LMFDB - Embedded newform 3822.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3822,2,Mod(1,3822)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3822, 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("3822.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3822.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:	$30.5188236525$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3822.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

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

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	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	−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
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	1.00000	0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	−4.00000	−0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.00000	−0.204124
$25$	−1.00000	−0.200000
$26$	1.00000	0.196116
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	2.00000	0.365148
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	4.00000	0.696311
$34$	−6.00000	−1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	4.00000	0.648886
$39$	−1.00000	−0.160128
$40$	−2.00000	−0.316228
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−4.00000	−0.603023
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−1.00000	−0.141421
$51$	6.00000	0.840168
$52$	1.00000	0.138675
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	−1.00000	−0.136083
$55$	8.00000	1.07872
$56$	0	0
$57$	−4.00000	−0.529813
$58$	−6.00000	−0.787839
$59$	−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	$0.374508\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	4.00000	0.492366
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000	0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	10.0000	1.16248
$75$	1.00000	0.115470
$76$	4.00000	0.458831
$77$	0	0
$78$	−1.00000	−0.113228
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	4.00000	0.431331
$87$	6.00000	0.643268
$88$	−4.00000	−0.426401
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	0	0
$93$	−8.00000	−0.829561
$94$	−4.00000	−0.412568
$95$	−8.00000	−0.820783
$96$	−1.00000	−0.102062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	−1.00000	−0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	6.00000	0.594089
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	10.0000	0.971286
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	−1.00000	−0.0962250
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	8.00000	0.762770
$111$	−10.0000	−0.949158
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	−6.00000	−0.557086
$117$	1.00000	0.0924500
$118$	−4.00000	−0.368230
$119$	0	0
$120$	2.00000	0.182574
$121$	5.00000	0.454545
$122$	6.00000	0.543214
$123$	−6.00000	−0.541002
$124$	8.00000	0.718421
$125$	12.0000	1.07331
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	−2.00000	−0.175412
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	−8.00000	−0.691095
$135$	2.00000	0.172133
$136$	−6.00000	−0.514496
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	0	0
$143$	−4.00000	−0.334497
$144$	1.00000	0.0833333
$145$	12.0000	0.996546
$146$	10.0000	0.827606
$147$	0	0
$148$	10.0000	0.821995
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	1.00000	0.0816497
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	4.00000	0.324443
$153$	−6.00000	−0.485071
$154$	0	0
$155$	−16.0000	−1.28515
$156$	−1.00000	−0.0800641
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	−8.00000	−0.636446
$159$	−10.0000	−0.793052
$160$	−2.00000	−0.158114
$161$	0	0
$162$	1.00000	0.0785674
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	6.00000	0.468521
$165$	−8.00000	−0.622799
$166$	−4.00000	−0.310460
$167$	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
$169$	1.00000	0.0769231
$170$	12.0000	0.920358
$171$	4.00000	0.305888
$172$	4.00000	0.304997
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	−4.00000	−0.301511
$177$	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
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	0	0
$185$	−20.0000	−1.47043
$186$	−8.00000	−0.586588
$187$	24.0000	1.75505
$188$	−4.00000	−0.291730
$189$	0	0
$190$	−8.00000	−0.580381
