631
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	−1.00000	−0.0510310
$385$	−15.0000	−0.764471
$386$	−22.0000	−1.11977
$387$	−7.00000	−0.355830
$388$	10.0000	0.507673
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	−6.00000	−0.303822
$391$	−20.0000	−1.01144
$392$	2.00000	0.101015
$393$	−2.00000	−0.100887
$394$	−22.0000	−1.10834
$395$	7.00000	0.352208
$396$	5.00000	0.251259
$397$	13.0000	0.652451	0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000	0.652451	0.326226	+	0.945292i	$0.394223\pi$
$398$	1.00000	0.0501255
$399$	−15.0000	−0.750939
$400$	1.00000	0.0500000
$401$	−1.00000	−0.0499376	−0.0249688	−	0.999688i	$-0.507949\pi$
$401$	−1.00000	−0.0499376	−0.0249688	+	0.999688i	$0.507949\pi$
$402$	−10.0000	−0.498755
$403$	6.00000	0.298881
$404$	−1.00000	−0.0497519
$405$	−1.00000	−0.0496904
$406$	24.0000	1.19110
$407$	20.0000	0.991363
$408$	4.00000	0.198030
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	−16.0000	−0.788263
$413$	42.0000	2.06668
$414$	5.00000	0.245737
$415$	12.0000	0.589057
$416$	−6.00000	−0.294174
$417$	−14.0000	−0.685583
$418$	25.0000	1.22279
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	3.00000	0.146385
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	−19.0000	−0.924906
$423$	6.00000	0.291730
$424$	11.0000	0.534207
$425$	−4.00000	−0.194029
$426$	9.00000	0.436051
$427$	−18.0000	−0.871081
$428$	−17.0000	−0.821726
$429$	30.0000	1.44841
$430$	7.00000	0.337570
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	−1.00000	−0.0481125
$433$	−1.00000	−0.0480569	−0.0240285	−	0.999711i	$-0.507649\pi$
$433$	−1.00000	−0.0480569	−0.0240285	+	0.999711i	$0.507649\pi$
$434$	−3.00000	−0.144005
$435$	8.00000	0.383571
$436$	−12.0000	−0.574696
$437$	25.0000	1.19591
$438$	3.00000	0.143346
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	−5.00000	−0.238366
$441$	2.00000	0.0952381
$442$	24.0000	1.14156
$443$	−31.0000	−1.47285	−0.736427	−	0.676517i	$-0.763489\pi$
$443$	−31.0000	−1.47285	−0.736427	+	0.676517i	$0.763489\pi$
$444$	−4.00000	−0.189832
$445$	9.00000	0.426641
$446$	10.0000	0.473514
$447$	−13.0000	−0.614879
$448$	3.00000	0.141737
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	−1.00000	−0.0470360
$453$	8.00000	0.375873
$454$	−13.0000	−0.610120
$455$	18.0000	0.843853
$456$	−5.00000	−0.234146
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−5.00000	−0.233635
$459$	4.00000	0.186704
$460$	−5.00000	−0.233126
$461$	28.0000	1.30409	0.652045	−	0.758180i	$-0.273911\pi$
$461$	28.0000	1.30409	0.652045	+	0.758180i	$0.273911\pi$
$462$	−15.0000	−0.697863
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	8.00000	0.371391
$465$	−1.00000	−0.0463739
$466$	1.00000	0.0463241
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	−6.00000	−0.277350
$469$	30.0000	1.38527
$470$	−6.00000	−0.276759
$471$	3.00000	0.138233
$472$	14.0000	0.644402
$473$	−35.0000	−1.60930
$474$	7.00000	0.321521
$475$	5.00000	0.229416
$476$	−12.0000	−0.550019
$477$	11.0000	0.503655
$478$	−12.0000	−0.548867
$479$	3.00000	0.137073	0.0685367	−	0.997649i	$-0.478167\pi$
