LMFDB - Embedded newform 786.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [786,2,Mod(1,786)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(786, 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("786.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 786 = 2 \cdot 3 \cdot 131 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	786.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:	$6.27624159887$
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	$\chi$	$=$	786.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{10} -6.00000 q^{11} -1.00000 q^{12} -6.00000 q^{13} +1.00000 q^{14} -4.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -5.00000 q^{19} +4.00000 q^{20} +1.00000 q^{21} +6.00000 q^{22} -2.00000 q^{23} +1.00000 q^{24} +11.0000 q^{25} +6.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -3.00000 q^{29} +4.00000 q^{30} -2.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} +2.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} +7.00000 q^{37} +5.00000 q^{38} +6.00000 q^{39} -4.00000 q^{40} -9.00000 q^{41} -1.00000 q^{42} +12.0000 q^{43} -6.00000 q^{44} +4.00000 q^{45} +2.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} -11.0000 q^{50} +2.00000 q^{51} -6.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -24.0000 q^{55} +1.00000 q^{56} +5.00000 q^{57} +3.00000 q^{58} -8.00000 q^{59} -4.00000 q^{60} +2.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -24.0000 q^{65} -6.00000 q^{66} +3.00000 q^{67} -2.00000 q^{68} +2.00000 q^{69} +4.00000 q^{70} +10.0000 q^{71} -1.00000 q^{72} +10.0000 q^{73} -7.00000 q^{74} -11.0000 q^{75} -5.00000 q^{76} +6.00000 q^{77} -6.00000 q^{78} -8.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} +9.00000 q^{82} +1.00000 q^{83} +1.00000 q^{84} -8.00000 q^{85} -12.0000 q^{86} +3.00000 q^{87} +6.00000 q^{88} -11.0000 q^{89} -4.00000 q^{90} +6.00000 q^{91} -2.00000 q^{92} +2.00000 q^{93} +8.00000 q^{94} -20.0000 q^{95} +1.00000 q^{96} +6.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−4.00000	−1.26491
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	−1.00000	−0.288675
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	1.00000	0.267261
$15$	−4.00000	−1.03280
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−1.00000	−0.235702
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	4.00000	0.894427
$21$	1.00000	0.218218
$22$	6.00000	1.27920
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	1.00000	0.204124
$25$	11.0000	2.20000
$26$	6.00000	1.17670
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	4.00000	0.730297
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−1.00000	−0.176777
$33$	6.00000	1.04447
$34$	2.00000	0.342997
$35$	−4.00000	−0.676123
$36$	1.00000	0.166667
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	5.00000	0.811107
$39$	6.00000	0.960769
$40$	−4.00000	−0.632456
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	−1.00000	−0.154303
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	−6.00000	−0.904534
$45$	4.00000	0.596285
$46$	2.00000	0.294884
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	−1.00000	−0.144338
$49$	−6.00000	−0.857143
$50$	−11.0000	−1.55563
$51$	2.00000	0.280056
$52$	−6.00000	−0.832050
$53$	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
$55$	−24.0000	−3.23616
$56$	1.00000	0.133631
$57$	5.00000	0.662266
$58$	3.00000	0.393919
$59$	−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
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	2.00000	0.254000
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−24.0000	−2.97683
$66$	−6.00000	−0.738549
$67$	3.00000	0.366508	0.183254	−	0.983066i	$-0.441337\pi$
$67$	3.00000	0.366508	0.183254	+	0.983066i	$0.441337\pi$
$68$	−2.00000	−0.242536
$69$	2.00000	0.240772
$70$	4.00000	0.478091
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	−1.00000	−0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	−7.00000	−0.813733
$75$	−11.0000	−1.27017
$76$	−5.00000	−0.573539
$77$	6.00000	0.683763
$78$	−6.00000	−0.679366
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	9.00000	0.993884
$83$	1.00000	0.109764	0.0548821	−	0.998493i	$-0.482522\pi$
$83$	1.00000	0.109764	0.0548821	+	0.998493i	$0.482522\pi$
$84$	1.00000	0.109109
$85$	−8.00000	−0.867722
$86$	−12.0000	−1.29399
$87$	3.00000	0.321634
$88$	6.00000	0.639602
$89$	−11.0000	−1.16600	−0.582999	−	0.812473i	$-0.698121\pi$
$89$	−11.0000	−1.16600	−0.582999	+	0.812473i	$0.698121\pi$
$90$	−4.00000	−0.421637
$91$	6.00000	0.628971
$92$	−2.00000	−0.208514
$93$	2.00000	0.207390