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−1.00000	−0.0721688
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	2.00000	0.143592
$195$	2.00000	0.143223
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−4.00000	−0.284268
$199$	−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
$201$	8.00000	0.564276
$202$	−10.0000	−0.703598
$203$	0	0
$204$	6.00000	0.420084
$205$	−12.0000	−0.838116
$206$	12.0000	0.836080
$207$	0	0
$208$	1.00000	0.0693375
$209$	−16.0000	−1.10674
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	10.0000	0.686803
$213$	0	0
$214$	16.0000	1.09374
$215$	−8.00000	−0.545595
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−6.00000	−0.406371
$219$	−10.0000	−0.675737
$220$	8.00000	0.539360
$221$	−6.00000	−0.403604
$222$	−10.0000	−0.671156
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	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	$0.369623\pi$
$228$	−4.00000	−0.264906
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	1.00000	0.0653720
$235$	8.00000	0.521862
$236$	−4.00000	−0.260378
$237$	8.00000	0.519656
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	2.00000	0.129099
$241$	−30.0000	−1.93247	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000	−1.93247	−0.966235	+	0.257663i	$0.917048\pi$
$242$	5.00000	0.321412
$243$	−1.00000	−0.0641500
$244$	6.00000	0.384111
$245$	0	0
$246$	−6.00000	−0.382546
$247$	4.00000	0.254514
$248$	8.00000	0.508001
$249$	4.00000	0.253490
$250$	12.0000	0.758947
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	0	0
$254$	−16.0000	−1.00393
$255$	−12.0000	−0.751469
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	−2.00000	−0.124035
$261$	−6.00000	−0.371391
$262$	20.0000	1.23560
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	4.00000	0.246183
$265$	−20.0000	−1.22859
$266$	0	0
$267$	−6.00000	−0.367194
$268$	−8.00000	−0.488678
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	2.00000	0.121716
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	10.0000	0.604122
$275$	4.00000	0.241209
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	12.0000	0.719712
$279$	8.00000	0.478947
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	4.00000	0.238197
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	0	0
$285$	8.00000	0.473879
$286$	−4.00000	−0.236525
$287$	0	0
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	12.0000	0.704664
$291$	−2.00000	−0.117242
$292$	10.0000	0.585206
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	10.0000	0.581238
$297$	4.00000	0.232104
$298$	14.0000	0.810998
$299$	0	0
$300$	1.00000	0.0577350
$301$	0	0
$302$	16.0000	0.920697
$303$	10.0000	0.574485
$304$	4.00000	0.229416
$305$	−12.0000	−0.687118
$306$	−6.00000	−0.342997
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	−12.0000	−0.682656
$310$	−16.0000	−0.908739
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	−1.00000	−0.0566139
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	−18.0000	−1.01580
$315$	0	0
$316$	−8.00000	−0.450035
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−10.0000	−0.560772
$319$	24.0000	1.34374
$320$	−2.00000	−0.111803
$321$	−16.0000	−0.893033
$322$	0	0
$323$	−24.0000	−1.33540
$324$	1.00000	0.0555556
$325$	−1.00000	−0.0554700
$326$	16.0000	0.886158
$327$	6.00000	0.331801
$328$	6.00000	0.331295
$329$	0	0
$330$	−8.00000	−0.440386
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	−4.00000	−0.219529
$333$	10.0000	0.547997
$334$	12.0000	0.656611
$335$	16.0000	0.874173
$336$	0	0
$337$	34.0000	1.85210	0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000	1.85210	0.926049	+	0.377403i	$0.123183\pi$
$338$	1.00000	0.0543928
$339$	−2.00000	−0.108625
$340$	12.0000	0.650791
$341$	−32.0000	−1.73290
$342$	4.00000	0.216295
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	−18.0000	−0.967686
$347$	8.00000	0.429463	0.214731	−	0.976673i	$-0.431112\pi$
$347$	8.00000	0.429463	0.214731	+	0.976673i	$0.431112\pi$
$348$	6.00000	0.321634
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−4.00000	−0.213201
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	4.00000	0.212598
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	24.0000	1.26844
