LMFDB - Embedded newform 6762.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6762,2,Mod(1,6762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6762, 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("6762.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6762.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:	$53.9948418468$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6762.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +2.00000 q^{11} -1.00000 q^{12} -7.00000 q^{13} -3.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -6.00000 q^{19} +3.00000 q^{20} -2.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} +7.00000 q^{26} -1.00000 q^{27} +7.00000 q^{29} +3.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} -5.00000 q^{37} +6.00000 q^{38} +7.00000 q^{39} -3.00000 q^{40} +5.00000 q^{41} -5.00000 q^{43} +2.00000 q^{44} +3.00000 q^{45} -1.00000 q^{46} +3.00000 q^{47} -1.00000 q^{48} -4.00000 q^{50} -4.00000 q^{51} -7.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +6.00000 q^{55} +6.00000 q^{57} -7.00000 q^{58} -8.00000 q^{59} -3.00000 q^{60} +12.0000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -21.0000 q^{65} +2.00000 q^{66} +8.00000 q^{67} +4.00000 q^{68} -1.00000 q^{69} -12.0000 q^{71} -1.00000 q^{72} -14.0000 q^{73} +5.00000 q^{74} -4.00000 q^{75} -6.00000 q^{76} -7.00000 q^{78} -6.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} -5.00000 q^{82} +12.0000 q^{83} +12.0000 q^{85} +5.00000 q^{86} -7.00000 q^{87} -2.00000 q^{88} -8.00000 q^{89} -3.00000 q^{90} +1.00000 q^{92} +8.00000 q^{93} -3.00000 q^{94} -18.0000 q^{95} +1.00000 q^{96} -3.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−3.00000	−0.948683
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	−1.00000	−0.288675
$13$	−7.00000	−1.94145	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−7.00000	−1.94145	−0.970725	+	0.240192i	$0.922790\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−1.00000	−0.235702
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	−2.00000	−0.426401
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	4.00000	0.800000
$26$	7.00000	1.37281
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.00000	1.29987	0.649934	−	0.759991i	$-0.274797\pi$
$29$	7.00000	1.29987	0.649934	+	0.759991i	$0.274797\pi$
$30$	3.00000	0.547723
$31$	−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
$33$	−2.00000	−0.348155
$34$	−4.00000	−0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	6.00000	0.973329
$39$	7.00000	1.12090
$40$	−3.00000	−0.474342
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	2.00000	0.301511
$45$	3.00000	0.447214
$46$	−1.00000	−0.147442
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−4.00000	−0.565685
$51$	−4.00000	−0.560112
$52$	−7.00000	−0.970725
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	1.00000	0.136083
$55$	6.00000	0.809040
$56$	0	0
$57$	6.00000	0.794719
$58$	−7.00000	−0.919145
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	−3.00000	−0.387298
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	−21.0000	−2.60473
$66$	2.00000	0.246183
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	4.00000	0.485071
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	−1.00000	−0.117851
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	5.00000	0.581238
$75$	−4.00000	−0.461880
$76$	−6.00000	−0.688247
$77$	0	0
$78$	−7.00000	−0.792594
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	−5.00000	−0.552158
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	5.00000	0.539164
$87$	−7.00000	−0.750479
$88$	−2.00000	−0.213201
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	−3.00000	−0.316228
$91$	0	0
$92$	1.00000	0.104257
$93$	8.00000	0.829561
$94$	−3.00000	−0.309426
$95$	−18.0000	−1.84676
$96$	1.00000	0.102062
$97$	−3.00000	−0.304604	−0.152302	−	0.988334i	$-0.548669\pi$
$97$	−3.00000	−0.304604	−0.152302	+	0.988334i	$0.548669\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	4.00000	0.400000
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	4.00000	0.396059
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	7.00000	0.686406
