LMFDB - Embedded newform 1638.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.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:	$13.0794958511$
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	$\chi$	$=$	1638.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -5.00000 q^{11} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{19} -1.00000 q^{20} -5.00000 q^{22} -5.00000 q^{23} -4.00000 q^{25} +1.00000 q^{26} -1.00000 q^{28} -1.00000 q^{29} -6.00000 q^{31} +1.00000 q^{32} -1.00000 q^{34} +1.00000 q^{35} +7.00000 q^{37} -1.00000 q^{38} -1.00000 q^{40} -2.00000 q^{41} +1.00000 q^{43} -5.00000 q^{44} -5.00000 q^{46} +1.00000 q^{49} -4.00000 q^{50} +1.00000 q^{52} -6.00000 q^{53} +5.00000 q^{55} -1.00000 q^{56} -1.00000 q^{58} -6.00000 q^{59} -7.00000 q^{61} -6.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} -8.00000 q^{67} -1.00000 q^{68} +1.00000 q^{70} -10.0000 q^{71} +9.00000 q^{73} +7.00000 q^{74} -1.00000 q^{76} +5.00000 q^{77} +2.00000 q^{79} -1.00000 q^{80} -2.00000 q^{82} -2.00000 q^{83} +1.00000 q^{85} +1.00000 q^{86} -5.00000 q^{88} +6.00000 q^{89} -1.00000 q^{91} -5.00000 q^{92} +1.00000 q^{95} +14.0000 q^{97} +1.00000 q^{98} +O(q^{100})$

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
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−5.00000	−1.06600
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	1.00000	0.196116
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	1.00000	0.169031
$36$	0	0
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	−5.00000	−0.753778
$45$	0	0
$46$	−5.00000	−0.737210
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−4.00000	−0.565685
$51$	0	0
$52$	1.00000	0.138675
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	5.00000	0.674200
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−1.00000	−0.131306
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	−6.00000	−0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−1.00000	−0.121268
$69$	0	0
$70$	1.00000	0.119523
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	7.00000	0.813733
$75$	0	0
$76$	−1.00000	−0.114708
$77$	5.00000	0.569803
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−2.00000	−0.220863
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	1.00000	0.107833
$87$	0	0
$88$	−5.00000	−0.533002
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	−5.00000	−0.521286
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	−4.00000	−0.400000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	7.00000	0.689730	0.344865	−	0.938652i	$-0.387925\pi$
$103$	7.00000	0.689730	0.344865	+	0.938652i	$0.387925\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−6.00000	−0.582772
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	5.00000	0.476731
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−16.0000	−1.50515	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000	−1.50515	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	5.00000	0.466252
$116$	−1.00000	−0.0928477
$117$	0	0
$118$	−6.00000	−0.552345
$119$	1.00000	0.0916698
$120$	0	0
$121$	14.0000	1.27273
$122$	−7.00000	−0.633750
$123$	0	0
$124$	−6.00000	−0.538816
$125$	9.00000	0.804984
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−1.00000	−0.0877058
$131$	9.00000	0.786334	0.393167	−	0.919467i	$-0.371379\pi$
$131$	9.00000	0.786334	0.393167	+	0.919467i	$0.371379\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	−13.0000	−1.11066	−0.555332	−	0.831628i	$-0.687409\pi$
$137$	−13.0000	−1.11066	−0.555332	+	0.831628i	$0.687409\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	−10.0000	−0.839181
