LMFDB - Embedded newform 7600.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 380)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.73205 q^{3} +2.00000 q^{7} +4.46410 q^{9} +3.46410 q^{11} +2.73205 q^{13} +3.46410 q^{17} -1.00000 q^{19} +5.46410 q^{21} -3.46410 q^{23} +4.00000 q^{27} +3.46410 q^{29} +1.46410 q^{31} +9.46410 q^{33} -6.73205 q^{37} +7.46410 q^{39} -6.00000 q^{41} -4.92820 q^{43} +12.9282 q^{47} -3.00000 q^{49} +9.46410 q^{51} +10.7321 q^{53} -2.73205 q^{57} -6.92820 q^{59} +12.3923 q^{61} +8.92820 q^{63} +6.73205 q^{67} -9.46410 q^{69} +2.53590 q^{71} +0.535898 q^{73} +6.92820 q^{77} -2.92820 q^{79} -2.46410 q^{81} +3.46410 q^{83} +9.46410 q^{87} -15.4641 q^{89} +5.46410 q^{91} +4.00000 q^{93} +16.5885 q^{97} +15.4641 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 4 q^{7} + 2 q^{9} + 2 q^{13} - 2 q^{19} + 4 q^{21} + 8 q^{27} - 4 q^{31} + 12 q^{33} - 10 q^{37} + 8 q^{39} - 12 q^{41} + 4 q^{43} + 12 q^{47} - 6 q^{49} + 12 q^{51} + 18 q^{53} - 2 q^{57}+ \cdots + 24 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 4 * q^7 + 2 * q^9 + 2 * q^13 - 2 * q^19 + 4 * q^21 + 8 * q^27 - 4 * q^31 + 12 * q^33 - 10 * q^37 + 8 * q^39 - 12 * q^41 + 4 * q^43 + 12 * q^47 - 6 * q^49 + 12 * q^51 + 18 * q^53 - 2 * q^57 + 4 * q^61 + 4 * q^63 + 10 * q^67 - 12 * q^69 + 12 * q^71 + 8 * q^73 + 8 * q^79 + 2 * q^81 + 12 * q^87 - 24 * q^89 + 4 * q^91 + 8 * q^93 + 2 * q^97 + 24 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	2.73205	1.57735	0.788675	−	0.614810i	$-0.210767\pi$
$3$	2.73205	1.57735	0.788675	+	0.614810i	$0.210767\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	4.46410	1.48803
$10$	0	0
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	2.73205	0.757735	0.378867	−	0.925451i	$-0.376314\pi$
$13$	2.73205	0.757735	0.378867	+	0.925451i	$0.376314\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	5.46410	1.19236
$22$	0	0
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	0	0
$31$	1.46410	0.262960	0.131480	−	0.991319i	$-0.458027\pi$
$31$	1.46410	0.262960	0.131480	+	0.991319i	$0.458027\pi$
$32$	0	0
$33$	9.46410	1.64749
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.73205	−1.10674	−0.553371	−	0.832935i	$-0.686659\pi$
$37$	−6.73205	−1.10674	−0.553371	+	0.832935i	$0.686659\pi$
$38$	0	0
$39$	7.46410	1.19521
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−4.92820	−0.751544	−0.375772	−	0.926712i	$-0.622622\pi$
$43$	−4.92820	−0.751544	−0.375772	+	0.926712i	$0.622622\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.9282	1.88577	0.942886	−	0.333115i	$-0.108100\pi$
$47$	12.9282	1.88577	0.942886	+	0.333115i	$0.108100\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	9.46410	1.32524
$52$	0	0
$53$	10.7321	1.47416	0.737080	−	0.675805i	$-0.236204\pi$
$53$	10.7321	1.47416	0.737080	+	0.675805i	$0.236204\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.73205	−0.361869
$58$	0	0
$59$	−6.92820	−0.901975	−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820	−0.901975	−0.450988	+	0.892530i	$0.648928\pi$
$60$	0	0
$61$	12.3923	1.58667	0.793336	−	0.608784i	$-0.208342\pi$
$61$	12.3923	1.58667	0.793336	+	0.608784i	$0.208342\pi$
$62$	0	0
$63$	8.92820	1.12485
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.73205	0.822451	0.411225	−	0.911534i	$-0.365101\pi$
$67$	6.73205	0.822451	0.411225	+	0.911534i	$0.365101\pi$
$68$	0	0
$69$	−9.46410	−1.13934
$70$	0	0
$71$	2.53590	0.300956	0.150478	−	0.988613i	$-0.451919\pi$
$71$	2.53590	0.300956	0.150478	+	0.988613i	$0.451919\pi$
$72$	0	0
$73$	0.535898	0.0627222	0.0313611	−	0.999508i	$-0.490016\pi$
$73$	0.535898	0.0627222	0.0313611	+	0.999508i	$0.490016\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.92820	0.789542
$78$	0	0
$79$	−2.92820	−0.329449	−0.164724	−	0.986340i	$-0.552673\pi$
$79$	−2.92820	−0.329449	−0.164724	+	0.986340i	$0.552673\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	3.46410	0.380235	0.190117	−	0.981761i	$-0.439113\pi$
$83$	3.46410	0.380235	0.190117	+	0.981761i	$0.439113\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.46410	1.01466
$88$	0	0
$89$	−15.4641	−1.63919	−0.819596	−	0.572942i	$-0.805802\pi$
$89$	−15.4641	−1.63919	−0.819596	+	0.572942i	$0.805802\pi$
$90$	0	0
$91$	5.46410	0.572793
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.5885	1.68430	0.842151	−	0.539241i	$-0.181289\pi$
