LMFDB - Embedded newform 5586.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5586,2,Mod(1,5586)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5586, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5586.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5586.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:	$44.6044345691$
Analytic rank:	$2$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 798)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5586.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.00000 q^{11} -1.00000 q^{12} -6.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{20} +5.00000 q^{22} -9.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} +6.00000 q^{26} -1.00000 q^{27} -8.00000 q^{29} -1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +5.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} +1.00000 q^{38} +6.00000 q^{39} +1.00000 q^{40} +4.00000 q^{41} +1.00000 q^{43} -5.00000 q^{44} -1.00000 q^{45} +9.00000 q^{46} +1.00000 q^{47} -1.00000 q^{48} +4.00000 q^{50} +2.00000 q^{51} -6.00000 q^{52} -8.00000 q^{53} +1.00000 q^{54} +5.00000 q^{55} +1.00000 q^{57} +8.00000 q^{58} +6.00000 q^{59} +1.00000 q^{60} +15.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} -5.00000 q^{66} -10.0000 q^{67} -2.00000 q^{68} +9.00000 q^{69} -6.00000 q^{71} -1.00000 q^{72} -9.00000 q^{73} +4.00000 q^{74} +4.00000 q^{75} -1.00000 q^{76} -6.00000 q^{78} +12.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -4.00000 q^{82} -11.0000 q^{83} +2.00000 q^{85} -1.00000 q^{86} +8.00000 q^{87} +5.00000 q^{88} -2.00000 q^{89} +1.00000 q^{90} -9.00000 q^{92} +4.00000 q^{93} -1.00000 q^{94} +1.00000 q^{95} +1.00000 q^{96} +8.00000 q^{97} -5.00000 q^{99} +O(q^{100})\)

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
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	−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	$0.812833\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−1.00000	−0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.00000	−0.223607
$21$	0	0
$22$	5.00000	1.06600
$23$	−9.00000	−1.87663	−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000	−1.87663	−0.938315	+	0.345782i	$0.887614\pi$
$24$	1.00000	0.204124
$25$	−4.00000	−0.800000
$26$	6.00000	1.17670
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	−1.00000	−0.182574
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	5.00000	0.870388
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	1.00000	0.162221
$39$	6.00000	0.960769
$40$	1.00000	0.158114
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	−5.00000	−0.753778
$45$	−1.00000	−0.149071
$46$	9.00000	1.32698
$47$	1.00000	0.145865	0.0729325	−	0.997337i	$-0.476764\pi$
$47$	1.00000	0.145865	0.0729325	+	0.997337i	$0.476764\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	4.00000	0.565685
$51$	2.00000	0.280056
$52$	−6.00000	−0.832050
$53$	−8.00000	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$54$	1.00000	0.136083
$55$	5.00000	0.674200
$56$	0	0
$57$	1.00000	0.132453
$58$	8.00000	1.05045
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	1.00000	0.129099
$61$	15.0000	1.92055	0.960277	−	0.279050i	$-0.0900195\pi$
$61$	15.0000	1.92055	0.960277	+	0.279050i	$0.0900195\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	−5.00000	−0.615457
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	−2.00000	−0.242536
$69$	9.00000	1.08347
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−1.00000	−0.117851
$73$	−9.00000	−1.05337	−0.526685	−	0.850060i	$-0.676565\pi$
$73$	−9.00000	−1.05337	−0.526685	+	0.850060i	$0.676565\pi$
$74$	4.00000	0.464991
$75$	4.00000	0.461880
$76$	−1.00000	−0.114708
$77$	0	0
$78$	−6.00000	−0.679366
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−4.00000	−0.441726
$83$	−11.0000	−1.20741	−0.603703	−	0.797209i	$-0.706309\pi$
$83$	−11.0000	−1.20741	−0.603703	+	0.797209i	$0.706309\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	−1.00000	−0.107833
$87$	8.00000	0.857690
$88$	5.00000	0.533002
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−9.00000	−0.938315
$93$	4.00000	0.414781
