LMFDB - Embedded newform 7938.2.a.co.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7938,2,Mod(1,7938)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7938, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7938.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7938 = 2 \cdot 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7938.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:	$63.3852491245$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 882)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.93185$ of defining polynomial
Character	$\chi$	$=$	7938.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.93185 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +1.93185 q^{5} +1.00000 q^{8} +1.93185 q^{10} -5.46410 q^{11} +2.44949 q^{13} +1.00000 q^{16} -6.31319 q^{17} -1.93185 q^{19} +1.93185 q^{20} -5.46410 q^{22} +5.92820 q^{23} -1.26795 q^{25} +2.44949 q^{26} +0.732051 q^{29} -7.34847 q^{31} +1.00000 q^{32} -6.31319 q^{34} -8.00000 q^{37} -1.93185 q^{38} +1.93185 q^{40} -5.65685 q^{41} -1.80385 q^{43} -5.46410 q^{44} +5.92820 q^{46} +9.52056 q^{47} -1.26795 q^{50} +2.44949 q^{52} -3.26795 q^{53} -10.5558 q^{55} +0.732051 q^{58} -5.00052 q^{59} -2.96713 q^{61} -7.34847 q^{62} +1.00000 q^{64} +4.73205 q^{65} -14.1962 q^{67} -6.31319 q^{68} +11.1962 q^{71} +9.52056 q^{73} -8.00000 q^{74} -1.93185 q^{76} -10.1244 q^{79} +1.93185 q^{80} -5.65685 q^{82} +9.89949 q^{83} -12.1962 q^{85} -1.80385 q^{86} -5.46410 q^{88} -13.2827 q^{89} +5.92820 q^{92} +9.52056 q^{94} -3.73205 q^{95} +3.86370 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} - 8 q^{11} + 4 q^{16} - 8 q^{22} - 4 q^{23} - 12 q^{25} - 4 q^{29} + 4 q^{32} - 32 q^{37} - 28 q^{43} - 8 q^{44} - 4 q^{46} - 12 q^{50} - 20 q^{53} - 4 q^{58} + 4 q^{64} + 12 q^{65} - 36 q^{67} + 24 q^{71} - 32 q^{74} + 8 q^{79} - 28 q^{85} - 28 q^{86} - 8 q^{88} - 4 q^{92} - 8 q^{95}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 - 8 * q^11 + 4 * q^16 - 8 * q^22 - 4 * q^23 - 12 * q^25 - 4 * q^29 + 4 * q^32 - 32 * q^37 - 28 * q^43 - 8 * q^44 - 4 * q^46 - 12 * q^50 - 20 * q^53 - 4 * q^58 + 4 * q^64 + 12 * q^65 - 36 * q^67 + 24 * q^71 - 32 * q^74 + 8 * q^79 - 28 * q^85 - 28 * q^86 - 8 * q^88 - 4 * q^92 - 8 * q^95

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
$5$	1.93185	0.863950	0.431975	−	0.901886i	$-0.357817\pi$
$5$	1.93185	0.863950	0.431975	+	0.901886i	$0.357817\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	1.93185	0.610905
$11$	−5.46410	−1.64749	−0.823744	−	0.566961i	$-0.808119\pi$
$11$	−5.46410	−1.64749	−0.823744	+	0.566961i	$0.808119\pi$
$12$	0	0
$13$	2.44949	0.679366	0.339683	−	0.940540i	$-0.389680\pi$
$13$	2.44949	0.679366	0.339683	+	0.940540i	$0.389680\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.31319	−1.53117	−0.765587	−	0.643332i	$-0.777551\pi$
$17$	−6.31319	−1.53117	−0.765587	+	0.643332i	$0.777551\pi$
$18$	0	0
$19$	−1.93185	−0.443197	−0.221599	−	0.975138i	$-0.571127\pi$
$19$	−1.93185	−0.443197	−0.221599	+	0.975138i	$0.571127\pi$
$20$	1.93185	0.431975
$21$	0	0
$22$	−5.46410	−1.16495
$23$	5.92820	1.23612	0.618058	−	0.786133i	$-0.287920\pi$
$23$	5.92820	1.23612	0.618058	+	0.786133i	$0.287920\pi$
$24$	0	0
$25$	−1.26795	−0.253590
$26$	2.44949	0.480384
$27$	0	0
$28$	0	0
$29$	0.732051	0.135938	0.0679692	−	0.997687i	$-0.478348\pi$
$29$	0.732051	0.135938	0.0679692	+	0.997687i	$0.478348\pi$
$30$	0	0
$31$	−7.34847	−1.31982	−0.659912	−	0.751343i	$-0.729406\pi$
$31$	−7.34847	−1.31982	−0.659912	+	0.751343i	$0.729406\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.31319	−1.08270
$35$	0	0
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	−1.93185	−0.313388
$39$	0	0
$40$	1.93185	0.305453
$41$	−5.65685	−0.883452	−0.441726	−	0.897150i	$-0.645634\pi$
$41$	−5.65685	−0.883452	−0.441726	+	0.897150i	$0.645634\pi$
$42$	0	0
$43$	−1.80385	−0.275084	−0.137542	−	0.990496i	$-0.543920\pi$
$43$	−1.80385	−0.275084	−0.137542	+	0.990496i	$0.543920\pi$
$44$	−5.46410	−0.823744
$45$	0	0
$46$	5.92820	0.874066
$47$	9.52056	1.38872	0.694358	−	0.719630i	$-0.255688\pi$
$47$	9.52056	1.38872	0.694358	+	0.719630i	$0.255688\pi$
$48$	0	0
$49$	0	0
$50$	−1.26795	−0.179315
$51$	0	0
$52$	2.44949	0.339683
$53$	−3.26795	−0.448887	−0.224444	−	0.974487i	$-0.572056\pi$
$53$	−3.26795	−0.448887	−0.224444	+	0.974487i	$0.572056\pi$
$54$	0	0
$55$	−10.5558	−1.42335
$56$	0	0
$57$	0	0
$58$	0.732051	0.0961230
$59$	−5.00052	−0.651012	−0.325506	−	0.945540i	$-0.605535\pi$
$59$	−5.00052	−0.651012	−0.325506	+	0.945540i	$0.605535\pi$
$60$	0	0
$61$	−2.96713	−0.379902	−0.189951	−	0.981794i	$-0.560833\pi$
$61$	−2.96713	−0.379902	−0.189951	+	0.981794i	$0.560833\pi$
$62$	−7.34847	−0.933257
$63$	0	0
$64$	1.00000	0.125000
$65$	4.73205	0.586939
$66$	0	0
$67$	−14.1962	−1.73434	−0.867168	−	0.498016i	$-0.834062\pi$
$67$	−14.1962	−1.73434	−0.867168	+	0.498016i	$0.834062\pi$
$68$	−6.31319	−0.765587
$69$	0	0
$70$	0	0
$71$	11.1962	1.32874	0.664369	−	0.747404i	$-0.268700\pi$
