LMFDB - Embedded newform 5586.2.a.bb.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:	$0$
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)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} +2.00000 q^{20} +2.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} -2.00000 q^{29} +2.00000 q^{30} +6.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} -1.00000 q^{38} +6.00000 q^{39} +2.00000 q^{40} -6.00000 q^{41} -4.00000 q^{43} +2.00000 q^{44} +2.00000 q^{45} -4.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} -1.00000 q^{50} +4.00000 q^{51} +6.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +4.00000 q^{55} -1.00000 q^{57} -2.00000 q^{58} -4.00000 q^{59} +2.00000 q^{60} +6.00000 q^{61} +6.00000 q^{62} +1.00000 q^{64} +12.0000 q^{65} +2.00000 q^{66} -14.0000 q^{67} +4.00000 q^{68} -4.00000 q^{69} +8.00000 q^{71} +1.00000 q^{72} -10.0000 q^{73} -4.00000 q^{74} -1.00000 q^{75} -1.00000 q^{76} +6.00000 q^{78} +2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -8.00000 q^{83} +8.00000 q^{85} -4.00000 q^{86} -2.00000 q^{87} +2.00000 q^{88} -6.00000 q^{89} +2.00000 q^{90} -4.00000 q^{92} +6.00000 q^{93} +6.00000 q^{94} -2.00000 q^{95} +1.00000 q^{96} -16.0000 q^{97} +2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	1.00000	0.288675
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	2.00000	0.447214
$21$	0	0
$22$	2.00000	0.426401
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	1.00000	0.204124
$25$	−1.00000	−0.200000
$26$	6.00000	1.17670
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	2.00000	0.365148
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	1.00000	0.176777
$33$	2.00000	0.348155
$34$	4.00000	0.685994
$35$	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
$40$	2.00000	0.316228
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	2.00000	0.301511
$45$	2.00000	0.298142
$46$	−4.00000	−0.589768
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−1.00000	−0.141421
$51$	4.00000	0.560112
$52$	6.00000	0.832050
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	1.00000	0.136083
$55$	4.00000	0.539360
$56$	0	0
$57$	−1.00000	−0.132453
$58$	−2.00000	−0.262613
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	2.00000	0.258199
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	12.0000	1.48842
$66$	2.00000	0.246183
$67$	−14.0000	−1.71037	−0.855186	−	0.518321i	$-0.826557\pi$
$67$	−14.0000	−1.71037	−0.855186	+	0.518321i	$0.826557\pi$
$68$	4.00000	0.485071
$69$	−4.00000	−0.481543
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	1.00000	0.117851
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−4.00000	−0.464991
$75$	−1.00000	−0.115470
$76$	−1.00000	−0.114708
$77$	0	0
$78$	6.00000	0.679366
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	8.00000	0.867722
$86$	−4.00000	−0.431331
$87$	−2.00000	−0.214423
$88$	2.00000	0.213201
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	−4.00000	−0.417029
$93$	6.00000	0.622171
$94$	6.00000	0.618853
$95$	−2.00000	−0.205196
$96$	1.00000	0.102062
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	−1.00000	−0.100000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	4.00000	0.396059
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	6.00000	0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	1.00000	0.0962250
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	4.00000	0.381385
$111$	−4.00000	−0.379663
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	−1.00000	−0.0936586
$115$	−8.00000	−0.746004
$116$	−2.00000	−0.185695
$117$	6.00000	0.554700