$94$	−1.00000	−0.103142
$95$	1.00000	0.102598
$96$	1.00000	0.102062
$97$	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
$99$	−5.00000	−0.502519
$100$	−4.00000	−0.400000
$101$	−15.0000	−1.49256	−0.746278	−	0.665635i	$-0.768161\pi$
$101$	−15.0000	−1.49256	−0.746278	+	0.665635i	$0.768161\pi$
$102$	−2.00000	−0.198030
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	8.00000	0.777029
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−1.00000	−0.0962250
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	−5.00000	−0.476731
$111$	4.00000	0.379663
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	−1.00000	−0.0936586
$115$	9.00000	0.839254
$116$	−8.00000	−0.742781
$117$	−6.00000	−0.554700
$118$	−6.00000	−0.552345
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	14.0000	1.27273
$122$	−15.0000	−1.35804
$123$	−4.00000	−0.360668
$124$	−4.00000	−0.359211
$125$	9.00000	0.804984
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	−1.00000	−0.0883883
$129$	−1.00000	−0.0880451
$130$	−6.00000	−0.526235
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	5.00000	0.435194
$133$	0	0
$134$	10.0000	0.863868
$135$	1.00000	0.0860663
$136$	2.00000	0.171499
$137$	−15.0000	−1.28154	−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000	−1.28154	−0.640768	+	0.767734i	$0.721384\pi$
$138$	−9.00000	−0.766131
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	−1.00000	−0.0842152
$142$	6.00000	0.503509
$143$	30.0000	2.50873
$144$	1.00000	0.0833333
$145$	8.00000	0.664364
$146$	9.00000	0.744845
$147$	0	0
$148$	−4.00000	−0.328798
$149$	7.00000	0.573462	0.286731	−	0.958011i	$-0.407431\pi$
$149$	7.00000	0.573462	0.286731	+	0.958011i	$0.407431\pi$
$150$	−4.00000	−0.326599
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	1.00000	0.0811107
$153$	−2.00000	−0.161690
$154$	0	0
$155$	4.00000	0.321288
$156$	6.00000	0.480384
$157$	−7.00000	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$158$	−12.0000	−0.954669
$159$	8.00000	0.634441
$160$	1.00000	0.0790569
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	4.00000	0.312348
$165$	−5.00000	−0.389249
$166$	11.0000	0.853766
$167$	14.0000	1.08335	0.541676	−	0.840587i	$-0.317790\pi$
$167$	14.0000	1.08335	0.541676	+	0.840587i	$0.317790\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−2.00000	−0.153393
$171$	−1.00000	−0.0764719
$172$	1.00000	0.0762493
$173$	8.00000	0.608229	0.304114	−	0.952636i	$-0.401639\pi$
$173$	8.00000	0.608229	0.304114	+	0.952636i	$0.401639\pi$
$174$	−8.00000	−0.606478
$175$	0	0
$176$	−5.00000	−0.376889
$177$	−6.00000	−0.450988
$178$	2.00000	0.149906
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	−1.00000	−0.0745356
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	−15.0000	−1.10883
$184$	9.00000	0.663489
$185$	4.00000	0.294086
$186$	−4.00000	−0.293294
$187$	10.0000	0.731272
$188$	1.00000	0.0729325
$189$	0	0
$190$	−1.00000	−0.0725476
$191$	7.00000	0.506502	0.253251	−	0.967401i	$-0.418500\pi$
$191$	7.00000	0.506502	0.253251	+	0.967401i	$0.418500\pi$
$192$	−1.00000	−0.0721688
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	−8.00000	−0.574367
$195$	−6.00000	−0.429669
$196$	0	0
$197$	−15.0000	−1.06871	−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000	−1.06871	−0.534353	+	0.845262i	$0.679445\pi$
$198$	5.00000	0.355335
$199$	21.0000	1.48865	0.744325	−	0.667817i	$-0.232771\pi$
$199$	21.0000	1.48865	0.744325	+	0.667817i	$0.232771\pi$
$200$	4.00000	0.282843
$201$	10.0000	0.705346
$202$	15.0000	1.05540
$203$	0	0
$204$	2.00000	0.140028
$205$	−4.00000	−0.279372
$206$	−8.00000	−0.557386
$207$	−9.00000	−0.625543
$208$	−6.00000	−0.416025
$209$	5.00000	0.345857
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	−8.00000	−0.549442
$213$	6.00000	0.411113
$214$	−12.0000	−0.820303
$215$	−1.00000	−0.0681994
$216$	1.00000	0.0680414
$217$	0	0
$218$	0	0
$219$	9.00000	0.608164
$220$	5.00000	0.337100
$221$	12.0000	0.807207
$222$	−4.00000	−0.268462
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	1.00000	0.0662266
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−9.00000	−0.593442
$231$	0	0
$232$	8.00000	0.525226
