LMFDB - Embedded newform 8281.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.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:	$66.1241179138$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 637)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8281.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +2.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -4.00000 q^{6} +1.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +2.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -4.00000 q^{6} +1.00000 q^{9} -2.00000 q^{10} +2.00000 q^{11} +4.00000 q^{12} +2.00000 q^{15} -4.00000 q^{16} -6.00000 q^{17} -2.00000 q^{18} -3.00000 q^{19} +2.00000 q^{20} -4.00000 q^{22} +3.00000 q^{23} -4.00000 q^{25} -4.00000 q^{27} +3.00000 q^{29} -4.00000 q^{30} -3.00000 q^{31} +8.00000 q^{32} +4.00000 q^{33} +12.0000 q^{34} +2.00000 q^{36} -6.00000 q^{37} +6.00000 q^{38} +10.0000 q^{41} -1.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} -6.00000 q^{46} +11.0000 q^{47} -8.00000 q^{48} +8.00000 q^{50} -12.0000 q^{51} -9.00000 q^{53} +8.00000 q^{54} +2.00000 q^{55} -6.00000 q^{57} -6.00000 q^{58} -8.00000 q^{59} +4.00000 q^{60} +8.00000 q^{61} +6.00000 q^{62} -8.00000 q^{64} -8.00000 q^{66} +12.0000 q^{67} -12.0000 q^{68} +6.00000 q^{69} +14.0000 q^{71} -9.00000 q^{73} +12.0000 q^{74} -8.00000 q^{75} -6.00000 q^{76} -9.00000 q^{79} -4.00000 q^{80} -11.0000 q^{81} -20.0000 q^{82} +11.0000 q^{83} -6.00000 q^{85} +2.00000 q^{86} +6.00000 q^{87} -5.00000 q^{89} -2.00000 q^{90} +6.00000 q^{92} -6.00000 q^{93} -22.0000 q^{94} -3.00000 q^{95} +16.0000 q^{96} -9.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	2.00000	1.00000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−4.00000	−1.63299
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	4.00000	1.15470
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.00000	0.516398
$16$	−4.00000	−1.00000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	−2.00000	−0.471405
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	−4.00000	−0.852803
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	−4.00000	−0.730297
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	8.00000	1.41421
$33$	4.00000	0.696311
$34$	12.0000	2.05798
$35$	0	0
$36$	2.00000	0.333333
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	4.00000	0.603023
$45$	1.00000	0.149071
$46$	−6.00000	−0.884652
$47$	11.0000	1.60451	0.802257	−	0.596978i	$-0.203632\pi$
$47$	11.0000	1.60451	0.802257	+	0.596978i	$0.203632\pi$
$48$	−8.00000	−1.15470
$49$	0	0
$50$	8.00000	1.13137
$51$	−12.0000	−1.68034
$52$	0	0
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	8.00000	1.08866
$55$	2.00000	0.269680
$56$	0	0
$57$	−6.00000	−0.794719
$58$	−6.00000	−0.787839
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	4.00000	0.516398
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	−8.00000	−0.984732
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−12.0000	−1.45521
$69$	6.00000	0.722315
$70$	0	0
$71$	14.0000	1.66149	0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000	1.66149	0.830747	+	0.556650i	$0.187914\pi$
$72$	0	0
$73$	−9.00000	−1.05337	−0.526685	−	0.850060i	$-0.676565\pi$
$73$	−9.00000	−1.05337	−0.526685	+	0.850060i	$0.676565\pi$
$74$	12.0000	1.39497
$75$	−8.00000	−0.923760
$76$	−6.00000	−0.688247
$77$	0	0
$78$	0	0
$79$	−9.00000	−1.01258	−0.506290	−	0.862364i	$-0.668983\pi$
$79$	−9.00000	−1.01258	−0.506290	+	0.862364i	$0.668983\pi$
$80$	−4.00000	−0.447214
$81$	−11.0000	−1.22222
$82$	−20.0000	−2.20863
$83$	11.0000	1.20741	0.603703	−	0.797209i	$-0.293691\pi$
$83$	11.0000	1.20741	0.603703	+	0.797209i	$0.293691\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	2.00000	0.215666
$87$	6.00000	0.643268
$88$	0	0
$89$	−5.00000	−0.529999	−0.264999	−	0.964249i	$-0.585372\pi$
$89$	−5.00000	−0.529999	−0.264999	+	0.964249i	$0.585372\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	6.00000	0.625543
$93$	−6.00000	−0.622171
$94$	−22.0000	−2.26913
$95$	−3.00000	−0.307794