$97$	16.5885	1.68430	0.842151	+	0.539241i	$0.181289\pi$
$98$	0	0
$99$	15.4641	1.55420
$100$	0	0
$101$	−16.3923	−1.63110	−0.815548	−	0.578690i	$-0.803564\pi$
$101$	−16.3923	−1.63110	−0.815548	+	0.578690i	$0.803564\pi$
$102$	0	0
$103$	13.6603	1.34598	0.672992	−	0.739649i	$-0.265009\pi$
$103$	13.6603	1.34598	0.672992	+	0.739649i	$0.265009\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.66025	−0.547197	−0.273599	−	0.961844i	$-0.588214\pi$
$107$	−5.66025	−0.547197	−0.273599	+	0.961844i	$0.588214\pi$
$108$	0	0
$109$	−14.3923	−1.37853	−0.689266	−	0.724508i	$-0.742067\pi$
$109$	−14.3923	−1.37853	−0.689266	+	0.724508i	$0.742067\pi$
$110$	0	0
$111$	−18.3923	−1.74572
$112$	0	0
$113$	−12.5885	−1.18422	−0.592111	−	0.805856i	$-0.701705\pi$
$113$	−12.5885	−1.18422	−0.592111	+	0.805856i	$0.701705\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	12.1962	1.12753
$118$	0	0
$119$	6.92820	0.635107
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−16.3923	−1.47804
$124$	0	0
$125$	0	0
$126$	0	0
$127$	20.5885	1.82693	0.913465	−	0.406917i	$-0.133396\pi$
$127$	20.5885	1.82693	0.913465	+	0.406917i	$0.133396\pi$
$128$	0	0
$129$	−13.4641	−1.18545
$130$	0	0
$131$	18.9282	1.65376	0.826882	−	0.562375i	$-0.190112\pi$
$131$	18.9282	1.65376	0.826882	+	0.562375i	$0.190112\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−17.3205	−1.47979	−0.739895	−	0.672722i	$-0.765125\pi$
$137$	−17.3205	−1.47979	−0.739895	+	0.672722i	$0.765125\pi$
$138$	0	0
$139$	−11.4641	−0.972372	−0.486186	−	0.873855i	$-0.661612\pi$
$139$	−11.4641	−0.972372	−0.486186	+	0.873855i	$0.661612\pi$
$140$	0	0
$141$	35.3205	2.97452
$142$	0	0
$143$	9.46410	0.791428
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−8.19615	−0.676007
$148$	0	0
$149$	−4.39230	−0.359832	−0.179916	−	0.983682i	$-0.557583\pi$
$149$	−4.39230	−0.359832	−0.179916	+	0.983682i	$0.557583\pi$
$150$	0	0
$151$	8.39230	0.682956	0.341478	−	0.939890i	$-0.389073\pi$
$151$	8.39230	0.682956	0.341478	+	0.939890i	$0.389073\pi$
$152$	0	0
$153$	15.4641	1.25020
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−23.4641	−1.87264	−0.936320	−	0.351149i	$-0.885791\pi$
$157$	−23.4641	−1.87264	−0.936320	+	0.351149i	$0.885791\pi$
$158$	0	0
$159$	29.3205	2.32527
$160$	0	0
$161$	−6.92820	−0.546019
$162$	0	0
$163$	7.07180	0.553906	0.276953	−	0.960883i	$-0.410675\pi$
$163$	7.07180	0.553906	0.276953	+	0.960883i	$0.410675\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.7321	−0.830471	−0.415236	−	0.909714i	$-0.636301\pi$
$167$	−10.7321	−0.830471	−0.415236	+	0.909714i	$0.636301\pi$
$168$	0	0
$169$	−5.53590	−0.425838
$170$	0	0
$171$	−4.46410	−0.341378
$172$	0	0
$173$	−3.80385	−0.289201	−0.144601	−	0.989490i	$-0.546190\pi$
$173$	−3.80385	−0.289201	−0.144601	+	0.989490i	$0.546190\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−18.9282	−1.42273
$178$	0	0
$179$	−20.7846	−1.55351	−0.776757	−	0.629800i	$-0.783137\pi$
$179$	−20.7846	−1.55351	−0.776757	+	0.629800i	$0.783137\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	33.8564	2.50274
$184$	0	0
$185$	0	0
$186$	0	0
$187$	12.0000	0.877527
$188$	0	0
$189$	8.00000	0.581914
$190$	0	0
$191$	−6.92820	−0.501307	−0.250654	−	0.968077i	$-0.580646\pi$
$191$	−6.92820	−0.501307	−0.250654	+	0.968077i	$0.580646\pi$
$192$	0	0
$193$	7.80385	0.561733	0.280867	−	0.959747i	$-0.409378\pi$
$193$	7.80385	0.561733	0.280867	+	0.959747i	$0.409378\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.928203	−0.0661317	−0.0330659	−	0.999453i	$-0.510527\pi$
$197$	−0.928203	−0.0661317	−0.0330659	+	0.999453i	$0.510527\pi$
$198$	0	0
$199$	−26.9282	−1.90889	−0.954445	−	0.298387i	$-0.903551\pi$
$199$	−26.9282	−1.90889	−0.954445	+	0.298387i	$0.903551\pi$
$200$	0	0
$201$	18.3923	1.29729
$202$	0	0
$203$	6.92820	0.486265
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−15.4641	−1.07483
$208$	0	0
$209$	−3.46410	−0.239617
$210$	0	0
$211$	−19.3205	−1.33008	−0.665039	−	0.746808i	$-0.731585\pi$
$211$	−19.3205	−1.33008	−0.665039	+	0.746808i	$0.731585\pi$
$212$	0	0
$213$	6.92820	0.474713
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.92820	0.198779
$218$	0	0
$219$	1.46410	0.0989348
$220$	0	0
$221$	9.46410	0.636624
$222$	0	0
$223$	−9.66025	−0.646898	−0.323449	−	0.946246i	$-0.604842\pi$
$223$	−9.66025	−0.646898	−0.323449	+	0.946246i	$0.604842\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.19615	0.543998	0.271999	−	0.962298i	$-0.412315\pi$