$143$	−5.00000	−0.418121
$144$	0	0
$145$	1.00000	0.0830455
$146$	9.00000	0.744845
$147$	0	0
$148$	7.00000	0.575396
$149$	−24.0000	−1.96616	−0.983078	−	0.183186i	$-0.941359\pi$
$149$	−24.0000	−1.96616	−0.983078	+	0.183186i	$0.941359\pi$
$150$	0	0
$151$	−7.00000	−0.569652	−0.284826	−	0.958579i	$-0.591936\pi$
$151$	−7.00000	−0.569652	−0.284826	+	0.958579i	$0.591936\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	5.00000	0.402911
$155$	6.00000	0.481932
$156$	0	0
$157$	13.0000	1.03751	0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000	1.03751	0.518756	+	0.854922i	$0.326395\pi$
$158$	2.00000	0.159111
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	5.00000	0.394055
$162$	0	0
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−2.00000	−0.155230
$167$	−3.00000	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	1.00000	0.0766965
$171$	0	0
$172$	1.00000	0.0762493
$173$	−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
$176$	−5.00000	−0.376889
$177$	0	0
$178$	6.00000	0.449719
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	−5.00000	−0.368605
$185$	−7.00000	−0.514650
$186$	0	0
$187$	5.00000	0.365636
$188$	0	0
$189$	0	0
$190$	1.00000	0.0725476
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	1.00000	0.0714286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	25.0000	1.77220	0.886102	−	0.463491i	$-0.153403\pi$
$199$	25.0000	1.77220	0.886102	+	0.463491i	$0.153403\pi$
$200$	−4.00000	−0.282843
$201$	0	0
$202$	6.00000	0.422159
$203$	1.00000	0.0701862
$204$	0	0
$205$	2.00000	0.139686
$206$	7.00000	0.487713
$207$	0	0
$208$	1.00000	0.0693375
$209$	5.00000	0.345857
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	6.00000	0.410152
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	6.00000	0.407307
$218$	−7.00000	−0.474100
$219$	0	0
$220$	5.00000	0.337100
$221$	−1.00000	−0.0672673
$222$	0	0
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−16.0000	−1.06430
$227$	−10.0000	−0.663723	−0.331862	−	0.943328i	$-0.607677\pi$
$227$	−10.0000	−0.663723	−0.331862	+	0.943328i	$0.607677\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	5.00000	0.329690
$231$	0	0
$232$	−1.00000	−0.0656532
$233$	4.00000	0.262049	0.131024	−	0.991379i	$-0.458173\pi$
$233$	4.00000	0.262049	0.131024	+	0.991379i	$0.458173\pi$
$234$	0	0
$235$	0	0
$236$	−6.00000	−0.390567
$237$	0	0
$238$	1.00000	0.0648204
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	14.0000	0.899954
$243$	0	0
$244$	−7.00000	−0.448129
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−1.00000	−0.0636285
$248$	−6.00000	−0.381000
$249$	0	0
$250$	9.00000	0.569210
$251$	25.0000	1.57799	0.788993	−	0.614402i	$-0.210603\pi$
$251$	25.0000	1.57799	0.788993	+	0.614402i	$0.210603\pi$
$252$	0	0
$253$	25.0000	1.57174
$254$	4.00000	0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−7.00000	−0.434959
$260$	−1.00000	−0.0620174
$261$	0	0
$262$	9.00000	0.556022
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	1.00000	0.0613139
$267$	0	0
$268$	−8.00000	−0.488678
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	−30.0000	−1.82237	−0.911185	−	0.411997i	$-0.864831\pi$
$271$	−30.0000	−1.82237	−0.911185	+	0.411997i	$0.864831\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	−13.0000	−0.785359
$275$	20.0000	1.20605
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	−2.00000	−0.119952
$279$	0	0
$280$	1.00000	0.0597614
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−10.0000	−0.593391
$285$	0	0
$286$	−5.00000	−0.295656
$287$	2.00000	0.118056