$479$	3.00000	0.137073	0.0685367	+	0.997649i	$0.478167\pi$
$480$	1.00000	0.0456435
$481$	−24.0000	−1.09431
$482$	−14.0000	−0.637683
$483$	−15.0000	−0.682524
$484$	14.0000	0.636364
$485$	−10.0000	−0.454077
$486$	−1.00000	−0.0453609
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	−6.00000	−0.271607
$489$	−6.00000	−0.271329
$490$	−2.00000	−0.0903508
$491$	−19.0000	−0.857458	−0.428729	−	0.903433i	$-0.641038\pi$
$491$	−19.0000	−0.857458	−0.428729	+	0.903433i	$0.641038\pi$
$492$	0	0
$493$	−32.0000	−1.44121
$494$	−30.0000	−1.34976
$495$	−5.00000	−0.224733
$496$	−1.00000	−0.0449013
$497$	−27.0000	−1.21112
$498$	12.0000	0.537733
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	−1.00000	−0.0447214
$501$	3.00000	0.134030
$502$	20.0000	0.892644
$503$	−40.0000	−1.78351	−0.891756	−	0.452517i	$-0.850526\pi$
$503$	−40.0000	−1.78351	−0.891756	+	0.452517i	$0.850526\pi$
$504$	3.00000	0.133631
$505$	1.00000	0.0444994
$506$	25.0000	1.11139
$507$	−23.0000	−1.02147
$508$	−20.0000	−0.887357
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	−4.00000	−0.177123
$511$	−9.00000	−0.398137
$512$	1.00000	0.0441942
$513$	−5.00000	−0.220755
$514$	7.00000	0.308757
$515$	16.0000	0.705044
$516$	7.00000	0.308158
$517$	30.0000	1.31940
$518$	12.0000	0.527250
$519$	−10.0000	−0.438951
$520$	6.00000	0.263117
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	8.00000	0.350150
$523$	29.0000	1.26808	0.634041	−	0.773300i	$-0.281395\pi$
$523$	29.0000	1.26808	0.634041	+	0.773300i	$0.281395\pi$
$524$	2.00000	0.0873704
$525$	−3.00000	−0.130931
$526$	−24.0000	−1.04645
$527$	4.00000	0.174243
$528$	−5.00000	−0.217597
$529$	2.00000	0.0869565
$530$	−11.0000	−0.477809
$531$	14.0000	0.607548
$532$	15.0000	0.650332
$533$	0	0
$534$	9.00000	0.389468
$535$	17.0000	0.734974
$536$	10.0000	0.431934
$537$	−20.0000	−0.863064
$538$	−6.00000	−0.258678
$539$	10.0000	0.430730
$540$	1.00000	0.0430331
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	19.0000	0.816120
$543$	21.0000	0.901196
$544$	−4.00000	−0.171499
$545$	12.0000	0.514024
$546$	18.0000	0.770329
$547$	−14.0000	−0.598597	−0.299298	−	0.954160i	$-0.596753\pi$
$547$	−14.0000	−0.598597	−0.299298	+	0.954160i	$0.596753\pi$
$548$	−6.00000	−0.256307
$549$	−6.00000	−0.256074
$550$	5.00000	0.213201
$551$	40.0000	1.70406
$552$	−5.00000	−0.212814
$553$	−21.0000	−0.893011
$554$	−24.0000	−1.01966
$555$	4.00000	0.169791
$556$	14.0000	0.593732
$557$	15.0000	0.635570	0.317785	−	0.948163i	$-0.397061\pi$
$557$	15.0000	0.635570	0.317785	+	0.948163i	$0.397061\pi$
$558$	−1.00000	−0.0423334
$559$	42.0000	1.77641
$560$	−3.00000	−0.126773
$561$	20.0000	0.844401
$562$	−20.0000	−0.843649
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	−6.00000	−0.252646
$565$	1.00000	0.0420703
$566$	24.0000	1.00880
$567$	3.00000	0.125988
$568$	−9.00000	−0.377632
$569$	39.0000	1.63497	0.817483	−	0.575953i	$-0.195369\pi$
$569$	39.0000	1.63497	0.817483	+	0.575953i	$0.195369\pi$
$570$	5.00000	0.209427
$571$	−14.0000	−0.585882	−0.292941	−	0.956131i	$-0.594634\pi$