$94$	8.00000	0.825137
$95$	−20.0000	−2.05196
$96$	1.00000	0.102062
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	6.00000	0.606092
$99$	−6.00000	−0.603023
$100$	11.0000	1.10000
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	−2.00000	−0.198030
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	6.00000	0.588348
$105$	4.00000	0.390360
$106$	−6.00000	−0.582772
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	−1.00000	−0.0962250
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	24.0000	2.28831
$111$	−7.00000	−0.664411
$112$	−1.00000	−0.0944911
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	−5.00000	−0.468293
$115$	−8.00000	−0.746004
$116$	−3.00000	−0.278543
$117$	−6.00000	−0.554700
$118$	8.00000	0.736460
$119$	2.00000	0.183340
$120$	4.00000	0.365148
$121$	25.0000	2.27273
$122$	0	0
$123$	9.00000	0.811503
$124$	−2.00000	−0.179605
$125$	24.0000	2.14663
$126$	1.00000	0.0890871
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	−1.00000	−0.0883883
$129$	−12.0000	−1.05654
$130$	24.0000	2.10494
$131$	−1.00000	−0.0873704
$131$	−1.00000	−0.0873704
$132$	6.00000	0.522233
$133$	5.00000	0.433555
$134$	−3.00000	−0.259161
$135$	−4.00000	−0.344265
$136$	2.00000	0.171499
$137$	−16.0000	−1.36697	−0.683486	−	0.729964i	$-0.739537\pi$
$137$	−16.0000	−1.36697	−0.683486	+	0.729964i	$0.739537\pi$
$138$	−2.00000	−0.170251
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	−4.00000	−0.338062
$141$	8.00000	0.673722
$142$	−10.0000	−0.839181
$143$	36.0000	3.01047
$144$	1.00000	0.0833333
$145$	−12.0000	−0.996546
$146$	−10.0000	−0.827606
$147$	6.00000	0.494872
$148$	7.00000	0.575396
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	11.0000	0.898146
$151$	−3.00000	−0.244137	−0.122068	−	0.992522i	$-0.538953\pi$
$151$	−3.00000	−0.244137	−0.122068	+	0.992522i	$0.538953\pi$
$152$	5.00000	0.405554
$153$	−2.00000	−0.161690
$154$	−6.00000	−0.483494
$155$	−8.00000	−0.642575
$156$	6.00000	0.480384
$157$	23.0000	1.83560	0.917800	−	0.397043i	$-0.129964\pi$
$157$	23.0000	1.83560	0.917800	+	0.397043i	$0.129964\pi$
$158$	8.00000	0.636446
$159$	−6.00000	−0.475831
$160$	−4.00000	−0.316228
$161$	2.00000	0.157622
$162$	−1.00000	−0.0785674
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	−9.00000	−0.702782
$165$	24.0000	1.86840
$166$	−1.00000	−0.0776151
$167$	−15.0000	−1.16073	−0.580367	−	0.814355i	$-0.697091\pi$
$167$	−15.0000	−1.16073	−0.580367	+	0.814355i	$0.697091\pi$
$168$	−1.00000	−0.0771517
$169$	23.0000	1.76923
$170$	8.00000	0.613572
$171$	−5.00000	−0.382360
$172$	12.0000	0.914991
$173$	−5.00000	−0.380143	−0.190071	−	0.981770i	$-0.560872\pi$
$173$	−5.00000	−0.380143	−0.190071	+	0.981770i	$0.560872\pi$
$174$	−3.00000	−0.227429
$175$	−11.0000	−0.831522
$176$	−6.00000	−0.452267
$177$	8.00000	0.601317
$178$	11.0000	0.824485
$179$	14.0000	1.04641	0.523205	−	0.852207i	$-0.324736\pi$
$179$	14.0000	1.04641	0.523205	+	0.852207i	$0.324736\pi$
$180$	4.00000	0.298142
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	−6.00000	−0.444750
$183$	0	0
$184$	2.00000	0.147442
$185$	28.0000	2.05860
$186$	−2.00000	−0.146647
$187$	12.0000	0.877527
$188$	−8.00000	−0.583460
$189$	1.00000	0.0727393
$190$	20.0000	1.45095
$191$	−5.00000	−0.361787	−0.180894	−	0.983503i	$-0.557899\pi$
$191$	−5.00000	−0.361787	−0.180894	+	0.983503i	$0.557899\pi$
$192$	−1.00000	−0.0721688
$193$	−23.0000	−1.65558	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−23.0000	−1.65558	−0.827788	+	0.561041i	$0.810401\pi$
$194$	0	0
$195$	24.0000	1.71868
$196$	−6.00000	−0.428571
$197$	3.00000	0.213741	0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000	0.213741	0.106871	+	0.994273i	$0.465917\pi$
$198$	6.00000	0.426401
$199$	28.0000	1.98487	0.992434	−	0.122782i	$-0.0391815\pi$
$199$	28.0000	1.98487	0.992434	+	0.122782i	$0.0391815\pi$
$200$	−11.0000	−0.777817
$201$	−3.00000	−0.211604
$202$	4.00000	0.281439
$203$	3.00000	0.210559
$204$	2.00000	0.140028
$205$	−36.0000	−2.51435
$206$	8.00000	0.557386
$207$	−2.00000	−0.139010
$208$	−6.00000	−0.416025
$209$	30.0000	2.07514
$210$	−4.00000	−0.276026
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	6.00000	0.412082
$213$	−10.0000	−0.685189
$214$	0	0
$215$	48.0000	3.27357
$216$	1.00000	0.0680414
$217$	2.00000	0.135769
$218$	−8.00000	−0.541828
$219$	−10.0000	−0.675737
$220$	−24.0000	−1.61808
$221$	12.0000	0.807207
$222$	7.00000	0.469809
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	1.00000	0.0668153
$225$	11.0000	0.733333
$226$	−1.00000	−0.0665190
$227$	21.0000	1.39382	0.696909	−	0.717159i	$-0.254558\pi$
