LMFDB - Embedded newform 1638.2.a.o.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:	no (minimal twist has level 546)
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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -3.00000 q^{11} -1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -5.00000 q^{17} +1.00000 q^{19} -1.00000 q^{20} -3.00000 q^{22} -3.00000 q^{23} -4.00000 q^{25} -1.00000 q^{26} -1.00000 q^{28} -5.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} -5.00000 q^{34} +1.00000 q^{35} -5.00000 q^{37} +1.00000 q^{38} -1.00000 q^{40} +8.00000 q^{41} -1.00000 q^{43} -3.00000 q^{44} -3.00000 q^{46} -8.00000 q^{47} +1.00000 q^{49} -4.00000 q^{50} -1.00000 q^{52} -6.00000 q^{53} +3.00000 q^{55} -1.00000 q^{56} -5.00000 q^{58} +13.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} -10.0000 q^{67} -5.00000 q^{68} +1.00000 q^{70} -8.00000 q^{71} -15.0000 q^{73} -5.00000 q^{74} +1.00000 q^{76} +3.00000 q^{77} +6.00000 q^{79} -1.00000 q^{80} +8.00000 q^{82} +2.00000 q^{83} +5.00000 q^{85} -1.00000 q^{86} -3.00000 q^{88} +2.00000 q^{89} +1.00000 q^{91} -3.00000 q^{92} -8.00000 q^{94} -1.00000 q^{95} -2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
$2$	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
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	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
$17$	−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
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−3.00000	−0.639602
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−1.00000	−0.196116
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.00000	−0.857493
$35$	1.00000	0.169031
$36$	0	0
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−1.00000	−0.158114
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	−3.00000	−0.442326
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	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
$55$	3.00000	0.404520
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−5.00000	−0.656532
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	13.0000	1.66448	0.832240	−	0.554416i	$-0.187058\pi$
$61$	13.0000	1.66448	0.832240	+	0.554416i	$0.187058\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	−5.00000	−0.606339
$69$	0	0
$70$	1.00000	0.119523
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−15.0000	−1.75562	−0.877809	−	0.479012i	$-0.840995\pi$
$73$	−15.0000	−1.75562	−0.877809	+	0.479012i	$0.840995\pi$
$74$	−5.00000	−0.581238
$75$	0	0
$76$	1.00000	0.114708
$77$	3.00000	0.341882
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	8.00000	0.883452
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	5.00000	0.542326
$86$	−1.00000	−0.107833
$87$	0	0
$88$	−3.00000	−0.319801
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−3.00000	−0.312772
$93$	0	0
$94$	−8.00000	−0.825137
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−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
$100$	−4.00000	−0.400000
$101$	16.0000	1.59206	0.796030	−	0.605257i	$-0.206930\pi$
$101$	16.0000	1.59206	0.796030	+	0.605257i	$0.206930\pi$
$102$	0	0
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	−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	$0.608817\pi$
$110$	3.00000	0.286039
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	−5.00000	−0.464238
$117$	0	0
$118$	0	0
$119$	5.00000	0.458349
$120$	0	0
$121$	−2.00000	−0.181818
$122$	13.0000	1.17696
$123$	0	0
$124$	4.00000	0.359211
$125$	9.00000	0.804984
$126$	0	0
$127$	−18.0000	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	1.00000	0.0877058
$131$	−17.0000	−1.48530	−0.742648	−	0.669681i	$-0.766431\pi$