$71$	11.1962	1.32874	0.664369	+	0.747404i	$0.268700\pi$
$72$	0	0
$73$	9.52056	1.11430	0.557148	−	0.830413i	$-0.311895\pi$
$73$	9.52056	1.11430	0.557148	+	0.830413i	$0.311895\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	−1.93185	−0.221599
$77$	0	0
$78$	0	0
$79$	−10.1244	−1.13908	−0.569540	−	0.821964i	$-0.692878\pi$
$79$	−10.1244	−1.13908	−0.569540	+	0.821964i	$0.692878\pi$
$80$	1.93185	0.215988
$81$	0	0
$82$	−5.65685	−0.624695
$83$	9.89949	1.08661	0.543305	−	0.839535i	$-0.317173\pi$
$83$	9.89949	1.08661	0.543305	+	0.839535i	$0.317173\pi$
$84$	0	0
$85$	−12.1962	−1.32286
$86$	−1.80385	−0.194514
$87$	0	0
$88$	−5.46410	−0.582475
$89$	−13.2827	−1.40797	−0.703983	−	0.710217i	$-0.748597\pi$
$89$	−13.2827	−1.40797	−0.703983	+	0.710217i	$0.748597\pi$
$90$	0	0
$91$	0	0
$92$	5.92820	0.618058
$93$	0	0
$94$	9.52056	0.981971
$95$	−3.73205	−0.382900
$96$	0	0
$97$	3.86370	0.392300	0.196150	−	0.980574i	$-0.437156\pi$
$97$	3.86370	0.392300	0.196150	+	0.980574i	$0.437156\pi$
$98$	0	0
$99$	0	0
$100$	−1.26795	−0.126795
$101$	1.55291	0.154521	0.0772604	−	0.997011i	$-0.475383\pi$
$101$	1.55291	0.154521	0.0772604	+	0.997011i	$0.475383\pi$
$102$	0	0
$103$	8.38375	0.826075	0.413037	−	0.910714i	$-0.364468\pi$
$103$	8.38375	0.826075	0.413037	+	0.910714i	$0.364468\pi$
$104$	2.44949	0.240192
$105$	0	0
$106$	−3.26795	−0.317411
$107$	−16.9282	−1.63651	−0.818256	−	0.574855i	$-0.805059\pi$
$107$	−16.9282	−1.63651	−0.818256	+	0.574855i	$0.805059\pi$
$108$	0	0
$109$	−20.5885	−1.97202	−0.986008	−	0.166696i	$-0.946690\pi$
$109$	−20.5885	−1.97202	−0.986008	+	0.166696i	$0.946690\pi$
$110$	−10.5558	−1.00646
$111$	0	0
$112$	0	0
$113$	−2.66025	−0.250256	−0.125128	−	0.992141i	$-0.539934\pi$
$113$	−2.66025	−0.250256	−0.125128	+	0.992141i	$0.539934\pi$
$114$	0	0
$115$	11.4524	1.06794
$116$	0.732051	0.0679692
$117$	0	0
$118$	−5.00052	−0.460335
$119$	0	0
$120$	0	0
$121$	18.8564	1.71422
$122$	−2.96713	−0.268631
$123$	0	0
$124$	−7.34847	−0.659912
$125$	−12.1087	−1.08304
$126$	0	0
$127$	7.92820	0.703514	0.351757	−	0.936091i	$-0.385584\pi$
$127$	7.92820	0.703514	0.351757	+	0.936091i	$0.385584\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	4.73205	0.415028
$131$	−0.240237	−0.0209896	−0.0104948	−	0.999945i	$-0.503341\pi$
$131$	−0.240237	−0.0209896	−0.0104948	+	0.999945i	$0.503341\pi$
$132$	0	0
$133$	0	0
$134$	−14.1962	−1.22636
$135$	0	0
$136$	−6.31319	−0.541352
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	1.83032	0.155245	0.0776227	−	0.996983i	$-0.475267\pi$
$139$	1.83032	0.155245	0.0776227	+	0.996983i	$0.475267\pi$
$140$	0	0
$141$	0	0
$142$	11.1962	0.939560
$143$	−13.3843	−1.11925
$144$	0	0
$145$	1.41421	0.117444
$146$	9.52056	0.787927
$147$	0	0
$148$	−8.00000	−0.657596
$149$	−18.3923	−1.50676	−0.753378	−	0.657588i	$-0.771577\pi$
$149$	−18.3923	−1.50676	−0.753378	+	0.657588i	$0.771577\pi$
$150$	0	0
$151$	3.39230	0.276062	0.138031	−	0.990428i	$-0.455923\pi$
$151$	3.39230	0.276062	0.138031	+	0.990428i	$0.455923\pi$
$152$	−1.93185	−0.156694
$153$	0	0
$154$	0	0
$155$	−14.1962	−1.14026
$156$	0	0
$157$	24.3562	1.94384	0.971918	−	0.235320i	$-0.0756137\pi$
$157$	24.3562	1.94384	0.971918	+	0.235320i	$0.0756137\pi$
$158$	−10.1244	−0.805450
$159$	0	0
$160$	1.93185	0.152726
$161$	0	0
$162$	0	0
$163$	−20.9282	−1.63922	−0.819612	−	0.572919i	$-0.805811\pi$
$163$	−20.9282	−1.63922	−0.819612	+	0.572919i	$0.805811\pi$
$164$	−5.65685	−0.441726
$165$	0	0
$166$	9.89949	0.768350
$167$	16.1112	1.24672	0.623359	−	0.781936i	$-0.285767\pi$
$167$	16.1112	1.24672	0.623359	+	0.781936i	$0.285767\pi$
$168$	0	0
$169$	−7.00000	−0.538462
$170$	−12.1962	−0.935402
$171$	0	0
$172$	−1.80385	−0.137542
$173$	14.5211	1.10402	0.552008	−	0.833839i	$-0.313862\pi$
$173$	14.5211	1.10402	0.552008	+	0.833839i	$0.313862\pi$
$174$	0	0
$175$	0	0
$176$	−5.46410	−0.411872
$177$	0	0
$178$	−13.2827	−0.995582
$179$	−2.19615	−0.164148	−0.0820741	−	0.996626i	$-0.526154\pi$
$179$	−2.19615	−0.164148	−0.0820741	+	0.996626i	$0.526154\pi$
$180$	0	0
$181$	−8.72552	−0.648563	−0.324281	−	0.945961i	$-0.605122\pi$
$181$	−8.72552	−0.648563	−0.324281	+	0.945961i	$0.605122\pi$
$182$	0	0
$183$	0	0
$184$	5.92820	0.437033
$185$	−15.4548	−1.13626
$186$	0	0
$187$	34.4959	2.52259
$188$	9.52056	0.694358
$189$	0	0
$190$	−3.73205	−0.270751
$191$	−3.19615	−0.231265	−0.115633	−	0.993292i	$-0.536890\pi$
$191$	−3.19615	−0.231265	−0.115633	+	0.993292i	$0.536890\pi$
$192$	0	0
$193$	−20.2679	−1.45892	−0.729459	−	0.684024i	$-0.760228\pi$
$193$	−20.2679	−1.45892	−0.729459	+	0.684024i	$0.760228\pi$
$194$	3.86370	0.277398
$195$	0	0
$196$	0	0
$197$	7.66025	0.545771	0.272885	−	0.962047i	$-0.412022\pi$
$197$	7.66025	0.545771	0.272885	+	0.962047i	$0.412022\pi$
$198$	0	0
$199$	−3.10583	−0.220166	−0.110083	−	0.993922i	$-0.535112\pi$
$199$	−3.10583	−0.220166	−0.110083	+	0.993922i	$0.535112\pi$
$200$	−1.26795	−0.0896575
$201$	0	0
$202$	1.55291	0.109263
$203$	0	0
$204$	0	0
$205$	−10.9282	−0.763259
$206$	8.38375	0.584123
$207$	0	0
$208$	2.44949	0.169842
$209$	10.5558	0.730162
