LMFDB - Embedded newform 722.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [722,2,Mod(1,722)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(722, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("722.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 722 = 2 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	722.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:	$5.76519902594$
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	$\chi$	$=$	722.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +3.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +3.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} -2.00000 q^{10} -2.00000 q^{11} -3.00000 q^{12} +3.00000 q^{13} +3.00000 q^{14} -6.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -6.00000 q^{18} +2.00000 q^{20} +9.00000 q^{21} +2.00000 q^{22} +5.00000 q^{23} +3.00000 q^{24} -1.00000 q^{25} -3.00000 q^{26} -9.00000 q^{27} -3.00000 q^{28} +3.00000 q^{29} +6.00000 q^{30} +6.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} +1.00000 q^{34} -6.00000 q^{35} +6.00000 q^{36} -6.00000 q^{37} -9.00000 q^{39} -2.00000 q^{40} -12.0000 q^{41} -9.00000 q^{42} -10.0000 q^{43} -2.00000 q^{44} +12.0000 q^{45} -5.00000 q^{46} -8.00000 q^{47} -3.00000 q^{48} +2.00000 q^{49} +1.00000 q^{50} +3.00000 q^{51} +3.00000 q^{52} +3.00000 q^{53} +9.00000 q^{54} -4.00000 q^{55} +3.00000 q^{56} -3.00000 q^{58} -3.00000 q^{59} -6.00000 q^{60} -6.00000 q^{62} -18.0000 q^{63} +1.00000 q^{64} +6.00000 q^{65} -6.00000 q^{66} -15.0000 q^{67} -1.00000 q^{68} -15.0000 q^{69} +6.00000 q^{70} -6.00000 q^{72} -11.0000 q^{73} +6.00000 q^{74} +3.00000 q^{75} +6.00000 q^{77} +9.00000 q^{78} +12.0000 q^{79} +2.00000 q^{80} +9.00000 q^{81} +12.0000 q^{82} +2.00000 q^{83} +9.00000 q^{84} -2.00000 q^{85} +10.0000 q^{86} -9.00000 q^{87} +2.00000 q^{88} -6.00000 q^{89} -12.0000 q^{90} -9.00000 q^{91} +5.00000 q^{92} -18.0000 q^{93} +8.00000 q^{94} +3.00000 q^{96} -12.0000 q^{97} -2.00000 q^{98} -12.0000 q^{99} +O(q^{100})\)

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
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	1.00000	0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	3.00000	1.22474
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−1.00000	−0.353553
$9$	6.00000	2.00000
$10$	−2.00000	−0.632456
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−3.00000	−0.866025
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	3.00000	0.801784
$15$	−6.00000	−1.54919
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	−6.00000	−1.41421
$19$	0	0
$19$	0	0
$20$	2.00000	0.447214
$21$	9.00000	1.96396
$22$	2.00000	0.426401
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	3.00000	0.612372
$25$	−1.00000	−0.200000
$26$	−3.00000	−0.588348
$27$	−9.00000	−1.73205
$28$	−3.00000	−0.566947
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	6.00000	1.09545
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−1.00000	−0.176777
$33$	6.00000	1.04447
$34$	1.00000	0.171499
$35$	−6.00000	−1.01419
$36$	6.00000	1.00000
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	−9.00000	−1.44115
$40$	−2.00000	−0.316228
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	−9.00000	−1.38873
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	−2.00000	−0.301511
$45$	12.0000	1.78885
$46$	−5.00000	−0.737210
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	−3.00000	−0.433013
$49$	2.00000	0.285714
$50$	1.00000	0.141421
$51$	3.00000	0.420084
$52$	3.00000	0.416025
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	9.00000	1.22474
$55$	−4.00000	−0.539360
$56$	3.00000	0.400892
$57$	0	0
$58$	−3.00000	−0.393919
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	−6.00000	−0.774597
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	−6.00000	−0.762001
$63$	−18.0000	−2.26779
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	−6.00000	−0.738549
$67$	−15.0000	−1.83254	−0.916271	−	0.400559i	$-0.868816\pi$
$67$	−15.0000	−1.83254	−0.916271	+	0.400559i	$0.868816\pi$
$68$	−1.00000	−0.121268
$69$	−15.0000	−1.80579
$70$	6.00000	0.717137
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−6.00000	−0.707107
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	6.00000	0.697486
$75$	3.00000	0.346410
$76$	0	0
$77$	6.00000	0.683763
$78$	9.00000	1.01905
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	2.00000	0.223607
$81$	9.00000	1.00000
$82$	12.0000	1.32518
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	9.00000	0.981981
$85$	−2.00000	−0.216930
