LMFDB - Embedded newform 2023.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2023,2,Mod(1,2023)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2023, 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("2023.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2023 = 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2023.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:	$16.1537363289$
Analytic rank:	$2$
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	$\chi$	$=$	2023.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -3.00000 q^{3} -1.00000 q^{4} -4.00000 q^{5} +3.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -3.00000 q^{3} -1.00000 q^{4} -4.00000 q^{5} +3.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +6.00000 q^{9} +4.00000 q^{10} +3.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +12.0000 q^{15} -1.00000 q^{16} -6.00000 q^{18} -7.00000 q^{19} +4.00000 q^{20} -3.00000 q^{21} -4.00000 q^{23} -9.00000 q^{24} +11.0000 q^{25} +2.00000 q^{26} -9.00000 q^{27} -1.00000 q^{28} -3.00000 q^{29} -12.0000 q^{30} -7.00000 q^{31} -5.00000 q^{32} -4.00000 q^{35} -6.00000 q^{36} -10.0000 q^{37} +7.00000 q^{38} +6.00000 q^{39} -12.0000 q^{40} +3.00000 q^{42} -2.00000 q^{43} -24.0000 q^{45} +4.00000 q^{46} +3.00000 q^{47} +3.00000 q^{48} +1.00000 q^{49} -11.0000 q^{50} +2.00000 q^{52} -3.00000 q^{53} +9.00000 q^{54} +3.00000 q^{56} +21.0000 q^{57} +3.00000 q^{58} -9.00000 q^{59} -12.0000 q^{60} -8.00000 q^{61} +7.00000 q^{62} +6.00000 q^{63} +7.00000 q^{64} +8.00000 q^{65} -8.00000 q^{67} +12.0000 q^{69} +4.00000 q^{70} -2.00000 q^{71} +18.0000 q^{72} -6.00000 q^{73} +10.0000 q^{74} -33.0000 q^{75} +7.00000 q^{76} -6.00000 q^{78} -6.00000 q^{79} +4.00000 q^{80} +9.00000 q^{81} -1.00000 q^{83} +3.00000 q^{84} +2.00000 q^{86} +9.00000 q^{87} -8.00000 q^{89} +24.0000 q^{90} -2.00000 q^{91} +4.00000 q^{92} +21.0000 q^{93} -3.00000 q^{94} +28.0000 q^{95} +15.0000 q^{96} -2.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−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
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	3.00000	1.22474
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	3.00000	1.06066
$9$	6.00000	2.00000
$10$	4.00000	1.26491
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	3.00000	0.866025
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−1.00000	−0.267261
$15$	12.0000	3.09839
$16$	−1.00000	−0.250000
$17$	0	0
$17$	0	0
$18$	−6.00000	−1.41421
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	4.00000	0.894427
$21$	−3.00000	−0.654654
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−9.00000	−1.83712
$25$	11.0000	2.20000
$26$	2.00000	0.392232
$27$	−9.00000	−1.73205
$28$	−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	$0.589851\pi$
$30$	−12.0000	−2.19089
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	0	0
$35$	−4.00000	−0.676123
$36$	−6.00000	−1.00000
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	7.00000	1.13555
$39$	6.00000	0.960769
$40$	−12.0000	−1.89737
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	3.00000	0.462910
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	−24.0000	−3.57771
$46$	4.00000	0.589768
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	3.00000	0.433013
$49$	1.00000	0.142857
$50$	−11.0000	−1.55563
$51$	0	0
$52$	2.00000	0.277350
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	9.00000	1.22474
$55$	0	0
$56$	3.00000	0.400892
$57$	21.0000	2.78152
$58$	3.00000	0.393919
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	−12.0000	−1.54919
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	7.00000	0.889001
$63$	6.00000	0.755929
$64$	7.00000	0.875000
$65$	8.00000	0.992278
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	12.0000	1.44463
$70$	4.00000	0.478091
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	18.0000	2.12132
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	10.0000	1.16248
$75$	−33.0000	−3.81051
$76$	7.00000	0.802955
$77$	0	0
$78$	−6.00000	−0.679366
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	4.00000	0.447214
$81$	9.00000	1.00000
$82$	0	0
$83$	−1.00000	−0.109764	−0.0548821	−	0.998493i	$-0.517478\pi$