$210$	0	0
$211$	4.73205	0.325768	0.162884	−	0.986645i	$-0.447920\pi$
$211$	4.73205	0.325768	0.162884	+	0.986645i	$0.447920\pi$
$212$	−3.26795	−0.224444
$213$	0	0
$214$	−16.9282	−1.15719
$215$	−3.48477	−0.237659
$216$	0	0
$217$	0	0
$218$	−20.5885	−1.39443
$219$	0	0
$220$	−10.5558	−0.711674
$221$	−15.4641	−1.04023
$222$	0	0
$223$	−20.7327	−1.38837	−0.694183	−	0.719798i	$-0.744234\pi$
$223$	−20.7327	−1.38837	−0.694183	+	0.719798i	$0.744234\pi$
$224$	0	0
$225$	0	0
$226$	−2.66025	−0.176957
$227$	−0.896575	−0.0595078	−0.0297539	−	0.999557i	$-0.509472\pi$
$227$	−0.896575	−0.0595078	−0.0297539	+	0.999557i	$0.509472\pi$
$228$	0	0
$229$	−18.0430	−1.19232	−0.596158	−	0.802867i	$-0.703307\pi$
$229$	−18.0430	−1.19232	−0.596158	+	0.802867i	$0.703307\pi$
$230$	11.4524	0.755150
$231$	0	0
$232$	0.732051	0.0480615
$233$	19.3923	1.27043	0.635216	−	0.772334i	$-0.280911\pi$
$233$	19.3923	1.27043	0.635216	+	0.772334i	$0.280911\pi$
$234$	0	0
$235$	18.3923	1.19978
$236$	−5.00052	−0.325506
$237$	0	0
$238$	0	0
$239$	5.53590	0.358087	0.179044	−	0.983841i	$-0.442700\pi$
$239$	5.53590	0.358087	0.179044	+	0.983841i	$0.442700\pi$
$240$	0	0
$241$	12.7279	0.819878	0.409939	−	0.912113i	$-0.365550\pi$
$241$	12.7279	0.819878	0.409939	+	0.912113i	$0.365550\pi$
$242$	18.8564	1.21214
$243$	0	0
$244$	−2.96713	−0.189951
$245$	0	0
$246$	0	0
$247$	−4.73205	−0.301093
$248$	−7.34847	−0.466628
$249$	0	0
$250$	−12.1087	−0.765824
$251$	−1.93185	−0.121937	−0.0609687	−	0.998140i	$-0.519419\pi$
$251$	−1.93185	−0.121937	−0.0609687	+	0.998140i	$0.519419\pi$
$252$	0	0
$253$	−32.3923	−2.03649
$254$	7.92820	0.497460
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.7327	1.29327	0.646636	−	0.762799i	$-0.276175\pi$
$257$	20.7327	1.29327	0.646636	+	0.762799i	$0.276175\pi$
$258$	0	0
$259$	0	0
$260$	4.73205	0.293469
$261$	0	0
$262$	−0.240237	−0.0148419
$263$	14.6603	0.903990	0.451995	−	0.892020i	$-0.350712\pi$
$263$	14.6603	0.903990	0.451995	+	0.892020i	$0.350712\pi$
$264$	0	0
$265$	−6.31319	−0.387816
$266$	0	0
$267$	0	0
$268$	−14.1962	−0.867168
$269$	0.0371647	0.00226597	0.00113299	−	0.999999i	$-0.499639\pi$
$269$	0.0371647	0.00226597	0.00113299	+	0.999999i	$0.499639\pi$
$270$	0	0
$271$	5.00052	0.303760	0.151880	−	0.988399i	$-0.451467\pi$
$271$	5.00052	0.303760	0.151880	+	0.988399i	$0.451467\pi$
$272$	−6.31319	−0.382794
$273$	0	0
$274$	0	0
$275$	6.92820	0.417786
$276$	0	0
$277$	−5.80385	−0.348719	−0.174360	−	0.984682i	$-0.555786\pi$
$277$	−5.80385	−0.348719	−0.174360	+	0.984682i	$0.555786\pi$
$278$	1.83032	0.109775
$279$	0	0
$280$	0	0
$281$	−3.39230	−0.202368	−0.101184	−	0.994868i	$-0.532263\pi$
$281$	−3.39230	−0.202368	−0.101184	+	0.994868i	$0.532263\pi$
$282$	0	0
$283$	17.1093	1.01704	0.508520	−	0.861050i	$-0.330193\pi$
$283$	17.1093	1.01704	0.508520	+	0.861050i	$0.330193\pi$
$284$	11.1962	0.664369
$285$	0	0
$286$	−13.3843	−0.791428
$287$	0	0
$288$	0	0
$289$	22.8564	1.34449
$290$	1.41421	0.0830455
$291$	0	0
$292$	9.52056	0.557148
$293$	−4.38134	−0.255961	−0.127980	−	0.991777i	$-0.540849\pi$
$293$	−4.38134	−0.255961	−0.127980	+	0.991777i	$0.540849\pi$
$294$	0	0
$295$	−9.66025	−0.562442
$296$	−8.00000	−0.464991
$297$	0	0
$298$	−18.3923	−1.06544
$299$	14.5211	0.839775
$300$	0	0
$301$	0	0
$302$	3.39230	0.195205
$303$	0	0
$304$	−1.93185	−0.110799
$305$	−5.73205	−0.328216
$306$	0	0
$307$	−29.0793	−1.65964	−0.829822	−	0.558028i	$-0.811558\pi$
$307$	−29.0793	−1.65964	−0.829822	+	0.558028i	$0.811558\pi$
$308$	0	0
$309$	0	0
$310$	−14.1962	−0.806287
$311$	21.0101	1.19138	0.595688	−	0.803216i	$-0.296880\pi$
$311$	21.0101	1.19138	0.595688	+	0.803216i	$0.296880\pi$
$312$	0	0
$313$	−1.79315	−0.101355	−0.0506774	−	0.998715i	$-0.516138\pi$
$313$	−1.79315	−0.101355	−0.0506774	+	0.998715i	$0.516138\pi$
$314$	24.3562	1.37450
$315$	0	0
$316$	−10.1244	−0.569540
$317$	−1.41154	−0.0792801	−0.0396401	−	0.999214i	$-0.512621\pi$
$317$	−1.41154	−0.0792801	−0.0396401	+	0.999214i	$0.512621\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	1.93185	0.107994
$321$	0	0
$322$	0	0
$323$	12.1962	0.678612
$324$	0	0
$325$	−3.10583	−0.172280
$326$	−20.9282	−1.15911
$327$	0	0
$328$	−5.65685	−0.312348
$329$	0	0
$330$	0	0
$331$	26.0526	1.43198	0.715989	−	0.698111i	$-0.245976\pi$
$331$	26.0526	1.43198	0.715989	+	0.698111i	$0.245976\pi$
$332$	9.89949	0.543305
$333$	0	0
$334$	16.1112	0.881563
$335$	−27.4249	−1.49838
$336$	0	0
$337$	13.3205	0.725614	0.362807	−	0.931864i	$-0.381818\pi$
$337$	13.3205	0.725614	0.362807	+	0.931864i	$0.381818\pi$
$338$	−7.00000	−0.380750
$339$	0	0
$340$	−12.1962	−0.661429
$341$	40.1528	2.17440
$342$	0	0
$343$	0	0
$344$	−1.80385	−0.0972569
$345$	0	0
$346$	14.5211	0.780658
$347$	33.7128	1.80980	0.904899	−	0.425626i	$-0.139946\pi$
$347$	33.7128	1.80980	0.904899	+	0.425626i	$0.139946\pi$
$348$	0	0
$349$	−24.7995	−1.32749	−0.663744	−	0.747960i	$-0.731033\pi$
$349$	−24.7995	−1.32749	−0.663744	+	0.747960i	$0.731033\pi$
$350$	0	0
$351$	0	0
$352$	−5.46410	−0.291238
$353$	−19.6975	−1.04839	−0.524195	−	0.851598i	$-0.675634\pi$
