LMFDB - Embedded newform 123.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [123,2,Mod(1,123)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(123, 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("123.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 123 = 3 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	123.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:	$0.982159944862$
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	$\chi$	$=$	123.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -2.00000 q^{4} -2.00000 q^{5} -4.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{3} -2.00000 q^{4} -2.00000 q^{5} -4.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} +2.00000 q^{12} -4.00000 q^{13} +2.00000 q^{15} +4.00000 q^{16} -5.00000 q^{17} -2.00000 q^{19} +4.00000 q^{20} +4.00000 q^{21} +4.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} +8.00000 q^{28} +1.00000 q^{29} -5.00000 q^{31} -5.00000 q^{33} +8.00000 q^{35} -2.00000 q^{36} -7.00000 q^{37} +4.00000 q^{39} -1.00000 q^{41} +7.00000 q^{43} -10.0000 q^{44} -2.00000 q^{45} +7.00000 q^{47} -4.00000 q^{48} +9.00000 q^{49} +5.00000 q^{51} +8.00000 q^{52} -14.0000 q^{53} -10.0000 q^{55} +2.00000 q^{57} -12.0000 q^{59} -4.00000 q^{60} -3.00000 q^{61} -4.00000 q^{63} -8.00000 q^{64} +8.00000 q^{65} -2.00000 q^{67} +10.0000 q^{68} -4.00000 q^{69} -3.00000 q^{71} +13.0000 q^{73} +1.00000 q^{75} +4.00000 q^{76} -20.0000 q^{77} -2.00000 q^{79} -8.00000 q^{80} +1.00000 q^{81} -2.00000 q^{83} -8.00000 q^{84} +10.0000 q^{85} -1.00000 q^{87} +18.0000 q^{89} +16.0000 q^{91} -8.00000 q^{92} +5.00000 q^{93} +4.00000 q^{95} -14.0000 q^{97} +5.00000 q^{99} +O(q^{100})\)

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 $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−2.00000	−1.00000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	2.00000	0.577350
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	4.00000	1.00000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	4.00000	0.894427
$21$	4.00000	0.872872
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	8.00000	1.51186
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	0	0
$33$	−5.00000	−0.870388
$34$	0	0
$35$	8.00000	1.35225
$36$	−2.00000	−0.333333
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	−10.0000	−1.50756
$45$	−2.00000	−0.298142
$46$	0	0
$47$	7.00000	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$48$	−4.00000	−0.577350
$49$	9.00000	1.28571
$50$	0	0
$51$	5.00000	0.700140
$52$	8.00000	1.10940
$53$	−14.0000	−1.92305	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−14.0000	−1.92305	−0.961524	+	0.274721i	$0.911414\pi$
$54$	0	0
$55$	−10.0000	−1.34840
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	−4.00000	−0.516398
$61$	−3.00000	−0.384111	−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000	−0.384111	−0.192055	+	0.981384i	$0.561515\pi$
$62$	0	0
$63$	−4.00000	−0.503953
$64$	−8.00000	−1.00000
$65$	8.00000	0.992278
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	10.0000	1.21268
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	0	0
$73$	13.0000	1.52153	0.760767	−	0.649025i	$-0.224823\pi$
$73$	13.0000	1.52153	0.760767	+	0.649025i	$0.224823\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	4.00000	0.458831
$77$	−20.0000	−2.27921
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	−8.00000	−0.894427
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	−8.00000	−0.872872
$85$	10.0000	1.08465
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	0	0
$91$	16.0000	1.67726
$92$	−8.00000	−0.834058
$93$	5.00000	0.518476
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	5.00000	0.502519
$100$	2.00000	0.200000
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	7.00000	0.689730	0.344865	−	0.938652i	$-0.387925\pi$
