LMFDB - Embedded newform 1638.2.a.f.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 182)
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)

$f(q)$

$=$

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

$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +3.00000 q^{11} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{19} -3.00000 q^{22} +3.00000 q^{23} -5.00000 q^{25} -1.00000 q^{26} +1.00000 q^{28} +5.00000 q^{31} -1.00000 q^{32} -7.00000 q^{37} -2.00000 q^{38} -3.00000 q^{41} +8.00000 q^{43} +3.00000 q^{44} -3.00000 q^{46} +3.00000 q^{47} +1.00000 q^{49} +5.00000 q^{50} +1.00000 q^{52} +12.0000 q^{53} -1.00000 q^{56} -6.00000 q^{59} -1.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} +5.00000 q^{67} -12.0000 q^{71} +11.0000 q^{73} +7.00000 q^{74} +2.00000 q^{76} +3.00000 q^{77} -1.00000 q^{79} +3.00000 q^{82} -12.0000 q^{83} -8.00000 q^{86} -3.00000 q^{88} +18.0000 q^{89} +1.00000 q^{91} +3.00000 q^{92} -3.00000 q^{94} +17.0000 q^{97} -1.00000 q^{98} +O(q^{100})$

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
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	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
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	−3.00000	−0.639602
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	−1.00000	−0.196116
$27$	0	0
$28$	1.00000	0.188982
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	−3.00000	−0.442326
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	5.00000	0.707107
$51$	0	0
$52$	1.00000	0.138675
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	0	0
$59$	−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
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	−5.00000	−0.635001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	7.00000	0.813733
$75$	0	0
$76$	2.00000	0.229416
$77$	3.00000	0.341882
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	0	0
$81$	0	0
$82$	3.00000	0.331295
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	0	0
$86$	−8.00000	−0.862662
$87$	0	0
$88$	−3.00000	−0.319801
$89$	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
$91$	1.00000	0.104828
$92$	3.00000	0.312772
$93$	0	0
$94$	−3.00000	−0.309426
$95$	0	0
$96$	0	0
$97$	17.0000	1.72609	0.863044	−	0.505128i	$-0.168555\pi$
$97$	17.0000	1.72609	0.863044	+	0.505128i	$0.168555\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−5.00000	−0.500000
$101$	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
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−12.0000	−1.16554
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	1.00000	0.0944911
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	6.00000	0.552345
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	1.00000	0.0905357
$123$	0	0
$124$	5.00000	0.449013
$125$	0	0
$126$	0	0
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	−5.00000	−0.431934
$135$	0	0
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	12.0000	1.00702
$143$	3.00000	0.250873
$144$	0	0
$145$	0	0
$146$	−11.0000	−0.910366
$147$	0	0
$148$	−7.00000	−0.575396
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−2.00000	−0.162221
$153$	0	0
$154$	−3.00000	−0.241747
$155$	0	0
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	1.00000	0.0795557
$159$	0	0
$160$	0	0
$161$	3.00000	0.236433
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	12.0000	0.931381
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	8.00000	0.609994
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	−5.00000	−0.377964
$176$	3.00000	0.226134
$177$	0	0
$178$	−18.0000	−1.34916
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	−3.00000	−0.221163
$185$	0	0
$186$	0	0
$187$	0	0
$188$	3.00000	0.218797
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−17.0000	−1.22053
$195$	0	0
$196$	1.00000	0.0714286