$288$	0	0
$289$	−16.0000	−0.941176
$290$	1.00000	0.0587220
$291$	0	0
$292$	9.00000	0.526685
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	7.00000	0.406867
$297$	0	0
$298$	−24.0000	−1.39028
$299$	−5.00000	−0.289157
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	−7.00000	−0.402805
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	7.00000	0.400819
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	5.00000	0.284901
$309$	0	0
$310$	6.00000	0.340777
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	−30.0000	−1.69570	−0.847850	−	0.530236i	$-0.822103\pi$
$313$	−30.0000	−1.69570	−0.847850	+	0.530236i	$0.822103\pi$
$314$	13.0000	0.733632
$315$	0	0
$316$	2.00000	0.112509
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	0	0
$319$	5.00000	0.279946
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	5.00000	0.278639
$323$	1.00000	0.0556415
$324$	0	0
$325$	−4.00000	−0.221880
$326$	−6.00000	−0.332309
$327$	0	0
$328$	−2.00000	−0.110432
$329$	0	0
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	−2.00000	−0.109764
$333$	0	0
$334$	−3.00000	−0.164153
$335$	8.00000	0.437087
$336$	0	0
$337$	−9.00000	−0.490261	−0.245131	−	0.969490i	$-0.578831\pi$
$337$	−9.00000	−0.490261	−0.245131	+	0.969490i	$0.578831\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	1.00000	0.0542326
$341$	30.0000	1.62459
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	1.00000	0.0539164
$345$	0	0
$346$	−2.00000	−0.107521
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	−4.00000	−0.214115	−0.107058	−	0.994253i	$-0.534143\pi$
$349$	−4.00000	−0.214115	−0.107058	+	0.994253i	$0.534143\pi$
$350$	4.00000	0.213809
$351$	0	0
$352$	−5.00000	−0.266501
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	0	0
$355$	10.0000	0.530745
$356$	6.00000	0.317999
$357$	0	0
$358$	0	0
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	10.0000	0.525588
$363$	0	0
$364$	−1.00000	−0.0524142
$365$	−9.00000	−0.471082
$366$	0	0
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	−5.00000	−0.260643
$369$	0	0
$370$	−7.00000	−0.363913
$371$	6.00000	0.311504
$372$	0	0
$373$	34.0000	1.76045	0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000	1.76045	0.880227	+	0.474554i	$0.157390\pi$
$374$	5.00000	0.258544
$375$	0	0
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	1.00000	0.0512989
$381$	0	0
$382$	15.0000	0.767467
$383$	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
$385$	−5.00000	−0.254824
$386$	14.0000	0.712581
$387$	0	0
$388$	14.0000	0.710742
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	5.00000	0.252861
$392$	1.00000	0.0505076
$393$	0	0
$394$	−6.00000	−0.302276
$395$	−2.00000	−0.100631
$396$	0	0
$397$	−4.00000	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$398$	25.0000	1.25314
$399$	0	0
$400$	−4.00000	−0.200000
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	−6.00000	−0.298881
$404$	6.00000	0.298511
$405$	0	0
$406$	1.00000	0.0496292
$407$	−35.0000	−1.73489
$408$	0	0
$409$	−13.0000	−0.642809	−0.321404	−	0.946942i	$-0.604155\pi$
$409$	−13.0000	−0.642809	−0.321404	+	0.946942i	$0.604155\pi$
$410$	2.00000	0.0987730
$411$	0	0
$412$	7.00000	0.344865
$413$	6.00000	0.295241
$414$	0	0
$415$	2.00000	0.0981761
$416$	1.00000	0.0490290
$417$	0	0
$418$	5.00000	0.244558
$419$	19.0000	0.928211	0.464105	−	0.885780i	$-0.346376\pi$
$419$	19.0000	0.928211	0.464105	+	0.885780i	$0.346376\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−5.00000	−0.243396
$423$	0	0
$424$	−6.00000	−0.291386
$425$	4.00000	0.194029
$426$	0	0
$427$	7.00000	0.338754