$86$	10.0000	1.07833
$87$	−9.00000	−0.964901
$88$	2.00000	0.213201
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	−12.0000	−1.26491
$91$	−9.00000	−0.943456
$92$	5.00000	0.521286
$93$	−18.0000	−1.86651
$94$	8.00000	0.825137
$95$	0	0
$96$	3.00000	0.306186
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	−2.00000	−0.202031
$99$	−12.0000	−1.20605
$100$	−1.00000	−0.100000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	−3.00000	−0.297044
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	−3.00000	−0.294174
$105$	18.0000	1.75662
$106$	−3.00000	−0.291386
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	−9.00000	−0.866025
$109$	−3.00000	−0.287348	−0.143674	−	0.989625i	$-0.545892\pi$
$109$	−3.00000	−0.287348	−0.143674	+	0.989625i	$0.545892\pi$
$110$	4.00000	0.381385
$111$	18.0000	1.70848
$112$	−3.00000	−0.283473
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	10.0000	0.932505
$116$	3.00000	0.278543
$117$	18.0000	1.66410
$118$	3.00000	0.276172
$119$	3.00000	0.275010
$120$	6.00000	0.547723
$121$	−7.00000	−0.636364
$122$	0	0
$123$	36.0000	3.24601
$124$	6.00000	0.538816
$125$	−12.0000	−1.07331
$126$	18.0000	1.60357
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	−1.00000	−0.0883883
$129$	30.0000	2.64135
$130$	−6.00000	−0.526235
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	6.00000	0.522233
$133$	0	0
$134$	15.0000	1.29580
$135$	−18.0000	−1.54919
$136$	1.00000	0.0857493
$137$	19.0000	1.62328	0.811640	−	0.584158i	$-0.198575\pi$
$137$	19.0000	1.62328	0.811640	+	0.584158i	$0.198575\pi$
$138$	15.0000	1.27688
$139$	6.00000	0.508913	0.254457	−	0.967084i	$-0.418103\pi$
$139$	6.00000	0.508913	0.254457	+	0.967084i	$0.418103\pi$
$140$	−6.00000	−0.507093
$141$	24.0000	2.02116
$142$	0	0
$143$	−6.00000	−0.501745
$144$	6.00000	0.500000
$145$	6.00000	0.498273
$146$	11.0000	0.910366
$147$	−6.00000	−0.494872
$148$	−6.00000	−0.493197
$149$	−8.00000	−0.655386	−0.327693	−	0.944784i	$-0.606271\pi$
$149$	−8.00000	−0.655386	−0.327693	+	0.944784i	$0.606271\pi$
$150$	−3.00000	−0.244949
$151$	18.0000	1.46482	0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000	1.46482	0.732410	+	0.680864i	$0.238396\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	−6.00000	−0.483494
$155$	12.0000	0.963863
$156$	−9.00000	−0.720577
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	−12.0000	−0.954669
$159$	−9.00000	−0.713746
$160$	−2.00000	−0.158114
$161$	−15.0000	−1.18217
$162$	−9.00000	−0.707107
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	−12.0000	−0.937043
$165$	12.0000	0.934199
$166$	−2.00000	−0.155230
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	−9.00000	−0.694365
$169$	−4.00000	−0.307692
$170$	2.00000	0.153393
$171$	0	0
$172$	−10.0000	−0.762493
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	9.00000	0.682288
$175$	3.00000	0.226779
$176$	−2.00000	−0.150756
$177$	9.00000	0.676481
$178$	6.00000	0.449719
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	12.0000	0.894427
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	9.00000	0.667124
$183$	0	0
$184$	−5.00000	−0.368605
$185$	−12.0000	−0.882258
$186$	18.0000	1.31982
$187$	2.00000	0.146254
$188$	−8.00000	−0.583460
$189$	27.0000	1.96396
$190$	0	0
$191$	−11.0000	−0.795932	−0.397966	−	0.917400i	$-0.630284\pi$
$191$	−11.0000	−0.795932	−0.397966	+	0.917400i	$0.630284\pi$
$192$	−3.00000	−0.216506
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	12.0000	0.861550
$195$	−18.0000	−1.28901
$196$	2.00000	0.142857
$197$	4.00000	0.284988	0.142494	−	0.989796i	$-0.454488\pi$
$197$	4.00000	0.284988	0.142494	+	0.989796i	$0.454488\pi$
$198$	12.0000	0.852803
$199$	−7.00000	−0.496217	−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000	−0.496217	−0.248108	+	0.968732i	$0.579809\pi$
$200$	1.00000	0.0707107
$201$	45.0000	3.17406
$202$	−10.0000	−0.703598
$203$	−9.00000	−0.631676
$204$	3.00000	0.210042
$205$	−24.0000	−1.67623
$206$	6.00000	0.418040
$207$	30.0000	2.08514
$208$	3.00000	0.208013
$209$	0	0
$210$	−18.0000	−1.24212
$211$	−3.00000	−0.206529	−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000	−0.206529	−0.103264	+	0.994654i	$0.532929\pi$
$212$	3.00000	0.206041
$213$	0	0
$214$	−3.00000	−0.205076
$215$	−20.0000	−1.36399
$216$	9.00000	0.612372
$217$	−18.0000	−1.22192
$218$	3.00000	0.203186
$219$	33.0000	2.22993
$220$	−4.00000	−0.269680
$221$	−3.00000	−0.201802
$222$	−18.0000	−1.20808
$223$	−18.0000	−1.20537	−0.602685	−	0.797980i	$-0.705902\pi$
$223$	−18.0000	−1.20537	−0.602685	+	0.797980i	$0.705902\pi$