$227$	8.19615	0.543998	0.271999	+	0.962298i	$0.412315\pi$
$228$	0	0
$229$	17.4641	1.15406	0.577030	−	0.816723i	$-0.304212\pi$
$229$	17.4641	1.15406	0.577030	+	0.816723i	$0.304212\pi$
$230$	0	0
$231$	18.9282	1.24538
$232$	0	0
$233$	0.928203	0.0608086	0.0304043	−	0.999538i	$-0.490321\pi$
$233$	0.928203	0.0608086	0.0304043	+	0.999538i	$0.490321\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.00000	−0.519656
$238$	0	0
$239$	13.8564	0.896296	0.448148	−	0.893959i	$-0.352084\pi$
$239$	13.8564	0.896296	0.448148	+	0.893959i	$0.352084\pi$
$240$	0	0
$241$	−11.8564	−0.763738	−0.381869	−	0.924216i	$-0.624719\pi$
$241$	−11.8564	−0.763738	−0.381869	+	0.924216i	$0.624719\pi$
$242$	0	0
$243$	−18.7321	−1.20166
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.73205	−0.173836
$248$	0	0
$249$	9.46410	0.599763
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.1244	0.943431	0.471716	−	0.881751i	$-0.343635\pi$
$257$	15.1244	0.943431	0.471716	+	0.881751i	$0.343635\pi$
$258$	0	0
$259$	−13.4641	−0.836619
$260$	0	0
$261$	15.4641	0.957204
$262$	0	0
$263$	8.53590	0.526346	0.263173	−	0.964749i	$-0.415231\pi$
$263$	8.53590	0.526346	0.263173	+	0.964749i	$0.415231\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−42.2487	−2.58558
$268$	0	0
$269$	10.3923	0.633630	0.316815	−	0.948487i	$-0.397387\pi$
$269$	10.3923	0.633630	0.316815	+	0.948487i	$0.397387\pi$
$270$	0	0
$271$	2.39230	0.145322	0.0726611	−	0.997357i	$-0.476851\pi$
$271$	2.39230	0.145322	0.0726611	+	0.997357i	$0.476851\pi$
$272$	0	0
$273$	14.9282	0.903496
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−18.3923	−1.10509	−0.552543	−	0.833484i	$-0.686343\pi$
$277$	−18.3923	−1.10509	−0.552543	+	0.833484i	$0.686343\pi$
$278$	0	0
$279$	6.53590	0.391294
$280$	0	0
$281$	0.928203	0.0553720	0.0276860	−	0.999617i	$-0.491186\pi$
$281$	0.928203	0.0553720	0.0276860	+	0.999617i	$0.491186\pi$
$282$	0	0
$283$	18.3923	1.09331	0.546655	−	0.837358i	$-0.315901\pi$
$283$	18.3923	1.09331	0.546655	+	0.837358i	$0.315901\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	45.3205	2.65674
$292$	0	0
$293$	5.66025	0.330676	0.165338	−	0.986237i	$-0.447129\pi$
$293$	5.66025	0.330676	0.165338	+	0.986237i	$0.447129\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	13.8564	0.804030
$298$	0	0
$299$	−9.46410	−0.547323
$300$	0	0
$301$	−9.85641	−0.568114
$302$	0	0
$303$	−44.7846	−2.57281
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−7.80385	−0.445389	−0.222695	−	0.974888i	$-0.571485\pi$
$307$	−7.80385	−0.445389	−0.222695	+	0.974888i	$0.571485\pi$
$308$	0	0
$309$	37.3205	2.12309
$310$	0	0
$311$	1.60770	0.0911640	0.0455820	−	0.998961i	$-0.485486\pi$
$311$	1.60770	0.0911640	0.0455820	+	0.998961i	$0.485486\pi$
$312$	0	0
$313$	−10.7846	−0.609582	−0.304791	−	0.952419i	$-0.598587\pi$
$313$	−10.7846	−0.609582	−0.304791	+	0.952419i	$0.598587\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.12436	−0.175481	−0.0877406	−	0.996143i	$-0.527965\pi$
$317$	−3.12436	−0.175481	−0.0877406	+	0.996143i	$0.527965\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	0	0
$321$	−15.4641	−0.863122
$322$	0	0
$323$	−3.46410	−0.192748
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−39.3205	−2.17443
$328$	0	0
$329$	25.8564	1.42551
$330$	0	0
$331$	22.2487	1.22290	0.611450	−	0.791283i	$-0.290587\pi$
$331$	22.2487	1.22290	0.611450	+	0.791283i	$0.290587\pi$
$332$	0	0
$333$	−30.0526	−1.64687
$334$	0	0
$335$	0	0
$336$	0	0
$337$	17.2679	0.940645	0.470323	−	0.882495i	$-0.344138\pi$
$337$	17.2679	0.940645	0.470323	+	0.882495i	$0.344138\pi$
$338$	0	0
$339$	−34.3923	−1.86793
$340$	0	0
$341$	5.07180	0.274653
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	31.1769	1.67366	0.836832	−	0.547459i	$-0.184405\pi$
$347$	31.1769	1.67366	0.836832	+	0.547459i	$0.184405\pi$
$348$	0	0
$349$	20.9282	1.12026	0.560131	−	0.828404i	$-0.310751\pi$
$349$	20.9282	1.12026	0.560131	+	0.828404i	$0.310751\pi$
$350$	0	0
$351$	10.9282	0.583304
$352$	0	0
$353$	−1.60770	−0.0855690	−0.0427845	−	0.999084i	$-0.513623\pi$
$353$	−1.60770	−0.0855690	−0.0427845	+	0.999084i	$0.513623\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	18.9282	1.00179
$358$	0	0
$359$	8.53590	0.450507	0.225254	−	0.974300i	$-0.427679\pi$
$359$	8.53590	0.450507	0.225254	+	0.974300i	$0.427679\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.73205	0.143395