$571$	−14.0000	−0.585882	−0.292941	+	0.956131i	$0.594634\pi$
$572$	−30.0000	−1.25436
$573$	−16.0000	−0.668410
$574$	0	0
$575$	5.00000	0.208514
$576$	1.00000	0.0416667
$577$	24.0000	0.999133	0.499567	−	0.866276i	$-0.333493\pi$
$577$	24.0000	0.999133	0.499567	+	0.866276i	$0.333493\pi$
$578$	−1.00000	−0.0415945
$579$	22.0000	0.914289
$580$	−8.00000	−0.332182
$581$	−36.0000	−1.49353
$582$	−10.0000	−0.414513
$583$	55.0000	2.27787
$584$	−3.00000	−0.124141
$585$	6.00000	0.248069
$586$	6.00000	0.247858
$587$	−8.00000	−0.330195	−0.165098	−	0.986277i	$-0.552794\pi$
$587$	−8.00000	−0.330195	−0.165098	+	0.986277i	$0.552794\pi$
$588$	−2.00000	−0.0824786
$589$	−5.00000	−0.206021
$590$	−14.0000	−0.576371
$591$	22.0000	0.904959
$592$	4.00000	0.164399
$593$	26.0000	1.06769	0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000	1.06769	0.533846	+	0.845582i	$0.320746\pi$
$594$	−5.00000	−0.205152
$595$	12.0000	0.491952
$596$	13.0000	0.532501
$597$	−1.00000	−0.0409273
$598$	−30.0000	−1.22679
$599$	−43.0000	−1.75693	−0.878466	−	0.477805i	$-0.841433\pi$
$599$	−43.0000	−1.75693	−0.878466	+	0.477805i	$0.841433\pi$
$600$	−1.00000	−0.0408248
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	−21.0000	−0.855896
$603$	10.0000	0.407231
$604$	−8.00000	−0.325515
$605$	−14.0000	−0.569181
$606$	1.00000	0.0406222
$607$	27.0000	1.09590	0.547948	−	0.836512i	$-0.315409\pi$
$607$	27.0000	1.09590	0.547948	+	0.836512i	$0.315409\pi$
$608$	5.00000	0.202777
$609$	−24.0000	−0.972529
$610$	6.00000	0.242933
$611$	−36.0000	−1.45640
$612$	−4.00000	−0.161690
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	15.0000	0.604367
$617$	43.0000	1.73111	0.865557	−	0.500810i	$-0.166964\pi$
$617$	43.0000	1.73111	0.865557	+	0.500810i	$0.166964\pi$
$618$	16.0000	0.643614
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	1.00000	0.0401610
$621$	−5.00000	−0.200643
$622$	8.00000	0.320771
$623$	−27.0000	−1.08173
$624$	6.00000	0.240192
$625$	1.00000	0.0400000
$626$	14.0000	0.559553
$627$	−25.0000	−0.998404
$628$	−3.00000	−0.119713
$629$	−16.0000	−0.637962
$630$	−3.00000	−0.119523
$631$	47.0000	1.87104	0.935520	−	0.353273i	$-0.114931\pi$
$631$	47.0000	1.87104	0.935520	+	0.353273i	$0.114931\pi$
$632$	−7.00000	−0.278445
$633$	19.0000	0.755182
$634$	20.0000	0.794301
$635$	20.0000	0.793676
$636$	−11.0000	−0.436178
$637$	−12.0000	−0.475457
$638$	40.0000	1.58362
$639$	−9.00000	−0.356034
$640$	−1.00000	−0.0395285
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	17.0000	0.670936
$643$	19.0000	0.749287	0.374643	−	0.927169i	$-0.377765\pi$
$643$	19.0000	0.749287	0.374643	+	0.927169i	$0.377765\pi$
$644$	15.0000	0.591083
$645$	−7.00000	−0.275625
$646$	−20.0000	−0.786889
$647$	−21.0000	−0.825595	−0.412798	−	0.910823i	$-0.635448\pi$
$647$	−21.0000	−0.825595	−0.412798	+	0.910823i	$0.635448\pi$
$648$	1.00000	0.0392837
$649$	70.0000	2.74774
$650$	−6.00000	−0.235339
$651$	3.00000	0.117579
$652$	6.00000	0.234978
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	12.0000	0.469237
$655$	−2.00000	−0.0781465
$656$	0	0
$657$	−3.00000	−0.117041