$428$	6.00000	0.290021
$429$	0	0
$430$	−1.00000	−0.0482243
$431$	−26.0000	−1.25238	−0.626188	−	0.779672i	$-0.715386\pi$
$431$	−26.0000	−1.25238	−0.626188	+	0.779672i	$0.715386\pi$
$432$	0	0
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	6.00000	0.288009
$435$	0	0
$436$	−7.00000	−0.335239
$437$	5.00000	0.239182
$438$	0	0
$439$	−41.0000	−1.95682	−0.978412	−	0.206666i	$-0.933739\pi$
$439$	−41.0000	−1.95682	−0.978412	+	0.206666i	$0.933739\pi$
$440$	5.00000	0.238366
$441$	0	0
$442$	−1.00000	−0.0475651
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	−12.0000	−0.568216
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−33.0000	−1.55737	−0.778683	−	0.627417i	$-0.784112\pi$
$449$	−33.0000	−1.55737	−0.778683	+	0.627417i	$0.784112\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	−16.0000	−0.752577
$453$	0	0
$454$	−10.0000	−0.469323
$455$	1.00000	0.0468807
$456$	0	0
$457$	12.0000	0.561336	0.280668	−	0.959805i	$-0.409444\pi$
$457$	12.0000	0.561336	0.280668	+	0.959805i	$0.409444\pi$
$458$	22.0000	1.02799
$459$	0	0
$460$	5.00000	0.233126
$461$	29.0000	1.35066	0.675332	−	0.737514i	$-0.264000\pi$
$461$	29.0000	1.35066	0.675332	+	0.737514i	$0.264000\pi$
$462$	0	0
$463$	21.0000	0.975953	0.487976	−	0.872857i	$-0.337735\pi$
$463$	21.0000	0.975953	0.487976	+	0.872857i	$0.337735\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	4.00000	0.185296
$467$	29.0000	1.34196	0.670980	−	0.741475i	$-0.265874\pi$
$467$	29.0000	1.34196	0.670980	+	0.741475i	$0.265874\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	0	0
$472$	−6.00000	−0.276172
$473$	−5.00000	−0.229900
$474$	0	0
$475$	4.00000	0.183533
$476$	1.00000	0.0458349
$477$	0	0
$478$	−12.0000	−0.548867
$479$	−33.0000	−1.50781	−0.753904	−	0.656984i	$-0.771832\pi$
$479$	−33.0000	−1.50781	−0.753904	+	0.656984i	$0.771832\pi$
$480$	0	0
$481$	7.00000	0.319173
$482$	−10.0000	−0.455488
$483$	0	0
$484$	14.0000	0.636364
$485$	−14.0000	−0.635707
$486$	0	0
$487$	−40.0000	−1.81257	−0.906287	−	0.422664i	$-0.861095\pi$
$487$	−40.0000	−1.81257	−0.906287	+	0.422664i	$0.861095\pi$
$488$	−7.00000	−0.316875
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	−16.0000	−0.722070	−0.361035	−	0.932552i	$-0.617576\pi$
$491$	−16.0000	−0.722070	−0.361035	+	0.932552i	$0.617576\pi$
$492$	0	0
$493$	1.00000	0.0450377
$494$	−1.00000	−0.0449921
$495$	0	0
$496$	−6.00000	−0.269408
$497$	10.0000	0.448561
$498$	0	0
$499$	−8.00000	−0.358129	−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000	−0.358129	−0.179065	+	0.983837i	$0.557307\pi$
$500$	9.00000	0.402492
$501$	0	0
$502$	25.0000	1.11580
$503$	28.0000	1.24846	0.624229	−	0.781241i	$-0.285413\pi$
$503$	28.0000	1.24846	0.624229	+	0.781241i	$0.285413\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	25.0000	1.11139
$507$	0	0
$508$	4.00000	0.177471
$509$	21.0000	0.930809	0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000	0.930809	0.465404	+	0.885098i	$0.345909\pi$
$510$	0	0
$511$	−9.00000	−0.398137
$512$	1.00000	0.0441942
$513$	0	0
$514$	18.0000	0.793946
$515$	−7.00000	−0.308457
$516$	0	0
$517$	0	0
$518$	−7.00000	−0.307562
$519$	0	0
$520$	−1.00000	−0.0438529
$521$	−15.0000	−0.657162	−0.328581	−	0.944476i	$-0.606570\pi$
$521$	−15.0000	−0.657162	−0.328581	+	0.944476i	$0.606570\pi$
$522$	0	0
$523$	−42.0000	−1.83653	−0.918266	−	0.395964i	$-0.870410\pi$
$523$	−42.0000	−1.83653	−0.918266	+	0.395964i	$0.870410\pi$
$524$	9.00000	0.393167
$525$	0	0
$526$	8.00000	0.348817
$527$	6.00000	0.261364
$528$	0	0
$529$	2.00000	0.0869565
$530$	6.00000	0.260623
$531$	0	0
$532$	1.00000	0.0433555