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	−2.00000	−0.105409
$361$	−3.00000	−0.157895
$362$	22.0000	1.15629
$363$	−5.00000	−0.262432
$364$	0	0
$365$	−20.0000	−1.04685
$366$	−6.00000	−0.313625
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	−20.0000	−1.03975
$371$	0	0
$372$	−8.00000	−0.414781
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	24.0000	1.24101
$375$	−12.0000	−0.619677
$376$	−4.00000	−0.206284
$377$	−6.00000	−0.309016
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	−8.00000	−0.410391
$381$	16.0000	0.819705
$382$	0	0
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−6.00000	−0.305392
$387$	4.00000	0.203331
$388$	2.00000	0.101535
$389$	34.0000	1.72387	0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000	1.72387	0.861934	+	0.507020i	$0.169253\pi$
$390$	2.00000	0.101274
$391$	0	0
$392$	0	0
$393$	−20.0000	−1.00887
$394$	6.00000	0.302276
$395$	16.0000	0.805047
$396$	−4.00000	−0.201008
$397$	−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
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	8.00000	0.399004
$403$	8.00000	0.398508
$404$	−10.0000	−0.497519
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	−40.0000	−1.98273
$408$	6.00000	0.297044
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	−12.0000	−0.592638
$411$	−10.0000	−0.493264
$412$	12.0000	0.591198
$413$	0	0
$414$	0	0
$415$	8.00000	0.392705
$416$	1.00000	0.0490290
$417$	−12.0000	−0.587643
$418$	−16.0000	−0.782586
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	−30.0000	−1.46211	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000	−1.46211	−0.731055	+	0.682318i	$0.760972\pi$
$422$	12.0000	0.584151
$423$	−4.00000	−0.194487
$424$	10.0000	0.485643
$425$	6.00000	0.291043
$426$	0	0
$427$	0	0
$428$	16.0000	0.773389
$429$	4.00000	0.193122
$430$	−8.00000	−0.385794
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	−1.00000	−0.0481125
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	−12.0000	−0.575356
$436$	−6.00000	−0.287348
$437$	0	0
$438$	−10.0000	−0.477818
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	8.00000	0.381385
$441$	0	0
$442$	−6.00000	−0.285391
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	−10.0000	−0.474579
$445$	−12.0000	−0.568855
$446$	8.00000	0.378811
$447$	−14.0000	−0.662177
$448$	0	0
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	−1.00000	−0.0471405
$451$	−24.0000	−1.13012
$452$	2.00000	0.0940721
$453$	−16.0000	−0.751746
$454$	12.0000	0.563188
$455$	0	0
$456$	−4.00000	−0.187317
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	6.00000	0.280362
$459$	6.00000	0.280056
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	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
$465$	16.0000	0.741982
$466$	18.0000	0.833834
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	1.00000	0.0462250
$469$	0	0
$470$	8.00000	0.369012
$471$	18.0000	0.829396
$472$	−4.00000	−0.184115
$473$	−16.0000	−0.735681
$474$	8.00000	0.367452
$475$	−4.00000	−0.183533
$476$	0	0
$477$	10.0000	0.457869
$478$	−24.0000	−1.09773
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	2.00000	0.0912871
$481$	10.0000	0.455961
$482$	−30.0000	−1.36646
$483$	0	0
$484$	5.00000	0.227273
$485$	−4.00000	−0.181631
$486$	−1.00000	−0.0453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	6.00000	0.271607
$489$	−16.0000	−0.723545
$490$	0	0
$491$	−32.0000	−1.44414	−0.722070	−	0.691820i	$-0.756809\pi$
$491$	−32.0000	−1.44414	−0.722070	+	0.691820i	$0.756809\pi$
$492$	−6.00000	−0.270501
$493$	36.0000	1.62136
$494$	4.00000	0.179969
$495$	8.00000	0.359573
$496$	8.00000	0.359211
$497$	0	0
$498$	4.00000	0.179244
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	12.0000	0.536656
$501$	−12.0000	−0.536120
$502$	4.00000	0.178529
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	20.0000	0.889988
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	−16.0000	−0.709885
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	−12.0000	−0.531369
$511$	0	0
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	−6.00000	−0.264649
$515$	−24.0000	−1.05757
$516$	−4.00000	−0.176090
$517$	16.0000	0.703679
$518$	0	0
$519$	18.0000	0.790112
$520$	−2.00000	−0.0877058