$658$	18.0000	0.701713
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	5.00000	0.194625
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	−12.0000	−0.466393
$663$	−24.0000	−0.932083
$664$	−12.0000	−0.465690
$665$	−15.0000	−0.581675
$666$	4.00000	0.154997
$667$	40.0000	1.54881
$668$	−3.00000	−0.116073
$669$	−10.0000	−0.386622
$670$	−10.0000	−0.386334
$671$	−30.0000	−1.15814
$672$	−3.00000	−0.115728
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	14.0000	0.539260
$675$	−1.00000	−0.0384900
$676$	23.0000	0.884615
$677$	3.00000	0.115299	0.0576497	−	0.998337i	$-0.481639\pi$
$677$	3.00000	0.115299	0.0576497	+	0.998337i	$0.481639\pi$
$678$	1.00000	0.0384048
$679$	30.0000	1.15129
$680$	4.00000	0.153393
$681$	13.0000	0.498161
$682$	−5.00000	−0.191460
$683$	27.0000	1.03313	0.516563	−	0.856249i	$-0.327211\pi$
$683$	27.0000	1.03313	0.516563	+	0.856249i	$0.327211\pi$
$684$	5.00000	0.191180
$685$	6.00000	0.229248
$686$	−15.0000	−0.572703
$687$	5.00000	0.190762
$688$	−7.00000	−0.266872
$689$	−66.0000	−2.51440
$690$	5.00000	0.190347
$691$	1.00000	0.0380418	0.0190209	−	0.999819i	$-0.493945\pi$
$691$	1.00000	0.0380418	0.0190209	+	0.999819i	$0.493945\pi$
$692$	10.0000	0.380143
$693$	15.0000	0.569803
$694$	20.0000	0.759190
$695$	−14.0000	−0.531050
$696$	−8.00000	−0.303239
$697$	0	0
$698$	−8.00000	−0.302804
$699$	−1.00000	−0.0378235
$700$	3.00000	0.113389
$701$	−5.00000	−0.188847	−0.0944237	−	0.995532i	$-0.530101\pi$
$701$	−5.00000	−0.188847	−0.0944237	+	0.995532i	$0.530101\pi$
$702$	6.00000	0.226455
$703$	20.0000	0.754314
$704$	5.00000	0.188445
$705$	6.00000	0.225973
$706$	−6.00000	−0.225813
$707$	−3.00000	−0.112827
$708$	−14.0000	−0.526152
$709$	41.0000	1.53979	0.769894	−	0.638172i	$-0.220309\pi$
$709$	41.0000	1.53979	0.769894	+	0.638172i	$0.220309\pi$
$710$	9.00000	0.337764
$711$	−7.00000	−0.262521
$712$	−9.00000	−0.337289
$713$	−5.00000	−0.187251
$714$	12.0000	0.449089
$715$	30.0000	1.12194
$716$	20.0000	0.747435
$717$	12.0000	0.448148
$718$	−17.0000	−0.634434
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	−1.00000	−0.0372678
$721$	−48.0000	−1.78761
$722$	6.00000	0.223297
$723$	14.0000	0.520666
$724$	−21.0000	−0.780459
$725$	8.00000	0.297113
$726$	−14.0000	−0.519589
$727$	−5.00000	−0.185440	−0.0927199	−	0.995692i	$-0.529556\pi$
$727$	−5.00000	−0.185440	−0.0927199	+	0.995692i	$0.529556\pi$
$728$	−18.0000	−0.667124
$729$	1.00000	0.0370370
$730$	3.00000	0.111035
$731$	28.0000	1.03562
$732$	6.00000	0.221766
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	34.0000	1.25496
$735$	2.00000	0.0737711
$736$	5.00000	0.184302
$737$	50.0000	1.84177
$738$	0	0
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	−4.00000	−0.147043
$741$	30.0000	1.10208
$742$	33.0000	1.21147
$743$	−21.0000	−0.770415	−0.385208	−	0.922830i	$-0.625870\pi$
$743$	−21.0000	−0.770415	−0.385208	+	0.922830i	$0.625870\pi$
$744$	1.00000	0.0366618
$745$	−13.0000	−0.476283
$746$	1.00000	0.0366126
$747$	−12.0000	−0.439057
$748$	−20.0000	−0.731272
$749$	−51.0000	−1.86350
$750$	1.00000	0.0365148
