LMFDB - Embedded newform 3822.2.a.ba.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:	$1$
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)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

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

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
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−4.00000	−1.26491
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−4.00000	−1.03280
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	−4.00000	−0.894427
$21$	0	0
$22$	−1.00000	−0.213201
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	1.00000	0.204124
$25$	11.0000	2.20000
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	0	0
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	−4.00000	−0.730297
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	3.00000	0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	−1.00000	−0.162221
$39$	−1.00000	−0.160128
$40$	−4.00000	−0.632456
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	−1.00000	−0.150756
$45$	−4.00000	−0.596285
$46$	6.00000	0.884652
$47$	−11.0000	−1.60451	−0.802257	−	0.596978i	$-0.796368\pi$
$47$	−11.0000	−1.60451	−0.802257	+	0.596978i	$0.796368\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	11.0000	1.55563
$51$	3.00000	0.420084
$52$	−1.00000	−0.138675
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	1.00000	0.136083
$55$	4.00000	0.539360
$56$	0	0
$57$	−1.00000	−0.132453
$58$	−9.00000	−1.18176
$59$	−5.00000	−0.650945	−0.325472	−	0.945552i	$-0.605523\pi$
$59$	−5.00000	−0.650945	−0.325472	+	0.945552i	$0.605523\pi$
$60$	−4.00000	−0.516398
$61$	−15.0000	−1.92055	−0.960277	−	0.279050i	$-0.909981\pi$
$61$	−15.0000	−1.92055	−0.960277	+	0.279050i	$0.909981\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	−1.00000	−0.123091
$67$	−5.00000	−0.610847	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−5.00000	−0.610847	−0.305424	+	0.952217i	$0.598798\pi$
$68$	3.00000	0.363803
$69$	6.00000	0.722315
$70$	0	0
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	1.00000	0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−8.00000	−0.929981
$75$	11.0000	1.27017
$76$	−1.00000	−0.114708
$77$	0	0
$78$	−1.00000	−0.113228
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	0	0
$83$	−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
$85$	−12.0000	−1.30158
$86$	10.0000	1.07833
$87$	−9.00000	−0.964901
$88$	−1.00000	−0.106600
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	−4.00000	−0.421637
$91$	0	0
$92$	6.00000	0.625543
$93$	−8.00000	−0.829561
$94$	−11.0000	−1.13456
$95$	4.00000	0.410391
$96$	1.00000	0.102062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	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	$0.745272\pi$
$102$	3.00000	0.297044
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	1.00000	0.0971286
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	1.00000	0.0962250
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	4.00000	0.381385
$111$	−8.00000	−0.759326
$112$	0	0
$113$	5.00000	0.470360	0.235180	−	0.971952i	$-0.424432\pi$
$113$	5.00000	0.470360	0.235180	+	0.971952i	$0.424432\pi$
$114$	−1.00000	−0.0936586
$115$	−24.0000	−2.23801
$116$	−9.00000	−0.835629
$117$	−1.00000	−0.0924500
$118$	−5.00000	−0.460287
$119$	0	0
$120$	−4.00000	−0.365148
$121$	−10.0000	−0.909091
$122$	−15.0000	−1.35804
$123$	0	0
$124$	−8.00000	−0.718421
$125$	−24.0000	−2.14663
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	1.00000	0.0883883
$129$	10.0000	0.880451
$130$	4.00000	0.350823
$131$	−2.00000	−0.174741	−0.0873704	−	0.996176i	$-0.527846\pi$
$131$	−2.00000	−0.174741	−0.0873704	+	0.996176i	$0.527846\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	−5.00000	−0.431934
$135$	−4.00000	−0.344265
$136$	3.00000	0.257248
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	6.00000	0.510754
$139$	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
$141$	−11.0000	−0.926367