$83$	−1.00000	−0.109764	−0.0548821	+	0.998493i	$0.517478\pi$
$84$	3.00000	0.327327
$85$	0	0
$86$	2.00000	0.215666
$87$	9.00000	0.964901
$88$	0	0
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	24.0000	2.52982
$91$	−2.00000	−0.209657
$92$	4.00000	0.417029
$93$	21.0000	2.17760
$94$	−3.00000	−0.309426
$95$	28.0000	2.87274
$96$	15.0000	1.53093
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−11.0000	−1.10000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	−6.00000	−0.588348
$105$	12.0000	1.17108
$106$	3.00000	0.291386
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	9.00000	0.866025
$109$	9.00000	0.862044	0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000	0.862044	0.431022	+	0.902342i	$0.358153\pi$
$110$	0	0
$111$	30.0000	2.84747
$112$	−1.00000	−0.0944911
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−21.0000	−1.96683
$115$	16.0000	1.49201
$116$	3.00000	0.278543
$117$	−12.0000	−1.10940
$118$	9.00000	0.828517
$119$	0	0
$120$	36.0000	3.28634
$121$	−11.0000	−1.00000
$122$	8.00000	0.724286
$123$	0	0
$124$	7.00000	0.628619
$125$	−24.0000	−2.14663
$126$	−6.00000	−0.534522
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	3.00000	0.265165
$129$	6.00000	0.528271
$130$	−8.00000	−0.701646
$131$	−19.0000	−1.66004	−0.830019	−	0.557735i	$-0.811670\pi$
$131$	−19.0000	−1.66004	−0.830019	+	0.557735i	$0.811670\pi$
$132$	0	0
$133$	−7.00000	−0.606977
$134$	8.00000	0.691095
$135$	36.0000	3.09839
$136$	0	0
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	−12.0000	−1.02151
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	4.00000	0.338062
$141$	−9.00000	−0.757937
$142$	2.00000	0.167836
$143$	0	0
$144$	−6.00000	−0.500000
$145$	12.0000	0.996546
$146$	6.00000	0.496564
$147$	−3.00000	−0.247436
$148$	10.0000	0.821995
$149$	−7.00000	−0.573462	−0.286731	−	0.958011i	$-0.592569\pi$
$149$	−7.00000	−0.573462	−0.286731	+	0.958011i	$0.592569\pi$
$150$	33.0000	2.69444
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	−21.0000	−1.70332
$153$	0	0
$154$	0	0
$155$	28.0000	2.24901
$156$	−6.00000	−0.480384
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	6.00000	0.477334
$159$	9.00000	0.713746
$160$	20.0000	1.58114
$161$	−4.00000	−0.315244
$162$	−9.00000	−0.707107
$163$	−18.0000	−1.40987	−0.704934	−	0.709273i	$-0.749024\pi$
$163$	−18.0000	−1.40987	−0.704934	+	0.709273i	$0.749024\pi$
$164$	0	0
$165$	0	0
$166$	1.00000	0.0776151
$167$	1.00000	0.0773823	0.0386912	−	0.999251i	$-0.487681\pi$
$167$	1.00000	0.0773823	0.0386912	+	0.999251i	$0.487681\pi$
$168$	−9.00000	−0.694365
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−42.0000	−3.21182
$172$	2.00000	0.152499
$173$	20.0000	1.52057	0.760286	−	0.649589i	$-0.225059\pi$
$173$	20.0000	1.52057	0.760286	+	0.649589i	$0.225059\pi$
$174$	−9.00000	−0.682288
$175$	11.0000	0.831522
$176$	0	0
$177$	27.0000	2.02944
$178$	8.00000	0.599625
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	24.0000	1.78885
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	2.00000	0.148250
$183$	24.0000	1.77413
$184$	−12.0000	−0.884652
$185$	40.0000	2.94086
$186$	−21.0000	−1.53979
$187$	0	0
$188$	−3.00000	−0.218797
$189$	−9.00000	−0.654654
$190$	−28.0000	−2.03133
$191$	−22.0000	−1.59186	−0.795932	−	0.605386i	$-0.793019\pi$
$191$	−22.0000	−1.59186	−0.795932	+	0.605386i	$0.793019\pi$
$192$	−21.0000	−1.51554
$193$	−23.0000	−1.65558	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−23.0000	−1.65558	−0.827788	+	0.561041i	$0.810401\pi$
$194$	2.00000	0.143592
$195$	−24.0000	−1.71868
$196$	−1.00000	−0.0714286
$197$	15.0000	1.06871	0.534353	−	0.845262i	$-0.320555\pi$
$197$	15.0000	1.06871	0.534353	+	0.845262i	$0.320555\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	33.0000	2.33345
$201$	24.0000	1.69283
$202$	−2.00000	−0.140720
$203$	−3.00000	−0.210559
$204$	0	0
$205$	0	0
$206$	7.00000	0.487713
$207$	−24.0000	−1.66812
$208$	2.00000	0.138675
$209$	0	0
$210$	−12.0000	−0.828079
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	3.00000	0.206041
$213$	6.00000	0.411113
$214$	−16.0000	−1.09374
$215$	8.00000	0.545595
$216$	−27.0000	−1.83712
$217$	−7.00000	−0.475191
$218$	−9.00000	−0.609557
$219$	18.0000	1.21633
$220$	0	0
$221$	0	0
$222$	−30.0000	−2.01347