$131$	−17.0000	−1.48530	−0.742648	+	0.669681i	$0.766431\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	−10.0000	−0.863868
$135$	0	0
$136$	−5.00000	−0.428746
$137$	15.0000	1.28154	0.640768	−	0.767734i	$-0.278616\pi$
$137$	15.0000	1.28154	0.640768	+	0.767734i	$0.278616\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	−8.00000	−0.671345
$143$	3.00000	0.250873
$144$	0	0
$145$	5.00000	0.415227
$146$	−15.0000	−1.24141
$147$	0	0
$148$	−5.00000	−0.410997
$149$	16.0000	1.31077	0.655386	−	0.755295i	$-0.272506\pi$
$149$	16.0000	1.31077	0.655386	+	0.755295i	$0.272506\pi$
$150$	0	0
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	3.00000	0.241747
$155$	−4.00000	−0.321288
$156$	0	0
$157$	9.00000	0.718278	0.359139	−	0.933284i	$-0.383070\pi$
$157$	9.00000	0.718278	0.359139	+	0.933284i	$0.383070\pi$
$158$	6.00000	0.477334
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	3.00000	0.236433
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	8.00000	0.624695
$165$	0	0
$166$	2.00000	0.155230
$167$	15.0000	1.16073	0.580367	−	0.814355i	$-0.302909\pi$
$167$	15.0000	1.16073	0.580367	+	0.814355i	$0.302909\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	5.00000	0.383482
$171$	0	0
$172$	−1.00000	−0.0762493
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	−3.00000	−0.226134
$177$	0	0
$178$	2.00000	0.149906
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	1.00000	0.0741249
$183$	0	0
$184$	−3.00000	−0.221163
$185$	5.00000	0.367607
$186$	0	0
$187$	15.0000	1.09691
$188$	−8.00000	−0.583460
$189$	0	0
$190$	−1.00000	−0.0725476
$191$	−11.0000	−0.795932	−0.397966	−	0.917400i	$-0.630284\pi$
$191$	−11.0000	−0.795932	−0.397966	+	0.917400i	$0.630284\pi$
$192$	0	0
$193$	−20.0000	−1.43963	−0.719816	−	0.694165i	$-0.755774\pi$
$193$	−20.0000	−1.43963	−0.719816	+	0.694165i	$0.755774\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	1.00000	0.0714286
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−5.00000	−0.354441	−0.177220	−	0.984171i	$-0.556711\pi$
$199$	−5.00000	−0.354441	−0.177220	+	0.984171i	$0.556711\pi$
$200$	−4.00000	−0.282843
$201$	0	0
$202$	16.0000	1.12576
$203$	5.00000	0.350931
$204$	0	0
$205$	−8.00000	−0.558744
$206$	1.00000	0.0696733
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	−3.00000	−0.207514
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	6.00000	0.410152
$215$	1.00000	0.0681994
$216$	0	0
$217$	−4.00000	−0.271538
$218$	−7.00000	−0.474100
$219$	0	0
$220$	3.00000	0.202260
$221$	5.00000	0.336336
$222$	0	0
$223$	26.0000	1.74109	0.870544	−	0.492090i	$-0.163767\pi$
$223$	26.0000	1.74109	0.870544	+	0.492090i	$0.163767\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	6.00000	0.399114
$227$	−2.00000	−0.132745	−0.0663723	−	0.997795i	$-0.521143\pi$
$227$	−2.00000	−0.132745	−0.0663723	+	0.997795i	$0.521143\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	3.00000	0.197814
$231$	0	0
$232$	−5.00000	−0.328266
$233$	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
$235$	8.00000	0.521862
$236$	0	0
$237$	0	0
$238$	5.00000	0.324102
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	−2.00000	−0.128565
$243$	0	0
$244$	13.0000	0.832240
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−1.00000	−0.0636285
$248$	4.00000	0.254000
$249$	0	0
$250$	9.00000	0.569210
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	0	0
$253$	9.00000	0.565825
$254$	−18.0000	−1.12942
$255$	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
$259$	5.00000	0.310685
$260$	1.00000	0.0620174
$261$	0	0
$262$	−17.0000	−1.05026
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	−1.00000	−0.0613139
$267$	0	0