$96$	16.0000	1.63299
$97$	−9.00000	−0.913812	−0.456906	−	0.889515i	$-0.651042\pi$
$97$	−9.00000	−0.913812	−0.456906	+	0.889515i	$0.651042\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	−8.00000	−0.800000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	24.0000	2.37635
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	18.0000	1.74831
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−8.00000	−0.769800
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	−4.00000	−0.381385
$111$	−12.0000	−1.13899
$112$	0	0
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	12.0000	1.12390
$115$	3.00000	0.279751
$116$	6.00000	0.557086
$117$	0	0
$118$	16.0000	1.47292
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−16.0000	−1.44857
$123$	20.0000	1.80334
$124$	−6.00000	−0.538816
$125$	−9.00000	−0.804984
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	8.00000	0.696311
$133$	0	0
$134$	−24.0000	−2.07328
$135$	−4.00000	−0.344265
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	−12.0000	−1.02151
$139$	18.0000	1.52674	0.763370	−	0.645961i	$-0.223543\pi$
$139$	18.0000	1.52674	0.763370	+	0.645961i	$0.223543\pi$
$140$	0	0
$141$	22.0000	1.85273
$142$	−28.0000	−2.34971
$143$	0	0
$144$	−4.00000	−0.333333
$145$	3.00000	0.249136
$146$	18.0000	1.48969
$147$	0	0
$148$	−12.0000	−0.986394
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	16.0000	1.30639
$151$	−6.00000	−0.488273	−0.244137	−	0.969741i	$-0.578505\pi$
$151$	−6.00000	−0.488273	−0.244137	+	0.969741i	$0.578505\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	−3.00000	−0.240966
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	18.0000	1.43200
$159$	−18.0000	−1.42749
$160$	8.00000	0.632456
$161$	0	0
$162$	22.0000	1.72848
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	20.0000	1.56174
$165$	4.00000	0.311400
$166$	−22.0000	−1.70753
$167$	1.00000	0.0773823	0.0386912	−	0.999251i	$-0.487681\pi$
$167$	1.00000	0.0773823	0.0386912	+	0.999251i	$0.487681\pi$
$168$	0	0
$169$	0	0
$170$	12.0000	0.920358
$171$	−3.00000	−0.229416
$172$	−2.00000	−0.152499
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−12.0000	−0.909718
$175$	0	0
$176$	−8.00000	−0.603023
$177$	−16.0000	−1.20263
$178$	10.0000	0.749532
$179$	−9.00000	−0.672692	−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000	−0.672692	−0.336346	+	0.941739i	$0.609191\pi$
$180$	2.00000	0.149071
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	16.0000	1.18275
$184$	0	0
$185$	−6.00000	−0.441129
$186$	12.0000	0.879883
$187$	−12.0000	−0.877527
$188$	22.0000	1.60451
$189$	0	0
$190$	6.00000	0.435286
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−16.0000	−1.15470
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	18.0000	1.29232
$195$	0	0
$196$	0	0
$197$	4.00000	0.284988	0.142494	−	0.989796i	$-0.454488\pi$
$197$	4.00000	0.284988	0.142494	+	0.989796i	$0.454488\pi$
$198$	−4.00000	−0.284268
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	24.0000	1.69283
$202$	12.0000	0.844317
$203$	0	0
$204$	−24.0000	−1.68034
$205$	10.0000	0.698430
$206$	0	0
$207$	3.00000	0.208514
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−9.00000	−0.619586	−0.309793	−	0.950804i	$-0.600260\pi$
$211$	−9.00000	−0.619586	−0.309793	+	0.950804i	$0.600260\pi$
$212$	−18.0000	−1.23625
$213$	28.0000	1.91853
$214$	24.0000	1.64061
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	0	0
$218$	36.0000	2.43823
$219$	−18.0000	−1.21633
$220$	4.00000	0.269680
$221$	0	0
$222$	24.0000	1.61077
$223$	−21.0000	−1.40626	−0.703132	−	0.711059i	$-0.748216\pi$
$223$	−21.0000	−1.40626	−0.703132	+	0.711059i	$0.748216\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	30.0000	1.99557
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	−12.0000	−0.794719
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	−6.00000	−0.395628
$231$	0	0
$232$	0	0
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	0	0
$235$	11.0000	0.717561
$236$	−16.0000	−1.04151
$237$	−18.0000	−1.16923