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	−6.00000	−0.572078
$111$	5.00000	0.474579
$112$	0	0
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	−6.00000	−0.561951
$115$	3.00000	0.279751
$116$	7.00000	0.649934
$117$	−7.00000	−0.647150
$118$	8.00000	0.736460
$119$	0	0
$120$	3.00000	0.273861
$121$	−7.00000	−0.636364
$122$	−12.0000	−1.08643
$123$	−5.00000	−0.450835
$124$	−8.00000	−0.718421
$125$	−3.00000	−0.268328
$126$	0	0
$127$	13.0000	1.15356	0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000	1.15356	0.576782	+	0.816898i	$0.304308\pi$
$128$	−1.00000	−0.0883883
$129$	5.00000	0.440225
$130$	21.0000	1.84182
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−2.00000	−0.174078
$133$	0	0
$134$	−8.00000	−0.691095
$135$	−3.00000	−0.258199
$136$	−4.00000	−0.342997
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	1.00000	0.0851257
$139$	7.00000	0.593732	0.296866	−	0.954919i	$-0.404058\pi$
$139$	7.00000	0.593732	0.296866	+	0.954919i	$0.404058\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	12.0000	1.00702
$143$	−14.0000	−1.17074
$144$	1.00000	0.0833333
$145$	21.0000	1.74396
$146$	14.0000	1.15865
$147$	0	0
$148$	−5.00000	−0.410997
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	4.00000	0.326599
$151$	−19.0000	−1.54620	−0.773099	−	0.634285i	$-0.781294\pi$
$151$	−19.0000	−1.54620	−0.773099	+	0.634285i	$0.781294\pi$
$152$	6.00000	0.486664
$153$	4.00000	0.323381
$154$	0	0
$155$	−24.0000	−1.92773
$156$	7.00000	0.560449
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	6.00000	0.477334
$159$	−6.00000	−0.475831
$160$	−3.00000	−0.237171
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	5.00000	0.390434
$165$	−6.00000	−0.467099
$166$	−12.0000	−0.931381
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	36.0000	2.76923
$170$	−12.0000	−0.920358
$171$	−6.00000	−0.458831
$172$	−5.00000	−0.381246
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	7.00000	0.530669
$175$	0	0
$176$	2.00000	0.150756
$177$	8.00000	0.601317
$178$	8.00000	0.599625
$179$	9.00000	0.672692	0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000	0.672692	0.336346	+	0.941739i	$0.390809\pi$
$180$	3.00000	0.223607
$181$	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
$183$	−12.0000	−0.887066
$184$	−1.00000	−0.0737210
$185$	−15.0000	−1.10282
$186$	−8.00000	−0.586588
$187$	8.00000	0.585018
$188$	3.00000	0.218797
$189$	0	0
$190$	18.0000	1.30586
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	−1.00000	−0.0721688
$193$	−21.0000	−1.51161	−0.755807	−	0.654795i	$-0.772755\pi$
$193$	−21.0000	−1.51161	−0.755807	+	0.654795i	$0.772755\pi$
$194$	3.00000	0.215387
$195$	21.0000	1.50384
$196$	0	0
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	−2.00000	−0.142134
$199$	1.00000	0.0708881	0.0354441	−	0.999372i	$-0.488715\pi$
$199$	1.00000	0.0708881	0.0354441	+	0.999372i	$0.488715\pi$
$200$	−4.00000	−0.282843
$201$	−8.00000	−0.564276
$202$	4.00000	0.281439
$203$	0	0
$204$	−4.00000	−0.280056
$205$	15.0000	1.04765
$206$	13.0000	0.905753
$207$	1.00000	0.0695048
$208$	−7.00000	−0.485363
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−2.00000	−0.137686	−0.0688428	−	0.997628i	$-0.521931\pi$
$211$	−2.00000	−0.137686	−0.0688428	+	0.997628i	$0.521931\pi$
$212$	6.00000	0.412082
$213$	12.0000	0.822226
$214$	12.0000	0.820303
$215$	−15.0000	−1.02299
$216$	1.00000	0.0680414
$217$	0	0
$218$	1.00000	0.0677285
$219$	14.0000	0.946032
$220$	6.00000	0.404520
$221$	−28.0000	−1.88348
$222$	−5.00000	−0.335578
$223$	−26.0000	−1.74109	−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000	−1.74109	−0.870544	+	0.492090i	$0.836233\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	−1.00000	−0.0665190
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	6.00000	0.397360
$229$	16.0000	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$230$	−3.00000	−0.197814
$231$	0	0
$232$	−7.00000	−0.459573
$233$	28.0000	1.83434	0.917170	−	0.398495i	$-0.130467\pi$
$233$	28.0000	1.83434	0.917170	+	0.398495i	$0.130467\pi$
$234$	7.00000	0.457604
$235$	9.00000	0.587095
$236$	−8.00000	−0.520756
$237$	6.00000	0.389742
$238$	0	0
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	−3.00000	−0.193649