$197$	3.00000	0.213741	0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000	0.213741	0.106871	+	0.994273i	$0.465917\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	5.00000	0.353553
$201$	0	0
$202$	−3.00000	−0.211079
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−14.0000	−0.975426
$207$	0	0
$208$	1.00000	0.0693375
$209$	6.00000	0.415029
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	12.0000	0.824163
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5.00000	0.339422
$218$	−2.00000	−0.135457
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−1.00000	−0.0669650	−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000	−0.0669650	−0.0334825	+	0.999439i	$0.510660\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	9.00000	0.598671
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.00000	−0.196537	−0.0982683	−	0.995160i	$-0.531330\pi$
$233$	−3.00000	−0.196537	−0.0982683	+	0.995160i	$0.531330\pi$
$234$	0	0
$235$	0	0
$236$	−6.00000	−0.390567
$237$	0	0
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	2.00000	0.128565
$243$	0	0
$244$	−1.00000	−0.0640184
$245$	0	0
$246$	0	0
$247$	2.00000	0.127257
$248$	−5.00000	−0.317500
$249$	0	0
$250$	0	0
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	0	0
$253$	9.00000	0.565825
$254$	7.00000	0.439219
$255$	0	0
$256$	1.00000	0.0625000
$257$	−24.0000	−1.49708	−0.748539	−	0.663090i	$-0.769245\pi$
$257$	−24.0000	−1.49708	−0.748539	+	0.663090i	$0.769245\pi$
$258$	0	0
$259$	−7.00000	−0.434959
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	0	0
$266$	−2.00000	−0.122628
$267$	0	0
$268$	5.00000	0.305424
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	0	0
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	0	0
$273$	0	0
$274$	−6.00000	−0.362473
$275$	−15.0000	−0.904534
$276$	0	0
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	−31.0000	−1.84276	−0.921379	−	0.388664i	$-0.872937\pi$
$283$	−31.0000	−1.84276	−0.921379	+	0.388664i	$0.872937\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	−3.00000	−0.177394
$287$	−3.00000	−0.177084
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	11.0000	0.643726
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	0	0
$296$	7.00000	0.406867
$297$	0	0
$298$	−15.0000	−0.868927
$299$	3.00000	0.173494
$300$	0	0
$301$	8.00000	0.461112
$302$	−8.00000	−0.460348
$303$	0	0
$304$	2.00000	0.114708
$305$	0	0
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	3.00000	0.170941
$309$	0	0
$310$	0	0
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	13.0000	0.733632
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	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
$319$	0	0
$320$	0	0
$321$	0	0
$322$	−3.00000	−0.167183
$323$	0	0
$324$	0	0
$325$	−5.00000	−0.277350
$326$	−20.0000	−1.10770
$327$	0	0
$328$	3.00000	0.165647
$329$	3.00000	0.165395
$330$	0	0
$331$	−19.0000	−1.04433	−0.522167	−	0.852843i	$-0.674876\pi$
$331$	−19.0000	−1.04433	−0.522167	+	0.852843i	$0.674876\pi$
$332$	−12.0000	−0.658586
$333$	0	0
$334$	−24.0000	−1.31322
$335$	0	0
$336$	0	0
$337$	−31.0000	−1.68868	−0.844339	−	0.535810i	$-0.820006\pi$
$337$	−31.0000	−1.68868	−0.844339	+	0.535810i	$0.820006\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	0	0
$341$	15.0000	0.812296
$342$	0	0
$343$	1.00000	0.0539949
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−18.0000	−0.967686
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	5.00000	0.267261
$351$	0	0
$352$	−3.00000	−0.159901
$353$	−15.0000	−0.798369	−0.399185	−	0.916871i	$-0.630707\pi$
$353$	−15.0000	−0.798369	−0.399185	+	0.916871i	$0.630707\pi$
$354$	0	0
$355$	0	0
$356$	18.0000	0.953998
$357$	0	0
$358$	−6.00000	−0.317110
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	7.00000	0.367912
$363$	0	0
$364$	1.00000	0.0524142
$365$	0	0
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	3.00000	0.156386
$369$	0	0
$370$	0	0
$371$	12.0000	0.623009
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	0	0
$375$	0	0
$376$	−3.00000	−0.154713