$118$	−4.00000	−0.368230
$119$	0	0
$120$	2.00000	0.182574
$121$	−7.00000	−0.636364
$122$	6.00000	0.543214
$123$	−6.00000	−0.541002
$124$	6.00000	0.538816
$125$	−12.0000	−1.07331
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	12.0000	1.05247
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	2.00000	0.174078
$133$	0	0
$134$	−14.0000	−1.20942
$135$	2.00000	0.172133
$136$	4.00000	0.342997
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	−4.00000	−0.340503
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	8.00000	0.671345
$143$	12.0000	1.00349
$144$	1.00000	0.0833333
$145$	−4.00000	−0.332182
$146$	−10.0000	−0.827606
$147$	0	0
$148$	−4.00000	−0.328798
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	−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	$0.605538\pi$
$152$	−1.00000	−0.0811107
$153$	4.00000	0.323381
$154$	0	0
$155$	12.0000	0.963863
$156$	6.00000	0.480384
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	2.00000	0.158114
$161$	0	0
$162$	1.00000	0.0785674
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	−6.00000	−0.468521
$165$	4.00000	0.311400
$166$	−8.00000	−0.620920
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	8.00000	0.613572
$171$	−1.00000	−0.0764719
$172$	−4.00000	−0.304997
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	2.00000	0.150756
$177$	−4.00000	−0.300658
$178$	−6.00000	−0.449719
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	2.00000	0.149071
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	−4.00000	−0.294884
$185$	−8.00000	−0.588172
$186$	6.00000	0.439941
$187$	8.00000	0.585018
$188$	6.00000	0.437595
$189$	0	0
$190$	−2.00000	−0.145095
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	1.00000	0.0721688
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−16.0000	−1.14873
$195$	12.0000	0.859338
$196$	0	0
$197$	−16.0000	−1.13995	−0.569976	−	0.821661i	$-0.693048\pi$
$197$	−16.0000	−1.13995	−0.569976	+	0.821661i	$0.693048\pi$
$198$	2.00000	0.142134
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	−1.00000	−0.0707107
$201$	−14.0000	−0.987484
$202$	2.00000	0.140720
$203$	0	0
$204$	4.00000	0.280056
$205$	−12.0000	−0.838116
$206$	10.0000	0.696733
$207$	−4.00000	−0.278019
$208$	6.00000	0.416025
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	6.00000	0.412082
$213$	8.00000	0.548151
$214$	−12.0000	−0.820303
$215$	−8.00000	−0.545595
$216$	1.00000	0.0680414
$217$	0	0
$218$	16.0000	1.08366
$219$	−10.0000	−0.675737
$220$	4.00000	0.269680
$221$	24.0000	1.61441
$222$	−4.00000	−0.268462
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	−2.00000	−0.133038
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−1.00000	−0.0662266
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	−2.00000	−0.131306
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	6.00000	0.392232
$235$	12.0000	0.782794
$236$	−4.00000	−0.260378
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	2.00000	0.129099
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	−7.00000	−0.449977
$243$	1.00000	0.0641500
$244$	6.00000	0.384111
$245$	0	0
$246$	−6.00000	−0.382546
$247$	−6.00000	−0.381771
$248$	6.00000	0.381000
$249$	−8.00000	−0.506979
$250$	−12.0000	−0.758947
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	0	0
$255$	8.00000	0.500979
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	12.0000	0.744208
$261$	−2.00000	−0.123797
$262$	−4.00000	−0.247121
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	2.00000	0.123091
$265$	12.0000	0.737154
$266$	0	0
$267$	−6.00000	−0.367194
$268$	−14.0000	−0.855186