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.85641	0.201303	0.100651	−	0.994922i	$-0.467907\pi$
$367$	3.85641	0.201303	0.100651	+	0.994922i	$0.467907\pi$
$368$	0	0
$369$	−26.7846	−1.39435
$370$	0	0
$371$	21.4641	1.11436
$372$	0	0
$373$	16.5885	0.858918	0.429459	−	0.903086i	$-0.358704\pi$
$373$	16.5885	0.858918	0.429459	+	0.903086i	$0.358704\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.46410	0.487426
$378$	0	0
$379$	−14.9282	−0.766810	−0.383405	−	0.923580i	$-0.625249\pi$
$379$	−14.9282	−0.766810	−0.383405	+	0.923580i	$0.625249\pi$
$380$	0	0
$381$	56.2487	2.88171
$382$	0	0
$383$	6.33975	0.323946	0.161973	−	0.986795i	$-0.448214\pi$
$383$	6.33975	0.323946	0.161973	+	0.986795i	$0.448214\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−22.0000	−1.11832
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	51.7128	2.60857
$394$	0	0
$395$	0	0
$396$	0	0
$397$	19.4641	0.976875	0.488438	−	0.872599i	$-0.337567\pi$
$397$	19.4641	0.976875	0.488438	+	0.872599i	$0.337567\pi$
$398$	0	0
$399$	−5.46410	−0.273547
$400$	0	0
$401$	38.7846	1.93681	0.968405	−	0.249381i	$-0.0802271\pi$
$401$	38.7846	1.93681	0.968405	+	0.249381i	$0.0802271\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−23.3205	−1.15595
$408$	0	0
$409$	−14.3923	−0.711654	−0.355827	−	0.934552i	$-0.615801\pi$
$409$	−14.3923	−0.711654	−0.355827	+	0.934552i	$0.615801\pi$
$410$	0	0
$411$	−47.3205	−2.33415
$412$	0	0
$413$	−13.8564	−0.681829
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−31.3205	−1.53377
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	0	0
$423$	57.7128	2.80609
$424$	0	0
$425$	0	0
$426$	0	0
$427$	24.7846	1.19941
$428$	0	0
$429$	25.8564	1.24836
$430$	0	0
$431$	33.4641	1.61191	0.805955	−	0.591977i	$-0.201653\pi$
$431$	33.4641	1.61191	0.805955	+	0.591977i	$0.201653\pi$
$432$	0	0
$433$	22.3397	1.07358	0.536790	−	0.843716i	$-0.319637\pi$
$433$	22.3397	1.07358	0.536790	+	0.843716i	$0.319637\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.46410	0.165710
$438$	0	0
$439$	31.7128	1.51357	0.756785	−	0.653664i	$-0.226769\pi$
$439$	31.7128	1.51357	0.756785	+	0.653664i	$0.226769\pi$
$440$	0	0
$441$	−13.3923	−0.637729
$442$	0	0
$443$	13.6077	0.646521	0.323261	−	0.946310i	$-0.395221\pi$
$443$	13.6077	0.646521	0.323261	+	0.946310i	$0.395221\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−12.0000	−0.567581
$448$	0	0
$449$	−3.46410	−0.163481	−0.0817405	−	0.996654i	$-0.526048\pi$
$449$	−3.46410	−0.163481	−0.0817405	+	0.996654i	$0.526048\pi$
$450$	0	0
$451$	−20.7846	−0.978709
$452$	0	0
$453$	22.9282	1.07726
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−34.7846	−1.62716	−0.813578	−	0.581456i	$-0.802483\pi$
$457$	−34.7846	−1.62716	−0.813578	+	0.581456i	$0.802483\pi$
$458$	0	0
$459$	13.8564	0.646762
$460$	0	0
$461$	−21.7128	−1.01127	−0.505633	−	0.862749i	$-0.668741\pi$
$461$	−21.7128	−1.01127	−0.505633	+	0.862749i	$0.668741\pi$
$462$	0	0
$463$	4.53590	0.210801	0.105401	−	0.994430i	$-0.466388\pi$
$463$	4.53590	0.210801	0.105401	+	0.994430i	$0.466388\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−29.3205	−1.35679	−0.678396	−	0.734697i	$-0.737324\pi$
$467$	−29.3205	−1.35679	−0.678396	+	0.734697i	$0.737324\pi$
$468$	0	0
$469$	13.4641	0.621714
$470$	0	0
$471$	−64.1051	−2.95381
$472$	0	0
$473$	−17.0718	−0.784962
$474$	0	0
$475$	0	0
$476$	0	0
$477$	47.9090	2.19360
$478$	0	0
$479$	22.3923	1.02313	0.511565	−	0.859244i	$-0.329066\pi$
$479$	22.3923	1.02313	0.511565	+	0.859244i	$0.329066\pi$
$480$	0	0
$481$	−18.3923	−0.838617
$482$	0	0
$483$	−18.9282	−0.861263
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−12.1962	−0.552660	−0.276330	−	0.961063i	$-0.589118\pi$
$487$	−12.1962	−0.552660	−0.276330	+	0.961063i	$0.589118\pi$
$488$	0	0
$489$	19.3205	0.873704
$490$	0	0
$491$	−18.9282	−0.854218	−0.427109	−	0.904200i	$-0.640468\pi$
$491$	−18.9282	−0.854218	−0.427109	+	0.904200i	$0.640468\pi$
$492$	0	0
$493$	12.0000	0.540453
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.07180	0.227501
$498$	0	0
$499$	−23.4641	−1.05040	−0.525199	−	0.850980i	$-0.676009\pi$
$499$	−23.4641	−1.05040	−0.525199	+	0.850980i	$0.676009\pi$
$500$	0	0
$501$	−29.3205	−1.30994
$502$	0	0
$503$	15.4641	0.689510	0.344755	−	0.938693i	$-0.387962\pi$
$503$	15.4641	0.689510	0.344755	+	0.938693i	$0.387962\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−15.1244	−0.671696