$224$	3.00000	0.200446
$225$	−6.00000	−0.400000
$226$	12.0000	0.798228
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	0	0
$229$	−12.0000	−0.792982	−0.396491	−	0.918039i	$-0.629772\pi$
$229$	−12.0000	−0.792982	−0.396491	+	0.918039i	$0.629772\pi$
$230$	−10.0000	−0.659380
$231$	−18.0000	−1.18431
$232$	−3.00000	−0.196960
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	−18.0000	−1.17670
$235$	−16.0000	−1.04372
$236$	−3.00000	−0.195283
$237$	−36.0000	−2.33845
$238$	−3.00000	−0.194461
$239$	1.00000	0.0646846	0.0323423	−	0.999477i	$-0.489703\pi$
$239$	1.00000	0.0646846	0.0323423	+	0.999477i	$0.489703\pi$
$240$	−6.00000	−0.387298
$241$	−24.0000	−1.54598	−0.772988	−	0.634421i	$-0.781239\pi$
$241$	−24.0000	−1.54598	−0.772988	+	0.634421i	$0.781239\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	0	0
$245$	4.00000	0.255551
$246$	−36.0000	−2.29528
$247$	0	0
$248$	−6.00000	−0.381000
$249$	−6.00000	−0.380235
$250$	12.0000	0.758947
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	−18.0000	−1.13389
$253$	−10.0000	−0.628695
$254$	12.0000	0.752947
$255$	6.00000	0.375735
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	−30.0000	−1.86772
$259$	18.0000	1.11847
$260$	6.00000	0.372104
$261$	18.0000	1.11417
$262$	−14.0000	−0.864923
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	−6.00000	−0.369274
$265$	6.00000	0.368577
$266$	0	0
$267$	18.0000	1.10158
$268$	−15.0000	−0.916271
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	18.0000	1.09545
$271$	−11.0000	−0.668202	−0.334101	−	0.942537i	$-0.608433\pi$
$271$	−11.0000	−0.668202	−0.334101	+	0.942537i	$0.608433\pi$
$272$	−1.00000	−0.0606339
$273$	27.0000	1.63411
$274$	−19.0000	−1.14783
$275$	2.00000	0.120605
$276$	−15.0000	−0.902894
$277$	30.0000	1.80253	0.901263	−	0.433273i	$-0.142641\pi$
$277$	30.0000	1.80253	0.901263	+	0.433273i	$0.142641\pi$
$278$	−6.00000	−0.359856
$279$	36.0000	2.15526
$280$	6.00000	0.358569
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	−24.0000	−1.42918
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	0	0
$285$	0	0
$286$	6.00000	0.354787
$287$	36.0000	2.12501
$288$	−6.00000	−0.353553
$289$	−16.0000	−0.941176
$290$	−6.00000	−0.352332
$291$	36.0000	2.11036
$292$	−11.0000	−0.643726
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	6.00000	0.349927
$295$	−6.00000	−0.349334
$296$	6.00000	0.348743
$297$	18.0000	1.04447
$298$	8.00000	0.463428
$299$	15.0000	0.867472
$300$	3.00000	0.173205
$301$	30.0000	1.72917
$302$	−18.0000	−1.03578
$303$	−30.0000	−1.72345
$304$	0	0
$305$	0	0
$306$	6.00000	0.342997
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	6.00000	0.341882
$309$	18.0000	1.02398
$310$	−12.0000	−0.681554
$311$	−11.0000	−0.623753	−0.311876	−	0.950123i	$-0.600957\pi$
$311$	−11.0000	−0.623753	−0.311876	+	0.950123i	$0.600957\pi$
$312$	9.00000	0.509525
$313$	21.0000	1.18699	0.593495	−	0.804838i	$-0.297748\pi$
$313$	21.0000	1.18699	0.593495	+	0.804838i	$0.297748\pi$
$314$	0	0
$315$	−36.0000	−2.02837
$316$	12.0000	0.675053
$317$	33.0000	1.85346	0.926732	−	0.375722i	$-0.122605\pi$
$317$	33.0000	1.85346	0.926732	+	0.375722i	$0.122605\pi$
$318$	9.00000	0.504695
$319$	−6.00000	−0.335936
$320$	2.00000	0.111803
$321$	−9.00000	−0.502331
$322$	15.0000	0.835917
$323$	0	0
$324$	9.00000	0.500000
$325$	−3.00000	−0.166410
$326$	6.00000	0.332309
$327$	9.00000	0.497701
$328$	12.0000	0.662589
$329$	24.0000	1.32316
$330$	−12.0000	−0.660578
$331$	−9.00000	−0.494685	−0.247342	−	0.968928i	$-0.579557\pi$
$331$	−9.00000	−0.494685	−0.247342	+	0.968928i	$0.579557\pi$
$332$	2.00000	0.109764
$333$	−36.0000	−1.97279
$334$	−12.0000	−0.656611
$335$	−30.0000	−1.63908
$336$	9.00000	0.490990
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	4.00000	0.217571
$339$	36.0000	1.95525
$340$	−2.00000	−0.108465
$341$	−12.0000	−0.649836
$342$	0	0
$343$	15.0000	0.809924
$344$	10.0000	0.539164
$345$	−30.0000	−1.61515
$346$	−18.0000	−0.967686
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	−9.00000	−0.482451
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	−3.00000	−0.160357
$351$	−27.0000	−1.44115
$352$	2.00000	0.106600
$353$	−31.0000	−1.64996	−0.824982	−	0.565159i	$-0.808815\pi$
$353$	−31.0000	−1.64996	−0.824982	+	0.565159i	$0.808815\pi$
$354$	−9.00000	−0.478345
$355$	0	0
$356$	−6.00000	−0.317999
$357$	−9.00000	−0.476331
$358$	−12.0000	−0.634220
$359$	19.0000	1.00278	0.501391	−	0.865221i	$-0.332822\pi$
$359$	19.0000	1.00278	0.501391	+	0.865221i	$0.332822\pi$
$360$	−12.0000	−0.632456
$361$	0	0
$362$	18.0000	0.946059