$103$	7.00000	0.689730	0.344865	+	0.938652i	$0.387925\pi$
$104$	0	0
$105$	−8.00000	−0.780720
$106$	0	0
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	2.00000	0.192450
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	0	0
$111$	7.00000	0.664411
$112$	−16.0000	−1.51186
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	−2.00000	−0.185695
$117$	−4.00000	−0.369800
$118$	0	0
$119$	20.0000	1.83340
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	1.00000	0.0901670
$124$	10.0000	0.898027
$125$	12.0000	1.07331
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	0	0
$129$	−7.00000	−0.616316
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	10.0000	0.870388
$133$	8.00000	0.693688
$134$	0	0
$135$	2.00000	0.172133
$136$	0	0
$137$	−1.00000	−0.0854358	−0.0427179	−	0.999087i	$-0.513602\pi$
$137$	−1.00000	−0.0854358	−0.0427179	+	0.999087i	$0.513602\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	−16.0000	−1.35225
$141$	−7.00000	−0.589506
$142$	0	0
$143$	−20.0000	−1.67248
$144$	4.00000	0.333333
$145$	−2.00000	−0.166091
$146$	0	0
$147$	−9.00000	−0.742307
$148$	14.0000	1.15079
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−5.00000	−0.404226
$154$	0	0
$155$	10.0000	0.803219
$156$	−8.00000	−0.640513
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	0	0
$159$	14.0000	1.11027
$160$	0	0
$161$	−16.0000	−1.26098
$162$	0	0
$163$	−9.00000	−0.704934	−0.352467	−	0.935824i	$-0.614657\pi$
$163$	−9.00000	−0.704934	−0.352467	+	0.935824i	$0.614657\pi$
$164$	2.00000	0.156174
$165$	10.0000	0.778499
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−2.00000	−0.152944
$172$	−14.0000	−1.06749
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	20.0000	1.50756
$177$	12.0000	0.901975
$178$	0	0
$179$	−25.0000	−1.86859	−0.934294	−	0.356504i	$-0.883969\pi$
$179$	−25.0000	−1.86859	−0.934294	+	0.356504i	$0.883969\pi$
$180$	4.00000	0.298142
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	0	0
$183$	3.00000	0.221766
$184$	0	0
$185$	14.0000	1.02930
$186$	0	0
$187$	−25.0000	−1.82818
$188$	−14.0000	−1.02105
$189$	4.00000	0.290957
$190$	0	0
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	8.00000	0.577350
$193$	−20.0000	−1.43963	−0.719816	−	0.694165i	$-0.755774\pi$
$193$	−20.0000	−1.43963	−0.719816	+	0.694165i	$0.755774\pi$
$194$	0	0
$195$	−8.00000	−0.572892
$196$	−18.0000	−1.28571
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	6.00000	0.425329	0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000	0.425329	0.212664	+	0.977125i	$0.431786\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	0	0
$203$	−4.00000	−0.280745
$204$	−10.0000	−0.700140
$205$	2.00000	0.139686
$206$	0	0
$207$	4.00000	0.278019
$208$	−16.0000	−1.10940
$209$	−10.0000	−0.691714
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	28.0000	1.92305
$213$	3.00000	0.205557
$214$	0	0
$215$	−14.0000	−0.954792
$216$	0	0
$217$	20.0000	1.35769
$218$	0	0
$219$	−13.0000	−0.878459
$220$	20.0000	1.34840
$221$	20.0000	1.34535
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	3.00000	0.199117	0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000	0.199117	0.0995585	+	0.995032i	$0.468257\pi$
$228$	−4.00000	−0.264906
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	0	0
$231$	20.0000	1.31590
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−14.0000	−0.913259
$236$	24.0000	1.56227
$237$	2.00000	0.129914
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	8.00000	0.516398
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	6.00000	0.384111
$245$	−18.0000	−1.14998
$246$	0	0
$247$	8.00000	0.509028
$248$	0	0
$249$	2.00000	0.126745
$250$	0	0
$251$	−14.0000	−0.883672	−0.441836	−	0.897096i	$-0.645673\pi$
$251$	−14.0000	−0.883672	−0.441836	+	0.897096i	$0.645673\pi$