$142$	−15.0000	−1.25877
$143$	1.00000	0.0836242
$144$	1.00000	0.0833333
$145$	36.0000	2.98964
$146$	2.00000	0.165521
$147$	0	0
$148$	−8.00000	−0.657596
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	11.0000	0.898146
$151$	19.0000	1.54620	0.773099	−	0.634285i	$-0.218706\pi$
$151$	19.0000	1.54620	0.773099	+	0.634285i	$0.218706\pi$
$152$	−1.00000	−0.0811107
$153$	3.00000	0.242536
$154$	0	0
$155$	32.0000	2.57030
$156$	−1.00000	−0.0800641
$157$	15.0000	1.19713	0.598565	−	0.801074i	$-0.295738\pi$
$157$	15.0000	1.19713	0.598565	+	0.801074i	$0.295738\pi$
$158$	−2.00000	−0.159111
$159$	1.00000	0.0793052
$160$	−4.00000	−0.316228
$161$	0	0
$162$	1.00000	0.0785674
$163$	13.0000	1.01824	0.509119	−	0.860696i	$-0.329971\pi$
$163$	13.0000	1.01824	0.509119	+	0.860696i	$0.329971\pi$
$164$	0	0
$165$	4.00000	0.311400
$166$	−8.00000	−0.620920
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−12.0000	−0.920358
$171$	−1.00000	−0.0764719
$172$	10.0000	0.762493
$173$	3.00000	0.228086	0.114043	−	0.993476i	$-0.463620\pi$
$173$	3.00000	0.228086	0.114043	+	0.993476i	$0.463620\pi$
$174$	−9.00000	−0.682288
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−5.00000	−0.375823
$178$	0	0
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	−4.00000	−0.298142
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	0	0
$183$	−15.0000	−1.10883
$184$	6.00000	0.442326
$185$	32.0000	2.35269
$186$	−8.00000	−0.586588
$187$	−3.00000	−0.219382
$188$	−11.0000	−0.802257
$189$	0	0
$190$	4.00000	0.290191
$191$	−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
$193$	−12.0000	−0.863779	−0.431889	−	0.901927i	$-0.642153\pi$
$193$	−12.0000	−0.863779	−0.431889	+	0.901927i	$0.642153\pi$
$194$	10.0000	0.717958
$195$	4.00000	0.286446
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	−1.00000	−0.0710669
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	11.0000	0.777817
$201$	−5.00000	−0.352673
$202$	−14.0000	−0.985037
$203$	0	0
$204$	3.00000	0.210042
$205$	0	0
$206$	6.00000	0.418040
$207$	6.00000	0.417029
$208$	−1.00000	−0.0693375
$209$	1.00000	0.0691714
$210$	0	0
$211$	18.0000	1.23917	0.619586	−	0.784929i	$-0.287301\pi$
$211$	18.0000	1.23917	0.619586	+	0.784929i	$0.287301\pi$
$212$	1.00000	0.0686803
$213$	−15.0000	−1.02778
$214$	−8.00000	−0.546869
$215$	−40.0000	−2.72798
$216$	1.00000	0.0680414
$217$	0	0
$218$	0	0
$219$	2.00000	0.135147
$220$	4.00000	0.269680
$221$	−3.00000	−0.201802
$222$	−8.00000	−0.536925
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	0	0
$225$	11.0000	0.733333
$226$	5.00000	0.332595
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−1.00000	−0.0662266
$229$	18.0000	1.18947	0.594737	−	0.803921i	$-0.297256\pi$
$229$	18.0000	1.18947	0.594737	+	0.803921i	$0.297256\pi$
$230$	−24.0000	−1.58251
$231$	0	0
$232$	−9.00000	−0.590879
$233$	−15.0000	−0.982683	−0.491341	−	0.870967i	$-0.663493\pi$
$233$	−15.0000	−0.982683	−0.491341	+	0.870967i	$0.663493\pi$
$234$	−1.00000	−0.0653720
$235$	44.0000	2.87024
$236$	−5.00000	−0.325472
$237$	−2.00000	−0.129914
$238$	0	0
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	−4.00000	−0.258199
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	−10.0000	−0.642824
$243$	1.00000	0.0641500
$244$	−15.0000	−0.960277
$245$	0	0
$246$	0	0
$247$	1.00000	0.0636285
$248$	−8.00000	−0.508001
$249$	−8.00000	−0.506979
$250$	−24.0000	−1.51789
$251$	−22.0000	−1.38863	−0.694314	−	0.719672i	$-0.744292\pi$
$251$	−22.0000	−1.38863	−0.694314	+	0.719672i	$0.744292\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	2.00000	0.125491
$255$	−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	$0.559920\pi$
$258$	10.0000	0.622573
$259$	0	0
$260$	4.00000	0.248069
$261$	−9.00000	−0.557086
$262$	−2.00000	−0.123560
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	−1.00000	−0.0615457
$265$	−4.00000	−0.245718
$266$	0	0
$267$	0	0
$268$	−5.00000	−0.305424
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	−4.00000	−0.243432