$241$	−5.00000	−0.322078	−0.161039	−	0.986948i	$-0.551485\pi$
$241$	−5.00000	−0.322078	−0.161039	+	0.986948i	$0.551485\pi$
$242$	7.00000	0.449977
$243$	−1.00000	−0.0641500
$244$	12.0000	0.768221
$245$	0	0
$246$	5.00000	0.318788
$247$	42.0000	2.67240
$248$	8.00000	0.508001
$249$	−12.0000	−0.760469
$250$	3.00000	0.189737
$251$	−11.0000	−0.694314	−0.347157	−	0.937807i	$-0.612853\pi$
$251$	−11.0000	−0.694314	−0.347157	+	0.937807i	$0.612853\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	−13.0000	−0.815693
$255$	−12.0000	−0.751469
$256$	1.00000	0.0625000
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	−5.00000	−0.311286
$259$	0	0
$260$	−21.0000	−1.30236
$261$	7.00000	0.433289
$262$	0	0
$263$	−1.00000	−0.0616626	−0.0308313	−	0.999525i	$-0.509815\pi$
$263$	−1.00000	−0.0616626	−0.0308313	+	0.999525i	$0.509815\pi$
$264$	2.00000	0.123091
$265$	18.0000	1.10573
$266$	0	0
$267$	8.00000	0.489592
$268$	8.00000	0.488678
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	3.00000	0.182574
$271$	18.0000	1.09342	0.546711	−	0.837321i	$-0.315880\pi$
$271$	18.0000	1.09342	0.546711	+	0.837321i	$0.315880\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	3.00000	0.181237
$275$	8.00000	0.482418
$276$	−1.00000	−0.0601929
$277$	6.00000	0.360505	0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000	0.360505	0.180253	+	0.983620i	$0.442309\pi$
$278$	−7.00000	−0.419832
$279$	−8.00000	−0.478947
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	3.00000	0.178647
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	−12.0000	−0.712069
$285$	18.0000	1.06623
$286$	14.0000	0.827837
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	−21.0000	−1.23316
$291$	3.00000	0.175863
$292$	−14.0000	−0.819288
$293$	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
$295$	−24.0000	−1.39733
$296$	5.00000	0.290619
$297$	−2.00000	−0.116052
$298$	14.0000	0.810998
$299$	−7.00000	−0.404820
$300$	−4.00000	−0.230940
$301$	0	0
$302$	19.0000	1.09333
$303$	4.00000	0.229794
$304$	−6.00000	−0.344124
$305$	36.0000	2.06135
$306$	−4.00000	−0.228665
$307$	−11.0000	−0.627803	−0.313902	−	0.949456i	$-0.601636\pi$
$307$	−11.0000	−0.627803	−0.313902	+	0.949456i	$0.601636\pi$
$308$	0	0
$309$	13.0000	0.739544
$310$	24.0000	1.36311
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	−7.00000	−0.396297
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	22.0000	1.24153
$315$	0	0
$316$	−6.00000	−0.337526
$317$	17.0000	0.954815	0.477408	−	0.878682i	$-0.341577\pi$
$317$	17.0000	0.954815	0.477408	+	0.878682i	$0.341577\pi$
$318$	6.00000	0.336463
$319$	14.0000	0.783850
$320$	3.00000	0.167705
$321$	12.0000	0.669775
$322$	0	0
$323$	−24.0000	−1.33540
$324$	1.00000	0.0555556
$325$	−28.0000	−1.55316
$326$	8.00000	0.443079
$327$	1.00000	0.0553001
$328$	−5.00000	−0.276079
$329$	0	0
$330$	6.00000	0.330289
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	12.0000	0.658586
$333$	−5.00000	−0.273998
$334$	8.00000	0.437741
$335$	24.0000	1.31126
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−36.0000	−1.95814
$339$	−1.00000	−0.0543125
$340$	12.0000	0.650791
$341$	−16.0000	−0.866449
$342$	6.00000	0.324443
$343$	0	0
$344$	5.00000	0.269582
$345$	−3.00000	−0.161515
$346$	6.00000	0.322562
$347$	29.0000	1.55680	0.778401	−	0.627768i	$-0.216031\pi$
$347$	29.0000	1.55680	0.778401	+	0.627768i	$0.216031\pi$
$348$	−7.00000	−0.375239
$349$	−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
$351$	7.00000	0.373632
$352$	−2.00000	−0.106600
$353$	15.0000	0.798369	0.399185	−	0.916871i	$-0.369293\pi$
$353$	15.0000	0.798369	0.399185	+	0.916871i	$0.369293\pi$
$354$	−8.00000	−0.425195
$355$	−36.0000	−1.91068
$356$	−8.00000	−0.423999
$357$	0	0
$358$	−9.00000	−0.475665
$359$	−19.0000	−1.00278	−0.501391	−	0.865221i	$-0.667178\pi$
$359$	−19.0000	−1.00278	−0.501391	+	0.865221i	$0.667178\pi$
$360$	−3.00000	−0.158114
$361$	17.0000	0.894737
$362$	−22.0000	−1.15629
$363$	7.00000	0.367405
$364$	0	0
$365$	−42.0000	−2.19838
$366$	12.0000	0.627250
$367$	1.00000	0.0521996	0.0260998	−	0.999659i	$-0.491691\pi$
$367$	1.00000	0.0521996	0.0260998	+	0.999659i	$0.491691\pi$
$368$	1.00000	0.0521286
$369$	5.00000	0.260290
$370$	15.0000	0.779813