$252$	8.00000	0.503953
$253$	20.0000	1.25739
$254$	0	0
$255$	−10.0000	−0.626224
$256$	16.0000	1.00000
$257$	3.00000	0.187135	0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000	0.187135	0.0935674	+	0.995613i	$0.470173\pi$
$258$	0	0
$259$	28.0000	1.73984
$260$	−16.0000	−0.992278
$261$	1.00000	0.0618984
$262$	0	0
$263$	−23.0000	−1.41824	−0.709120	−	0.705087i	$-0.750908\pi$
$263$	−23.0000	−1.41824	−0.709120	+	0.705087i	$0.750908\pi$
$264$	0	0
$265$	28.0000	1.72003
$266$	0	0
$267$	−18.0000	−1.10158
$268$	4.00000	0.244339
$269$	16.0000	0.975537	0.487769	−	0.872973i	$-0.337811\pi$
$269$	16.0000	0.975537	0.487769	+	0.872973i	$0.337811\pi$
$270$	0	0
$271$	−27.0000	−1.64013	−0.820067	−	0.572268i	$-0.806064\pi$
$271$	−27.0000	−1.64013	−0.820067	+	0.572268i	$0.806064\pi$
$272$	−20.0000	−1.21268
$273$	−16.0000	−0.968364
$274$	0	0
$275$	−5.00000	−0.301511
$276$	8.00000	0.481543
$277$	19.0000	1.14160	0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000	1.14160	0.570800	+	0.821089i	$0.306633\pi$
$278$	0	0
$279$	−5.00000	−0.299342
$280$	0	0
$281$	15.0000	0.894825	0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000	0.894825	0.447412	+	0.894328i	$0.352346\pi$
$282$	0	0
$283$	17.0000	1.01055	0.505273	−	0.862960i	$-0.331392\pi$
$283$	17.0000	1.01055	0.505273	+	0.862960i	$0.331392\pi$
$284$	6.00000	0.356034
$285$	−4.00000	−0.236940
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	14.0000	0.820695
$292$	−26.0000	−1.52153
$293$	−23.0000	−1.34367	−0.671837	−	0.740699i	$-0.734495\pi$
$293$	−23.0000	−1.34367	−0.671837	+	0.740699i	$0.734495\pi$
$294$	0	0
$295$	24.0000	1.39733
$296$	0	0
$297$	−5.00000	−0.290129
$298$	0	0
$299$	−16.0000	−0.925304
$300$	−2.00000	−0.115470
$301$	−28.0000	−1.61389
$302$	0	0
$303$	−3.00000	−0.172345
$304$	−8.00000	−0.458831
$305$	6.00000	0.343559
$306$	0	0
$307$	17.0000	0.970241	0.485121	−	0.874447i	$-0.338776\pi$
$307$	17.0000	0.970241	0.485121	+	0.874447i	$0.338776\pi$
$308$	40.0000	2.27921
$309$	−7.00000	−0.398216
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	16.0000	0.904373	0.452187	−	0.891923i	$-0.350644\pi$
$313$	16.0000	0.904373	0.452187	+	0.891923i	$0.350644\pi$
$314$	0	0
$315$	8.00000	0.450749
$316$	4.00000	0.225018
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	0	0
$319$	5.00000	0.279946
$320$	16.0000	0.894427
$321$	−10.0000	−0.558146
$322$	0	0
$323$	10.0000	0.556415
$324$	−2.00000	−0.111111
$325$	4.00000	0.221880
$326$	0	0
$327$	8.00000	0.442401
$328$	0	0
$329$	−28.0000	−1.54369
$330$	0	0
$331$	−22.0000	−1.20923	−0.604615	−	0.796518i	$-0.706673\pi$
$331$	−22.0000	−1.20923	−0.604615	+	0.796518i	$0.706673\pi$
$332$	4.00000	0.219529
$333$	−7.00000	−0.383598
$334$	0	0
$335$	4.00000	0.218543
$336$	16.0000	0.872872
$337$	21.0000	1.14394	0.571971	−	0.820274i	$-0.306179\pi$
$337$	21.0000	1.14394	0.571971	+	0.820274i	$0.306179\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	−20.0000	−1.08465
$341$	−25.0000	−1.35383
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	8.00000	0.430706
$346$	0	0
$347$	−15.0000	−0.805242	−0.402621	−	0.915367i	$-0.631901\pi$
$347$	−15.0000	−0.805242	−0.402621	+	0.915367i	$0.631901\pi$
$348$	2.00000	0.107211
$349$	7.00000	0.374701	0.187351	−	0.982293i	$-0.440010\pi$
$349$	7.00000	0.374701	0.187351	+	0.982293i	$0.440010\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	−36.0000	−1.90800
$357$	−20.0000	−1.05851
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−14.0000	−0.734809
$364$	−32.0000	−1.67726
$365$	−26.0000	−1.36090
$366$	0	0
$367$	−25.0000	−1.30499	−0.652495	−	0.757793i	$-0.726278\pi$
$367$	−25.0000	−1.30499	−0.652495	+	0.757793i	$0.726278\pi$
$368$	16.0000	0.834058
$369$	−1.00000	−0.0520579
$370$	0	0
$371$	56.0000	2.90738
$372$	−10.0000	−0.518476
$373$	11.0000	0.569558	0.284779	−	0.958593i	$-0.408080\pi$