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	6.00000	0.392232
$235$	−1.00000	−0.0652328
$236$	6.00000	0.390567
$237$	−12.0000	−0.779484
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	1.00000	0.0645497
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	−14.0000	−0.899954
$243$	−1.00000	−0.0641500
$244$	15.0000	0.960277
$245$	0	0
$246$	4.00000	0.255031
$247$	6.00000	0.381771
$248$	4.00000	0.254000
$249$	11.0000	0.697097
$250$	−9.00000	−0.569210
$251$	17.0000	1.07303	0.536515	−	0.843891i	$-0.319740\pi$
$251$	17.0000	1.07303	0.536515	+	0.843891i	$0.319740\pi$
$252$	0	0
$253$	45.0000	2.82913
$254$	2.00000	0.125491
$255$	−2.00000	−0.125245
$256$	1.00000	0.0625000
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	1.00000	0.0622573
$259$	0	0
$260$	6.00000	0.372104
$261$	−8.00000	−0.495188
$262$	12.0000	0.741362
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	−5.00000	−0.307729
$265$	8.00000	0.491436
$266$	0	0
$267$	2.00000	0.122398
$268$	−10.0000	−0.610847
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	−1.00000	−0.0608581
$271$	−29.0000	−1.76162	−0.880812	−	0.473466i	$-0.843003\pi$
$271$	−29.0000	−1.76162	−0.880812	+	0.473466i	$0.843003\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	15.0000	0.906183
$275$	20.0000	1.20605
$276$	9.00000	0.541736
$277$	31.0000	1.86261	0.931305	−	0.364241i	$-0.118672\pi$
$277$	31.0000	1.86261	0.931305	+	0.364241i	$0.118672\pi$
$278$	−5.00000	−0.299880
$279$	−4.00000	−0.239474
$280$	0	0
$281$	20.0000	1.19310	0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000	1.19310	0.596550	+	0.802576i	$0.296538\pi$
$282$	1.00000	0.0595491
$283$	7.00000	0.416107	0.208053	−	0.978117i	$-0.433287\pi$
$283$	7.00000	0.416107	0.208053	+	0.978117i	$0.433287\pi$
$284$	−6.00000	−0.356034
$285$	−1.00000	−0.0592349
$286$	−30.0000	−1.77394
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	−8.00000	−0.469776
$291$	−8.00000	−0.468968
$292$	−9.00000	−0.526685
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	4.00000	0.232495
$297$	5.00000	0.290129
$298$	−7.00000	−0.405499
$299$	54.0000	3.12290
$300$	4.00000	0.230940
$301$	0	0
$302$	16.0000	0.920697
$303$	15.0000	0.861727
$304$	−1.00000	−0.0573539
$305$	−15.0000	−0.858898
$306$	2.00000	0.114332
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	−4.00000	−0.227185
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	−6.00000	−0.339683
$313$	−19.0000	−1.07394	−0.536972	−	0.843600i	$-0.680432\pi$
$313$	−19.0000	−1.07394	−0.536972	+	0.843600i	$0.680432\pi$
$314$	7.00000	0.395033
$315$	0	0
$316$	12.0000	0.675053
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	−8.00000	−0.448618
$319$	40.0000	2.23957
$320$	−1.00000	−0.0559017
$321$	−12.0000	−0.669775
$322$	0	0
$323$	2.00000	0.111283
$324$	1.00000	0.0555556
$325$	24.0000	1.33128
$326$	11.0000	0.609234
$327$	0	0
$328$	−4.00000	−0.220863
$329$	0	0
$330$	5.00000	0.275241
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	−11.0000	−0.603703
$333$	−4.00000	−0.219199
$334$	−14.0000	−0.766046
$335$	10.0000	0.546358
$336$	0	0
$337$	−34.0000	−1.85210	−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−34.0000	−1.85210	−0.926049	+	0.377403i	$0.876817\pi$
$338$	−23.0000	−1.25104
$339$	0	0
$340$	2.00000	0.108465
$341$	20.0000	1.08306
$342$	1.00000	0.0540738
$343$	0	0
$344$	−1.00000	−0.0539164
$345$	−9.00000	−0.484544
$346$	−8.00000	−0.430083
$347$	−37.0000	−1.98626	−0.993132	−	0.116999i	$-0.962673\pi$
$347$	−37.0000	−1.98626	−0.993132	+	0.116999i	$0.962673\pi$
$348$	8.00000	0.428845
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	5.00000	0.266501
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	6.00000	0.318896
$355$	6.00000	0.318447
$356$	−2.00000	−0.106000
$357$	0	0
$358$	10.0000	0.528516
$359$	−3.00000	−0.158334	−0.0791670	−	0.996861i	$-0.525226\pi$
$359$	−3.00000	−0.158334	−0.0791670	+	0.996861i	$0.525226\pi$
$360$	1.00000	0.0527046
$361$	1.00000	0.0526316
$362$	14.0000	0.735824
$363$	−14.0000	−0.734809
$364$	0	0
$365$	9.00000	0.471082
$366$	15.0000	0.784063
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	−9.00000	−0.469157