$271$	−1.00000	−0.0607457	−0.0303728	−	0.999539i	$-0.509669\pi$
$271$	−1.00000	−0.0607457	−0.0303728	+	0.999539i	$0.509669\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	−8.00000	−0.483298
$275$	−11.0000	−0.663325
$276$	6.00000	0.361158
$277$	−1.00000	−0.0600842	−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000	−0.0600842	−0.0300421	+	0.999549i	$0.509564\pi$
$278$	12.0000	0.719712
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−32.0000	−1.90896	−0.954480	−	0.298275i	$-0.903589\pi$
$281$	−32.0000	−1.90896	−0.954480	+	0.298275i	$0.903589\pi$
$282$	−11.0000	−0.655040
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	−15.0000	−0.890086
$285$	4.00000	0.236940
$286$	1.00000	0.0591312
$287$	0	0
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	36.0000	2.11399
$291$	10.0000	0.586210
$292$	2.00000	0.117041
$293$	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
$295$	20.0000	1.16445
$296$	−8.00000	−0.464991
$297$	−1.00000	−0.0580259
$298$	−10.0000	−0.579284
$299$	−6.00000	−0.346989
$300$	11.0000	0.635085
$301$	0	0
$302$	19.0000	1.09333
$303$	−14.0000	−0.804279
$304$	−1.00000	−0.0573539
$305$	60.0000	3.43559
$306$	3.00000	0.171499
$307$	27.0000	1.54097	0.770486	−	0.637457i	$-0.220014\pi$
$307$	27.0000	1.54097	0.770486	+	0.637457i	$0.220014\pi$
$308$	0	0
$309$	6.00000	0.341328
$310$	32.0000	1.81748
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	−1.00000	−0.0566139
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	15.0000	0.846499
$315$	0	0
$316$	−2.00000	−0.112509
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	1.00000	0.0560772
$319$	9.00000	0.503903
$320$	−4.00000	−0.223607
$321$	−8.00000	−0.446516
$322$	0	0
$323$	−3.00000	−0.166924
$324$	1.00000	0.0555556
$325$	−11.0000	−0.610170
$326$	13.0000	0.720003
$327$	0	0
$328$	0	0
$329$	0	0
$330$	4.00000	0.220193
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	−8.00000	−0.439057
$333$	−8.00000	−0.438397
$334$	3.00000	0.164153
$335$	20.0000	1.09272
$336$	0	0
$337$	31.0000	1.68868	0.844339	−	0.535810i	$-0.179994\pi$
$337$	31.0000	1.68868	0.844339	+	0.535810i	$0.179994\pi$
$338$	1.00000	0.0543928
$339$	5.00000	0.271563
$340$	−12.0000	−0.650791
$341$	8.00000	0.433224
$342$	−1.00000	−0.0540738
$343$	0	0
$344$	10.0000	0.539164
$345$	−24.0000	−1.29212
$346$	3.00000	0.161281
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	−9.00000	−0.482451
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−1.00000	−0.0533002
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	−5.00000	−0.265747
$355$	60.0000	3.18447
$356$	0	0
$357$	0	0
$358$	−18.0000	−0.951330
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	−4.00000	−0.210819
$361$	−18.0000	−0.947368
$362$	−7.00000	−0.367912
$363$	−10.0000	−0.524864
$364$	0	0
$365$	−8.00000	−0.418739
$366$	−15.0000	−0.784063
$367$	26.0000	1.35719	0.678594	−	0.734513i	$-0.262589\pi$
$367$	26.0000	1.35719	0.678594	+	0.734513i	$0.262589\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	32.0000	1.66360
$371$	0	0
$372$	−8.00000	−0.414781
$373$	−1.00000	−0.0517780	−0.0258890	−	0.999665i	$-0.508242\pi$
$373$	−1.00000	−0.0517780	−0.0258890	+	0.999665i	$0.508242\pi$
$374$	−3.00000	−0.155126
$375$	−24.0000	−1.23935
$376$	−11.0000	−0.567282
$377$	9.00000	0.463524
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	4.00000	0.205196
$381$	2.00000	0.102463
$382$	−12.0000	−0.613973
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−12.0000	−0.610784
$387$	10.0000	0.508329
$388$	10.0000	0.507673
$389$	−23.0000	−1.16615	−0.583073	−	0.812420i	$-0.698150\pi$
$389$	−23.0000	−1.16615	−0.583073	+	0.812420i	$0.698150\pi$
$390$	4.00000	0.202548
$391$	18.0000	0.910299
$392$	0	0
$393$	−2.00000	−0.100887
$394$	18.0000	0.906827
$395$	8.00000	0.402524
$396$	−1.00000	−0.0502519
$397$	−30.0000	−1.50566	−0.752828	−	0.658217i	$-0.771311\pi$
$397$	−30.0000	−1.50566	−0.752828	+	0.658217i	$0.771311\pi$