$373$	11.0000	0.569558	0.284779	+	0.958593i	$0.408080\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	0	0
$377$	−4.00000	−0.206010
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	−8.00000	−0.410391
$381$	−12.0000	−0.614779
$382$	0	0
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	0	0
$385$	40.0000	2.03859
$386$	0	0
$387$	7.00000	0.355830
$388$	28.0000	1.42148
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−20.0000	−1.01144
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	4.00000	0.201262
$396$	−10.0000	−0.502519
$397$	12.0000	0.602263	0.301131	−	0.953583i	$-0.402636\pi$
$397$	12.0000	0.602263	0.301131	+	0.953583i	$0.402636\pi$
$398$	0	0
$399$	−8.00000	−0.400501
$400$	−4.00000	−0.200000
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	0	0
$403$	20.0000	0.996271
$404$	−6.00000	−0.298511
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	−35.0000	−1.73489
$408$	0	0
$409$	7.00000	0.346128	0.173064	−	0.984911i	$-0.444633\pi$
$409$	7.00000	0.346128	0.173064	+	0.984911i	$0.444633\pi$
$410$	0	0
$411$	1.00000	0.0493264
$412$	−14.0000	−0.689730
$413$	48.0000	2.36193
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	4.00000	0.195881
$418$	0	0
$419$	32.0000	1.56330	0.781651	−	0.623716i	$-0.214378\pi$
$419$	32.0000	1.56330	0.781651	+	0.623716i	$0.214378\pi$
$420$	16.0000	0.780720
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	0	0
$423$	7.00000	0.340352
$424$	0	0
$425$	5.00000	0.242536
$426$	0	0
$427$	12.0000	0.580721
$428$	−20.0000	−0.966736
$429$	20.0000	0.965609
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	−4.00000	−0.192450
$433$	−5.00000	−0.240285	−0.120142	−	0.992757i	$-0.538335\pi$
$433$	−5.00000	−0.240285	−0.120142	+	0.992757i	$0.538335\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	16.0000	0.766261
$437$	−8.00000	−0.382692
$438$	0	0
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	−14.0000	−0.664411
$445$	−36.0000	−1.70656
$446$	0	0
$447$	10.0000	0.472984
$448$	32.0000	1.51186
$449$	40.0000	1.88772	0.943858	−	0.330350i	$-0.107167\pi$
$449$	40.0000	1.88772	0.943858	+	0.330350i	$0.107167\pi$
$450$	0	0
$451$	−5.00000	−0.235441
$452$	12.0000	0.564433
$453$	0	0
$454$	0	0
$455$	−32.0000	−1.50018
$456$	0	0
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	0	0
$459$	5.00000	0.233380
$460$	16.0000	0.746004
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	−12.0000	−0.557687	−0.278844	−	0.960337i	$-0.589951\pi$
$463$	−12.0000	−0.557687	−0.278844	+	0.960337i	$0.589951\pi$
$464$	4.00000	0.185695
$465$	−10.0000	−0.463739
$466$	0	0
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	8.00000	0.369800
$469$	8.00000	0.369406
$470$	0	0
$471$	22.0000	1.01371
$472$	0	0
$473$	35.0000	1.60930
$474$	0	0
$475$	2.00000	0.0917663
$476$	−40.0000	−1.83340
$477$	−14.0000	−0.641016
$478$	0	0
$479$	13.0000	0.593985	0.296993	−	0.954880i	$-0.404016\pi$
$479$	13.0000	0.593985	0.296993	+	0.954880i	$0.404016\pi$
$480$	0	0
$481$	28.0000	1.27669
$482$	0	0
$483$	16.0000	0.728025
$484$	−28.0000	−1.27273
$485$	28.0000	1.27141
$486$	0	0
$487$	19.0000	0.860972	0.430486	−	0.902597i	$-0.358342\pi$
$487$	19.0000	0.860972	0.430486	+	0.902597i	$0.358342\pi$
$488$	0	0
$489$	9.00000	0.406994
$490$	0	0
$491$	22.0000	0.992846	0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000	0.992846	0.496423	+	0.868081i	$0.334646\pi$
$492$	−2.00000	−0.0901670
$493$	−5.00000	−0.225189
$494$	0	0
$495$	−10.0000	−0.449467
$496$	−20.0000	−0.898027
$497$	12.0000	0.538274
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	−24.0000	−1.07331
$501$	0	0
$502$	0	0
$503$	27.0000	1.20387	0.601935	−	0.798545i	$-0.294397\pi$
$503$	27.0000	1.20387	0.601935	+	0.798545i	$0.294397\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	−3.00000	−0.133235