$223$	−15.0000	−1.00447	−0.502237	−	0.864730i	$-0.667490\pi$
$223$	−15.0000	−1.00447	−0.502237	+	0.864730i	$0.667490\pi$
$224$	−5.00000	−0.334077
$225$	66.0000	4.40000
$226$	−2.00000	−0.133038
$227$	9.00000	0.597351	0.298675	−	0.954355i	$-0.403455\pi$
$227$	9.00000	0.597351	0.298675	+	0.954355i	$0.403455\pi$
$228$	−21.0000	−1.39076
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	−16.0000	−1.05501
$231$	0	0
$232$	−9.00000	−0.590879
$233$	−9.00000	−0.589610	−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000	−0.589610	−0.294805	+	0.955557i	$0.595255\pi$
$234$	12.0000	0.784465
$235$	−12.0000	−0.782794
$236$	9.00000	0.585850
$237$	18.0000	1.16923
$238$	0	0
$239$	2.00000	0.129369	0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000	0.129369	0.0646846	+	0.997906i	$0.479396\pi$
$240$	−12.0000	−0.774597
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	11.0000	0.707107
$243$	0	0
$244$	8.00000	0.512148
$245$	−4.00000	−0.255551
$246$	0	0
$247$	14.0000	0.890799
$248$	−21.0000	−1.33350
$249$	3.00000	0.190117
$250$	24.0000	1.51789
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	−6.00000	−0.377964
$253$	0	0
$254$	4.00000	0.250982
$255$	0	0
$256$	−17.0000	−1.06250
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	−6.00000	−0.373544
$259$	−10.0000	−0.621370
$260$	−8.00000	−0.496139
$261$	−18.0000	−1.11417
$262$	19.0000	1.17382
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	7.00000	0.429198
$267$	24.0000	1.46878
$268$	8.00000	0.488678
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	−36.0000	−2.19089
$271$	17.0000	1.03268	0.516338	−	0.856385i	$-0.327295\pi$
$271$	17.0000	1.03268	0.516338	+	0.856385i	$0.327295\pi$
$272$	0	0
$273$	6.00000	0.363137
$274$	−9.00000	−0.543710
$275$	0	0
$276$	−12.0000	−0.722315
$277$	−5.00000	−0.300421	−0.150210	−	0.988654i	$-0.547995\pi$
$277$	−5.00000	−0.300421	−0.150210	+	0.988654i	$0.547995\pi$
$278$	−20.0000	−1.19952
$279$	−42.0000	−2.51447
$280$	−12.0000	−0.717137
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	9.00000	0.535942
$283$	−5.00000	−0.297219	−0.148610	−	0.988896i	$-0.547480\pi$
$283$	−5.00000	−0.297219	−0.148610	+	0.988896i	$0.547480\pi$
$284$	2.00000	0.118678
$285$	−84.0000	−4.97573
$286$	0	0
$287$	0	0
$288$	−30.0000	−1.76777
$289$	0	0
$290$	−12.0000	−0.704664
$291$	6.00000	0.351726
$292$	6.00000	0.351123
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	3.00000	0.174964
$295$	36.0000	2.09600
$296$	−30.0000	−1.74371
$297$	0	0
$298$	7.00000	0.405499
$299$	8.00000	0.462652
$300$	33.0000	1.90526
$301$	−2.00000	−0.115278
$302$	−2.00000	−0.115087
$303$	−6.00000	−0.344691
$304$	7.00000	0.401478
$305$	32.0000	1.83231
$306$	0	0
$307$	15.0000	0.856095	0.428048	−	0.903756i	$-0.359202\pi$
$307$	15.0000	0.856095	0.428048	+	0.903756i	$0.359202\pi$
$308$	0	0
$309$	21.0000	1.19465
$310$	−28.0000	−1.59029
$311$	−31.0000	−1.75785	−0.878924	−	0.476961i	$-0.841738\pi$
$311$	−31.0000	−1.75785	−0.878924	+	0.476961i	$0.841738\pi$
$312$	18.0000	1.01905
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	14.0000	0.790066
$315$	−24.0000	−1.35225
$316$	6.00000	0.337526
$317$	31.0000	1.74113	0.870567	−	0.492050i	$-0.163752\pi$
$317$	31.0000	1.74113	0.870567	+	0.492050i	$0.163752\pi$
$318$	−9.00000	−0.504695
$319$	0	0
$320$	−28.0000	−1.56525
$321$	−48.0000	−2.67910
$322$	4.00000	0.222911
$323$	0	0
$324$	−9.00000	−0.500000
$325$	−22.0000	−1.22034
$326$	18.0000	0.996928
$327$	−27.0000	−1.49310
$328$	0	0
$329$	3.00000	0.165395
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	1.00000	0.0548821
$333$	−60.0000	−3.28798
$334$	−1.00000	−0.0547176
$335$	32.0000	1.74835
$336$	3.00000	0.163663
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	9.00000	0.489535
$339$	−6.00000	−0.325875
$340$	0	0
$341$	0	0
$342$	42.0000	2.27110
$343$	1.00000	0.0539949
$344$	−6.00000	−0.323498
$345$	−48.0000	−2.58423
$346$	−20.0000	−1.07521
$347$	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
$349$	−36.0000	−1.92704	−0.963518	−	0.267644i	$-0.913755\pi$
$349$	−36.0000	−1.92704	−0.963518	+	0.267644i	$0.913755\pi$
$350$	−11.0000	−0.587975
$351$	18.0000	0.960769
$352$	0	0
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	−27.0000	−1.43503