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	−8.00000	−0.516398
$241$	−21.0000	−1.35273	−0.676364	−	0.736567i	$-0.736446\pi$
$241$	−21.0000	−1.35273	−0.676364	+	0.736567i	$0.736446\pi$
$242$	14.0000	0.899954
$243$	−10.0000	−0.641500
$244$	16.0000	1.02430
$245$	0	0
$246$	−40.0000	−2.55031
$247$	0	0
$248$	0	0
$249$	22.0000	1.39419
$250$	18.0000	1.13842
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	0	0
$255$	−12.0000	−0.751469
$256$	16.0000	1.00000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	4.00000	0.249029
$259$	0	0
$260$	0	0
$261$	3.00000	0.185695
$262$	24.0000	1.48272
$263$	9.00000	0.554964	0.277482	−	0.960731i	$-0.410500\pi$
$263$	9.00000	0.554964	0.277482	+	0.960731i	$0.410500\pi$
$264$	0	0
$265$	−9.00000	−0.552866
$266$	0	0
$267$	−10.0000	−0.611990
$268$	24.0000	1.46603
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	8.00000	0.486864
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	24.0000	1.45521
$273$	0	0
$274$	4.00000	0.241649
$275$	−8.00000	−0.482418
$276$	12.0000	0.722315
$277$	17.0000	1.02143	0.510716	−	0.859750i	$-0.329381\pi$
$277$	17.0000	1.02143	0.510716	+	0.859750i	$0.329381\pi$
$278$	−36.0000	−2.15914
$279$	−3.00000	−0.179605
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	−44.0000	−2.62016
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	28.0000	1.66149
$285$	−6.00000	−0.355409
$286$	0	0
$287$	0	0
$288$	8.00000	0.471405
$289$	19.0000	1.11765
$290$	−6.00000	−0.352332
$291$	−18.0000	−1.05518
$292$	−18.0000	−1.05337
$293$	1.00000	0.0584206	0.0292103	−	0.999573i	$-0.490701\pi$
$293$	1.00000	0.0584206	0.0292103	+	0.999573i	$0.490701\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	−8.00000	−0.464207
$298$	20.0000	1.15857
$299$	0	0
$300$	−16.0000	−0.923760
$301$	0	0
$302$	12.0000	0.690522
$303$	−12.0000	−0.689382
$304$	12.0000	0.688247
$305$	8.00000	0.458079
$306$	12.0000	0.685994
$307$	−33.0000	−1.88341	−0.941705	−	0.336440i	$-0.890777\pi$
$307$	−33.0000	−1.88341	−0.941705	+	0.336440i	$0.890777\pi$
$308$	0	0
$309$	0	0
$310$	6.00000	0.340777
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	−36.0000	−2.03160
$315$	0	0
$316$	−18.0000	−1.01258
$317$	−20.0000	−1.12331	−0.561656	−	0.827371i	$-0.689836\pi$
$317$	−20.0000	−1.12331	−0.561656	+	0.827371i	$0.689836\pi$
$318$	36.0000	2.01878
$319$	6.00000	0.335936
$320$	−8.00000	−0.447214
$321$	−24.0000	−1.33955
$322$	0	0
$323$	18.0000	1.00155
$324$	−22.0000	−1.22222
$325$	0	0
$326$	−24.0000	−1.32924
$327$	−36.0000	−1.99080
$328$	0	0
$329$	0	0
$330$	−8.00000	−0.440386
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	22.0000	1.20741
$333$	−6.00000	−0.328798
$334$	−2.00000	−0.109435
$335$	12.0000	0.655630
$336$	0	0
$337$	−27.0000	−1.47078	−0.735392	−	0.677642i	$-0.763002\pi$
$337$	−27.0000	−1.47078	−0.735392	+	0.677642i	$0.763002\pi$
$338$	0	0
$339$	−30.0000	−1.62938
$340$	−12.0000	−0.650791
$341$	−6.00000	−0.324918
$342$	6.00000	0.324443
$343$	0	0
$344$	0	0
$345$	6.00000	0.323029
$346$	12.0000	0.645124
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	12.0000	0.643268
$349$	27.0000	1.44528	0.722638	−	0.691226i	$-0.242929\pi$
$349$	27.0000	1.44528	0.722638	+	0.691226i	$0.242929\pi$
$350$	0	0
$351$	0	0
$352$	16.0000	0.852803
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	32.0000	1.70078
$355$	14.0000	0.743043
$356$	−10.0000	−0.529999
$357$	0	0
$358$	18.0000	0.951330
$359$	−22.0000	−1.16112	−0.580558	−	0.814219i	$-0.697165\pi$
$359$	−22.0000	−1.16112	−0.580558	+	0.814219i	$0.697165\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	36.0000	1.89212
$363$	−14.0000	−0.734809
$364$	0	0
$365$	−9.00000	−0.471082
$366$	−32.0000	−1.67267
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	−12.0000	−0.625543
$369$	10.0000	0.520579
$370$	12.0000	0.623850
$371$	0	0
$372$	−12.0000	−0.622171
$373$	−18.0000	−0.932005	−0.466002	−	0.884783i	$-0.654306\pi$
$373$	−18.0000	−0.932005	−0.466002	+	0.884783i	$0.654306\pi$
$374$	24.0000	1.24101
$375$	−18.0000	−0.929516
$376$	0	0
$377$	0	0
$378$	0	0