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	2.00000	0.121716
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	10.0000	0.604122
$275$	−2.00000	−0.120605
$276$	−4.00000	−0.240772
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	20.0000	1.19952
$279$	6.00000	0.359211
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	6.00000	0.357295
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	8.00000	0.474713
$285$	−2.00000	−0.118470
$286$	12.0000	0.709575
$287$	0	0
$288$	1.00000	0.0589256
$289$	−1.00000	−0.0588235
$290$	−4.00000	−0.234888
$291$	−16.0000	−0.937937
$292$	−10.0000	−0.585206
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	−4.00000	−0.232495
$297$	2.00000	0.116052
$298$	0	0
$299$	−24.0000	−1.38796
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	−8.00000	−0.460348
$303$	2.00000	0.114897
$304$	−1.00000	−0.0573539
$305$	12.0000	0.687118
$306$	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	$0.306660\pi$
$308$	0	0
$309$	10.0000	0.568880
$310$	12.0000	0.681554
$311$	2.00000	0.113410	0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000	0.113410	0.0567048	+	0.998391i	$0.481941\pi$
$312$	6.00000	0.339683
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	0	0
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	6.00000	0.336463
$319$	−4.00000	−0.223957
$320$	2.00000	0.111803
$321$	−12.0000	−0.669775
$322$	0	0
$323$	−4.00000	−0.222566
$324$	1.00000	0.0555556
$325$	−6.00000	−0.332820
$326$	−16.0000	−0.886158
$327$	16.0000	0.884802
$328$	−6.00000	−0.331295
$329$	0	0
$330$	4.00000	0.220193
$331$	−34.0000	−1.86881	−0.934405	−	0.356214i	$-0.884068\pi$
$331$	−34.0000	−1.86881	−0.934405	+	0.356214i	$0.884068\pi$
$332$	−8.00000	−0.439057
$333$	−4.00000	−0.219199
$334$	−16.0000	−0.875481
$335$	−28.0000	−1.52980
$336$	0	0
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	23.0000	1.25104
$339$	−2.00000	−0.108625
$340$	8.00000	0.433861
$341$	12.0000	0.649836
$342$	−1.00000	−0.0540738
$343$	0	0
$344$	−4.00000	−0.215666
$345$	−8.00000	−0.430706
$346$	14.0000	0.752645
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	−2.00000	−0.107211
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	2.00000	0.106600
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	−4.00000	−0.212598
$355$	16.0000	0.849192
$356$	−6.00000	−0.317999
$357$	0	0
$358$	24.0000	1.26844
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	2.00000	0.105409
$361$	1.00000	0.0526316
$362$	6.00000	0.315353
$363$	−7.00000	−0.367405
$364$	0	0
$365$	−20.0000	−1.04685
$366$	6.00000	0.313625
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	−4.00000	−0.208514
$369$	−6.00000	−0.312348
$370$	−8.00000	−0.415900
$371$	0	0
$372$	6.00000	0.311086
$373$	−8.00000	−0.414224	−0.207112	−	0.978317i	$-0.566407\pi$
$373$	−8.00000	−0.414224	−0.207112	+	0.978317i	$0.566407\pi$
$374$	8.00000	0.413670
$375$	−12.0000	−0.619677
$376$	6.00000	0.309426
$377$	−12.0000	−0.618031
$378$	0	0
$379$	−26.0000	−1.33553	−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000	−1.33553	−0.667765	+	0.744372i	$0.732749\pi$
$380$	−2.00000	−0.102598
$381$	0	0
$382$	−12.0000	−0.613973
$383$	−32.0000	−1.63512	−0.817562	−	0.575841i	$-0.804675\pi$
$383$	−32.0000	−1.63512	−0.817562	+	0.575841i	$0.804675\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	14.0000	0.712581
$387$	−4.00000	−0.203331
$388$	−16.0000	−0.812277
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	12.0000	0.607644
$391$	−16.0000	−0.809155
$392$	0	0
$393$	−4.00000	−0.201773
$394$	−16.0000	−0.806068
$395$	0	0