$268$	−10.0000	−0.610847
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	−5.00000	−0.303170
$273$	0	0
$274$	15.0000	0.906183
$275$	12.0000	0.723627
$276$	0	0
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	−16.0000	−0.959616
$279$	0	0
$280$	1.00000	0.0597614
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	−8.00000	−0.475551	−0.237775	−	0.971320i	$-0.576418\pi$
$283$	−8.00000	−0.475551	−0.237775	+	0.971320i	$0.576418\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	3.00000	0.177394
$287$	−8.00000	−0.472225
$288$	0	0
$289$	8.00000	0.470588
$290$	5.00000	0.293610
$291$	0	0
$292$	−15.0000	−0.877809
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	0	0
$296$	−5.00000	−0.290619
$297$	0	0
$298$	16.0000	0.926855
$299$	3.00000	0.173494
$300$	0	0
$301$	1.00000	0.0576390
$302$	−5.00000	−0.287718
$303$	0	0
$304$	1.00000	0.0573539
$305$	−13.0000	−0.744378
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	3.00000	0.170941
$309$	0	0
$310$	−4.00000	−0.227185
$311$	−26.0000	−1.47432	−0.737162	−	0.675716i	$-0.763835\pi$
$311$	−26.0000	−1.47432	−0.737162	+	0.675716i	$0.763835\pi$
$312$	0	0
$313$	28.0000	1.58265	0.791327	−	0.611393i	$-0.209391\pi$
$313$	28.0000	1.58265	0.791327	+	0.611393i	$0.209391\pi$
$314$	9.00000	0.507899
$315$	0	0
$316$	6.00000	0.337526
$317$	24.0000	1.34797	0.673987	−	0.738743i	$-0.264580\pi$
$317$	24.0000	1.34797	0.673987	+	0.738743i	$0.264580\pi$
$318$	0	0
$319$	15.0000	0.839839
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	3.00000	0.167183
$323$	−5.00000	−0.278207
$324$	0	0
$325$	4.00000	0.221880
$326$	10.0000	0.553849
$327$	0	0
$328$	8.00000	0.441726
$329$	8.00000	0.441054
$330$	0	0
$331$	34.0000	1.86881	0.934405	−	0.356214i	$-0.115932\pi$
$331$	34.0000	1.86881	0.934405	+	0.356214i	$0.115932\pi$
$332$	2.00000	0.109764
$333$	0	0
$334$	15.0000	0.820763
$335$	10.0000	0.546358
$336$	0	0
$337$	−29.0000	−1.57973	−0.789865	−	0.613280i	$-0.789850\pi$
$337$	−29.0000	−1.57973	−0.789865	+	0.613280i	$0.789850\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	5.00000	0.271163
$341$	−12.0000	−0.649836
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−32.0000	−1.71785	−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000	−1.71785	−0.858925	+	0.512101i	$0.828867\pi$
$348$	0	0
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	4.00000	0.213809
$351$	0	0
$352$	−3.00000	−0.159901
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	2.00000	0.106000
$357$	0	0
$358$	16.0000	0.845626
$359$	−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
$361$	−18.0000	−0.947368
$362$	6.00000	0.315353
$363$	0	0
$364$	1.00000	0.0524142
$365$	15.0000	0.785136
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−3.00000	−0.156386
$369$	0	0
$370$	5.00000	0.259938
$371$	6.00000	0.311504
$372$	0	0
$373$	−32.0000	−1.65690	−0.828449	−	0.560065i	$-0.810776\pi$
$373$	−32.0000	−1.65690	−0.828449	+	0.560065i	$0.810776\pi$
$374$	15.0000	0.775632
$375$	0	0
$376$	−8.00000	−0.412568
$377$	5.00000	0.257513
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	−1.00000	−0.0512989
$381$	0	0
$382$	−11.0000	−0.562809
$383$	−1.00000	−0.0510976	−0.0255488	−	0.999674i	$-0.508133\pi$
$383$	−1.00000	−0.0510976	−0.0255488	+	0.999674i	$0.508133\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	−20.0000	−1.01797
$387$	0	0
$388$	−2.00000	−0.101535
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	15.0000	0.758583
$392$	1.00000	0.0505076
$393$	0	0
$394$	0	0
$395$	−6.00000	−0.301893
$396$	0	0
$397$	30.0000	1.50566	0.752828	−	0.658217i	$-0.228689\pi$
$397$	30.0000	1.50566	0.752828	+	0.658217i	$0.228689\pi$
$398$	−5.00000	−0.250627
$399$	0	0