$533$	−2.00000	−0.0866296
$534$	0	0
$535$	−6.00000	−0.259403
$536$	−8.00000	−0.345547
$537$	0	0
$538$	30.0000	1.29339
$539$	−5.00000	−0.215365
$540$	0	0
$541$	25.0000	1.07483	0.537417	−	0.843317i	$-0.319400\pi$
$541$	25.0000	1.07483	0.537417	+	0.843317i	$0.319400\pi$
$542$	−30.0000	−1.28861
$543$	0	0
$544$	−1.00000	−0.0428746
$545$	7.00000	0.299847
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	−13.0000	−0.555332
$549$	0	0
$550$	20.0000	0.852803
$551$	1.00000	0.0426014
$552$	0	0
$553$	−2.00000	−0.0850487
$554$	−8.00000	−0.339887
$555$	0	0
$556$	−2.00000	−0.0848189
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	0	0
$559$	1.00000	0.0422955
$560$	1.00000	0.0422577
$561$	0	0
$562$	2.00000	0.0843649
$563$	−23.0000	−0.969334	−0.484667	−	0.874699i	$-0.661059\pi$
$563$	−23.0000	−0.969334	−0.484667	+	0.874699i	$0.661059\pi$
$564$	0	0
$565$	16.0000	0.673125
$566$	−4.00000	−0.168133
$567$	0	0
$568$	−10.0000	−0.419591
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	−5.00000	−0.209061
$573$	0	0
$574$	2.00000	0.0834784
$575$	20.0000	0.834058
$576$	0	0
$577$	−42.0000	−1.74848	−0.874241	−	0.485491i	$-0.838641\pi$
$577$	−42.0000	−1.74848	−0.874241	+	0.485491i	$0.838641\pi$
$578$	−16.0000	−0.665512
$579$	0	0
$580$	1.00000	0.0415227
$581$	2.00000	0.0829740
$582$	0	0
$583$	30.0000	1.24247
$584$	9.00000	0.372423
$585$	0	0
$586$	−26.0000	−1.07405
$587$	−42.0000	−1.73353	−0.866763	−	0.498721i	$-0.833803\pi$
$587$	−42.0000	−1.73353	−0.866763	+	0.498721i	$0.833803\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	6.00000	0.247016
$591$	0	0
$592$	7.00000	0.287698
$593$	40.0000	1.64260	0.821302	−	0.570494i	$-0.193248\pi$
$593$	40.0000	1.64260	0.821302	+	0.570494i	$0.193248\pi$
$594$	0	0
$595$	−1.00000	−0.0409960
$596$	−24.0000	−0.983078
$597$	0	0
$598$	−5.00000	−0.204465
$599$	−15.0000	−0.612883	−0.306442	−	0.951889i	$-0.599138\pi$
$599$	−15.0000	−0.612883	−0.306442	+	0.951889i	$0.599138\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	−1.00000	−0.0407570
$603$	0	0
$604$	−7.00000	−0.284826
$605$	−14.0000	−0.569181
$606$	0	0
$607$	−37.0000	−1.50178	−0.750892	−	0.660425i	$-0.770376\pi$
$607$	−37.0000	−1.50178	−0.750892	+	0.660425i	$0.770376\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	7.00000	0.283422
$611$	0	0
$612$	0	0
$613$	−7.00000	−0.282727	−0.141364	−	0.989958i	$-0.545149\pi$
$613$	−7.00000	−0.282727	−0.141364	+	0.989958i	$0.545149\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	5.00000	0.201456
$617$	−43.0000	−1.73111	−0.865557	−	0.500810i	$-0.833036\pi$
$617$	−43.0000	−1.73111	−0.865557	+	0.500810i	$0.833036\pi$
$618$	0	0
$619$	−37.0000	−1.48716	−0.743578	−	0.668649i	$-0.766873\pi$
$619$	−37.0000	−1.48716	−0.743578	+	0.668649i	$0.766873\pi$
$620$	6.00000	0.240966
$621$	0	0
$622$	−12.0000	−0.481156
$623$	−6.00000	−0.240385
$624$	0	0
$625$	11.0000	0.440000
$626$	−30.0000	−1.19904
$627$	0	0
$628$	13.0000	0.518756
$629$	−7.00000	−0.279108
$630$	0	0
$631$	25.0000	0.995234	0.497617	−	0.867397i	$-0.334208\pi$
$631$	25.0000	0.995234	0.497617	+	0.867397i	$0.334208\pi$
$632$	2.00000	0.0795557
$633$	0	0
$634$	10.0000	0.397151
$635$	−4.00000	−0.158735
$636$	0	0
$637$	1.00000	0.0396214
$638$	5.00000	0.197952
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−14.0000	−0.552967	−0.276483	−	0.961019i	$-0.589169\pi$
$641$	−14.0000	−0.552967	−0.276483	+	0.961019i	$0.589169\pi$
$642$	0	0
$643$	5.00000	0.197181	0.0985904	−	0.995128i	$-0.468567\pi$
$643$	5.00000	0.197181	0.0985904	+	0.995128i	$0.468567\pi$
$644$	5.00000	0.197028
$645$	0	0
$646$	1.00000	0.0393445