$379$	18.0000	0.924598	0.462299	−	0.886724i	$-0.347025\pi$
$379$	18.0000	0.924598	0.462299	+	0.886724i	$0.347025\pi$
$380$	−6.00000	−0.307794
$381$	0	0
$382$	0	0
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	0	0
$386$	−24.0000	−1.22157
$387$	−1.00000	−0.0508329
$388$	−18.0000	−0.913812
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	0	0
$393$	−24.0000	−1.21064
$394$	−8.00000	−0.403034
$395$	−9.00000	−0.452839
$396$	4.00000	0.201008
$397$	−21.0000	−1.05396	−0.526980	−	0.849878i	$-0.676676\pi$
$397$	−21.0000	−1.05396	−0.526980	+	0.849878i	$0.676676\pi$
$398$	−4.00000	−0.200502
$399$	0	0
$400$	16.0000	0.800000
$401$	−28.0000	−1.39825	−0.699127	−	0.714998i	$-0.746428\pi$
$401$	−28.0000	−1.39825	−0.699127	+	0.714998i	$0.746428\pi$
$402$	−48.0000	−2.39402
$403$	0	0
$404$	−12.0000	−0.597022
$405$	−11.0000	−0.546594
$406$	0	0
$407$	−12.0000	−0.594818
$408$	0	0
$409$	−9.00000	−0.445021	−0.222511	−	0.974930i	$-0.571425\pi$
$409$	−9.00000	−0.445021	−0.222511	+	0.974930i	$0.571425\pi$
$410$	−20.0000	−0.987730
$411$	−4.00000	−0.197305
$412$	0	0
$413$	0	0
$414$	−6.00000	−0.294884
$415$	11.0000	0.539969
$416$	0	0
$417$	36.0000	1.76293
$418$	12.0000	0.586939
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	18.0000	0.876226
$423$	11.0000	0.534838
$424$	0	0
$425$	24.0000	1.16417
$426$	−56.0000	−2.71321
$427$	0	0
$428$	−24.0000	−1.16008
$429$	0	0
$430$	2.00000	0.0964486
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	16.0000	0.769800
$433$	−20.0000	−0.961139	−0.480569	−	0.876957i	$-0.659570\pi$
$433$	−20.0000	−0.961139	−0.480569	+	0.876957i	$0.659570\pi$
$434$	0	0
$435$	6.00000	0.287678
$436$	−36.0000	−1.72409
$437$	−9.00000	−0.430528
$438$	36.0000	1.72015
$439$	36.0000	1.71819	0.859093	−	0.511819i	$-0.171028\pi$
$439$	36.0000	1.71819	0.859093	+	0.511819i	$0.171028\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	27.0000	1.28281	0.641404	−	0.767203i	$-0.278352\pi$
$443$	27.0000	1.28281	0.641404	+	0.767203i	$0.278352\pi$
$444$	−24.0000	−1.13899
$445$	−5.00000	−0.237023
$446$	42.0000	1.98876
$447$	−20.0000	−0.945968
$448$	0	0
$449$	−4.00000	−0.188772	−0.0943858	−	0.995536i	$-0.530089\pi$
$449$	−4.00000	−0.188772	−0.0943858	+	0.995536i	$0.530089\pi$
$450$	8.00000	0.377124
$451$	20.0000	0.941763
$452$	−30.0000	−1.41108
$453$	−12.0000	−0.563809
$454$	40.0000	1.87729
$455$	0	0
$456$	0	0
$457$	42.0000	1.96468	0.982339	−	0.187112i	$-0.0599128\pi$
$457$	42.0000	1.96468	0.982339	+	0.187112i	$0.0599128\pi$
$458$	12.0000	0.560723
$459$	24.0000	1.12022
$460$	6.00000	0.279751
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	30.0000	1.39422	0.697109	−	0.716965i	$-0.254469\pi$
$463$	30.0000	1.39422	0.697109	+	0.716965i	$0.254469\pi$
$464$	−12.0000	−0.557086
$465$	−6.00000	−0.278243
$466$	−6.00000	−0.277945
$467$	−42.0000	−1.94353	−0.971764	−	0.235954i	$-0.924178\pi$
$467$	−42.0000	−1.94353	−0.971764	+	0.235954i	$0.924178\pi$
$468$	0	0
$469$	0	0
$470$	−22.0000	−1.01478
$471$	36.0000	1.65879
$472$	0	0
$473$	−2.00000	−0.0919601
$474$	36.0000	1.65353
$475$	12.0000	0.550598
$476$	0	0
$477$	−9.00000	−0.412082
$478$	16.0000	0.731823
$479$	29.0000	1.32504	0.662522	−	0.749043i	$-0.269486\pi$
$479$	29.0000	1.32504	0.662522	+	0.749043i	$0.269486\pi$
$480$	16.0000	0.730297
$481$	0	0
$482$	42.0000	1.91305
$483$	0	0
$484$	−14.0000	−0.636364
$485$	−9.00000	−0.408669
$486$	20.0000	0.907218
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	0	0
$489$	24.0000	1.08532
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	40.0000	1.80334
$493$	−18.0000	−0.810679
$494$	0	0
$495$	2.00000	0.0898933
$496$	12.0000	0.538816
$497$	0	0
$498$	−44.0000	−1.97169
$499$	−6.00000	−0.268597	−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000	−0.268597	−0.134298	+	0.990941i	$0.542878\pi$
$500$	−18.0000	−0.804984
$501$	2.00000	0.0893534
$502$	12.0000	0.535586
$503$	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
$505$	−6.00000	−0.266996
$506$	−12.0000	−0.533465
$507$	0	0
$508$	0	0
$509$	37.0000	1.64000	0.819998	−	0.572366i	$-0.193974\pi$
