LMFDB - Embedded newform 7938.2.a.bk.1.1

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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 1134)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{8} -4.24264 q^{11} +6.24264 q^{13} +1.00000 q^{16} +6.24264 q^{19} +4.24264 q^{22} -7.24264 q^{23} -5.00000 q^{25} -6.24264 q^{26} +4.24264 q^{29} +0.757359 q^{31} -1.00000 q^{32} -4.00000 q^{37} -6.24264 q^{38} +5.48528 q^{41} -6.48528 q^{43} -4.24264 q^{44} +7.24264 q^{46} -13.2426 q^{47} +5.00000 q^{50} +6.24264 q^{52} -4.24264 q^{53} -4.24264 q^{58} +6.24264 q^{61} -0.757359 q^{62} +1.00000 q^{64} -8.24264 q^{67} +7.24264 q^{71} -7.00000 q^{73} +4.00000 q^{74} +6.24264 q^{76} +9.24264 q^{79} -5.48528 q^{82} -7.75736 q^{83} +6.48528 q^{86} +4.24264 q^{88} +5.48528 q^{89} -7.24264 q^{92} +13.2426 q^{94} -12.4853 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 4 q^{13} + 2 q^{16} + 4 q^{19} - 6 q^{23} - 10 q^{25} - 4 q^{26} + 10 q^{31} - 2 q^{32} - 8 q^{37} - 4 q^{38} - 6 q^{41} + 4 q^{43} + 6 q^{46} - 18 q^{47} + 10 q^{50}+ \cdots - 8 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 4 * q^13 + 2 * q^16 + 4 * q^19 - 6 * q^23 - 10 * q^25 - 4 * q^26 + 10 * q^31 - 2 * q^32 - 8 * q^37 - 4 * q^38 - 6 * q^41 + 4 * q^43 + 6 * q^46 - 18 * q^47 + 10 * q^50 + 4 * q^52 + 4 * q^61 - 10 * q^62 + 2 * q^64 - 8 * q^67 + 6 * q^71 - 14 * q^73 + 8 * q^74 + 4 * q^76 + 10 * q^79 + 6 * q^82 - 24 * q^83 - 4 * q^86 - 6 * q^89 - 6 * q^92 + 18 * q^94 - 8 * q^97

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
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−4.24264	−1.27920	−0.639602	−	0.768706i	$-0.720901\pi$
$11$	−4.24264	−1.27920	−0.639602	+	0.768706i	$0.720901\pi$
$12$	0	0
$13$	6.24264	1.73140	0.865699	−	0.500566i	$-0.166875\pi$
$13$	6.24264	1.73140	0.865699	+	0.500566i	$0.166875\pi$
$14$	0	0
$15$	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
$19$	6.24264	1.43216	0.716080	−	0.698018i	$-0.245935\pi$
$19$	6.24264	1.43216	0.716080	+	0.698018i	$0.245935\pi$
$20$	0	0
$21$	0	0
$22$	4.24264	0.904534
$23$	−7.24264	−1.51019	−0.755097	−	0.655613i	$-0.772410\pi$
$23$	−7.24264	−1.51019	−0.755097	+	0.655613i	$0.772410\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	−6.24264	−1.22428
$27$	0	0
$28$	0	0
$29$	4.24264	0.787839	0.393919	−	0.919145i	$-0.371119\pi$
$29$	4.24264	0.787839	0.393919	+	0.919145i	$0.371119\pi$
$30$	0	0
$31$	0.757359	0.136026	0.0680129	−	0.997684i	$-0.478334\pi$
$31$	0.757359	0.136026	0.0680129	+	0.997684i	$0.478334\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	−6.24264	−1.01269
$39$	0	0
$40$	0	0
$41$	5.48528	0.856657	0.428329	−	0.903623i	$-0.359103\pi$
$41$	5.48528	0.856657	0.428329	+	0.903623i	$0.359103\pi$
$42$	0	0
$43$	−6.48528	−0.988996	−0.494498	−	0.869179i	$-0.664648\pi$
$43$	−6.48528	−0.988996	−0.494498	+	0.869179i	$0.664648\pi$
$44$	−4.24264	−0.639602
$45$	0	0
$46$	7.24264	1.06787
$47$	−13.2426	−1.93164	−0.965819	−	0.259218i	$-0.916535\pi$
$47$	−13.2426	−1.93164	−0.965819	+	0.259218i	$0.916535\pi$
$48$	0	0
$49$	0	0
$50$	5.00000	0.707107
$51$	0	0
$52$	6.24264	0.865699
$53$	−4.24264	−0.582772	−0.291386	−	0.956606i	$-0.594116\pi$
$53$	−4.24264	−0.582772	−0.291386	+	0.956606i	$0.594116\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−4.24264	−0.557086
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	6.24264	0.799288	0.399644	−	0.916670i	$-0.369134\pi$
$61$	6.24264	0.799288	0.399644	+	0.916670i	$0.369134\pi$
$62$	−0.757359	−0.0961847
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−8.24264	−1.00700	−0.503499	−	0.863996i	$-0.667954\pi$
$67$	−8.24264	−1.00700	−0.503499	+	0.863996i	$0.667954\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.24264	0.859543	0.429772	−	0.902938i	$-0.358594\pi$
$71$	7.24264	0.859543	0.429772	+	0.902938i	$0.358594\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	6.24264	0.716080
$77$	0	0
$78$	0	0
$79$	9.24264	1.03988	0.519939	−	0.854203i	$-0.325955\pi$
$79$	9.24264	1.03988	0.519939	+	0.854203i	$0.325955\pi$
$80$	0	0
$81$	0	0
$82$	−5.48528	−0.605748
$83$	−7.75736	−0.851481	−0.425740	−	0.904845i	$-0.639986\pi$
$83$	−7.75736	−0.851481	−0.425740	+	0.904845i	$0.639986\pi$
$84$	0	0
$85$	0	0
$86$	6.48528	0.699326
$87$	0	0
$88$	4.24264	0.452267
$89$	5.48528	0.581439	0.290719	−	0.956808i	$-0.406105\pi$
$89$	5.48528	0.581439	0.290719	+	0.956808i	$0.406105\pi$
$90$	0	0
$91$	0	0
$92$	−7.24264	−0.755097
$93$	0	0