$398$	−10.0000	−0.501255
$399$	0	0
$400$	11.0000	0.550000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	−5.00000	−0.249377
$403$	8.00000	0.398508
$404$	−14.0000	−0.696526
$405$	−4.00000	−0.198762
$406$	0	0
$407$	8.00000	0.396545
$408$	3.00000	0.148522
$409$	−8.00000	−0.395575	−0.197787	−	0.980245i	$-0.563376\pi$
$409$	−8.00000	−0.395575	−0.197787	+	0.980245i	$0.563376\pi$
$410$	0	0
$411$	−8.00000	−0.394611
$412$	6.00000	0.295599
$413$	0	0
$414$	6.00000	0.294884
$415$	32.0000	1.57082
$416$	−1.00000	−0.0490290
$417$	12.0000	0.587643
$418$	1.00000	0.0489116
$419$	−14.0000	−0.683945	−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000	−0.683945	−0.341972	+	0.939710i	$0.611095\pi$
$420$	0	0
$421$	−30.0000	−1.46211	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000	−1.46211	−0.731055	+	0.682318i	$0.760972\pi$
$422$	18.0000	0.876226
$423$	−11.0000	−0.534838
$424$	1.00000	0.0485643
$425$	33.0000	1.60074
$426$	−15.0000	−0.726752
$427$	0	0
$428$	−8.00000	−0.386695
$429$	1.00000	0.0482805
$430$	−40.0000	−1.92897
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	1.00000	0.0481125
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	0	0
$435$	36.0000	1.72607
$436$	0	0
$437$	−6.00000	−0.287019
$438$	2.00000	0.0955637
$439$	−2.00000	−0.0954548	−0.0477274	−	0.998860i	$-0.515198\pi$
$439$	−2.00000	−0.0954548	−0.0477274	+	0.998860i	$0.515198\pi$
$440$	4.00000	0.190693
$441$	0	0
$442$	−3.00000	−0.142695
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	19.0000	0.899676
$447$	−10.0000	−0.472984
$448$	0	0
$449$	−4.00000	−0.188772	−0.0943858	−	0.995536i	$-0.530089\pi$
$449$	−4.00000	−0.188772	−0.0943858	+	0.995536i	$0.530089\pi$
$450$	11.0000	0.518545
$451$	0	0
$452$	5.00000	0.235180
$453$	19.0000	0.892698
$454$	−12.0000	−0.563188
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	34.0000	1.59045	0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000	1.59045	0.795226	+	0.606313i	$0.207352\pi$
$458$	18.0000	0.841085
$459$	3.00000	0.140028
$460$	−24.0000	−1.11901
$461$	40.0000	1.86299	0.931493	−	0.363760i	$-0.118507\pi$
$461$	40.0000	1.86299	0.931493	+	0.363760i	$0.118507\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	−9.00000	−0.417815
$465$	32.0000	1.48396
$466$	−15.0000	−0.694862
$467$	30.0000	1.38823	0.694117	−	0.719862i	$-0.255795\pi$
$467$	30.0000	1.38823	0.694117	+	0.719862i	$0.255795\pi$
$468$	−1.00000	−0.0462250
$469$	0	0
$470$	44.0000	2.02957
$471$	15.0000	0.691164
$472$	−5.00000	−0.230144
$473$	−10.0000	−0.459800
$474$	−2.00000	−0.0918630
$475$	−11.0000	−0.504715
$476$	0	0
$477$	1.00000	0.0457869
$478$	15.0000	0.686084
$479$	5.00000	0.228456	0.114228	−	0.993455i	$-0.463561\pi$
$479$	5.00000	0.228456	0.114228	+	0.993455i	$0.463561\pi$
$480$	−4.00000	−0.182574
$481$	8.00000	0.364769
$482$	0	0
$483$	0	0
$484$	−10.0000	−0.454545
$485$	−40.0000	−1.81631
$486$	1.00000	0.0453609
$487$	−1.00000	−0.0453143	−0.0226572	−	0.999743i	$-0.507213\pi$
$487$	−1.00000	−0.0453143	−0.0226572	+	0.999743i	$0.507213\pi$
$488$	−15.0000	−0.679018
$489$	13.0000	0.587880
$490$	0	0
$491$	−38.0000	−1.71492	−0.857458	−	0.514554i	$-0.827958\pi$
$491$	−38.0000	−1.71492	−0.857458	+	0.514554i	$0.827958\pi$
$492$	0	0
$493$	−27.0000	−1.21602
$494$	1.00000	0.0449921
$495$	4.00000	0.179787
$496$	−8.00000	−0.359211
$497$	0	0
$498$	−8.00000	−0.358489
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	−24.0000	−1.07331
$501$	3.00000	0.134030
$502$	−22.0000	−0.981908
$503$	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
$505$	56.0000	2.49197
$506$	−6.00000	−0.266733
$507$	1.00000	0.0444116
$508$	2.00000	0.0887357
$509$	40.0000	1.77297	0.886484	−	0.462758i	$-0.153140\pi$
$509$	40.0000	1.77297	0.886484	+	0.462758i	$0.153140\pi$
$510$	−12.0000	−0.531369
$511$	0	0
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	−6.00000	−0.264649
$515$	−24.0000	−1.05757
$516$	10.0000	0.440225
$517$	11.0000	0.483779
$518$	0	0
$519$	3.00000	0.131685