$509$	37.0000	1.64000	0.819998	+	0.572366i	$0.193974\pi$
$510$	24.0000	1.06274
$511$	0	0
$512$	−32.0000	−1.41421
$513$	12.0000	0.529813
$514$	−12.0000	−0.529297
$515$	0	0
$516$	−4.00000	−0.176090
$517$	22.0000	0.967559
$518$	0	0
$519$	−12.0000	−0.526742
$520$	0	0
$521$	24.0000	1.05146	0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000	1.05146	0.525730	+	0.850652i	$0.323792\pi$
$522$	−6.00000	−0.262613
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	−24.0000	−1.04844
$525$	0	0
$526$	−18.0000	−0.784837
$527$	18.0000	0.784092
$528$	−16.0000	−0.696311
$529$	−14.0000	−0.608696
$530$	18.0000	0.781870
$531$	−8.00000	−0.347170
$532$	0	0
$533$	0	0
$534$	20.0000	0.865485
$535$	−12.0000	−0.518805
$536$	0	0
$537$	−18.0000	−0.776757
$538$	48.0000	2.06943
$539$	0	0
$540$	−8.00000	−0.344265
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	0	0
$543$	−36.0000	−1.54491
$544$	−48.0000	−2.05798
$545$	−18.0000	−0.771035
$546$	0	0
$547$	17.0000	0.726868	0.363434	−	0.931620i	$-0.381604\pi$
$547$	17.0000	0.726868	0.363434	+	0.931620i	$0.381604\pi$
$548$	−4.00000	−0.170872
$549$	8.00000	0.341432
$550$	16.0000	0.682242
$551$	−9.00000	−0.383413
$552$	0	0
$553$	0	0
$554$	−34.0000	−1.44452
$555$	−12.0000	−0.509372
$556$	36.0000	1.52674
$557$	−32.0000	−1.35588	−0.677942	−	0.735116i	$-0.737128\pi$
$557$	−32.0000	−1.35588	−0.677942	+	0.735116i	$0.737128\pi$
$558$	6.00000	0.254000
$559$	0	0
$560$	0	0
$561$	−24.0000	−1.01328
$562$	4.00000	0.168730
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	44.0000	1.85273
$565$	−15.0000	−0.631055
$566$	−8.00000	−0.336265
$567$	0	0
$568$	0	0
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	12.0000	0.502625
$571$	−9.00000	−0.376638	−0.188319	−	0.982108i	$-0.560304\pi$
$571$	−9.00000	−0.376638	−0.188319	+	0.982108i	$0.560304\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12.0000	−0.500435
$576$	−8.00000	−0.333333
$577$	−42.0000	−1.74848	−0.874241	−	0.485491i	$-0.838641\pi$
$577$	−42.0000	−1.74848	−0.874241	+	0.485491i	$0.838641\pi$
$578$	−38.0000	−1.58059
$579$	24.0000	0.997406
$580$	6.00000	0.249136
$581$	0	0
$582$	36.0000	1.49225
$583$	−18.0000	−0.745484
$584$	0	0
$585$	0	0
$586$	−2.00000	−0.0826192
$587$	43.0000	1.77480	0.887400	−	0.461000i	$-0.152509\pi$
$587$	43.0000	1.77480	0.887400	+	0.461000i	$0.152509\pi$
$588$	0	0
$589$	9.00000	0.370839
$590$	16.0000	0.658710
$591$	8.00000	0.329076
$592$	24.0000	0.986394
$593$	13.0000	0.533846	0.266923	−	0.963718i	$-0.413993\pi$
$593$	13.0000	0.533846	0.266923	+	0.963718i	$0.413993\pi$
$594$	16.0000	0.656488
$595$	0	0
$596$	−20.0000	−0.819232
$597$	4.00000	0.163709
$598$	0	0
$599$	−21.0000	−0.858037	−0.429018	−	0.903296i	$-0.641140\pi$
$599$	−21.0000	−0.858037	−0.429018	+	0.903296i	$0.641140\pi$
$600$	0	0
$601$	−44.0000	−1.79480	−0.897399	−	0.441221i	$-0.854546\pi$
$601$	−44.0000	−1.79480	−0.897399	+	0.441221i	$0.854546\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	−12.0000	−0.488273
$605$	−7.00000	−0.284590
$606$	24.0000	0.974933
$607$	−18.0000	−0.730597	−0.365299	−	0.930890i	$-0.619033\pi$
$607$	−18.0000	−0.730597	−0.365299	+	0.930890i	$0.619033\pi$
$608$	−24.0000	−0.973329
$609$	0	0
$610$	−16.0000	−0.647821
$611$	0	0
$612$	−12.0000	−0.485071
$613$	−24.0000	−0.969351	−0.484675	−	0.874694i	$-0.661062\pi$
$613$	−24.0000	−0.969351	−0.484675	+	0.874694i	$0.661062\pi$
$614$	66.0000	2.66354
$615$	20.0000	0.806478
$616$	0	0
$617$	−8.00000	−0.322068	−0.161034	−	0.986949i	$-0.551483\pi$
$617$	−8.00000	−0.322068	−0.161034	+	0.986949i	$0.551483\pi$
$618$	0	0
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	−6.00000	−0.240966
$621$	−12.0000	−0.481543
$622$	−60.0000	−2.40578
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	20.0000	0.799361
$627$	−12.0000	−0.479234
$628$	36.0000	1.43656
$629$	36.0000	1.43541
$630$	0	0
$631$	12.0000	0.477712	0.238856	−	0.971055i	$-0.423228\pi$
$631$	12.0000	0.477712	0.238856	+	0.971055i	$0.423228\pi$
$632$	0	0
$633$	−18.0000	−0.715436
$634$	40.0000	1.58860