$508$	0	0
$509$	−5.32051	−0.235827	−0.117914	−	0.993024i	$-0.537621\pi$
$509$	−5.32051	−0.235827	−0.117914	+	0.993024i	$0.537621\pi$
$510$	0	0
$511$	1.07180	0.0474135
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	0	0
$516$	0	0
$517$	44.7846	1.96962
$518$	0	0
$519$	−10.3923	−0.456172
$520$	0	0
$521$	−14.7846	−0.647726	−0.323863	−	0.946104i	$-0.604982\pi$
$521$	−14.7846	−0.647726	−0.323863	+	0.946104i	$0.604982\pi$
$522$	0	0
$523$	−37.3731	−1.63421	−0.817105	−	0.576489i	$-0.804422\pi$
$523$	−37.3731	−1.63421	−0.817105	+	0.576489i	$0.804422\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.07180	0.220931
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	−30.9282	−1.34217
$532$	0	0
$533$	−16.3923	−0.710030
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−56.7846	−2.45044
$538$	0	0
$539$	−10.3923	−0.447628
$540$	0	0
$541$	−29.1769	−1.25441	−0.627207	−	0.778853i	$-0.715802\pi$
$541$	−29.1769	−1.25441	−0.627207	+	0.778853i	$0.715802\pi$
$542$	0	0
$543$	38.2487	1.64141
$544$	0	0
$545$	0	0
$546$	0	0
$547$	6.73205	0.287842	0.143921	−	0.989589i	$-0.454029\pi$
$547$	6.73205	0.287842	0.143921	+	0.989589i	$0.454029\pi$
$548$	0	0
$549$	55.3205	2.36102
$550$	0	0
$551$	−3.46410	−0.147576
$552$	0	0
$553$	−5.85641	−0.249040
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.32051	−0.225437	−0.112719	−	0.993627i	$-0.535956\pi$
$557$	−5.32051	−0.225437	−0.112719	+	0.993627i	$0.535956\pi$
$558$	0	0
$559$	−13.4641	−0.569471
$560$	0	0
$561$	32.7846	1.38417
$562$	0	0
$563$	44.1962	1.86265	0.931323	−	0.364195i	$-0.118656\pi$
$563$	44.1962	1.86265	0.931323	+	0.364195i	$0.118656\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−4.92820	−0.206965
$568$	0	0
$569$	−29.3205	−1.22918	−0.614590	−	0.788847i	$-0.710678\pi$
$569$	−29.3205	−1.22918	−0.614590	+	0.788847i	$0.710678\pi$
$570$	0	0
$571$	−18.3923	−0.769694	−0.384847	−	0.922980i	$-0.625746\pi$
$571$	−18.3923	−0.769694	−0.384847	+	0.922980i	$0.625746\pi$
$572$	0	0
$573$	−18.9282	−0.790737
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6.78461	0.282447	0.141223	−	0.989978i	$-0.454896\pi$
$577$	6.78461	0.282447	0.141223	+	0.989978i	$0.454896\pi$
$578$	0	0
$579$	21.3205	0.886050
$580$	0	0
$581$	6.92820	0.287430
$582$	0	0
$583$	37.1769	1.53971
$584$	0	0
$585$	0	0
$586$	0	0
$587$	43.1769	1.78210	0.891051	−	0.453903i	$-0.149969\pi$
$587$	43.1769	1.78210	0.891051	+	0.453903i	$0.149969\pi$
$588$	0	0
$589$	−1.46410	−0.0603273
$590$	0	0
$591$	−2.53590	−0.104313
$592$	0	0
$593$	−31.8564	−1.30819	−0.654093	−	0.756414i	$-0.726949\pi$
$593$	−31.8564	−1.30819	−0.654093	+	0.756414i	$0.726949\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−73.5692	−3.01099
$598$	0	0
$599$	20.7846	0.849236	0.424618	−	0.905373i	$-0.360408\pi$
$599$	20.7846	0.849236	0.424618	+	0.905373i	$0.360408\pi$
$600$	0	0
$601$	24.6410	1.00513	0.502564	−	0.864540i	$-0.332390\pi$
$601$	24.6410	1.00513	0.502564	+	0.864540i	$0.332390\pi$
$602$	0	0
$603$	30.0526	1.22383
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−39.9090	−1.61985	−0.809927	−	0.586530i	$-0.800494\pi$
$607$	−39.9090	−1.61985	−0.809927	+	0.586530i	$0.800494\pi$
$608$	0	0
$609$	18.9282	0.767010
$610$	0	0
$611$	35.3205	1.42891
$612$	0	0
$613$	−6.39230	−0.258183	−0.129091	−	0.991633i	$-0.541206\pi$
$613$	−6.39230	−0.258183	−0.129091	+	0.991633i	$0.541206\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.60770	−0.0647234	−0.0323617	−	0.999476i	$-0.510303\pi$
$617$	−1.60770	−0.0647234	−0.0323617	+	0.999476i	$0.510303\pi$
$618$	0	0
$619$	2.39230	0.0961549	0.0480774	−	0.998844i	$-0.484691\pi$
$619$	2.39230	0.0961549	0.0480774	+	0.998844i	$0.484691\pi$
$620$	0	0
$621$	−13.8564	−0.556038
$622$	0	0
$623$	−30.9282	−1.23911
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−9.46410	−0.377960
$628$	0	0
$629$	−23.3205	−0.929850
$630$	0	0
$631$	−11.4641	−0.456379	−0.228189	−	0.973617i	$-0.573281\pi$
$631$	−11.4641	−0.456379	−0.228189	+	0.973617i	$0.573281\pi$
$632$	0	0
$633$	−52.7846	−2.09800
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−8.19615	−0.324743
$638$	0	0
$639$	11.3205	0.447832
$640$	0	0
$641$	−24.9282	−0.984605	−0.492302	−	0.870424i	$-0.663845\pi$
$641$	−24.9282	−0.984605	−0.492302	+	0.870424i	$0.663845\pi$
$642$	0	0
$643$	−14.3923	−0.567577	−0.283789	−	0.958887i	$-0.591591\pi$
