LMFDB - Embedded newform 1386.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(1,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, 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("1386.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.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:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1386.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -4.00000 q^{10} +1.00000 q^{11} -6.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -2.00000 q^{19} +4.00000 q^{20} -1.00000 q^{22} +8.00000 q^{23} +11.0000 q^{25} +6.00000 q^{26} +1.00000 q^{28} +6.00000 q^{29} +6.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} +4.00000 q^{35} -6.00000 q^{37} +2.00000 q^{38} -4.00000 q^{40} -12.0000 q^{41} +4.00000 q^{43} +1.00000 q^{44} -8.00000 q^{46} -6.00000 q^{47} +1.00000 q^{49} -11.0000 q^{50} -6.00000 q^{52} -2.00000 q^{53} +4.00000 q^{55} -1.00000 q^{56} -6.00000 q^{58} +10.0000 q^{61} -6.00000 q^{62} +1.00000 q^{64} -24.0000 q^{65} +4.00000 q^{67} +4.00000 q^{68} -4.00000 q^{70} +12.0000 q^{71} +6.00000 q^{74} -2.00000 q^{76} +1.00000 q^{77} -16.0000 q^{79} +4.00000 q^{80} +12.0000 q^{82} +14.0000 q^{83} +16.0000 q^{85} -4.00000 q^{86} -1.00000 q^{88} +14.0000 q^{89} -6.00000 q^{91} +8.00000 q^{92} +6.00000 q^{94} -8.00000 q^{95} -14.0000 q^{97} -1.00000 q^{98} +O(q^{100})$

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−4.00000	−1.26491
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	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	$0.338794\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	4.00000	0.894427
$21$	0	0
$22$	−1.00000	−0.213201
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	6.00000	1.17670
$27$	0	0
$28$	1.00000	0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	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
$33$	0	0
$34$	−4.00000	−0.685994
$35$	4.00000	0.676123
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	−4.00000	−0.632456
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−8.00000	−1.17954
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−11.0000	−1.55563
$51$	0	0
$52$	−6.00000	−0.832050
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−6.00000	−0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−6.00000	−0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	−24.0000	−2.97683
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	−4.00000	−0.478091
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	−2.00000	−0.229416
$77$	1.00000	0.113961
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	4.00000	0.447214
$81$	0	0
$82$	12.0000	1.32518
$83$	14.0000	1.53670	0.768350	−	0.640030i	$-0.221078\pi$
$83$	14.0000	1.53670	0.768350	+	0.640030i	$0.221078\pi$
$84$	0	0
$85$	16.0000	1.73544
$86$	−4.00000	−0.431331
$87$	0	0
$88$	−1.00000	−0.106600
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	8.00000	0.834058
$93$	0	0
$94$	6.00000	0.618853
$95$	−8.00000	−0.820783
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	11.0000	1.10000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	2.00000	0.194257
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	−4.00000	−0.381385
$111$	0	0
$112$	1.00000	0.0944911
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	32.0000	2.98402
$116$	6.00000	0.557086
$117$	0	0
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	1.00000	0.0909091
$122$	−10.0000	−0.905357
$123$	0	0
$124$	6.00000	0.538816
$125$	24.0000	2.14663