$520$	4.00000	0.175412
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	−9.00000	−0.393919
$523$	−14.0000	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	−2.00000	−0.0873704
$525$	0	0
$526$	24.0000	1.04645
$527$	−24.0000	−1.04546
$528$	−1.00000	−0.0435194
$529$	13.0000	0.565217
$530$	−4.00000	−0.173749
$531$	−5.00000	−0.216982
$532$	0	0
$533$	0	0
$534$	0	0
$535$	32.0000	1.38348
$536$	−5.00000	−0.215967
$537$	−18.0000	−0.776757
$538$	−9.00000	−0.388018
$539$	0	0
$540$	−4.00000	−0.172133
$541$	−28.0000	−1.20381	−0.601907	−	0.798566i	$-0.705592\pi$
$541$	−28.0000	−1.20381	−0.601907	+	0.798566i	$0.705592\pi$
$542$	−1.00000	−0.0429537
$543$	−7.00000	−0.300399
$544$	3.00000	0.128624
$545$	0	0
$546$	0	0
$547$	−42.0000	−1.79579	−0.897895	−	0.440209i	$-0.854904\pi$
$547$	−42.0000	−1.79579	−0.897895	+	0.440209i	$0.854904\pi$
$548$	−8.00000	−0.341743
$549$	−15.0000	−0.640184
$550$	−11.0000	−0.469042
$551$	9.00000	0.383413
$552$	6.00000	0.255377
$553$	0	0
$554$	−1.00000	−0.0424859
$555$	32.0000	1.35832
$556$	12.0000	0.508913
$557$	34.0000	1.44063	0.720313	−	0.693649i	$-0.243998\pi$
$557$	34.0000	1.44063	0.720313	+	0.693649i	$0.243998\pi$
$558$	−8.00000	−0.338667
$559$	−10.0000	−0.422955
$560$	0	0
$561$	−3.00000	−0.126660
$562$	−32.0000	−1.34984
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	−11.0000	−0.463184
$565$	−20.0000	−0.841406
$566$	−14.0000	−0.588464
$567$	0	0
$568$	−15.0000	−0.629386
$569$	−19.0000	−0.796521	−0.398261	−	0.917272i	$-0.630386\pi$
$569$	−19.0000	−0.796521	−0.398261	+	0.917272i	$0.630386\pi$
$570$	4.00000	0.167542
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	1.00000	0.0418121
$573$	−12.0000	−0.501307
$574$	0	0
$575$	66.0000	2.75239
$576$	1.00000	0.0416667
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	−8.00000	−0.332756
$579$	−12.0000	−0.498703
$580$	36.0000	1.49482
$581$	0	0
$582$	10.0000	0.414513
$583$	−1.00000	−0.0414158
$584$	2.00000	0.0827606
$585$	4.00000	0.165380
$586$	14.0000	0.578335
$587$	47.0000	1.93990	0.969949	−	0.243309i	$-0.0782329\pi$
$587$	47.0000	1.93990	0.969949	+	0.243309i	$0.0782329\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	20.0000	0.823387
$591$	18.0000	0.740421
$592$	−8.00000	−0.328798
$593$	−12.0000	−0.492781	−0.246390	−	0.969171i	$-0.579245\pi$
$593$	−12.0000	−0.492781	−0.246390	+	0.969171i	$0.579245\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	−10.0000	−0.409616
$597$	−10.0000	−0.409273
$598$	−6.00000	−0.245358
$599$	−4.00000	−0.163436	−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000	−0.163436	−0.0817178	+	0.996656i	$0.526041\pi$
$600$	11.0000	0.449073
$601$	17.0000	0.693444	0.346722	−	0.937968i	$-0.387295\pi$
$601$	17.0000	0.693444	0.346722	+	0.937968i	$0.387295\pi$
$602$	0	0
$603$	−5.00000	−0.203616
$604$	19.0000	0.773099
$605$	40.0000	1.62623
$606$	−14.0000	−0.568711
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	60.0000	2.42933
$611$	11.0000	0.445012
$612$	3.00000	0.121268
$613$	−24.0000	−0.969351	−0.484675	−	0.874694i	$-0.661062\pi$
$613$	−24.0000	−0.969351	−0.484675	+	0.874694i	$0.661062\pi$
$614$	27.0000	1.08963
$615$	0	0
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	6.00000	0.241355
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	32.0000	1.28515
$621$	6.00000	0.240772
$622$	8.00000	0.320771
$623$	0	0
$624$	−1.00000	−0.0400320
$625$	41.0000	1.64000
$626$	−14.0000	−0.559553
$627$	1.00000	0.0399362
$628$	15.0000	0.598565
$629$	−24.0000	−0.956943
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	−2.00000	−0.0795557
$633$	18.0000	0.715436
$634$	−6.00000	−0.238290
$635$	−8.00000	−0.317470
$636$	1.00000	0.0396526
$637$	0	0
$638$	9.00000	0.356313
$639$	−15.0000	−0.593391
$640$	−4.00000	−0.158114
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−8.00000	−0.315735
$643$	−49.0000	−1.93237	−0.966186	−	0.257847i	$-0.916987\pi$
$643$	−49.0000	−1.93237	−0.966186	+	0.257847i	$0.916987\pi$
$644$	0	0