$227$	21.0000	1.39382	0.696909	+	0.717159i	$0.254558\pi$
$228$	5.00000	0.331133
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	8.00000	0.527504
$231$	−6.00000	−0.394771
$232$	3.00000	0.196960
$233$	−25.0000	−1.63780	−0.818902	−	0.573933i	$-0.805417\pi$
$233$	−25.0000	−1.63780	−0.818902	+	0.573933i	$0.805417\pi$
$234$	6.00000	0.392232
$235$	−32.0000	−2.08745
$236$	−8.00000	−0.520756
$237$	8.00000	0.519656
$238$	−2.00000	−0.129641
$239$	5.00000	0.323423	0.161712	−	0.986838i	$-0.448299\pi$
$239$	5.00000	0.323423	0.161712	+	0.986838i	$0.448299\pi$
$240$	−4.00000	−0.258199
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	−25.0000	−1.60706
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−24.0000	−1.53330
$246$	−9.00000	−0.573819
$247$	30.0000	1.90885
$248$	2.00000	0.127000
$249$	−1.00000	−0.0633724
$250$	−24.0000	−1.51789
$251$	11.0000	0.694314	0.347157	−	0.937807i	$-0.387147\pi$
$251$	11.0000	0.694314	0.347157	+	0.937807i	$0.387147\pi$
$252$	−1.00000	−0.0629941
$253$	12.0000	0.754434
$254$	−12.0000	−0.752947
$255$	8.00000	0.500979
$256$	1.00000	0.0625000
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	12.0000	0.747087
$259$	−7.00000	−0.434959
$260$	−24.0000	−1.48842
$261$	−3.00000	−0.185695
$262$	1.00000	0.0617802
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	−6.00000	−0.369274
$265$	24.0000	1.47431
$266$	−5.00000	−0.306570
$267$	11.0000	0.673189
$268$	3.00000	0.183254
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	4.00000	0.243432
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	−2.00000	−0.121268
$273$	−6.00000	−0.363137
$274$	16.0000	0.966595
$275$	−66.0000	−3.97995
$276$	2.00000	0.120386
$277$	12.0000	0.721010	0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000	0.721010	0.360505	+	0.932757i	$0.382604\pi$
$278$	0	0
$279$	−2.00000	−0.119737
$280$	4.00000	0.239046
$281$	14.0000	0.835170	0.417585	−	0.908638i	$-0.362877\pi$
$281$	14.0000	0.835170	0.417585	+	0.908638i	$0.362877\pi$
$282$	−8.00000	−0.476393
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	10.0000	0.593391
$285$	20.0000	1.18470
$286$	−36.0000	−2.12872
$287$	9.00000	0.531253
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	12.0000	0.704664
$291$	0	0
$292$	10.0000	0.585206
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	−6.00000	−0.349927
$295$	−32.0000	−1.86311
$296$	−7.00000	−0.406867
$297$	6.00000	0.348155
$298$	6.00000	0.347571
$299$	12.0000	0.693978
$300$	−11.0000	−0.635085
$301$	−12.0000	−0.691669
$302$	3.00000	0.172631
$303$	4.00000	0.229794
$304$	−5.00000	−0.286770
$305$	0	0
$306$	2.00000	0.114332
$307$	−26.0000	−1.48390	−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000	−1.48390	−0.741949	+	0.670456i	$0.766098\pi$
$308$	6.00000	0.341882
$309$	8.00000	0.455104
$310$	8.00000	0.454369
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−6.00000	−0.339683
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−23.0000	−1.29797
$315$	−4.00000	−0.225374
$316$	−8.00000	−0.450035
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	6.00000	0.336463
$319$	18.0000	1.00781
$320$	4.00000	0.223607
$321$	0	0
$322$	−2.00000	−0.111456
$323$	10.0000	0.556415
$324$	1.00000	0.0555556
$325$	−66.0000	−3.66102
$326$	16.0000	0.886158
$327$	−8.00000	−0.442401
$328$	9.00000	0.496942
$329$	8.00000	0.441054
$330$	−24.0000	−1.32116
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	1.00000	0.0548821
$333$	7.00000	0.383598
$334$	15.0000	0.820763
$335$	12.0000	0.655630
$336$	1.00000	0.0545545
$337$	19.0000	1.03500	0.517498	−	0.855684i	$-0.326864\pi$
$337$	19.0000	1.03500	0.517498	+	0.855684i	$0.326864\pi$
$338$	−23.0000	−1.25104
$339$	−1.00000	−0.0543125
$340$	−8.00000	−0.433861
$341$	12.0000	0.649836
$342$	5.00000	0.270369
$343$	13.0000	0.701934
$344$	−12.0000	−0.646997
$345$	8.00000	0.430706
$346$	5.00000	0.268802
$347$	9.00000	0.483145	0.241573	−	0.970383i	$-0.422337\pi$
$347$	9.00000	0.483145	0.241573	+	0.970383i	$0.422337\pi$
$348$	3.00000	0.160817
$349$	−9.00000	−0.481759	−0.240879	−	0.970555i	$-0.577436\pi$
$349$	−9.00000	−0.481759	−0.240879	+	0.970555i	$0.577436\pi$
$350$	11.0000	0.587975
$351$	6.00000	0.320256
$352$	6.00000	0.319801
$353$	7.00000	0.372572	0.186286	−	0.982496i	$-0.440355\pi$
$353$	7.00000	0.372572	0.186286	+	0.982496i	$0.440355\pi$
$354$	−8.00000	−0.425195
$355$	40.0000	2.12298
$356$	−11.0000	−0.582999
$357$	−2.00000	−0.105851
$358$	−14.0000	−0.739923
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	−4.00000	−0.210819
$361$	6.00000	0.315789
$362$	10.0000	0.525588
$363$	−25.0000	−1.31216