$371$	0	0
$372$	8.00000	0.414781
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	−8.00000	−0.413670
$375$	3.00000	0.154919
$376$	−3.00000	−0.154713
$377$	−49.0000	−2.52363
$378$	0	0
$379$	25.0000	1.28416	0.642082	−	0.766636i	$-0.278071\pi$
$379$	25.0000	1.28416	0.642082	+	0.766636i	$0.278071\pi$
$380$	−18.0000	−0.923381
$381$	−13.0000	−0.666010
$382$	−20.0000	−1.02329
$383$	−30.0000	−1.53293	−0.766464	−	0.642287i	$-0.777986\pi$
$383$	−30.0000	−1.53293	−0.766464	+	0.642287i	$0.777986\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	21.0000	1.06887
$387$	−5.00000	−0.254164
$388$	−3.00000	−0.152302
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	−21.0000	−1.06338
$391$	4.00000	0.202289
$392$	0	0
$393$	0	0
$394$	3.00000	0.151138
$395$	−18.0000	−0.905678
$396$	2.00000	0.100504
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	−1.00000	−0.0501255
$399$	0	0
$400$	4.00000	0.200000
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	8.00000	0.399004
$403$	56.0000	2.78956
$404$	−4.00000	−0.199007
$405$	3.00000	0.149071
$406$	0	0
$407$	−10.0000	−0.495682
$408$	4.00000	0.198030
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	−15.0000	−0.740797
$411$	3.00000	0.147979
$412$	−13.0000	−0.640464
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	36.0000	1.76717
$416$	7.00000	0.343203
$417$	−7.00000	−0.342791
$418$	12.0000	0.586939
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	15.0000	0.731055	0.365528	−	0.930800i	$-0.380889\pi$
$421$	15.0000	0.731055	0.365528	+	0.930800i	$0.380889\pi$
$422$	2.00000	0.0973585
$423$	3.00000	0.145865
$424$	−6.00000	−0.291386
$425$	16.0000	0.776114
$426$	−12.0000	−0.581402
$427$	0	0
$428$	−12.0000	−0.580042
$429$	14.0000	0.675926
$430$	15.0000	0.723364
$431$	−25.0000	−1.20421	−0.602104	−	0.798418i	$-0.705671\pi$
$431$	−25.0000	−1.20421	−0.602104	+	0.798418i	$0.705671\pi$
$432$	−1.00000	−0.0481125
$433$	−9.00000	−0.432512	−0.216256	−	0.976337i	$-0.569385\pi$
$433$	−9.00000	−0.432512	−0.216256	+	0.976337i	$0.569385\pi$
$434$	0	0
$435$	−21.0000	−1.00687
$436$	−1.00000	−0.0478913
$437$	−6.00000	−0.287019
$438$	−14.0000	−0.668946
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	−6.00000	−0.286039
$441$	0	0
$442$	28.0000	1.33182
$443$	−37.0000	−1.75792	−0.878962	−	0.476893i	$-0.841763\pi$
$443$	−37.0000	−1.75792	−0.878962	+	0.476893i	$0.841763\pi$
$444$	5.00000	0.237289
$445$	−24.0000	−1.13771
$446$	26.0000	1.23114
$447$	14.0000	0.662177
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	−4.00000	−0.188562
$451$	10.0000	0.470882
$452$	1.00000	0.0470360
$453$	19.0000	0.892698
$454$	3.00000	0.140797
$455$	0	0
$456$	−6.00000	−0.280976
$457$	−16.0000	−0.748448	−0.374224	−	0.927338i	$-0.622091\pi$
$457$	−16.0000	−0.748448	−0.374224	+	0.927338i	$0.622091\pi$
$458$	−16.0000	−0.747631
$459$	−4.00000	−0.186704
$460$	3.00000	0.139876
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	−17.0000	−0.790057	−0.395029	−	0.918669i	$-0.629265\pi$
$463$	−17.0000	−0.790057	−0.395029	+	0.918669i	$0.629265\pi$
$464$	7.00000	0.324967
$465$	24.0000	1.11297
$466$	−28.0000	−1.29707
$467$	−9.00000	−0.416470	−0.208235	−	0.978079i	$-0.566772\pi$
$467$	−9.00000	−0.416470	−0.208235	+	0.978079i	$0.566772\pi$
$468$	−7.00000	−0.323575
$469$	0	0
$470$	−9.00000	−0.415139
$471$	22.0000	1.01371
$472$	8.00000	0.368230
$473$	−10.0000	−0.459800
$474$	−6.00000	−0.275589
$475$	−24.0000	−1.10120
$476$	0	0
$477$	6.00000	0.274721
$478$	−20.0000	−0.914779
$479$	34.0000	1.55350	0.776750	−	0.629809i	$-0.216867\pi$
$479$	34.0000	1.55350	0.776750	+	0.629809i	$0.216867\pi$
$480$	3.00000	0.136931
$481$	35.0000	1.59586
$482$	5.00000	0.227744
$483$	0	0
$484$	−7.00000	−0.318182
$485$	−9.00000	−0.408669
$486$	1.00000	0.0453609
$487$	−17.0000	−0.770344	−0.385172	−	0.922845i	$-0.625858\pi$
$487$	−17.0000	−0.770344	−0.385172	+	0.922845i	$0.625858\pi$
$488$	−12.0000	−0.543214
$489$	8.00000	0.361773
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−5.00000	−0.225417
$493$	28.0000	1.26106
$494$	−42.0000	−1.88967
$495$	6.00000	0.269680
$496$	−8.00000	−0.359211
$497$	0	0
$498$	12.0000	0.537733
$499$	−38.0000	−1.70111	−0.850557	−	0.525883i	$-0.823735\pi$