$400$	−4.00000	−0.200000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	16.0000	0.796030
$405$	0	0
$406$	5.00000	0.248146
$407$	15.0000	0.743522
$408$	0	0
$409$	3.00000	0.148340	0.0741702	−	0.997246i	$-0.476369\pi$
$409$	3.00000	0.148340	0.0741702	+	0.997246i	$0.476369\pi$
$410$	−8.00000	−0.395092
$411$	0	0
$412$	1.00000	0.0492665
$413$	0	0
$414$	0	0
$415$	−2.00000	−0.0981761
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	−3.00000	−0.146735
$419$	5.00000	0.244266	0.122133	−	0.992514i	$-0.461027\pi$
$419$	5.00000	0.244266	0.122133	+	0.992514i	$0.461027\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	5.00000	0.243396
$423$	0	0
$424$	−6.00000	−0.291386
$425$	20.0000	0.970143
$426$	0	0
$427$	−13.0000	−0.629114
$428$	6.00000	0.290021
$429$	0	0
$430$	1.00000	0.0482243
$431$	−10.0000	−0.481683	−0.240842	−	0.970564i	$-0.577423\pi$
$431$	−10.0000	−0.481683	−0.240842	+	0.970564i	$0.577423\pi$
$432$	0	0
$433$	20.0000	0.961139	0.480569	−	0.876957i	$-0.340430\pi$
$433$	20.0000	0.961139	0.480569	+	0.876957i	$0.340430\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−7.00000	−0.335239
$437$	−3.00000	−0.143509
$438$	0	0
$439$	−11.0000	−0.525001	−0.262501	−	0.964932i	$-0.584547\pi$
$439$	−11.0000	−0.525001	−0.262501	+	0.964932i	$0.584547\pi$
$440$	3.00000	0.143019
$441$	0	0
$442$	5.00000	0.237826
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	26.0000	1.23114
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−13.0000	−0.613508	−0.306754	−	0.951789i	$-0.599243\pi$
$449$	−13.0000	−0.613508	−0.306754	+	0.951789i	$0.599243\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	6.00000	0.282216
$453$	0	0
$454$	−2.00000	−0.0938647
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	−6.00000	−0.280362
$459$	0	0
$460$	3.00000	0.139876
$461$	13.0000	0.605470	0.302735	−	0.953075i	$-0.402100\pi$
$461$	13.0000	0.605470	0.302735	+	0.953075i	$0.402100\pi$
$462$	0	0
$463$	−13.0000	−0.604161	−0.302081	−	0.953282i	$-0.597681\pi$
$463$	−13.0000	−0.604161	−0.302081	+	0.953282i	$0.597681\pi$
$464$	−5.00000	−0.232119
$465$	0	0
$466$	4.00000	0.185296
$467$	−21.0000	−0.971764	−0.485882	−	0.874024i	$-0.661502\pi$
$467$	−21.0000	−0.971764	−0.485882	+	0.874024i	$0.661502\pi$
$468$	0	0
$469$	10.0000	0.461757
$470$	8.00000	0.369012
$471$	0	0
$472$	0	0
$473$	3.00000	0.137940
$474$	0	0
$475$	−4.00000	−0.183533
$476$	5.00000	0.229175
$477$	0	0
$478$	12.0000	0.548867
$479$	−27.0000	−1.23366	−0.616831	−	0.787096i	$-0.711584\pi$
$479$	−27.0000	−1.23366	−0.616831	+	0.787096i	$0.711584\pi$
$480$	0	0
$481$	5.00000	0.227980
$482$	22.0000	1.00207
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	2.00000	0.0908153
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	13.0000	0.588482
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	38.0000	1.71492	0.857458	−	0.514554i	$-0.172042\pi$
$491$	38.0000	1.71492	0.857458	+	0.514554i	$0.172042\pi$
$492$	0	0
$493$	25.0000	1.12594
$494$	−1.00000	−0.0449921
$495$	0	0
$496$	4.00000	0.179605
$497$	8.00000	0.358849
$498$	0	0
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	9.00000	0.402492
$501$	0	0
$502$	−21.0000	−0.937276
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	−16.0000	−0.711991
$506$	9.00000	0.400099
$507$	0	0
$508$	−18.0000	−0.798621
$509$	−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
$511$	15.0000	0.663561
$512$	1.00000	0.0441942
$513$	0	0
$514$	18.0000	0.793946
$515$	−1.00000	−0.0440653
$516$	0	0
$517$	24.0000	1.05552
$518$	5.00000	0.219687
$519$	0	0
$520$	1.00000	0.0438529
$521$	−39.0000	−1.70862	−0.854311	−	0.519763i	$-0.826020\pi$
$521$	−39.0000	−1.70862	−0.854311	+	0.519763i	$0.826020\pi$