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	24.0000	2.10494
$131$	2.00000	0.174741	0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000	0.174741	0.0873704	+	0.996176i	$0.472154\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−4.00000	−0.342997
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	−12.0000	−1.00702
$143$	−6.00000	−0.501745
$144$	0	0
$145$	24.0000	1.99309
$146$	0	0
$147$	0	0
$148$	−6.00000	−0.493197
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	24.0000	1.92773
$156$	0	0
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	16.0000	1.27289
$159$	0	0
$160$	−4.00000	−0.316228
$161$	8.00000	0.630488
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−12.0000	−0.937043
$165$	0	0
$166$	−14.0000	−1.08661
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−16.0000	−1.22714
$171$	0	0
$172$	4.00000	0.304997
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	11.0000	0.831522
$176$	1.00000	0.0753778
$177$	0	0
$178$	−14.0000	−1.04934
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	6.00000	0.444750
$183$	0	0
$184$	−8.00000	−0.589768
$185$	−24.0000	−1.76452
$186$	0	0
$187$	4.00000	0.292509
$188$	−6.00000	−0.437595
$189$	0	0
$190$	8.00000	0.580381
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	1.00000	0.0714286
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	−11.0000	−0.777817
$201$	0	0
$202$	14.0000	0.985037
$203$	6.00000	0.421117
$204$	0	0
$205$	−48.0000	−3.35247
$206$	6.00000	0.418040
$207$	0	0
$208$	−6.00000	−0.416025
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	−8.00000	−0.546869
$215$	16.0000	1.09119
$216$	0	0
$217$	6.00000	0.407307
$218$	−6.00000	−0.406371
$219$	0	0
$220$	4.00000	0.269680
$221$	−24.0000	−1.61441
$222$	0	0
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	14.0000	0.931266
$227$	−6.00000	−0.398234	−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000	−0.398234	−0.199117	+	0.979976i	$0.563807\pi$
$228$	0	0
$229$	8.00000	0.528655	0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000	0.528655	0.264327	+	0.964433i	$0.414850\pi$
$230$	−32.0000	−2.11002
$231$	0	0
$232$	−6.00000	−0.393919
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	0	0
$235$	−24.0000	−1.56559
$236$	0	0
$237$	0	0
$238$	−4.00000	−0.259281
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	10.0000	0.640184
$245$	4.00000	0.255551
$246$	0	0
$247$	12.0000	0.763542
$248$	−6.00000	−0.381000
$249$	0	0
$250$	−24.0000	−1.51789
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	−26.0000	−1.62184	−0.810918	−	0.585160i	$-0.801032\pi$
$257$	−26.0000	−1.62184	−0.810918	+	0.585160i	$0.801032\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	−24.0000	−1.48842
$261$	0	0
$262$	−2.00000	−0.123560
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−8.00000	−0.491436
$266$	2.00000	0.122628
$267$	0	0
$268$	4.00000	0.244339
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	14.0000	0.845771
$275$	11.0000	0.663325
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	−2.00000	−0.119952
$279$	0	0
$280$	−4.00000	−0.239046
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	0	0
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	6.00000	0.354787
$287$	−12.0000	−0.708338
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	−24.0000	−1.40933
$291$	0	0
$292$	0	0
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	0	0
$296$	6.00000	0.348743
$297$	0	0
$298$	−10.0000	−0.579284
$299$	−48.0000	−2.77591
$300$	0	0
$301$	4.00000	0.230556
$302$	8.00000	0.460348
$303$	0	0
