LMFDB - Embedded newform 7488.2.a.dc.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7488,2,Mod(1,7488)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7488.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7488, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7488 = 2^{6} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7488.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,8,0,4,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$59.7919810335$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 3744)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.93185$ of defining polynomial
Character	$\chi$	$=$	7488.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.86370 q^{5} +1.03528 q^{7} +5.46410 q^{11} +1.00000 q^{13} +5.65685 q^{17} -6.69213 q^{19} +6.92820 q^{23} +9.92820 q^{25} +7.72741 q^{29} +1.03528 q^{31} +4.00000 q^{35} +2.00000 q^{37} +3.86370 q^{41} -5.65685 q^{43} -9.46410 q^{47} -5.92820 q^{49} -5.65685 q^{53} +21.1117 q^{55} +2.53590 q^{59} -4.92820 q^{61} +3.86370 q^{65} +6.69213 q^{67} -9.46410 q^{71} -15.8564 q^{73} +5.65685 q^{77} +13.4641 q^{83} +21.8564 q^{85} +0.277401 q^{89} +1.03528 q^{91} -25.8564 q^{95} +6.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{11} + 4 q^{13} + 12 q^{25} + 16 q^{35} + 8 q^{37} - 24 q^{47} + 4 q^{49} + 24 q^{59} + 8 q^{61} - 24 q^{71} - 8 q^{73} + 40 q^{83} + 32 q^{85} - 48 q^{95} + 24 q^{97}+O(q^{100}) $ 4 * q + 8 * q^11 + 4 * q^13 + 12 * q^25 + 16 * q^35 + 8 * q^37 - 24 * q^47 + 4 * q^49 + 24 * q^59 + 8 * q^61 - 24 * q^71 - 8 * q^73 + 40 * q^83 + 32 * q^85 - 48 * q^95 + 24 * 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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.86370	1.72790	0.863950	−	0.503577i	$-0.167983\pi$
$5$	3.86370	1.72790	0.863950	+	0.503577i	$0.167983\pi$
$6$	0	0
$7$	1.03528	0.391298	0.195649	−	0.980674i	$-0.437319\pi$
$7$	1.03528	0.391298	0.195649	+	0.980674i	$0.437319\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.46410	1.64749	0.823744	−	0.566961i	$-0.191881\pi$
$11$	5.46410	1.64749	0.823744	+	0.566961i	$0.191881\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.65685	1.37199	0.685994	−	0.727607i	$-0.259367\pi$
$17$	5.65685	1.37199	0.685994	+	0.727607i	$0.259367\pi$
$18$	0	0
$19$	−6.69213	−1.53528	−0.767640	−	0.640881i	$-0.778569\pi$
$19$	−6.69213	−1.53528	−0.767640	+	0.640881i	$0.778569\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.92820	1.44463	0.722315	−	0.691564i	$-0.243078\pi$
$23$	6.92820	1.44463	0.722315	+	0.691564i	$0.243078\pi$
$24$	0	0
$25$	9.92820	1.98564
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.72741	1.43494	0.717472	−	0.696588i	$-0.245299\pi$
$29$	7.72741	1.43494	0.717472	+	0.696588i	$0.245299\pi$
$30$	0	0
$31$	1.03528	0.185941	0.0929705	−	0.995669i	$-0.470364\pi$
$31$	1.03528	0.185941	0.0929705	+	0.995669i	$0.470364\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.00000	0.676123
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.86370	0.603409	0.301705	−	0.953401i	$-0.402444\pi$
$41$	3.86370	0.603409	0.301705	+	0.953401i	$0.402444\pi$
$42$	0	0
$43$	−5.65685	−0.862662	−0.431331	−	0.902194i	$-0.641956\pi$
$43$	−5.65685	−0.862662	−0.431331	+	0.902194i	$0.641956\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.46410	−1.38048	−0.690241	−	0.723580i	$-0.742495\pi$
$47$	−9.46410	−1.38048	−0.690241	+	0.723580i	$0.742495\pi$
$48$	0	0