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	30.0000	1.17760
$650$	−4.00000	−0.156893
$651$	0	0
$652$	−6.00000	−0.234978
$653$	−3.00000	−0.117399	−0.0586995	−	0.998276i	$-0.518695\pi$
$653$	−3.00000	−0.117399	−0.0586995	+	0.998276i	$0.518695\pi$
$654$	0	0
$655$	−9.00000	−0.351659
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−20.0000	−0.777910	−0.388955	−	0.921257i	$-0.627164\pi$
$661$	−20.0000	−0.777910	−0.388955	+	0.921257i	$0.627164\pi$
$662$	−18.0000	−0.699590
$663$	0	0
$664$	−2.00000	−0.0776151
$665$	−1.00000	−0.0387783
$666$	0	0
$667$	5.00000	0.193601
$668$	−3.00000	−0.116073
$669$	0	0
$670$	8.00000	0.309067
$671$	35.0000	1.35116
$672$	0	0
$673$	−21.0000	−0.809491	−0.404745	−	0.914429i	$-0.632640\pi$
$673$	−21.0000	−0.809491	−0.404745	+	0.914429i	$0.632640\pi$
$674$	−9.00000	−0.346667
$675$	0	0
$676$	1.00000	0.0384615
$677$	2.00000	0.0768662	0.0384331	−	0.999261i	$-0.487763\pi$
$677$	2.00000	0.0768662	0.0384331	+	0.999261i	$0.487763\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	1.00000	0.0383482
$681$	0	0
$682$	30.0000	1.14876
$683$	5.00000	0.191320	0.0956598	−	0.995414i	$-0.469504\pi$
$683$	5.00000	0.191320	0.0956598	+	0.995414i	$0.469504\pi$
$684$	0	0
$685$	13.0000	0.496704
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	1.00000	0.0381246
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	20.0000	0.759190
$695$	2.00000	0.0758643
$696$	0	0
$697$	2.00000	0.0757554
$698$	−4.00000	−0.151402
$699$	0	0
$700$	4.00000	0.151186
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	−7.00000	−0.264010
$704$	−5.00000	−0.188445
$705$	0	0
$706$	−10.0000	−0.376355
$707$	−6.00000	−0.225653
$708$	0	0
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	10.0000	0.375293
$711$	0	0
$712$	6.00000	0.224860
$713$	30.0000	1.12351
$714$	0	0
$715$	5.00000	0.186989
$716$	0	0
$717$	0	0
$718$	30.0000	1.11959
$719$	−44.0000	−1.64092	−0.820462	−	0.571702i	$-0.806283\pi$
$719$	−44.0000	−1.64092	−0.820462	+	0.571702i	$0.806283\pi$
$720$	0	0
$721$	−7.00000	−0.260694
$722$	−18.0000	−0.669891
$723$	0	0
$724$	10.0000	0.371647
$725$	4.00000	0.148556
$726$	0	0
$727$	3.00000	0.111264	0.0556319	−	0.998451i	$-0.482283\pi$
$727$	3.00000	0.111264	0.0556319	+	0.998451i	$0.482283\pi$
$728$	−1.00000	−0.0370625
$729$	0	0
$730$	−9.00000	−0.333105
$731$	−1.00000	−0.0369863
$732$	0	0
$733$	12.0000	0.443230	0.221615	−	0.975134i	$-0.428867\pi$
$733$	12.0000	0.443230	0.221615	+	0.975134i	$0.428867\pi$
$734$	−24.0000	−0.885856
$735$	0	0
$736$	−5.00000	−0.184302
$737$	40.0000	1.47342
$738$	0	0
$739$	46.0000	1.69214	0.846069	−	0.533074i	$-0.178963\pi$
$739$	46.0000	1.69214	0.846069	+	0.533074i	$0.178963\pi$
$740$	−7.00000	−0.257325
$741$	0	0
$742$	6.00000	0.220267
$743$	−28.0000	−1.02722	−0.513610	−	0.858024i	$-0.671692\pi$
$743$	−28.0000	−1.02722	−0.513610	+	0.858024i	$0.671692\pi$
$744$	0	0
$745$	24.0000	0.879292
$746$	34.0000	1.24483
$747$	0	0
$748$	5.00000	0.182818
$749$	−6.00000	−0.219235
$750$	0	0
$751$	28.0000	1.02173	0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000	1.02173	0.510867	+	0.859660i	$0.329324\pi$
$752$	0	0
$753$	0	0
$754$	−1.00000	−0.0364179
$755$	7.00000	0.254756
$756$	0	0
$757$	−32.0000	−1.16306	−0.581530	−	0.813525i	$-0.697546\pi$
$757$	−32.0000	−1.16306	−0.581530	+	0.813525i	$0.697546\pi$
$758$	−2.00000	−0.0726433
$759$	0	0
$760$	1.00000	0.0362738
$761$	−28.0000	−1.01500	−0.507500	−	0.861652i	$-0.669430\pi$
$761$	−28.0000	−1.01500	−0.507500	+	0.861652i	$0.669430\pi$
$762$	0	0
$763$	7.00000	0.253417
$764$	15.0000	0.542681
$765$	0	0