$396$	2.00000	0.100504
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	−20.0000	−1.00251
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	−14.0000	−0.698257
$403$	36.0000	1.79329
$404$	2.00000	0.0995037
$405$	2.00000	0.0993808
$406$	0	0
$407$	−8.00000	−0.396545
$408$	4.00000	0.198030
$409$	16.0000	0.791149	0.395575	−	0.918434i	$-0.370545\pi$
$409$	16.0000	0.791149	0.395575	+	0.918434i	$0.370545\pi$
$410$	−12.0000	−0.592638
$411$	10.0000	0.493264
$412$	10.0000	0.492665
$413$	0	0
$414$	−4.00000	−0.196589
$415$	−16.0000	−0.785409
$416$	6.00000	0.294174
$417$	20.0000	0.979404
$418$	−2.00000	−0.0978232
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	−10.0000	−0.486792
$423$	6.00000	0.291730
$424$	6.00000	0.291386
$425$	−4.00000	−0.194029
$426$	8.00000	0.387601
$427$	0	0
$428$	−12.0000	−0.580042
$429$	12.0000	0.579365
$430$	−8.00000	−0.385794
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	1.00000	0.0481125
$433$	−32.0000	−1.53782	−0.768911	−	0.639356i	$-0.779201\pi$
$433$	−32.0000	−1.53782	−0.768911	+	0.639356i	$0.779201\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	16.0000	0.766261
$437$	4.00000	0.191346
$438$	−10.0000	−0.477818
$439$	6.00000	0.286364	0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000	0.286364	0.143182	+	0.989696i	$0.454267\pi$
$440$	4.00000	0.190693
$441$	0	0
$442$	24.0000	1.14156
$443$	26.0000	1.23530	0.617649	−	0.786454i	$-0.288085\pi$
$443$	26.0000	1.23530	0.617649	+	0.786454i	$0.288085\pi$
$444$	−4.00000	−0.189832
$445$	−12.0000	−0.568855
$446$	−14.0000	−0.662919
$447$	0	0
$448$	0	0
$449$	34.0000	1.60456	0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000	1.60456	0.802280	+	0.596948i	$0.203620\pi$
$450$	−1.00000	−0.0471405
$451$	−12.0000	−0.565058
$452$	−2.00000	−0.0940721
$453$	−8.00000	−0.375873
$454$	12.0000	0.563188
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	26.0000	1.21490
$459$	4.00000	0.186704
$460$	−8.00000	−0.373002
$461$	−34.0000	−1.58354	−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000	−1.58354	−0.791769	+	0.610821i	$0.790840\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	−2.00000	−0.0928477
$465$	12.0000	0.556487
$466$	14.0000	0.648537
$467$	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
$469$	0	0
$470$	12.0000	0.553519
$471$	10.0000	0.460776
$472$	−4.00000	−0.184115
$473$	−8.00000	−0.367840
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	−2.00000	−0.0913823	−0.0456912	−	0.998956i	$-0.514549\pi$
$479$	−2.00000	−0.0913823	−0.0456912	+	0.998956i	$0.514549\pi$
$480$	2.00000	0.0912871
$481$	−24.0000	−1.09431
$482$	4.00000	0.182195
$483$	0	0
$484$	−7.00000	−0.318182
$485$	−32.0000	−1.45305
$486$	1.00000	0.0453609
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	6.00000	0.271607
$489$	−16.0000	−0.723545
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	−6.00000	−0.270501
$493$	−8.00000	−0.360302
$494$	−6.00000	−0.269953
$495$	4.00000	0.179787
$496$	6.00000	0.269408
$497$	0	0
$498$	−8.00000	−0.358489
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	−12.0000	−0.536656
$501$	−16.0000	−0.714827
$502$	−12.0000	−0.535586
$503$	42.0000	1.87269	0.936344	−	0.351085i	$-0.114187\pi$
$503$	42.0000	1.87269	0.936344	+	0.351085i	$0.114187\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	−8.00000	−0.355643
$507$	23.0000	1.02147
$508$	0	0
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	8.00000	0.354246
$511$	0	0
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	6.00000	0.264649
$515$	20.0000	0.881305
$516$	−4.00000	−0.176090
$517$	12.0000	0.527759