$522$	0	0
$523$	−24.0000	−1.04945	−0.524723	−	0.851273i	$-0.675831\pi$
$523$	−24.0000	−1.04945	−0.524723	+	0.851273i	$0.675831\pi$
$524$	−17.0000	−0.742648
$525$	0	0
$526$	−12.0000	−0.523225
$527$	−20.0000	−0.871214
$528$	0	0
$529$	−14.0000	−0.608696
$530$	6.00000	0.260623
$531$	0	0
$532$	−1.00000	−0.0433555
$533$	−8.00000	−0.346518
$534$	0	0
$535$	−6.00000	−0.259403
$536$	−10.0000	−0.431934
$537$	0	0
$538$	−6.00000	−0.258678
$539$	−3.00000	−0.129219
$540$	0	0
$541$	41.0000	1.76273	0.881364	−	0.472438i	$-0.156626\pi$
$541$	41.0000	1.76273	0.881364	+	0.472438i	$0.156626\pi$
$542$	2.00000	0.0859074
$543$	0	0
$544$	−5.00000	−0.214373
$545$	7.00000	0.299847
$546$	0	0
$547$	−44.0000	−1.88130	−0.940652	−	0.339372i	$-0.889785\pi$
$547$	−44.0000	−1.88130	−0.940652	+	0.339372i	$0.889785\pi$
$548$	15.0000	0.640768
$549$	0	0
$550$	12.0000	0.511682
$551$	−5.00000	−0.213007
$552$	0	0
$553$	−6.00000	−0.255146
$554$	−18.0000	−0.764747
$555$	0	0
$556$	−16.0000	−0.678551
$557$	24.0000	1.01691	0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000	1.01691	0.508456	+	0.861088i	$0.330216\pi$
$558$	0	0
$559$	1.00000	0.0422955
$560$	1.00000	0.0422577
$561$	0	0
$562$	−10.0000	−0.421825
$563$	−9.00000	−0.379305	−0.189652	−	0.981851i	$-0.560736\pi$
$563$	−9.00000	−0.379305	−0.189652	+	0.981851i	$0.560736\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	−8.00000	−0.336265
$567$	0	0
$568$	−8.00000	−0.335673
$569$	−34.0000	−1.42535	−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000	−1.42535	−0.712677	+	0.701492i	$0.752517\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	3.00000	0.125436
$573$	0	0
$574$	−8.00000	−0.333914
$575$	12.0000	0.500435
$576$	0	0
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	8.00000	0.332756
$579$	0	0
$580$	5.00000	0.207614
$581$	−2.00000	−0.0829740
$582$	0	0
$583$	18.0000	0.745484
$584$	−15.0000	−0.620704
$585$	0	0
$586$	6.00000	0.247858
$587$	26.0000	1.07313	0.536567	−	0.843857i	$-0.319721\pi$
$587$	26.0000	1.07313	0.536567	+	0.843857i	$0.319721\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	0	0
$592$	−5.00000	−0.205499
$593$	−38.0000	−1.56047	−0.780236	−	0.625485i	$-0.784901\pi$
$593$	−38.0000	−1.56047	−0.780236	+	0.625485i	$0.784901\pi$
$594$	0	0
$595$	−5.00000	−0.204980
$596$	16.0000	0.655386
$597$	0	0
$598$	3.00000	0.122679
$599$	−37.0000	−1.51178	−0.755890	−	0.654699i	$-0.772795\pi$
$599$	−37.0000	−1.51178	−0.755890	+	0.654699i	$0.772795\pi$
$600$	0	0
$601$	20.0000	0.815817	0.407909	−	0.913023i	$-0.366258\pi$
$601$	20.0000	0.815817	0.407909	+	0.913023i	$0.366258\pi$
$602$	1.00000	0.0407570
$603$	0	0
$604$	−5.00000	−0.203447
$605$	2.00000	0.0813116
$606$	0	0
$607$	−27.0000	−1.09590	−0.547948	−	0.836512i	$-0.684591\pi$
$607$	−27.0000	−1.09590	−0.547948	+	0.836512i	$0.684591\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	−13.0000	−0.526355
$611$	8.00000	0.323645
$612$	0	0
$613$	41.0000	1.65597	0.827987	−	0.560747i	$-0.189486\pi$
$613$	41.0000	1.65597	0.827987	+	0.560747i	$0.189486\pi$
$614$	0	0
$615$	0	0
$616$	3.00000	0.120873
$617$	−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
$619$	−31.0000	−1.24600	−0.622998	−	0.782224i	$-0.714085\pi$
$619$	−31.0000	−1.24600	−0.622998	+	0.782224i	$0.714085\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	−26.0000	−1.04251
$623$	−2.00000	−0.0801283
$624$	0	0
$625$	11.0000	0.440000
$626$	28.0000	1.11911
$627$	0	0
$628$	9.00000	0.359139
$629$	25.0000	0.996815
$630$	0	0
$631$	−9.00000	−0.358284	−0.179142	−	0.983823i	$-0.557332\pi$
$631$	−9.00000	−0.358284	−0.179142	+	0.983823i	$0.557332\pi$
$632$	6.00000	0.238667
$633$	0	0
$634$	24.0000	0.953162