$643$	−14.3923	−0.567577	−0.283789	+	0.958887i	$0.591591\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.4641	0.607957	0.303978	−	0.952679i	$-0.401685\pi$
$647$	15.4641	0.607957	0.303978	+	0.952679i	$0.401685\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	8.00000	0.313545
$652$	0	0
$653$	−27.4641	−1.07475	−0.537377	−	0.843342i	$-0.680585\pi$
$653$	−27.4641	−1.07475	−0.537377	+	0.843342i	$0.680585\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.39230	0.0933327
$658$	0	0
$659$	−8.78461	−0.342200	−0.171100	−	0.985254i	$-0.554732\pi$
$659$	−8.78461	−0.342200	−0.171100	+	0.985254i	$0.554732\pi$
$660$	0	0
$661$	5.21539	0.202855	0.101428	−	0.994843i	$-0.467659\pi$
$661$	5.21539	0.202855	0.101428	+	0.994843i	$0.467659\pi$
$662$	0	0
$663$	25.8564	1.00418
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12.0000	−0.464642
$668$	0	0
$669$	−26.3923	−1.02039
$670$	0	0
$671$	42.9282	1.65722
$672$	0	0
$673$	50.7321	1.95558	0.977788	−	0.209594i	$-0.0672143\pi$
$673$	50.7321	1.95558	0.977788	+	0.209594i	$0.0672143\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−16.9808	−0.652624	−0.326312	−	0.945262i	$-0.605806\pi$
$677$	−16.9808	−0.652624	−0.326312	+	0.945262i	$0.605806\pi$
$678$	0	0
$679$	33.1769	1.27321
$680$	0	0
$681$	22.3923	0.858075
$682$	0	0
$683$	−17.6603	−0.675751	−0.337875	−	0.941191i	$-0.609708\pi$
$683$	−17.6603	−0.675751	−0.337875	+	0.941191i	$0.609708\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	47.7128	1.82036
$688$	0	0
$689$	29.3205	1.11702
$690$	0	0
$691$	−51.1769	−1.94686	−0.973431	−	0.228981i	$-0.926460\pi$
$691$	−51.1769	−1.94686	−0.973431	+	0.228981i	$0.926460\pi$
$692$	0	0
$693$	30.9282	1.17487
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−20.7846	−0.787273
$698$	0	0
$699$	2.53590	0.0959165
$700$	0	0
$701$	−26.5359	−1.00225	−0.501124	−	0.865376i	$-0.667080\pi$
$701$	−26.5359	−1.00225	−0.501124	+	0.865376i	$0.667080\pi$
$702$	0	0
$703$	6.73205	0.253904
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−32.7846	−1.23299
$708$	0	0
$709$	−30.7846	−1.15614	−0.578070	−	0.815987i	$-0.696194\pi$
$709$	−30.7846	−1.15614	−0.578070	+	0.815987i	$0.696194\pi$
$710$	0	0
$711$	−13.0718	−0.490231
$712$	0	0
$713$	−5.07180	−0.189940
$714$	0	0
$715$	0	0
$716$	0	0
$717$	37.8564	1.41377
$718$	0	0
$719$	−17.3205	−0.645946	−0.322973	−	0.946408i	$-0.604682\pi$
$719$	−17.3205	−0.645946	−0.322973	+	0.946408i	$0.604682\pi$
$720$	0	0
$721$	27.3205	1.01747
$722$	0	0
$723$	−32.3923	−1.20468
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0.143594	0.00532559	0.00266279	−	0.999996i	$-0.499152\pi$
$727$	0.143594	0.00532559	0.00266279	+	0.999996i	$0.499152\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	−17.0718	−0.631423
$732$	0	0
$733$	−10.7846	−0.398339	−0.199169	−	0.979965i	$-0.563824\pi$
$733$	−10.7846	−0.398339	−0.199169	+	0.979965i	$0.563824\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	23.3205	0.859022
$738$	0	0
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	0	0
$741$	−7.46410	−0.274201
$742$	0	0
$743$	20.8756	0.765853	0.382927	−	0.923779i	$-0.374916\pi$
$743$	20.8756	0.765853	0.382927	+	0.923779i	$0.374916\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	15.4641	0.565802
$748$	0	0
$749$	−11.3205	−0.413642
$750$	0	0
$751$	25.4641	0.929198	0.464599	−	0.885521i	$-0.346198\pi$
$751$	25.4641	0.929198	0.464599	+	0.885521i	$0.346198\pi$
$752$	0	0
$753$	65.5692	2.38948
$754$	0	0
$755$	0	0
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	0	0
$759$	−32.7846	−1.19001
$760$	0	0
$761$	19.6077	0.710778	0.355389	−	0.934718i	$-0.384348\pi$
$761$	19.6077	0.710778	0.355389	+	0.934718i	$0.384348\pi$
$762$	0	0
$763$	−28.7846	−1.04207
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−18.9282	−0.683458
$768$	0	0
$769$	−30.5359	−1.10115	−0.550576	−	0.834785i	$-0.685592\pi$
$769$	−30.5359	−1.10115	−0.550576	+	0.834785i	$0.685592\pi$
$770$	0	0
$771$	41.3205	1.48812
$772$	0	0
$773$	−22.0526	−0.793175	−0.396588	−	0.917997i	$-0.629806\pi$
$773$	−22.0526	−0.793175	−0.396588	+	0.917997i	$0.629806\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−36.7846	−1.31964
$778$	0	0
$779$	6.00000	0.214972
$780$	0	0
$781$	8.78461	0.314338
$782$	0	0
$783$	13.8564	0.495188
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−12.1962	−0.434746	−0.217373	−	0.976089i	$-0.569749\pi$
$787$	−12.1962	−0.434746	−0.217373	+	0.976089i	$0.569749\pi$