$635$	0	0
$636$	−36.0000	−1.42749
$637$	0	0
$638$	−12.0000	−0.475085
$639$	14.0000	0.553831
$640$	0	0
$641$	−39.0000	−1.54041	−0.770204	−	0.637798i	$-0.779845\pi$
$641$	−39.0000	−1.54041	−0.770204	+	0.637798i	$0.779845\pi$
$642$	48.0000	1.89441
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	0	0
$645$	−2.00000	−0.0787499
$646$	−36.0000	−1.41640
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	24.0000	0.939913
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	72.0000	2.81542
$655$	−12.0000	−0.468879
$656$	−40.0000	−1.56174
$657$	−9.00000	−0.351123
$658$	0	0
$659$	−39.0000	−1.51922	−0.759612	−	0.650376i	$-0.774611\pi$
$659$	−39.0000	−1.51922	−0.759612	+	0.650376i	$0.774611\pi$
$660$	8.00000	0.311400
$661$	3.00000	0.116686	0.0583432	−	0.998297i	$-0.481418\pi$
$661$	3.00000	0.116686	0.0583432	+	0.998297i	$0.481418\pi$
$662$	24.0000	0.932786
$663$	0	0
$664$	0	0
$665$	0	0
$666$	12.0000	0.464991
$667$	9.00000	0.348481
$668$	2.00000	0.0773823
$669$	−42.0000	−1.62381
$670$	−24.0000	−0.927201
$671$	16.0000	0.617673
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	54.0000	2.08000
$675$	16.0000	0.615840
$676$	0	0
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	60.0000	2.30429
$679$	0	0
$680$	0	0
$681$	−40.0000	−1.53280
$682$	12.0000	0.459504
$683$	40.0000	1.53056	0.765279	−	0.643699i	$-0.222601\pi$
$683$	40.0000	1.53056	0.765279	+	0.643699i	$0.222601\pi$
$684$	−6.00000	−0.229416
$685$	−2.00000	−0.0764161
$686$	0	0
$687$	−12.0000	−0.457829
$688$	4.00000	0.152499
$689$	0	0
$690$	−12.0000	−0.456832
$691$	15.0000	0.570627	0.285313	−	0.958434i	$-0.407902\pi$
$691$	15.0000	0.570627	0.285313	+	0.958434i	$0.407902\pi$
$692$	−12.0000	−0.456172
$693$	0	0
$694$	48.0000	1.82206
$695$	18.0000	0.682779
$696$	0	0
$697$	−60.0000	−2.27266
$698$	−54.0000	−2.04393
$699$	6.00000	0.226941
$700$	0	0
$701$	−15.0000	−0.566542	−0.283271	−	0.959040i	$-0.591420\pi$
$701$	−15.0000	−0.566542	−0.283271	+	0.959040i	$0.591420\pi$
$702$	0	0
$703$	18.0000	0.678883
$704$	−16.0000	−0.603023
$705$	22.0000	0.828568
$706$	−4.00000	−0.150542
$707$	0	0
$708$	−32.0000	−1.20263
$709$	30.0000	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$710$	−28.0000	−1.05082
$711$	−9.00000	−0.337526
$712$	0	0
$713$	−9.00000	−0.337053
$714$	0	0
$715$	0	0
$716$	−18.0000	−0.672692
$717$	−16.0000	−0.597531
$718$	44.0000	1.64207
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	−4.00000	−0.149071
$721$	0	0
$722$	20.0000	0.744323
$723$	−42.0000	−1.56200
$724$	−36.0000	−1.33793
$725$	−12.0000	−0.445669
$726$	28.0000	1.03918
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	18.0000	0.666210
$731$	6.00000	0.221918
$732$	32.0000	1.18275
$733$	15.0000	0.554038	0.277019	−	0.960864i	$-0.410654\pi$
$733$	15.0000	0.554038	0.277019	+	0.960864i	$0.410654\pi$
$734$	−36.0000	−1.32878
$735$	0	0
$736$	24.0000	0.884652
$737$	24.0000	0.884051
$738$	−20.0000	−0.736210
$739$	−30.0000	−1.10357	−0.551784	−	0.833987i	$-0.686053\pi$
$739$	−30.0000	−1.10357	−0.551784	+	0.833987i	$0.686053\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	0	0
$743$	−44.0000	−1.61420	−0.807102	−	0.590412i	$-0.798965\pi$
$743$	−44.0000	−1.61420	−0.807102	+	0.590412i	$0.798965\pi$
$744$	0	0
$745$	−10.0000	−0.366372
$746$	36.0000	1.31805
$747$	11.0000	0.402469
$748$	−24.0000	−0.877527
$749$	0	0
$750$	36.0000	1.31453
$751$	−13.0000	−0.474377	−0.237188	−	0.971464i	$-0.576226\pi$
$751$	−13.0000	−0.474377	−0.237188	+	0.971464i	$0.576226\pi$
$752$	−44.0000	−1.60451
$753$	−12.0000	−0.437304
$754$	0	0
$755$	−6.00000	−0.218362
$756$	0	0
$757$	9.00000	0.327111	0.163555	−	0.986534i	$-0.447704\pi$
$757$	9.00000	0.327111	0.163555	+	0.986534i	$0.447704\pi$
$758$	−36.0000	−1.30758
$759$	12.0000	0.435572
$760$	0	0
$761$	−19.0000	−0.688749	−0.344375	−	0.938832i	$-0.611909\pi$
$761$	−19.0000	−0.688749	−0.344375	+	0.938832i	$0.611909\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−6.00000	−0.216930
$766$	16.0000	0.578103
$767$	0	0
$768$	32.0000	1.15470