$377$	0	0
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.00000	−0.153293	−0.0766464	−	0.997058i	$-0.524421\pi$
$383$	−3.00000	−0.153293	−0.0766464	+	0.997058i	$0.524421\pi$
$384$	0	0
$385$	0	0
$386$	4.00000	0.203595
$387$	0	0
$388$	17.0000	0.863044
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	0	0
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−3.00000	−0.151138
$395$	0	0
$396$	0	0
$397$	20.0000	1.00377	0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000	1.00377	0.501886	+	0.864934i	$0.332640\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−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
$403$	5.00000	0.249068
$404$	3.00000	0.149256
$405$	0	0
$406$	0	0
$407$	−21.0000	−1.04093
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	0	0
$412$	14.0000	0.689730
$413$	−6.00000	−0.295241
$414$	0	0
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	−6.00000	−0.293470
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	−37.0000	−1.80327	−0.901635	−	0.432498i	$-0.857632\pi$
$421$	−37.0000	−1.80327	−0.901635	+	0.432498i	$0.857632\pi$
$422$	22.0000	1.07094
$423$	0	0
$424$	−12.0000	−0.582772
$425$	0	0
$426$	0	0
$427$	−1.00000	−0.0483934
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	0	0
$433$	38.0000	1.82616	0.913082	−	0.407777i	$-0.133696\pi$
$433$	38.0000	1.82616	0.913082	+	0.407777i	$0.133696\pi$
$434$	−5.00000	−0.240008
$435$	0	0
$436$	2.00000	0.0957826
$437$	6.00000	0.287019
$438$	0	0
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	0	0
$446$	1.00000	0.0473514
$447$	0	0
$448$	1.00000	0.0472456
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−9.00000	−0.423793
$452$	−9.00000	−0.423324
$453$	0	0
$454$	−24.0000	−1.12638
$455$	0	0
$456$	0	0
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	4.00000	0.186908
$459$	0	0
$460$	0	0
$461$	−24.0000	−1.11779	−0.558896	−	0.829238i	$-0.688775\pi$
$461$	−24.0000	−1.11779	−0.558896	+	0.829238i	$0.688775\pi$
$462$	0	0
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	0	0
$465$	0	0
$466$	3.00000	0.138972
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	5.00000	0.230879
$470$	0	0
$471$	0	0
$472$	6.00000	0.276172
$473$	24.0000	1.10352
$474$	0	0
$475$	−10.0000	−0.458831
$476$	0	0
$477$	0	0
$478$	6.00000	0.274434
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−7.00000	−0.319173
$482$	−26.0000	−1.18427
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	1.00000	0.0452679
$489$	0	0
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	0	0
$494$	−2.00000	−0.0899843
$495$	0	0
$496$	5.00000	0.224507
$497$	−12.0000	−0.538274
$498$	0	0
$499$	23.0000	1.02962	0.514811	−	0.857304i	$-0.327862\pi$
$499$	23.0000	1.02962	0.514811	+	0.857304i	$0.327862\pi$
$500$	0	0
$501$	0	0
$502$	−15.0000	−0.669483
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	0	0
$505$	0	0
$506$	−9.00000	−0.400099
$507$	0	0
$508$	−7.00000	−0.310575
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	11.0000	0.486611
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	24.0000	1.05859
$515$	0	0
$516$	0	0
$517$	9.00000	0.395820
$518$	7.00000	0.307562
$519$	0	0
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	−7.00000	−0.306089	−0.153044	−	0.988219i	$-0.548908\pi$
$523$	−7.00000	−0.306089	−0.153044	+	0.988219i	$0.548908\pi$
$524$	0	0
$525$	0	0
$526$	24.0000	1.04645
$527$	0	0
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	0	0
$532$	2.00000	0.0867110
$533$	−3.00000	−0.129944
$534$	0	0
$535$	0	0
$536$	−5.00000	−0.215967
$537$	0	0
$538$	−9.00000	−0.388018
$539$	3.00000	0.129219
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	−11.0000	−0.472490
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	15.0000	0.639602
$551$	0	0
$552$	0	0
$553$	−1.00000	−0.0425243
$554$	−26.0000	−1.10463
$555$	0	0
$556$	−4.00000	−0.169638
$557$	15.0000	0.635570	0.317785	−	0.948163i	$-0.397061\pi$