$364$	6.00000	0.314485
$365$	40.0000	2.09370
$366$	0	0
$367$	−17.0000	−0.887393	−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000	−0.887393	−0.443696	+	0.896177i	$0.646333\pi$
$368$	−2.00000	−0.104257
$369$	−9.00000	−0.468521
$370$	−28.0000	−1.45565
$371$	−6.00000	−0.311504
$372$	2.00000	0.103695
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	−12.0000	−0.620505
$375$	−24.0000	−1.23935
$376$	8.00000	0.412568
$377$	18.0000	0.927047
$378$	−1.00000	−0.0514344
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	−20.0000	−1.02598
$381$	−12.0000	−0.614779
$382$	5.00000	0.255822
$383$	−36.0000	−1.83951	−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000	−1.83951	−0.919757	+	0.392488i	$0.871614\pi$
$384$	1.00000	0.0510310
$385$	24.0000	1.22315
$386$	23.0000	1.17067
$387$	12.0000	0.609994
$388$	0	0
$389$	21.0000	1.06474	0.532371	−	0.846511i	$-0.321301\pi$
$389$	21.0000	1.06474	0.532371	+	0.846511i	$0.321301\pi$
$390$	−24.0000	−1.21529
$391$	4.00000	0.202289
$392$	6.00000	0.303046
$393$	1.00000	0.0504433
$394$	−3.00000	−0.151138
$395$	−32.0000	−1.61009
$396$	−6.00000	−0.301511
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	−28.0000	−1.40351
$399$	−5.00000	−0.250313
$400$	11.0000	0.550000
$401$	16.0000	0.799002	0.399501	−	0.916733i	$-0.369183\pi$
$401$	16.0000	0.799002	0.399501	+	0.916733i	$0.369183\pi$
$402$	3.00000	0.149626
$403$	12.0000	0.597763
$404$	−4.00000	−0.199007
$405$	4.00000	0.198762
$406$	−3.00000	−0.148888
$407$	−42.0000	−2.08186
$408$	−2.00000	−0.0990148
$409$	−37.0000	−1.82953	−0.914766	−	0.403984i	$-0.867625\pi$
$409$	−37.0000	−1.82953	−0.914766	+	0.403984i	$0.867625\pi$
$410$	36.0000	1.77791
$411$	16.0000	0.789222
$412$	−8.00000	−0.394132
$413$	8.00000	0.393654
$414$	2.00000	0.0982946
$415$	4.00000	0.196352
$416$	6.00000	0.294174
$417$	0	0
$418$	−30.0000	−1.46735
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	4.00000	0.195180
$421$	−30.0000	−1.46211	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000	−1.46211	−0.731055	+	0.682318i	$0.760972\pi$
$422$	22.0000	1.07094
$423$	−8.00000	−0.388973
$424$	−6.00000	−0.291386
$425$	−22.0000	−1.06716
$426$	10.0000	0.484502
$427$	0	0
$428$	0	0
$429$	−36.0000	−1.73810
$430$	−48.0000	−2.31477
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	−1.00000	−0.0481125
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	−2.00000	−0.0960031
$435$	12.0000	0.575356
$436$	8.00000	0.383131
$437$	10.0000	0.478365
$438$	10.0000	0.477818
$439$	7.00000	0.334092	0.167046	−	0.985949i	$-0.446577\pi$
$439$	7.00000	0.334092	0.167046	+	0.985949i	$0.446577\pi$
$440$	24.0000	1.14416
$441$	−6.00000	−0.285714
$442$	−12.0000	−0.570782
$443$	−27.0000	−1.28281	−0.641404	−	0.767203i	$-0.721648\pi$
$443$	−27.0000	−1.28281	−0.641404	+	0.767203i	$0.721648\pi$
$444$	−7.00000	−0.332205
$445$	−44.0000	−2.08580
$446$	−2.00000	−0.0947027
$447$	6.00000	0.283790
$448$	−1.00000	−0.0472456
$449$	−28.0000	−1.32140	−0.660701	−	0.750649i	$-0.729741\pi$
$449$	−28.0000	−1.32140	−0.660701	+	0.750649i	$0.729741\pi$
$450$	−11.0000	−0.518545
$451$	54.0000	2.54276
$452$	1.00000	0.0470360
$453$	3.00000	0.140952
$454$	−21.0000	−0.985579
$455$	24.0000	1.12514
$456$	−5.00000	−0.234146
$457$	35.0000	1.63723	0.818615	−	0.574342i	$-0.194742\pi$
$457$	35.0000	1.63723	0.818615	+	0.574342i	$0.194742\pi$
$458$	22.0000	1.02799
$459$	2.00000	0.0933520
$460$	−8.00000	−0.373002
$461$	−27.0000	−1.25752	−0.628758	−	0.777601i	$-0.716436\pi$
$461$	−27.0000	−1.25752	−0.628758	+	0.777601i	$0.716436\pi$
$462$	6.00000	0.279145
$463$	34.0000	1.58011	0.790057	−	0.613033i	$-0.210051\pi$
$463$	34.0000	1.58011	0.790057	+	0.613033i	$0.210051\pi$
$464$	−3.00000	−0.139272
$465$	8.00000	0.370991
$466$	25.0000	1.15810
$467$	10.0000	0.462745	0.231372	−	0.972865i	$-0.425678\pi$
$467$	10.0000	0.462745	0.231372	+	0.972865i	$0.425678\pi$
$468$	−6.00000	−0.277350
$469$	−3.00000	−0.138527
$470$	32.0000	1.47605
$471$	−23.0000	−1.05978
$472$	8.00000	0.368230
$473$	−72.0000	−3.31056
$474$	−8.00000	−0.367452
$475$	−55.0000	−2.52357
$476$	2.00000	0.0916698
$477$	6.00000	0.274721
$478$	−5.00000	−0.228695
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	4.00000	0.182574
$481$	−42.0000	−1.91504
$482$	28.0000	1.27537
$483$	−2.00000	−0.0910032
$484$	25.0000	1.13636
$485$	0	0
$486$	1.00000	0.0453609
$487$	4.00000	0.181257	0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000	0.181257	0.0906287	+	0.995885i	$0.471112\pi$
$488$	0	0
$489$	16.0000	0.723545
$490$	24.0000	1.08421