$369$	4.00000	0.208232
$370$	−4.00000	−0.207950
$371$	0	0
$372$	4.00000	0.207390
$373$	24.0000	1.24267	0.621336	−	0.783544i	$-0.286590\pi$
$373$	24.0000	1.24267	0.621336	+	0.783544i	$0.286590\pi$
$374$	−10.0000	−0.517088
$375$	−9.00000	−0.464758
$376$	−1.00000	−0.0515711
$377$	48.0000	2.47213
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	1.00000	0.0512989
$381$	2.00000	0.102463
$382$	−7.00000	−0.358151
$383$	22.0000	1.12415	0.562074	−	0.827087i	$-0.310004\pi$
$383$	22.0000	1.12415	0.562074	+	0.827087i	$0.310004\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−2.00000	−0.101797
$387$	1.00000	0.0508329
$388$	8.00000	0.406138
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	6.00000	0.303822
$391$	18.0000	0.910299
$392$	0	0
$393$	12.0000	0.605320
$394$	15.0000	0.755689
$395$	−12.0000	−0.603786
$396$	−5.00000	−0.251259
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	−21.0000	−1.05263
$399$	0	0
$400$	−4.00000	−0.200000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	−10.0000	−0.498755
$403$	24.0000	1.19553
$404$	−15.0000	−0.746278
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	20.0000	0.991363
$408$	−2.00000	−0.0990148
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	4.00000	0.197546
$411$	15.0000	0.739895
$412$	8.00000	0.394132
$413$	0	0
$414$	9.00000	0.442326
$415$	11.0000	0.539969
$416$	6.00000	0.294174
$417$	−5.00000	−0.244851
$418$	−5.00000	−0.244558
$419$	5.00000	0.244266	0.122133	−	0.992514i	$-0.461027\pi$
$419$	5.00000	0.244266	0.122133	+	0.992514i	$0.461027\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	22.0000	1.07094
$423$	1.00000	0.0486217
$424$	8.00000	0.388514
$425$	8.00000	0.388057
$426$	−6.00000	−0.290701
$427$	0	0
$428$	12.0000	0.580042
$429$	−30.0000	−1.44841
$430$	1.00000	0.0482243
$431$	−10.0000	−0.481683	−0.240842	−	0.970564i	$-0.577423\pi$
$431$	−10.0000	−0.481683	−0.240842	+	0.970564i	$0.577423\pi$
$432$	−1.00000	−0.0481125
$433$	24.0000	1.15337	0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000	1.15337	0.576683	+	0.816968i	$0.304347\pi$
$434$	0	0
$435$	−8.00000	−0.383571
$436$	0	0
$437$	9.00000	0.430528
$438$	−9.00000	−0.430037
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	−5.00000	−0.238366
$441$	0	0
$442$	−12.0000	−0.570782
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	4.00000	0.189832
$445$	2.00000	0.0948091
$446$	−14.0000	−0.662919
$447$	−7.00000	−0.331089
$448$	0	0
$449$	−24.0000	−1.13263	−0.566315	−	0.824189i	$-0.691631\pi$
$449$	−24.0000	−1.13263	−0.566315	+	0.824189i	$0.691631\pi$
$450$	4.00000	0.188562
$451$	−20.0000	−0.941763
$452$	0	0
$453$	16.0000	0.751746
$454$	18.0000	0.844782
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	19.0000	0.888783	0.444391	−	0.895833i	$-0.353420\pi$
$457$	19.0000	0.888783	0.444391	+	0.895833i	$0.353420\pi$
$458$	10.0000	0.467269
$459$	2.00000	0.0933520
$460$	9.00000	0.419627
$461$	1.00000	0.0465746	0.0232873	−	0.999729i	$-0.492587\pi$
$461$	1.00000	0.0465746	0.0232873	+	0.999729i	$0.492587\pi$
$462$	0	0
$463$	23.0000	1.06890	0.534450	−	0.845200i	$-0.320519\pi$
$463$	23.0000	1.06890	0.534450	+	0.845200i	$0.320519\pi$
$464$	−8.00000	−0.371391
$465$	−4.00000	−0.185496
$466$	18.0000	0.833834
$467$	27.0000	1.24941	0.624705	−	0.780860i	$-0.285219\pi$
$467$	27.0000	1.24941	0.624705	+	0.780860i	$0.285219\pi$
$468$	−6.00000	−0.277350
$469$	0	0
$470$	1.00000	0.0461266
$471$	7.00000	0.322543
$472$	−6.00000	−0.276172
$473$	−5.00000	−0.229900
$474$	12.0000	0.551178
$475$	4.00000	0.183533
$476$	0	0
$477$	−8.00000	−0.366295
$478$	16.0000	0.731823
$479$	35.0000	1.59919	0.799595	−	0.600539i	$-0.205047\pi$
$479$	35.0000	1.59919	0.799595	+	0.600539i	$0.205047\pi$
$480$	−1.00000	−0.0456435
$481$	24.0000	1.09431
$482$	4.00000	0.182195
$483$	0	0
$484$	14.0000	0.636364
$485$	−8.00000	−0.363261
$486$	1.00000	0.0453609
$487$	18.0000	0.815658	0.407829	−	0.913058i	$-0.366286\pi$
$487$	18.0000	0.815658	0.407829	+	0.913058i	$0.366286\pi$
$488$	−15.0000	−0.679018
$489$	11.0000	0.497437
$490$	0	0
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	−4.00000	−0.180334
$493$	16.0000	0.720604