$355$	8.00000	0.424596
$356$	8.00000	0.423999
$357$	0	0
$358$	6.00000	0.317110
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	−72.0000	−3.79473
$361$	30.0000	1.57895
$362$	16.0000	0.840941
$363$	33.0000	1.73205
$364$	2.00000	0.104828
$365$	24.0000	1.25622
$366$	−24.0000	−1.25450
$367$	23.0000	1.20059	0.600295	−	0.799779i	$-0.295050\pi$
$367$	23.0000	1.20059	0.600295	+	0.799779i	$0.295050\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	−40.0000	−2.07950
$371$	−3.00000	−0.155752
$372$	−21.0000	−1.08880
$373$	−15.0000	−0.776671	−0.388335	−	0.921518i	$-0.626950\pi$
$373$	−15.0000	−0.776671	−0.388335	+	0.921518i	$0.626950\pi$
$374$	0	0
$375$	72.0000	3.71806
$376$	9.00000	0.464140
$377$	6.00000	0.309016
$378$	9.00000	0.462910
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	−28.0000	−1.43637
$381$	12.0000	0.614779
$382$	22.0000	1.12562
$383$	21.0000	1.07305	0.536525	−	0.843884i	$-0.319737\pi$
$383$	21.0000	1.07305	0.536525	+	0.843884i	$0.319737\pi$
$384$	−9.00000	−0.459279
$385$	0	0
$386$	23.0000	1.17067
$387$	−12.0000	−0.609994
$388$	2.00000	0.101535
$389$	27.0000	1.36895	0.684477	−	0.729034i	$-0.260031\pi$
$389$	27.0000	1.36895	0.684477	+	0.729034i	$0.260031\pi$
$390$	24.0000	1.21529
$391$	0	0
$392$	3.00000	0.151523
$393$	57.0000	2.87527
$394$	−15.0000	−0.755689
$395$	24.0000	1.20757
$396$	0	0
$397$	24.0000	1.20453	0.602263	−	0.798298i	$-0.294266\pi$
$397$	24.0000	1.20453	0.602263	+	0.798298i	$0.294266\pi$
$398$	16.0000	0.802008
$399$	21.0000	1.05131
$400$	−11.0000	−0.550000
$401$	−1.00000	−0.0499376	−0.0249688	−	0.999688i	$-0.507949\pi$
$401$	−1.00000	−0.0499376	−0.0249688	+	0.999688i	$0.507949\pi$
$402$	−24.0000	−1.19701
$403$	14.0000	0.697390
$404$	−2.00000	−0.0995037
$405$	−36.0000	−1.78885
$406$	3.00000	0.148888
$407$	0	0
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	0	0
$411$	−27.0000	−1.33181
$412$	7.00000	0.344865
$413$	−9.00000	−0.442861
$414$	24.0000	1.17954
$415$	4.00000	0.196352
$416$	10.0000	0.490290
$417$	−60.0000	−2.93821
$418$	0	0
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	−12.0000	−0.585540
$421$	13.0000	0.633581	0.316791	−	0.948495i	$-0.397395\pi$
$421$	13.0000	0.633581	0.316791	+	0.948495i	$0.397395\pi$
$422$	−10.0000	−0.486792
$423$	18.0000	0.875190
$424$	−9.00000	−0.437079
$425$	0	0
$426$	−6.00000	−0.290701
$427$	−8.00000	−0.387147
$428$	−16.0000	−0.773389
$429$	0	0
$430$	−8.00000	−0.385794
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	9.00000	0.433013
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	7.00000	0.336011
$435$	−36.0000	−1.72607
$436$	−9.00000	−0.431022
$437$	28.0000	1.33942
$438$	−18.0000	−0.860073
$439$	−1.00000	−0.0477274	−0.0238637	−	0.999715i	$-0.507597\pi$
$439$	−1.00000	−0.0477274	−0.0238637	+	0.999715i	$0.507597\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	−26.0000	−1.23530	−0.617649	−	0.786454i	$-0.711915\pi$
$443$	−26.0000	−1.23530	−0.617649	+	0.786454i	$0.711915\pi$
$444$	−30.0000	−1.42374
$445$	32.0000	1.51695
$446$	15.0000	0.710271
$447$	21.0000	0.993266
$448$	7.00000	0.330719
$449$	−7.00000	−0.330350	−0.165175	−	0.986264i	$-0.552819\pi$
$449$	−7.00000	−0.330350	−0.165175	+	0.986264i	$0.552819\pi$
$450$	−66.0000	−3.11127
$451$	0	0
$452$	−2.00000	−0.0940721
$453$	−6.00000	−0.281905
$454$	−9.00000	−0.422391
$455$	8.00000	0.375046
$456$	63.0000	2.95025
$457$	3.00000	0.140334	0.0701670	−	0.997535i	$-0.477647\pi$
$457$	3.00000	0.140334	0.0701670	+	0.997535i	$0.477647\pi$
$458$	6.00000	0.280362
$459$	0	0
$460$	−16.0000	−0.746004
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	3.00000	0.139272
$465$	−84.0000	−3.89541
$466$	9.00000	0.416917
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	12.0000	0.554700
$469$	−8.00000	−0.369406
$470$	12.0000	0.553519
$471$	42.0000	1.93526
$472$	−27.0000	−1.24278
$473$	0	0
$474$	−18.0000	−0.826767
$475$	−77.0000	−3.53300
$476$	0	0
$477$	−18.0000	−0.824163
$478$	−2.00000	−0.0914779
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	−60.0000	−2.73861
$481$	20.0000	0.911922
$482$	−12.0000	−0.546585
$483$	12.0000	0.546019
$484$	11.0000	0.500000
$485$	8.00000	0.363261
$486$	0	0
$487$	10.0000	0.453143	0.226572	−	0.973995i	$-0.427248\pi$