$491$	9.00000	0.406164	0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000	0.406164	0.203082	+	0.979162i	$0.434904\pi$
$492$	9.00000	0.405751
$493$	6.00000	0.270226
$494$	−30.0000	−1.34976
$495$	−24.0000	−1.07872
$496$	−2.00000	−0.0898027
$497$	−10.0000	−0.448561
$498$	1.00000	0.0448111
$499$	−41.0000	−1.83541	−0.917706	−	0.397260i	$-0.869961\pi$
$499$	−41.0000	−1.83541	−0.917706	+	0.397260i	$0.869961\pi$
$500$	24.0000	1.07331
$501$	15.0000	0.670151
$502$	−11.0000	−0.490954
$503$	−8.00000	−0.356702	−0.178351	−	0.983967i	$-0.557076\pi$
$503$	−8.00000	−0.356702	−0.178351	+	0.983967i	$0.557076\pi$
$504$	1.00000	0.0445435
$505$	−16.0000	−0.711991
$506$	−12.0000	−0.533465
$507$	−23.0000	−1.02147
$508$	12.0000	0.532414
$509$	3.00000	0.132973	0.0664863	−	0.997787i	$-0.478821\pi$
$509$	3.00000	0.132973	0.0664863	+	0.997787i	$0.478821\pi$
$510$	−8.00000	−0.354246
$511$	−10.0000	−0.442374
$512$	−1.00000	−0.0441942
$513$	5.00000	0.220755
$514$	22.0000	0.970378
$515$	−32.0000	−1.41009
$516$	−12.0000	−0.528271
$517$	48.0000	2.11104
$518$	7.00000	0.307562
$519$	5.00000	0.219476
$520$	24.0000	1.05247
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	3.00000	0.131306
$523$	40.0000	1.74908	0.874539	−	0.484955i	$-0.161164\pi$
$523$	40.0000	1.74908	0.874539	+	0.484955i	$0.161164\pi$
$524$	−1.00000	−0.0436852
$525$	11.0000	0.480079
$526$	−12.0000	−0.523225
$527$	4.00000	0.174243
$528$	6.00000	0.261116
$529$	−19.0000	−0.826087
$530$	−24.0000	−1.04249
$531$	−8.00000	−0.347170
$532$	5.00000	0.216777
$533$	54.0000	2.33900
$534$	−11.0000	−0.476017
$535$	0	0
$536$	−3.00000	−0.129580
$537$	−14.0000	−0.604145
$538$	−10.0000	−0.431131
$539$	36.0000	1.55063
$540$	−4.00000	−0.172133
$541$	−23.0000	−0.988847	−0.494424	−	0.869221i	$-0.664621\pi$
$541$	−23.0000	−0.988847	−0.494424	+	0.869221i	$0.664621\pi$
$542$	−16.0000	−0.687259
$543$	10.0000	0.429141
$544$	2.00000	0.0857493
$545$	32.0000	1.37073
$546$	6.00000	0.256776
$547$	17.0000	0.726868	0.363434	−	0.931620i	$-0.381604\pi$
$547$	17.0000	0.726868	0.363434	+	0.931620i	$0.381604\pi$
$548$	−16.0000	−0.683486
$549$	0	0
$550$	66.0000	2.81425
$551$	15.0000	0.639021
$552$	−2.00000	−0.0851257
$553$	8.00000	0.340195
$554$	−12.0000	−0.509831
$555$	−28.0000	−1.18853
$556$	0	0
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	2.00000	0.0846668
$559$	−72.0000	−3.04528
$560$	−4.00000	−0.169031
$561$	−12.0000	−0.506640
$562$	−14.0000	−0.590554
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	8.00000	0.336861
$565$	4.00000	0.168281
$566$	14.0000	0.588464
$567$	−1.00000	−0.0419961
$568$	−10.0000	−0.419591
$569$	11.0000	0.461144	0.230572	−	0.973055i	$-0.425940\pi$
$569$	11.0000	0.461144	0.230572	+	0.973055i	$0.425940\pi$
$570$	−20.0000	−0.837708
$571$	41.0000	1.71580	0.857898	−	0.513820i	$-0.171770\pi$
$571$	41.0000	1.71580	0.857898	+	0.513820i	$0.171770\pi$
$572$	36.0000	1.50524
$573$	5.00000	0.208878
$574$	−9.00000	−0.375653
$575$	−22.0000	−0.917463
$576$	1.00000	0.0416667
$577$	17.0000	0.707719	0.353860	−	0.935299i	$-0.384869\pi$
$577$	17.0000	0.707719	0.353860	+	0.935299i	$0.384869\pi$
$578$	13.0000	0.540729
$579$	23.0000	0.955847
$580$	−12.0000	−0.498273
$581$	−1.00000	−0.0414870
$582$	0	0
$583$	−36.0000	−1.49097
$584$	−10.0000	−0.413803
$585$	−24.0000	−0.992278
$586$	−14.0000	−0.578335
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	6.00000	0.247436
$589$	10.0000	0.412043
$590$	32.0000	1.31742
$591$	−3.00000	−0.123404
$592$	7.00000	0.287698
$593$	−20.0000	−0.821302	−0.410651	−	0.911793i	$-0.634698\pi$
$593$	−20.0000	−0.821302	−0.410651	+	0.911793i	$0.634698\pi$
$594$	−6.00000	−0.246183
$595$	8.00000	0.327968
$596$	−6.00000	−0.245770
$597$	−28.0000	−1.14596
$598$	−12.0000	−0.490716
$599$	9.00000	0.367730	0.183865	−	0.982952i	$-0.441139\pi$
$599$	9.00000	0.367730	0.183865	+	0.982952i	$0.441139\pi$
$600$	11.0000	0.449073
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	12.0000	0.489083
$603$	3.00000	0.122169
$604$	−3.00000	−0.122068
$605$	100.000	4.06558
$606$	−4.00000	−0.162489
$607$	2.00000	0.0811775	0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000	0.0811775	0.0405887	+	0.999176i	$0.487077\pi$
$608$	5.00000	0.202777
$609$	−3.00000	−0.121566
$610$	0	0
$611$	48.0000	1.94187
$612$	−2.00000	−0.0808452
$613$	−18.0000	−0.727013	−0.363507	−	0.931592i	$-0.618421\pi$
$613$	−18.0000	−0.727013	−0.363507	+	0.931592i	$0.618421\pi$
$614$	26.0000	1.04927
$615$	36.0000	1.45166
$616$	−6.00000	−0.241747
$617$	32.0000	1.28827	0.644136	−	0.764911i	$-0.277217\pi$
$617$	32.0000	1.28827	0.644136	+	0.764911i	$0.277217\pi$