$49$	−5.92820	−0.846886
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.65685	−0.777029	−0.388514	−	0.921443i	$-0.627012\pi$
$53$	−5.65685	−0.777029	−0.388514	+	0.921443i	$0.627012\pi$
$54$	0	0
$55$	21.1117	2.84670
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.53590	0.330146	0.165073	−	0.986281i	$-0.447214\pi$
$59$	2.53590	0.330146	0.165073	+	0.986281i	$0.447214\pi$
$60$	0	0
$61$	−4.92820	−0.630992	−0.315496	−	0.948927i	$-0.602171\pi$
$61$	−4.92820	−0.630992	−0.315496	+	0.948927i	$0.602171\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.86370	0.479233
$66$	0	0
$67$	6.69213	0.817574	0.408787	−	0.912630i	$-0.365952\pi$
$67$	6.69213	0.817574	0.408787	+	0.912630i	$0.365952\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.46410	−1.12318	−0.561591	−	0.827415i	$-0.689811\pi$
$71$	−9.46410	−1.12318	−0.561591	+	0.827415i	$0.689811\pi$
$72$	0	0
$73$	−15.8564	−1.85585	−0.927926	−	0.372764i	$-0.878410\pi$
$73$	−15.8564	−1.85585	−0.927926	+	0.372764i	$0.878410\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.65685	0.644658
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.4641	1.47788	0.738939	−	0.673773i	$-0.235327\pi$
$83$	13.4641	1.47788	0.738939	+	0.673773i	$0.235327\pi$
$84$	0	0
$85$	21.8564	2.37066
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0.277401	0.0294045	0.0147022	−	0.999892i	$-0.495320\pi$
$89$	0.277401	0.0294045	0.0147022	+	0.999892i	$0.495320\pi$
$90$	0	0
$91$	1.03528	0.108526
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−25.8564	−2.65281
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.07055	0.206028	0.103014	−	0.994680i	$-0.467151\pi$
$101$	2.07055	0.206028	0.103014	+	0.994680i	$0.467151\pi$
$102$	0	0
$103$	−16.9706	−1.67216	−0.836080	−	0.548608i	$-0.815158\pi$
$103$	−16.9706	−1.67216	−0.836080	+	0.548608i	$0.815158\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.92820	−0.669775	−0.334887	−	0.942258i	$-0.608698\pi$
$107$	−6.92820	−0.669775	−0.334887	+	0.942258i	$0.608698\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−9.24316	−0.869523	−0.434761	−	0.900546i	$-0.643167\pi$
$113$	−9.24316	−0.869523	−0.434761	+	0.900546i	$0.643167\pi$
$114$	0	0
$115$	26.7685	2.49618
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.85641	0.536856
$120$	0	0
$121$	18.8564	1.71422
$122$	0	0
$123$	0	0
$124$	0	0
$125$	19.0411	1.70309
$126$	0	0
$127$	−17.5254	−1.55512	−0.777562	−	0.628806i	$-0.783544\pi$
$127$	−17.5254	−1.55512	−0.777562	+	0.628806i	$0.783544\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.07180	0.0936433	0.0468217	−	0.998903i	$-0.485091\pi$
$131$	1.07180	0.0936433	0.0468217	+	0.998903i	$0.485091\pi$
$132$	0	0
$133$	−6.92820	−0.600751
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.3185	1.65049	0.825246	−	0.564773i	$-0.191036\pi$
$137$	19.3185	1.65049	0.825246	+	0.564773i	$0.191036\pi$
$138$	0	0
$139$	−3.58630	−0.304186	−0.152093	−	0.988366i	$-0.548601\pi$
$139$	−3.58630	−0.304186	−0.152093	+	0.988366i	$0.548601\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.46410	0.456931
$144$	0	0
$145$	29.8564	2.47944
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−23.4596	−1.92189	−0.960944	−	0.276745i	$-0.910744\pi$
$149$	−23.4596	−1.92189	−0.960944	+	0.276745i	$0.910744\pi$
$150$	0	0
$151$	−16.4901	−1.34194	−0.670972	−	0.741482i	$-0.734123\pi$