$494$	−6.00000	−0.269953
$495$	5.00000	0.224733
$496$	−4.00000	−0.179605
$497$	0	0
$498$	−11.0000	−0.492922
$499$	−21.0000	−0.940089	−0.470045	−	0.882643i	$-0.655762\pi$
$499$	−21.0000	−0.940089	−0.470045	+	0.882643i	$0.655762\pi$
$500$	9.00000	0.402492
$501$	−14.0000	−0.625474
$502$	−17.0000	−0.758747
$503$	17.0000	0.757993	0.378996	−	0.925398i	$-0.376269\pi$
$503$	17.0000	0.757993	0.378996	+	0.925398i	$0.376269\pi$
$504$	0	0
$505$	15.0000	0.667491
$506$	−45.0000	−2.00049
$507$	−23.0000	−1.02147
$508$	−2.00000	−0.0887357
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	2.00000	0.0885615
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	1.00000	0.0441511
$514$	22.0000	0.970378
$515$	−8.00000	−0.352522
$516$	−1.00000	−0.0440225
$517$	−5.00000	−0.219900
$518$	0	0
$519$	−8.00000	−0.351161
$520$	−6.00000	−0.263117
$521$	16.0000	0.700973	0.350486	−	0.936568i	$-0.386016\pi$
$521$	16.0000	0.700973	0.350486	+	0.936568i	$0.386016\pi$
$522$	8.00000	0.350150
$523$	−8.00000	−0.349816	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000	−0.349816	−0.174908	+	0.984585i	$0.555963\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−4.00000	−0.174408
$527$	8.00000	0.348485
$528$	5.00000	0.217597
$529$	58.0000	2.52174
$530$	−8.00000	−0.347498
$531$	6.00000	0.260378
$532$	0	0
$533$	−24.0000	−1.03956
$534$	−2.00000	−0.0865485
$535$	−12.0000	−0.518805
$536$	10.0000	0.431934
$537$	10.0000	0.431532
$538$	12.0000	0.517357
$539$	0	0
$540$	1.00000	0.0430331
$541$	−43.0000	−1.84871	−0.924357	−	0.381528i	$-0.875398\pi$
$541$	−43.0000	−1.84871	−0.924357	+	0.381528i	$0.875398\pi$
$542$	29.0000	1.24566
$543$	14.0000	0.600798
$544$	2.00000	0.0857493
$545$	0	0
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	−15.0000	−0.640768
$549$	15.0000	0.640184
$550$	−20.0000	−0.852803
$551$	8.00000	0.340811
$552$	−9.00000	−0.383065
$553$	0	0
$554$	−31.0000	−1.31706
$555$	−4.00000	−0.169791
$556$	5.00000	0.212047
$557$	−21.0000	−0.889799	−0.444899	−	0.895581i	$-0.646761\pi$
$557$	−21.0000	−0.889799	−0.444899	+	0.895581i	$0.646761\pi$
$558$	4.00000	0.169334
$559$	−6.00000	−0.253773
$560$	0	0
$561$	−10.0000	−0.422200
$562$	−20.0000	−0.843649
$563$	−34.0000	−1.43293	−0.716465	−	0.697623i	$-0.754241\pi$
$563$	−34.0000	−1.43293	−0.716465	+	0.697623i	$0.754241\pi$
$564$	−1.00000	−0.0421076
$565$	0	0
$566$	−7.00000	−0.294232
$567$	0	0
$568$	6.00000	0.251754
$569$	−14.0000	−0.586911	−0.293455	−	0.955973i	$-0.594805\pi$
$569$	−14.0000	−0.586911	−0.293455	+	0.955973i	$0.594805\pi$
$570$	1.00000	0.0418854
$571$	−45.0000	−1.88319	−0.941596	−	0.336746i	$-0.890674\pi$
$571$	−45.0000	−1.88319	−0.941596	+	0.336746i	$0.890674\pi$
$572$	30.0000	1.25436
$573$	−7.00000	−0.292429
$574$	0	0
$575$	36.0000	1.50130
$576$	1.00000	0.0416667
$577$	−7.00000	−0.291414	−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000	−0.291414	−0.145707	+	0.989328i	$0.546546\pi$
$578$	13.0000	0.540729
$579$	−2.00000	−0.0831172
$580$	8.00000	0.332182
$581$	0	0
$582$	8.00000	0.331611
$583$	40.0000	1.65663
$584$	9.00000	0.372423
$585$	6.00000	0.248069
$586$	24.0000	0.991431
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	6.00000	0.247016
$591$	15.0000	0.617018
$592$	−4.00000	−0.164399
$593$	−19.0000	−0.780236	−0.390118	−	0.920765i	$-0.627566\pi$
$593$	−19.0000	−0.780236	−0.390118	+	0.920765i	$0.627566\pi$
$594$	−5.00000	−0.205152
$595$	0	0
$596$	7.00000	0.286731
$597$	−21.0000	−0.859473
$598$	−54.0000	−2.20822
$599$	22.0000	0.898896	0.449448	−	0.893307i	$-0.351621\pi$
$599$	22.0000	0.898896	0.449448	+	0.893307i	$0.351621\pi$
$600$	−4.00000	−0.163299
$601$	−16.0000	−0.652654	−0.326327	−	0.945257i	$-0.605811\pi$
$601$	−16.0000	−0.652654	−0.326327	+	0.945257i	$0.605811\pi$
$602$	0	0
$603$	−10.0000	−0.407231
$604$	−16.0000	−0.651031
$605$	−14.0000	−0.569181
$606$	−15.0000	−0.609333
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	15.0000	0.607332
$611$	−6.00000	−0.242734
$612$	−2.00000	−0.0808452
$613$	−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
$615$	4.00000	0.161296
$616$	0	0
$617$	13.0000	0.523360	0.261680	−	0.965155i	$-0.415723\pi$