$508$	−24.0000	−1.06483
$509$	−11.0000	−0.487566	−0.243783	−	0.969830i	$-0.578389\pi$
$509$	−11.0000	−0.487566	−0.243783	+	0.969830i	$0.578389\pi$
$510$	0	0
$511$	−52.0000	−2.30034
$512$	0	0
$513$	2.00000	0.0883022
$514$	0	0
$515$	−14.0000	−0.616914
$516$	14.0000	0.616316
$517$	35.0000	1.53930
$518$	0	0
$519$	2.00000	0.0877903
$520$	0	0
$521$	−9.00000	−0.394297	−0.197149	−	0.980374i	$-0.563168\pi$
$521$	−9.00000	−0.394297	−0.197149	+	0.980374i	$0.563168\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	−24.0000	−1.04844
$525$	−4.00000	−0.174574
$526$	0	0
$527$	25.0000	1.08902
$528$	−20.0000	−0.870388
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−12.0000	−0.520756
$532$	−16.0000	−0.693688
$533$	4.00000	0.173259
$534$	0	0
$535$	−20.0000	−0.864675
$536$	0	0
$537$	25.0000	1.07883
$538$	0	0
$539$	45.0000	1.93829
$540$	−4.00000	−0.172133
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	0	0
$543$	8.00000	0.343313
$544$	0	0
$545$	16.0000	0.685365
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	2.00000	0.0854358
$549$	−3.00000	−0.128037
$550$	0	0
$551$	−2.00000	−0.0852029
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	−14.0000	−0.594267
$556$	8.00000	0.339276
$557$	35.0000	1.48300	0.741499	−	0.670954i	$-0.234115\pi$
$557$	35.0000	1.48300	0.741499	+	0.670954i	$0.234115\pi$
$558$	0	0
$559$	−28.0000	−1.18427
$560$	32.0000	1.35225
$561$	25.0000	1.05550
$562$	0	0
$563$	−45.0000	−1.89652	−0.948262	−	0.317489i	$-0.897160\pi$
$563$	−45.0000	−1.89652	−0.948262	+	0.317489i	$0.897160\pi$
$564$	14.0000	0.589506
$565$	12.0000	0.504844
$566$	0	0
$567$	−4.00000	−0.167984
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	40.0000	1.67248
$573$	−20.0000	−0.835512
$574$	0	0
$575$	−4.00000	−0.166812
$576$	−8.00000	−0.333333
$577$	24.0000	0.999133	0.499567	−	0.866276i	$-0.333493\pi$
$577$	24.0000	0.999133	0.499567	+	0.866276i	$0.333493\pi$
$578$	0	0
$579$	20.0000	0.831172
$580$	4.00000	0.166091
$581$	8.00000	0.331896
$582$	0	0
$583$	−70.0000	−2.89910
$584$	0	0
$585$	8.00000	0.330759
$586$	0	0
$587$	−33.0000	−1.36206	−0.681028	−	0.732257i	$-0.738467\pi$
$587$	−33.0000	−1.36206	−0.681028	+	0.732257i	$0.738467\pi$
$588$	18.0000	0.742307
$589$	10.0000	0.412043
$590$	0	0
$591$	−10.0000	−0.411345
$592$	−28.0000	−1.15079
$593$	−39.0000	−1.60154	−0.800769	−	0.598973i	$-0.795576\pi$
$593$	−39.0000	−1.60154	−0.800769	+	0.598973i	$0.795576\pi$
$594$	0	0
$595$	−40.0000	−1.63984
$596$	20.0000	0.819232
$597$	−6.00000	−0.245564
$598$	0	0
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	0	0
$605$	−28.0000	−1.13836
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	4.00000	0.162088
$610$	0	0
$611$	−28.0000	−1.13276
$612$	10.0000	0.404226
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	0	0
$615$	−2.00000	−0.0806478
$616$	0	0
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	0	0
$619$	19.0000	0.763674	0.381837	−	0.924230i	$-0.375291\pi$
$619$	19.0000	0.763674	0.381837	+	0.924230i	$0.375291\pi$
$620$	−20.0000	−0.803219
$621$	−4.00000	−0.160514
$622$	0	0
$623$	−72.0000	−2.88462
$624$	16.0000	0.640513
$625$	−19.0000	−0.760000
$626$	0	0
$627$	10.0000	0.399362
$628$	44.0000	1.75579
$629$	35.0000	1.39554
$630$	0	0
$631$	−5.00000	−0.199047	−0.0995234	−	0.995035i	$-0.531732\pi$
$631$	−5.00000	−0.199047	−0.0995234	+	0.995035i	$0.531732\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	0	0
$635$	−24.0000	−0.952411
$636$	−28.0000	−1.11027
$637$	−36.0000	−1.42637
$638$	0	0
$639$	−3.00000	−0.118678
$640$	0	0
$641$	3.00000	0.118493	0.0592464	−	0.998243i	$-0.481130\pi$
$641$	3.00000	0.118493	0.0592464	+	0.998243i	$0.481130\pi$
$642$	0	0
$643$	40.0000	1.57745	0.788723	−	0.614749i	$-0.210743\pi$
$643$	40.0000	1.57745	0.788723	+	0.614749i	$0.210743\pi$
$644$	32.0000	1.26098