$94$	13.2426	1.36587
$95$	0	0
$96$	0	0
$97$	−12.4853	−1.26769	−0.633844	−	0.773461i	$-0.718524\pi$
$97$	−12.4853	−1.26769	−0.633844	+	0.773461i	$0.718524\pi$
$98$	0	0
$99$	0	0
$100$	−5.00000	−0.500000
$101$	−7.75736	−0.771886	−0.385943	−	0.922523i	$-0.626124\pi$
$101$	−7.75736	−0.771886	−0.385943	+	0.922523i	$0.626124\pi$
$102$	0	0
$103$	0.757359	0.0746248	0.0373124	−	0.999304i	$-0.488120\pi$
$103$	0.757359	0.0746248	0.0373124	+	0.999304i	$0.488120\pi$
$104$	−6.24264	−0.612141
$105$	0	0
$106$	4.24264	0.412082
$107$	2.48528	0.240261	0.120131	−	0.992758i	$-0.461669\pi$
$107$	2.48528	0.240261	0.120131	+	0.992758i	$0.461669\pi$
$108$	0	0
$109$	−2.24264	−0.214806	−0.107403	−	0.994216i	$-0.534254\pi$
$109$	−2.24264	−0.214806	−0.107403	+	0.994216i	$0.534254\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	20.4853	1.92709	0.963547	−	0.267541i	$-0.0862110\pi$
$113$	20.4853	1.92709	0.963547	+	0.267541i	$0.0862110\pi$
$114$	0	0
$115$	0	0
$116$	4.24264	0.393919
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.00000	0.636364
$122$	−6.24264	−0.565182
$123$	0	0
$124$	0.757359	0.0680129
$125$	0	0
$126$	0	0
$127$	6.75736	0.599619	0.299809	−	0.953999i	$-0.403077\pi$
$127$	6.75736	0.599619	0.299809	+	0.953999i	$0.403077\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	7.75736	0.677764	0.338882	−	0.940829i	$-0.389951\pi$
$131$	7.75736	0.677764	0.338882	+	0.940829i	$0.389951\pi$
$132$	0	0
$133$	0	0
$134$	8.24264	0.712056
$135$	0	0
$136$	0	0
$137$	19.9706	1.70620	0.853100	−	0.521747i	$-0.174720\pi$
$137$	19.9706	1.70620	0.853100	+	0.521747i	$0.174720\pi$
$138$	0	0
$139$	−4.72792	−0.401017	−0.200509	−	0.979692i	$-0.564259\pi$
$139$	−4.72792	−0.401017	−0.200509	+	0.979692i	$0.564259\pi$
$140$	0	0
$141$	0	0
$142$	−7.24264	−0.607789
$143$	−26.4853	−2.21481
$144$	0	0
$145$	0	0
$146$	7.00000	0.579324
$147$	0	0
$148$	−4.00000	−0.328798
$149$	−16.2426	−1.33065	−0.665324	−	0.746554i	$-0.731707\pi$
$149$	−16.2426	−1.33065	−0.665324	+	0.746554i	$0.731707\pi$
$150$	0	0
$151$	−2.75736	−0.224391	−0.112195	−	0.993686i	$-0.535788\pi$
$151$	−2.75736	−0.224391	−0.112195	+	0.993686i	$0.535788\pi$
$152$	−6.24264	−0.506345
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	−9.24264	−0.735305
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−11.7574	−0.920907	−0.460454	−	0.887684i	$-0.652313\pi$
$163$	−11.7574	−0.920907	−0.460454	+	0.887684i	$0.652313\pi$
$164$	5.48528	0.428329
$165$	0	0
$166$	7.75736	0.602088
$167$	−24.2132	−1.87367	−0.936837	−	0.349766i	$-0.886261\pi$
$167$	−24.2132	−1.87367	−0.936837	+	0.349766i	$0.886261\pi$
$168$	0	0
$169$	25.9706	1.99774
$170$	0	0
$171$	0	0
$172$	−6.48528	−0.494498
$173$	10.9706	0.834076	0.417038	−	0.908889i	$-0.363068\pi$
$173$	10.9706	0.834076	0.417038	+	0.908889i	$0.363068\pi$
$174$	0	0
$175$	0	0
$176$	−4.24264	−0.319801
$177$	0	0
$178$	−5.48528	−0.411139
$179$	16.2426	1.21403	0.607016	−	0.794690i	$-0.292366\pi$
$179$	16.2426	1.21403	0.607016	+	0.794690i	$0.292366\pi$
$180$	0	0
$181$	−20.2426	−1.50462	−0.752312	−	0.658807i	$-0.771061\pi$
$181$	−20.2426	−1.50462	−0.752312	+	0.658807i	$0.771061\pi$
$182$	0	0
$183$	0	0
$184$	7.24264	0.533935
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−13.2426	−0.965819
$189$	0	0
$190$	0	0
$191$	−8.48528	−0.613973	−0.306987	−	0.951714i	$-0.599321\pi$
$191$	−8.48528	−0.613973	−0.306987	+	0.951714i	$0.599321\pi$
$192$	0	0
$193$	−4.51472	−0.324977	−0.162488	−	0.986710i	$-0.551952\pi$
$193$	−4.51472	−0.324977	−0.162488	+	0.986710i	$0.551952\pi$
$194$	12.4853	0.896391
$195$	0	0
$196$	0	0
$197$	−16.9706	−1.20910	−0.604551	−	0.796566i	$-0.706648\pi$
$197$	−16.9706	−1.20910	−0.604551	+	0.796566i	$0.706648\pi$
$198$	0	0
$199$	−14.7574	−1.04612	−0.523061	−	0.852295i	$-0.675210\pi$
$199$	−14.7574	−1.04612	−0.523061	+	0.852295i	$0.675210\pi$
$200$	5.00000	0.353553
$201$	0	0
$202$	7.75736	0.545806
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−0.757359	−0.0527677
$207$	0	0
$208$	6.24264	0.432849
$209$	−26.4853	−1.83203
$210$	0	0
$211$	−6.48528	−0.446465	−0.223233	−	0.974765i	$-0.571661\pi$
$211$	−6.48528	−0.446465	−0.223233	+	0.974765i	$0.571661\pi$
$212$	−4.24264	−0.291386
$213$	0	0
$214$	−2.48528	−0.169890
$215$	0	0
$216$	0	0
$217$	0	0
$218$	2.24264	0.151891
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	11.7279	0.785360	0.392680	−	0.919675i	$-0.371548\pi$
$223$	11.7279	0.785360	0.392680	+	0.919675i	$0.371548\pi$
$224$	0	0