$617$	13.0000	0.523360	0.261680	+	0.965155i	$0.415723\pi$
$618$	8.00000	0.321807
$619$	17.0000	0.683288	0.341644	−	0.939829i	$-0.389016\pi$
$619$	17.0000	0.683288	0.341644	+	0.939829i	$0.389016\pi$
$620$	4.00000	0.160644
$621$	9.00000	0.361158
$622$	8.00000	0.320771
$623$	0	0
$624$	6.00000	0.240192
$625$	11.0000	0.440000
$626$	19.0000	0.759393
$627$	−5.00000	−0.199681
$628$	−7.00000	−0.279330
$629$	8.00000	0.318981
$630$	0	0
$631$	−25.0000	−0.995234	−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000	−0.995234	−0.497617	+	0.867397i	$0.665792\pi$
$632$	−12.0000	−0.477334
$633$	22.0000	0.874421
$634$	6.00000	0.238290
$635$	2.00000	0.0793676
$636$	8.00000	0.317221
$637$	0	0
$638$	−40.0000	−1.58362
$639$	−6.00000	−0.237356
$640$	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	$0.462194\pi$
$642$	12.0000	0.473602
$643$	−40.0000	−1.57745	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−40.0000	−1.57745	−0.788723	+	0.614749i	$0.789257\pi$
$644$	0	0
$645$	1.00000	0.0393750
$646$	−2.00000	−0.0786889
$647$	21.0000	0.825595	0.412798	−	0.910823i	$-0.364552\pi$
$647$	21.0000	0.825595	0.412798	+	0.910823i	$0.364552\pi$
$648$	−1.00000	−0.0392837
$649$	−30.0000	−1.17760
$650$	−24.0000	−0.941357
$651$	0	0
$652$	−11.0000	−0.430793
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	4.00000	0.156174
$657$	−9.00000	−0.351123
$658$	0	0
$659$	32.0000	1.24654	0.623272	−	0.782006i	$-0.285803\pi$
$659$	32.0000	1.24654	0.623272	+	0.782006i	$0.285803\pi$
$660$	−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	$0.235292\pi$
$662$	−10.0000	−0.388661
$663$	−12.0000	−0.466041
$664$	11.0000	0.426883
$665$	0	0
$666$	4.00000	0.154997
$667$	72.0000	2.78785
$668$	14.0000	0.541676
$669$	−14.0000	−0.541271
$670$	−10.0000	−0.386334
$671$	−75.0000	−2.89534
$672$	0	0
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	34.0000	1.30963
$675$	4.00000	0.153960
$676$	23.0000	0.884615
$677$	16.0000	0.614930	0.307465	−	0.951559i	$-0.400519\pi$
$677$	16.0000	0.614930	0.307465	+	0.951559i	$0.400519\pi$
$678$	0	0
$679$	0	0
$680$	−2.00000	−0.0766965
$681$	18.0000	0.689761
$682$	−20.0000	−0.765840
$683$	−38.0000	−1.45403	−0.727015	−	0.686622i	$-0.759093\pi$
$683$	−38.0000	−1.45403	−0.727015	+	0.686622i	$0.759093\pi$
$684$	−1.00000	−0.0382360
$685$	15.0000	0.573121
$686$	0	0
$687$	10.0000	0.381524
$688$	1.00000	0.0381246
$689$	48.0000	1.82865
$690$	9.00000	0.342624
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	8.00000	0.304114
$693$	0	0
$694$	37.0000	1.40450
$695$	−5.00000	−0.189661
$696$	−8.00000	−0.303239
$697$	−8.00000	−0.303022
$698$	10.0000	0.378506
$699$	18.0000	0.680823
$700$	0	0
$701$	−33.0000	−1.24639	−0.623196	−	0.782065i	$-0.714166\pi$
$701$	−33.0000	−1.24639	−0.623196	+	0.782065i	$0.714166\pi$
$702$	−6.00000	−0.226455
$703$	4.00000	0.150863
$704$	−5.00000	−0.188445
$705$	1.00000	0.0376622
$706$	30.0000	1.12906
$707$	0	0
$708$	−6.00000	−0.225494
$709$	39.0000	1.46468	0.732338	−	0.680941i	$-0.238429\pi$
$709$	39.0000	1.46468	0.732338	+	0.680941i	$0.238429\pi$
$710$	−6.00000	−0.225176
$711$	12.0000	0.450035
$712$	2.00000	0.0749532
$713$	36.0000	1.34821
$714$	0	0
$715$	−30.0000	−1.12194
$716$	−10.0000	−0.373718
$717$	16.0000	0.597531
$718$	3.00000	0.111959
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	−1.00000	−0.0372161
$723$	4.00000	0.148762
$724$	−14.0000	−0.520306
$725$	32.0000	1.18845
$726$	14.0000	0.519589
$727$	19.0000	0.704671	0.352335	−	0.935874i	$-0.385388\pi$
$727$	19.0000	0.704671	0.352335	+	0.935874i	$0.385388\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−9.00000	−0.333105
$731$	−2.00000	−0.0739727
$732$	−15.0000	−0.554416
$733$	−54.0000	−1.99454	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$	−54.0000	−1.99454	−0.997268	+	0.0738717i	$0.976464\pi$
$734$	−20.0000	−0.738213
$735$	0	0
$736$	9.00000	0.331744
$737$	50.0000	1.84177
$738$	−4.00000	−0.147242
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	4.00000	0.147043
$741$	−6.00000	−0.220416
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	−4.00000	−0.146647
$745$	−7.00000	−0.256460
$746$	−24.0000	−0.878702