$304$	−2.00000	−0.114708
$305$	40.0000	2.29039
$306$	0	0
$307$	−14.0000	−0.799022	−0.399511	−	0.916728i	$-0.630820\pi$
$307$	−14.0000	−0.799022	−0.399511	+	0.916728i	$0.630820\pi$
$308$	1.00000	0.0569803
$309$	0	0
$310$	−24.0000	−1.36311
$311$	−10.0000	−0.567048	−0.283524	−	0.958965i	$-0.591504\pi$
$311$	−10.0000	−0.567048	−0.283524	+	0.958965i	$0.591504\pi$
$312$	0	0
$313$	−34.0000	−1.92179	−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−34.0000	−1.92179	−0.960897	+	0.276907i	$0.910691\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	−16.0000	−0.900070
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	4.00000	0.223607
$321$	0	0
$322$	−8.00000	−0.445823
$323$	−8.00000	−0.445132
$324$	0	0
$325$	−66.0000	−3.66102
$326$	−4.00000	−0.221540
$327$	0	0
$328$	12.0000	0.662589
$329$	−6.00000	−0.330791
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	14.0000	0.768350
$333$	0	0
$334$	−12.0000	−0.656611
$335$	16.0000	0.874173
$336$	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
$339$	0	0
$340$	16.0000	0.867722
$341$	6.00000	0.324918
$342$	0	0
$343$	1.00000	0.0539949
$344$	−4.00000	−0.215666
$345$	0	0
$346$	2.00000	0.107521
$347$	−8.00000	−0.429463	−0.214731	−	0.976673i	$-0.568888\pi$
$347$	−8.00000	−0.429463	−0.214731	+	0.976673i	$0.568888\pi$
$348$	0	0
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	−11.0000	−0.587975
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	48.0000	2.54758
$356$	14.0000	0.741999
$357$	0	0
$358$	−4.00000	−0.211407
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	8.00000	0.420471
$363$	0	0
$364$	−6.00000	−0.314485
$365$	0	0
$366$	0	0
$367$	−34.0000	−1.77479	−0.887393	−	0.461014i	$-0.847486\pi$
$367$	−34.0000	−1.77479	−0.887393	+	0.461014i	$0.847486\pi$
$368$	8.00000	0.417029
$369$	0	0
$370$	24.0000	1.24770
$371$	−2.00000	−0.103835
$372$	0	0
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	6.00000	0.309426
$377$	−36.0000	−1.85409
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	−8.00000	−0.410391
$381$	0	0
$382$	0	0
$383$	22.0000	1.12415	0.562074	−	0.827087i	$-0.310004\pi$
$383$	22.0000	1.12415	0.562074	+	0.827087i	$0.310004\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	14.0000	0.712581
$387$	0	0
$388$	−14.0000	−0.710742
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	32.0000	1.61831
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	10.0000	0.503793
$395$	−64.0000	−3.22019
$396$	0	0
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	−2.00000	−0.100251
$399$	0	0
$400$	11.0000	0.550000
$401$	26.0000	1.29838	0.649189	−	0.760627i	$-0.275108\pi$
$401$	26.0000	1.29838	0.649189	+	0.760627i	$0.275108\pi$
$402$	0	0
$403$	−36.0000	−1.79329
$404$	−14.0000	−0.696526
$405$	0	0
$406$	−6.00000	−0.297775
$407$	−6.00000	−0.297409
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	48.0000	2.37055
$411$	0	0
$412$	−6.00000	−0.295599
$413$	0	0
$414$	0	0
$415$	56.0000	2.74893
$416$	6.00000	0.294174
$417$	0	0
$418$	2.00000	0.0978232
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	2.00000	0.0971286
$425$	44.0000	2.13431
$426$	0	0
$427$	10.0000	0.483934
$428$	8.00000	0.386695
$429$	0	0
$430$	−16.0000	−0.771589
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	−6.00000	−0.288009
$435$	0	0
$436$	6.00000	0.287348
$437$	−16.0000	−0.765384
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	−4.00000	−0.190693
$441$	0	0
$442$	24.0000	1.14156
$443$	−28.0000	−1.33032	−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000	−1.33032	−0.665160	+	0.746701i	$0.731637\pi$