$151$	−16.4901	−1.34194	−0.670972	+	0.741482i	$0.734123\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	16.9282	1.35102	0.675509	−	0.737352i	$-0.263924\pi$
$157$	16.9282	1.35102	0.675509	+	0.737352i	$0.263924\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.17260	0.565280
$162$	0	0
$163$	−12.9038	−1.01070	−0.505351	−	0.862914i	$-0.668637\pi$
$163$	−12.9038	−1.01070	−0.505351	+	0.862914i	$0.668637\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.46410	−0.732354	−0.366177	−	0.930545i	$-0.619334\pi$
$167$	−9.46410	−0.732354	−0.366177	+	0.930545i	$0.619334\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.21166	0.472264	0.236132	−	0.971721i	$-0.424120\pi$
$173$	6.21166	0.472264	0.236132	+	0.971721i	$0.424120\pi$
$174$	0	0
$175$	10.2784	0.776976
$176$	0	0
$177$	0	0
$178$	0	0
$179$	17.8564	1.33465	0.667325	−	0.744766i	$-0.267439\pi$
$179$	17.8564	1.33465	0.667325	+	0.744766i	$0.267439\pi$
$180$	0	0
$181$	−22.7846	−1.69357	−0.846783	−	0.531938i	$-0.821464\pi$
$181$	−22.7846	−1.69357	−0.846783	+	0.531938i	$0.821464\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.72741	0.568130
$186$	0	0
$187$	30.9096	2.26034
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.5911	−0.825832	−0.412916	−	0.910769i	$-0.635490\pi$
$197$	−11.5911	−0.825832	−0.412916	+	0.910769i	$0.635490\pi$
$198$	0	0
$199$	13.3843	0.948785	0.474393	−	0.880313i	$-0.342668\pi$
$199$	13.3843	0.948785	0.474393	+	0.880313i	$0.342668\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.00000	0.561490
$204$	0	0
$205$	14.9282	1.04263
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−36.5665	−2.52936
$210$	0	0
$211$	−15.4548	−1.06395	−0.531977	−	0.846759i	$-0.678551\pi$
$211$	−15.4548	−1.06395	−0.531977	+	0.846759i	$0.678551\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−21.8564	−1.49059
$216$	0	0
$217$	1.07180	0.0727583
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.65685	0.380521
$222$	0	0
$223$	−18.5606	−1.24291	−0.621456	−	0.783449i	$-0.713459\pi$
$223$	−18.5606	−1.24291	−0.621456	+	0.783449i	$0.713459\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.53590	0.168313	0.0841567	−	0.996453i	$-0.473180\pi$
$227$	2.53590	0.168313	0.0841567	+	0.996453i	$0.473180\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.58630	0.234946	0.117473	−	0.993076i	$-0.462521\pi$
$233$	3.58630	0.234946	0.117473	+	0.993076i	$0.462521\pi$
$234$	0	0
$235$	−36.5665	−2.38533
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3.60770	−0.233362	−0.116681	−	0.993169i	$-0.537226\pi$
$239$	−3.60770	−0.233362	−0.116681	+	0.993169i	$0.537226\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−22.9048	−1.46334
$246$	0	0
$247$	−6.69213	−0.425810
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.7846	−1.56439	−0.782195	−	0.623033i	$-0.785900\pi$
$251$	−24.7846	−1.56439	−0.782195	+	0.623033i	$0.785900\pi$
$252$	0	0
$253$	37.8564	2.38001
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.8685	−0.740337	−0.370169	−	0.928965i	$-0.620700\pi$
$257$	−11.8685	−0.740337	−0.370169	+	0.928965i	$0.620700\pi$
$258$	0	0
$259$	2.07055	0.128658
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.92820	−0.180561	−0.0902804	−	0.995916i	$-0.528776\pi$
$263$	−2.92820	−0.180561	−0.0902804	+	0.995916i	$0.528776\pi$
$264$	0	0
$265$	−21.8564	−1.34263
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.58630	0.218661	0.109330	−	0.994005i	$-0.465129\pi$
$269$	3.58630	0.218661	0.109330	+	0.994005i	$0.465129\pi$
$270$	0	0