$751$	18.0000	0.656829	0.328415	−	0.944534i	$-0.393486\pi$
$751$	18.0000	0.656829	0.328415	+	0.944534i	$0.393486\pi$
$752$	6.00000	0.218797
$753$	−20.0000	−0.728841
$754$	−48.0000	−1.74806
$755$	8.00000	0.291150
$756$	−3.00000	−0.109109
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	−17.0000	−0.617468
$759$	−25.0000	−0.907443
$760$	−5.00000	−0.181369
$761$	−5.00000	−0.181250	−0.0906249	−	0.995885i	$-0.528886\pi$
$761$	−5.00000	−0.181250	−0.0906249	+	0.995885i	$0.528886\pi$
$762$	20.0000	0.724524
$763$	−36.0000	−1.30329
$764$	16.0000	0.578860
$765$	4.00000	0.144620
$766$	−20.0000	−0.722629
$767$	−84.0000	−3.03306
$768$	−1.00000	−0.0360844
$769$	25.0000	0.901523	0.450762	−	0.892644i	$-0.351152\pi$
$769$	25.0000	0.901523	0.450762	+	0.892644i	$0.351152\pi$
$770$	−15.0000	−0.540562
$771$	−7.00000	−0.252099
$772$	−22.0000	−0.791797
$773$	−31.0000	−1.11499	−0.557496	−	0.830179i	$-0.688238\pi$
$773$	−31.0000	−1.11499	−0.557496	+	0.830179i	$0.688238\pi$
$774$	−7.00000	−0.251610
$775$	−1.00000	−0.0359211
$776$	10.0000	0.358979
$777$	−12.0000	−0.430498
$778$	18.0000	0.645331
$779$	0	0
$780$	−6.00000	−0.214834
$781$	−45.0000	−1.61023
$782$	−20.0000	−0.715199
$783$	−8.00000	−0.285897
$784$	2.00000	0.0714286
$785$	3.00000	0.107075
$786$	−2.00000	−0.0713376
$787$	−11.0000	−0.392108	−0.196054	−	0.980593i	$-0.562813\pi$
$787$	−11.0000	−0.392108	−0.196054	+	0.980593i	$0.562813\pi$
$788$	−22.0000	−0.783718
$789$	24.0000	0.854423
$790$	7.00000	0.249049
$791$	−3.00000	−0.106668
$792$	5.00000	0.177667
$793$	36.0000	1.27840
$794$	13.0000	0.461353
$795$	11.0000	0.390130
$796$	1.00000	0.0354441
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	−15.0000	−0.530994
$799$	−24.0000	−0.849059
$800$	1.00000	0.0353553
$801$	−9.00000	−0.317999
$802$	−1.00000	−0.0353112
$803$	−15.0000	−0.529339
$804$	−10.0000	−0.352673
$805$	−15.0000	−0.528681
$806$	6.00000	0.211341
$807$	6.00000	0.211210
$808$	−1.00000	−0.0351799
$809$	−1.00000	−0.0351581	−0.0175791	−	0.999845i	$-0.505596\pi$
$809$	−1.00000	−0.0351581	−0.0175791	+	0.999845i	$0.505596\pi$
$810$	−1.00000	−0.0351364
$811$	21.0000	0.737410	0.368705	−	0.929547i	$-0.379801\pi$
$811$	21.0000	0.737410	0.368705	+	0.929547i	$0.379801\pi$
$812$	24.0000	0.842235
$813$	−19.0000	−0.666359
$814$	20.0000	0.701000
$815$	−6.00000	−0.210171
$816$	4.00000	0.140028
$817$	−35.0000	−1.22449
$818$	0	0
$819$	−18.0000	−0.628971
$820$	0	0
$821$	−28.0000	−0.977207	−0.488603	−	0.872506i	$-0.662493\pi$
$821$	−28.0000	−0.977207	−0.488603	+	0.872506i	$0.662493\pi$
$822$	6.00000	0.209274
$823$	−50.0000	−1.74289	−0.871445	−	0.490493i	$-0.836817\pi$
$823$	−50.0000	−1.74289	−0.871445	+	0.490493i	$0.836817\pi$
$824$	−16.0000	−0.557386
$825$	−5.00000	−0.174078
$826$	42.0000	1.46137
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	5.00000	0.173762
$829$	−13.0000	−0.451509	−0.225754	−	0.974184i	$-0.572485\pi$
$829$	−13.0000	−0.451509	−0.225754	+	0.974184i	$0.572485\pi$
$830$	12.0000	0.416526
$831$	24.0000	0.832551
$832$	−6.00000	−0.208013
$833$	−8.00000	−0.277184