$353$	−19.6975	−1.04839	−0.524195	+	0.851598i	$0.675634\pi$
$354$	0	0
$355$	21.6293	1.14796
$356$	−13.2827	−0.703983
$357$	0	0
$358$	−2.19615	−0.116070
$359$	0.0717968	0.00378929	0.00189464	−	0.999998i	$-0.499397\pi$
$359$	0.0717968	0.00378929	0.00189464	+	0.999998i	$0.499397\pi$
$360$	0	0
$361$	−15.2679	−0.803576
$362$	−8.72552	−0.458603
$363$	0	0
$364$	0	0
$365$	18.3923	0.962697
$366$	0	0
$367$	14.4195	0.752694	0.376347	−	0.926479i	$-0.377180\pi$
$367$	14.4195	0.752694	0.376347	+	0.926479i	$0.377180\pi$
$368$	5.92820	0.309029
$369$	0	0
$370$	−15.4548	−0.803457
$371$	0	0
$372$	0	0
$373$	−5.12436	−0.265329	−0.132665	−	0.991161i	$-0.542353\pi$
$373$	−5.12436	−0.265329	−0.132665	+	0.991161i	$0.542353\pi$
$374$	34.4959	1.78374
$375$	0	0
$376$	9.52056	0.490985
$377$	1.79315	0.0923520
$378$	0	0
$379$	17.5167	0.899770	0.449885	−	0.893086i	$-0.351465\pi$
$379$	17.5167	0.899770	0.449885	+	0.893086i	$0.351465\pi$
$380$	−3.73205	−0.191450
$381$	0	0
$382$	−3.19615	−0.163529
$383$	19.5216	0.997507	0.498753	−	0.866744i	$-0.333791\pi$
$383$	19.5216	0.997507	0.498753	+	0.866744i	$0.333791\pi$
$384$	0	0
$385$	0	0
$386$	−20.2679	−1.03161
$387$	0	0
$388$	3.86370	0.196150
$389$	29.1244	1.47666	0.738332	−	0.674438i	$-0.235614\pi$
$389$	29.1244	1.47666	0.738332	+	0.674438i	$0.235614\pi$
$390$	0	0
$391$	−37.4259	−1.89271
$392$	0	0
$393$	0	0
$394$	7.66025	0.385918
$395$	−19.5588	−0.984108
$396$	0	0
$397$	1.89469	0.0950916	0.0475458	−	0.998869i	$-0.484860\pi$
$397$	1.89469	0.0950916	0.0475458	+	0.998869i	$0.484860\pi$
$398$	−3.10583	−0.155681
$399$	0	0
$400$	−1.26795	−0.0633975
$401$	−13.0526	−0.651814	−0.325907	−	0.945402i	$-0.605670\pi$
$401$	−13.0526	−0.651814	−0.325907	+	0.945402i	$0.605670\pi$
$402$	0	0
$403$	−18.0000	−0.896644
$404$	1.55291	0.0772604
$405$	0	0
$406$	0	0
$407$	43.7128	2.16676
$408$	0	0
$409$	−26.2880	−1.29986	−0.649930	−	0.759994i	$-0.725202\pi$
$409$	−26.2880	−1.29986	−0.649930	+	0.759994i	$0.725202\pi$
$410$	−10.9282	−0.539705
$411$	0	0
$412$	8.38375	0.413037
$413$	0	0
$414$	0	0
$415$	19.1244	0.938778
$416$	2.44949	0.120096
$417$	0	0
$418$	10.5558	0.516303
$419$	−4.27981	−0.209082	−0.104541	−	0.994521i	$-0.533337\pi$
$419$	−4.27981	−0.209082	−0.104541	+	0.994521i	$0.533337\pi$
$420$	0	0
$421$	−10.0526	−0.489932	−0.244966	−	0.969532i	$-0.578777\pi$
$421$	−10.0526	−0.489932	−0.244966	+	0.969532i	$0.578777\pi$
$422$	4.73205	0.230353
$423$	0	0
$424$	−3.26795	−0.158706
$425$	8.00481	0.388290
$426$	0	0
$427$	0	0
$428$	−16.9282	−0.818256
$429$	0	0
$430$	−3.48477	−0.168050
$431$	−14.7846	−0.712150	−0.356075	−	0.934457i	$-0.615885\pi$
$431$	−14.7846	−0.712150	−0.356075	+	0.934457i	$0.615885\pi$
$432$	0	0
$433$	28.7375	1.38104	0.690519	−	0.723314i	$-0.257382\pi$
$433$	28.7375	1.38104	0.690519	+	0.723314i	$0.257382\pi$
$434$	0	0
$435$	0	0
$436$	−20.5885	−0.986008
$437$	−11.4524	−0.547843
$438$	0	0
$439$	−1.31268	−0.0626507	−0.0313253	−	0.999509i	$-0.509973\pi$
$439$	−1.31268	−0.0626507	−0.0313253	+	0.999509i	$0.509973\pi$
$440$	−10.5558	−0.503230
$441$	0	0
$442$	−15.4641	−0.735552
$443$	−14.9808	−0.711757	−0.355879	−	0.934532i	$-0.615818\pi$
$443$	−14.9808	−0.711757	−0.355879	+	0.934532i	$0.615818\pi$
$444$	0	0
$445$	−25.6603	−1.21641
$446$	−20.7327	−0.981723
$447$	0	0
$448$	0	0
$449$	−17.7846	−0.839308	−0.419654	−	0.907684i	$-0.637849\pi$
$449$	−17.7846	−0.839308	−0.419654	+	0.907684i	$0.637849\pi$
$450$	0	0
$451$	30.9096	1.45548
$452$	−2.66025	−0.125128
$453$	0	0
$454$	−0.896575	−0.0420784
$455$	0	0
$456$	0	0
$457$	−25.7321	−1.20369	−0.601847	−	0.798611i	$-0.705568\pi$
$457$	−25.7321	−1.20369	−0.601847	+	0.798611i	$0.705568\pi$
$458$	−18.0430	−0.843094
$459$	0	0
$460$	11.4524	0.533971
$461$	−21.8324	−1.01684	−0.508418	−	0.861111i	$-0.669769\pi$
$461$	−21.8324	−1.01684	−0.508418	+	0.861111i	$0.669769\pi$
$462$	0	0
$463$	−10.6603	−0.495424	−0.247712	−	0.968834i	$-0.579679\pi$
$463$	−10.6603	−0.495424	−0.247712	+	0.968834i	$0.579679\pi$
$464$	0.732051	0.0339846
$465$	0	0
$466$	19.3923	0.898331
$467$	4.79744	0.221999	0.111000	−	0.993820i	$-0.464595\pi$
$467$	4.79744	0.221999	0.111000	+	0.993820i	$0.464595\pi$
$468$	0	0
$469$	0	0
$470$	18.3923	0.848374
$471$	0	0
$472$	−5.00052	−0.230167
$473$	9.85641	0.453198
$474$	0	0
$475$	2.44949	0.112390
$476$	0	0
$477$	0	0
$478$	5.53590	0.253206
$479$	−16.1112	−0.736137	−0.368069	−	0.929799i	$-0.619981\pi$
$479$	−16.1112	−0.736137	−0.368069	+	0.929799i	$0.619981\pi$
$480$	0	0
$481$	−19.5959	−0.893497
$482$	12.7279	0.579741
$483$	0	0
$484$	18.8564	0.857109
$485$	7.46410	0.338927
$486$	0	0
$487$	2.32051	0.105152	0.0525761	−	0.998617i	$-0.483257\pi$
$487$	2.32051	0.105152	0.0525761	+	0.998617i	$0.483257\pi$
$488$	−2.96713	−0.134316
$489$	0	0
$490$	0	0
$491$	18.9282	0.854218	0.427109	−	0.904200i	$-0.359532\pi$
$491$	18.9282	0.854218	0.427109	+	0.904200i	$0.359532\pi$
$492$	0	0
$493$	−4.62158	−0.208145
$494$	−4.73205	−0.212905
$495$	0	0
$496$	−7.34847	−0.329956
$497$	0	0
$498$	0	0