$645$	14.0000	0.551249
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	−60.0000	−2.35521
$650$	0	0
$651$	−20.0000	−0.783862
$652$	18.0000	0.704934
$653$	27.0000	1.05659	0.528296	−	0.849060i	$-0.322831\pi$
$653$	27.0000	1.05659	0.528296	+	0.849060i	$0.322831\pi$
$654$	0	0
$655$	−24.0000	−0.937758
$656$	−4.00000	−0.156174
$657$	13.0000	0.507178
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	−20.0000	−0.778499
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	−20.0000	−0.776736
$664$	0	0
$665$	−16.0000	−0.620453
$666$	0	0
$667$	4.00000	0.154881
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	−15.0000	−0.579069
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	−6.00000	−0.230769
$677$	−2.00000	−0.0768662	−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	−2.00000	−0.0768662	−0.0384331	+	0.999261i	$0.512237\pi$
$678$	0	0
$679$	56.0000	2.14908
$680$	0	0
$681$	−3.00000	−0.114960
$682$	0	0
$683$	28.0000	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$683$	28.0000	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$684$	4.00000	0.152944
$685$	2.00000	0.0764161
$686$	0	0
$687$	18.0000	0.686743
$688$	28.0000	1.06749
$689$	56.0000	2.13343
$690$	0	0
$691$	36.0000	1.36950	0.684752	−	0.728776i	$-0.259910\pi$
$691$	36.0000	1.36950	0.684752	+	0.728776i	$0.259910\pi$
$692$	4.00000	0.152057
$693$	−20.0000	−0.759737
$694$	0	0
$695$	8.00000	0.303457
$696$	0	0
$697$	5.00000	0.189389
$698$	0	0
$699$	−6.00000	−0.226941
$700$	−8.00000	−0.302372
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	14.0000	0.528020
$704$	−40.0000	−1.50756
$705$	14.0000	0.527271
$706$	0	0
$707$	−12.0000	−0.451306
$708$	−24.0000	−0.901975
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	0	0
$711$	−2.00000	−0.0750059
$712$	0	0
$713$	−20.0000	−0.749006
$714$	0	0
$715$	40.0000	1.49592
$716$	50.0000	1.86859
$717$	−8.00000	−0.298765
$718$	0	0
$719$	17.0000	0.633993	0.316997	−	0.948427i	$-0.397326\pi$
$719$	17.0000	0.633993	0.316997	+	0.948427i	$0.397326\pi$
$720$	−8.00000	−0.298142
$721$	−28.0000	−1.04277
$722$	0	0
$723$	−7.00000	−0.260333
$724$	16.0000	0.594635
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	42.0000	1.55769	0.778847	−	0.627214i	$-0.215805\pi$
$727$	42.0000	1.55769	0.778847	+	0.627214i	$0.215805\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−35.0000	−1.29452
$732$	−6.00000	−0.221766
$733$	−25.0000	−0.923396	−0.461698	−	0.887037i	$-0.652760\pi$
$733$	−25.0000	−0.923396	−0.461698	+	0.887037i	$0.652760\pi$
$734$	0	0
$735$	18.0000	0.663940
$736$	0	0
$737$	−10.0000	−0.368355
$738$	0	0
$739$	41.0000	1.50821	0.754105	−	0.656754i	$-0.228071\pi$
$739$	41.0000	1.50821	0.754105	+	0.656754i	$0.228071\pi$
$740$	−28.0000	−1.02930
$741$	−8.00000	−0.293887
$742$	0	0
$743$	−10.0000	−0.366864	−0.183432	−	0.983032i	$-0.558721\pi$
$743$	−10.0000	−0.366864	−0.183432	+	0.983032i	$0.558721\pi$
$744$	0	0
$745$	20.0000	0.732743
$746$	0	0
$747$	−2.00000	−0.0731762
$748$	50.0000	1.82818
$749$	−40.0000	−1.46157
$750$	0	0
$751$	−38.0000	−1.38664	−0.693320	−	0.720630i	$-0.743853\pi$
$751$	−38.0000	−1.38664	−0.693320	+	0.720630i	$0.743853\pi$
$752$	28.0000	1.02105
$753$	14.0000	0.510188
$754$	0	0
$755$	0	0
$756$	−8.00000	−0.290957
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	0	0
$759$	−20.0000	−0.725954
$760$	0	0
$761$	20.0000	0.724999	0.362500	−	0.931984i	$-0.381923\pi$
$761$	20.0000	0.724999	0.362500	+	0.931984i	$0.381923\pi$
$762$	0	0
$763$	32.0000	1.15848
$764$	−40.0000	−1.44715
$765$	10.0000	0.361551
$766$	0	0
$767$	48.0000	1.73318
$768$	−16.0000	−0.577350
$769$	−33.0000	−1.19001	−0.595005	−	0.803722i	$-0.702850\pi$
$769$	−33.0000	−1.19001	−0.595005	+	0.803722i	$0.702850\pi$
$770$	0	0
$771$	−3.00000	−0.108042
$772$	40.0000	1.43963
$773$	−31.0000	−1.11499	−0.557496	−	0.830179i	$-0.688238\pi$