$499$	−38.0000	−1.70111	−0.850557	+	0.525883i	$0.823735\pi$
$500$	−3.00000	−0.134164
$501$	8.00000	0.357414
$502$	11.0000	0.490954
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	−2.00000	−0.0889108
$507$	−36.0000	−1.59882
$508$	13.0000	0.576782
$509$	−26.0000	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$510$	12.0000	0.531369
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	6.00000	0.264906
$514$	14.0000	0.617514
$515$	−39.0000	−1.71855
$516$	5.00000	0.220113
$517$	6.00000	0.263880
$518$	0	0
$519$	6.00000	0.263371
$520$	21.0000	0.920911
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	−7.00000	−0.306382
$523$	−10.0000	−0.437269	−0.218635	−	0.975807i	$-0.570160\pi$
$523$	−10.0000	−0.437269	−0.218635	+	0.975807i	$0.570160\pi$
$524$	0	0
$525$	0	0
$526$	1.00000	0.0436021
$527$	−32.0000	−1.39394
$528$	−2.00000	−0.0870388
$529$	1.00000	0.0434783
$530$	−18.0000	−0.781870
$531$	−8.00000	−0.347170
$532$	0	0
$533$	−35.0000	−1.51602
$534$	−8.00000	−0.346194
$535$	−36.0000	−1.55642
$536$	−8.00000	−0.345547
$537$	−9.00000	−0.388379
$538$	12.0000	0.517357
$539$	0	0
$540$	−3.00000	−0.129099
$541$	32.0000	1.37579	0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000	1.37579	0.687894	+	0.725811i	$0.258536\pi$
$542$	−18.0000	−0.773166
$543$	−22.0000	−0.944110
$544$	−4.00000	−0.171499
$545$	−3.00000	−0.128506
$546$	0	0
$547$	14.0000	0.598597	0.299298	−	0.954160i	$-0.403247\pi$
$547$	14.0000	0.598597	0.299298	+	0.954160i	$0.403247\pi$
$548$	−3.00000	−0.128154
$549$	12.0000	0.512148
$550$	−8.00000	−0.341121
$551$	−42.0000	−1.78926
$552$	1.00000	0.0425628
$553$	0	0
$554$	−6.00000	−0.254916
$555$	15.0000	0.636715
$556$	7.00000	0.296866
$557$	24.0000	1.01691	0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000	1.01691	0.508456	+	0.861088i	$0.330216\pi$
$558$	8.00000	0.338667
$559$	35.0000	1.48034
$560$	0	0
$561$	−8.00000	−0.337760
$562$	−27.0000	−1.13893
$563$	−15.0000	−0.632175	−0.316087	−	0.948730i	$-0.602369\pi$
$563$	−15.0000	−0.632175	−0.316087	+	0.948730i	$0.602369\pi$
$564$	−3.00000	−0.126323
$565$	3.00000	0.126211
$566$	28.0000	1.17693
$567$	0	0
$568$	12.0000	0.503509
$569$	35.0000	1.46728	0.733638	−	0.679540i	$-0.237821\pi$
$569$	35.0000	1.46728	0.733638	+	0.679540i	$0.237821\pi$
$570$	−18.0000	−0.753937
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	−14.0000	−0.585369
$573$	−20.0000	−0.835512
$574$	0	0
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	1.00000	0.0415945
$579$	21.0000	0.872730
$580$	21.0000	0.871978
$581$	0	0
$582$	−3.00000	−0.124354
$583$	12.0000	0.496989
$584$	14.0000	0.579324
$585$	−21.0000	−0.868243
$586$	−14.0000	−0.578335
$587$	−6.00000	−0.247647	−0.123823	−	0.992304i	$-0.539516\pi$
$587$	−6.00000	−0.247647	−0.123823	+	0.992304i	$0.539516\pi$
$588$	0	0
$589$	48.0000	1.97781
$590$	24.0000	0.988064
$591$	3.00000	0.123404
$592$	−5.00000	−0.205499
$593$	−27.0000	−1.10876	−0.554379	−	0.832265i	$-0.687044\pi$
$593$	−27.0000	−1.10876	−0.554379	+	0.832265i	$0.687044\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	−14.0000	−0.573462
$597$	−1.00000	−0.0409273
$598$	7.00000	0.286251
$599$	26.0000	1.06233	0.531166	−	0.847268i	$-0.321754\pi$
$599$	26.0000	1.06233	0.531166	+	0.847268i	$0.321754\pi$
$600$	4.00000	0.163299
$601$	4.00000	0.163163	0.0815817	−	0.996667i	$-0.474003\pi$
$601$	4.00000	0.163163	0.0815817	+	0.996667i	$0.474003\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	−19.0000	−0.773099
$605$	−21.0000	−0.853771
$606$	−4.00000	−0.162489
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	6.00000	0.243332
$609$	0	0
$610$	−36.0000	−1.45760
$611$	−21.0000	−0.849569
$612$	4.00000	0.161690
$613$	9.00000	0.363507	0.181753	−	0.983344i	$-0.441823\pi$
$613$	9.00000	0.363507	0.181753	+	0.983344i	$0.441823\pi$
$614$	11.0000	0.443924
$615$	−15.0000	−0.604858
$616$	0	0
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	−13.0000	−0.522937
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	−24.0000	−0.963863
$621$	−1.00000	−0.0401286
$622$	−12.0000	−0.481156
$623$	0	0
$624$	7.00000	0.280224
$625$	−29.0000	−1.16000
$626$	14.0000	0.559553
$627$	12.0000	0.479234
$628$	−22.0000	−0.877896