$518$	0	0
$519$	14.0000	0.614532
$520$	12.0000	0.526235
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	−2.00000	−0.0875376
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	12.0000	0.523225
$527$	24.0000	1.04546
$528$	2.00000	0.0870388
$529$	−7.00000	−0.304348
$530$	12.0000	0.521247
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−36.0000	−1.55933
$534$	−6.00000	−0.259645
$535$	−24.0000	−1.03761
$536$	−14.0000	−0.604708
$537$	24.0000	1.03568
$538$	26.0000	1.12094
$539$	0	0
$540$	2.00000	0.0860663
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	24.0000	1.03089
$543$	6.00000	0.257485
$544$	4.00000	0.171499
$545$	32.0000	1.37073
$546$	0	0
$547$	−6.00000	−0.256541	−0.128271	−	0.991739i	$-0.540943\pi$
$547$	−6.00000	−0.256541	−0.128271	+	0.991739i	$0.540943\pi$
$548$	10.0000	0.427179
$549$	6.00000	0.256074
$550$	−2.00000	−0.0852803
$551$	2.00000	0.0852029
$552$	−4.00000	−0.170251
$553$	0	0
$554$	14.0000	0.594803
$555$	−8.00000	−0.339581
$556$	20.0000	0.848189
$557$	−32.0000	−1.35588	−0.677942	−	0.735116i	$-0.737128\pi$
$557$	−32.0000	−1.35588	−0.677942	+	0.735116i	$0.737128\pi$
$558$	6.00000	0.254000
$559$	−24.0000	−1.01509
$560$	0	0
$561$	8.00000	0.337760
$562$	−10.0000	−0.421825
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	6.00000	0.252646
$565$	−4.00000	−0.168281
$566$	−4.00000	−0.168133
$567$	0	0
$568$	8.00000	0.335673
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	−2.00000	−0.0837708
$571$	24.0000	1.00437	0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000	1.00437	0.502184	+	0.864761i	$0.332530\pi$
$572$	12.0000	0.501745
$573$	−12.0000	−0.501307
$574$	0	0
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	−1.00000	−0.0415945
$579$	14.0000	0.581820
$580$	−4.00000	−0.166091
$581$	0	0
$582$	−16.0000	−0.663221
$583$	12.0000	0.496989
$584$	−10.0000	−0.413803
$585$	12.0000	0.496139
$586$	−6.00000	−0.247858
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	0	0
$589$	−6.00000	−0.247226
$590$	−8.00000	−0.329355
$591$	−16.0000	−0.658152
$592$	−4.00000	−0.164399
$593$	36.0000	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	0	0
$597$	−20.0000	−0.818546
$598$	−24.0000	−0.981433
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	−1.00000	−0.0408248
$601$	−44.0000	−1.79480	−0.897399	−	0.441221i	$-0.854546\pi$
$601$	−44.0000	−1.79480	−0.897399	+	0.441221i	$0.854546\pi$
$602$	0	0
$603$	−14.0000	−0.570124
$604$	−8.00000	−0.325515
$605$	−14.0000	−0.569181
$606$	2.00000	0.0812444
$607$	−42.0000	−1.70473	−0.852364	−	0.522949i	$-0.824832\pi$
$607$	−42.0000	−1.70473	−0.852364	+	0.522949i	$0.824832\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	12.0000	0.485866
$611$	36.0000	1.45640
$612$	4.00000	0.161690
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	20.0000	0.807134
$615$	−12.0000	−0.483887
$616$	0	0
$617$	−34.0000	−1.36879	−0.684394	−	0.729112i	$-0.739933\pi$
$617$	−34.0000	−1.36879	−0.684394	+	0.729112i	$0.739933\pi$
$618$	10.0000	0.402259
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	12.0000	0.481932
$621$	−4.00000	−0.160514
$622$	2.00000	0.0801927
$623$	0	0
$624$	6.00000	0.240192
$625$	−19.0000	−0.760000
$626$	−10.0000	−0.399680
$627$	−2.00000	−0.0798723
$628$	10.0000	0.399043
$629$	−16.0000	−0.637962
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	−10.0000	−0.397464
$634$	−6.00000	−0.238290
$635$	0	0
$636$	6.00000	0.237915
$637$	0	0
$638$	−4.00000	−0.158362
$639$	8.00000	0.316475
$640$	2.00000	0.0790569
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	−12.0000	−0.473602