$487$	10.0000	0.453143	0.226572	+	0.973995i	$0.427248\pi$
$488$	−24.0000	−1.08643
$489$	54.0000	2.44196
$490$	4.00000	0.180702
$491$	−22.0000	−0.992846	−0.496423	−	0.868081i	$-0.665354\pi$
$491$	−22.0000	−0.992846	−0.496423	+	0.868081i	$0.665354\pi$
$492$	0	0
$493$	0	0
$494$	−14.0000	−0.629890
$495$	0	0
$496$	7.00000	0.314309
$497$	−2.00000	−0.0897123
$498$	−3.00000	−0.134433
$499$	−2.00000	−0.0895323	−0.0447661	−	0.998997i	$-0.514254\pi$
$499$	−2.00000	−0.0895323	−0.0447661	+	0.998997i	$0.514254\pi$
$500$	24.0000	1.07331
$501$	−3.00000	−0.134030
$502$	24.0000	1.07117
$503$	−9.00000	−0.401290	−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000	−0.401290	−0.200645	+	0.979664i	$0.564304\pi$
$504$	18.0000	0.801784
$505$	−8.00000	−0.355995
$506$	0	0
$507$	27.0000	1.19911
$508$	4.00000	0.177471
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	11.0000	0.486136
$513$	63.0000	2.78152
$514$	−30.0000	−1.32324
$515$	28.0000	1.23383
$516$	−6.00000	−0.264135
$517$	0	0
$518$	10.0000	0.439375
$519$	−60.0000	−2.63371
$520$	24.0000	1.05247
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	18.0000	0.787839
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	19.0000	0.830019
$525$	−33.0000	−1.44024
$526$	16.0000	0.697633
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−12.0000	−0.521247
$531$	−54.0000	−2.34340
$532$	7.00000	0.303488
$533$	0	0
$534$	−24.0000	−1.03858
$535$	−64.0000	−2.76696
$536$	−24.0000	−1.03664
$537$	18.0000	0.776757
$538$	10.0000	0.431131
$539$	0	0
$540$	−36.0000	−1.54919
$541$	39.0000	1.67674	0.838370	−	0.545101i	$-0.183509\pi$
$541$	39.0000	1.67674	0.838370	+	0.545101i	$0.183509\pi$
$542$	−17.0000	−0.730213
$543$	48.0000	2.05988
$544$	0	0
$545$	−36.0000	−1.54207
$546$	−6.00000	−0.256776
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	−9.00000	−0.384461
$549$	−48.0000	−2.04859
$550$	0	0
$551$	21.0000	0.894630
$552$	36.0000	1.53226
$553$	−6.00000	−0.255146
$554$	5.00000	0.212430
$555$	−120.000	−5.09372
$556$	−20.0000	−0.848189
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	42.0000	1.77800
$559$	4.00000	0.169182
$560$	4.00000	0.169031
$561$	0	0
$562$	−18.0000	−0.759284
$563$	11.0000	0.463595	0.231797	−	0.972764i	$-0.425539\pi$
$563$	11.0000	0.463595	0.231797	+	0.972764i	$0.425539\pi$
$564$	9.00000	0.378968
$565$	−8.00000	−0.336563
$566$	5.00000	0.210166
$567$	9.00000	0.377964
$568$	−6.00000	−0.251754
$569$	−23.0000	−0.964210	−0.482105	−	0.876113i	$-0.660128\pi$
$569$	−23.0000	−0.964210	−0.482105	+	0.876113i	$0.660128\pi$
$570$	84.0000	3.51837
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	0	0
$573$	66.0000	2.75719
$574$	0	0
$575$	−44.0000	−1.83493
$576$	42.0000	1.75000
$577$	16.0000	0.666089	0.333044	−	0.942911i	$-0.391924\pi$
$577$	16.0000	0.666089	0.333044	+	0.942911i	$0.391924\pi$
$578$	0	0
$579$	69.0000	2.86754
$580$	−12.0000	−0.498273
$581$	−1.00000	−0.0414870
$582$	−6.00000	−0.248708
$583$	0	0
$584$	−18.0000	−0.744845
$585$	48.0000	1.98456
$586$	−12.0000	−0.495715
$587$	−17.0000	−0.701665	−0.350833	−	0.936438i	$-0.614101\pi$
$587$	−17.0000	−0.701665	−0.350833	+	0.936438i	$0.614101\pi$
$588$	3.00000	0.123718
$589$	49.0000	2.01901
$590$	−36.0000	−1.48210
$591$	−45.0000	−1.85105
$592$	10.0000	0.410997
$593$	42.0000	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$594$	0	0
$595$	0	0
$596$	7.00000	0.286731
$597$	48.0000	1.96451
$598$	−8.00000	−0.327144
$599$	18.0000	0.735460	0.367730	−	0.929933i	$-0.380135\pi$
$599$	18.0000	0.735460	0.367730	+	0.929933i	$0.380135\pi$
$600$	−99.0000	−4.04166
$601$	−16.0000	−0.652654	−0.326327	−	0.945257i	$-0.605811\pi$
$601$	−16.0000	−0.652654	−0.326327	+	0.945257i	$0.605811\pi$
$602$	2.00000	0.0815139
$603$	−48.0000	−1.95471
$604$	−2.00000	−0.0813788
$605$	44.0000	1.78885
$606$	6.00000	0.243733
$607$	7.00000	0.284121	0.142061	−	0.989858i	$-0.454627\pi$
$607$	7.00000	0.284121	0.142061	+	0.989858i	$0.454627\pi$
$608$	35.0000	1.41944
$609$	9.00000	0.364698
$610$	−32.0000	−1.29564
$611$	−6.00000	−0.242734
$612$	0	0
$613$	34.0000	1.37325	0.686624	−	0.727013i	$-0.259092\pi$
$613$	34.0000	1.37325	0.686624	+	0.727013i	$0.259092\pi$
$614$	−15.0000	−0.605351
$615$	0	0
$616$	0	0