$645$	−40.0000	−1.57500
$646$	−3.00000	−0.118033
$647$	2.00000	0.0786281	0.0393141	−	0.999227i	$-0.487483\pi$
$647$	2.00000	0.0786281	0.0393141	+	0.999227i	$0.487483\pi$
$648$	1.00000	0.0392837
$649$	5.00000	0.196267
$650$	−11.0000	−0.431455
$651$	0	0
$652$	13.0000	0.509119
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	0	0
$655$	8.00000	0.312586
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	4.00000	0.155700
$661$	16.0000	0.622328	0.311164	−	0.950356i	$-0.399281\pi$
$661$	16.0000	0.622328	0.311164	+	0.950356i	$0.399281\pi$
$662$	28.0000	1.08825
$663$	−3.00000	−0.116510
$664$	−8.00000	−0.310460
$665$	0	0
$666$	−8.00000	−0.309994
$667$	−54.0000	−2.09089
$668$	3.00000	0.116073
$669$	19.0000	0.734582
$670$	20.0000	0.772667
$671$	15.0000	0.579069
$672$	0	0
$673$	−18.0000	−0.693849	−0.346925	−	0.937893i	$-0.612774\pi$
$673$	−18.0000	−0.693849	−0.346925	+	0.937893i	$0.612774\pi$
$674$	31.0000	1.19408
$675$	11.0000	0.423390
$676$	1.00000	0.0384615
$677$	1.00000	0.0384331	0.0192166	−	0.999815i	$-0.493883\pi$
$677$	1.00000	0.0384331	0.0192166	+	0.999815i	$0.493883\pi$
$678$	5.00000	0.192024
$679$	0	0
$680$	−12.0000	−0.460179
$681$	−12.0000	−0.459841
$682$	8.00000	0.306336
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	−1.00000	−0.0382360
$685$	32.0000	1.22266
$686$	0	0
$687$	18.0000	0.686743
$688$	10.0000	0.381246
$689$	−1.00000	−0.0380970
$690$	−24.0000	−0.913664
$691$	5.00000	0.190209	0.0951045	−	0.995467i	$-0.469681\pi$
$691$	5.00000	0.190209	0.0951045	+	0.995467i	$0.469681\pi$
$692$	3.00000	0.114043
$693$	0	0
$694$	−4.00000	−0.151838
$695$	−48.0000	−1.82074
$696$	−9.00000	−0.341144
$697$	0	0
$698$	−10.0000	−0.378506
$699$	−15.0000	−0.567352
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	−1.00000	−0.0377426
$703$	8.00000	0.301726
$704$	−1.00000	−0.0376889
$705$	44.0000	1.65714
$706$	18.0000	0.677439
$707$	0	0
$708$	−5.00000	−0.187912
$709$	30.0000	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$710$	60.0000	2.25176
$711$	−2.00000	−0.0750059
$712$	0	0
$713$	−48.0000	−1.79761
$714$	0	0
$715$	−4.00000	−0.149592
$716$	−18.0000	−0.672692
$717$	15.0000	0.560185
$718$	8.00000	0.298557
$719$	−26.0000	−0.969636	−0.484818	−	0.874615i	$-0.661114\pi$
$719$	−26.0000	−0.969636	−0.484818	+	0.874615i	$0.661114\pi$
$720$	−4.00000	−0.149071
$721$	0	0
$722$	−18.0000	−0.669891
$723$	0	0
$724$	−7.00000	−0.260153
$725$	−99.0000	−3.67677
$726$	−10.0000	−0.371135
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−8.00000	−0.296093
$731$	30.0000	1.10959
$732$	−15.0000	−0.554416
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	26.0000	0.959678
$735$	0	0
$736$	6.00000	0.221163
$737$	5.00000	0.184177
$738$	0	0
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	32.0000	1.17634
$741$	1.00000	0.0367359
$742$	0	0
$743$	−5.00000	−0.183432	−0.0917161	−	0.995785i	$-0.529235\pi$
$743$	−5.00000	−0.183432	−0.0917161	+	0.995785i	$0.529235\pi$
$744$	−8.00000	−0.293294
$745$	40.0000	1.46549
$746$	−1.00000	−0.0366126
$747$	−8.00000	−0.292705
$748$	−3.00000	−0.109691
$749$	0	0
$750$	−24.0000	−0.876356
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	−11.0000	−0.401129
$753$	−22.0000	−0.801725
$754$	9.00000	0.327761
$755$	−76.0000	−2.76592
$756$	0	0
$757$	−47.0000	−1.70824	−0.854122	−	0.520073i	$-0.825905\pi$
$757$	−47.0000	−1.70824	−0.854122	+	0.520073i	$0.825905\pi$
$758$	−20.0000	−0.726433
$759$	−6.00000	−0.217786
$760$	4.00000	0.145095
$761$	−16.0000	−0.580000	−0.290000	−	0.957027i	$-0.593655\pi$
$761$	−16.0000	−0.580000	−0.290000	+	0.957027i	$0.593655\pi$
$762$	2.00000	0.0724524
$763$	0	0
$764$	−12.0000	−0.434145
$765$	−12.0000	−0.433861
$766$	16.0000	0.578103
$767$	5.00000	0.180540
$768$	1.00000	0.0360844
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	−12.0000	−0.431889
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	10.0000	0.359443
$775$	−88.0000	−3.16105
$776$	10.0000	0.358979
$777$	0	0