$444$	0	0
$445$	56.0000	2.65465
$446$	−6.00000	−0.284108
$447$	0	0
$448$	1.00000	0.0472456
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	−14.0000	−0.658505
$453$	0	0
$454$	6.00000	0.281594
$455$	−24.0000	−1.12514
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	−8.00000	−0.373815
$459$	0	0
$460$	32.0000	1.49201
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−14.0000	−0.648537
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	24.0000	1.10704
$471$	0	0
$472$	0	0
$473$	4.00000	0.183920
$474$	0	0
$475$	−22.0000	−1.00943
$476$	4.00000	0.183340
$477$	0	0
$478$	−16.0000	−0.731823
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	36.0000	1.64146
$482$	8.00000	0.364390
$483$	0	0
$484$	1.00000	0.0454545
$485$	−56.0000	−2.54283
$486$	0	0
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	−10.0000	−0.452679
$489$	0	0
$490$	−4.00000	−0.180702
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	0	0
$493$	24.0000	1.08091
$494$	−12.0000	−0.539906
$495$	0	0
$496$	6.00000	0.269408
$497$	12.0000	0.538274
$498$	0	0
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	24.0000	1.07331
$501$	0	0
$502$	−16.0000	−0.714115
$503$	−20.0000	−0.891756	−0.445878	−	0.895094i	$-0.647108\pi$
$503$	−20.0000	−0.891756	−0.445878	+	0.895094i	$0.647108\pi$
$504$	0	0
$505$	−56.0000	−2.49197
$506$	−8.00000	−0.355643
$507$	0	0
$508$	0	0
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	26.0000	1.14681
$515$	−24.0000	−1.05757
$516$	0	0
$517$	−6.00000	−0.263880
$518$	6.00000	0.263625
$519$	0	0
$520$	24.0000	1.05247
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	2.00000	0.0873704
$525$	0	0
$526$	0	0
$527$	24.0000	1.04546
$528$	0	0
$529$	41.0000	1.78261
$530$	8.00000	0.347498
$531$	0	0
$532$	−2.00000	−0.0867110
$533$	72.0000	3.11867
$534$	0	0
$535$	32.0000	1.38348
$536$	−4.00000	−0.172774
$537$	0	0
$538$	−4.00000	−0.172452
$539$	1.00000	0.0430730
$540$	0	0
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	4.00000	0.171815
$543$	0	0
$544$	−4.00000	−0.171499
$545$	24.0000	1.02805
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−14.0000	−0.598050
$549$	0	0
$550$	−11.0000	−0.469042
$551$	−12.0000	−0.511217
$552$	0	0
$553$	−16.0000	−0.680389
$554$	10.0000	0.424859
$555$	0	0
$556$	2.00000	0.0848189
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	4.00000	0.169031
$561$	0	0
$562$	26.0000	1.09674
$563$	26.0000	1.09577	0.547885	−	0.836554i	$-0.315433\pi$
$563$	26.0000	1.09577	0.547885	+	0.836554i	$0.315433\pi$
$564$	0	0
$565$	−56.0000	−2.35594
$566$	14.0000	0.588464
$567$	0	0
$568$	−12.0000	−0.503509
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	−6.00000	−0.250873
$573$	0	0
$574$	12.0000	0.500870
$575$	88.0000	3.66985
$576$	0	0
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	24.0000	0.996546
$581$	14.0000	0.580818
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	0	0
$585$	0	0
$586$	−14.0000	−0.578335
$587$	−24.0000	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	0	0
$591$	0	0
$592$	−6.00000	−0.246598
$593$	12.0000	0.492781	0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000	0.492781	0.246390	+	0.969171i	$0.420755\pi$
$594$	0	0
$595$	16.0000	0.655936
$596$	10.0000	0.409616
$597$	0	0
$598$	48.0000	1.96287
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	4.00000	0.163163	0.0815817	−	0.996667i	$-0.474003\pi$
$601$	4.00000	0.163163	0.0815817	+	0.996667i	$0.474003\pi$
$602$	−4.00000	−0.163028
$603$	0	0
$604$	−8.00000	−0.325515