$834$	−14.0000	−0.484780
$835$	3.00000	0.103819
$836$	25.0000	0.864643
$837$	1.00000	0.0345651
$838$	24.0000	0.829066
$839$	−5.00000	−0.172619	−0.0863096	−	0.996268i	$-0.527507\pi$
$839$	−5.00000	−0.172619	−0.0863096	+	0.996268i	$0.527507\pi$
$840$	3.00000	0.103510
$841$	35.0000	1.20690
$842$	14.0000	0.482472
$843$	20.0000	0.688837
$844$	−19.0000	−0.654007
$845$	−23.0000	−0.791224
$846$	6.00000	0.206284
$847$	42.0000	1.44314
$848$	11.0000	0.377742
$849$	−24.0000	−0.823678
$850$	−4.00000	−0.137199
$851$	20.0000	0.685591
$852$	9.00000	0.308335
$853$	−47.0000	−1.60925	−0.804625	−	0.593784i	$-0.797633\pi$
$853$	−47.0000	−1.60925	−0.804625	+	0.593784i	$0.797633\pi$
$854$	−18.0000	−0.615947
$855$	−5.00000	−0.170996
$856$	−17.0000	−0.581048
$857$	30.0000	1.02478	0.512390	−	0.858753i	$-0.328760\pi$
$857$	30.0000	1.02478	0.512390	+	0.858753i	$0.328760\pi$
$858$	30.0000	1.02418
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	7.00000	0.238698
$861$	0	0
$862$	24.0000	0.817443
$863$	−45.0000	−1.53182	−0.765909	−	0.642949i	$-0.777711\pi$
$863$	−45.0000	−1.53182	−0.765909	+	0.642949i	$0.777711\pi$
$864$	−1.00000	−0.0340207
$865$	−10.0000	−0.340010
$866$	−1.00000	−0.0339814
$867$	1.00000	0.0339618
$868$	−3.00000	−0.101827
$869$	−35.0000	−1.18729
$870$	8.00000	0.271225
$871$	−60.0000	−2.03302
$872$	−12.0000	−0.406371
$873$	10.0000	0.338449
$874$	25.0000	0.845638
$875$	−3.00000	−0.101419
$876$	3.00000	0.101361
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	−26.0000	−0.877457
$879$	−6.00000	−0.202375
$880$	−5.00000	−0.168550
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	2.00000	0.0673435
$883$	43.0000	1.44707	0.723533	−	0.690290i	$-0.242517\pi$
$883$	43.0000	1.44707	0.723533	+	0.690290i	$0.242517\pi$
$884$	24.0000	0.807207
$885$	14.0000	0.470605
$886$	−31.0000	−1.04147
$887$	−6.00000	−0.201460	−0.100730	−	0.994914i	$-0.532118\pi$
$887$	−6.00000	−0.201460	−0.100730	+	0.994914i	$0.532118\pi$
$888$	−4.00000	−0.134231
$889$	−60.0000	−2.01234
$890$	9.00000	0.301681
$891$	5.00000	0.167506
$892$	10.0000	0.334825
$893$	30.0000	1.00391
$894$	−13.0000	−0.434785
$895$	−20.0000	−0.668526
$896$	3.00000	0.100223
$897$	30.0000	1.00167
$898$	30.0000	1.00111
$899$	−8.00000	−0.266815
$900$	1.00000	0.0333333
$901$	−44.0000	−1.46585
$902$	0	0
$903$	21.0000	0.698836
$904$	−1.00000	−0.0332595
$905$	21.0000	0.698064
$906$	8.00000	0.265782
$907$	52.0000	1.72663	0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000	1.72663	0.863316	+	0.504664i	$0.168384\pi$
$908$	−13.0000	−0.431420
$909$	−1.00000	−0.0331679
$910$	18.0000	0.596694
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	−5.00000	−0.165567
$913$	−60.0000	−1.98571
$914$	−22.0000	−0.727695
$915$	−6.00000	−0.198354
$916$	−5.00000	−0.165205
$917$	6.00000	0.198137
$918$	4.00000	0.132020
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	−5.00000	−0.164845
$921$	−20.0000	−0.659022
$922$	28.0000	0.922131
$923$	54.0000	1.77743
$924$	−15.0000	−0.493464
$925$	4.00000	0.131519
$926$	0	0
$927$	−16.0000	−0.525509