$363$	21.0000	1.10221
$364$	−9.00000	−0.471728
$365$	−22.0000	−1.15153
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	5.00000	0.260643
$369$	−72.0000	−3.74817
$370$	12.0000	0.623850
$371$	−9.00000	−0.467257
$372$	−18.0000	−0.933257
$373$	21.0000	1.08734	0.543669	−	0.839299i	$-0.317035\pi$
$373$	21.0000	1.08734	0.543669	+	0.839299i	$0.317035\pi$
$374$	−2.00000	−0.103418
$375$	36.0000	1.85903
$376$	8.00000	0.412568
$377$	9.00000	0.463524
$378$	−27.0000	−1.38873
$379$	−3.00000	−0.154100	−0.0770498	−	0.997027i	$-0.524550\pi$
$379$	−3.00000	−0.154100	−0.0770498	+	0.997027i	$0.524550\pi$
$380$	0	0
$381$	36.0000	1.84434
$382$	11.0000	0.562809
$383$	−18.0000	−0.919757	−0.459879	−	0.887982i	$-0.652107\pi$
$383$	−18.0000	−0.919757	−0.459879	+	0.887982i	$0.652107\pi$
$384$	3.00000	0.153093
$385$	12.0000	0.611577
$386$	6.00000	0.305392
$387$	−60.0000	−3.04997
$388$	−12.0000	−0.609208
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	18.0000	0.911465
$391$	−5.00000	−0.252861
$392$	−2.00000	−0.101015
$393$	−42.0000	−2.11862
$394$	−4.00000	−0.201517
$395$	24.0000	1.20757
$396$	−12.0000	−0.603023
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	7.00000	0.350878
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−36.0000	−1.79775	−0.898877	−	0.438201i	$-0.855616\pi$
$401$	−36.0000	−1.79775	−0.898877	+	0.438201i	$0.855616\pi$
$402$	−45.0000	−2.24440
$403$	18.0000	0.896644
$404$	10.0000	0.497519
$405$	18.0000	0.894427
$406$	9.00000	0.446663
$407$	12.0000	0.594818
$408$	−3.00000	−0.148522
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	24.0000	1.18528
$411$	−57.0000	−2.81160
$412$	−6.00000	−0.295599
$413$	9.00000	0.442861
$414$	−30.0000	−1.47442
$415$	4.00000	0.196352
$416$	−3.00000	−0.147087
$417$	−18.0000	−0.881464
$418$	0	0
$419$	−14.0000	−0.683945	−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000	−0.683945	−0.341972	+	0.939710i	$0.611095\pi$
$420$	18.0000	0.878310
$421$	27.0000	1.31590	0.657950	−	0.753062i	$-0.271424\pi$
$421$	27.0000	1.31590	0.657950	+	0.753062i	$0.271424\pi$
$422$	3.00000	0.146038
$423$	−48.0000	−2.33384
$424$	−3.00000	−0.145693
$425$	1.00000	0.0485071
$426$	0	0
$427$	0	0
$428$	3.00000	0.145010
$429$	18.0000	0.869048
$430$	20.0000	0.964486
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	−9.00000	−0.433013
$433$	−30.0000	−1.44171	−0.720854	−	0.693087i	$-0.756250\pi$
$433$	−30.0000	−1.44171	−0.720854	+	0.693087i	$0.756250\pi$
$434$	18.0000	0.864028
$435$	−18.0000	−0.863034
$436$	−3.00000	−0.143674
$437$	0	0
$438$	−33.0000	−1.57680
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	4.00000	0.190693
$441$	12.0000	0.571429
$442$	3.00000	0.142695
$443$	−22.0000	−1.04525	−0.522626	−	0.852562i	$-0.675047\pi$
$443$	−22.0000	−1.04525	−0.522626	+	0.852562i	$0.675047\pi$
$444$	18.0000	0.854242
$445$	−12.0000	−0.568855
$446$	18.0000	0.852325
$447$	24.0000	1.13516
$448$	−3.00000	−0.141737
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	6.00000	0.282843
$451$	24.0000	1.13012
$452$	−12.0000	−0.564433
$453$	−54.0000	−2.53714
$454$	3.00000	0.140797
$455$	−18.0000	−0.843853
$456$	0	0
$457$	1.00000	0.0467780	0.0233890	−	0.999726i	$-0.492554\pi$
$457$	1.00000	0.0467780	0.0233890	+	0.999726i	$0.492554\pi$
$458$	12.0000	0.560723
$459$	9.00000	0.420084
$460$	10.0000	0.466252
$461$	4.00000	0.186299	0.0931493	−	0.995652i	$-0.470307\pi$
$461$	4.00000	0.186299	0.0931493	+	0.995652i	$0.470307\pi$
$462$	18.0000	0.837436
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	3.00000	0.139272
$465$	−36.0000	−1.66946
$466$	−14.0000	−0.648537
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	18.0000	0.832050
$469$	45.0000	2.07791
$470$	16.0000	0.738025
$471$	0	0
$472$	3.00000	0.138086
$473$	20.0000	0.919601
$474$	36.0000	1.65353
$475$	0	0
$476$	3.00000	0.137505
$477$	18.0000	0.824163
$478$	−1.00000	−0.0457389
$479$	−40.0000	−1.82765	−0.913823	−	0.406112i	$-0.866884\pi$
$479$	−40.0000	−1.82765	−0.913823	+	0.406112i	$0.866884\pi$
$480$	6.00000	0.273861
$481$	−18.0000	−0.820729
$482$	24.0000	1.09317
$483$	45.0000	2.04757
$484$	−7.00000	−0.318182
$485$	−24.0000	−1.08978
$486$	0	0
$487$	18.0000	0.815658	0.407829	−	0.913058i	$-0.366286\pi$
$487$	18.0000	0.815658	0.407829	+	0.913058i	$0.366286\pi$
$488$	0	0
$489$	18.0000	0.813988
$490$	−4.00000	−0.180702
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	36.0000	1.62301
$493$	−3.00000	−0.135113
$494$	0	0
$495$	−24.0000	−1.07872
$496$	6.00000	0.269408
$497$	0	0
$498$	6.00000	0.268866
$499$	18.0000	0.805791	0.402895	−	0.915246i	$-0.368004\pi$