$225$	0	0
$226$	−20.4853	−1.36266
$227$	−26.4853	−1.75789	−0.878945	−	0.476923i	$-0.841752\pi$
$227$	−26.4853	−1.75789	−0.878945	+	0.476923i	$0.841752\pi$
$228$	0	0
$229$	24.9706	1.65010	0.825051	−	0.565059i	$-0.191147\pi$
$229$	24.9706	1.65010	0.825051	+	0.565059i	$0.191147\pi$
$230$	0	0
$231$	0	0
$232$	−4.24264	−0.278543
$233$	−20.4853	−1.34204	−0.671018	−	0.741441i	$-0.734143\pi$
$233$	−20.4853	−1.34204	−0.671018	+	0.741441i	$0.734143\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3.72792	−0.241139	−0.120570	−	0.992705i	$-0.538472\pi$
$239$	−3.72792	−0.241139	−0.120570	+	0.992705i	$0.538472\pi$
$240$	0	0
$241$	8.51472	0.548481	0.274241	−	0.961661i	$-0.411574\pi$
$241$	8.51472	0.548481	0.274241	+	0.961661i	$0.411574\pi$
$242$	−7.00000	−0.449977
$243$	0	0
$244$	6.24264	0.399644
$245$	0	0
$246$	0	0
$247$	38.9706	2.47964
$248$	−0.757359	−0.0480924
$249$	0	0
$250$	0	0
$251$	−18.7279	−1.18210	−0.591048	−	0.806636i	$-0.701286\pi$
$251$	−18.7279	−1.18210	−0.591048	+	0.806636i	$0.701286\pi$
$252$	0	0
$253$	30.7279	1.93185
$254$	−6.75736	−0.423994
$255$	0	0
$256$	1.00000	0.0625000
$257$	−5.48528	−0.342162	−0.171081	−	0.985257i	$-0.554726\pi$
$257$	−5.48528	−0.342162	−0.171081	+	0.985257i	$0.554726\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−7.75736	−0.479251
$263$	−22.9706	−1.41643	−0.708213	−	0.705999i	$-0.750498\pi$
$263$	−22.9706	−1.41643	−0.708213	+	0.705999i	$0.750498\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−8.24264	−0.503499
$269$	10.9706	0.668887	0.334444	−	0.942416i	$-0.391452\pi$
$269$	10.9706	0.668887	0.334444	+	0.942416i	$0.391452\pi$
$270$	0	0
$271$	−12.4853	−0.758427	−0.379213	−	0.925309i	$-0.623805\pi$
$271$	−12.4853	−0.758427	−0.379213	+	0.925309i	$0.623805\pi$
$272$	0	0
$273$	0	0
$274$	−19.9706	−1.20647
$275$	21.2132	1.27920
$276$	0	0
$277$	−19.2132	−1.15441	−0.577205	−	0.816599i	$-0.695857\pi$
$277$	−19.2132	−1.15441	−0.577205	+	0.816599i	$0.695857\pi$
$278$	4.72792	0.283562
$279$	0	0
$280$	0	0
$281$	−22.4558	−1.33960	−0.669802	−	0.742540i	$-0.733621\pi$
$281$	−22.4558	−1.33960	−0.669802	+	0.742540i	$0.733621\pi$
$282$	0	0
$283$	24.9706	1.48435	0.742173	−	0.670208i	$-0.233795\pi$
$283$	24.9706	1.48435	0.742173	+	0.670208i	$0.233795\pi$
$284$	7.24264	0.429772
$285$	0	0
$286$	26.4853	1.56611
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	−7.00000	−0.409644
$293$	10.9706	0.640907	0.320454	−	0.947264i	$-0.396165\pi$
$293$	10.9706	0.640907	0.320454	+	0.947264i	$0.396165\pi$
$294$	0	0
$295$	0	0
$296$	4.00000	0.232495
$297$	0	0
$298$	16.2426	0.940911
$299$	−45.2132	−2.61475
$300$	0	0
$301$	0	0
$302$	2.75736	0.158668
$303$	0	0
$304$	6.24264	0.358040
$305$	0	0
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	10.9706	0.622084	0.311042	−	0.950396i	$-0.399322\pi$
$311$	10.9706	0.622084	0.311042	+	0.950396i	$0.399322\pi$
$312$	0	0
$313$	−17.9706	−1.01576	−0.507878	−	0.861429i	$-0.669570\pi$
$313$	−17.9706	−1.01576	−0.507878	+	0.861429i	$0.669570\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	9.24264	0.519939
$317$	16.9706	0.953162	0.476581	−	0.879131i	$-0.341876\pi$
$317$	16.9706	0.953162	0.476581	+	0.879131i	$0.341876\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−31.2132	−1.73140
$326$	11.7574	0.651180
$327$	0	0
$328$	−5.48528	−0.302874
$329$	0	0
$330$	0	0
$331$	−18.4853	−1.01604	−0.508021	−	0.861344i	$-0.669623\pi$
$331$	−18.4853	−1.01604	−0.508021	+	0.861344i	$0.669623\pi$
$332$	−7.75736	−0.425740
$333$	0	0
$334$	24.2132	1.32489
$335$	0	0
$336$	0	0
$337$	4.48528	0.244329	0.122164	−	0.992510i	$-0.461016\pi$
$337$	4.48528	0.244329	0.122164	+	0.992510i	$0.461016\pi$
$338$	−25.9706	−1.41261
$339$	0	0
$340$	0	0
$341$	−3.21320	−0.174005
$342$	0	0
$343$	0	0
$344$	6.48528	0.349663
$345$	0	0
$346$	−10.9706	−0.589781
$347$	6.72792	0.361174	0.180587	−	0.983559i	$-0.442200\pi$
$347$	6.72792	0.361174	0.180587	+	0.983559i	$0.442200\pi$
$348$	0	0
$349$	24.9706	1.33664	0.668322	−	0.743872i	$-0.267013\pi$
$349$	24.9706	1.33664	0.668322	+	0.743872i	$0.267013\pi$
$350$	0	0
$351$	0	0
$352$	4.24264	0.226134
$353$	−21.0000	−1.11772	−0.558859	−	0.829263i	$-0.688761\pi$
$353$	−21.0000	−1.11772	−0.558859	+	0.829263i	$0.688761\pi$
$354$	0	0
$355$	0	0
$356$	5.48528	0.290719
$357$	0	0
$358$	−16.2426	−0.858450
$359$	−10.7574	−0.567752	−0.283876	−	0.958861i	$-0.591620\pi$
$359$	−10.7574	−0.567752	−0.283876	+	0.958861i	$0.591620\pi$
$360$	0	0
$361$	19.9706	1.05108