$629$	−20.0000	−0.797452
$630$	0	0
$631$	18.0000	0.716569	0.358284	−	0.933613i	$-0.383362\pi$
$631$	18.0000	0.716569	0.358284	+	0.933613i	$0.383362\pi$
$632$	6.00000	0.238667
$633$	2.00000	0.0794929
$634$	−17.0000	−0.675156
$635$	39.0000	1.54767
$636$	−6.00000	−0.237915
$637$	0	0
$638$	−14.0000	−0.554265
$639$	−12.0000	−0.474713
$640$	−3.00000	−0.118585
$641$	35.0000	1.38242	0.691208	−	0.722655i	$-0.257079\pi$
$641$	35.0000	1.38242	0.691208	+	0.722655i	$0.257079\pi$
$642$	−12.0000	−0.473602
$643$	−26.0000	−1.02534	−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000	−1.02534	−0.512670	+	0.858586i	$0.671344\pi$
$644$	0	0
$645$	15.0000	0.590624
$646$	24.0000	0.944267
$647$	28.0000	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$648$	−1.00000	−0.0392837
$649$	−16.0000	−0.628055
$650$	28.0000	1.09825
$651$	0	0
$652$	−8.00000	−0.313304
$653$	−21.0000	−0.821794	−0.410897	−	0.911682i	$-0.634784\pi$
$653$	−21.0000	−0.821794	−0.410897	+	0.911682i	$0.634784\pi$
$654$	−1.00000	−0.0391031
$655$	0	0
$656$	5.00000	0.195217
$657$	−14.0000	−0.546192
$658$	0	0
$659$	−34.0000	−1.32445	−0.662226	−	0.749304i	$-0.730388\pi$
$659$	−34.0000	−1.32445	−0.662226	+	0.749304i	$0.730388\pi$
$660$	−6.00000	−0.233550
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	20.0000	0.777322
$663$	28.0000	1.08743
$664$	−12.0000	−0.465690
$665$	0	0
$666$	5.00000	0.193746
$667$	7.00000	0.271041
$668$	−8.00000	−0.309529
$669$	26.0000	1.00522
$670$	−24.0000	−0.927201
$671$	24.0000	0.926510
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	−14.0000	−0.539260
$675$	−4.00000	−0.153960
$676$	36.0000	1.38462
$677$	26.0000	0.999261	0.499631	−	0.866239i	$-0.333469\pi$
$677$	26.0000	0.999261	0.499631	+	0.866239i	$0.333469\pi$
$678$	1.00000	0.0384048
$679$	0	0
$680$	−12.0000	−0.460179
$681$	3.00000	0.114960
$682$	16.0000	0.612672
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	−6.00000	−0.229416
$685$	−9.00000	−0.343872
$686$	0	0
$687$	−16.0000	−0.610438
$688$	−5.00000	−0.190623
$689$	−42.0000	−1.60007
$690$	3.00000	0.114208
$691$	−39.0000	−1.48363	−0.741815	−	0.670605i	$-0.766035\pi$
$691$	−39.0000	−1.48363	−0.741815	+	0.670605i	$0.766035\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	−29.0000	−1.10082
$695$	21.0000	0.796575
$696$	7.00000	0.265334
$697$	20.0000	0.757554
$698$	10.0000	0.378506
$699$	−28.0000	−1.05906
$700$	0	0
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	−7.00000	−0.264198
$703$	30.0000	1.13147
$704$	2.00000	0.0753778
$705$	−9.00000	−0.338960
$706$	−15.0000	−0.564532
$707$	0	0
$708$	8.00000	0.300658
$709$	46.0000	1.72757	0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000	1.72757	0.863783	+	0.503864i	$0.168089\pi$
$710$	36.0000	1.35106
$711$	−6.00000	−0.225018
$712$	8.00000	0.299813
$713$	−8.00000	−0.299602
$714$	0	0
$715$	−42.0000	−1.57071
$716$	9.00000	0.336346
$717$	−20.0000	−0.746914
$718$	19.0000	0.709074
$719$	47.0000	1.75280	0.876402	−	0.481580i	$-0.159937\pi$
$719$	47.0000	1.75280	0.876402	+	0.481580i	$0.159937\pi$
$720$	3.00000	0.111803
$721$	0	0
$722$	−17.0000	−0.632674
$723$	5.00000	0.185952
$724$	22.0000	0.817624
$725$	28.0000	1.03989
$726$	−7.00000	−0.259794
$727$	−12.0000	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	42.0000	1.55449
$731$	−20.0000	−0.739727
$732$	−12.0000	−0.443533
$733$	−46.0000	−1.69905	−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000	−1.69905	−0.849524	+	0.527549i	$0.823111\pi$
$734$	−1.00000	−0.0369107
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	16.0000	0.589368
$738$	−5.00000	−0.184053
$739$	−44.0000	−1.61857	−0.809283	−	0.587419i	$-0.800144\pi$
$739$	−44.0000	−1.61857	−0.809283	+	0.587419i	$0.800144\pi$
$740$	−15.0000	−0.551411
$741$	−42.0000	−1.54291
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	−8.00000	−0.293294
$745$	−42.0000	−1.53876
$746$	22.0000	0.805477
$747$	12.0000	0.439057
$748$	8.00000	0.292509
$749$	0	0
$750$	−3.00000	−0.109545
$751$	2.00000	0.0729810	0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000	0.0729810	0.0364905	+	0.999334i	$0.488382\pi$
$752$	3.00000	0.109399
$753$	11.0000	0.400862
$754$	49.0000	1.78447
$755$	−57.0000	−2.07444
$756$	0	0