$499$	18.0000	0.805791	0.402895	+	0.915246i	$0.368004\pi$
$500$	−12.0000	−0.536656
$501$	−36.0000	−1.60836
$502$	20.0000	0.892644
$503$	1.00000	0.0445878	0.0222939	−	0.999751i	$-0.492903\pi$
$503$	1.00000	0.0445878	0.0222939	+	0.999751i	$0.492903\pi$
$504$	18.0000	0.801784
$505$	20.0000	0.889988
$506$	10.0000	0.444554
$507$	12.0000	0.532939
$508$	−12.0000	−0.532414
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	−6.00000	−0.265684
$511$	33.0000	1.45983
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−18.0000	−0.793946
$515$	−12.0000	−0.528783
$516$	30.0000	1.32068
$517$	16.0000	0.703679
$518$	−18.0000	−0.790875
$519$	−54.0000	−2.37034
$520$	−6.00000	−0.263117
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	−18.0000	−0.787839
$523$	−9.00000	−0.393543	−0.196771	−	0.980449i	$-0.563046\pi$
$523$	−9.00000	−0.393543	−0.196771	+	0.980449i	$0.563046\pi$
$524$	14.0000	0.611593
$525$	−9.00000	−0.392792
$526$	8.00000	0.348817
$527$	−6.00000	−0.261364
$528$	6.00000	0.261116
$529$	2.00000	0.0869565
$530$	−6.00000	−0.260623
$531$	−18.0000	−0.781133
$532$	0	0
$533$	−36.0000	−1.55933
$534$	−18.0000	−0.778936
$535$	6.00000	0.259403
$536$	15.0000	0.647901
$537$	−36.0000	−1.55351
$538$	−6.00000	−0.258678
$539$	−4.00000	−0.172292
$540$	−18.0000	−0.774597
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	11.0000	0.472490
$543$	54.0000	2.31736
$544$	1.00000	0.0428746
$545$	−6.00000	−0.257012
$546$	−27.0000	−1.15549
$547$	36.0000	1.53925	0.769624	−	0.638497i	$-0.220443\pi$
$547$	36.0000	1.53925	0.769624	+	0.638497i	$0.220443\pi$
$548$	19.0000	0.811640
$549$	0	0
$550$	−2.00000	−0.0852803
$551$	0	0
$552$	15.0000	0.638442
$553$	−36.0000	−1.53088
$554$	−30.0000	−1.27458
$555$	36.0000	1.52811
$556$	6.00000	0.254457
$557$	22.0000	0.932170	0.466085	−	0.884740i	$-0.345664\pi$
$557$	22.0000	0.932170	0.466085	+	0.884740i	$0.345664\pi$
$558$	−36.0000	−1.52400
$559$	−30.0000	−1.26886
$560$	−6.00000	−0.253546
$561$	−6.00000	−0.253320
$562$	12.0000	0.506189
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	24.0000	1.01058
$565$	−24.0000	−1.00969
$566$	14.0000	0.588464
$567$	−27.0000	−1.13389
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	−6.00000	−0.250873
$573$	33.0000	1.37859
$574$	−36.0000	−1.50261
$575$	−5.00000	−0.208514
$576$	6.00000	0.250000
$577$	15.0000	0.624458	0.312229	−	0.950007i	$-0.398924\pi$
$577$	15.0000	0.624458	0.312229	+	0.950007i	$0.398924\pi$
$578$	16.0000	0.665512
$579$	18.0000	0.748054
$580$	6.00000	0.249136
$581$	−6.00000	−0.248922
$582$	−36.0000	−1.49225
$583$	−6.00000	−0.248495
$584$	11.0000	0.455183
$585$	36.0000	1.48842
$586$	−9.00000	−0.371787
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	−6.00000	−0.247436
$589$	0	0
$590$	6.00000	0.247016
$591$	−12.0000	−0.493614
$592$	−6.00000	−0.246598
$593$	−2.00000	−0.0821302	−0.0410651	−	0.999156i	$-0.513075\pi$
$593$	−2.00000	−0.0821302	−0.0410651	+	0.999156i	$0.513075\pi$
$594$	−18.0000	−0.738549
$595$	6.00000	0.245976
$596$	−8.00000	−0.327693
$597$	21.0000	0.859473
$598$	−15.0000	−0.613396
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	−3.00000	−0.122474
$601$	6.00000	0.244745	0.122373	−	0.992484i	$-0.460950\pi$
$601$	6.00000	0.244745	0.122373	+	0.992484i	$0.460950\pi$
$602$	−30.0000	−1.22271
$603$	−90.0000	−3.66508
$604$	18.0000	0.732410
$605$	−14.0000	−0.569181
$606$	30.0000	1.21867
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	0	0
$609$	27.0000	1.09410
$610$	0	0
$611$	−24.0000	−0.970936
$612$	−6.00000	−0.242536
$613$	18.0000	0.727013	0.363507	−	0.931592i	$-0.381579\pi$
$613$	18.0000	0.727013	0.363507	+	0.931592i	$0.381579\pi$
$614$	12.0000	0.484281
$615$	72.0000	2.90332
$616$	−6.00000	−0.241747
$617$	10.0000	0.402585	0.201292	−	0.979531i	$-0.435486\pi$
$617$	10.0000	0.402585	0.201292	+	0.979531i	$0.435486\pi$
$618$	−18.0000	−0.724066
$619$	−24.0000	−0.964641	−0.482321	−	0.875995i	$-0.660206\pi$
$619$	−24.0000	−0.964641	−0.482321	+	0.875995i	$0.660206\pi$
$620$	12.0000	0.481932
$621$	−45.0000	−1.80579
$622$	11.0000	0.441060
$623$	18.0000	0.721155
$624$	−9.00000	−0.360288
$625$	−19.0000	−0.760000
$626$	−21.0000	−0.839329
$627$	0	0
$628$	0	0
$629$	6.00000	0.239236
$630$	36.0000	1.43427
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	−12.0000	−0.477334
$633$	9.00000	0.357718
$634$	−33.0000	−1.31060
$635$	−24.0000	−0.952411
$636$	−9.00000	−0.356873
$637$	6.00000	0.237729
$638$	6.00000	0.237542
$639$	0	0
$640$	−2.00000	−0.0790569
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	9.00000	0.355202