$643$	−20.0000	−0.788723	−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	−4.00000	−0.157378
$647$	6.00000	0.235884	0.117942	−	0.993020i	$-0.462370\pi$
$647$	6.00000	0.235884	0.117942	+	0.993020i	$0.462370\pi$
$648$	1.00000	0.0392837
$649$	−8.00000	−0.314027
$650$	−6.00000	−0.235339
$651$	0	0
$652$	−16.0000	−0.626608
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	16.0000	0.625650
$655$	−8.00000	−0.312586
$656$	−6.00000	−0.234261
$657$	−10.0000	−0.390137
$658$	0	0
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	4.00000	0.155700
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	−34.0000	−1.32145
$663$	24.0000	0.932083
$664$	−8.00000	−0.310460
$665$	0	0
$666$	−4.00000	−0.154997
$667$	8.00000	0.309761
$668$	−16.0000	−0.619059
$669$	−14.0000	−0.541271
$670$	−28.0000	−1.08173
$671$	12.0000	0.463255
$672$	0	0
$673$	50.0000	1.92736	0.963679	−	0.267063i	$-0.0860531\pi$
$673$	50.0000	1.92736	0.963679	+	0.267063i	$0.0860531\pi$
$674$	6.00000	0.231111
$675$	−1.00000	−0.0384900
$676$	23.0000	0.884615
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	−2.00000	−0.0768095
$679$	0	0
$680$	8.00000	0.306786
$681$	12.0000	0.459841
$682$	12.0000	0.459504
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	−1.00000	−0.0382360
$685$	20.0000	0.764161
$686$	0	0
$687$	26.0000	0.991962
$688$	−4.00000	−0.152499
$689$	36.0000	1.37149
$690$	−8.00000	−0.304555
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	14.0000	0.532200
$693$	0	0
$694$	−18.0000	−0.683271
$695$	40.0000	1.51729
$696$	−2.00000	−0.0758098
$697$	−24.0000	−0.909065
$698$	−22.0000	−0.832712
$699$	14.0000	0.529529
$700$	0	0
$701$	16.0000	0.604312	0.302156	−	0.953259i	$-0.402294\pi$
$701$	16.0000	0.604312	0.302156	+	0.953259i	$0.402294\pi$
$702$	6.00000	0.226455
$703$	4.00000	0.150863
$704$	2.00000	0.0753778
$705$	12.0000	0.451946
$706$	24.0000	0.903252
$707$	0	0
$708$	−4.00000	−0.150329
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	16.0000	0.600469
$711$	0	0
$712$	−6.00000	−0.224860
$713$	−24.0000	−0.898807
$714$	0	0
$715$	24.0000	0.897549
$716$	24.0000	0.896922
$717$	0	0
$718$	−32.0000	−1.19423
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	2.00000	0.0745356
$721$	0	0
$722$	1.00000	0.0372161
$723$	4.00000	0.148762
$724$	6.00000	0.222988
$725$	2.00000	0.0742781
$726$	−7.00000	−0.259794
$727$	44.0000	1.63187	0.815935	−	0.578144i	$-0.196223\pi$
$727$	44.0000	1.63187	0.815935	+	0.578144i	$0.196223\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−20.0000	−0.740233
$731$	−16.0000	−0.591781
$732$	6.00000	0.221766
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	−4.00000	−0.147442
$737$	−28.0000	−1.03139
$738$	−6.00000	−0.220863
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	−8.00000	−0.294086
$741$	−6.00000	−0.220416
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	6.00000	0.219971
$745$	0	0
$746$	−8.00000	−0.292901
$747$	−8.00000	−0.292705
$748$	8.00000	0.292509
$749$	0	0
$750$	−12.0000	−0.438178
$751$	12.0000	0.437886	0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000	0.437886	0.218943	+	0.975738i	$0.429739\pi$
$752$	6.00000	0.218797
$753$	−12.0000	−0.437304
$754$	−12.0000	−0.437014
$755$	−16.0000	−0.582300
$756$	0	0
$757$	−6.00000	−0.218074	−0.109037	−	0.994038i	$-0.534777\pi$
$757$	−6.00000	−0.218074	−0.109037	+	0.994038i	$0.534777\pi$
$758$	−26.0000	−0.944363
$759$	−8.00000	−0.290382
$760$	−2.00000	−0.0725476
$761$	−8.00000	−0.290000	−0.145000	−	0.989432i	$-0.546318\pi$
$761$	−8.00000	−0.290000	−0.145000	+	0.989432i	$0.546318\pi$
$762$	0	0
$763$	0	0
$764$	−12.0000	−0.434145