$499$	−27.3205	−1.22303	−0.611517	−	0.791231i	$-0.709440\pi$
$499$	−27.3205	−1.22303	−0.611517	+	0.791231i	$0.709440\pi$
$500$	−12.1087	−0.541520
$501$	0	0
$502$	−1.93185	−0.0862228
$503$	−19.1427	−0.853529	−0.426764	−	0.904363i	$-0.640347\pi$
$503$	−19.1427	−0.853529	−0.426764	+	0.904363i	$0.640347\pi$
$504$	0	0
$505$	3.00000	0.133498
$506$	−32.3923	−1.44001
$507$	0	0
$508$	7.92820	0.351757
$509$	−21.7680	−0.964850	−0.482425	−	0.875937i	$-0.660244\pi$
$509$	−21.7680	−0.964850	−0.482425	+	0.875937i	$0.660244\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	20.7327	0.914482
$515$	16.1962	0.713688
$516$	0	0
$517$	−52.0213	−2.28789
$518$	0	0
$519$	0	0
$520$	4.73205	0.207514
$521$	32.5269	1.42503	0.712515	−	0.701657i	$-0.247556\pi$
$521$	32.5269	1.42503	0.712515	+	0.701657i	$0.247556\pi$
$522$	0	0
$523$	7.41284	0.324141	0.162070	−	0.986779i	$-0.448183\pi$
$523$	7.41284	0.324141	0.162070	+	0.986779i	$0.448183\pi$
$524$	−0.240237	−0.0104948
$525$	0	0
$526$	14.6603	0.639217
$527$	46.3923	2.02088
$528$	0	0
$529$	12.1436	0.527982
$530$	−6.31319	−0.274228
$531$	0	0
$532$	0	0
$533$	−13.8564	−0.600188
$534$	0	0
$535$	−32.7028	−1.41386
$536$	−14.1962	−0.613180
$537$	0	0
$538$	0.0371647	0.00160229
$539$	0	0
$540$	0	0
$541$	17.2679	0.742407	0.371204	−	0.928552i	$-0.378945\pi$
$541$	17.2679	0.742407	0.371204	+	0.928552i	$0.378945\pi$
$542$	5.00052	0.214791
$543$	0	0
$544$	−6.31319	−0.270676
$545$	−39.7738	−1.70372
$546$	0	0
$547$	−4.53590	−0.193941	−0.0969705	−	0.995287i	$-0.530915\pi$
$547$	−4.53590	−0.193941	−0.0969705	+	0.995287i	$0.530915\pi$
$548$	0	0
$549$	0	0
$550$	6.92820	0.295420
$551$	−1.41421	−0.0602475
$552$	0	0
$553$	0	0
$554$	−5.80385	−0.246582
$555$	0	0
$556$	1.83032	0.0776227
$557$	−27.1244	−1.14930	−0.574648	−	0.818401i	$-0.694861\pi$
$557$	−27.1244	−1.14930	−0.574648	+	0.818401i	$0.694861\pi$
$558$	0	0
$559$	−4.41851	−0.186883
$560$	0	0
$561$	0	0
$562$	−3.39230	−0.143096
$563$	8.62398	0.363458	0.181729	−	0.983349i	$-0.441831\pi$
$563$	8.62398	0.363458	0.181729	+	0.983349i	$0.441831\pi$
$564$	0	0
$565$	−5.13922	−0.216208
$566$	17.1093	0.719156
$567$	0	0
$568$	11.1962	0.469780
$569$	−13.0718	−0.547998	−0.273999	−	0.961730i	$-0.588347\pi$
$569$	−13.0718	−0.547998	−0.273999	+	0.961730i	$0.588347\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	−13.3843	−0.559624
$573$	0	0
$574$	0	0
$575$	−7.51666	−0.313466
$576$	0	0
$577$	−17.8028	−0.741139	−0.370569	−	0.928805i	$-0.620837\pi$
$577$	−17.8028	−0.741139	−0.370569	+	0.928805i	$0.620837\pi$
$578$	22.8564	0.950701
$579$	0	0
$580$	1.41421	0.0587220
$581$	0	0
$582$	0	0
$583$	17.8564	0.739537
$584$	9.52056	0.393963
$585$	0	0
$586$	−4.38134	−0.180992
$587$	33.2204	1.37115	0.685577	−	0.728000i	$-0.259550\pi$
$587$	33.2204	1.37115	0.685577	+	0.728000i	$0.259550\pi$
$588$	0	0
$589$	14.1962	0.584942
$590$	−9.66025	−0.397706
$591$	0	0
$592$	−8.00000	−0.328798
$593$	21.4906	0.882513	0.441257	−	0.897381i	$-0.354533\pi$
$593$	21.4906	0.882513	0.441257	+	0.897381i	$0.354533\pi$
$594$	0	0
$595$	0	0
$596$	−18.3923	−0.753378
$597$	0	0
$598$	14.5211	0.593811
$599$	−8.39230	−0.342900	−0.171450	−	0.985193i	$-0.554845\pi$
$599$	−8.39230	−0.342900	−0.171450	+	0.985193i	$0.554845\pi$
$600$	0	0
$601$	−17.4510	−0.711843	−0.355921	−	0.934516i	$-0.615833\pi$
$601$	−17.4510	−0.711843	−0.355921	+	0.934516i	$0.615833\pi$
$602$	0	0
$603$	0	0
$604$	3.39230	0.138031
$605$	36.4278	1.48100
$606$	0	0
$607$	12.9038	0.523749	0.261874	−	0.965102i	$-0.415659\pi$
$607$	12.9038	0.523749	0.261874	+	0.965102i	$0.415659\pi$
$608$	−1.93185	−0.0783469
$609$	0	0
$610$	−5.73205	−0.232084
$611$	23.3205	0.943447
$612$	0	0
$613$	−38.4449	−1.55277	−0.776387	−	0.630257i	$-0.782950\pi$
$613$	−38.4449	−1.55277	−0.776387	+	0.630257i	$0.782950\pi$
$614$	−29.0793	−1.17355
$615$	0	0
$616$	0	0
$617$	−21.6077	−0.869893	−0.434947	−	0.900456i	$-0.643233\pi$
$617$	−21.6077	−0.869893	−0.434947	+	0.900456i	$0.643233\pi$
$618$	0	0
$619$	14.5582	0.585145	0.292572	−	0.956243i	$-0.405489\pi$
$619$	14.5582	0.585145	0.292572	+	0.956243i	$0.405489\pi$
$620$	−14.1962	−0.570131
$621$	0	0
$622$	21.0101	0.842430
$623$	0	0
$624$	0	0
$625$	−17.0526	−0.682102
$626$	−1.79315	−0.0716687
$627$	0	0
$628$	24.3562	0.971918
$629$	50.5055	2.01379
$630$	0	0
$631$	−28.1244	−1.11961	−0.559806	−	0.828623i	$-0.689125\pi$
$631$	−28.1244	−1.11961	−0.559806	+	0.828623i	$0.689125\pi$
$632$	−10.1244	−0.402725
$633$	0	0
$634$	−1.41154	−0.0560595
$635$	15.3161	0.607801
$636$	0	0
$637$	0	0
$638$	−4.00000	−0.158362
$639$	0	0
$640$	1.93185	0.0763631
$641$	−35.0526	−1.38449	−0.692246	−	0.721661i	$-0.743379\pi$
$641$	−35.0526	−1.38449	−0.692246	+	0.721661i	$0.743379\pi$
$642$	0	0
$643$	13.0053	0.512880	0.256440	−	0.966560i	$-0.417450\pi$
$643$	13.0053	0.512880	0.256440	+	0.966560i	$0.417450\pi$
$644$	0	0
$645$	0	0
$646$	12.1962	0.479851
$647$	1.13681	0.0446927	0.0223463	−	0.999750i	$-0.492886\pi$
$647$	1.13681	0.0446927	0.0223463	+	0.999750i	$0.492886\pi$
$648$	0	0
$649$	27.3233	1.07253