$747$	−11.0000	−0.402469
$748$	10.0000	0.365636
$749$	0	0
$750$	9.00000	0.328634
$751$	18.0000	0.656829	0.328415	−	0.944534i	$-0.393486\pi$
$751$	18.0000	0.656829	0.328415	+	0.944534i	$0.393486\pi$
$752$	1.00000	0.0364662
$753$	−17.0000	−0.619514
$754$	−48.0000	−1.74806
$755$	16.0000	0.582300
$756$	0	0
$757$	−39.0000	−1.41748	−0.708740	−	0.705470i	$-0.750736\pi$
$757$	−39.0000	−1.41748	−0.708740	+	0.705470i	$0.750736\pi$
$758$	12.0000	0.435860
$759$	−45.0000	−1.63340
$760$	−1.00000	−0.0362738
$761$	3.00000	0.108750	0.0543750	−	0.998521i	$-0.482683\pi$
$761$	3.00000	0.108750	0.0543750	+	0.998521i	$0.482683\pi$
$762$	−2.00000	−0.0724524
$763$	0	0
$764$	7.00000	0.253251
$765$	2.00000	0.0723102
$766$	−22.0000	−0.794892
$767$	−36.0000	−1.29988
$768$	−1.00000	−0.0360844
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	2.00000	0.0719816
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	−1.00000	−0.0359443
$775$	16.0000	0.574737
$776$	−8.00000	−0.287183
$777$	0	0
$778$	−26.0000	−0.932145
$779$	−4.00000	−0.143315
$780$	−6.00000	−0.214834
$781$	30.0000	1.07348
$782$	−18.0000	−0.643679
$783$	8.00000	0.285897
$784$	0	0
$785$	7.00000	0.249841
$786$	−12.0000	−0.428026
$787$	22.0000	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	−15.0000	−0.534353
$789$	−4.00000	−0.142404
$790$	12.0000	0.426941
$791$	0	0
$792$	5.00000	0.177667
$793$	−90.0000	−3.19599
$794$	−22.0000	−0.780751
$795$	−8.00000	−0.283731
$796$	21.0000	0.744325
$797$	−24.0000	−0.850124	−0.425062	−	0.905164i	$-0.639748\pi$
$797$	−24.0000	−0.850124	−0.425062	+	0.905164i	$0.639748\pi$
$798$	0	0
$799$	−2.00000	−0.0707549
$800$	4.00000	0.141421
$801$	−2.00000	−0.0706665
$802$	−18.0000	−0.635602
$803$	45.0000	1.58802
$804$	10.0000	0.352673
$805$	0	0
$806$	−24.0000	−0.845364
$807$	12.0000	0.422420
$808$	15.0000	0.527698
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	1.00000	0.0351364
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	0	0
$813$	29.0000	1.01707
$814$	−20.0000	−0.701000
$815$	11.0000	0.385313
$816$	2.00000	0.0700140
$817$	−1.00000	−0.0349856
$818$	14.0000	0.489499
$819$	0	0
$820$	−4.00000	−0.139686
$821$	−33.0000	−1.15171	−0.575854	−	0.817553i	$-0.695330\pi$
$821$	−33.0000	−1.15171	−0.575854	+	0.817553i	$0.695330\pi$
$822$	−15.0000	−0.523185
$823$	−7.00000	−0.244005	−0.122002	−	0.992530i	$-0.538932\pi$
$823$	−7.00000	−0.244005	−0.122002	+	0.992530i	$0.538932\pi$
$824$	−8.00000	−0.278693
$825$	−20.0000	−0.696311
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	−9.00000	−0.312772
$829$	−32.0000	−1.11141	−0.555703	−	0.831381i	$-0.687551\pi$
$829$	−32.0000	−1.11141	−0.555703	+	0.831381i	$0.687551\pi$
$830$	−11.0000	−0.381816
$831$	−31.0000	−1.07538
$832$	−6.00000	−0.208013
$833$	0	0
$834$	5.00000	0.173136
$835$	−14.0000	−0.484490
$836$	5.00000	0.172929
$837$	4.00000	0.138260
$838$	−5.00000	−0.172722
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	−10.0000	−0.344623
$843$	−20.0000	−0.688837
$844$	−22.0000	−0.757271
$845$	−23.0000	−0.791224
$846$	−1.00000	−0.0343807
$847$	0	0
$848$	−8.00000	−0.274721
$849$	−7.00000	−0.240239
$850$	−8.00000	−0.274398
$851$	36.0000	1.23406
$852$	6.00000	0.205557
$853$	−33.0000	−1.12990	−0.564949	−	0.825126i	$-0.691104\pi$
$853$	−33.0000	−1.12990	−0.564949	+	0.825126i	$0.691104\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	−12.0000	−0.410152
$857$	−40.0000	−1.36637	−0.683187	−	0.730243i	$-0.739407\pi$
$857$	−40.0000	−1.36637	−0.683187	+	0.730243i	$0.739407\pi$
$858$	30.0000	1.02418
$859$	−41.0000	−1.39890	−0.699451	−	0.714681i	$-0.746572\pi$
$859$	−41.0000	−1.39890	−0.699451	+	0.714681i	$0.746572\pi$
$860$	−1.00000	−0.0340997
$861$	0	0
$862$	10.0000	0.340601
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	1.00000	0.0340207
$865$	−8.00000	−0.272008
$866$	−24.0000	−0.815553
$867$	13.0000	0.441503
$868$	0	0
$869$	−60.0000	−2.03536
$870$	8.00000	0.271225
$871$	60.0000	2.03302
$872$	0	0
$873$	8.00000	0.270759
$874$	−9.00000	−0.304430
$875$	0	0
$876$	9.00000	0.304082
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	−22.0000	−0.742464
$879$	24.0000	0.809500