$557$	15.0000	0.635570	0.317785	+	0.948163i	$0.397061\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	12.0000	0.506189
$563$	39.0000	1.64365	0.821827	−	0.569737i	$-0.192955\pi$
$563$	39.0000	1.64365	0.821827	+	0.569737i	$0.192955\pi$
$564$	0	0
$565$	0	0
$566$	31.0000	1.30303
$567$	0	0
$568$	12.0000	0.503509
$569$	−21.0000	−0.880366	−0.440183	−	0.897908i	$-0.645086\pi$
$569$	−21.0000	−0.880366	−0.440183	+	0.897908i	$0.645086\pi$
$570$	0	0
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	3.00000	0.125436
$573$	0	0
$574$	3.00000	0.125218
$575$	−15.0000	−0.625543
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	17.0000	0.707107
$579$	0	0
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	36.0000	1.49097
$584$	−11.0000	−0.455183
$585$	0	0
$586$	18.0000	0.743573
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	0	0
$589$	10.0000	0.412043
$590$	0	0
$591$	0	0
$592$	−7.00000	−0.287698
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	0	0
$596$	15.0000	0.614424
$597$	0	0
$598$	−3.00000	−0.122679
$599$	−39.0000	−1.59350	−0.796748	−	0.604311i	$-0.793448\pi$
$599$	−39.0000	−1.59350	−0.796748	+	0.604311i	$0.793448\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	−8.00000	−0.326056
$603$	0	0
$604$	8.00000	0.325515
$605$	0	0
$606$	0	0
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	−2.00000	−0.0811107
$609$	0	0
$610$	0	0
$611$	3.00000	0.121367
$612$	0	0
$613$	−25.0000	−1.00974	−0.504870	−	0.863195i	$-0.668460\pi$
$613$	−25.0000	−1.00974	−0.504870	+	0.863195i	$0.668460\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	−3.00000	−0.120873
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	0	0
$621$	0	0
$622$	30.0000	1.20289
$623$	18.0000	0.721155
$624$	0	0
$625$	25.0000	1.00000
$626$	−26.0000	−1.03917
$627$	0	0
$628$	−13.0000	−0.518756
$629$	0	0
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	1.00000	0.0397779
$633$	0	0
$634$	−3.00000	−0.119145
$635$	0	0
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	0	0
$640$	0	0
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	3.00000	0.118217
$645$	0	0
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	−18.0000	−0.706562
$650$	5.00000	0.196116
$651$	0	0
$652$	20.0000	0.783260
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	0	0
$655$	0	0
$656$	−3.00000	−0.117130
$657$	0	0
$658$	−3.00000	−0.116952
$659$	−42.0000	−1.63609	−0.818044	−	0.575156i	$-0.804941\pi$
$659$	−42.0000	−1.63609	−0.818044	+	0.575156i	$0.804941\pi$
$660$	0	0
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	19.0000	0.738456
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	0	0
$667$	0	0
$668$	24.0000	0.928588
$669$	0	0
$670$	0	0
$671$	−3.00000	−0.115814
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	31.0000	1.19408
$675$	0	0
$676$	1.00000	0.0384615
$677$	39.0000	1.49889	0.749446	−	0.662066i	$-0.230320\pi$
$677$	39.0000	1.49889	0.749446	+	0.662066i	$0.230320\pi$
$678$	0	0
$679$	17.0000	0.652400
$680$	0	0
$681$	0	0
$682$	−15.0000	−0.574380
$683$	27.0000	1.03313	0.516563	−	0.856249i	$-0.327211\pi$
$683$	27.0000	1.03313	0.516563	+	0.856249i	$0.327211\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	8.00000	0.304997
$689$	12.0000	0.457164
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	24.0000	0.911028
$695$	0	0
$696$	0	0
$697$	0	0
$698$	10.0000	0.378506
$699$	0	0
$700$	−5.00000	−0.188982
$701$	36.0000	1.35970	0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000	1.35970	0.679851	+	0.733351i	$0.262045\pi$
$702$	0	0
$703$	−14.0000	−0.528020
$704$	3.00000	0.113067
$705$	0	0
$706$	15.0000	0.564532
$707$	3.00000	0.112827
$708$	0	0
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	0	0
$711$	0	0
$712$	−18.0000	−0.674579
$713$	15.0000	0.561754
$714$	0	0
$715$	0	0
$716$	6.00000	0.224231
$717$	0	0
$718$	−24.0000	−0.895672
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	0	0
$721$	14.0000	0.521387
$722$	15.0000	0.558242
$723$	0	0
$724$	−7.00000	−0.260153