$773$	−31.0000	−1.11499	−0.557496	+	0.830179i	$0.688238\pi$
$774$	0	0
$775$	5.00000	0.179605
$776$	0	0
$777$	−28.0000	−1.00449
$778$	0	0
$779$	2.00000	0.0716574
$780$	16.0000	0.572892
$781$	−15.0000	−0.536742
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	36.0000	1.28571
$785$	44.0000	1.57043
$786$	0	0
$787$	−17.0000	−0.605985	−0.302992	−	0.952993i	$-0.597986\pi$
$787$	−17.0000	−0.605985	−0.302992	+	0.952993i	$0.597986\pi$
$788$	−20.0000	−0.712470
$789$	23.0000	0.818822
$790$	0	0
$791$	24.0000	0.853342
$792$	0	0
$793$	12.0000	0.426132
$794$	0	0
$795$	−28.0000	−0.993058
$796$	−12.0000	−0.425329
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	0	0
$799$	−35.0000	−1.23821
$800$	0	0
$801$	18.0000	0.635999
$802$	0	0
$803$	65.0000	2.29380
$804$	−4.00000	−0.141069
$805$	32.0000	1.12785
$806$	0	0
$807$	−16.0000	−0.563227
$808$	0	0
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	0	0
$811$	1.00000	0.0351147	0.0175574	−	0.999846i	$-0.494411\pi$
$811$	1.00000	0.0351147	0.0175574	+	0.999846i	$0.494411\pi$
$812$	8.00000	0.280745
$813$	27.0000	0.946931
$814$	0	0
$815$	18.0000	0.630512
$816$	20.0000	0.700140
$817$	−14.0000	−0.489798
$818$	0	0
$819$	16.0000	0.559085
$820$	−4.00000	−0.139686
$821$	−44.0000	−1.53561	−0.767805	−	0.640683i	$-0.778651\pi$
$821$	−44.0000	−1.53561	−0.767805	+	0.640683i	$0.778651\pi$
$822$	0	0
$823$	−6.00000	−0.209147	−0.104573	−	0.994517i	$-0.533348\pi$
$823$	−6.00000	−0.209147	−0.104573	+	0.994517i	$0.533348\pi$
$824$	0	0
$825$	5.00000	0.174078
$826$	0	0
$827$	5.00000	0.173867	0.0869335	−	0.996214i	$-0.472293\pi$
$827$	5.00000	0.173867	0.0869335	+	0.996214i	$0.472293\pi$
$828$	−8.00000	−0.278019
$829$	13.0000	0.451509	0.225754	−	0.974184i	$-0.427515\pi$
$829$	13.0000	0.451509	0.225754	+	0.974184i	$0.427515\pi$
$830$	0	0
$831$	−19.0000	−0.659103
$832$	32.0000	1.10940
$833$	−45.0000	−1.55916
$834$	0	0
$835$	0	0
$836$	20.0000	0.691714
$837$	5.00000	0.172825
$838$	0	0
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	−15.0000	−0.516627
$844$	24.0000	0.826114
$845$	−6.00000	−0.206406
$846$	0	0
$847$	−56.0000	−1.92418
$848$	−56.0000	−1.92305
$849$	−17.0000	−0.583438
$850$	0	0
$851$	−28.0000	−0.959828
$852$	−6.00000	−0.205557
$853$	−18.0000	−0.616308	−0.308154	−	0.951336i	$-0.599711\pi$
$853$	−18.0000	−0.616308	−0.308154	+	0.951336i	$0.599711\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	12.0000	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$858$	0	0
$859$	3.00000	0.102359	0.0511793	−	0.998689i	$-0.483702\pi$
$859$	3.00000	0.102359	0.0511793	+	0.998689i	$0.483702\pi$
$860$	28.0000	0.954792
$861$	−4.00000	−0.136320
$862$	0	0
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	0	0
$865$	4.00000	0.136004
$866$	0	0
$867$	−8.00000	−0.271694
$868$	−40.0000	−1.35769
$869$	−10.0000	−0.339227
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	−14.0000	−0.473828
$874$	0	0
$875$	−48.0000	−1.62270
$876$	26.0000	0.878459
$877$	−57.0000	−1.92475	−0.962377	−	0.271719i	$-0.912408\pi$
$877$	−57.0000	−1.92475	−0.962377	+	0.271719i	$0.912408\pi$
$878$	0	0
$879$	23.0000	0.775771
$880$	−40.0000	−1.34840
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	6.00000	0.201916	0.100958	−	0.994891i	$-0.467809\pi$
$883$	6.00000	0.201916	0.100958	+	0.994891i	$0.467809\pi$
$884$	−40.0000	−1.34535
$885$	−24.0000	−0.806751
$886$	0	0
$887$	33.0000	1.10803	0.554016	−	0.832506i	$-0.313095\pi$
$887$	33.0000	1.10803	0.554016	+	0.832506i	$0.313095\pi$
$888$	0	0
$889$	−48.0000	−1.60987
$890$	0	0
$891$	5.00000	0.167506
$892$	32.0000	1.07144
$893$	−14.0000	−0.468492
$894$	0	0
$895$	50.0000	1.67132
$896$	0	0
$897$	16.0000	0.534224
$898$	0	0
$899$	−5.00000	−0.166759
$900$	2.00000	0.0666667
$901$	70.0000	2.33204
$902$	0	0
$903$	28.0000	0.931782