$635$	18.0000	0.714308
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	15.0000	0.593856
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	−1.00000	−0.0394362	−0.0197181	−	0.999806i	$-0.506277\pi$
$643$	−1.00000	−0.0394362	−0.0197181	+	0.999806i	$0.506277\pi$
$644$	3.00000	0.118217
$645$	0	0
$646$	−5.00000	−0.196722
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	0	0
$649$	0	0
$650$	4.00000	0.156893
$651$	0	0
$652$	10.0000	0.391630
$653$	−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
$655$	17.0000	0.664245
$656$	8.00000	0.312348
$657$	0	0
$658$	8.00000	0.311872
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	34.0000	1.32145
$663$	0	0
$664$	2.00000	0.0776151
$665$	1.00000	0.0387783
$666$	0	0
$667$	15.0000	0.580802
$668$	15.0000	0.580367
$669$	0	0
$670$	10.0000	0.386334
$671$	−39.0000	−1.50558
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	−29.0000	−1.11704
$675$	0	0
$676$	1.00000	0.0384615
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	5.00000	0.191741
$681$	0	0
$682$	−12.0000	−0.459504
$683$	−33.0000	−1.26271	−0.631355	−	0.775494i	$-0.717501\pi$
$683$	−33.0000	−1.26271	−0.631355	+	0.775494i	$0.717501\pi$
$684$	0	0
$685$	−15.0000	−0.573121
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−1.00000	−0.0381246
$689$	6.00000	0.228582
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	−32.0000	−1.21470
$695$	16.0000	0.606915
$696$	0	0
$697$	−40.0000	−1.51511
$698$	6.00000	0.227103
$699$	0	0
$700$	4.00000	0.151186
$701$	14.0000	0.528773	0.264386	−	0.964417i	$-0.414831\pi$
$701$	14.0000	0.528773	0.264386	+	0.964417i	$0.414831\pi$
$702$	0	0
$703$	−5.00000	−0.188579
$704$	−3.00000	−0.113067
$705$	0	0
$706$	−2.00000	−0.0752710
$707$	−16.0000	−0.601742
$708$	0	0
$709$	50.0000	1.87779	0.938895	−	0.344204i	$-0.111851\pi$
$709$	50.0000	1.87779	0.938895	+	0.344204i	$0.111851\pi$
$710$	8.00000	0.300235
$711$	0	0
$712$	2.00000	0.0749532
$713$	−12.0000	−0.449404
$714$	0	0
$715$	−3.00000	−0.112194
$716$	16.0000	0.597948
$717$	0	0
$718$	−24.0000	−0.895672
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	−1.00000	−0.0372419
$722$	−18.0000	−0.669891
$723$	0	0
$724$	6.00000	0.222988
$725$	20.0000	0.742781
$726$	0	0
$727$	−27.0000	−1.00137	−0.500687	−	0.865628i	$-0.666919\pi$
$727$	−27.0000	−1.00137	−0.500687	+	0.865628i	$0.666919\pi$
$728$	1.00000	0.0370625
$729$	0	0
$730$	15.0000	0.555175
$731$	5.00000	0.184932
$732$	0	0
$733$	20.0000	0.738717	0.369358	−	0.929287i	$-0.379577\pi$
$733$	20.0000	0.738717	0.369358	+	0.929287i	$0.379577\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	−3.00000	−0.110581
$737$	30.0000	1.10506
$738$	0	0
$739$	−26.0000	−0.956425	−0.478213	−	0.878244i	$-0.658715\pi$
$739$	−26.0000	−0.956425	−0.478213	+	0.878244i	$0.658715\pi$
$740$	5.00000	0.183804
$741$	0	0
$742$	6.00000	0.220267
$743$	42.0000	1.54083	0.770415	−	0.637542i	$-0.220049\pi$
$743$	42.0000	1.54083	0.770415	+	0.637542i	$0.220049\pi$
$744$	0	0
$745$	−16.0000	−0.586195
$746$	−32.0000	−1.17160
$747$	0	0
$748$	15.0000	0.548454
$749$	−6.00000	−0.219235
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	5.00000	0.182089
$755$	5.00000	0.181969
$756$	0	0
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	−8.00000	−0.290573
$759$	0	0
$760$	−1.00000	−0.0362738
$761$	24.0000	0.869999	0.435000	−	0.900431i	$-0.356748\pi$
$761$	24.0000	0.869999	0.435000	+	0.900431i	$0.356748\pi$
$762$	0	0
$763$	7.00000	0.253417
$764$	−11.0000	−0.397966
$765$	0	0
$766$	−1.00000	−0.0361315
$767$	0	0
$768$	0	0
$769$	43.0000	1.55062	0.775310	−	0.631581i	$-0.217594\pi$
$769$	43.0000	1.55062	0.775310	+	0.631581i	$0.217594\pi$