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	−6.00000	−0.262613
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	20.0000	0.873704
$525$	0	0
$526$	24.0000	1.04645
$527$	−48.0000	−2.09091
$528$	4.00000	0.174078
$529$	−23.0000	−1.00000
$530$	−20.0000	−0.868744
$531$	−4.00000	−0.173585
$532$	0	0
$533$	6.00000	0.259889
$534$	−6.00000	−0.259645
$535$	−32.0000	−1.38348
$536$	−8.00000	−0.345547
$537$	−24.0000	−1.03568
$538$	−18.0000	−0.776035
$539$	0	0
$540$	2.00000	0.0860663
$541$	−46.0000	−1.97769	−0.988847	−	0.148933i	$-0.952416\pi$
$541$	−46.0000	−1.97769	−0.988847	+	0.148933i	$0.952416\pi$
$542$	−8.00000	−0.343629
$543$	−22.0000	−0.944110
$544$	−6.00000	−0.257248
$545$	12.0000	0.514024
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	10.0000	0.427179
$549$	6.00000	0.256074
$550$	4.00000	0.170561
$551$	−24.0000	−1.02243
$552$	0	0
$553$	0	0
$554$	−10.0000	−0.424859
$555$	20.0000	0.848953
$556$	12.0000	0.508913
$557$	22.0000	0.932170	0.466085	−	0.884740i	$-0.345664\pi$
$557$	22.0000	0.932170	0.466085	+	0.884740i	$0.345664\pi$
$558$	8.00000	0.338667
$559$	4.00000	0.169182
$560$	0	0
$561$	−24.0000	−1.01328
$562$	10.0000	0.421825
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	4.00000	0.168430
$565$	−4.00000	−0.168281
$566$	−28.0000	−1.17693
$567$	0	0
$568$	0	0
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	8.00000	0.335083
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	−4.00000	−0.167248
$573$	0	0
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	19.0000	0.790296
$579$	6.00000	0.249351
$580$	12.0000	0.498273
$581$	0	0
$582$	−2.00000	−0.0829027
$583$	−40.0000	−1.65663
$584$	10.0000	0.413803
$585$	−2.00000	−0.0826898
$586$	−2.00000	−0.0826192
$587$	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
$589$	32.0000	1.31854
$590$	8.00000	0.329355
$591$	−6.00000	−0.246807
$592$	10.0000	0.410997
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	14.0000	0.573462
$597$	20.0000	0.818546
$598$	0	0
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	1.00000	0.0408248
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	16.0000	0.651031
$605$	−10.0000	−0.406558
$606$	10.0000	0.406222
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	−12.0000	−0.485866
$611$	−4.00000	−0.161823
$612$	−6.00000	−0.242536
$613$	−30.0000	−1.21169	−0.605844	−	0.795583i	$-0.707165\pi$
$613$	−30.0000	−1.21169	−0.605844	+	0.795583i	$0.707165\pi$
$614$	12.0000	0.484281
$615$	12.0000	0.483887
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	−12.0000	−0.482711
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	−16.0000	−0.642575
$621$	0	0
$622$	16.0000	0.641542
$623$	0	0
$624$	−1.00000	−0.0400320
$625$	−19.0000	−0.760000
$626$	−10.0000	−0.399680
$627$	16.0000	0.638978
$628$	−18.0000	−0.718278
$629$	−60.0000	−2.39236
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	−8.00000	−0.318223
$633$	−12.0000	−0.476957
$634$	−18.0000	−0.714871
$635$	32.0000	1.26988
$636$	−10.0000	−0.396526
$637$	0	0
$638$	24.0000	0.950169
$639$	0	0
$640$	−2.00000	−0.0790569
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−16.0000	−0.631470
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	−24.0000	−0.944267
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	1.00000	0.0392837
$649$	16.0000	0.628055
$650$	−1.00000	−0.0392232
$651$	0	0
$652$	16.0000	0.626608
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	6.00000	0.234619
$655$	−40.0000	−1.56293
$656$	6.00000	0.234261
$657$	10.0000	0.390137
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	−8.00000	−0.311400
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	−8.00000	−0.310929
$663$	6.00000	0.233021
$664$	−4.00000	−0.155230
$665$	0	0
$666$	10.0000	0.387492
$667$	0	0
$668$	12.0000	0.464294
$669$	−8.00000	−0.309298
$670$	16.0000	0.618134
$671$	−24.0000	−0.926510
$672$	0	0
$673$	18.0000	0.693849	0.346925	−	0.937893i	$-0.387226\pi$
$673$	18.0000	0.693849	0.346925	+	0.937893i	$0.387226\pi$
$674$	34.0000	1.30963
$675$	1.00000	0.0384900
$676$	1.00000	0.0384615
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	−2.00000	−0.0768095
$679$	0	0
$680$	12.0000	0.460179
$681$	−12.0000	−0.459841
$682$	−32.0000	−1.22534
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	4.00000	0.152944