$765$	8.00000	0.289241
$766$	−32.0000	−1.15621
$767$	−24.0000	−0.866590
$768$	1.00000	0.0360844
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	14.0000	0.503871
$773$	26.0000	0.935155	0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000	0.935155	0.467578	+	0.883952i	$0.345127\pi$
$774$	−4.00000	−0.143777
$775$	−6.00000	−0.215526
$776$	−16.0000	−0.574367
$777$	0	0
$778$	−8.00000	−0.286814
$779$	6.00000	0.214972
$780$	12.0000	0.429669
$781$	16.0000	0.572525
$782$	−16.0000	−0.572159
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	20.0000	0.713831
$786$	−4.00000	−0.142675
$787$	−48.0000	−1.71102	−0.855508	−	0.517790i	$-0.826755\pi$
$787$	−48.0000	−1.71102	−0.855508	+	0.517790i	$0.826755\pi$
$788$	−16.0000	−0.569976
$789$	12.0000	0.427211
$790$	0	0
$791$	0	0
$792$	2.00000	0.0710669
$793$	36.0000	1.27840
$794$	−18.0000	−0.638796
$795$	12.0000	0.425596
$796$	−20.0000	−0.708881
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	−1.00000	−0.0353553
$801$	−6.00000	−0.212000
$802$	18.0000	0.635602
$803$	−20.0000	−0.705785
$804$	−14.0000	−0.493742
$805$	0	0
$806$	36.0000	1.26805
$807$	26.0000	0.915243
$808$	2.00000	0.0703598
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	2.00000	0.0702728
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	0	0
$813$	24.0000	0.841717
$814$	−8.00000	−0.280400
$815$	−32.0000	−1.12091
$816$	4.00000	0.140028
$817$	4.00000	0.139942
$818$	16.0000	0.559427
$819$	0	0
$820$	−12.0000	−0.419058
$821$	20.0000	0.698005	0.349002	−	0.937122i	$-0.386521\pi$
$821$	20.0000	0.698005	0.349002	+	0.937122i	$0.386521\pi$
$822$	10.0000	0.348790
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	10.0000	0.348367
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	48.0000	1.66912	0.834562	−	0.550914i	$-0.185721\pi$
$827$	48.0000	1.66912	0.834562	+	0.550914i	$0.185721\pi$
$828$	−4.00000	−0.139010
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	−16.0000	−0.555368
$831$	14.0000	0.485655
$832$	6.00000	0.208013
$833$	0	0
$834$	20.0000	0.692543
$835$	−32.0000	−1.10741
$836$	−2.00000	−0.0691714
$837$	6.00000	0.207390
$838$	24.0000	0.829066
$839$	−32.0000	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−10.0000	−0.344418
$844$	−10.0000	−0.344214
$845$	46.0000	1.58245
$846$	6.00000	0.206284
$847$	0	0
$848$	6.00000	0.206041
$849$	−4.00000	−0.137280
$850$	−4.00000	−0.137199
$851$	16.0000	0.548473
$852$	8.00000	0.274075
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	−2.00000	−0.0683986
$856$	−12.0000	−0.410152
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	12.0000	0.409673
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	−24.0000	−0.817443
$863$	−8.00000	−0.272323	−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000	−0.272323	−0.136162	+	0.990687i	$0.543477\pi$
$864$	1.00000	0.0340207
$865$	28.0000	0.952029
$866$	−32.0000	−1.08740
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	0	0
$870$	−4.00000	−0.135613
$871$	−84.0000	−2.84623
$872$	16.0000	0.541828
$873$	−16.0000	−0.541518
$874$	4.00000	0.135302
$875$	0	0
$876$	−10.0000	−0.337869
$877$	−20.0000	−0.675352	−0.337676	−	0.941262i	$-0.609641\pi$
$877$	−20.0000	−0.675352	−0.337676	+	0.941262i	$0.609641\pi$
$878$	6.00000	0.202490
$879$	−6.00000	−0.202375
$880$	4.00000	0.134840
$881$	−24.0000	−0.808581	−0.404290	−	0.914631i	$-0.632481\pi$
$881$	−24.0000	−0.808581	−0.404290	+	0.914631i	$0.632481\pi$
$882$	0	0
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	24.0000	0.807207
$885$	−8.00000	−0.268917
$886$	26.0000	0.873487