$757$	−46.0000	−1.67190	−0.835949	−	0.548807i	$-0.815082\pi$
$757$	−46.0000	−1.67190	−0.835949	+	0.548807i	$0.815082\pi$
$758$	−25.0000	−0.908041
$759$	−2.00000	−0.0725954
$760$	18.0000	0.652929
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	13.0000	0.470940
$763$	0	0
$764$	20.0000	0.723575
$765$	12.0000	0.433861
$766$	30.0000	1.08394
$767$	56.0000	2.02204
$768$	−1.00000	−0.0360844
$769$	49.0000	1.76699	0.883493	−	0.468445i	$-0.155186\pi$
$769$	49.0000	1.76699	0.883493	+	0.468445i	$0.155186\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	−21.0000	−0.755807
$773$	−33.0000	−1.18693	−0.593464	−	0.804861i	$-0.702240\pi$
$773$	−33.0000	−1.18693	−0.593464	+	0.804861i	$0.702240\pi$
$774$	5.00000	0.179721
$775$	−32.0000	−1.14947
$776$	3.00000	0.107694
$777$	0	0
$778$	30.0000	1.07555
$779$	−30.0000	−1.07486
$780$	21.0000	0.751921
$781$	−24.0000	−0.858788
$782$	−4.00000	−0.143040
$783$	−7.00000	−0.250160
$784$	0	0
$785$	−66.0000	−2.35564
$786$	0	0
$787$	−22.0000	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$788$	−3.00000	−0.106871
$789$	1.00000	0.0356009
$790$	18.0000	0.640411
$791$	0	0
$792$	−2.00000	−0.0710669
$793$	−84.0000	−2.98293
$794$	22.0000	0.780751
$795$	−18.0000	−0.638394
$796$	1.00000	0.0354441
$797$	35.0000	1.23976	0.619882	−	0.784695i	$-0.287181\pi$
$797$	35.0000	1.23976	0.619882	+	0.784695i	$0.287181\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	−4.00000	−0.141421
$801$	−8.00000	−0.282666
$802$	2.00000	0.0706225
$803$	−28.0000	−0.988099
$804$	−8.00000	−0.282138
$805$	0	0
$806$	−56.0000	−1.97252
$807$	12.0000	0.422420
$808$	4.00000	0.140720
$809$	46.0000	1.61727	0.808637	−	0.588308i	$-0.200206\pi$
$809$	46.0000	1.61727	0.808637	+	0.588308i	$0.200206\pi$
$810$	−3.00000	−0.105409
$811$	−33.0000	−1.15879	−0.579393	−	0.815048i	$-0.696710\pi$
$811$	−33.0000	−1.15879	−0.579393	+	0.815048i	$0.696710\pi$
$812$	0	0
$813$	−18.0000	−0.631288
$814$	10.0000	0.350500
$815$	−24.0000	−0.840683
$816$	−4.00000	−0.140028
$817$	30.0000	1.04957
$818$	30.0000	1.04893
$819$	0	0
$820$	15.0000	0.523823
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	−3.00000	−0.104637
$823$	−3.00000	−0.104573	−0.0522867	−	0.998632i	$-0.516651\pi$
$823$	−3.00000	−0.104573	−0.0522867	+	0.998632i	$0.516651\pi$
$824$	13.0000	0.452876
$825$	−8.00000	−0.278524
$826$	0	0
$827$	−42.0000	−1.46048	−0.730242	−	0.683189i	$-0.760592\pi$
$827$	−42.0000	−1.46048	−0.730242	+	0.683189i	$0.760592\pi$
$828$	1.00000	0.0347524
$829$	54.0000	1.87550	0.937749	−	0.347314i	$-0.112906\pi$
$829$	54.0000	1.87550	0.937749	+	0.347314i	$0.112906\pi$
$830$	−36.0000	−1.24958
$831$	−6.00000	−0.208138
$832$	−7.00000	−0.242681
$833$	0	0
$834$	7.00000	0.242390
$835$	−24.0000	−0.830554
$836$	−12.0000	−0.415029
$837$	8.00000	0.276520
$838$	−20.0000	−0.690889
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	−15.0000	−0.516934
$843$	−27.0000	−0.929929
$844$	−2.00000	−0.0688428
$845$	108.000	3.71531
$846$	−3.00000	−0.103142
$847$	0	0
$848$	6.00000	0.206041
$849$	28.0000	0.960958
$850$	−16.0000	−0.548795
$851$	−5.00000	−0.171398
$852$	12.0000	0.411113
$853$	−25.0000	−0.855984	−0.427992	−	0.903783i	$-0.640779\pi$
$853$	−25.0000	−0.855984	−0.427992	+	0.903783i	$0.640779\pi$
$854$	0	0
$855$	−18.0000	−0.615587
$856$	12.0000	0.410152
$857$	5.00000	0.170797	0.0853984	−	0.996347i	$-0.472784\pi$
$857$	5.00000	0.170797	0.0853984	+	0.996347i	$0.472784\pi$
$858$	−14.0000	−0.477952
$859$	−45.0000	−1.53538	−0.767690	−	0.640821i	$-0.778594\pi$
$859$	−45.0000	−1.53538	−0.767690	+	0.640821i	$0.778594\pi$
$860$	−15.0000	−0.511496
$861$	0	0
$862$	25.0000	0.851503
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	1.00000	0.0340207
$865$	−18.0000	−0.612018
$866$	9.00000	0.305832
$867$	1.00000	0.0339618
$868$	0	0
$869$	−12.0000	−0.407072
$870$	21.0000	0.711967
$871$	−56.0000	−1.89749
$872$	1.00000	0.0338643
$873$	−3.00000	−0.101535
$874$	6.00000	0.202953
$875$	0	0
$876$	14.0000	0.473016
$877$	32.0000	1.08056	0.540282	−	0.841484i	$-0.318318\pi$
$877$	32.0000	1.08056	0.540282	+	0.841484i	$0.318318\pi$
$878$	−4.00000	−0.134993
$879$	−14.0000	−0.472208
$880$	6.00000	0.202260