$271$	−29.8744	−1.81474	−0.907369	−	0.420335i	$-0.861912\pi$
$271$	−29.8744	−1.81474	−0.907369	+	0.420335i	$0.861912\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	54.2487	3.27132
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−0.277401	−0.0165484	−0.00827419	−	0.999966i	$-0.502634\pi$
$281$	−0.277401	−0.0165484	−0.00827419	+	0.999966i	$0.502634\pi$
$282$	0	0
$283$	24.6980	1.46814	0.734071	−	0.679073i	$-0.237618\pi$
$283$	24.6980	1.46814	0.734071	+	0.679073i	$0.237618\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	15.0000	0.882353
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.45001	0.435234	0.217617	−	0.976034i	$-0.430172\pi$
$293$	7.45001	0.435234	0.217617	+	0.976034i	$0.430172\pi$
$294$	0	0
$295$	9.79796	0.570459
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.92820	0.400668
$300$	0	0
$301$	−5.85641	−0.337558
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−19.0411	−1.09029
$306$	0	0
$307$	−26.2880	−1.50034	−0.750169	−	0.661246i	$-0.770028\pi$
$307$	−26.2880	−1.50034	−0.750169	+	0.661246i	$0.770028\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6.92820	0.392862	0.196431	−	0.980518i	$-0.437065\pi$
$311$	6.92820	0.392862	0.196431	+	0.980518i	$0.437065\pi$
$312$	0	0
$313$	−19.8564	−1.12235	−0.561175	−	0.827697i	$-0.689651\pi$
$313$	−19.8564	−1.12235	−0.561175	+	0.827697i	$0.689651\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.1459	0.682182	0.341091	−	0.940030i	$-0.389203\pi$
$317$	12.1459	0.682182	0.341091	+	0.940030i	$0.389203\pi$
$318$	0	0
$319$	42.2233	2.36405
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−37.8564	−2.10639
$324$	0	0
$325$	9.92820	0.550718
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9.79796	−0.540179
$330$	0	0
$331$	33.4607	1.83916	0.919582	−	0.392898i	$-0.128528\pi$
$331$	33.4607	1.83916	0.919582	+	0.392898i	$0.128528\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	25.8564	1.41269
$336$	0	0
$337$	0.143594	0.00782204	0.00391102	−	0.999992i	$-0.498755\pi$
$337$	0.143594	0.00782204	0.00391102	+	0.999992i	$0.498755\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.65685	0.306336
$342$	0	0
$343$	−13.3843	−0.722682
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	0	0
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−27.0459	−1.43951	−0.719755	−	0.694229i	$-0.755746\pi$
$353$	−27.0459	−1.43951	−0.719755	+	0.694229i	$0.755746\pi$
$354$	0	0
$355$	−36.5665	−1.94075
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.46410	−0.0772723	−0.0386362	−	0.999253i	$-0.512301\pi$
$359$	−1.46410	−0.0772723	−0.0386362	+	0.999253i	$0.512301\pi$
$360$	0	0
$361$	25.7846	1.35708
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−61.2645	−3.20673
$366$	0	0
$367$	23.1822	1.21010	0.605051	−	0.796187i	$-0.293153\pi$
$367$	23.1822	1.21010	0.605051	+	0.796187i	$0.293153\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.85641	−0.304049
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.72741	0.397982
$378$	0	0
$379$	4.62158	0.237395	0.118697	−	0.992930i	$-0.462128\pi$
$379$	4.62158	0.237395	0.118697	+	0.992930i	$0.462128\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	9.46410	0.483593	0.241797	−	0.970327i	$-0.422263\pi$
$383$	9.46410	0.483593	0.241797	+	0.970327i	$0.422263\pi$
$384$	0	0
$385$	21.8564	1.11391
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4.14110	−0.209962	−0.104981	−	0.994474i	$-0.533478\pi$