$362$	20.2426	1.06393
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11.7279	0.612193	0.306096	−	0.952001i	$-0.400977\pi$
$367$	11.7279	0.612193	0.306096	+	0.952001i	$0.400977\pi$
$368$	−7.24264	−0.377549
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−7.21320	−0.373486	−0.186743	−	0.982409i	$-0.559793\pi$
$373$	−7.21320	−0.373486	−0.186743	+	0.982409i	$0.559793\pi$
$374$	0	0
$375$	0	0
$376$	13.2426	0.682937
$377$	26.4853	1.36406
$378$	0	0
$379$	1.27208	0.0653423	0.0326711	−	0.999466i	$-0.489599\pi$
$379$	1.27208	0.0653423	0.0326711	+	0.999466i	$0.489599\pi$
$380$	0	0
$381$	0	0
$382$	8.48528	0.434145
$383$	24.2132	1.23724	0.618618	−	0.785692i	$-0.287693\pi$
$383$	24.2132	1.23724	0.618618	+	0.785692i	$0.287693\pi$
$384$	0	0
$385$	0	0
$386$	4.51472	0.229793
$387$	0	0
$388$	−12.4853	−0.633844
$389$	−10.2426	−0.519322	−0.259661	−	0.965700i	$-0.583611\pi$
$389$	−10.2426	−0.519322	−0.259661	+	0.965700i	$0.583611\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	16.9706	0.854965
$395$	0	0
$396$	0	0
$397$	−20.2426	−1.01595	−0.507975	−	0.861372i	$-0.669606\pi$
$397$	−20.2426	−1.01595	−0.507975	+	0.861372i	$0.669606\pi$
$398$	14.7574	0.739720
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−29.4853	−1.47242	−0.736212	−	0.676751i	$-0.763388\pi$
$401$	−29.4853	−1.47242	−0.736212	+	0.676751i	$0.763388\pi$
$402$	0	0
$403$	4.72792	0.235515
$404$	−7.75736	−0.385943
$405$	0	0
$406$	0	0
$407$	16.9706	0.841200
$408$	0	0
$409$	35.0000	1.73064	0.865319	−	0.501221i	$-0.167116\pi$
$409$	35.0000	1.73064	0.865319	+	0.501221i	$0.167116\pi$
$410$	0	0
$411$	0	0
$412$	0.757359	0.0373124
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−6.24264	−0.306071
$417$	0	0
$418$	26.4853	1.29544
$419$	7.75736	0.378972	0.189486	−	0.981883i	$-0.439318\pi$
$419$	7.75736	0.378972	0.189486	+	0.981883i	$0.439318\pi$
$420$	0	0
$421$	−14.2426	−0.694144	−0.347072	−	0.937839i	$-0.612824\pi$
$421$	−14.2426	−0.694144	−0.347072	+	0.937839i	$0.612824\pi$
$422$	6.48528	0.315699
$423$	0	0
$424$	4.24264	0.206041
$425$	0	0
$426$	0	0
$427$	0	0
$428$	2.48528	0.120131
$429$	0	0
$430$	0	0
$431$	37.2426	1.79391	0.896957	−	0.442117i	$-0.145772\pi$
$431$	37.2426	1.79391	0.896957	+	0.442117i	$0.145772\pi$
$432$	0	0
$433$	3.97056	0.190813	0.0954065	−	0.995438i	$-0.469585\pi$
$433$	3.97056	0.190813	0.0954065	+	0.995438i	$0.469585\pi$
$434$	0	0
$435$	0	0
$436$	−2.24264	−0.107403
$437$	−45.2132	−2.16284
$438$	0	0
$439$	11.7279	0.559743	0.279872	−	0.960037i	$-0.409708\pi$
$439$	11.7279	0.559743	0.279872	+	0.960037i	$0.409708\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9.51472	−0.452058	−0.226029	−	0.974121i	$-0.572574\pi$
$443$	−9.51472	−0.452058	−0.226029	+	0.974121i	$0.572574\pi$
$444$	0	0
$445$	0	0
$446$	−11.7279	−0.555333
$447$	0	0
$448$	0	0
$449$	4.97056	0.234575	0.117288	−	0.993098i	$-0.462580\pi$
$449$	4.97056	0.234575	0.117288	+	0.993098i	$0.462580\pi$
$450$	0	0
$451$	−23.2721	−1.09584
$452$	20.4853	0.963547
$453$	0	0
$454$	26.4853	1.24302
$455$	0	0
$456$	0	0
$457$	11.5147	0.538636	0.269318	−	0.963051i	$-0.413202\pi$
$457$	11.5147	0.538636	0.269318	+	0.963051i	$0.413202\pi$
$458$	−24.9706	−1.16680
$459$	0	0
$460$	0	0
$461$	−34.2426	−1.59484	−0.797419	−	0.603425i	$-0.793802\pi$
$461$	−34.2426	−1.59484	−0.797419	+	0.603425i	$0.793802\pi$
$462$	0	0
$463$	−8.75736	−0.406989	−0.203495	−	0.979076i	$-0.565230\pi$
$463$	−8.75736	−0.406989	−0.203495	+	0.979076i	$0.565230\pi$
$464$	4.24264	0.196960
$465$	0	0
$466$	20.4853	0.948962
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	27.5147	1.26513
$474$	0	0
$475$	−31.2132	−1.43216
$476$	0	0
$477$	0	0
$478$	3.72792	0.170511
$479$	13.2426	0.605072	0.302536	−	0.953138i	$-0.402167\pi$
$479$	13.2426	0.605072	0.302536	+	0.953138i	$0.402167\pi$
$480$	0	0
$481$	−24.9706	−1.13856
$482$	−8.51472	−0.387835
$483$	0	0
$484$	7.00000	0.318182
$485$	0	0
$486$	0	0
$487$	−2.75736	−0.124948	−0.0624739	−	0.998047i	$-0.519899\pi$
$487$	−2.75736	−0.124948	−0.0624739	+	0.998047i	$0.519899\pi$
$488$	−6.24264	−0.282591
$489$	0	0
$490$	0	0
$491$	−6.72792	−0.303627	−0.151813	−	0.988409i	$-0.548511\pi$
$491$	−6.72792	−0.303627	−0.151813	+	0.988409i	$0.548511\pi$
$492$	0	0
$493$	0	0
$494$	−38.9706	−1.75337
$495$	0	0
$496$	0.757359	0.0340064
$497$	0	0
$498$	0	0
$499$	−10.7279	−0.480248	−0.240124	−	0.970742i	$-0.577188\pi$
$499$	−10.7279	−0.480248	−0.240124	+	0.970742i	$0.577188\pi$
$500$	0	0