$725$	0	0
$726$	0	0
$727$	26.0000	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$728$	−1.00000	−0.0370625
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	28.0000	1.03350
$735$	0	0
$736$	−3.00000	−0.110581
$737$	15.0000	0.552532
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	0	0
$741$	0	0
$742$	−12.0000	−0.440534
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	0	0
$746$	4.00000	0.146450
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−31.0000	−1.13121	−0.565603	−	0.824678i	$-0.691357\pi$
$751$	−31.0000	−1.13121	−0.565603	+	0.824678i	$0.691357\pi$
$752$	3.00000	0.109399
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	16.0000	0.581146
$759$	0	0
$760$	0	0
$761$	−3.00000	−0.108750	−0.0543750	−	0.998521i	$-0.517317\pi$
$761$	−3.00000	−0.108750	−0.0543750	+	0.998521i	$0.517317\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	0	0
$765$	0	0
$766$	3.00000	0.108394
$767$	−6.00000	−0.216647
$768$	0	0
$769$	−13.0000	−0.468792	−0.234396	−	0.972141i	$-0.575311\pi$
$769$	−13.0000	−0.468792	−0.234396	+	0.972141i	$0.575311\pi$
$770$	0	0
$771$	0	0
$772$	−4.00000	−0.143963
$773$	54.0000	1.94225	0.971123	−	0.238581i	$-0.0766824\pi$
$773$	54.0000	1.94225	0.971123	+	0.238581i	$0.0766824\pi$
$774$	0	0
$775$	−25.0000	−0.898027
$776$	−17.0000	−0.610264
$777$	0	0
$778$	−36.0000	−1.29066
$779$	−6.00000	−0.214972
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	0	0
$784$	1.00000	0.0357143
$785$	0	0
$786$	0	0
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	3.00000	0.106871
$789$	0	0
$790$	0	0
$791$	−9.00000	−0.320003
$792$	0	0
$793$	−1.00000	−0.0355110
$794$	−20.0000	−0.709773
$795$	0	0
$796$	−16.0000	−0.567105
$797$	−9.00000	−0.318796	−0.159398	−	0.987214i	$-0.550955\pi$
$797$	−9.00000	−0.318796	−0.159398	+	0.987214i	$0.550955\pi$
$798$	0	0
$799$	0	0
$800$	5.00000	0.176777
$801$	0	0
$802$	24.0000	0.847469
$803$	33.0000	1.16454
$804$	0	0
$805$	0	0
$806$	−5.00000	−0.176117
$807$	0	0
$808$	−3.00000	−0.105540
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	0	0
$814$	21.0000	0.736050
$815$	0	0
$816$	0	0
$817$	16.0000	0.559769
$818$	22.0000	0.769212
$819$	0	0
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	41.0000	1.42917	0.714585	−	0.699549i	$-0.246616\pi$
$823$	41.0000	1.42917	0.714585	+	0.699549i	$0.246616\pi$
$824$	−14.0000	−0.487713
$825$	0	0
$826$	6.00000	0.208767
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	0	0
$832$	1.00000	0.0346688
$833$	0	0
$834$	0	0
$835$	0	0
$836$	6.00000	0.207514
$837$	0	0
$838$	21.0000	0.725433
$839$	21.0000	0.725001	0.362500	−	0.931984i	$-0.381923\pi$
$839$	21.0000	0.725001	0.362500	+	0.931984i	$0.381923\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	37.0000	1.27510
$843$	0	0
$844$	−22.0000	−0.757271
$845$	0	0
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	12.0000	0.412082
$849$	0	0
$850$	0	0
$851$	−21.0000	−0.719871
$852$	0	0
$853$	44.0000	1.50653	0.753266	−	0.657716i	$-0.228477\pi$
$853$	44.0000	1.50653	0.753266	+	0.657716i	$0.228477\pi$
$854$	1.00000	0.0342193
$855$	0	0
$856$	0	0
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	0	0
$859$	23.0000	0.784750	0.392375	−	0.919805i	$-0.371654\pi$
$859$	23.0000	0.784750	0.392375	+	0.919805i	$0.371654\pi$
$860$	0	0
$861$	0	0
$862$	−6.00000	−0.204361
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	0	0
$865$	0	0
$866$	−38.0000	−1.29129
$867$	0	0
$868$	5.00000	0.169711
$869$	−3.00000	−0.101768
$870$	0	0
$871$	5.00000	0.169419
$872$	−2.00000	−0.0677285
$873$	0	0
$874$	−6.00000	−0.202953
$875$	0	0
$876$	0	0
$877$	−13.0000	−0.438979	−0.219489	−	0.975615i	$-0.570439\pi$
$877$	−13.0000	−0.438979	−0.219489	+	0.975615i	$0.570439\pi$
$878$	−26.0000	−0.877457
$879$	0	0
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	0	0
$883$	2.00000	0.0673054	0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000	0.0673054	0.0336527	+	0.999434i	$0.489286\pi$
$884$	0	0