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	−15.0000	−0.591083
$645$	60.0000	2.36250
$646$	0	0
$647$	−23.0000	−0.904223	−0.452112	−	0.891961i	$-0.649329\pi$
$647$	−23.0000	−0.904223	−0.452112	+	0.891961i	$0.649329\pi$
$648$	−9.00000	−0.353553
$649$	6.00000	0.235521
$650$	3.00000	0.117670
$651$	54.0000	2.11643
$652$	−6.00000	−0.234978
$653$	−10.0000	−0.391330	−0.195665	−	0.980671i	$-0.562687\pi$
$653$	−10.0000	−0.391330	−0.195665	+	0.980671i	$0.562687\pi$
$654$	−9.00000	−0.351928
$655$	28.0000	1.09405
$656$	−12.0000	−0.468521
$657$	−66.0000	−2.57491
$658$	−24.0000	−0.935617
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405627\pi$
$660$	12.0000	0.467099
$661$	−15.0000	−0.583432	−0.291716	−	0.956505i	$-0.594226\pi$
$661$	−15.0000	−0.583432	−0.291716	+	0.956505i	$0.594226\pi$
$662$	9.00000	0.349795
$663$	9.00000	0.349531
$664$	−2.00000	−0.0776151
$665$	0	0
$666$	36.0000	1.39497
$667$	15.0000	0.580802
$668$	12.0000	0.464294
$669$	54.0000	2.08776
$670$	30.0000	1.15900
$671$	0	0
$672$	−9.00000	−0.347183
$673$	48.0000	1.85026	0.925132	−	0.379646i	$-0.123954\pi$
$673$	48.0000	1.85026	0.925132	+	0.379646i	$0.123954\pi$
$674$	18.0000	0.693334
$675$	9.00000	0.346410
$676$	−4.00000	−0.153846
$677$	−3.00000	−0.115299	−0.0576497	−	0.998337i	$-0.518361\pi$
$677$	−3.00000	−0.115299	−0.0576497	+	0.998337i	$0.518361\pi$
$678$	−36.0000	−1.38257
$679$	36.0000	1.38155
$680$	2.00000	0.0766965
$681$	9.00000	0.344881
$682$	12.0000	0.459504
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	38.0000	1.45191
$686$	−15.0000	−0.572703
$687$	36.0000	1.37349
$688$	−10.0000	−0.381246
$689$	9.00000	0.342873
$690$	30.0000	1.14208
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	18.0000	0.684257
$693$	36.0000	1.36753
$694$	−16.0000	−0.607352
$695$	12.0000	0.455186
$696$	9.00000	0.341144
$697$	12.0000	0.454532
$698$	−28.0000	−1.05982
$699$	−42.0000	−1.58859
$700$	3.00000	0.113389
$701$	−40.0000	−1.51078	−0.755390	−	0.655276i	$-0.772552\pi$
$701$	−40.0000	−1.51078	−0.755390	+	0.655276i	$0.772552\pi$
$702$	27.0000	1.01905
$703$	0	0
$704$	−2.00000	−0.0753778
$705$	48.0000	1.80778
$706$	31.0000	1.16670
$707$	−30.0000	−1.12827
$708$	9.00000	0.338241
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	0	0
$711$	72.0000	2.70021
$712$	6.00000	0.224860
$713$	30.0000	1.12351
$714$	9.00000	0.336817
$715$	−12.0000	−0.448775
$716$	12.0000	0.448461
$717$	−3.00000	−0.112037
$718$	−19.0000	−0.709074
$719$	−43.0000	−1.60363	−0.801815	−	0.597573i	$-0.796132\pi$
$719$	−43.0000	−1.60363	−0.801815	+	0.597573i	$0.796132\pi$
$720$	12.0000	0.447214
$721$	18.0000	0.670355
$722$	0	0
$723$	72.0000	2.67771
$724$	−18.0000	−0.668965
$725$	−3.00000	−0.111417
$726$	−21.0000	−0.779383
$727$	−35.0000	−1.29808	−0.649039	−	0.760755i	$-0.724829\pi$
$727$	−35.0000	−1.29808	−0.649039	+	0.760755i	$0.724829\pi$
$728$	9.00000	0.333562
$729$	−27.0000	−1.00000
$730$	22.0000	0.814257
$731$	10.0000	0.369863
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	−8.00000	−0.295285
$735$	−12.0000	−0.442627
$736$	−5.00000	−0.184302
$737$	30.0000	1.10506
$738$	72.0000	2.65036
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	9.00000	0.330400
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	18.0000	0.659912
$745$	−16.0000	−0.586195
$746$	−21.0000	−0.768865
$747$	12.0000	0.439057
$748$	2.00000	0.0731272
$749$	−9.00000	−0.328853
$750$	−36.0000	−1.31453
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	−8.00000	−0.291730
$753$	60.0000	2.18652
$754$	−9.00000	−0.327761
$755$	36.0000	1.31017
$756$	27.0000	0.981981
$757$	12.0000	0.436147	0.218074	−	0.975932i	$-0.430023\pi$
$757$	12.0000	0.436147	0.218074	+	0.975932i	$0.430023\pi$
$758$	3.00000	0.108965
$759$	30.0000	1.08893
$760$	0	0
$761$	−13.0000	−0.471250	−0.235625	−	0.971844i	$-0.575714\pi$
$761$	−13.0000	−0.471250	−0.235625	+	0.971844i	$0.575714\pi$
$762$	−36.0000	−1.30414
$763$	9.00000	0.325822
$764$	−11.0000	−0.397966
$765$	−12.0000	−0.433861
$766$	18.0000	0.650366
$767$	−9.00000	−0.324971
$768$	−3.00000	−0.108253
$769$	−15.0000	−0.540914	−0.270457	−	0.962732i	$-0.587175\pi$
$769$	−15.0000	−0.540914	−0.270457	+	0.962732i	$0.587175\pi$
$770$	−12.0000	−0.432450
$771$	−54.0000	−1.94476
$772$	−6.00000	−0.215945
$773$	−15.0000	−0.539513	−0.269756	−	0.962929i	$-0.586943\pi$
$773$	−15.0000	−0.539513	−0.269756	+	0.962929i	$0.586943\pi$
$774$	60.0000	2.15666
$775$	−6.00000	−0.215526
$776$	12.0000	0.430775
$777$	−54.0000	−1.93724
$778$	−26.0000	−0.932145
$779$	0	0
$780$	−18.0000	−0.644503