$770$	−3.00000	−0.108112
$771$	0	0
$772$	−20.0000	−0.719816
$773$	−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
$775$	−16.0000	−0.574737
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	18.0000	0.645331
$779$	8.00000	0.286630
$780$	0	0
$781$	24.0000	0.858788
$782$	15.0000	0.536399
$783$	0	0
$784$	1.00000	0.0357143
$785$	−9.00000	−0.321224
$786$	0	0
$787$	−47.0000	−1.67537	−0.837685	−	0.546154i	$-0.816091\pi$
$787$	−47.0000	−1.67537	−0.837685	+	0.546154i	$0.816091\pi$
$788$	0	0
$789$	0	0
$790$	−6.00000	−0.213470
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−13.0000	−0.461644
$794$	30.0000	1.06466
$795$	0	0
$796$	−5.00000	−0.177220
$797$	28.0000	0.991811	0.495905	−	0.868377i	$-0.334836\pi$
$797$	28.0000	0.991811	0.495905	+	0.868377i	$0.334836\pi$
$798$	0	0
$799$	40.0000	1.41510
$800$	−4.00000	−0.141421
$801$	0	0
$802$	30.0000	1.05934
$803$	45.0000	1.58802
$804$	0	0
$805$	−3.00000	−0.105736
$806$	−4.00000	−0.140894
$807$	0	0
$808$	16.0000	0.562878
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	−19.0000	−0.667180	−0.333590	−	0.942718i	$-0.608260\pi$
$811$	−19.0000	−0.667180	−0.333590	+	0.942718i	$0.608260\pi$
$812$	5.00000	0.175466
$813$	0	0
$814$	15.0000	0.525750
$815$	−10.0000	−0.350285
$816$	0	0
$817$	−1.00000	−0.0349856
$818$	3.00000	0.104893
$819$	0	0
$820$	−8.00000	−0.279372
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	0	0
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	1.00000	0.0348367
$825$	0	0
$826$	0	0
$827$	−43.0000	−1.49526	−0.747628	−	0.664117i	$-0.768807\pi$
$827$	−43.0000	−1.49526	−0.747628	+	0.664117i	$0.768807\pi$
$828$	0	0
$829$	−19.0000	−0.659897	−0.329949	−	0.943999i	$-0.607031\pi$
$829$	−19.0000	−0.659897	−0.329949	+	0.943999i	$0.607031\pi$
$830$	−2.00000	−0.0694210
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	−5.00000	−0.173240
$834$	0	0
$835$	−15.0000	−0.519096
$836$	−3.00000	−0.103757
$837$	0	0
$838$	5.00000	0.172722
$839$	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
$841$	−4.00000	−0.137931
$842$	−22.0000	−0.758170
$843$	0	0
$844$	5.00000	0.172107
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	2.00000	0.0687208
$848$	−6.00000	−0.206041
$849$	0	0
$850$	20.0000	0.685994
$851$	15.0000	0.514193
$852$	0	0
$853$	−28.0000	−0.958702	−0.479351	−	0.877623i	$-0.659128\pi$
$853$	−28.0000	−0.958702	−0.479351	+	0.877623i	$0.659128\pi$
$854$	−13.0000	−0.444851
$855$	0	0
$856$	6.00000	0.205076
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	1.00000	0.0340997
$861$	0	0
$862$	−10.0000	−0.340601
$863$	2.00000	0.0680808	0.0340404	−	0.999420i	$-0.489163\pi$
$863$	2.00000	0.0680808	0.0340404	+	0.999420i	$0.489163\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	20.0000	0.679628
$867$	0	0
$868$	−4.00000	−0.135769
$869$	−18.0000	−0.610608
$870$	0	0
$871$	10.0000	0.338837
$872$	−7.00000	−0.237050
$873$	0	0
$874$	−3.00000	−0.101477
$875$	−9.00000	−0.304256
$876$	0	0
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	−11.0000	−0.371232
$879$	0	0
$880$	3.00000	0.101130
$881$	−37.0000	−1.24656	−0.623281	−	0.781998i	$-0.714201\pi$
$881$	−37.0000	−1.24656	−0.623281	+	0.781998i	$0.714201\pi$
$882$	0	0
$883$	−41.0000	−1.37976	−0.689880	−	0.723924i	$-0.742337\pi$
$883$	−41.0000	−1.37976	−0.689880	+	0.723924i	$0.742337\pi$
$884$	5.00000	0.168168
$885$	0	0
$886$	18.0000	0.604722
$887$	20.0000	0.671534	0.335767	−	0.941945i	$-0.391004\pi$
$887$	20.0000	0.671534	0.335767	+	0.941945i	$0.391004\pi$
$888$	0	0
$889$	18.0000	0.603701
$890$	−2.00000	−0.0670402
$891$	0	0
$892$	26.0000	0.870544
$893$	−8.00000	−0.267710
$894$	0	0
$895$	−16.0000	−0.534821
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−13.0000	−0.433816