$650$	−3.10583	−0.121821
$651$	0	0
$652$	−20.9282	−0.819612
$653$	6.67949	0.261389	0.130694	−	0.991423i	$-0.458279\pi$
$653$	6.67949	0.261389	0.130694	+	0.991423i	$0.458279\pi$
$654$	0	0
$655$	−0.464102	−0.0181340
$656$	−5.65685	−0.220863
$657$	0	0
$658$	0	0
$659$	14.8756	0.579473	0.289736	−	0.957106i	$-0.406432\pi$
$659$	14.8756	0.579473	0.289736	+	0.957106i	$0.406432\pi$
$660$	0	0
$661$	33.7009	1.31081	0.655406	−	0.755276i	$-0.272497\pi$
$661$	33.7009	1.31081	0.655406	+	0.755276i	$0.272497\pi$
$662$	26.0526	1.01256
$663$	0	0
$664$	9.89949	0.384175
$665$	0	0
$666$	0	0
$667$	4.33975	0.168036
$668$	16.1112	0.623359
$669$	0	0
$670$	−27.4249	−1.05951
$671$	16.2127	0.625884
$672$	0	0
$673$	22.1769	0.854857	0.427429	−	0.904049i	$-0.359420\pi$
$673$	22.1769	0.854857	0.427429	+	0.904049i	$0.359420\pi$
$674$	13.3205	0.513087
$675$	0	0
$676$	−7.00000	−0.269231
$677$	−15.8338	−0.608540	−0.304270	−	0.952586i	$-0.598413\pi$
$677$	−15.8338	−0.608540	−0.304270	+	0.952586i	$0.598413\pi$
$678$	0	0
$679$	0	0
$680$	−12.1962	−0.467701
$681$	0	0
$682$	40.1528	1.53753
$683$	34.3923	1.31598	0.657992	−	0.753024i	$-0.271406\pi$
$683$	34.3923	1.31598	0.657992	+	0.753024i	$0.271406\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−1.80385	−0.0687710
$689$	−8.00481	−0.304959
$690$	0	0
$691$	24.2547	0.922691	0.461345	−	0.887221i	$-0.347367\pi$
$691$	24.2547	0.922691	0.461345	+	0.887221i	$0.347367\pi$
$692$	14.5211	0.552008
$693$	0	0
$694$	33.7128	1.27972
$695$	3.53590	0.134124
$696$	0	0
$697$	35.7128	1.35272
$698$	−24.7995	−0.938675
$699$	0	0
$700$	0	0
$701$	−27.4641	−1.03730	−0.518652	−	0.854985i	$-0.673566\pi$
$701$	−27.4641	−1.03730	−0.518652	+	0.854985i	$0.673566\pi$
$702$	0	0
$703$	15.4548	0.582889
$704$	−5.46410	−0.205936
$705$	0	0
$706$	−19.6975	−0.741323
$707$	0	0
$708$	0	0
$709$	−29.4641	−1.10655	−0.553274	−	0.832999i	$-0.686622\pi$
$709$	−29.4641	−1.10655	−0.553274	+	0.832999i	$0.686622\pi$
$710$	21.6293	0.811733
$711$	0	0
$712$	−13.2827	−0.497791
$713$	−43.5632	−1.63146
$714$	0	0
$715$	−25.8564	−0.966975
$716$	−2.19615	−0.0820741
$717$	0	0
$718$	0.0717968	0.00267943
$719$	−19.8733	−0.741150	−0.370575	−	0.928803i	$-0.620839\pi$
$719$	−19.8733	−0.741150	−0.370575	+	0.928803i	$0.620839\pi$
$720$	0	0
$721$	0	0
$722$	−15.2679	−0.568214
$723$	0	0
$724$	−8.72552	−0.324281
$725$	−0.928203	−0.0344726
$726$	0	0
$727$	24.1160	0.894411	0.447206	−	0.894431i	$-0.352419\pi$
$727$	24.1160	0.894411	0.447206	+	0.894431i	$0.352419\pi$
$728$	0	0
$729$	0	0
$730$	18.3923	0.680730
$731$	11.3880	0.421202
$732$	0	0
$733$	8.06918	0.298042	0.149021	−	0.988834i	$-0.452388\pi$
$733$	8.06918	0.298042	0.149021	+	0.988834i	$0.452388\pi$
$734$	14.4195	0.532235
$735$	0	0
$736$	5.92820	0.218516
$737$	77.5692	2.85730
$738$	0	0
$739$	−43.8564	−1.61328	−0.806642	−	0.591040i	$-0.798717\pi$
$739$	−43.8564	−1.61328	−0.806642	+	0.591040i	$0.798717\pi$
$740$	−15.4548	−0.568130
$741$	0	0
$742$	0	0
$743$	−20.2487	−0.742853	−0.371427	−	0.928462i	$-0.621131\pi$
$743$	−20.2487	−0.742853	−0.371427	+	0.928462i	$0.621131\pi$
$744$	0	0
$745$	−35.5312	−1.30176
$746$	−5.12436	−0.187616
$747$	0	0
$748$	34.4959	1.26130
$749$	0	0
$750$	0	0
$751$	12.1769	0.444342	0.222171	−	0.975008i	$-0.428686\pi$
$751$	12.1769	0.444342	0.222171	+	0.975008i	$0.428686\pi$
$752$	9.52056	0.347179
$753$	0	0
$754$	1.79315	0.0653027
$755$	6.55343	0.238504
$756$	0	0
$757$	−28.2487	−1.02672	−0.513358	−	0.858174i	$-0.671599\pi$
$757$	−28.2487	−1.02672	−0.513358	+	0.858174i	$0.671599\pi$
$758$	17.5167	0.636234
$759$	0	0
$760$	−3.73205	−0.135376
$761$	5.48099	0.198686	0.0993428	−	0.995053i	$-0.468326\pi$
$761$	5.48099	0.198686	0.0993428	+	0.995053i	$0.468326\pi$
$762$	0	0
$763$	0	0
$764$	−3.19615	−0.115633
$765$	0	0
$766$	19.5216	0.705344
$767$	−12.2487	−0.442275
$768$	0	0
$769$	−4.41851	−0.159335	−0.0796677	−	0.996821i	$-0.525386\pi$
$769$	−4.41851	−0.159335	−0.0796677	+	0.996821i	$0.525386\pi$
$770$	0	0
$771$	0	0
$772$	−20.2679	−0.729459
$773$	3.62347	0.130327	0.0651635	−	0.997875i	$-0.479243\pi$
$773$	3.62347	0.130327	0.0651635	+	0.997875i	$0.479243\pi$
$774$	0	0
$775$	9.31749	0.334694
$776$	3.86370	0.138699
$777$	0	0
$778$	29.1244	1.04416
$779$	10.9282	0.391544
$780$	0	0
$781$	−61.1769	−2.18908
$782$	−37.4259	−1.33835
$783$	0	0
$784$	0	0
$785$	47.0526	1.67938
$786$	0	0
$787$	7.90327	0.281721	0.140861	−	0.990029i	$-0.455013\pi$
$787$	7.90327	0.281721	0.140861	+	0.990029i	$0.455013\pi$
$788$	7.66025	0.272885
$789$	0	0
$790$	−19.5588	−0.695869
$791$	0	0
$792$	0	0
$793$	−7.26795	−0.258092
$794$	1.89469	0.0672399
$795$	0	0
$796$	−3.10583	−0.110083
$797$	−22.1098	−0.783169	−0.391584	−	0.920142i	$-0.628073\pi$
$797$	−22.1098	−0.783169	−0.391584	+	0.920142i	$0.628073\pi$
$798$	0	0
$799$	−60.1051	−2.12637
$800$	−1.26795	−0.0448288
$801$	0	0
$802$	−13.0526	−0.460902
$803$	−52.0213	−1.83579
$804$	0	0
$805$	0	0
$806$	−18.0000	−0.634023
$807$	0	0
$808$	1.55291	0.0546313
$809$	1.32051	0.0464266	0.0232133	−	0.999731i	$-0.492610\pi$