$685$	−20.0000	−0.764161
$686$	0	0
$687$	−6.00000	−0.228914
$688$	4.00000	0.152499
$689$	10.0000	0.380970
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	8.00000	0.303676
$695$	−24.0000	−0.910372
$696$	6.00000	0.227429
$697$	−36.0000	−1.36360
$698$	−2.00000	−0.0757011
$699$	−18.0000	−0.680823
$700$	0	0
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	−1.00000	−0.0377426
$703$	40.0000	1.50863
$704$	−4.00000	−0.150756
$705$	−8.00000	−0.301297
$706$	6.00000	0.225813
$707$	0	0
$708$	4.00000	0.150329
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	6.00000	0.224860
$713$	0	0
$714$	0	0
$715$	8.00000	0.299183
$716$	24.0000	0.896922
$717$	24.0000	0.896296
$718$	−16.0000	−0.597115
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	−3.00000	−0.111648
$723$	30.0000	1.11571
$724$	22.0000	0.817624
$725$	6.00000	0.222834
$726$	−5.00000	−0.185567
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−20.0000	−0.740233
$731$	−24.0000	−0.887672
$732$	−6.00000	−0.221766
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	28.0000	1.03350
$735$	0	0
$736$	0	0
$737$	32.0000	1.17874
$738$	6.00000	0.220863
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	−20.0000	−0.735215
$741$	−4.00000	−0.146944
$742$	0	0
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	−8.00000	−0.293294
$745$	−28.0000	−1.02584
$746$	14.0000	0.512576
$747$	−4.00000	−0.146352
$748$	24.0000	0.877527
$749$	0	0
$750$	−12.0000	−0.438178
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	−4.00000	−0.145865
$753$	−4.00000	−0.145768
$754$	−6.00000	−0.218507
$755$	−32.0000	−1.16460
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	16.0000	0.581146
$759$	0	0
$760$	−8.00000	−0.290191
$761$	−26.0000	−0.942499	−0.471250	−	0.882000i	$-0.656197\pi$
$761$	−26.0000	−0.942499	−0.471250	+	0.882000i	$0.656197\pi$
$762$	16.0000	0.579619
$763$	0	0
$764$	0	0
$765$	12.0000	0.433861
$766$	−4.00000	−0.144526
$767$	−4.00000	−0.144432
$768$	−1.00000	−0.0360844
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−6.00000	−0.215945
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	4.00000	0.143777
$775$	−8.00000	−0.287368
$776$	2.00000	0.0717958
$777$	0	0
$778$	34.0000	1.21896
$779$	24.0000	0.859889
$780$	2.00000	0.0716115
$781$	0	0
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	36.0000	1.28490
$786$	−20.0000	−0.713376
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	6.00000	0.213741
$789$	−24.0000	−0.854423
$790$	16.0000	0.569254
$791$	0	0
$792$	−4.00000	−0.142134
$793$	6.00000	0.213066
$794$	−18.0000	−0.638796
$795$	20.0000	0.709327
$796$	−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
$802$	−30.0000	−1.05934
$803$	−40.0000	−1.41157
$804$	8.00000	0.282138
$805$	0	0
$806$	8.00000	0.281788
$807$	18.0000	0.633630
$808$	−10.0000	−0.351799
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	−2.00000	−0.0702728
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	8.00000	0.280572
$814$	−40.0000	−1.40200
$815$	−32.0000	−1.12091
$816$	6.00000	0.210042
$817$	16.0000	0.559769
$818$	−22.0000	−0.769212
$819$	0	0
$820$	−12.0000	−0.419058
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	−10.0000	−0.348790
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	12.0000	0.418040
$825$	−4.00000	−0.139262
$826$	0	0
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	8.00000	0.277684
$831$	10.0000	0.346896
$832$	1.00000	0.0346688
$833$	0	0
$834$	−12.0000	−0.415526
$835$	−24.0000	−0.830554
$836$	−16.0000	−0.553372
$837$	−8.00000	−0.276520
$838$	−28.0000	−0.967244
$839$	4.00000	0.138095	0.0690477	−	0.997613i	$-0.478004\pi$
$839$	4.00000	0.138095	0.0690477	+	0.997613i	$0.478004\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−30.0000	−1.03387
$843$	−10.0000	−0.344418
$844$	12.0000	0.413057
$845$	−2.00000	−0.0688021
$846$	−4.00000	−0.137523
$847$	0	0
$848$	10.0000	0.343401
$849$	28.0000	0.960958
$850$	6.00000	0.205798
$851$	0	0
$852$	0	0
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	16.0000	0.546869
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	4.00000	0.136558
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	−32.0000	−1.08992
$863$	56.0000	1.90626	0.953131	−	0.302558i	$-0.0978405\pi$