$904$	0	0
$905$	16.0000	0.531858
$906$	0	0
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	−6.00000	−0.199117
$909$	3.00000	0.0995037
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	8.00000	0.264906
$913$	−10.0000	−0.330952
$914$	0	0
$915$	−6.00000	−0.198354
$916$	36.0000	1.18947
$917$	−48.0000	−1.58510
$918$	0	0
$919$	44.0000	1.45143	0.725713	−	0.687998i	$-0.241510\pi$
$919$	44.0000	1.45143	0.725713	+	0.687998i	$0.241510\pi$
$920$	0	0
$921$	−17.0000	−0.560169
$922$	0	0
$923$	12.0000	0.394985
$924$	−40.0000	−1.31590
$925$	7.00000	0.230159
$926$	0	0
$927$	7.00000	0.229910
$928$	0	0
$929$	−45.0000	−1.47640	−0.738201	−	0.674581i	$-0.764324\pi$
$929$	−45.0000	−1.47640	−0.738201	+	0.674581i	$0.764324\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	−12.0000	−0.393073
$933$	−4.00000	−0.130954
$934$	0	0
$935$	50.0000	1.63517
$936$	0	0
$937$	32.0000	1.04539	0.522697	−	0.852518i	$-0.324926\pi$
$937$	32.0000	1.04539	0.522697	+	0.852518i	$0.324926\pi$
$938$	0	0
$939$	−16.0000	−0.522140
$940$	28.0000	0.913259
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	0	0
$943$	−4.00000	−0.130258
$944$	−48.0000	−1.56227
$945$	−8.00000	−0.260240
$946$	0	0
$947$	−26.0000	−0.844886	−0.422443	−	0.906389i	$-0.638827\pi$
$947$	−26.0000	−0.844886	−0.422443	+	0.906389i	$0.638827\pi$
$948$	−4.00000	−0.129914
$949$	−52.0000	−1.68799
$950$	0	0
$951$	−3.00000	−0.0972817
$952$	0	0
$953$	44.0000	1.42530	0.712650	−	0.701520i	$-0.247495\pi$
$953$	44.0000	1.42530	0.712650	+	0.701520i	$0.247495\pi$
$954$	0	0
$955$	−40.0000	−1.29437
$956$	−16.0000	−0.517477
$957$	−5.00000	−0.161627
$958$	0	0
$959$	4.00000	0.129167
$960$	−16.0000	−0.516398
$961$	−6.00000	−0.193548
$962$	0	0
$963$	10.0000	0.322245
$964$	−14.0000	−0.450910
$965$	40.0000	1.28765
$966$	0	0
$967$	−14.0000	−0.450210	−0.225105	−	0.974335i	$-0.572272\pi$
$967$	−14.0000	−0.450210	−0.225105	+	0.974335i	$0.572272\pi$
$968$	0	0
$969$	−10.0000	−0.321246
$970$	0	0
$971$	−25.0000	−0.802288	−0.401144	−	0.916015i	$-0.631387\pi$
$971$	−25.0000	−0.802288	−0.401144	+	0.916015i	$0.631387\pi$
$972$	2.00000	0.0641500
$973$	16.0000	0.512936
$974$	0	0
$975$	−4.00000	−0.128103
$976$	−12.0000	−0.384111
$977$	7.00000	0.223950	0.111975	−	0.993711i	$-0.464282\pi$
$977$	7.00000	0.223950	0.111975	+	0.993711i	$0.464282\pi$
$978$	0	0
$979$	90.0000	2.87641
$980$	36.0000	1.14998
$981$	−8.00000	−0.255420
$982$	0	0
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	0	0
$985$	−20.0000	−0.637253
$986$	0	0
$987$	28.0000	0.891250
$988$	−16.0000	−0.509028
$989$	28.0000	0.890348
$990$	0	0
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	0	0
$993$	22.0000	0.698149
$994$	0	0
$995$	−12.0000	−0.380426
$996$	−4.00000	−0.126745
$997$	20.0000	0.633406	0.316703	−	0.948525i	$-0.397424\pi$
$997$	20.0000	0.633406	0.316703	+	0.948525i	$0.397424\pi$
$998$	0	0
$999$	7.00000	0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	123.2.a.b.1.1	✓	1
3.2	odd	2		369.2.a.a.1.1		1
4.3	odd	2		1968.2.a.j.1.1		1
5.4	even	2		3075.2.a.i.1.1		1
7.6	odd	2		6027.2.a.e.1.1		1
8.3	odd	2		7872.2.a.q.1.1		1
8.5	even	2		7872.2.a.bc.1.1		1
12.11	even	2		5904.2.a.t.1.1		1
15.14	odd	2		9225.2.a.w.1.1		1
41.40	even	2		5043.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.2.a.b.1.1	✓	1	1.1	even	1	trivial
369.2.a.a.1.1		1	3.2	odd	2
1968.2.a.j.1.1		1	4.3	odd	2
3075.2.a.i.1.1		1	5.4	even	2
5043.2.a.b.1.1		1	41.40	even	2
5904.2.a.t.1.1		1	12.11	even	2
6027.2.a.e.1.1		1	7.6	odd	2
7872.2.a.q.1.1		1	8.3	odd	2
7872.2.a.bc.1.1		1	8.5	even	2
9225.2.a.w.1.1		1	15.14	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:	\( N \)	\(=\)	\( 123 = 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	123.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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