$501$	0	0
$502$	18.7279	0.835868
$503$	24.2132	1.07961	0.539807	−	0.841789i	$-0.318497\pi$
$503$	24.2132	1.07961	0.539807	+	0.841789i	$0.318497\pi$
$504$	0	0
$505$	0	0
$506$	−30.7279	−1.36602
$507$	0	0
$508$	6.75736	0.299809
$509$	−7.75736	−0.343839	−0.171919	−	0.985111i	$-0.554997\pi$
$509$	−7.75736	−0.343839	−0.171919	+	0.985111i	$0.554997\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	5.48528	0.241945
$515$	0	0
$516$	0	0
$517$	56.1838	2.47096
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−16.4558	−0.720944	−0.360472	−	0.932770i	$-0.617384\pi$
$521$	−16.4558	−0.720944	−0.360472	+	0.932770i	$0.617384\pi$
$522$	0	0
$523$	−20.2426	−0.885149	−0.442574	−	0.896732i	$-0.645935\pi$
$523$	−20.2426	−0.885149	−0.442574	+	0.896732i	$0.645935\pi$
$524$	7.75736	0.338882
$525$	0	0
$526$	22.9706	1.00156
$527$	0	0
$528$	0	0
$529$	29.4558	1.28069
$530$	0	0
$531$	0	0
$532$	0	0
$533$	34.2426	1.48321
$534$	0	0
$535$	0	0
$536$	8.24264	0.356028
$537$	0	0
$538$	−10.9706	−0.472975
$539$	0	0
$540$	0	0
$541$	−14.9706	−0.643635	−0.321817	−	0.946802i	$-0.604294\pi$
$541$	−14.9706	−0.643635	−0.321817	+	0.946802i	$0.604294\pi$
$542$	12.4853	0.536289
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−6.48528	−0.277291	−0.138645	−	0.990342i	$-0.544275\pi$
$547$	−6.48528	−0.277291	−0.138645	+	0.990342i	$0.544275\pi$
$548$	19.9706	0.853100
$549$	0	0
$550$	−21.2132	−0.904534
$551$	26.4853	1.12831
$552$	0	0
$553$	0	0
$554$	19.2132	0.816291
$555$	0	0
$556$	−4.72792	−0.200509
$557$	40.9706	1.73598	0.867989	−	0.496583i	$-0.165412\pi$
$557$	40.9706	1.73598	0.867989	+	0.496583i	$0.165412\pi$
$558$	0	0
$559$	−40.4853	−1.71234
$560$	0	0
$561$	0	0
$562$	22.4558	0.947243
$563$	−18.7279	−0.789288	−0.394644	−	0.918834i	$-0.629132\pi$
$563$	−18.7279	−0.789288	−0.394644	+	0.918834i	$0.629132\pi$
$564$	0	0
$565$	0	0
$566$	−24.9706	−1.04959
$567$	0	0
$568$	−7.24264	−0.303894
$569$	−24.9411	−1.04559	−0.522793	−	0.852460i	$-0.675110\pi$
$569$	−24.9411	−1.04559	−0.522793	+	0.852460i	$0.675110\pi$
$570$	0	0
$571$	−20.9706	−0.877591	−0.438795	−	0.898587i	$-0.644595\pi$
$571$	−20.9706	−0.877591	−0.438795	+	0.898587i	$0.644595\pi$
$572$	−26.4853	−1.10741
$573$	0	0
$574$	0	0
$575$	36.2132	1.51019
$576$	0	0
$577$	35.9411	1.49625	0.748124	−	0.663559i	$-0.230955\pi$
$577$	35.9411	1.49625	0.748124	+	0.663559i	$0.230955\pi$
$578$	17.0000	0.707107
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	18.0000	0.745484
$584$	7.00000	0.289662
$585$	0	0
$586$	−10.9706	−0.453190
$587$	3.21320	0.132623	0.0663115	−	0.997799i	$-0.478877\pi$
$587$	3.21320	0.132623	0.0663115	+	0.997799i	$0.478877\pi$
$588$	0	0
$589$	4.72792	0.194811
$590$	0	0
$591$	0	0
$592$	−4.00000	−0.164399
$593$	5.48528	0.225254	0.112627	−	0.993637i	$-0.464074\pi$
$593$	5.48528	0.225254	0.112627	+	0.993637i	$0.464074\pi$
$594$	0	0
$595$	0	0
$596$	−16.2426	−0.665324
$597$	0	0
$598$	45.2132	1.84891
$599$	−22.9706	−0.938552	−0.469276	−	0.883052i	$-0.655485\pi$
$599$	−22.9706	−0.938552	−0.469276	+	0.883052i	$0.655485\pi$
$600$	0	0
$601$	−17.9706	−0.733035	−0.366517	−	0.930411i	$-0.619450\pi$
$601$	−17.9706	−0.733035	−0.366517	+	0.930411i	$0.619450\pi$
$602$	0	0
$603$	0	0
$604$	−2.75736	−0.112195
$605$	0	0
$606$	0	0
$607$	−38.9706	−1.58177	−0.790883	−	0.611967i	$-0.790378\pi$
$607$	−38.9706	−1.58177	−0.790883	+	0.611967i	$0.790378\pi$
$608$	−6.24264	−0.253173
$609$	0	0
$610$	0	0
$611$	−82.6690	−3.34443
$612$	0	0
$613$	−37.9411	−1.53243	−0.766214	−	0.642586i	$-0.777862\pi$
$613$	−37.9411	−1.53243	−0.766214	+	0.642586i	$0.777862\pi$
$614$	28.0000	1.12999
$615$	0	0
$616$	0	0
$617$	28.4558	1.14559	0.572795	−	0.819699i	$-0.305859\pi$
$617$	28.4558	1.14559	0.572795	+	0.819699i	$0.305859\pi$
$618$	0	0
$619$	−1.51472	−0.0608817	−0.0304408	−	0.999537i	$-0.509691\pi$
$619$	−1.51472	−0.0608817	−0.0304408	+	0.999537i	$0.509691\pi$
$620$	0	0
$621$	0	0
$622$	−10.9706	−0.439879
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	17.9706	0.718248
$627$	0	0
$628$	14.0000	0.558661
$629$	0	0
$630$	0	0
$631$	−24.4853	−0.974744	−0.487372	−	0.873195i	$-0.662044\pi$
$631$	−24.4853	−0.974744	−0.487372	+	0.873195i	$0.662044\pi$
$632$	−9.24264	−0.367653
$633$	0	0
$634$	−16.9706	−0.673987
$635$	0	0
$636$	0	0
$637$	0	0
$638$	18.0000	0.712627
$639$	0	0
$640$	0	0
$641$	4.45584	0.175995	0.0879976	−	0.996121i	$-0.471953\pi$
$641$	4.45584	0.175995	0.0879976	+	0.996121i	$0.471953\pi$
$642$	0	0