$617$	27.0000	1.08698	0.543490	−	0.839416i	$-0.317103\pi$
$617$	27.0000	1.08698	0.543490	+	0.839416i	$0.317103\pi$
$618$	−21.0000	−0.844744
$619$	−15.0000	−0.602901	−0.301450	−	0.953482i	$-0.597471\pi$
$619$	−15.0000	−0.602901	−0.301450	+	0.953482i	$0.597471\pi$
$620$	−28.0000	−1.12451
$621$	36.0000	1.44463
$622$	31.0000	1.24299
$623$	−8.00000	−0.320513
$624$	−6.00000	−0.240192
$625$	41.0000	1.64000
$626$	−10.0000	−0.399680
$627$	0	0
$628$	14.0000	0.558661
$629$	0	0
$630$	24.0000	0.956183
$631$	−38.0000	−1.51276	−0.756378	−	0.654135i	$-0.773033\pi$
$631$	−38.0000	−1.51276	−0.756378	+	0.654135i	$0.773033\pi$
$632$	−18.0000	−0.716002
$633$	−30.0000	−1.19239
$634$	−31.0000	−1.23117
$635$	16.0000	0.634941
$636$	−9.00000	−0.356873
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	−12.0000	−0.474713
$640$	−12.0000	−0.474342
$641$	−45.0000	−1.77739	−0.888697	−	0.458496i	$-0.848388\pi$
$641$	−45.0000	−1.77739	−0.888697	+	0.458496i	$0.848388\pi$
$642$	48.0000	1.89441
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	4.00000	0.157622
$645$	−24.0000	−0.944999
$646$	0	0
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	27.0000	1.06066
$649$	0	0
$650$	22.0000	0.862911
$651$	21.0000	0.823055
$652$	18.0000	0.704934
$653$	−3.00000	−0.117399	−0.0586995	−	0.998276i	$-0.518695\pi$
$653$	−3.00000	−0.117399	−0.0586995	+	0.998276i	$0.518695\pi$
$654$	27.0000	1.05578
$655$	76.0000	2.96957
$656$	0	0
$657$	−36.0000	−1.40449
$658$	−3.00000	−0.116952
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	0	0
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	−3.00000	−0.116423
$665$	28.0000	1.08579
$666$	60.0000	2.32495
$667$	12.0000	0.464642
$668$	−1.00000	−0.0386912
$669$	45.0000	1.73980
$670$	−32.0000	−1.23627
$671$	0	0
$672$	15.0000	0.578638
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	13.0000	0.500741
$675$	−99.0000	−3.81051
$676$	9.00000	0.346154
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	6.00000	0.230429
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	−27.0000	−1.03464
$682$	0	0
$683$	−8.00000	−0.306111	−0.153056	−	0.988218i	$-0.548911\pi$
$683$	−8.00000	−0.306111	−0.153056	+	0.988218i	$0.548911\pi$
$684$	42.0000	1.60591
$685$	−36.0000	−1.37549
$686$	−1.00000	−0.0381802
$687$	18.0000	0.686743
$688$	2.00000	0.0762493
$689$	6.00000	0.228582
$690$	48.0000	1.82733
$691$	−1.00000	−0.0380418	−0.0190209	−	0.999819i	$-0.506055\pi$
$691$	−1.00000	−0.0380418	−0.0190209	+	0.999819i	$0.506055\pi$
$692$	−20.0000	−0.760286
$693$	0	0
$694$	−16.0000	−0.607352
$695$	−80.0000	−3.03457
$696$	27.0000	1.02343
$697$	0	0
$698$	36.0000	1.36262
$699$	27.0000	1.02123
$700$	−11.0000	−0.415761
$701$	−9.00000	−0.339925	−0.169963	−	0.985451i	$-0.554365\pi$
$701$	−9.00000	−0.339925	−0.169963	+	0.985451i	$0.554365\pi$
$702$	−18.0000	−0.679366
$703$	70.0000	2.64010
$704$	0	0
$705$	36.0000	1.35584
$706$	−24.0000	−0.903252
$707$	2.00000	0.0752177
$708$	−27.0000	−1.01472
$709$	46.0000	1.72757	0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000	1.72757	0.863783	+	0.503864i	$0.168089\pi$
$710$	−8.00000	−0.300235
$711$	−36.0000	−1.35011
$712$	−24.0000	−0.899438
$713$	28.0000	1.04861
$714$	0	0
$715$	0	0
$716$	6.00000	0.224231
$717$	−6.00000	−0.224074
$718$	12.0000	0.447836
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	24.0000	0.894427
$721$	−7.00000	−0.260694
$722$	−30.0000	−1.11648
$723$	−36.0000	−1.33885
$724$	16.0000	0.594635
$725$	−33.0000	−1.22559
$726$	−33.0000	−1.22474
$727$	−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
$729$	−27.0000	−1.00000
$730$	−24.0000	−0.888280
$731$	0	0
$732$	−24.0000	−0.887066
$733$	−8.00000	−0.295487	−0.147743	−	0.989026i	$-0.547201\pi$
$733$	−8.00000	−0.295487	−0.147743	+	0.989026i	$0.547201\pi$
$734$	−23.0000	−0.848945
$735$	12.0000	0.442627
$736$	20.0000	0.737210
$737$	0	0
$738$	0	0
$739$	34.0000	1.25071	0.625355	−	0.780340i	$-0.284954\pi$
$739$	34.0000	1.25071	0.625355	+	0.780340i	$0.284954\pi$
$740$	−40.0000	−1.47043
$741$	−42.0000	−1.54291
$742$	3.00000	0.110133
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	63.0000	2.30969
$745$	28.0000	1.02584
$746$	15.0000	0.549189
$747$	−6.00000	−0.219529
$748$	0	0
$749$	16.0000	0.584627