$809$	1.32051	0.0464266	0.0232133	+	0.999731i	$0.492610\pi$
$810$	0	0
$811$	17.6269	0.618964	0.309482	−	0.950905i	$-0.399844\pi$
$811$	17.6269	0.618964	0.309482	+	0.950905i	$0.399844\pi$
$812$	0	0
$813$	0	0
$814$	43.7128	1.53213
$815$	−40.4302	−1.41621
$816$	0	0
$817$	3.48477	0.121917
$818$	−26.2880	−0.919140
$819$	0	0
$820$	−10.9282	−0.381629
$821$	41.3205	1.44210	0.721048	−	0.692885i	$-0.243661\pi$
$821$	41.3205	1.44210	0.721048	+	0.692885i	$0.243661\pi$
$822$	0	0
$823$	27.3205	0.952333	0.476167	−	0.879355i	$-0.342026\pi$
$823$	27.3205	0.952333	0.476167	+	0.879355i	$0.342026\pi$
$824$	8.38375	0.292062
$825$	0	0
$826$	0	0
$827$	49.2679	1.71321	0.856607	−	0.515969i	$-0.172568\pi$
$827$	49.2679	1.71321	0.856607	+	0.515969i	$0.172568\pi$
$828$	0	0
$829$	−26.3896	−0.916548	−0.458274	−	0.888811i	$-0.651532\pi$
$829$	−26.3896	−0.916548	−0.458274	+	0.888811i	$0.651532\pi$
$830$	19.1244	0.663816
$831$	0	0
$832$	2.44949	0.0849208
$833$	0	0
$834$	0	0
$835$	31.1244	1.07710
$836$	10.5558	0.365081
$837$	0	0
$838$	−4.27981	−0.147843
$839$	−13.8375	−0.477725	−0.238862	−	0.971053i	$-0.576774\pi$
$839$	−13.8375	−0.477725	−0.238862	+	0.971053i	$0.576774\pi$
$840$	0	0
$841$	−28.4641	−0.981521
$842$	−10.0526	−0.346434
$843$	0	0
$844$	4.73205	0.162884
$845$	−13.5230	−0.465204
$846$	0	0
$847$	0	0
$848$	−3.26795	−0.112222
$849$	0	0
$850$	8.00481	0.274563
$851$	−47.4256	−1.62573
$852$	0	0
$853$	−27.8410	−0.953256	−0.476628	−	0.879105i	$-0.658141\pi$
$853$	−27.8410	−0.953256	−0.476628	+	0.879105i	$0.658141\pi$
$854$	0	0
$855$	0	0
$856$	−16.9282	−0.578594
$857$	33.7381	1.15247	0.576235	−	0.817284i	$-0.304521\pi$
$857$	33.7381	1.15247	0.576235	+	0.817284i	$0.304521\pi$
$858$	0	0
$859$	−29.4954	−1.00637	−0.503185	−	0.864179i	$-0.667839\pi$
$859$	−29.4954	−1.00637	−0.503185	+	0.864179i	$0.667839\pi$
$860$	−3.48477	−0.118830
$861$	0	0
$862$	−14.7846	−0.503566
$863$	−10.8038	−0.367767	−0.183884	−	0.982948i	$-0.558867\pi$
$863$	−10.8038	−0.367767	−0.183884	+	0.982948i	$0.558867\pi$
$864$	0	0
$865$	28.0526	0.953816
$866$	28.7375	0.976541
$867$	0	0
$868$	0	0
$869$	55.3205	1.87662
$870$	0	0
$871$	−34.7733	−1.17825
$872$	−20.5885	−0.697213
$873$	0	0
$874$	−11.4524	−0.387384
$875$	0	0
$876$	0	0
$877$	−14.0526	−0.474521	−0.237261	−	0.971446i	$-0.576250\pi$
$877$	−14.0526	−0.474521	−0.237261	+	0.971446i	$0.576250\pi$
$878$	−1.31268	−0.0443007
$879$	0	0
$880$	−10.5558	−0.355837
$881$	−24.9754	−0.841442	−0.420721	−	0.907190i	$-0.638223\pi$
$881$	−24.9754	−0.841442	−0.420721	+	0.907190i	$0.638223\pi$
$882$	0	0
$883$	−10.2487	−0.344897	−0.172448	−	0.985019i	$-0.555168\pi$
$883$	−10.2487	−0.344897	−0.172448	+	0.985019i	$0.555168\pi$
$884$	−15.4641	−0.520114
$885$	0	0
$886$	−14.9808	−0.503289
$887$	−46.5675	−1.56358	−0.781792	−	0.623539i	$-0.785694\pi$
$887$	−46.5675	−1.56358	−0.781792	+	0.623539i	$0.785694\pi$
$888$	0	0
$889$	0	0
$890$	−25.6603	−0.860134
$891$	0	0
$892$	−20.7327	−0.694183
$893$	−18.3923	−0.615475
$894$	0	0
$895$	−4.24264	−0.141816
$896$	0	0
$897$	0	0
$898$	−17.7846	−0.593480
$899$	−5.37945	−0.179415
$900$	0	0
$901$	20.6312	0.687325
$902$	30.9096	1.02918
$903$	0	0
$904$	−2.66025	−0.0884787
$905$	−16.8564	−0.560326
$906$	0	0
$907$	19.1244	0.635014	0.317507	−	0.948256i	$-0.397154\pi$
$907$	19.1244	0.635014	0.317507	+	0.948256i	$0.397154\pi$
$908$	−0.896575	−0.0297539
$909$	0	0
$910$	0	0
$911$	13.7846	0.456704	0.228352	−	0.973579i	$-0.426666\pi$
$911$	13.7846	0.456704	0.228352	+	0.973579i	$0.426666\pi$
$912$	0	0
$913$	−54.0918	−1.79018
$914$	−25.7321	−0.851141
$915$	0	0
$916$	−18.0430	−0.596158
$917$	0	0
$918$	0	0
$919$	−56.3731	−1.85958	−0.929788	−	0.368096i	$-0.880010\pi$
$919$	−56.3731	−1.85958	−0.929788	+	0.368096i	$0.880010\pi$
$920$	11.4524	0.377575
$921$	0	0
$922$	−21.8324	−0.719011
$923$	27.4249	0.902700
$924$	0	0
$925$	10.1436	0.333519
$926$	−10.6603	−0.350318
$927$	0	0
$928$	0.732051	0.0240307
$929$	35.1523	1.15331	0.576654	−	0.816988i	$-0.304358\pi$
$929$	35.1523	1.15331	0.576654	+	0.816988i	$0.304358\pi$
$930$	0	0
$931$	0	0
$932$	19.3923	0.635216
$933$	0	0
$934$	4.79744	0.156977
$935$	66.6410	2.17939
$936$	0	0
$937$	21.9711	0.717764	0.358882	−	0.933383i	$-0.383158\pi$
$937$	21.9711	0.717764	0.358882	+	0.933383i	$0.383158\pi$
$938$	0	0
$939$	0	0
$940$	18.3923	0.599891
$941$	36.3535	1.18509	0.592544	−	0.805538i	$-0.298124\pi$
$941$	36.3535	1.18509	0.592544	+	0.805538i	$0.298124\pi$
$942$	0	0
$943$	−33.5350	−1.09205
$944$	−5.00052	−0.162753
$945$	0	0
$946$	9.85641	0.320459
$947$	18.3397	0.595962	0.297981	−	0.954572i	$-0.403687\pi$
$947$	18.3397	0.595962	0.297981	+	0.954572i	$0.403687\pi$
$948$	0	0
$949$	23.3205	0.757016
$950$	2.44949	0.0794719
$951$	0	0
$952$	0	0
$953$	31.7128	1.02728	0.513639	−	0.858006i	$-0.328297\pi$
$953$	31.7128	1.02728	0.513639	+	0.858006i	$0.328297\pi$
$954$	0	0
$955$	−6.17449	−0.199802
$956$	5.53590	0.179044
$957$	0	0
$958$	−16.1112	−0.520528
$959$	0	0
$960$	0	0
$961$	23.0000	0.741935
$962$	−19.5959	−0.631798
$963$	0	0