$605$	4.00000	0.162623
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	−40.0000	−1.61955
$611$	36.0000	1.45640
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	14.0000	0.564994
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	−26.0000	−1.04672	−0.523360	−	0.852111i	$-0.675322\pi$
$617$	−26.0000	−1.04672	−0.523360	+	0.852111i	$0.675322\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	24.0000	0.963863
$621$	0	0
$622$	10.0000	0.400963
$623$	14.0000	0.560898
$624$	0	0
$625$	41.0000	1.64000
$626$	34.0000	1.35891
$627$	0	0
$628$	−4.00000	−0.159617
$629$	−24.0000	−0.956943
$630$	0	0
$631$	28.0000	1.11466	0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000	1.11466	0.557331	+	0.830290i	$0.311825\pi$
$632$	16.0000	0.636446
$633$	0	0
$634$	−6.00000	−0.238290
$635$	0	0
$636$	0	0
$637$	−6.00000	−0.237729
$638$	−6.00000	−0.237542
$639$	0	0
$640$	−4.00000	−0.158114
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	8.00000	0.314756
$647$	−26.0000	−1.02217	−0.511083	−	0.859532i	$-0.670755\pi$
$647$	−26.0000	−1.02217	−0.511083	+	0.859532i	$0.670755\pi$
$648$	0	0
$649$	0	0
$650$	66.0000	2.58873
$651$	0	0
$652$	4.00000	0.156652
$653$	−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
$655$	8.00000	0.312586
$656$	−12.0000	−0.468521
$657$	0	0
$658$	6.00000	0.233904
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	−14.0000	−0.543305
$665$	−8.00000	−0.310227
$666$	0	0
$667$	48.0000	1.85857
$668$	12.0000	0.464294
$669$	0	0
$670$	−16.0000	−0.618134
$671$	10.0000	0.386046
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	−6.00000	−0.231111
$675$	0	0
$676$	23.0000	0.884615
$677$	−34.0000	−1.30673	−0.653363	−	0.757045i	$-0.726642\pi$
$677$	−34.0000	−1.30673	−0.653363	+	0.757045i	$0.726642\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	−16.0000	−0.613572
$681$	0	0
$682$	−6.00000	−0.229752
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	−56.0000	−2.13965
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	4.00000	0.152499
$689$	12.0000	0.457164
$690$	0	0
$691$	24.0000	0.913003	0.456502	−	0.889723i	$-0.349102\pi$
$691$	24.0000	0.913003	0.456502	+	0.889723i	$0.349102\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	8.00000	0.303676
$695$	8.00000	0.303457
$696$	0	0
$697$	−48.0000	−1.81813
$698$	−6.00000	−0.227103
$699$	0	0
$700$	11.0000	0.415761
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	1.00000	0.0376889
$705$	0	0
$706$	−6.00000	−0.225813
$707$	−14.0000	−0.526524
$708$	0	0
$709$	−30.0000	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$710$	−48.0000	−1.80141
$711$	0	0
$712$	−14.0000	−0.524672
$713$	48.0000	1.79761
$714$	0	0
$715$	−24.0000	−0.897549
$716$	4.00000	0.149487
$717$	0	0
$718$	24.0000	0.895672
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	0	0
$721$	−6.00000	−0.223452
$722$	15.0000	0.558242
$723$	0	0
$724$	−8.00000	−0.297318
$725$	66.0000	2.45118
$726$	0	0
$727$	−42.0000	−1.55769	−0.778847	−	0.627214i	$-0.784195\pi$
$727$	−42.0000	−1.55769	−0.778847	+	0.627214i	$0.784195\pi$
$728$	6.00000	0.222375
$729$	0	0
$730$	0	0
$731$	16.0000	0.591781
$732$	0	0
$733$	50.0000	1.84679	0.923396	−	0.383849i	$-0.125402\pi$
$733$	50.0000	1.84679	0.923396	+	0.383849i	$0.125402\pi$
$734$	34.0000	1.25496
$735$	0	0
$736$	−8.00000	−0.294884
$737$	4.00000	0.147342
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	−24.0000	−0.882258
$741$	0	0
$742$	2.00000	0.0734223
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	40.0000	1.46549
$746$	14.0000	0.512576
$747$	0	0