$881$	−54.0000	−1.81931	−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000	−1.81931	−0.909653	+	0.415369i	$0.863653\pi$
$882$	0	0
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	−28.0000	−0.941742
$885$	24.0000	0.806751
$886$	37.0000	1.24304
$887$	20.0000	0.671534	0.335767	−	0.941945i	$-0.391004\pi$
$887$	20.0000	0.671534	0.335767	+	0.941945i	$0.391004\pi$
$888$	−5.00000	−0.167789
$889$	0	0
$890$	24.0000	0.804482
$891$	2.00000	0.0670025
$892$	−26.0000	−0.870544
$893$	−18.0000	−0.602347
$894$	−14.0000	−0.468230
$895$	27.0000	0.902510
$896$	0	0
$897$	7.00000	0.233723
$898$	−14.0000	−0.467186
$899$	−56.0000	−1.86770
$900$	4.00000	0.133333
$901$	24.0000	0.799556
$902$	−10.0000	−0.332964
$903$	0	0
$904$	−1.00000	−0.0332595
$905$	66.0000	2.19391
$906$	−19.0000	−0.631233
$907$	−7.00000	−0.232431	−0.116216	−	0.993224i	$-0.537076\pi$
$907$	−7.00000	−0.232431	−0.116216	+	0.993224i	$0.537076\pi$
$908$	−3.00000	−0.0995585
$909$	−4.00000	−0.132672
$910$	0	0
$911$	33.0000	1.09334	0.546669	−	0.837349i	$-0.315895\pi$
$911$	33.0000	1.09334	0.546669	+	0.837349i	$0.315895\pi$
$912$	6.00000	0.198680
$913$	24.0000	0.794284
$914$	16.0000	0.529233
$915$	−36.0000	−1.19012
$916$	16.0000	0.528655
$917$	0	0
$918$	4.00000	0.132020
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	−3.00000	−0.0989071
$921$	11.0000	0.362462
$922$	12.0000	0.395199
$923$	84.0000	2.76489
$924$	0	0
$925$	−20.0000	−0.657596
$926$	17.0000	0.558655
$927$	−13.0000	−0.426976
$928$	−7.00000	−0.229786
$929$	33.0000	1.08269	0.541347	−	0.840799i	$-0.317914\pi$
$929$	33.0000	1.08269	0.541347	+	0.840799i	$0.317914\pi$
$930$	−24.0000	−0.786991
$931$	0	0
$932$	28.0000	0.917170
$933$	−12.0000	−0.392862
$934$	9.00000	0.294489
$935$	24.0000	0.784884
$936$	7.00000	0.228802
$937$	−11.0000	−0.359354	−0.179677	−	0.983726i	$-0.557505\pi$
$937$	−11.0000	−0.359354	−0.179677	+	0.983726i	$0.557505\pi$
$938$	0	0
$939$	14.0000	0.456873
$940$	9.00000	0.293548
$941$	39.0000	1.27136	0.635682	−	0.771951i	$-0.280719\pi$
$941$	39.0000	1.27136	0.635682	+	0.771951i	$0.280719\pi$
$942$	−22.0000	−0.716799
$943$	5.00000	0.162822
$944$	−8.00000	−0.260378
$945$	0	0
$946$	10.0000	0.325128
$947$	−21.0000	−0.682408	−0.341204	−	0.939989i	$-0.610835\pi$
$947$	−21.0000	−0.682408	−0.341204	+	0.939989i	$0.610835\pi$
$948$	6.00000	0.194871
$949$	98.0000	3.18121
$950$	24.0000	0.778663
$951$	−17.0000	−0.551263
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	−6.00000	−0.194257
$955$	60.0000	1.94155
$956$	20.0000	0.646846
$957$	−14.0000	−0.452556
$958$	−34.0000	−1.09849
$959$	0	0
$960$	−3.00000	−0.0968246
$961$	33.0000	1.06452
$962$	−35.0000	−1.12845
$963$	−12.0000	−0.386695
$964$	−5.00000	−0.161039
$965$	−63.0000	−2.02804
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	7.00000	0.224989
$969$	24.0000	0.770991
$970$	9.00000	0.288973
$971$	4.00000	0.128366	0.0641831	−	0.997938i	$-0.479556\pi$
$971$	4.00000	0.128366	0.0641831	+	0.997938i	$0.479556\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	17.0000	0.544715
$975$	28.0000	0.896718
$976$	12.0000	0.384111
$977$	1.00000	0.0319928	0.0159964	−	0.999872i	$-0.494908\pi$
$977$	1.00000	0.0319928	0.0159964	+	0.999872i	$0.494908\pi$
$978$	−8.00000	−0.255812
$979$	−16.0000	−0.511362
$980$	0	0
$981$	−1.00000	−0.0319275
$982$	12.0000	0.382935
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	5.00000	0.159394
$985$	−9.00000	−0.286764
$986$	−28.0000	−0.891702
$987$	0	0
$988$	42.0000	1.33620
$989$	−5.00000	−0.158991
$990$	−6.00000	−0.190693
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	8.00000	0.254000
$993$	20.0000	0.634681
$994$	0	0
$995$	3.00000	0.0951064
$996$	−12.0000	−0.380235
$997$	50.0000	1.58352	0.791758	−	0.610835i	$-0.209166\pi$
$997$	50.0000	1.58352	0.791758	+	0.610835i	$0.209166\pi$
$998$	38.0000	1.20287
$999$	5.00000	0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6762.2.a.j.1.1	✓	1
7.6	odd	2		6762.2.a.k.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6762.2.a.j.1.1	✓	1	1.1	even	1	trivial
6762.2.a.k.1.1	yes	1	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6762.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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