$899$	−20.0000	−0.667037
$900$	0	0
$901$	30.0000	0.999445
$902$	−24.0000	−0.799113
$903$	0	0
$904$	6.00000	0.199557
$905$	−6.00000	−0.199447
$906$	0	0
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	−2.00000	−0.0663723
$909$	0	0
$910$	−1.00000	−0.0331497
$911$	−29.0000	−0.960813	−0.480406	−	0.877046i	$-0.659511\pi$
$911$	−29.0000	−0.960813	−0.480406	+	0.877046i	$0.659511\pi$
$912$	0	0
$913$	−6.00000	−0.198571
$914$	8.00000	0.264616
$915$	0	0
$916$	−6.00000	−0.198246
$917$	17.0000	0.561389
$918$	0	0
$919$	−50.0000	−1.64935	−0.824674	−	0.565608i	$-0.808641\pi$
$919$	−50.0000	−1.64935	−0.824674	+	0.565608i	$0.808641\pi$
$920$	3.00000	0.0989071
$921$	0	0
$922$	13.0000	0.428132
$923$	8.00000	0.263323
$924$	0	0
$925$	20.0000	0.657596
$926$	−13.0000	−0.427207
$927$	0	0
$928$	−5.00000	−0.164133
$929$	4.00000	0.131236	0.0656179	−	0.997845i	$-0.479098\pi$
$929$	4.00000	0.131236	0.0656179	+	0.997845i	$0.479098\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	4.00000	0.131024
$933$	0	0
$934$	−21.0000	−0.687141
$935$	−15.0000	−0.490552
$936$	0	0
$937$	−14.0000	−0.457360	−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000	−0.457360	−0.228680	+	0.973502i	$0.573441\pi$
$938$	10.0000	0.326512
$939$	0	0
$940$	8.00000	0.260931
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	0	0
$945$	0	0
$946$	3.00000	0.0975384
$947$	−47.0000	−1.52729	−0.763647	−	0.645634i	$-0.776593\pi$
$947$	−47.0000	−1.52729	−0.763647	+	0.645634i	$0.776593\pi$
$948$	0	0
$949$	15.0000	0.486921
$950$	−4.00000	−0.129777
$951$	0	0
$952$	5.00000	0.162051
$953$	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
$955$	11.0000	0.355952
$956$	12.0000	0.388108
$957$	0	0
$958$	−27.0000	−0.872330
$959$	−15.0000	−0.484375
$960$	0	0
$961$	−15.0000	−0.483871
$962$	5.00000	0.161206
$963$	0	0
$964$	22.0000	0.708572
$965$	20.0000	0.643823
$966$	0	0
$967$	−31.0000	−0.996893	−0.498446	−	0.866921i	$-0.666096\pi$
$967$	−31.0000	−0.996893	−0.498446	+	0.866921i	$0.666096\pi$
$968$	−2.00000	−0.0642824
$969$	0	0
$970$	2.00000	0.0642161
$971$	32.0000	1.02693	0.513464	−	0.858111i	$-0.328362\pi$
$971$	32.0000	1.02693	0.513464	+	0.858111i	$0.328362\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	8.00000	0.256337
$975$	0	0
$976$	13.0000	0.416120
$977$	25.0000	0.799821	0.399910	−	0.916554i	$-0.369041\pi$
$977$	25.0000	0.799821	0.399910	+	0.916554i	$0.369041\pi$
$978$	0	0
$979$	−6.00000	−0.191761
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	38.0000	1.21263
$983$	5.00000	0.159475	0.0797376	−	0.996816i	$-0.474592\pi$
$983$	5.00000	0.159475	0.0797376	+	0.996816i	$0.474592\pi$
$984$	0	0
$985$	0	0
$986$	25.0000	0.796162
$987$	0	0
$988$	−1.00000	−0.0318142
$989$	3.00000	0.0953945
$990$	0	0
$991$	−30.0000	−0.952981	−0.476491	−	0.879180i	$-0.658091\pi$
$991$	−30.0000	−0.952981	−0.476491	+	0.879180i	$0.658091\pi$
$992$	4.00000	0.127000
$993$	0	0
$994$	8.00000	0.253745
$995$	5.00000	0.158511
$996$	0	0
$997$	−50.0000	−1.58352	−0.791758	−	0.610835i	$-0.790834\pi$
$997$	−50.0000	−1.58352	−0.791758	+	0.610835i	$0.790834\pi$
$998$	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.o.1.1		1
3.2	odd	2		546.2.a.c.1.1	✓	1
12.11	even	2		4368.2.a.h.1.1		1
21.20	even	2		3822.2.a.e.1.1		1
39.38	odd	2		7098.2.a.z.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.a.c.1.1	✓	1	3.2	odd	2
1638.2.a.o.1.1		1	1.1	even	1	trivial
3822.2.a.e.1.1		1	21.20	even	2
4368.2.a.h.1.1		1	12.11	even	2
7098.2.a.z.1.1		1	39.38	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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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