$766$	9.00000	0.325183
$767$	−6.00000	−0.216647
$768$	0	0
$769$	−29.0000	−1.04577	−0.522883	−	0.852404i	$-0.675144\pi$
$769$	−29.0000	−1.04577	−0.522883	+	0.852404i	$0.675144\pi$
$770$	−5.00000	−0.180187
$771$	0	0
$772$	14.0000	0.503871
$773$	−11.0000	−0.395643	−0.197821	−	0.980238i	$-0.563387\pi$
$773$	−11.0000	−0.395643	−0.197821	+	0.980238i	$0.563387\pi$
$774$	0	0
$775$	24.0000	0.862105
$776$	14.0000	0.502571
$777$	0	0
$778$	10.0000	0.358517
$779$	2.00000	0.0716574
$780$	0	0
$781$	50.0000	1.78914
$782$	5.00000	0.178800
$783$	0	0
$784$	1.00000	0.0357143
$785$	−13.0000	−0.463990
$786$	0	0
$787$	−37.0000	−1.31891	−0.659454	−	0.751745i	$-0.729212\pi$
$787$	−37.0000	−1.31891	−0.659454	+	0.751745i	$0.729212\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	−2.00000	−0.0711568
$791$	16.0000	0.568895
$792$	0	0
$793$	−7.00000	−0.248577
$794$	−4.00000	−0.141955
$795$	0	0
$796$	25.0000	0.886102
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	0	0
$800$	−4.00000	−0.141421
$801$	0	0
$802$	10.0000	0.353112
$803$	−45.0000	−1.58802
$804$	0	0
$805$	−5.00000	−0.176227
$806$	−6.00000	−0.211341
$807$	0	0
$808$	6.00000	0.211079
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−37.0000	−1.29925	−0.649623	−	0.760257i	$-0.725073\pi$
$811$	−37.0000	−1.29925	−0.649623	+	0.760257i	$0.725073\pi$
$812$	1.00000	0.0350931
$813$	0	0
$814$	−35.0000	−1.22675
$815$	6.00000	0.210171
$816$	0	0
$817$	−1.00000	−0.0349856
$818$	−13.0000	−0.454534
$819$	0	0
$820$	2.00000	0.0698430
$821$	−4.00000	−0.139601	−0.0698005	−	0.997561i	$-0.522236\pi$
$821$	−4.00000	−0.139601	−0.0698005	+	0.997561i	$0.522236\pi$
$822$	0	0
$823$	−2.00000	−0.0697156	−0.0348578	−	0.999392i	$-0.511098\pi$
$823$	−2.00000	−0.0697156	−0.0348578	+	0.999392i	$0.511098\pi$
$824$	7.00000	0.243857
$825$	0	0
$826$	6.00000	0.208767
$827$	−13.0000	−0.452054	−0.226027	−	0.974121i	$-0.572574\pi$
$827$	−13.0000	−0.452054	−0.226027	+	0.974121i	$0.572574\pi$
$828$	0	0
$829$	33.0000	1.14614	0.573069	−	0.819507i	$-0.305753\pi$
$829$	33.0000	1.14614	0.573069	+	0.819507i	$0.305753\pi$
$830$	2.00000	0.0694210
$831$	0	0
$832$	1.00000	0.0346688
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	3.00000	0.103819
$836$	5.00000	0.172929
$837$	0	0
$838$	19.0000	0.656344
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	22.0000	0.758170
$843$	0	0
$844$	−5.00000	−0.172107
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−14.0000	−0.481046
$848$	−6.00000	−0.206041
$849$	0	0
$850$	4.00000	0.137199
$851$	−35.0000	−1.19978
$852$	0	0
$853$	16.0000	0.547830	0.273915	−	0.961754i	$-0.411681\pi$
$853$	16.0000	0.547830	0.273915	+	0.961754i	$0.411681\pi$
$854$	7.00000	0.239535
$855$	0	0
$856$	6.00000	0.205076
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	−1.00000	−0.0340997
$861$	0	0
$862$	−26.0000	−0.885564
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	2.00000	0.0680020
$866$	30.0000	1.01944
$867$	0	0
$868$	6.00000	0.203653
$869$	−10.0000	−0.339227
$870$	0	0
$871$	−8.00000	−0.271070
$872$	−7.00000	−0.237050
$873$	0	0
$874$	5.00000	0.169128
$875$	−9.00000	−0.304256
$876$	0	0
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	−41.0000	−1.38368
$879$	0	0
$880$	5.00000	0.168550
$881$	−1.00000	−0.0336909	−0.0168454	−	0.999858i	$-0.505362\pi$
$881$	−1.00000	−0.0336909	−0.0168454	+	0.999858i	$0.505362\pi$
$882$	0	0
$883$	29.0000	0.975928	0.487964	−	0.872864i	$-0.337740\pi$
$883$	29.0000	0.975928	0.487964	+	0.872864i	$0.337740\pi$
$884$	−1.00000	−0.0336336
$885$	0	0
$886$	6.00000	0.201574
$887$	10.0000	0.335767	0.167884	−	0.985807i	$-0.446307\pi$