$769$	9.00000	0.324548	0.162274	−	0.986746i	$-0.448117\pi$
$769$	9.00000	0.324548	0.162274	+	0.986746i	$0.448117\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	24.0000	0.863779
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	2.00000	0.0718885
$775$	12.0000	0.431053
$776$	0	0
$777$	0	0
$778$	−60.0000	−2.15110
$779$	−30.0000	−1.07486
$780$	0	0
$781$	28.0000	1.00192
$782$	36.0000	1.28736
$783$	−12.0000	−0.428845
$784$	0	0
$785$	18.0000	0.642448
$786$	48.0000	1.71210
$787$	21.0000	0.748569	0.374285	−	0.927314i	$-0.377888\pi$
$787$	21.0000	0.748569	0.374285	+	0.927314i	$0.377888\pi$
$788$	8.00000	0.284988
$789$	18.0000	0.640817
$790$	18.0000	0.640411
$791$	0	0
$792$	0	0
$793$	0	0
$794$	42.0000	1.49052
$795$	−18.0000	−0.638394
$796$	4.00000	0.141776
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	−66.0000	−2.33491
$800$	−32.0000	−1.13137
$801$	−5.00000	−0.176666
$802$	56.0000	1.97743
$803$	−18.0000	−0.635206
$804$	48.0000	1.69283
$805$	0	0
$806$	0	0
$807$	−48.0000	−1.68968
$808$	0	0
$809$	−15.0000	−0.527372	−0.263686	−	0.964609i	$-0.584938\pi$
$809$	−15.0000	−0.527372	−0.263686	+	0.964609i	$0.584938\pi$
$810$	22.0000	0.773001
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	0	0
$814$	24.0000	0.841200
$815$	12.0000	0.420342
$816$	48.0000	1.68034
$817$	3.00000	0.104957
$818$	18.0000	0.629355
$819$	0	0
$820$	20.0000	0.698430
$821$	−16.0000	−0.558404	−0.279202	−	0.960232i	$-0.590070\pi$
$821$	−16.0000	−0.558404	−0.279202	+	0.960232i	$0.590070\pi$
$822$	8.00000	0.279032
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	0	0
$825$	−16.0000	−0.557048
$826$	0	0
$827$	4.00000	0.139094	0.0695468	−	0.997579i	$-0.477845\pi$
$827$	4.00000	0.139094	0.0695468	+	0.997579i	$0.477845\pi$
$828$	6.00000	0.208514
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	−22.0000	−0.763631
$831$	34.0000	1.17945
$832$	0	0
$833$	0	0
$834$	−72.0000	−2.49316
$835$	1.00000	0.0346064
$836$	−12.0000	−0.415029
$837$	12.0000	0.414781
$838$	12.0000	0.414533
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−60.0000	−2.06774
$843$	−4.00000	−0.137767
$844$	−18.0000	−0.619586
$845$	0	0
$846$	−22.0000	−0.756376
$847$	0	0
$848$	36.0000	1.23625
$849$	8.00000	0.274559
$850$	−48.0000	−1.64639
$851$	−18.0000	−0.617032
$852$	56.0000	1.91853
$853$	45.0000	1.54077	0.770385	−	0.637579i	$-0.220064\pi$
$853$	45.0000	1.54077	0.770385	+	0.637579i	$0.220064\pi$
$854$	0	0
$855$	−3.00000	−0.102598
$856$	0	0
$857$	−36.0000	−1.22974	−0.614868	−	0.788630i	$-0.710791\pi$
$857$	−36.0000	−1.22974	−0.614868	+	0.788630i	$0.710791\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	−2.00000	−0.0681994
$861$	0	0
$862$	8.00000	0.272481
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	−32.0000	−1.08866
$865$	−6.00000	−0.204006
$866$	40.0000	1.35926
$867$	38.0000	1.29055
$868$	0	0
$869$	−18.0000	−0.610608
$870$	−12.0000	−0.406838
$871$	0	0
$872$	0	0
$873$	−9.00000	−0.304604
$874$	18.0000	0.608859
$875$	0	0
$876$	−36.0000	−1.21633
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	−72.0000	−2.42988
$879$	2.00000	0.0674583
$880$	−8.00000	−0.269680
$881$	12.0000	0.404290	0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000	0.404290	0.202145	+	0.979356i	$0.435209\pi$
$882$	0	0
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	0	0
$885$	−16.0000	−0.537834
$886$	−54.0000	−1.81417
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	0	0
$889$	0	0
$890$	10.0000	0.335201
$891$	−22.0000	−0.737028
$892$	−42.0000	−1.40626
$893$	−33.0000	−1.10430
$894$	40.0000	1.33780
$895$	−9.00000	−0.300837
$896$	0	0
$897$	0	0
$898$	8.00000	0.266963
$899$	−9.00000	−0.300167
$900$	−8.00000	−0.266667
$901$	54.0000	1.79900
$902$	−40.0000	−1.33185
$903$	0	0
$904$	0	0
$905$	−18.0000	−0.598340
$906$	24.0000	0.797347
$907$	−9.00000	−0.298840	−0.149420	−	0.988774i	$-0.547741\pi$
$907$	−9.00000	−0.298840	−0.149420	+	0.988774i	$0.547741\pi$
$908$	−40.0000	−1.32745
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−27.0000	−0.894550	−0.447275	−	0.894397i	$-0.647605\pi$