$643$	−46.7279	−1.84277	−0.921385	−	0.388652i	$-0.872941\pi$
$643$	−46.7279	−1.84277	−0.921385	+	0.388652i	$0.872941\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.27208	−0.0893246	−0.0446623	−	0.999002i	$-0.514221\pi$
$647$	−2.27208	−0.0893246	−0.0446623	+	0.999002i	$0.514221\pi$
$648$	0	0
$649$	0	0
$650$	31.2132	1.22428
$651$	0	0
$652$	−11.7574	−0.460454
$653$	−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
$656$	5.48528	0.214164
$657$	0	0
$658$	0	0
$659$	39.2132	1.52753	0.763765	−	0.645495i	$-0.223349\pi$
$659$	39.2132	1.52753	0.763765	+	0.645495i	$0.223349\pi$
$660$	0	0
$661$	−42.1838	−1.64076	−0.820379	−	0.571820i	$-0.806238\pi$
$661$	−42.1838	−1.64076	−0.820379	+	0.571820i	$0.806238\pi$
$662$	18.4853	0.718451
$663$	0	0
$664$	7.75736	0.301044
$665$	0	0
$666$	0	0
$667$	−30.7279	−1.18979
$668$	−24.2132	−0.936837
$669$	0	0
$670$	0	0
$671$	−26.4853	−1.02245
$672$	0	0
$673$	−27.4853	−1.05948	−0.529740	−	0.848160i	$-0.677710\pi$
$673$	−27.4853	−1.05948	−0.529740	+	0.848160i	$0.677710\pi$
$674$	−4.48528	−0.172767
$675$	0	0
$676$	25.9706	0.998868
$677$	−34.2426	−1.31605	−0.658026	−	0.752995i	$-0.728608\pi$
$677$	−34.2426	−1.31605	−0.658026	+	0.752995i	$0.728608\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	3.21320	0.123040
$683$	−41.6985	−1.59555	−0.797774	−	0.602956i	$-0.793989\pi$
$683$	−41.6985	−1.59555	−0.797774	+	0.602956i	$0.793989\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−6.48528	−0.247249
$689$	−26.4853	−1.00901
$690$	0	0
$691$	6.24264	0.237481	0.118741	−	0.992925i	$-0.462114\pi$
$691$	6.24264	0.237481	0.118741	+	0.992925i	$0.462114\pi$
$692$	10.9706	0.417038
$693$	0	0
$694$	−6.72792	−0.255388
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−24.9706	−0.945150
$699$	0	0
$700$	0	0
$701$	−13.7574	−0.519608	−0.259804	−	0.965661i	$-0.583658\pi$
$701$	−13.7574	−0.519608	−0.259804	+	0.965661i	$0.583658\pi$
$702$	0	0
$703$	−24.9706	−0.941783
$704$	−4.24264	−0.159901
$705$	0	0
$706$	21.0000	0.790345
$707$	0	0
$708$	0	0
$709$	25.6985	0.965127	0.482563	−	0.875861i	$-0.339706\pi$
$709$	25.6985	0.965127	0.482563	+	0.875861i	$0.339706\pi$
$710$	0	0
$711$	0	0
$712$	−5.48528	−0.205570
$713$	−5.48528	−0.205425
$714$	0	0
$715$	0	0
$716$	16.2426	0.607016
$717$	0	0
$718$	10.7574	0.401461
$719$	2.27208	0.0847342	0.0423671	−	0.999102i	$-0.486510\pi$
$719$	2.27208	0.0847342	0.0423671	+	0.999102i	$0.486510\pi$
$720$	0	0
$721$	0	0
$722$	−19.9706	−0.743227
$723$	0	0
$724$	−20.2426	−0.752312
$725$	−21.2132	−0.787839
$726$	0	0
$727$	−25.7279	−0.954196	−0.477098	−	0.878850i	$-0.658311\pi$
$727$	−25.7279	−0.954196	−0.477098	+	0.878850i	$0.658311\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−38.9706	−1.43941	−0.719705	−	0.694280i	$-0.755723\pi$
$733$	−38.9706	−1.43941	−0.719705	+	0.694280i	$0.755723\pi$
$734$	−11.7279	−0.432886
$735$	0	0
$736$	7.24264	0.266967
$737$	34.9706	1.28816
$738$	0	0
$739$	10.4853	0.385707	0.192854	−	0.981228i	$-0.438226\pi$
$739$	10.4853	0.385707	0.192854	+	0.981228i	$0.438226\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−34.7574	−1.27512	−0.637562	−	0.770399i	$-0.720057\pi$
$743$	−34.7574	−1.27512	−0.637562	+	0.770399i	$0.720057\pi$
$744$	0	0
$745$	0	0
$746$	7.21320	0.264094
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−17.2426	−0.629193	−0.314596	−	0.949226i	$-0.601869\pi$
$751$	−17.2426	−0.629193	−0.314596	+	0.949226i	$0.601869\pi$
$752$	−13.2426	−0.482909
$753$	0	0
$754$	−26.4853	−0.964537
$755$	0	0
$756$	0	0
$757$	12.9706	0.471423	0.235712	−	0.971823i	$-0.424258\pi$
$757$	12.9706	0.471423	0.235712	+	0.971823i	$0.424258\pi$
$758$	−1.27208	−0.0462040
$759$	0	0
$760$	0	0
$761$	−21.0000	−0.761249	−0.380625	−	0.924730i	$-0.624291\pi$
$761$	−21.0000	−0.761249	−0.380625	+	0.924730i	$0.624291\pi$
$762$	0	0
$763$	0	0
$764$	−8.48528	−0.306987
$765$	0	0
$766$	−24.2132	−0.874859
$767$	0	0
$768$	0	0
$769$	−12.4853	−0.450231	−0.225115	−	0.974332i	$-0.572276\pi$
$769$	−12.4853	−0.450231	−0.225115	+	0.974332i	$0.572276\pi$
$770$	0	0
$771$	0	0
$772$	−4.51472	−0.162488
$773$	45.2132	1.62621	0.813103	−	0.582120i	$-0.197777\pi$
$773$	45.2132	1.62621	0.813103	+	0.582120i	$0.197777\pi$
$774$	0	0
$775$	−3.78680	−0.136026
$776$	12.4853	0.448195
$777$	0	0
$778$	10.2426	0.367216
$779$	34.2426	1.22687
$780$	0	0
$781$	−30.7279	−1.09953
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−31.2132	−1.11263	−0.556315	−	0.830971i	$-0.687785\pi$