$618$	−8.00000	−0.321807
$619$	29.0000	1.16561	0.582804	−	0.812613i	$-0.301955\pi$
$619$	29.0000	1.16561	0.582804	+	0.812613i	$0.301955\pi$
$620$	−8.00000	−0.321288
$621$	2.00000	0.0802572
$622$	0	0
$623$	11.0000	0.440706
$624$	6.00000	0.240192
$625$	41.0000	1.64000
$626$	−14.0000	−0.559553
$627$	−30.0000	−1.19808
$628$	23.0000	0.917800
$629$	−14.0000	−0.558217
$630$	4.00000	0.159364
$631$	−13.0000	−0.517522	−0.258761	−	0.965941i	$-0.583314\pi$
$631$	−13.0000	−0.517522	−0.258761	+	0.965941i	$0.583314\pi$
$632$	8.00000	0.318223
$633$	22.0000	0.874421
$634$	−18.0000	−0.714871
$635$	48.0000	1.90482
$636$	−6.00000	−0.237915
$637$	36.0000	1.42637
$638$	−18.0000	−0.712627
$639$	10.0000	0.395594
$640$	−4.00000	−0.158114
$641$	−9.00000	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$642$	0	0
$643$	−36.0000	−1.41970	−0.709851	−	0.704352i	$-0.751238\pi$
$643$	−36.0000	−1.41970	−0.709851	+	0.704352i	$0.751238\pi$
$644$	2.00000	0.0788110
$645$	−48.0000	−1.89000
$646$	−10.0000	−0.393445
$647$	−19.0000	−0.746967	−0.373484	−	0.927637i	$-0.621837\pi$
$647$	−19.0000	−0.746967	−0.373484	+	0.927637i	$0.621837\pi$
$648$	−1.00000	−0.0392837
$649$	48.0000	1.88416
$650$	66.0000	2.58873
$651$	−2.00000	−0.0783862
$652$	−16.0000	−0.626608
$653$	−26.0000	−1.01746	−0.508729	−	0.860927i	$-0.669885\pi$
$653$	−26.0000	−1.01746	−0.508729	+	0.860927i	$0.669885\pi$
$654$	8.00000	0.312825
$655$	−4.00000	−0.156293
$656$	−9.00000	−0.351391
$657$	10.0000	0.390137
$658$	−8.00000	−0.311872
$659$	−26.0000	−1.01282	−0.506408	−	0.862294i	$-0.669027\pi$
$659$	−26.0000	−1.01282	−0.506408	+	0.862294i	$0.669027\pi$
$660$	24.0000	0.934199
$661$	−23.0000	−0.894596	−0.447298	−	0.894385i	$-0.647614\pi$
$661$	−23.0000	−0.894596	−0.447298	+	0.894385i	$0.647614\pi$
$662$	−28.0000	−1.08825
$663$	−12.0000	−0.466041
$664$	−1.00000	−0.0388075
$665$	20.0000	0.775567
$666$	−7.00000	−0.271244
$667$	6.00000	0.232321
$668$	−15.0000	−0.580367
$669$	−2.00000	−0.0773245
$670$	−12.0000	−0.463600
$671$	0	0
$672$	−1.00000	−0.0385758
$673$	−20.0000	−0.770943	−0.385472	−	0.922720i	$-0.625961\pi$
$673$	−20.0000	−0.770943	−0.385472	+	0.922720i	$0.625961\pi$
$674$	−19.0000	−0.731853
$675$	−11.0000	−0.423390
$676$	23.0000	0.884615
$677$	−45.0000	−1.72949	−0.864745	−	0.502211i	$-0.832520\pi$
$677$	−45.0000	−1.72949	−0.864745	+	0.502211i	$0.832520\pi$
$678$	1.00000	0.0384048
$679$	0	0
$680$	8.00000	0.306786
$681$	−21.0000	−0.804722
$682$	−12.0000	−0.459504
$683$	−10.0000	−0.382639	−0.191320	−	0.981528i	$-0.561277\pi$
$683$	−10.0000	−0.382639	−0.191320	+	0.981528i	$0.561277\pi$
$684$	−5.00000	−0.191180
$685$	−64.0000	−2.44531
$686$	−13.0000	−0.496342
$687$	22.0000	0.839352
$688$	12.0000	0.457496
$689$	−36.0000	−1.37149
$690$	−8.00000	−0.304555
$691$	14.0000	0.532585	0.266293	−	0.963892i	$-0.414201\pi$
$691$	14.0000	0.532585	0.266293	+	0.963892i	$0.414201\pi$
$692$	−5.00000	−0.190071
$693$	6.00000	0.227921
$694$	−9.00000	−0.341635
$695$	0	0
$696$	−3.00000	−0.113715
$697$	18.0000	0.681799
$698$	9.00000	0.340655
$699$	25.0000	0.945587
$700$	−11.0000	−0.415761
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	−6.00000	−0.226455
$703$	−35.0000	−1.32005
$704$	−6.00000	−0.226134
$705$	32.0000	1.20519
$706$	−7.00000	−0.263448
$707$	4.00000	0.150435
$708$	8.00000	0.300658
$709$	42.0000	1.57734	0.788672	−	0.614815i	$-0.210769\pi$
$709$	42.0000	1.57734	0.788672	+	0.614815i	$0.210769\pi$
$710$	−40.0000	−1.50117
$711$	−8.00000	−0.300023
$712$	11.0000	0.412242
$713$	4.00000	0.149801
$714$	2.00000	0.0748481
$715$	144.000	5.38529
$716$	14.0000	0.523205
$717$	−5.00000	−0.186728
$718$	32.0000	1.19423
$719$	53.0000	1.97657	0.988283	−	0.152631i	$-0.0487746\pi$
$719$	53.0000	1.97657	0.988283	+	0.152631i	$0.0487746\pi$
$720$	4.00000	0.149071
$721$	8.00000	0.297936
$722$	−6.00000	−0.223297
$723$	28.0000	1.04133
$724$	−10.0000	−0.371647
$725$	−33.0000	−1.22559
$726$	25.0000	0.927837
$727$	−20.0000	−0.741759	−0.370879	−	0.928681i	$-0.620944\pi$
$727$	−20.0000	−0.741759	−0.370879	+	0.928681i	$0.620944\pi$
$728$	−6.00000	−0.222375
$729$	1.00000	0.0370370
$730$	−40.0000	−1.48047
$731$	−24.0000	−0.887672
$732$	0	0
$733$	39.0000	1.44050	0.720249	−	0.693716i	$-0.244028\pi$
$733$	39.0000	1.44050	0.720249	+	0.693716i	$0.244028\pi$
$734$	17.0000	0.627481
$735$	24.0000	0.885253
$736$	2.00000	0.0737210
$737$	−18.0000	−0.663039
$738$	9.00000	0.331295
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	28.0000	1.02930
$741$	−30.0000	−1.10208
$742$	6.00000	0.220267
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	−2.00000	−0.0733236