$788$	0	0
$789$	23.3205	0.830232
$790$	0	0
$791$	−25.1769	−0.895188
$792$	0	0
$793$	33.8564	1.20228
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.66025	0.200496	0.100248	−	0.994962i	$-0.468036\pi$
$797$	5.66025	0.200496	0.100248	+	0.994962i	$0.468036\pi$
$798$	0	0
$799$	44.7846	1.58437
$800$	0	0
$801$	−69.0333	−2.43917
$802$	0	0
$803$	1.85641	0.0655112
$804$	0	0
$805$	0	0
$806$	0	0
$807$	28.3923	0.999456
$808$	0	0
$809$	9.71281	0.341484	0.170742	−	0.985316i	$-0.445383\pi$
$809$	9.71281	0.341484	0.170742	+	0.985316i	$0.445383\pi$
$810$	0	0
$811$	−15.6077	−0.548060	−0.274030	−	0.961721i	$-0.588357\pi$
$811$	−15.6077	−0.548060	−0.274030	+	0.961721i	$0.588357\pi$
$812$	0	0
$813$	6.53590	0.229224
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.92820	0.172416
$818$	0	0
$819$	24.3923	0.852336
$820$	0	0
$821$	−16.1436	−0.563415	−0.281708	−	0.959500i	$-0.590901\pi$
$821$	−16.1436	−0.563415	−0.281708	+	0.959500i	$0.590901\pi$
$822$	0	0
$823$	−39.5692	−1.37930	−0.689648	−	0.724145i	$-0.742235\pi$
$823$	−39.5692	−1.37930	−0.689648	+	0.724145i	$0.742235\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−29.6603	−1.03139	−0.515694	−	0.856773i	$-0.672466\pi$
$827$	−29.6603	−1.03139	−0.515694	+	0.856773i	$0.672466\pi$
$828$	0	0
$829$	8.24871	0.286490	0.143245	−	0.989687i	$-0.454246\pi$
$829$	8.24871	0.286490	0.143245	+	0.989687i	$0.454246\pi$
$830$	0	0
$831$	−50.2487	−1.74311
$832$	0	0
$833$	−10.3923	−0.360072
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5.85641	0.202427
$838$	0	0
$839$	−18.9282	−0.653474	−0.326737	−	0.945115i	$-0.605949\pi$
$839$	−18.9282	−0.653474	−0.326737	+	0.945115i	$0.605949\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	2.53590	0.0873410
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	50.2487	1.72453
$850$	0	0
$851$	23.3205	0.799417
$852$	0	0
$853$	44.6410	1.52848	0.764240	−	0.644932i	$-0.223114\pi$
$853$	44.6410	1.52848	0.764240	+	0.644932i	$0.223114\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.80385	−0.129937	−0.0649685	−	0.997887i	$-0.520695\pi$
$857$	−3.80385	−0.129937	−0.0649685	+	0.997887i	$0.520695\pi$
$858$	0	0
$859$	−47.7128	−1.62794	−0.813970	−	0.580907i	$-0.802698\pi$
$859$	−47.7128	−1.62794	−0.813970	+	0.580907i	$0.802698\pi$
$860$	0	0
$861$	−32.7846	−1.11730
$862$	0	0
$863$	−37.2679	−1.26862	−0.634308	−	0.773081i	$-0.718715\pi$
$863$	−37.2679	−1.26862	−0.634308	+	0.773081i	$0.718715\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−13.6603	−0.463927
$868$	0	0
$869$	−10.1436	−0.344098
$870$	0	0
$871$	18.3923	0.623199
$872$	0	0
$873$	74.0526	2.50630
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−0.980762	−0.0331180	−0.0165590	−	0.999863i	$-0.505271\pi$
$877$	−0.980762	−0.0331180	−0.0165590	+	0.999863i	$0.505271\pi$
$878$	0	0
$879$	15.4641	0.521591
$880$	0	0
$881$	−0.679492	−0.0228927	−0.0114463	−	0.999934i	$-0.503644\pi$
$881$	−0.679492	−0.0228927	−0.0114463	+	0.999934i	$0.503644\pi$
$882$	0	0
$883$	−22.0000	−0.740359	−0.370179	−	0.928960i	$-0.620704\pi$
$883$	−22.0000	−0.740359	−0.370179	+	0.928960i	$0.620704\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	23.4115	0.786083	0.393041	−	0.919521i	$-0.371423\pi$
$887$	23.4115	0.786083	0.393041	+	0.919521i	$0.371423\pi$
$888$	0	0
$889$	41.1769	1.38103
$890$	0	0
$891$	−8.53590	−0.285963
$892$	0	0
$893$	−12.9282	−0.432626
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−25.8564	−0.863320
$898$	0	0
$899$	5.07180	0.169154
$900$	0	0
$901$	37.1769	1.23854
$902$	0	0
$903$	−26.9282	−0.896114
$904$	0	0
$905$	0	0
$906$	0	0
$907$	7.41154	0.246096	0.123048	−	0.992401i	$-0.460733\pi$
$907$	7.41154	0.246096	0.123048	+	0.992401i	$0.460733\pi$
$908$	0	0
$909$	−73.1769	−2.42713
$910$	0	0
$911$	−14.5359	−0.481596	−0.240798	−	0.970575i	$-0.577409\pi$
$911$	−14.5359	−0.481596	−0.240798	+	0.970575i	$0.577409\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	0	0
$916$	0	0
$917$	37.8564	1.25013
$918$	0	0
$919$	−44.4974	−1.46783	−0.733917	−	0.679239i	$-0.762310\pi$
$919$	−44.4974	−1.46783	−0.733917	+	0.679239i	$0.762310\pi$
$920$	0	0
$921$	−21.3205	−0.702535
$922$	0	0
$923$	6.92820	0.228045
$924$	0	0
$925$	0	0
$926$	0	0
$927$	60.9808	2.00287
$928$	0	0
$929$	−47.5692	−1.56070	−0.780348	−	0.625346i	$-0.784958\pi$
$929$	−47.5692	−1.56070	−0.780348	+	0.625346i	$0.784958\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	0	0