$887$	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
$890$	−12.0000	−0.402241
$891$	2.00000	0.0670025
$892$	−14.0000	−0.468755
$893$	−6.00000	−0.200782
$894$	0	0
$895$	48.0000	1.60446
$896$	0	0
$897$	−24.0000	−0.801337
$898$	34.0000	1.13459
$899$	−12.0000	−0.400222
$900$	−1.00000	−0.0333333
$901$	24.0000	0.799556
$902$	−12.0000	−0.399556
$903$	0	0
$904$	−2.00000	−0.0665190
$905$	12.0000	0.398893
$906$	−8.00000	−0.265782
$907$	−42.0000	−1.39459	−0.697294	−	0.716786i	$-0.745613\pi$
$907$	−42.0000	−1.39459	−0.697294	+	0.716786i	$0.745613\pi$
$908$	12.0000	0.398234
$909$	2.00000	0.0663358
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	−1.00000	−0.0331133
$913$	−16.0000	−0.529523
$914$	−26.0000	−0.860004
$915$	12.0000	0.396708
$916$	26.0000	0.859064
$917$	0	0
$918$	4.00000	0.132020
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	−8.00000	−0.263752
$921$	20.0000	0.659022
$922$	−34.0000	−1.11973
$923$	48.0000	1.57994
$924$	0	0
$925$	4.00000	0.131519
$926$	−8.00000	−0.262896
$927$	10.0000	0.328443
$928$	−2.00000	−0.0656532
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	12.0000	0.393496
$931$	0	0
$932$	14.0000	0.458585
$933$	2.00000	0.0654771
$934$	8.00000	0.261768
$935$	16.0000	0.523256
$936$	6.00000	0.196116
$937$	58.0000	1.89478	0.947389	−	0.320085i	$-0.103712\pi$
$937$	58.0000	1.89478	0.947389	+	0.320085i	$0.103712\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	12.0000	0.391397
$941$	50.0000	1.62995	0.814977	−	0.579494i	$-0.196750\pi$
$941$	50.0000	1.62995	0.814977	+	0.579494i	$0.196750\pi$
$942$	10.0000	0.325818
$943$	24.0000	0.781548
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−8.00000	−0.260102
$947$	−38.0000	−1.23483	−0.617417	−	0.786636i	$-0.711821\pi$
$947$	−38.0000	−1.23483	−0.617417	+	0.786636i	$0.711821\pi$
$948$	0	0
$949$	−60.0000	−1.94768
$950$	1.00000	0.0324443
$951$	−6.00000	−0.194563
$952$	0	0
$953$	−10.0000	−0.323932	−0.161966	−	0.986796i	$-0.551783\pi$
$953$	−10.0000	−0.323932	−0.161966	+	0.986796i	$0.551783\pi$
$954$	6.00000	0.194257
$955$	−24.0000	−0.776622
$956$	0	0
$957$	−4.00000	−0.129302
$958$	−2.00000	−0.0646171
$959$	0	0
$960$	2.00000	0.0645497
$961$	5.00000	0.161290
$962$	−24.0000	−0.773791
$963$	−12.0000	−0.386695
$964$	4.00000	0.128831
$965$	28.0000	0.901352
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	−7.00000	−0.224989
$969$	−4.00000	−0.128499
$970$	−32.0000	−1.02746
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	20.0000	0.640841
$975$	−6.00000	−0.192154
$976$	6.00000	0.192055
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	−16.0000	−0.511624
$979$	−12.0000	−0.383522
$980$	0	0
$981$	16.0000	0.510841
$982$	−30.0000	−0.957338
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	−6.00000	−0.191273
$985$	−32.0000	−1.01960
$986$	−8.00000	−0.254772
$987$	0	0
$988$	−6.00000	−0.190885
$989$	16.0000	0.508770
$990$	4.00000	0.127128
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	6.00000	0.190500
$993$	−34.0000	−1.07896
$994$	0	0
$995$	−40.0000	−1.26809
$996$	−8.00000	−0.253490
$997$	−38.0000	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$998$	−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	5586.2.a.bb.1.1		1
7.6	odd	2		798.2.a.g.1.1	✓	1
21.20	even	2		2394.2.a.e.1.1		1
28.27	even	2		6384.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
798.2.a.g.1.1	✓	1	7.6	odd	2
2394.2.a.e.1.1		1	21.20	even	2
5586.2.a.bb.1.1		1	1.1	even	1	trivial
6384.2.a.t.1.1		1	28.27	even	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

Related objects

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