$880$	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	$0.598061\pi$
$882$	0	0
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	12.0000	0.403604
$885$	6.00000	0.201688
$886$	12.0000	0.403148
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	−4.00000	−0.134231
$889$	0	0
$890$	−2.00000	−0.0670402
$891$	−5.00000	−0.167506
$892$	14.0000	0.468755
$893$	−1.00000	−0.0334637
$894$	7.00000	0.234115
$895$	10.0000	0.334263
$896$	0	0
$897$	−54.0000	−1.80301
$898$	24.0000	0.800890
$899$	32.0000	1.06726
$900$	−4.00000	−0.133333
$901$	16.0000	0.533037
$902$	20.0000	0.665927
$903$	0	0
$904$	0	0
$905$	14.0000	0.465376
$906$	−16.0000	−0.531564
$907$	52.0000	1.72663	0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000	1.72663	0.863316	+	0.504664i	$0.168384\pi$
$908$	−18.0000	−0.597351
$909$	−15.0000	−0.497519
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	1.00000	0.0331133
$913$	55.0000	1.82023
$914$	−19.0000	−0.628464
$915$	15.0000	0.495885
$916$	−10.0000	−0.330409
$917$	0	0
$918$	−2.00000	−0.0660098
$919$	−3.00000	−0.0989609	−0.0494804	−	0.998775i	$-0.515757\pi$
$919$	−3.00000	−0.0989609	−0.0494804	+	0.998775i	$0.515757\pi$
$920$	−9.00000	−0.296721
$921$	−20.0000	−0.659022
$922$	−1.00000	−0.0329332
$923$	36.0000	1.18495
$924$	0	0
$925$	16.0000	0.526077
$926$	−23.0000	−0.755827
$927$	8.00000	0.262754
$928$	8.00000	0.262613
$929$	−7.00000	−0.229663	−0.114831	−	0.993385i	$-0.536633\pi$
$929$	−7.00000	−0.229663	−0.114831	+	0.993385i	$0.536633\pi$
$930$	4.00000	0.131165
$931$	0	0
$932$	−18.0000	−0.589610
$933$	8.00000	0.261908
$934$	−27.0000	−0.883467
$935$	−10.0000	−0.327035
$936$	6.00000	0.196116
$937$	23.0000	0.751377	0.375689	−	0.926746i	$-0.377406\pi$
$937$	23.0000	0.751377	0.375689	+	0.926746i	$0.377406\pi$
$938$	0	0
$939$	19.0000	0.620042
$940$	−1.00000	−0.0326164
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	−7.00000	−0.228072
$943$	−36.0000	−1.17232
$944$	6.00000	0.195283
$945$	0	0
$946$	5.00000	0.162564
$947$	−44.0000	−1.42981	−0.714904	−	0.699223i	$-0.753530\pi$
$947$	−44.0000	−1.42981	−0.714904	+	0.699223i	$0.753530\pi$
$948$	−12.0000	−0.389742
$949$	54.0000	1.75291
$950$	−4.00000	−0.129777
$951$	6.00000	0.194563
$952$	0	0
$953$	−36.0000	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$954$	8.00000	0.259010
$955$	−7.00000	−0.226515
$956$	−16.0000	−0.517477
$957$	−40.0000	−1.29302
$958$	−35.0000	−1.13080
$959$	0	0
$960$	1.00000	0.0322749
$961$	−15.0000	−0.483871
$962$	−24.0000	−0.773791
$963$	12.0000	0.386695
$964$	−4.00000	−0.128831
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	−14.0000	−0.449977
$969$	−2.00000	−0.0642493
$970$	8.00000	0.256865
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−18.0000	−0.576757
$975$	−24.0000	−0.768615
$976$	15.0000	0.480138
$977$	26.0000	0.831814	0.415907	−	0.909407i	$-0.363464\pi$
$977$	26.0000	0.831814	0.415907	+	0.909407i	$0.363464\pi$
$978$	−11.0000	−0.351741
$979$	10.0000	0.319601
$980$	0	0
$981$	0	0
$982$	−15.0000	−0.478669
$983$	−28.0000	−0.893061	−0.446531	−	0.894768i	$-0.647341\pi$
$983$	−28.0000	−0.893061	−0.446531	+	0.894768i	$0.647341\pi$
$984$	4.00000	0.127515
$985$	15.0000	0.477940
$986$	−16.0000	−0.509544
$987$	0	0
$988$	6.00000	0.190885
$989$	−9.00000	−0.286183
$990$	−5.00000	−0.158910
$991$	38.0000	1.20711	0.603555	−	0.797321i	$-0.293750\pi$
$991$	38.0000	1.20711	0.603555	+	0.797321i	$0.293750\pi$
$992$	4.00000	0.127000
$993$	−10.0000	−0.317340
$994$	0	0
$995$	−21.0000	−0.665745
$996$	11.0000	0.348548
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	21.0000	0.664743
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5586.2.a.c.1.1		1
7.3	odd	6		798.2.j.d.457.1	✓	2
7.5	odd	6		798.2.j.d.571.1	yes	2
7.6	odd	2		5586.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
798.2.j.d.457.1	✓	2	7.3	odd	6
798.2.j.d.571.1	yes	2	7.5	odd	6
5586.2.a.c.1.1		1	1.1	even	1	trivial
5586.2.a.r.1.1		1	7.6	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:	\( N \)	\(=\)	\( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5586.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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