$748$	4.00000	0.146254
$749$	8.00000	0.292314
$750$	0	0
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	−6.00000	−0.218797
$753$	0	0
$754$	36.0000	1.31104
$755$	−32.0000	−1.16460
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−20.0000	−0.726433
$759$	0	0
$760$	8.00000	0.290191
$761$	−24.0000	−0.869999	−0.435000	−	0.900431i	$-0.643252\pi$
$761$	−24.0000	−0.869999	−0.435000	+	0.900431i	$0.643252\pi$
$762$	0	0
$763$	6.00000	0.217215
$764$	0	0
$765$	0	0
$766$	−22.0000	−0.794892
$767$	0	0
$768$	0	0
$769$	28.0000	1.00971	0.504853	−	0.863205i	$-0.331547\pi$
$769$	28.0000	1.00971	0.504853	+	0.863205i	$0.331547\pi$
$770$	−4.00000	−0.144150
$771$	0	0
$772$	−14.0000	−0.503871
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	0	0
$775$	66.0000	2.37079
$776$	14.0000	0.502571
$777$	0	0
$778$	6.00000	0.215110
$779$	24.0000	0.859889
$780$	0	0
$781$	12.0000	0.429394
$782$	−32.0000	−1.14432
$783$	0	0
$784$	1.00000	0.0357143
$785$	−16.0000	−0.571064
$786$	0	0
$787$	−50.0000	−1.78231	−0.891154	−	0.453701i	$-0.850103\pi$
$787$	−50.0000	−1.78231	−0.891154	+	0.453701i	$0.850103\pi$
$788$	−10.0000	−0.356235
$789$	0	0
$790$	64.0000	2.27702
$791$	−14.0000	−0.497783
$792$	0	0
$793$	−60.0000	−2.13066
$794$	20.0000	0.709773
$795$	0	0
$796$	2.00000	0.0708881
$797$	−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
$799$	−24.0000	−0.849059
$800$	−11.0000	−0.388909
$801$	0	0
$802$	−26.0000	−0.918092
$803$	0	0
$804$	0	0
$805$	32.0000	1.12785
$806$	36.0000	1.26805
$807$	0	0
$808$	14.0000	0.492518
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	0	0
$811$	26.0000	0.912983	0.456492	−	0.889728i	$-0.349106\pi$
$811$	26.0000	0.912983	0.456492	+	0.889728i	$0.349106\pi$
$812$	6.00000	0.210559
$813$	0	0
$814$	6.00000	0.210300
$815$	16.0000	0.560456
$816$	0	0
$817$	−8.00000	−0.279885
$818$	−4.00000	−0.139857
$819$	0	0
$820$	−48.0000	−1.67623
$821$	26.0000	0.907406	0.453703	−	0.891153i	$-0.350103\pi$
$821$	26.0000	0.907406	0.453703	+	0.891153i	$0.350103\pi$
$822$	0	0
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	6.00000	0.209020
$825$	0	0
$826$	0	0
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	0	0
$829$	16.0000	0.555703	0.277851	−	0.960624i	$-0.410378\pi$
$829$	16.0000	0.555703	0.277851	+	0.960624i	$0.410378\pi$
$830$	−56.0000	−1.94379
$831$	0	0
$832$	−6.00000	−0.208013
$833$	4.00000	0.138592
$834$	0	0
$835$	48.0000	1.66111
$836$	−2.00000	−0.0691714
$837$	0	0
$838$	12.0000	0.414533
$839$	54.0000	1.86429	0.932144	−	0.362089i	$-0.117936\pi$
$839$	54.0000	1.86429	0.932144	+	0.362089i	$0.117936\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−38.0000	−1.30957
$843$	0	0
$844$	−4.00000	−0.137686
$845$	92.0000	3.16490
$846$	0	0
$847$	1.00000	0.0343604
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	−44.0000	−1.50919
$851$	−48.0000	−1.64542
$852$	0	0
$853$	−6.00000	−0.205436	−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000	−0.205436	−0.102718	+	0.994711i	$0.532754\pi$
$854$	−10.0000	−0.342193
$855$	0	0
$856$	−8.00000	−0.273434
$857$	40.0000	1.36637	0.683187	−	0.730243i	$-0.260593\pi$
$857$	40.0000	1.36637	0.683187	+	0.730243i	$0.260593\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	16.0000	0.545595
$861$	0	0
$862$	−24.0000	−0.817443
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	0	0
$865$	−8.00000	−0.272008
$866$	−26.0000	−0.883516
$867$	0	0
$868$	6.00000	0.203653
$869$	−16.0000	−0.542763
$870$	0	0
$871$	−24.0000	−0.813209
$872$	−6.00000	−0.203186
$873$	0	0
$874$	16.0000	0.541208