$389$	−4.14110	−0.209962	−0.104981	+	0.994474i	$0.533478\pi$
$390$	0	0
$391$	39.1918	1.98202
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.14359	0.207961	0.103980	−	0.994579i	$-0.466842\pi$
$397$	4.14359	0.207961	0.103980	+	0.994579i	$0.466842\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−26.4911	−1.32290	−0.661452	−	0.749988i	$-0.730059\pi$
$401$	−26.4911	−1.32290	−0.661452	+	0.749988i	$0.730059\pi$
$402$	0	0
$403$	1.03528	0.0515708
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.9282	0.541691
$408$	0	0
$409$	27.8564	1.37741	0.688705	−	0.725041i	$-0.258179\pi$
$409$	27.8564	1.37741	0.688705	+	0.725041i	$0.258179\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.62536	0.129185
$414$	0	0
$415$	52.0213	2.55362
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12.7846	0.624569	0.312285	−	0.949989i	$-0.398906\pi$
$419$	12.7846	0.624569	0.312285	+	0.949989i	$0.398906\pi$
$420$	0	0
$421$	17.7128	0.863270	0.431635	−	0.902048i	$-0.357937\pi$
$421$	17.7128	0.863270	0.431635	+	0.902048i	$0.357937\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	56.1624	2.72428
$426$	0	0
$427$	−5.10205	−0.246906
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−20.3923	−0.982263	−0.491131	−	0.871086i	$-0.663417\pi$
$431$	−20.3923	−0.982263	−0.491131	+	0.871086i	$0.663417\pi$
$432$	0	0
$433$	3.07180	0.147621	0.0738106	−	0.997272i	$-0.476484\pi$
$433$	3.07180	0.147621	0.0738106	+	0.997272i	$0.476484\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−46.3644	−2.21791
$438$	0	0
$439$	−18.0802	−0.862919	−0.431460	−	0.902132i	$-0.642001\pi$
$439$	−18.0802	−0.862919	−0.431460	+	0.902132i	$0.642001\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	34.6410	1.64584	0.822922	−	0.568154i	$-0.192342\pi$
$443$	34.6410	1.64584	0.822922	+	0.568154i	$0.192342\pi$
$444$	0	0
$445$	1.07180	0.0508080
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.41851	0.208522	0.104261	−	0.994550i	$-0.466752\pi$
$449$	4.41851	0.208522	0.104261	+	0.994550i	$0.466752\pi$
$450$	0	0
$451$	21.1117	0.994110
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.89520	0.321142	0.160571	−	0.987024i	$-0.448667\pi$
$461$	6.89520	0.321142	0.160571	+	0.987024i	$0.448667\pi$
$462$	0	0
$463$	−7.24693	−0.336794	−0.168397	−	0.985719i	$-0.553859\pi$
$463$	−7.24693	−0.336794	−0.168397	+	0.985719i	$0.553859\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−21.0718	−0.975086	−0.487543	−	0.873099i	$-0.662107\pi$
$467$	−21.0718	−0.975086	−0.487543	+	0.873099i	$0.662107\pi$
$468$	0	0
$469$	6.92820	0.319915
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−30.9096	−1.42123
$474$	0	0
$475$	−66.4408	−3.04851
$476$	0	0
$477$	0	0
$478$	0	0
$479$	25.4641	1.16348	0.581742	−	0.813373i	$-0.302371\pi$
$479$	25.4641	1.16348	0.581742	+	0.813373i	$0.302371\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	23.1822	1.05265
$486$	0	0
$487$	5.17638	0.234564	0.117282	−	0.993099i	$-0.462582\pi$
$487$	5.17638	0.234564	0.117282	+	0.993099i	$0.462582\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	24.7846	1.11851	0.559257	−	0.828994i	$-0.311087\pi$
$491$	24.7846	1.11851	0.559257	+	0.828994i	$0.311087\pi$
$492$	0	0
$493$	43.7128	1.96873
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.79796	−0.439499
$498$	0	0
$499$	30.4292	1.36220	0.681098	−	0.732192i	$-0.261503\pi$