$911$	−27.0000	−0.894550	−0.447275	+	0.894397i	$0.647605\pi$
$912$	24.0000	0.794719
$913$	22.0000	0.728094
$914$	−84.0000	−2.77847
$915$	16.0000	0.528944
$916$	−12.0000	−0.396491
$917$	0	0
$918$	−48.0000	−1.58424
$919$	36.0000	1.18753	0.593765	−	0.804638i	$-0.297641\pi$
$919$	36.0000	1.18753	0.593765	+	0.804638i	$0.297641\pi$
$920$	0	0
$921$	−66.0000	−2.17477
$922$	−4.00000	−0.131733
$923$	0	0
$924$	0	0
$925$	24.0000	0.789115
$926$	−60.0000	−1.97172
$927$	0	0
$928$	24.0000	0.787839
$929$	−7.00000	−0.229663	−0.114831	−	0.993385i	$-0.536633\pi$
$929$	−7.00000	−0.229663	−0.114831	+	0.993385i	$0.536633\pi$
$930$	12.0000	0.393496
$931$	0	0
$932$	6.00000	0.196537
$933$	60.0000	1.96431
$934$	84.0000	2.74856
$935$	−12.0000	−0.392442
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	−20.0000	−0.652675
$940$	22.0000	0.717561
$941$	−1.00000	−0.0325991	−0.0162995	−	0.999867i	$-0.505189\pi$
$941$	−1.00000	−0.0325991	−0.0162995	+	0.999867i	$0.505189\pi$
$942$	−72.0000	−2.34589
$943$	30.0000	0.976934
$944$	32.0000	1.04151
$945$	0	0
$946$	4.00000	0.130051
$947$	16.0000	0.519930	0.259965	−	0.965618i	$-0.416289\pi$
$947$	16.0000	0.519930	0.259965	+	0.965618i	$0.416289\pi$
$948$	−36.0000	−1.16923
$949$	0	0
$950$	−24.0000	−0.778663
$951$	−40.0000	−1.29709
$952$	0	0
$953$	−3.00000	−0.0971795	−0.0485898	−	0.998819i	$-0.515473\pi$
$953$	−3.00000	−0.0971795	−0.0485898	+	0.998819i	$0.515473\pi$
$954$	18.0000	0.582772
$955$	0	0
$956$	−16.0000	−0.517477
$957$	12.0000	0.387905
$958$	−58.0000	−1.87389
$959$	0	0
$960$	−16.0000	−0.516398
$961$	−22.0000	−0.709677
$962$	0	0
$963$	−12.0000	−0.386695
$964$	−42.0000	−1.35273
$965$	12.0000	0.386294
$966$	0	0
$967$	54.0000	1.73652	0.868261	−	0.496107i	$-0.165238\pi$
$967$	54.0000	1.73652	0.868261	+	0.496107i	$0.165238\pi$
$968$	0	0
$969$	36.0000	1.15649
$970$	18.0000	0.577945
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	−20.0000	−0.641500
$973$	0	0
$974$	24.0000	0.769010
$975$	0	0
$976$	−32.0000	−1.02430
$977$	10.0000	0.319928	0.159964	−	0.987123i	$-0.448862\pi$
$977$	10.0000	0.319928	0.159964	+	0.987123i	$0.448862\pi$
$978$	−48.0000	−1.53487
$979$	−10.0000	−0.319601
$980$	0	0
$981$	−18.0000	−0.574696
$982$	24.0000	0.765871
$983$	29.0000	0.924956	0.462478	−	0.886631i	$-0.346960\pi$
$983$	29.0000	0.924956	0.462478	+	0.886631i	$0.346960\pi$
$984$	0	0
$985$	4.00000	0.127451
$986$	36.0000	1.14647
$987$	0	0
$988$	0	0
$989$	−3.00000	−0.0953945
$990$	−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	$0.848912\pi$
$992$	−24.0000	−0.762001
$993$	−24.0000	−0.761617
$994$	0	0
$995$	2.00000	0.0634043
$996$	44.0000	1.39419
$997$	−36.0000	−1.14013	−0.570066	−	0.821599i	$-0.693082\pi$
$997$	−36.0000	−1.14013	−0.570066	+	0.821599i	$0.693082\pi$
$998$	12.0000	0.379853
$999$	24.0000	0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.b.1.1		1
7.6	odd	2		8281.2.a.a.1.1		1
13.5	odd	4		637.2.c.c.246.2	yes	2
13.8	odd	4		637.2.c.c.246.1	yes	2
13.12	even	2		8281.2.a.m.1.1		1
91.5	even	12		637.2.r.c.116.2		4
91.18	odd	12		637.2.r.a.324.1		4
91.31	even	12		637.2.r.c.324.1		4
91.34	even	4		637.2.c.a.246.1	✓	2
91.44	odd	12		637.2.r.a.116.2		4
91.47	even	12		637.2.r.c.116.1		4
91.60	odd	12		637.2.r.a.324.2		4
91.73	even	12		637.2.r.c.324.2		4
91.83	even	4		637.2.c.a.246.2	yes	2
91.86	odd	12		637.2.r.a.116.1		4
91.90	odd	2		8281.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.c.a.246.1	✓	2	91.34	even	4
637.2.c.a.246.2	yes	2	91.83	even	4
637.2.c.c.246.1	yes	2	13.8	odd	4
637.2.c.c.246.2	yes	2	13.5	odd	4
637.2.r.a.116.1		4	91.86	odd	12
637.2.r.a.116.2		4	91.44	odd	12
637.2.r.a.324.1		4	91.18	odd	12
637.2.r.a.324.2		4	91.60	odd	12
637.2.r.c.116.1		4	91.47	even	12
637.2.r.c.116.2		4	91.5	even	12
637.2.r.c.324.1		4	91.31	even	12
637.2.r.c.324.2		4	91.73	even	12
8281.2.a.a.1.1		1	7.6	odd	2
8281.2.a.b.1.1		1	1.1	even	1	trivial
8281.2.a.k.1.1		1	91.90	odd	2
8281.2.a.m.1.1		1	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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