$875$	24.0000	0.811348
$876$	0	0
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	20.0000	0.674967
$879$	0	0
$880$	4.00000	0.134840
$881$	50.0000	1.68454	0.842271	−	0.539054i	$-0.181218\pi$
$881$	50.0000	1.68454	0.842271	+	0.539054i	$0.181218\pi$
$882$	0	0
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	28.0000	0.940678
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	0	0
$890$	−56.0000	−1.87712
$891$	0	0
$892$	6.00000	0.200895
$893$	12.0000	0.401565
$894$	0	0
$895$	16.0000	0.534821
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	14.0000	0.467186
$899$	36.0000	1.20067
$900$	0	0
$901$	−8.00000	−0.266519
$902$	12.0000	0.399556
$903$	0	0
$904$	14.0000	0.465633
$905$	−32.0000	−1.06372
$906$	0	0
$907$	52.0000	1.72663	0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000	1.72663	0.863316	+	0.504664i	$0.168384\pi$
$908$	−6.00000	−0.199117
$909$	0	0
$910$	24.0000	0.795592
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	14.0000	0.463332
$914$	−10.0000	−0.330771
$915$	0	0
$916$	8.00000	0.264327
$917$	2.00000	0.0660458
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	−32.0000	−1.05501
$921$	0	0
$922$	14.0000	0.461065
$923$	−72.0000	−2.36991
$924$	0	0
$925$	−66.0000	−2.17007
$926$	24.0000	0.788689
$927$	0	0
$928$	−6.00000	−0.196960
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	14.0000	0.458585
$933$	0	0
$934$	36.0000	1.17796
$935$	16.0000	0.523256
$936$	0	0
$937$	−32.0000	−1.04539	−0.522697	−	0.852518i	$-0.675074\pi$
$937$	−32.0000	−1.04539	−0.522697	+	0.852518i	$0.675074\pi$
$938$	−4.00000	−0.130605
$939$	0	0
$940$	−24.0000	−0.782794
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	−96.0000	−3.12619
$944$	0	0
$945$	0	0
$946$	−4.00000	−0.130051
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	0	0
$949$	0	0
$950$	22.0000	0.713774
$951$	0	0
$952$	−4.00000	−0.129641
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	0	0
$955$	0	0
$956$	16.0000	0.517477
$957$	0	0
$958$	24.0000	0.775405
$959$	−14.0000	−0.452084
$960$	0	0
$961$	5.00000	0.161290
$962$	−36.0000	−1.16069
$963$	0	0
$964$	−8.00000	−0.257663
$965$	−56.0000	−1.80270
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	56.0000	1.79805
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	0	0
$973$	2.00000	0.0641171
$974$	−20.0000	−0.640841
$975$	0	0
$976$	10.0000	0.320092
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	0	0
$979$	14.0000	0.447442
$980$	4.00000	0.127775
$981$	0	0
$982$	−28.0000	−0.893516
$983$	42.0000	1.33959	0.669796	−	0.742545i	$-0.266382\pi$
$983$	42.0000	1.33959	0.669796	+	0.742545i	$0.266382\pi$
$984$	0	0
$985$	−40.0000	−1.27451
$986$	−24.0000	−0.764316
$987$	0	0
$988$	12.0000	0.381771
$989$	32.0000	1.01754
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	−6.00000	−0.190500
$993$	0	0
$994$	−12.0000	−0.380617
$995$	8.00000	0.253617
$996$	0	0
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	28.0000	0.886325
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.a.e.1.1		1
3.2	odd	2		462.2.a.e.1.1	✓	1
7.6	odd	2		9702.2.a.b.1.1		1
12.11	even	2		3696.2.a.p.1.1		1
21.20	even	2		3234.2.a.v.1.1		1
33.32	even	2		5082.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.a.e.1.1	✓	1	3.2	odd	2
1386.2.a.e.1.1		1	1.1	even	1	trivial
3234.2.a.v.1.1		1	21.20	even	2
3696.2.a.p.1.1		1	12.11	even	2
5082.2.a.a.1.1		1	33.32	even	2
9702.2.a.b.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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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