$928$	8.00000	0.262613
$929$	−15.0000	−0.492134	−0.246067	−	0.969253i	$-0.579138\pi$
$929$	−15.0000	−0.492134	−0.246067	+	0.969253i	$0.579138\pi$
$930$	−1.00000	−0.0327913
$931$	10.0000	0.327737
$932$	1.00000	0.0327561
$933$	−8.00000	−0.261908
$934$	8.00000	0.261768
$935$	20.0000	0.654070
$936$	−6.00000	−0.196116
$937$	28.0000	0.914720	0.457360	−	0.889282i	$-0.348795\pi$
$937$	28.0000	0.914720	0.457360	+	0.889282i	$0.348795\pi$
$938$	30.0000	0.979535
$939$	−14.0000	−0.456873
$940$	−6.00000	−0.195698
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	3.00000	0.0977453
$943$	0	0
$944$	14.0000	0.455661
$945$	3.00000	0.0975900
$946$	−35.0000	−1.13795
$947$	−38.0000	−1.23483	−0.617417	−	0.786636i	$-0.711821\pi$
$947$	−38.0000	−1.23483	−0.617417	+	0.786636i	$0.711821\pi$
$948$	7.00000	0.227349
$949$	18.0000	0.584305
$950$	5.00000	0.162221
$951$	−20.0000	−0.648544
$952$	−12.0000	−0.388922
$953$	−36.0000	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$954$	11.0000	0.356138
$955$	−16.0000	−0.517748
$956$	−12.0000	−0.388108
$957$	−40.0000	−1.29302
$958$	3.00000	0.0969256
$959$	−18.0000	−0.581250
$960$	1.00000	0.0322749
$961$	1.00000	0.0322581
$962$	−24.0000	−0.773791
$963$	−17.0000	−0.547817
$964$	−14.0000	−0.450910
$965$	22.0000	0.708205
$966$	−15.0000	−0.482617
$967$	52.0000	1.67221	0.836104	−	0.548572i	$-0.184828\pi$
$967$	52.0000	1.67221	0.836104	+	0.548572i	$0.184828\pi$
$968$	14.0000	0.449977
$969$	20.0000	0.642493
$970$	−10.0000	−0.321081
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	−1.00000	−0.0320750
$973$	42.0000	1.34646
$974$	−28.0000	−0.897178
$975$	6.00000	0.192154
$976$	−6.00000	−0.192055
$977$	−38.0000	−1.21573	−0.607864	−	0.794041i	$-0.707973\pi$
$977$	−38.0000	−1.21573	−0.607864	+	0.794041i	$0.707973\pi$
$978$	−6.00000	−0.191859
$979$	−45.0000	−1.43821
$980$	−2.00000	−0.0638877
$981$	−12.0000	−0.383131
$982$	−19.0000	−0.606314
$983$	−40.0000	−1.27580	−0.637901	−	0.770118i	$-0.720197\pi$
$983$	−40.0000	−1.27580	−0.637901	+	0.770118i	$0.720197\pi$
$984$	0	0
$985$	22.0000	0.700978
$986$	−32.0000	−1.01909
$987$	−18.0000	−0.572946
$988$	−30.0000	−0.954427
$989$	−35.0000	−1.11294
$990$	−5.00000	−0.158910
$991$	−31.0000	−0.984747	−0.492374	−	0.870384i	$-0.663871\pi$
$991$	−31.0000	−0.984747	−0.492374	+	0.870384i	$0.663871\pi$
$992$	−1.00000	−0.0317500
$993$	12.0000	0.380808
$994$	−27.0000	−0.856388
$995$	−1.00000	−0.0317021
$996$	12.0000	0.380235
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	−20.0000	−0.633089
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.a.m.1.1	✓	1
3.2	odd	2		2790.2.a.l.1.1		1
4.3	odd	2		7440.2.a.p.1.1		1
5.2	odd	4		4650.2.d.j.3349.2		2
5.3	odd	4		4650.2.d.j.3349.1		2
5.4	even	2		4650.2.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.m.1.1	✓	1	1.1	even	1	trivial
2790.2.a.l.1.1		1	3.2	odd	2
4650.2.a.n.1.1		1	5.4	even	2
4650.2.d.j.3349.1		2	5.3	odd	4
4650.2.d.j.3349.2		2	5.2	odd	4
7440.2.a.p.1.1		1	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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