$887$	10.0000	0.335767	0.167884	+	0.985807i	$0.446307\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	−6.00000	−0.201120
$891$	0	0
$892$	−12.0000	−0.401790
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−33.0000	−1.10122
$899$	6.00000	0.200111
$900$	0	0
$901$	6.00000	0.199889
$902$	10.0000	0.332964
$903$	0	0
$904$	−16.0000	−0.532152
$905$	−10.0000	−0.332411
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	−10.0000	−0.331862
$909$	0	0
$910$	1.00000	0.0331497
$911$	45.0000	1.49092	0.745458	−	0.666552i	$-0.232231\pi$
$911$	45.0000	1.49092	0.745458	+	0.666552i	$0.232231\pi$
$912$	0	0
$913$	10.0000	0.330952
$914$	12.0000	0.396925
$915$	0	0
$916$	22.0000	0.726900
$917$	−9.00000	−0.297206
$918$	0	0
$919$	−10.0000	−0.329870	−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000	−0.329870	−0.164935	+	0.986304i	$0.552741\pi$
$920$	5.00000	0.164845
$921$	0	0
$922$	29.0000	0.955064
$923$	−10.0000	−0.329154
$924$	0	0
$925$	−28.0000	−0.920634
$926$	21.0000	0.690103
$927$	0	0
$928$	−1.00000	−0.0328266
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	4.00000	0.131024
$933$	0	0
$934$	29.0000	0.948909
$935$	−5.00000	−0.163517
$936$	0	0
$937$	6.00000	0.196011	0.0980057	−	0.995186i	$-0.468754\pi$
$937$	6.00000	0.196011	0.0980057	+	0.995186i	$0.468754\pi$
$938$	8.00000	0.261209
$939$	0	0
$940$	0	0
$941$	46.0000	1.49956	0.749779	−	0.661689i	$-0.230160\pi$
$941$	46.0000	1.49956	0.749779	+	0.661689i	$0.230160\pi$
$942$	0	0
$943$	10.0000	0.325645
$944$	−6.00000	−0.195283
$945$	0	0
$946$	−5.00000	−0.162564
$947$	−61.0000	−1.98223	−0.991117	−	0.132994i	$-0.957541\pi$
$947$	−61.0000	−1.98223	−0.991117	+	0.132994i	$0.957541\pi$
$948$	0	0
$949$	9.00000	0.292152
$950$	4.00000	0.129777
$951$	0	0
$952$	1.00000	0.0324102
$953$	−56.0000	−1.81402	−0.907009	−	0.421111i	$-0.861640\pi$
$953$	−56.0000	−1.81402	−0.907009	+	0.421111i	$0.861640\pi$
$954$	0	0
$955$	−15.0000	−0.485389
$956$	−12.0000	−0.388108
$957$	0	0
$958$	−33.0000	−1.06618
$959$	13.0000	0.419792
$960$	0	0
$961$	5.00000	0.161290
$962$	7.00000	0.225689
$963$	0	0
$964$	−10.0000	−0.322078
$965$	−14.0000	−0.450676
$966$	0	0
$967$	−21.0000	−0.675314	−0.337657	−	0.941269i	$-0.609634\pi$
$967$	−21.0000	−0.675314	−0.337657	+	0.941269i	$0.609634\pi$
$968$	14.0000	0.449977
$969$	0	0
$970$	−14.0000	−0.449513
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	0	0
$973$	2.00000	0.0641171
$974$	−40.0000	−1.28168
$975$	0	0
$976$	−7.00000	−0.224065
$977$	−51.0000	−1.63163	−0.815817	−	0.578310i	$-0.803713\pi$
$977$	−51.0000	−1.63163	−0.815817	+	0.578310i	$0.803713\pi$
$978$	0	0
$979$	−30.0000	−0.958804
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	−16.0000	−0.510581
$983$	43.0000	1.37149	0.685744	−	0.727843i	$-0.259477\pi$
$983$	43.0000	1.37149	0.685744	+	0.727843i	$0.259477\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	1.00000	0.0318465
$987$	0	0
$988$	−1.00000	−0.0318142
$989$	−5.00000	−0.158991
$990$	0	0
$991$	−6.00000	−0.190596	−0.0952981	−	0.995449i	$-0.530380\pi$
$991$	−6.00000	−0.190596	−0.0952981	+	0.995449i	$0.530380\pi$
$992$	−6.00000	−0.190500
$993$	0	0
$994$	10.0000	0.317181
$995$	−25.0000	−0.792553
$996$	0	0
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	−8.00000	−0.253236
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.a.n.1.1	yes	1
3.2	odd	2		1638.2.a.g.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1638.2.a.g.1.1	✓	1	3.2	odd	2
1638.2.a.n.1.1	yes	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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