$885$	0	0
$886$	24.0000	0.806296
$887$	−42.0000	−1.41022	−0.705111	−	0.709097i	$-0.749103\pi$
$887$	−42.0000	−1.41022	−0.705111	+	0.709097i	$0.749103\pi$
$888$	0	0
$889$	−7.00000	−0.234772
$890$	0	0
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	6.00000	0.200782
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−6.00000	−0.200223
$899$	0	0
$900$	0	0
$901$	0	0
$902$	9.00000	0.299667
$903$	0	0
$904$	9.00000	0.299336
$905$	0	0
$906$	0	0
$907$	26.0000	0.863316	0.431658	−	0.902037i	$-0.357929\pi$
$907$	26.0000	0.863316	0.431658	+	0.902037i	$0.357929\pi$
$908$	24.0000	0.796468
$909$	0	0
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	−36.0000	−1.19143
$914$	−26.0000	−0.860004
$915$	0	0
$916$	−4.00000	−0.132164
$917$	0	0
$918$	0	0
$919$	−25.0000	−0.824674	−0.412337	−	0.911031i	$-0.635287\pi$
$919$	−25.0000	−0.824674	−0.412337	+	0.911031i	$0.635287\pi$
$920$	0	0
$921$	0	0
$922$	24.0000	0.790398
$923$	−12.0000	−0.394985
$924$	0	0
$925$	35.0000	1.15079
$926$	40.0000	1.31448
$927$	0	0
$928$	0	0
$929$	9.00000	0.295280	0.147640	−	0.989041i	$-0.452832\pi$
$929$	9.00000	0.295280	0.147640	+	0.989041i	$0.452832\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	−3.00000	−0.0982683
$933$	0	0
$934$	12.0000	0.392652
$935$	0	0
$936$	0	0
$937$	−52.0000	−1.69877	−0.849383	−	0.527777i	$-0.823026\pi$
$937$	−52.0000	−1.69877	−0.849383	+	0.527777i	$0.823026\pi$
$938$	−5.00000	−0.163256
$939$	0	0
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	−9.00000	−0.293080
$944$	−6.00000	−0.195283
$945$	0	0
$946$	−24.0000	−0.780307
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	0	0
$949$	11.0000	0.357075
$950$	10.0000	0.324443
$951$	0	0
$952$	0	0
$953$	42.0000	1.36051	0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000	1.36051	0.680257	+	0.732974i	$0.261868\pi$
$954$	0	0
$955$	0	0
$956$	−6.00000	−0.194054
$957$	0	0
$958$	24.0000	0.775405
$959$	6.00000	0.193750
$960$	0	0
$961$	−6.00000	−0.193548
$962$	7.00000	0.225689
$963$	0	0
$964$	26.0000	0.837404
$965$	0	0
$966$	0	0
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	2.00000	0.0642824
$969$	0	0
$970$	0	0
$971$	21.0000	0.673922	0.336961	−	0.941519i	$-0.390601\pi$
$971$	21.0000	0.673922	0.336961	+	0.941519i	$0.390601\pi$
$972$	0	0
$973$	−4.00000	−0.128234
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	54.0000	1.72585
$980$	0	0
$981$	0	0
$982$	12.0000	0.382935
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	2.00000	0.0636285
$989$	24.0000	0.763156
$990$	0	0
$991$	11.0000	0.349427	0.174713	−	0.984619i	$-0.444100\pi$
$991$	11.0000	0.349427	0.174713	+	0.984619i	$0.444100\pi$
$992$	−5.00000	−0.158750
$993$	0	0
$994$	12.0000	0.380617
$995$	0	0
$996$	0	0
$997$	−37.0000	−1.17180	−0.585901	−	0.810383i	$-0.699259\pi$
$997$	−37.0000	−1.17180	−0.585901	+	0.810383i	$0.699259\pi$
$998$	−23.0000	−0.728052
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.a.f.1.1		1
3.2	odd	2		182.2.a.d.1.1	✓	1
12.11	even	2		1456.2.a.d.1.1		1
15.14	odd	2		4550.2.a.c.1.1		1
21.2	odd	6		1274.2.f.d.1145.1		2
21.5	even	6		1274.2.f.i.1145.1		2
21.11	odd	6		1274.2.f.d.79.1		2
21.17	even	6		1274.2.f.i.79.1		2
21.20	even	2		1274.2.a.j.1.1		1
24.5	odd	2		5824.2.a.k.1.1		1
24.11	even	2		5824.2.a.x.1.1		1
39.5	even	4		2366.2.d.e.337.1		2
39.8	even	4		2366.2.d.e.337.2		2
39.38	odd	2		2366.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.2.a.d.1.1	✓	1	3.2	odd	2
1274.2.a.j.1.1		1	21.20	even	2
1274.2.f.d.79.1		2	21.11	odd	6
1274.2.f.d.1145.1		2	21.2	odd	6
1274.2.f.i.79.1		2	21.17	even	6
1274.2.f.i.1145.1		2	21.5	even	6
1456.2.a.d.1.1		1	12.11	even	2
1638.2.a.f.1.1		1	1.1	even	1	trivial
2366.2.a.e.1.1		1	39.38	odd	2
2366.2.d.e.337.1		2	39.5	even	4
2366.2.d.e.337.2		2	39.8	even	4
4550.2.a.c.1.1		1	15.14	odd	2
5824.2.a.k.1.1		1	24.5	odd	2
5824.2.a.x.1.1		1	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

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