$778$	−23.0000	−0.824590
$779$	0	0
$780$	4.00000	0.143223
$781$	15.0000	0.536742
$782$	18.0000	0.643679
$783$	−9.00000	−0.321634
$784$	0	0
$785$	−60.0000	−2.14149
$786$	−2.00000	−0.0713376
$787$	−15.0000	−0.534692	−0.267346	−	0.963601i	$-0.586147\pi$
$787$	−15.0000	−0.534692	−0.267346	+	0.963601i	$0.586147\pi$
$788$	18.0000	0.641223
$789$	24.0000	0.854423
$790$	8.00000	0.284627
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	15.0000	0.532666
$794$	−30.0000	−1.06466
$795$	−4.00000	−0.141865
$796$	−10.0000	−0.354441
$797$	54.0000	1.91278	0.956389	−	0.292096i	$-0.0943526\pi$
$797$	54.0000	1.91278	0.956389	+	0.292096i	$0.0943526\pi$
$798$	0	0
$799$	−33.0000	−1.16746
$800$	11.0000	0.388909
$801$	0	0
$802$	30.0000	1.05934
$803$	−2.00000	−0.0705785
$804$	−5.00000	−0.176336
$805$	0	0
$806$	8.00000	0.281788
$807$	−9.00000	−0.316815
$808$	−14.0000	−0.492518
$809$	−3.00000	−0.105474	−0.0527372	−	0.998608i	$-0.516795\pi$
$809$	−3.00000	−0.105474	−0.0527372	+	0.998608i	$0.516795\pi$
$810$	−4.00000	−0.140546
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	−1.00000	−0.0350715
$814$	8.00000	0.280400
$815$	−52.0000	−1.82148
$816$	3.00000	0.105021
$817$	−10.0000	−0.349856
$818$	−8.00000	−0.279713
$819$	0	0
$820$	0	0
$821$	40.0000	1.39601	0.698005	−	0.716093i	$-0.254071\pi$
$821$	40.0000	1.39601	0.698005	+	0.716093i	$0.254071\pi$
$822$	−8.00000	−0.279032
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	6.00000	0.209020
$825$	−11.0000	−0.382971
$826$	0	0
$827$	17.0000	0.591148	0.295574	−	0.955320i	$-0.404489\pi$
$827$	17.0000	0.591148	0.295574	+	0.955320i	$0.404489\pi$
$828$	6.00000	0.208514
$829$	43.0000	1.49345	0.746726	−	0.665132i	$-0.231625\pi$
$829$	43.0000	1.49345	0.746726	+	0.665132i	$0.231625\pi$
$830$	32.0000	1.11074
$831$	−1.00000	−0.0346896
$832$	−1.00000	−0.0346688
$833$	0	0
$834$	12.0000	0.415526
$835$	−12.0000	−0.415277
$836$	1.00000	0.0345857
$837$	−8.00000	−0.276520
$838$	−14.0000	−0.483622
$839$	5.00000	0.172619	0.0863096	−	0.996268i	$-0.472493\pi$
$839$	5.00000	0.172619	0.0863096	+	0.996268i	$0.472493\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	−30.0000	−1.03387
$843$	−32.0000	−1.10214
$844$	18.0000	0.619586
$845$	−4.00000	−0.137604
$846$	−11.0000	−0.378188
$847$	0	0
$848$	1.00000	0.0343401
$849$	−14.0000	−0.480479
$850$	33.0000	1.13189
$851$	−48.0000	−1.64542
$852$	−15.0000	−0.513892
$853$	2.00000	0.0684787	0.0342393	−	0.999414i	$-0.489099\pi$
$853$	2.00000	0.0684787	0.0342393	+	0.999414i	$0.489099\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	−8.00000	−0.273434
$857$	9.00000	0.307434	0.153717	−	0.988115i	$-0.450876\pi$
$857$	9.00000	0.307434	0.153717	+	0.988115i	$0.450876\pi$
$858$	1.00000	0.0341394
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	−40.0000	−1.36399
$861$	0	0
$862$	−8.00000	−0.272481
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	1.00000	0.0340207
$865$	−12.0000	−0.408012
$866$	−11.0000	−0.373795
$867$	−8.00000	−0.271694
$868$	0	0
$869$	2.00000	0.0678454
$870$	36.0000	1.22051
$871$	5.00000	0.169419
$872$	0	0
$873$	10.0000	0.338449
$874$	−6.00000	−0.202953
$875$	0	0
$876$	2.00000	0.0675737
$877$	24.0000	0.810422	0.405211	−	0.914223i	$-0.367198\pi$
$877$	24.0000	0.810422	0.405211	+	0.914223i	$0.367198\pi$
$878$	−2.00000	−0.0674967
$879$	14.0000	0.472208
$880$	4.00000	0.134840
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	24.0000	0.807664	0.403832	−	0.914833i	$-0.367678\pi$
$883$	24.0000	0.807664	0.403832	+	0.914833i	$0.367678\pi$
$884$	−3.00000	−0.100901
$885$	20.0000	0.672293
$886$	0	0
$887$	−6.00000	−0.201460	−0.100730	−	0.994914i	$-0.532118\pi$
$887$	−6.00000	−0.201460	−0.100730	+	0.994914i	$0.532118\pi$
$888$	−8.00000	−0.268462
$889$	0	0
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	19.0000	0.636167
$893$	11.0000	0.368101
$894$	−10.0000	−0.334450
$895$	72.0000	2.40669
$896$	0	0
$897$	−6.00000	−0.200334
$898$	−4.00000	−0.133482
$899$	72.0000	2.40133