$787$	−31.2132	−1.11263	−0.556315	+	0.830971i	$0.687785\pi$
$788$	−16.9706	−0.604551
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	38.9706	1.38389
$794$	20.2426	0.718384
$795$	0	0
$796$	−14.7574	−0.523061
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	5.00000	0.176777
$801$	0	0
$802$	29.4853	1.04116
$803$	29.6985	1.04804
$804$	0	0
$805$	0	0
$806$	−4.72792	−0.166534
$807$	0	0
$808$	7.75736	0.272903
$809$	9.00000	0.316423	0.158212	−	0.987405i	$-0.449427\pi$
$809$	9.00000	0.316423	0.158212	+	0.987405i	$0.449427\pi$
$810$	0	0
$811$	−23.4558	−0.823646	−0.411823	−	0.911264i	$-0.635108\pi$
$811$	−23.4558	−0.823646	−0.411823	+	0.911264i	$0.635108\pi$
$812$	0	0
$813$	0	0
$814$	−16.9706	−0.594818
$815$	0	0
$816$	0	0
$817$	−40.4853	−1.41640
$818$	−35.0000	−1.22375
$819$	0	0
$820$	0	0
$821$	32.1838	1.12322	0.561611	−	0.827402i	$-0.310182\pi$
$821$	32.1838	1.12322	0.561611	+	0.827402i	$0.310182\pi$
$822$	0	0
$823$	40.6985	1.41866	0.709330	−	0.704877i	$-0.248998\pi$
$823$	40.6985	1.41866	0.709330	+	0.704877i	$0.248998\pi$
$824$	−0.757359	−0.0263839
$825$	0	0
$826$	0	0
$827$	−16.9706	−0.590124	−0.295062	−	0.955478i	$-0.595340\pi$
$827$	−16.9706	−0.590124	−0.295062	+	0.955478i	$0.595340\pi$
$828$	0	0
$829$	−49.9411	−1.73453	−0.867263	−	0.497849i	$-0.834123\pi$
$829$	−49.9411	−1.73453	−0.867263	+	0.497849i	$0.834123\pi$
$830$	0	0
$831$	0	0
$832$	6.24264	0.216425
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−26.4853	−0.916013
$837$	0	0
$838$	−7.75736	−0.267974
$839$	−15.5147	−0.535628	−0.267814	−	0.963471i	$-0.586301\pi$
$839$	−15.5147	−0.535628	−0.267814	+	0.963471i	$0.586301\pi$
$840$	0	0
$841$	−11.0000	−0.379310
$842$	14.2426	0.490834
$843$	0	0
$844$	−6.48528	−0.223233
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−4.24264	−0.145693
$849$	0	0
$850$	0	0
$851$	28.9706	0.993098
$852$	0	0
$853$	28.1838	0.964994	0.482497	−	0.875898i	$-0.339730\pi$
$853$	28.1838	0.964994	0.482497	+	0.875898i	$0.339730\pi$
$854$	0	0
$855$	0	0
$856$	−2.48528	−0.0849452
$857$	5.48528	0.187374	0.0936868	−	0.995602i	$-0.470135\pi$
$857$	5.48528	0.187374	0.0936868	+	0.995602i	$0.470135\pi$
$858$	0	0
$859$	43.6985	1.49097	0.745487	−	0.666521i	$-0.232217\pi$
$859$	43.6985	1.49097	0.745487	+	0.666521i	$0.232217\pi$
$860$	0	0
$861$	0	0
$862$	−37.2426	−1.26849
$863$	0.213203	0.00725753	0.00362876	−	0.999993i	$-0.498845\pi$
$863$	0.213203	0.00725753	0.00362876	+	0.999993i	$0.498845\pi$
$864$	0	0
$865$	0	0
$866$	−3.97056	−0.134925
$867$	0	0
$868$	0	0
$869$	−39.2132	−1.33022
$870$	0	0
$871$	−51.4558	−1.74351
$872$	2.24264	0.0759454
$873$	0	0
$874$	45.2132	1.52936
$875$	0	0
$876$	0	0
$877$	25.6985	0.867776	0.433888	−	0.900967i	$-0.357141\pi$
$877$	25.6985	0.867776	0.433888	+	0.900967i	$0.357141\pi$
$878$	−11.7279	−0.395798
$879$	0	0
$880$	0	0
$881$	36.5147	1.23021	0.615106	−	0.788444i	$-0.289113\pi$
$881$	36.5147	1.23021	0.615106	+	0.788444i	$0.289113\pi$
$882$	0	0
$883$	49.6985	1.67249	0.836244	−	0.548358i	$-0.184747\pi$
$883$	49.6985	1.67249	0.836244	+	0.548358i	$0.184747\pi$
$884$	0	0
$885$	0	0
$886$	9.51472	0.319653
$887$	−39.7279	−1.33393	−0.666967	−	0.745088i	$-0.732408\pi$
$887$	−39.7279	−1.33393	−0.666967	+	0.745088i	$0.732408\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	11.7279	0.392680
$893$	−82.6690	−2.76641
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−4.97056	−0.165870
$899$	3.21320	0.107166
$900$	0	0
$901$	0	0
$902$	23.2721	0.774875
$903$	0	0
$904$	−20.4853	−0.681330
$905$	0	0
$906$	0	0
$907$	−37.9411	−1.25981	−0.629907	−	0.776670i	$-0.716907\pi$
$907$	−37.9411	−1.25981	−0.629907	+	0.776670i	$0.716907\pi$
$908$	−26.4853	−0.878945
$909$	0	0
$910$	0	0
$911$	14.2721	0.472855	0.236428	−	0.971649i	$-0.424023\pi$
$911$	14.2721	0.472855	0.236428	+	0.971649i	$0.424023\pi$
$912$	0	0
$913$	32.9117	1.08922
$914$	−11.5147	−0.380873
$915$	0	0
$916$	24.9706	0.825051
$917$	0	0
$918$	0	0
$919$	33.4558	1.10361	0.551803	−	0.833974i	$-0.313940\pi$
$919$	33.4558	1.10361	0.551803	+	0.833974i	$0.313940\pi$
$920$	0	0
$921$	0	0
$922$	34.2426	1.12772
$923$	45.2132	1.48821
$924$	0	0
$925$	20.0000	0.657596
$926$	8.75736	0.287785
$927$	0	0
$928$	−4.24264	−0.139272
$929$	10.0294	0.329055	0.164528	−	0.986372i	$-0.447390\pi$
$929$	10.0294	0.329055	0.164528	+	0.986372i	$0.447390\pi$
$930$	0	0
$931$	0	0
$932$	−20.4853	−0.671018
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	30.4558	0.994949	0.497475	−	0.867479i	$-0.334261\pi$