$750$	−72.0000	−2.62907
$751$	−38.0000	−1.38664	−0.693320	−	0.720630i	$-0.743853\pi$
$751$	−38.0000	−1.38664	−0.693320	+	0.720630i	$0.743853\pi$
$752$	−3.00000	−0.109399
$753$	72.0000	2.62383
$754$	−6.00000	−0.218507
$755$	−8.00000	−0.291150
$756$	9.00000	0.327327
$757$	−5.00000	−0.181728	−0.0908640	−	0.995863i	$-0.528963\pi$
$757$	−5.00000	−0.181728	−0.0908640	+	0.995863i	$0.528963\pi$
$758$	4.00000	0.145287
$759$	0	0
$760$	84.0000	3.04700
$761$	38.0000	1.37750	0.688749	−	0.724999i	$-0.258160\pi$
$761$	38.0000	1.37750	0.688749	+	0.724999i	$0.258160\pi$
$762$	−12.0000	−0.434714
$763$	9.00000	0.325822
$764$	22.0000	0.795932
$765$	0	0
$766$	−21.0000	−0.758761
$767$	18.0000	0.649942
$768$	51.0000	1.84030
$769$	4.00000	0.144244	0.0721218	−	0.997396i	$-0.477023\pi$
$769$	4.00000	0.144244	0.0721218	+	0.997396i	$0.477023\pi$
$770$	0	0
$771$	−90.0000	−3.24127
$772$	23.0000	0.827788
$773$	40.0000	1.43870	0.719350	−	0.694648i	$-0.244440\pi$
$773$	40.0000	1.43870	0.719350	+	0.694648i	$0.244440\pi$
$774$	12.0000	0.431331
$775$	−77.0000	−2.76592
$776$	−6.00000	−0.215387
$777$	30.0000	1.07624
$778$	−27.0000	−0.967997
$779$	0	0
$780$	24.0000	0.859338
$781$	0	0
$782$	0	0
$783$	27.0000	0.964901
$784$	−1.00000	−0.0357143
$785$	56.0000	1.99873
$786$	−57.0000	−2.03312
$787$	20.0000	0.712923	0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000	0.712923	0.356462	+	0.934310i	$0.383983\pi$
$788$	−15.0000	−0.534353
$789$	48.0000	1.70885
$790$	−24.0000	−0.853882
$791$	2.00000	0.0711118
$792$	0	0
$793$	16.0000	0.568177
$794$	−24.0000	−0.851728
$795$	−36.0000	−1.27679
$796$	16.0000	0.567105
$797$	−8.00000	−0.283375	−0.141687	−	0.989911i	$-0.545253\pi$
$797$	−8.00000	−0.283375	−0.141687	+	0.989911i	$0.545253\pi$
$798$	−21.0000	−0.743392
$799$	0	0
$800$	−55.0000	−1.94454
$801$	−48.0000	−1.69600
$802$	1.00000	0.0353112
$803$	0	0
$804$	−24.0000	−0.846415
$805$	16.0000	0.563926
$806$	−14.0000	−0.493129
$807$	30.0000	1.05605
$808$	6.00000	0.211079
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	36.0000	1.26491
$811$	−49.0000	−1.72062	−0.860311	−	0.509769i	$-0.829731\pi$
$811$	−49.0000	−1.72062	−0.860311	+	0.509769i	$0.829731\pi$
$812$	3.00000	0.105279
$813$	−51.0000	−1.78865
$814$	0	0
$815$	72.0000	2.52205
$816$	0	0
$817$	14.0000	0.489798
$818$	−32.0000	−1.11885
$819$	−12.0000	−0.419314
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	27.0000	0.941733
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	−21.0000	−0.731570
$825$	0	0
$826$	9.00000	0.313150
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	24.0000	0.834058
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	−4.00000	−0.138842
$831$	15.0000	0.520344
$832$	−14.0000	−0.485363
$833$	0	0
$834$	60.0000	2.07763
$835$	−4.00000	−0.138426
$836$	0	0
$837$	63.0000	2.17760
$838$	−28.0000	−0.967244
$839$	−29.0000	−1.00119	−0.500596	−	0.865681i	$-0.666886\pi$
$839$	−29.0000	−1.00119	−0.500596	+	0.865681i	$0.666886\pi$
$840$	36.0000	1.24212
$841$	−20.0000	−0.689655
$842$	−13.0000	−0.448010
$843$	−54.0000	−1.85986
$844$	−10.0000	−0.344214
$845$	36.0000	1.23844
$846$	−18.0000	−0.618853
$847$	−11.0000	−0.377964
$848$	3.00000	0.103020
$849$	15.0000	0.514799
$850$	0	0
$851$	40.0000	1.37118
$852$	−6.00000	−0.205557
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	8.00000	0.273754
$855$	168.000	5.74548
$856$	48.0000	1.64061
$857$	−16.0000	−0.546550	−0.273275	−	0.961936i	$-0.588107\pi$
$857$	−16.0000	−0.546550	−0.273275	+	0.961936i	$0.588107\pi$
$858$	0	0
$859$	43.0000	1.46714	0.733571	−	0.679613i	$-0.237852\pi$
$859$	43.0000	1.46714	0.733571	+	0.679613i	$0.237852\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	−12.0000	−0.408722
$863$	−20.0000	−0.680808	−0.340404	−	0.940279i	$-0.610564\pi$
$863$	−20.0000	−0.680808	−0.340404	+	0.940279i	$0.610564\pi$
$864$	45.0000	1.53093
$865$	−80.0000	−2.72008
$866$	16.0000	0.543702
$867$	0	0
$868$	7.00000	0.237595
$869$	0	0
$870$	36.0000	1.22051
$871$	16.0000	0.542139
$872$	27.0000	0.914335
$873$	−12.0000	−0.406138
$874$	−28.0000	−0.947114
$875$	−24.0000	−0.811348
$876$	−18.0000	−0.608164
$877$	30.0000	1.01303	0.506514	−	0.862232i	$-0.330934\pi$
$877$	30.0000	1.01303	0.506514	+	0.862232i	$0.330934\pi$