$745$	−24.0000	−0.879292
$746$	14.0000	0.512576
$747$	1.00000	0.0365881
$748$	12.0000	0.438763
$749$	0	0
$750$	24.0000	0.876356
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	−8.00000	−0.291730
$753$	−11.0000	−0.400862
$754$	−18.0000	−0.655521
$755$	−12.0000	−0.436725
$756$	1.00000	0.0363696
$757$	16.0000	0.581530	0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000	0.581530	0.290765	+	0.956795i	$0.406090\pi$
$758$	−16.0000	−0.581146
$759$	−12.0000	−0.435572
$760$	20.0000	0.725476
$761$	26.0000	0.942499	0.471250	−	0.882000i	$-0.343803\pi$
$761$	26.0000	0.942499	0.471250	+	0.882000i	$0.343803\pi$
$762$	12.0000	0.434714
$763$	−8.00000	−0.289619
$764$	−5.00000	−0.180894
$765$	−8.00000	−0.289241
$766$	36.0000	1.30073
$767$	48.0000	1.73318
$768$	−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	$0.155186\pi$
$770$	−24.0000	−0.864900
$771$	22.0000	0.792311
$772$	−23.0000	−0.827788
$773$	7.00000	0.251773	0.125886	−	0.992045i	$-0.459823\pi$
$773$	7.00000	0.251773	0.125886	+	0.992045i	$0.459823\pi$
$774$	−12.0000	−0.431331
$775$	−22.0000	−0.790263
$776$	0	0
$777$	7.00000	0.251124
$778$	−21.0000	−0.752886
$779$	45.0000	1.61229
$780$	24.0000	0.859338
$781$	−60.0000	−2.14697
$782$	−4.00000	−0.143040
$783$	3.00000	0.107211
$784$	−6.00000	−0.214286
$785$	92.0000	3.28362
$786$	−1.00000	−0.0356688
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	3.00000	0.106871
$789$	−12.0000	−0.427211
$790$	32.0000	1.13851
$791$	−1.00000	−0.0355559
$792$	6.00000	0.213201
$793$	0	0
$794$	−34.0000	−1.20661
$795$	−24.0000	−0.851192
$796$	28.0000	0.992434
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	5.00000	0.176998
$799$	16.0000	0.566039
$800$	−11.0000	−0.388909
$801$	−11.0000	−0.388666
$802$	−16.0000	−0.564980
$803$	−60.0000	−2.11735
$804$	−3.00000	−0.105802
$805$	8.00000	0.281963
$806$	−12.0000	−0.422682
$807$	−10.0000	−0.352017
$808$	4.00000	0.140720
$809$	−40.0000	−1.40633	−0.703163	−	0.711029i	$-0.748229\pi$
$809$	−40.0000	−1.40633	−0.703163	+	0.711029i	$0.748229\pi$
$810$	−4.00000	−0.140546
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	3.00000	0.105279
$813$	−16.0000	−0.561144
$814$	42.0000	1.47210
$815$	−64.0000	−2.24182
$816$	2.00000	0.0700140
$817$	−60.0000	−2.09913
$818$	37.0000	1.29367
$819$	6.00000	0.209657
$820$	−36.0000	−1.25717
$821$	−28.0000	−0.977207	−0.488603	−	0.872506i	$-0.662493\pi$
$821$	−28.0000	−0.977207	−0.488603	+	0.872506i	$0.662493\pi$
$822$	−16.0000	−0.558064
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	8.00000	0.278693
$825$	66.0000	2.29783
$826$	−8.00000	−0.278356
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−2.00000	−0.0695048
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	−4.00000	−0.138842
$831$	−12.0000	−0.416275
$832$	−6.00000	−0.208013
$833$	12.0000	0.415775
$834$	0	0
$835$	−60.0000	−2.07639
$836$	30.0000	1.03757
$837$	2.00000	0.0691301
$838$	12.0000	0.414533
$839$	−35.0000	−1.20833	−0.604167	−	0.796858i	$-0.706494\pi$
$839$	−35.0000	−1.20833	−0.604167	+	0.796858i	$0.706494\pi$
$840$	−4.00000	−0.138013
$841$	−20.0000	−0.689655
$842$	30.0000	1.03387
$843$	−14.0000	−0.482186
$844$	−22.0000	−0.757271
$845$	92.0000	3.16490
$846$	8.00000	0.275046
$847$	−25.0000	−0.859010
$848$	6.00000	0.206041
$849$	14.0000	0.480479
$850$	22.0000	0.754594
$851$	−14.0000	−0.479914
$852$	−10.0000	−0.342594
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	0	0
$855$	−20.0000	−0.683986
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	36.0000	1.22902
$859$	7.00000	0.238837	0.119418	−	0.992844i	$-0.461897\pi$
$859$	7.00000	0.238837	0.119418	+	0.992844i	$0.461897\pi$
$860$	48.0000	1.63679
$861$	−9.00000	−0.306719
$862$	20.0000	0.681203
$863$	−43.0000	−1.46374	−0.731869	−	0.681446i	$-0.761351\pi$
$863$	−43.0000	−1.46374	−0.731869	+	0.681446i	$0.761351\pi$
$864$	1.00000	0.0340207
$865$	−20.0000	−0.680020
$866$	6.00000	0.203888
$867$	13.0000	0.441503
$868$	2.00000	0.0678844
$869$	48.0000	1.62829
$870$	−12.0000	−0.406838
$871$	−18.0000	−0.609907
$872$	−8.00000	−0.270914
$873$	0	0
$874$	−10.0000	−0.338255
$875$	−24.0000	−0.811348
$876$	−10.0000	−0.337869
$877$	−12.0000	−0.405211	−0.202606	−	0.979260i	$-0.564941\pi$
$877$	−12.0000	−0.405211	−0.202606	+	0.979260i	$0.564941\pi$
$878$	−7.00000	−0.236239
$879$	−14.0000	−0.472208
$880$	−24.0000	−0.809040
$881$	−32.0000	−1.07811	−0.539054	−	0.842271i	$-0.681218\pi$
$881$	−32.0000	−1.07811	−0.539054	+	0.842271i	$0.681218\pi$
$882$	6.00000	0.202031
$883$	1.00000	0.0336527	0.0168263	−	0.999858i	$-0.494644\pi$