$863$	56.0000	1.90626	0.953131	+	0.302558i	$0.0978405\pi$
$864$	−1.00000	−0.0340207
$865$	36.0000	1.22404
$866$	−34.0000	−1.15537
$867$	−19.0000	−0.645274
$868$	0	0
$869$	32.0000	1.08553
$870$	−12.0000	−0.406838
$871$	−8.00000	−0.271070
$872$	−6.00000	−0.203186
$873$	2.00000	0.0676897
$874$	0	0
$875$	0	0
$876$	−10.0000	−0.337869
$877$	−6.00000	−0.202606	−0.101303	−	0.994856i	$-0.532301\pi$
$877$	−6.00000	−0.202606	−0.101303	+	0.994856i	$0.532301\pi$
$878$	20.0000	0.674967
$879$	2.00000	0.0674583
$880$	8.00000	0.269680
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	−6.00000	−0.201802
$885$	−8.00000	−0.268917
$886$	24.0000	0.806296
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	−10.0000	−0.335578
$889$	0	0
$890$	−12.0000	−0.402241
$891$	−4.00000	−0.134005
$892$	8.00000	0.267860
$893$	−16.0000	−0.535420
$894$	−14.0000	−0.468230
$895$	−48.0000	−1.60446
$896$	0	0
$897$	0	0
$898$	2.00000	0.0667409
$899$	−48.0000	−1.60089
$900$	−1.00000	−0.0333333
$901$	−60.0000	−1.99889
$902$	−24.0000	−0.799113
$903$	0	0
$904$	2.00000	0.0665190
$905$	−44.0000	−1.46261
$906$	−16.0000	−0.531564
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	12.0000	0.398234
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	−4.00000	−0.132453
$913$	16.0000	0.529523
$914$	−14.0000	−0.463079
$915$	12.0000	0.396708
$916$	6.00000	0.198246
$917$	0	0
$918$	6.00000	0.198030
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	14.0000	0.461065
$923$	0	0
$924$	0	0
$925$	−10.0000	−0.328798
$926$	−24.0000	−0.788689
$927$	12.0000	0.394132
$928$	−6.00000	−0.196960
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	16.0000	0.524661
$931$	0	0
$932$	18.0000	0.589610
$933$	−16.0000	−0.523816
$934$	−12.0000	−0.392652
$935$	−48.0000	−1.56977
$936$	1.00000	0.0326860
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	8.00000	0.260931
$941$	54.0000	1.76035	0.880175	−	0.474650i	$-0.157425\pi$
$941$	54.0000	1.76035	0.880175	+	0.474650i	$0.157425\pi$
$942$	18.0000	0.586472
$943$	0	0
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−16.0000	−0.520205
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	8.00000	0.259828
$949$	10.0000	0.324614
$950$	−4.00000	−0.129777
$951$	18.0000	0.583690
$952$	0	0
$953$	−14.0000	−0.453504	−0.226752	−	0.973952i	$-0.572811\pi$
$953$	−14.0000	−0.453504	−0.226752	+	0.973952i	$0.572811\pi$
$954$	10.0000	0.323762
$955$	0	0
$956$	−24.0000	−0.776215
$957$	−24.0000	−0.775810
$958$	4.00000	0.129234
$959$	0	0
$960$	2.00000	0.0645497
$961$	33.0000	1.06452
$962$	10.0000	0.322413
$963$	16.0000	0.515593
$964$	−30.0000	−0.966235
$965$	12.0000	0.386294
$966$	0	0
$967$	16.0000	0.514525	0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000	0.514525	0.257263	+	0.966342i	$0.417179\pi$
$968$	5.00000	0.160706
$969$	24.0000	0.770991
$970$	−4.00000	−0.128432
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	8.00000	0.256337
$975$	1.00000	0.0320256
$976$	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
$979$	−24.0000	−0.767043
$980$	0	0
$981$	−6.00000	−0.191565
$982$	−32.0000	−1.02116
$983$	−4.00000	−0.127580	−0.0637901	−	0.997963i	$-0.520319\pi$
$983$	−4.00000	−0.127580	−0.0637901	+	0.997963i	$0.520319\pi$
$984$	−6.00000	−0.191273
$985$	−12.0000	−0.382352
$986$	36.0000	1.14647
$987$	0	0
$988$	4.00000	0.127257
$989$	0	0
$990$	8.00000	0.254257
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	8.00000	0.254000
$993$	8.00000	0.253872
$994$	0	0
$995$	40.0000	1.26809
$996$	4.00000	0.126745
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	32.0000	1.01294
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3822.2.a.t.1.1		1
7.6	odd	2		546.2.a.g.1.1	✓	1
21.20	even	2		1638.2.a.d.1.1		1
28.27	even	2		4368.2.a.k.1.1		1
91.90	odd	2		7098.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.a.g.1.1	✓	1	7.6	odd	2
1638.2.a.d.1.1		1	21.20	even	2
3822.2.a.t.1.1		1	1.1	even	1	trivial
4368.2.a.k.1.1		1	28.27	even	2
7098.2.a.j.1.1		1	91.90	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:	\( N \)	\(=\)	\( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3822.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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