$878$	1.00000	0.0337484
$879$	−36.0000	−1.21425
$880$	0	0
$881$	−50.0000	−1.68454	−0.842271	−	0.539054i	$-0.818782\pi$
$881$	−50.0000	−1.68454	−0.842271	+	0.539054i	$0.818782\pi$
$882$	−6.00000	−0.202031
$883$	−24.0000	−0.807664	−0.403832	−	0.914833i	$-0.632322\pi$
$883$	−24.0000	−0.807664	−0.403832	+	0.914833i	$0.632322\pi$
$884$	0	0
$885$	−108.000	−3.63038
$886$	26.0000	0.873487
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	90.0000	3.02020
$889$	−4.00000	−0.134156
$890$	−32.0000	−1.07264
$891$	0	0
$892$	15.0000	0.502237
$893$	−21.0000	−0.702738
$894$	−21.0000	−0.702345
$895$	24.0000	0.802232
$896$	3.00000	0.100223
$897$	−24.0000	−0.801337
$898$	7.00000	0.233593
$899$	21.0000	0.700389
$900$	−66.0000	−2.20000
$901$	0	0
$902$	0	0
$903$	6.00000	0.199667
$904$	6.00000	0.199557
$905$	64.0000	2.12743
$906$	6.00000	0.199337
$907$	−26.0000	−0.863316	−0.431658	−	0.902037i	$-0.642071\pi$
$907$	−26.0000	−0.863316	−0.431658	+	0.902037i	$0.642071\pi$
$908$	−9.00000	−0.298675
$909$	12.0000	0.398015
$910$	−8.00000	−0.265197
$911$	34.0000	1.12647	0.563235	−	0.826297i	$-0.309557\pi$
$911$	34.0000	1.12647	0.563235	+	0.826297i	$0.309557\pi$
$912$	−21.0000	−0.695379
$913$	0	0
$914$	−3.00000	−0.0992312
$915$	−96.0000	−3.17366
$916$	6.00000	0.198246
$917$	−19.0000	−0.627435
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	48.0000	1.58251
$921$	−45.0000	−1.48280
$922$	14.0000	0.461065
$923$	4.00000	0.131662
$924$	0	0
$925$	−110.000	−3.61678
$926$	40.0000	1.31448
$927$	−42.0000	−1.37946
$928$	15.0000	0.492399
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	84.0000	2.75447
$931$	−7.00000	−0.229416
$932$	9.00000	0.294805
$933$	93.0000	3.04468
$934$	12.0000	0.392652
$935$	0	0
$936$	−36.0000	−1.17670
$937$	28.0000	0.914720	0.457360	−	0.889282i	$-0.348795\pi$
$937$	28.0000	0.914720	0.457360	+	0.889282i	$0.348795\pi$
$938$	8.00000	0.261209
$939$	−30.0000	−0.979013
$940$	12.0000	0.391397
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	−42.0000	−1.36843
$943$	0	0
$944$	9.00000	0.292925
$945$	36.0000	1.17108
$946$	0	0
$947$	−44.0000	−1.42981	−0.714904	−	0.699223i	$-0.753530\pi$
$947$	−44.0000	−1.42981	−0.714904	+	0.699223i	$0.753530\pi$
$948$	−18.0000	−0.584613
$949$	12.0000	0.389536
$950$	77.0000	2.49821
$951$	−93.0000	−3.01573
$952$	0	0
$953$	11.0000	0.356325	0.178162	−	0.984001i	$-0.442985\pi$
$953$	11.0000	0.356325	0.178162	+	0.984001i	$0.442985\pi$
$954$	18.0000	0.582772
$955$	88.0000	2.84761
$956$	−2.00000	−0.0646846
$957$	0	0
$958$	36.0000	1.16311
$959$	9.00000	0.290625
$960$	84.0000	2.71109
$961$	18.0000	0.580645
$962$	−20.0000	−0.644826
$963$	96.0000	3.09356
$964$	−12.0000	−0.386494
$965$	92.0000	2.96158
$966$	−12.0000	−0.386094
$967$	54.0000	1.73652	0.868261	−	0.496107i	$-0.165238\pi$
$967$	54.0000	1.73652	0.868261	+	0.496107i	$0.165238\pi$
$968$	−33.0000	−1.06066
$969$	0	0
$970$	−8.00000	−0.256865
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	20.0000	0.641171
$974$	−10.0000	−0.320421
$975$	66.0000	2.11369
$976$	8.00000	0.256074
$977$	21.0000	0.671850	0.335925	−	0.941889i	$-0.390951\pi$
$977$	21.0000	0.671850	0.335925	+	0.941889i	$0.390951\pi$
$978$	−54.0000	−1.72673
$979$	0	0
$980$	4.00000	0.127775
$981$	54.0000	1.72409
$982$	22.0000	0.702048
$983$	9.00000	0.287055	0.143528	−	0.989646i	$-0.454155\pi$
$983$	9.00000	0.287055	0.143528	+	0.989646i	$0.454155\pi$
$984$	0	0
$985$	−60.0000	−1.91176
$986$	0	0
$987$	−9.00000	−0.286473
$988$	−14.0000	−0.445399
$989$	8.00000	0.254385
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	35.0000	1.11125
$993$	−12.0000	−0.380808
$994$	2.00000	0.0634361
$995$	64.0000	2.02894
$996$	−3.00000	−0.0950586
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	2.00000	0.0633089
$999$	90.0000	2.84747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2023.2.a.a.1.1	✓	1
17.16	even	2		2023.2.a.b.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2023.2.a.a.1.1	✓	1	1.1	even	1	trivial
2023.2.a.b.1.1	yes	1	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2023 = 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2023.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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