$781$	0	0
$782$	5.00000	0.178800
$783$	−27.0000	−0.964901
$784$	2.00000	0.0714286
$785$	0	0
$786$	42.0000	1.49809
$787$	9.00000	0.320815	0.160408	−	0.987051i	$-0.448719\pi$
$787$	9.00000	0.320815	0.160408	+	0.987051i	$0.448719\pi$
$788$	4.00000	0.142494
$789$	24.0000	0.854423
$790$	−24.0000	−0.853882
$791$	36.0000	1.28001
$792$	12.0000	0.426401
$793$	0	0
$794$	−2.00000	−0.0709773
$795$	−18.0000	−0.638394
$796$	−7.00000	−0.248108
$797$	−33.0000	−1.16892	−0.584460	−	0.811423i	$-0.698694\pi$
$797$	−33.0000	−1.16892	−0.584460	+	0.811423i	$0.698694\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	1.00000	0.0353553
$801$	−36.0000	−1.27200
$802$	36.0000	1.27120
$803$	22.0000	0.776363
$804$	45.0000	1.58703
$805$	−30.0000	−1.05736
$806$	−18.0000	−0.634023
$807$	−18.0000	−0.633630
$808$	−10.0000	−0.351799
$809$	−11.0000	−0.386739	−0.193370	−	0.981126i	$-0.561942\pi$
$809$	−11.0000	−0.386739	−0.193370	+	0.981126i	$0.561942\pi$
$810$	−18.0000	−0.632456
$811$	−33.0000	−1.15879	−0.579393	−	0.815048i	$-0.696710\pi$
$811$	−33.0000	−1.15879	−0.579393	+	0.815048i	$0.696710\pi$
$812$	−9.00000	−0.315838
$813$	33.0000	1.15736
$814$	−12.0000	−0.420600
$815$	−12.0000	−0.420342
$816$	3.00000	0.105021
$817$	0	0
$818$	6.00000	0.209785
$819$	−54.0000	−1.88691
$820$	−24.0000	−0.838116
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	57.0000	1.98810
$823$	3.00000	0.104573	0.0522867	−	0.998632i	$-0.483349\pi$
$823$	3.00000	0.104573	0.0522867	+	0.998632i	$0.483349\pi$
$824$	6.00000	0.209020
$825$	−6.00000	−0.208893
$826$	−9.00000	−0.313150
$827$	−15.0000	−0.521601	−0.260801	−	0.965393i	$-0.583986\pi$
$827$	−15.0000	−0.521601	−0.260801	+	0.965393i	$0.583986\pi$
$828$	30.0000	1.04257
$829$	−51.0000	−1.77130	−0.885652	−	0.464350i	$-0.846288\pi$
$829$	−51.0000	−1.77130	−0.885652	+	0.464350i	$0.846288\pi$
$830$	−4.00000	−0.138842
$831$	−90.0000	−3.12207
$832$	3.00000	0.104006
$833$	−2.00000	−0.0692959
$834$	18.0000	0.623289
$835$	24.0000	0.830554
$836$	0	0
$837$	−54.0000	−1.86651
$838$	14.0000	0.483622
$839$	18.0000	0.621429	0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000	0.621429	0.310715	+	0.950503i	$0.399432\pi$
$840$	−18.0000	−0.621059
$841$	−20.0000	−0.689655
$842$	−27.0000	−0.930481
$843$	36.0000	1.23991
$844$	−3.00000	−0.103264
$845$	−8.00000	−0.275208
$846$	48.0000	1.65027
$847$	21.0000	0.721569
$848$	3.00000	0.103020
$849$	42.0000	1.44144
$850$	−1.00000	−0.0342997
$851$	−30.0000	−1.02839
$852$	0	0
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	0	0
$855$	0	0
$856$	−3.00000	−0.102538
$857$	−24.0000	−0.819824	−0.409912	−	0.912125i	$-0.634441\pi$
$857$	−24.0000	−0.819824	−0.409912	+	0.912125i	$0.634441\pi$
$858$	−18.0000	−0.614510
$859$	54.0000	1.84246	0.921228	−	0.389023i	$-0.127187\pi$
$859$	54.0000	1.84246	0.921228	+	0.389023i	$0.127187\pi$
$860$	−20.0000	−0.681994
$861$	−108.000	−3.68063
$862$	24.0000	0.817443
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	9.00000	0.306186
$865$	36.0000	1.22404
$866$	30.0000	1.01944
$867$	48.0000	1.63017
$868$	−18.0000	−0.610960
$869$	−24.0000	−0.814144
$870$	18.0000	0.610257
$871$	−45.0000	−1.52477
$872$	3.00000	0.101593
$873$	−72.0000	−2.43683
$874$	0	0
$875$	36.0000	1.21702
$876$	33.0000	1.11497
$877$	−27.0000	−0.911725	−0.455863	−	0.890050i	$-0.650669\pi$
$877$	−27.0000	−0.911725	−0.455863	+	0.890050i	$0.650669\pi$
$878$	0	0
$879$	−27.0000	−0.910687
$880$	−4.00000	−0.134840
$881$	34.0000	1.14549	0.572745	−	0.819734i	$-0.305879\pi$
$881$	34.0000	1.14549	0.572745	+	0.819734i	$0.305879\pi$
$882$	−12.0000	−0.404061
$883$	−24.0000	−0.807664	−0.403832	−	0.914833i	$-0.632322\pi$
$883$	−24.0000	−0.807664	−0.403832	+	0.914833i	$0.632322\pi$
$884$	−3.00000	−0.100901
$885$	18.0000	0.605063
$886$	22.0000	0.739104
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	−18.0000	−0.604040
$889$	36.0000	1.20740
$890$	12.0000	0.402241
$891$	−18.0000	−0.603023
$892$	−18.0000	−0.602685
$893$	0	0
$894$	−24.0000	−0.802680
$895$	24.0000	0.802232
$896$	3.00000	0.100223
$897$	−45.0000	−1.50251
$898$	6.00000	0.200223
$899$	18.0000	0.600334
$900$	−6.00000	−0.200000
$901$	−3.00000	−0.0999445
$902$	−24.0000	−0.799113
$903$	−90.0000	−2.99501
$904$	12.0000	0.399114
$905$	−36.0000	−1.19668
$906$	54.0000	1.79403
$907$	15.0000	0.498067	0.249033	−	0.968495i	$-0.419887\pi$
$907$	15.0000	0.498067	0.249033	+	0.968495i	$0.419887\pi$
$908$	−3.00000	−0.0995585
$909$	60.0000	1.99007
$910$	18.0000	0.596694
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	0	0
$913$	−4.00000	−0.132381