$937$	30.4558	0.994949	0.497475	+	0.867479i	$0.334261\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	40.6690	1.32577	0.662887	−	0.748720i	$-0.269331\pi$
$941$	40.6690	1.32577	0.662887	+	0.748720i	$0.269331\pi$
$942$	0	0
$943$	−39.7279	−1.29372
$944$	0	0
$945$	0	0
$946$	−27.5147	−0.894581
$947$	21.5147	0.699134	0.349567	−	0.936911i	$-0.386329\pi$
$947$	21.5147	0.699134	0.349567	+	0.936911i	$0.386329\pi$
$948$	0	0
$949$	−43.6985	−1.41851
$950$	31.2132	1.01269
$951$	0	0
$952$	0	0
$953$	6.51472	0.211032	0.105516	−	0.994418i	$-0.466351\pi$
$953$	6.51472	0.211032	0.105516	+	0.994418i	$0.466351\pi$
$954$	0	0
$955$	0	0
$956$	−3.72792	−0.120570
$957$	0	0
$958$	−13.2426	−0.427850
$959$	0	0
$960$	0	0
$961$	−30.4264	−0.981497
$962$	24.9706	0.805083
$963$	0	0
$964$	8.51472	0.274241
$965$	0	0
$966$	0	0
$967$	−6.69848	−0.215409	−0.107704	−	0.994183i	$-0.534350\pi$
$967$	−6.69848	−0.215409	−0.107704	+	0.994183i	$0.534350\pi$
$968$	−7.00000	−0.224989
$969$	0	0
$970$	0	0
$971$	26.4853	0.849953	0.424977	−	0.905204i	$-0.360282\pi$
$971$	26.4853	0.849953	0.424977	+	0.905204i	$0.360282\pi$
$972$	0	0
$973$	0	0
$974$	2.75736	0.0883515
$975$	0	0
$976$	6.24264	0.199822
$977$	13.9706	0.446958	0.223479	−	0.974709i	$-0.428259\pi$
$977$	13.9706	0.446958	0.223479	+	0.974709i	$0.428259\pi$
$978$	0	0
$979$	−23.2721	−0.743779
$980$	0	0
$981$	0	0
$982$	6.72792	0.214697
$983$	15.5147	0.494843	0.247421	−	0.968908i	$-0.420417\pi$
$983$	15.5147	0.494843	0.247421	+	0.968908i	$0.420417\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	38.9706	1.23982
$989$	46.9706	1.49358
$990$	0	0
$991$	50.2132	1.59507	0.797537	−	0.603269i	$-0.206136\pi$
$991$	50.2132	1.59507	0.797537	+	0.603269i	$0.206136\pi$
$992$	−0.757359	−0.0240462
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	10.7279	0.339586
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7938.2.a.bk.1.1		2
3.2	odd	2		7938.2.a.bq.1.2		2
7.2	even	3		1134.2.g.j.487.1	yes	4
7.4	even	3		1134.2.g.j.163.1	yes	4
7.6	odd	2		7938.2.a.bj.1.1		2
21.2	odd	6		1134.2.g.i.487.1	yes	4
21.11	odd	6		1134.2.g.i.163.1	✓	4
21.20	even	2		7938.2.a.bp.1.2		2
63.2	odd	6		1134.2.h.r.109.2		4
63.4	even	3		1134.2.h.s.541.1		4
63.11	odd	6		1134.2.e.s.919.2		4
63.16	even	3		1134.2.h.s.109.2		4
63.23	odd	6		1134.2.e.s.865.2		4
63.25	even	3		1134.2.e.r.919.2		4
63.32	odd	6		1134.2.h.r.541.1		4
63.58	even	3		1134.2.e.r.865.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1134.2.e.r.865.2		4	63.58	even	3
1134.2.e.r.919.2		4	63.25	even	3
1134.2.e.s.865.2		4	63.23	odd	6
1134.2.e.s.919.2		4	63.11	odd	6
1134.2.g.i.163.1	✓	4	21.11	odd	6
1134.2.g.i.487.1	yes	4	21.2	odd	6
1134.2.g.j.163.1	yes	4	7.4	even	3
1134.2.g.j.487.1	yes	4	7.2	even	3
1134.2.h.r.109.2		4	63.2	odd	6
1134.2.h.r.541.1		4	63.32	odd	6
1134.2.h.s.109.2		4	63.16	even	3
1134.2.h.s.541.1		4	63.4	even	3
7938.2.a.bj.1.1		2	7.6	odd	2
7938.2.a.bk.1.1		2	1.1	even	1	trivial
7938.2.a.bp.1.2		2	21.20	even	2
7938.2.a.bq.1.2		2	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7938.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{8} -4.24264 q^{11} +6.24264 q^{13} +1.00000 q^{16} +6.24264 q^{19} +4.24264 q^{22} -7.24264 q^{23} -5.00000 q^{25} -6.24264 q^{26} +4.24264 q^{29} +0.757359 q^{31} -1.00000 q^{32} -4.00000 q^{37} -6.24264 q^{38} +5.48528 q^{41} -6.48528 q^{43} -4.24264 q^{44} +7.24264 q^{46} -13.2426 q^{47} +5.00000 q^{50} +6.24264 q^{52} -4.24264 q^{53} -4.24264 q^{58} +6.24264 q^{61} -0.757359 q^{62} +1.00000 q^{64} -8.24264 q^{67} +7.24264 q^{71} -7.00000 q^{73} +4.00000 q^{74} +6.24264 q^{76} +9.24264 q^{79} -5.48528 q^{82} -7.75736 q^{83} +6.48528 q^{86} +4.24264 q^{88} +5.48528 q^{89} -7.24264 q^{92} +13.2426 q^{94} -12.4853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 4 q^{13} + 2 q^{16} + 4 q^{19} - 6 q^{23} - 10 q^{25} - 4 q^{26} + 10 q^{31} - 2 q^{32} - 8 q^{37} - 4 q^{38} - 6 q^{41} + 4 q^{43} + 6 q^{46} - 18 q^{47} + 10 q^{50}+ \cdots - 8 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 4 * q^13 + 2 * q^16 + 4 * q^19 - 6 * q^23 - 10 * q^25 - 4 * q^26 + 10 * q^31 - 2 * q^32 - 8 * q^37 - 4 * q^38 - 6 * q^41 + 4 * q^43 + 6 * q^46 - 18 * q^47 + 10 * q^50 + 4 * q^52 + 4 * q^61 - 10 * q^62 + 2 * q^64 - 8 * q^67 + 6 * q^71 - 14 * q^73 + 8 * q^74 + 4 * q^76 + 10 * q^79 + 6 * q^82 - 24 * q^83 - 4 * q^86 - 6 * q^89 - 6 * q^92 + 18 * q^94 - 8 * q^97

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