$933$	4.39230	0.143798
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−51.1769	−1.67188	−0.835938	−	0.548823i	$-0.815076\pi$
$937$	−51.1769	−1.67188	−0.835938	+	0.548823i	$0.815076\pi$
$938$	0	0
$939$	−29.4641	−0.961525
$940$	0	0
$941$	52.6410	1.71605	0.858024	−	0.513610i	$-0.171692\pi$
$941$	52.6410	1.71605	0.858024	+	0.513610i	$0.171692\pi$
$942$	0	0
$943$	20.7846	0.676840
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.67949	0.217054	0.108527	−	0.994093i	$-0.465387\pi$
$947$	6.67949	0.217054	0.108527	+	0.994093i	$0.465387\pi$
$948$	0	0
$949$	1.46410	0.0475267
$950$	0	0
$951$	−8.53590	−0.276795
$952$	0	0
$953$	−60.5885	−1.96265	−0.981326	−	0.192350i	$-0.938389\pi$
$953$	−60.5885	−1.96265	−0.981326	+	0.192350i	$0.938389\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	32.7846	1.05978
$958$	0	0
$959$	−34.6410	−1.11862
$960$	0	0
$961$	−28.8564	−0.930852
$962$	0	0
$963$	−25.2679	−0.814248
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−45.3205	−1.45741	−0.728705	−	0.684828i	$-0.759877\pi$
$967$	−45.3205	−1.45741	−0.728705	+	0.684828i	$0.759877\pi$
$968$	0	0
$969$	−9.46410	−0.304031
$970$	0	0
$971$	14.5359	0.466479	0.233240	−	0.972419i	$-0.425067\pi$
$971$	14.5359	0.466479	0.233240	+	0.972419i	$0.425067\pi$
$972$	0	0
$973$	−22.9282	−0.735044
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−32.1962	−1.03005	−0.515023	−	0.857176i	$-0.672217\pi$
$977$	−32.1962	−1.03005	−0.515023	+	0.857176i	$0.672217\pi$
$978$	0	0
$979$	−53.5692	−1.71208
$980$	0	0
$981$	−64.2487	−2.05130
$982$	0	0
$983$	−2.44486	−0.0779790	−0.0389895	−	0.999240i	$-0.512414\pi$
$983$	−2.44486	−0.0779790	−0.0389895	+	0.999240i	$0.512414\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	70.6410	2.24853
$988$	0	0
$989$	17.0718	0.542852
$990$	0	0
$991$	8.39230	0.266590	0.133295	−	0.991076i	$-0.457444\pi$
$991$	8.39230	0.266590	0.133295	+	0.991076i	$0.457444\pi$
$992$	0	0
$993$	60.7846	1.92894
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−30.3923	−0.962534	−0.481267	−	0.876574i	$-0.659823\pi$
$997$	−30.3923	−0.962534	−0.481267	+	0.876574i	$0.659823\pi$
$998$	0	0
$999$	−26.9282	−0.851971

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.bf.1.2		2
4.3	odd	2		1900.2.a.d.1.1		2
5.4	even	2		1520.2.a.l.1.1		2
20.3	even	4		1900.2.c.e.1749.1		4
20.7	even	4		1900.2.c.e.1749.4		4
20.19	odd	2		380.2.a.d.1.2	✓	2
40.19	odd	2		6080.2.a.z.1.1		2
40.29	even	2		6080.2.a.bj.1.2		2
60.59	even	2		3420.2.a.h.1.2		2
380.379	even	2		7220.2.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.a.d.1.2	✓	2	20.19	odd	2
1520.2.a.l.1.1		2	5.4	even	2
1900.2.a.d.1.1		2	4.3	odd	2
1900.2.c.e.1749.1		4	20.3	even	4
1900.2.c.e.1749.4		4	20.7	even	4
3420.2.a.h.1.2		2	60.59	even	2
6080.2.a.z.1.1		2	40.19	odd	2
6080.2.a.bj.1.2		2	40.29	even	2
7220.2.a.h.1.1		2	380.379	even	2
7600.2.a.bf.1.2		2	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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q+2.73205 q^{3} +2.00000 q^{7} +4.46410 q^{9} +3.46410 q^{11} +2.73205 q^{13} +3.46410 q^{17} -1.00000 q^{19} +5.46410 q^{21} -3.46410 q^{23} +4.00000 q^{27} +3.46410 q^{29} +1.46410 q^{31} +9.46410 q^{33} -6.73205 q^{37} +7.46410 q^{39} -6.00000 q^{41} -4.92820 q^{43} +12.9282 q^{47} -3.00000 q^{49} +9.46410 q^{51} +10.7321 q^{53} -2.73205 q^{57} -6.92820 q^{59} +12.3923 q^{61} +8.92820 q^{63} +6.73205 q^{67} -9.46410 q^{69} +2.53590 q^{71} +0.535898 q^{73} +6.92820 q^{77} -2.92820 q^{79} -2.46410 q^{81} +3.46410 q^{83} +9.46410 q^{87} -15.4641 q^{89} +5.46410 q^{91} +4.00000 q^{93} +16.5885 q^{97} +15.4641 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 4 q^{7} + 2 q^{9} + 2 q^{13} - 2 q^{19} + 4 q^{21} + 8 q^{27} - 4 q^{31} + 12 q^{33} - 10 q^{37} + 8 q^{39} - 12 q^{41} + 4 q^{43} + 12 q^{47} - 6 q^{49} + 12 q^{51} + 18 q^{53} - 2 q^{57}+ \cdots + 24 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 4 * q^7 + 2 * q^9 + 2 * q^13 - 2 * q^19 + 4 * q^21 + 8 * q^27 - 4 * q^31 + 12 * q^33 - 10 * q^37 + 8 * q^39 - 12 * q^41 + 4 * q^43 + 12 * q^47 - 6 * q^49 + 12 * q^51 + 18 * q^53 - 2 * q^57 + 4 * q^61 + 4 * q^63 + 10 * q^67 - 12 * q^69 + 12 * q^71 + 8 * q^73 + 8 * q^79 + 2 * q^81 + 12 * q^87 - 24 * q^89 + 4 * q^91 + 8 * q^93 + 2 * q^97 + 24 * q^99

Introduction

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