$900$	11.0000	0.366667
$901$	3.00000	0.0999445
$902$	0	0
$903$	0	0
$904$	5.00000	0.166298
$905$	28.0000	0.930751
$906$	19.0000	0.631233
$907$	18.0000	0.597680	0.298840	−	0.954303i	$-0.403400\pi$
$907$	18.0000	0.597680	0.298840	+	0.954303i	$0.403400\pi$
$908$	−12.0000	−0.398234
$909$	−14.0000	−0.464351
$910$	0	0
$911$	2.00000	0.0662630	0.0331315	−	0.999451i	$-0.489452\pi$
$911$	2.00000	0.0662630	0.0331315	+	0.999451i	$0.489452\pi$
$912$	−1.00000	−0.0331133
$913$	8.00000	0.264761
$914$	34.0000	1.12462
$915$	60.0000	1.98354
$916$	18.0000	0.594737
$917$	0	0
$918$	3.00000	0.0990148
$919$	−22.0000	−0.725713	−0.362857	−	0.931845i	$-0.618198\pi$
$919$	−22.0000	−0.725713	−0.362857	+	0.931845i	$0.618198\pi$
$920$	−24.0000	−0.791257
$921$	27.0000	0.889680
$922$	40.0000	1.31733
$923$	15.0000	0.493731
$924$	0	0
$925$	−88.0000	−2.89342
$926$	0	0
$927$	6.00000	0.197066
$928$	−9.00000	−0.295439
$929$	10.0000	0.328089	0.164045	−	0.986453i	$-0.447546\pi$
$929$	10.0000	0.328089	0.164045	+	0.986453i	$0.447546\pi$
$930$	32.0000	1.04932
$931$	0	0
$932$	−15.0000	−0.491341
$933$	8.00000	0.261908
$934$	30.0000	0.981630
$935$	12.0000	0.392442
$936$	−1.00000	−0.0326860
$937$	−49.0000	−1.60076	−0.800380	−	0.599493i	$-0.795369\pi$
$937$	−49.0000	−1.60076	−0.800380	+	0.599493i	$0.795369\pi$
$938$	0	0
$939$	−14.0000	−0.456873
$940$	44.0000	1.43512
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	15.0000	0.488726
$943$	0	0
$944$	−5.00000	−0.162736
$945$	0	0
$946$	−10.0000	−0.325128
$947$	−33.0000	−1.07236	−0.536178	−	0.844105i	$-0.680132\pi$
$947$	−33.0000	−1.07236	−0.536178	+	0.844105i	$0.680132\pi$
$948$	−2.00000	−0.0649570
$949$	−2.00000	−0.0649227
$950$	−11.0000	−0.356887
$951$	−6.00000	−0.194563
$952$	0	0
$953$	13.0000	0.421111	0.210556	−	0.977582i	$-0.432473\pi$
$953$	13.0000	0.421111	0.210556	+	0.977582i	$0.432473\pi$
$954$	1.00000	0.0323762
$955$	48.0000	1.55324
$956$	15.0000	0.485135
$957$	9.00000	0.290929
$958$	5.00000	0.161543
$959$	0	0
$960$	−4.00000	−0.129099
$961$	33.0000	1.06452
$962$	8.00000	0.257930
$963$	−8.00000	−0.257796
$964$	0	0
$965$	48.0000	1.54517
$966$	0	0
$967$	13.0000	0.418052	0.209026	−	0.977910i	$-0.432971\pi$
$967$	13.0000	0.418052	0.209026	+	0.977910i	$0.432971\pi$
$968$	−10.0000	−0.321412
$969$	−3.00000	−0.0963739
$970$	−40.0000	−1.28432
$971$	40.0000	1.28366	0.641831	−	0.766846i	$-0.278175\pi$
$971$	40.0000	1.28366	0.641831	+	0.766846i	$0.278175\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−1.00000	−0.0320421
$975$	−11.0000	−0.352282
$976$	−15.0000	−0.480138
$977$	−6.00000	−0.191957	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000	−0.191957	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	13.0000	0.415694
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−38.0000	−1.21263
$983$	−5.00000	−0.159475	−0.0797376	−	0.996816i	$-0.525408\pi$
$983$	−5.00000	−0.159475	−0.0797376	+	0.996816i	$0.525408\pi$
$984$	0	0
$985$	−72.0000	−2.29411
$986$	−27.0000	−0.859855
$987$	0	0
$988$	1.00000	0.0318142
$989$	60.0000	1.90789
$990$	4.00000	0.127128
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	−8.00000	−0.254000
$993$	28.0000	0.888553
$994$	0	0
$995$	40.0000	1.26809
$996$	−8.00000	−0.253490
$997$	−31.0000	−0.981780	−0.490890	−	0.871222i	$-0.663328\pi$
$997$	−31.0000	−0.981780	−0.490890	+	0.871222i	$0.663328\pi$
$998$	8.00000	0.253236
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3822.2.a.ba.1.1		1
7.2	even	3		546.2.i.c.235.1	yes	2
7.4	even	3		546.2.i.c.79.1	✓	2
7.6	odd	2		3822.2.a.z.1.1		1
21.2	odd	6		1638.2.j.f.235.1		2
21.11	odd	6		1638.2.j.f.1171.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.i.c.79.1	✓	2	7.4	even	3
546.2.i.c.235.1	yes	2	7.2	even	3
1638.2.j.f.235.1		2	21.2	odd	6
1638.2.j.f.1171.1		2	21.11	odd	6
3822.2.a.z.1.1		1	7.6	odd	2
3822.2.a.ba.1.1		1	1.1	even	1	trivial

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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