$964$	12.7279	0.409939
$965$	−39.1547	−1.26043
$966$	0	0
$967$	6.46410	0.207871	0.103936	−	0.994584i	$-0.466856\pi$
$967$	6.46410	0.207871	0.103936	+	0.994584i	$0.466856\pi$
$968$	18.8564	0.606068
$969$	0	0
$970$	7.46410	0.239658
$971$	13.0425	0.418553	0.209277	−	0.977856i	$-0.432889\pi$
$971$	13.0425	0.418553	0.209277	+	0.977856i	$0.432889\pi$
$972$	0	0
$973$	0	0
$974$	2.32051	0.0743539
$975$	0	0
$976$	−2.96713	−0.0949754
$977$	43.8564	1.40309	0.701545	−	0.712625i	$-0.252494\pi$
$977$	43.8564	1.40309	0.701545	+	0.712625i	$0.252494\pi$
$978$	0	0
$979$	72.5782	2.31961
$980$	0	0
$981$	0	0
$982$	18.9282	0.604023
$983$	−54.7482	−1.74620	−0.873098	−	0.487545i	$-0.837893\pi$
$983$	−54.7482	−1.74620	−0.873098	+	0.487545i	$0.837893\pi$
$984$	0	0
$985$	14.7985	0.471519
$986$	−4.62158	−0.147181
$987$	0	0
$988$	−4.73205	−0.150547
$989$	−10.6936	−0.340036
$990$	0	0
$991$	49.3205	1.56672	0.783359	−	0.621570i	$-0.213505\pi$
$991$	49.3205	1.56672	0.783359	+	0.621570i	$0.213505\pi$
$992$	−7.34847	−0.233314
$993$	0	0
$994$	0	0
$995$	−6.00000	−0.190213
$996$	0	0
$997$	16.9334	0.536286	0.268143	−	0.963379i	$-0.413590\pi$
$997$	16.9334	0.536286	0.268143	+	0.963379i	$0.413590\pi$
$998$	−27.3205	−0.864816
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7938.2.a.co.1.4		4
3.2	odd	2		7938.2.a.cj.1.1		4
7.6	odd	2	inner	7938.2.a.co.1.1		4
9.2	odd	6		882.2.f.s.589.3	yes	8
9.4	even	3		2646.2.f.q.883.1		8
9.5	odd	6		882.2.f.s.295.4	yes	8
9.7	even	3		2646.2.f.q.1765.1		8
21.20	even	2		7938.2.a.cj.1.4		4
63.2	odd	6		882.2.h.t.67.3		8
63.4	even	3		2646.2.h.q.667.4		8
63.5	even	6		882.2.e.q.655.4		8
63.11	odd	6		882.2.e.q.373.1		8
63.13	odd	6		2646.2.f.q.883.4		8
63.16	even	3		2646.2.h.q.361.4		8
63.20	even	6		882.2.f.s.589.2	yes	8
63.23	odd	6		882.2.e.q.655.1		8
63.25	even	3		2646.2.e.t.1549.1		8
63.31	odd	6		2646.2.h.q.667.1		8
63.32	odd	6		882.2.h.t.79.3		8
63.34	odd	6		2646.2.f.q.1765.4		8
63.38	even	6		882.2.e.q.373.4		8
63.40	odd	6		2646.2.e.t.2125.4		8
63.41	even	6		882.2.f.s.295.1	✓	8
63.47	even	6		882.2.h.t.67.2		8
63.52	odd	6		2646.2.e.t.1549.4		8
63.58	even	3		2646.2.e.t.2125.1		8
63.59	even	6		882.2.h.t.79.2		8
63.61	odd	6		2646.2.h.q.361.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.e.q.373.1		8	63.11	odd	6
882.2.e.q.373.4		8	63.38	even	6
882.2.e.q.655.1		8	63.23	odd	6
882.2.e.q.655.4		8	63.5	even	6
882.2.f.s.295.1	✓	8	63.41	even	6
882.2.f.s.295.4	yes	8	9.5	odd	6
882.2.f.s.589.2	yes	8	63.20	even	6
882.2.f.s.589.3	yes	8	9.2	odd	6
882.2.h.t.67.2		8	63.47	even	6
882.2.h.t.67.3		8	63.2	odd	6
882.2.h.t.79.2		8	63.59	even	6
882.2.h.t.79.3		8	63.32	odd	6
2646.2.e.t.1549.1		8	63.25	even	3
2646.2.e.t.1549.4		8	63.52	odd	6
2646.2.e.t.2125.1		8	63.58	even	3
2646.2.e.t.2125.4		8	63.40	odd	6
2646.2.f.q.883.1		8	9.4	even	3
2646.2.f.q.883.4		8	63.13	odd	6
2646.2.f.q.1765.1		8	9.7	even	3
2646.2.f.q.1765.4		8	63.34	odd	6
2646.2.h.q.361.1		8	63.61	odd	6
2646.2.h.q.361.4		8	63.16	even	3
2646.2.h.q.667.1		8	63.31	odd	6
2646.2.h.q.667.4		8	63.4	even	3
7938.2.a.cj.1.1		4	3.2	odd	2
7938.2.a.cj.1.4		4	21.20	even	2
7938.2.a.co.1.1		4	7.6	odd	2	inner
7938.2.a.co.1.4		4	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:	\( N \)	\(=\)	\( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7938.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.93185 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.93185 q^{5} +1.00000 q^{8} +1.93185 q^{10} -5.46410 q^{11} +2.44949 q^{13} +1.00000 q^{16} -6.31319 q^{17} -1.93185 q^{19} +1.93185 q^{20} -5.46410 q^{22} +5.92820 q^{23} -1.26795 q^{25} +2.44949 q^{26} +0.732051 q^{29} -7.34847 q^{31} +1.00000 q^{32} -6.31319 q^{34} -8.00000 q^{37} -1.93185 q^{38} +1.93185 q^{40} -5.65685 q^{41} -1.80385 q^{43} -5.46410 q^{44} +5.92820 q^{46} +9.52056 q^{47} -1.26795 q^{50} +2.44949 q^{52} -3.26795 q^{53} -10.5558 q^{55} +0.732051 q^{58} -5.00052 q^{59} -2.96713 q^{61} -7.34847 q^{62} +1.00000 q^{64} +4.73205 q^{65} -14.1962 q^{67} -6.31319 q^{68} +11.1962 q^{71} +9.52056 q^{73} -8.00000 q^{74} -1.93185 q^{76} -10.1244 q^{79} +1.93185 q^{80} -5.65685 q^{82} +9.89949 q^{83} -12.1962 q^{85} -1.80385 q^{86} -5.46410 q^{88} -13.2827 q^{89} +5.92820 q^{92} +9.52056 q^{94} -3.73205 q^{95} +3.86370 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} - 8 q^{11} + 4 q^{16} - 8 q^{22} - 4 q^{23} - 12 q^{25} - 4 q^{29} + 4 q^{32} - 32 q^{37} - 28 q^{43} - 8 q^{44} - 4 q^{46} - 12 q^{50} - 20 q^{53} - 4 q^{58} + 4 q^{64} + 12 q^{65} - 36 q^{67} + 24 q^{71} - 32 q^{74} + 8 q^{79} - 28 q^{85} - 28 q^{86} - 8 q^{88} - 4 q^{92} - 8 q^{95}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 - 8 * q^11 + 4 * q^16 - 8 * q^22 - 4 * q^23 - 12 * q^25 - 4 * q^29 + 4 * q^32 - 32 * q^37 - 28 * q^43 - 8 * q^44 - 4 * q^46 - 12 * q^50 - 20 * q^53 - 4 * q^58 + 4 * q^64 + 12 * q^65 - 36 * q^67 + 24 * q^71 - 32 * q^74 + 8 * q^79 - 28 * q^85 - 28 * q^86 - 8 * q^88 - 4 * q^92 - 8 * q^95

Introduction

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