$914$	−1.00000	−0.0330771
$915$	0	0
$916$	−12.0000	−0.396491
$917$	−42.0000	−1.38696
$918$	−9.00000	−0.297044
$919$	15.0000	0.494804	0.247402	−	0.968913i	$-0.420423\pi$
$919$	15.0000	0.494804	0.247402	+	0.968913i	$0.420423\pi$
$920$	−10.0000	−0.329690
$921$	36.0000	1.18624
$922$	−4.00000	−0.131733
$923$	0	0
$924$	−18.0000	−0.592157
$925$	6.00000	0.197279
$926$	−32.0000	−1.05159
$927$	−36.0000	−1.18240
$928$	−3.00000	−0.0984798
$929$	41.0000	1.34517	0.672583	−	0.740022i	$-0.265185\pi$
$929$	41.0000	1.34517	0.672583	+	0.740022i	$0.265185\pi$
$930$	36.0000	1.18049
$931$	0	0
$932$	14.0000	0.458585
$933$	33.0000	1.08037
$934$	−8.00000	−0.261768
$935$	4.00000	0.130814
$936$	−18.0000	−0.588348
$937$	−47.0000	−1.53542	−0.767712	−	0.640796i	$-0.778605\pi$
$937$	−47.0000	−1.53542	−0.767712	+	0.640796i	$0.778605\pi$
$938$	−45.0000	−1.46930
$939$	−63.0000	−2.05593
$940$	−16.0000	−0.521862
$941$	27.0000	0.880175	0.440087	−	0.897955i	$-0.354947\pi$
$941$	27.0000	0.880175	0.440087	+	0.897955i	$0.354947\pi$
$942$	0	0
$943$	−60.0000	−1.95387
$944$	−3.00000	−0.0976417
$945$	54.0000	1.75662
$946$	−20.0000	−0.650256
$947$	−46.0000	−1.49480	−0.747400	−	0.664375i	$-0.768698\pi$
$947$	−46.0000	−1.49480	−0.747400	+	0.664375i	$0.768698\pi$
$948$	−36.0000	−1.16923
$949$	−33.0000	−1.07123
$950$	0	0
$951$	−99.0000	−3.21029
$952$	−3.00000	−0.0972306
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	−18.0000	−0.582772
$955$	−22.0000	−0.711903
$956$	1.00000	0.0323423
$957$	18.0000	0.581857
$958$	40.0000	1.29234
$959$	−57.0000	−1.84063
$960$	−6.00000	−0.193649
$961$	5.00000	0.161290
$962$	18.0000	0.580343
$963$	18.0000	0.580042
$964$	−24.0000	−0.772988
$965$	−12.0000	−0.386294
$966$	−45.0000	−1.44785
$967$	−24.0000	−0.771788	−0.385894	−	0.922543i	$-0.626107\pi$
$967$	−24.0000	−0.771788	−0.385894	+	0.922543i	$0.626107\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	24.0000	0.770594
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	0	0
$973$	−18.0000	−0.577054
$974$	−18.0000	−0.576757
$975$	9.00000	0.288231
$976$	0	0
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	−18.0000	−0.575577
$979$	12.0000	0.383522
$980$	4.00000	0.127775
$981$	−18.0000	−0.574696
$982$	−8.00000	−0.255290
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	−36.0000	−1.14764
$985$	8.00000	0.254901
$986$	3.00000	0.0955395
$987$	−72.0000	−2.29179
$988$	0	0
$989$	−50.0000	−1.58991
$990$	24.0000	0.762770
$991$	−6.00000	−0.190596	−0.0952981	−	0.995449i	$-0.530380\pi$
$991$	−6.00000	−0.190596	−0.0952981	+	0.995449i	$0.530380\pi$
$992$	−6.00000	−0.190500
$993$	27.0000	0.856819
$994$	0	0
$995$	−14.0000	−0.443830
$996$	−6.00000	−0.190117
$997$	6.00000	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$998$	−18.0000	−0.569780
$999$	54.0000	1.70848

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	722.2.a.a.1.1	✓	1
3.2	odd	2		6498.2.a.m.1.1		1
4.3	odd	2		5776.2.a.q.1.1		1
19.2	odd	18		722.2.e.g.99.1		6
19.3	odd	18		722.2.e.g.389.1		6
19.4	even	9		722.2.e.h.415.1		6
19.5	even	9		722.2.e.h.595.1		6
19.6	even	9		722.2.e.h.245.1		6
19.7	even	3		722.2.c.g.429.1		2
19.8	odd	6		722.2.c.a.653.1		2
19.9	even	9		722.2.e.h.423.1		6
19.10	odd	18		722.2.e.g.423.1		6
19.11	even	3		722.2.c.g.653.1		2
19.12	odd	6		722.2.c.a.429.1		2
19.13	odd	18		722.2.e.g.245.1		6
19.14	odd	18		722.2.e.g.595.1		6
19.15	odd	18		722.2.e.g.415.1		6
19.16	even	9		722.2.e.h.389.1		6
19.17	even	9		722.2.e.h.99.1		6
19.18	odd	2		722.2.a.f.1.1	yes	1
57.56	even	2		6498.2.a.a.1.1		1
76.75	even	2		5776.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
722.2.a.a.1.1	✓	1	1.1	even	1	trivial
722.2.a.f.1.1	yes	1	19.18	odd	2
722.2.c.a.429.1		2	19.12	odd	6
722.2.c.a.653.1		2	19.8	odd	6
722.2.c.g.429.1		2	19.7	even	3
722.2.c.g.653.1		2	19.11	even	3
722.2.e.g.99.1		6	19.2	odd	18
722.2.e.g.245.1		6	19.13	odd	18
722.2.e.g.389.1		6	19.3	odd	18
722.2.e.g.415.1		6	19.15	odd	18
722.2.e.g.423.1		6	19.10	odd	18
722.2.e.g.595.1		6	19.14	odd	18
722.2.e.h.99.1		6	19.17	even	9
722.2.e.h.245.1		6	19.6	even	9
722.2.e.h.389.1		6	19.16	even	9
722.2.e.h.415.1		6	19.4	even	9
722.2.e.h.423.1		6	19.9	even	9
722.2.e.h.595.1		6	19.5	even	9
5776.2.a.a.1.1		1	76.75	even	2
5776.2.a.q.1.1		1	4.3	odd	2
6498.2.a.a.1.1		1	57.56	even	2
6498.2.a.m.1.1		1	3.2	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:	\( N \)	\(=\)	\( 722 = 2 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	722.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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