$883$	1.00000	0.0336527	0.0168263	+	0.999858i	$0.494644\pi$
$884$	12.0000	0.403604
$885$	32.0000	1.07567
$886$	27.0000	0.907083
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	7.00000	0.234905
$889$	−12.0000	−0.402467
$890$	44.0000	1.47488
$891$	−6.00000	−0.201008
$892$	2.00000	0.0669650
$893$	40.0000	1.33855
$894$	−6.00000	−0.200670
$895$	56.0000	1.87187
$896$	1.00000	0.0334077
$897$	−12.0000	−0.400668
$898$	28.0000	0.934372
$899$	6.00000	0.200111
$900$	11.0000	0.366667
$901$	−12.0000	−0.399778
$902$	−54.0000	−1.79800
$903$	12.0000	0.399335
$904$	−1.00000	−0.0332595
$905$	−40.0000	−1.32964
$906$	−3.00000	−0.0996683
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	21.0000	0.696909
$909$	−4.00000	−0.132672
$910$	−24.0000	−0.795592
$911$	−5.00000	−0.165657	−0.0828287	−	0.996564i	$-0.526395\pi$
$911$	−5.00000	−0.165657	−0.0828287	+	0.996564i	$0.526395\pi$
$912$	5.00000	0.165567
$913$	−6.00000	−0.198571
$914$	−35.0000	−1.15770
$915$	0	0
$916$	−22.0000	−0.726900
$917$	1.00000	0.0330229
$918$	−2.00000	−0.0660098
$919$	−26.0000	−0.857661	−0.428830	−	0.903385i	$-0.641074\pi$
$919$	−26.0000	−0.857661	−0.428830	+	0.903385i	$0.641074\pi$
$920$	8.00000	0.263752
$921$	26.0000	0.856729
$922$	27.0000	0.889198
$923$	−60.0000	−1.97492
$924$	−6.00000	−0.197386
$925$	77.0000	2.53174
$926$	−34.0000	−1.11731
$927$	−8.00000	−0.262754
$928$	3.00000	0.0984798
$929$	−7.00000	−0.229663	−0.114831	−	0.993385i	$-0.536633\pi$
$929$	−7.00000	−0.229663	−0.114831	+	0.993385i	$0.536633\pi$
$930$	−8.00000	−0.262330
$931$	30.0000	0.983210
$932$	−25.0000	−0.818902
$933$	0	0
$934$	−10.0000	−0.327210
$935$	48.0000	1.56977
$936$	6.00000	0.196116
$937$	−7.00000	−0.228680	−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000	−0.228680	−0.114340	+	0.993442i	$0.536475\pi$
$938$	3.00000	0.0979535
$939$	−14.0000	−0.456873
$940$	−32.0000	−1.04372
$941$	−7.00000	−0.228193	−0.114097	−	0.993470i	$-0.536397\pi$
$941$	−7.00000	−0.228193	−0.114097	+	0.993470i	$0.536397\pi$
$942$	23.0000	0.749380
$943$	18.0000	0.586161
$944$	−8.00000	−0.260378
$945$	4.00000	0.130120
$946$	72.0000	2.34092
$947$	41.0000	1.33232	0.666160	−	0.745808i	$-0.267937\pi$
$947$	41.0000	1.33232	0.666160	+	0.745808i	$0.267937\pi$
$948$	8.00000	0.259828
$949$	−60.0000	−1.94768
$950$	55.0000	1.78444
$951$	−18.0000	−0.583690
$952$	−2.00000	−0.0648204
$953$	23.0000	0.745043	0.372522	−	0.928024i	$-0.378493\pi$
$953$	23.0000	0.745043	0.372522	+	0.928024i	$0.378493\pi$
$954$	−6.00000	−0.194257
$955$	−20.0000	−0.647185
$956$	5.00000	0.161712
$957$	−18.0000	−0.581857
$958$	0	0
$959$	16.0000	0.516667
$960$	−4.00000	−0.129099
$961$	−27.0000	−0.870968
$962$	42.0000	1.35413
$963$	0	0
$964$	−28.0000	−0.901819
$965$	−92.0000	−2.96158
$966$	2.00000	0.0643489
$967$	−6.00000	−0.192947	−0.0964735	−	0.995336i	$-0.530756\pi$
$967$	−6.00000	−0.192947	−0.0964735	+	0.995336i	$0.530756\pi$
$968$	−25.0000	−0.803530
$969$	−10.0000	−0.321246
$970$	0	0
$971$	−7.00000	−0.224641	−0.112320	−	0.993672i	$-0.535828\pi$
$971$	−7.00000	−0.224641	−0.112320	+	0.993672i	$0.535828\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−4.00000	−0.128168
$975$	66.0000	2.11369
$976$	0	0
$977$	−51.0000	−1.63163	−0.815817	−	0.578310i	$-0.803713\pi$
$977$	−51.0000	−1.63163	−0.815817	+	0.578310i	$0.803713\pi$
$978$	−16.0000	−0.511624
$979$	66.0000	2.10937
$980$	−24.0000	−0.766652
$981$	8.00000	0.255420
$982$	−9.00000	−0.287202
$983$	−54.0000	−1.72233	−0.861166	−	0.508323i	$-0.830265\pi$
$983$	−54.0000	−1.72233	−0.861166	+	0.508323i	$0.830265\pi$
$984$	−9.00000	−0.286910
$985$	12.0000	0.382352
$986$	−6.00000	−0.191079
$987$	−8.00000	−0.254643
$988$	30.0000	0.954427
$989$	−24.0000	−0.763156
$990$	24.0000	0.762770
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	2.00000	0.0635001
$993$	−28.0000	−0.888553
$994$	10.0000	0.317181
$995$	112.000	3.55064
$996$	−1.00000	−0.0316862
$997$	8.00000	0.253363	0.126681	−	0.991943i	$-0.459567\pi$
$997$	8.00000	0.253363	0.126681	+	0.991943i	$0.459567\pi$
$998$	41.0000	1.29783
$999$	−7.00000	−0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	786.2.a.e.1.1	✓	1
3.2	odd	2		2358.2.a.l.1.1		1
4.3	odd	2		6288.2.a.q.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
786.2.a.e.1.1	✓	1	1.1	even	1	trivial
2358.2.a.l.1.1		1	3.2	odd	2
6288.2.a.q.1.1		1	4.3	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:	\( N \)	\(=\)	\( 786 = 2 \cdot 3 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	786.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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