$499$	30.4292	1.36220	0.681098	+	0.732192i	$0.261503\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2.92820	0.130562	0.0652811	−	0.997867i	$-0.479206\pi$
$503$	2.92820	0.130562	0.0652811	+	0.997867i	$0.479206\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−15.7322	−0.697318	−0.348659	−	0.937250i	$-0.613363\pi$
$509$	−15.7322	−0.697318	−0.348659	+	0.937250i	$0.613363\pi$
$510$	0	0
$511$	−16.4158	−0.726190
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−65.5692	−2.88933
$516$	0	0
$517$	−51.7128	−2.27433
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−28.8391	−1.26346	−0.631731	−	0.775187i	$-0.717655\pi$
$521$	−28.8391	−1.26346	−0.631731	+	0.775187i	$0.717655\pi$
$522$	0	0
$523$	−11.8685	−0.518974	−0.259487	−	0.965747i	$-0.583553\pi$
$523$	−11.8685	−0.518974	−0.259487	+	0.965747i	$0.583553\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.85641	0.255109
$528$	0	0
$529$	25.0000	1.08696
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.86370	0.167356
$534$	0	0
$535$	−26.7685	−1.15730
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−32.3923	−1.39524
$540$	0	0
$541$	0.143594	0.00617357	0.00308678	−	0.999995i	$-0.499017\pi$
$541$	0.143594	0.00617357	0.00308678	+	0.999995i	$0.499017\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	23.1822	0.993017
$546$	0	0
$547$	−10.7589	−0.460018	−0.230009	−	0.973189i	$-0.573875\pi$
$547$	−10.7589	−0.460018	−0.230009	+	0.973189i	$0.573875\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−51.7128	−2.20304
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.30890	0.140203	0.0701013	−	0.997540i	$-0.477668\pi$
$557$	3.30890	0.140203	0.0701013	+	0.997540i	$0.477668\pi$
$558$	0	0
$559$	−5.65685	−0.239259
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−25.8564	−1.08972	−0.544859	−	0.838528i	$-0.683417\pi$
$563$	−25.8564	−1.08972	−0.544859	+	0.838528i	$0.683417\pi$
$564$	0	0
$565$	−35.7128	−1.50245
$566$	0	0
$567$	0	0
$568$	0	0
$569$	42.7781	1.79335	0.896676	−	0.442687i	$-0.145975\pi$
$569$	42.7781	1.79335	0.896676	+	0.442687i	$0.145975\pi$
$570$	0	0
$571$	−16.4158	−0.686978	−0.343489	−	0.939157i	$-0.611609\pi$
$571$	−16.4158	−0.686978	−0.343489	+	0.939157i	$0.611609\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	68.7846	2.86852
$576$	0	0
$577$	15.8564	0.660111	0.330055	−	0.943962i	$-0.392933\pi$
$577$	15.8564	0.660111	0.330055	+	0.943962i	$0.392933\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	13.9391	0.578290
$582$	0	0
$583$	−30.9096	−1.28015
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.60770	0.314003	0.157002	−	0.987598i	$-0.449817\pi$
$587$	7.60770	0.314003	0.157002	+	0.987598i	$0.449817\pi$
$588$	0	0
$589$	−6.92820	−0.285472
$590$	0	0
$591$	0	0
$592$	0	0
$593$	22.9048	0.940588	0.470294	−	0.882510i	$-0.344148\pi$
$593$	22.9048	0.940588	0.470294	+	0.882510i	$0.344148\pi$
$594$	0	0
$595$	22.6274	0.927634
$596$	0	0
$597$	0	0
$598$	0	0
$599$	10.9282	0.446514	0.223257	−	0.974760i	$-0.428331\pi$
$599$	10.9282	0.446514	0.223257	+	0.974760i	$0.428331\pi$
$600$	0	0
$601$	−35.8564	−1.46261	−0.731307	−	0.682049i	$-0.761089\pi$
$601$	−35.8564	−1.46261	−0.731307	+	0.682049i	$0.761089\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	72.8556	2.96200
$606$	0	0
$607$	−26.2137	−1.06398	−0.531991	−	0.846750i	$-0.678556\pi$
$607$	−26.2137	−1.06398	−0.531991	+	0.846750i	$0.678556\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9.46410	−0.382877
$612$	0	0
$613$	15.8564	0.640434	0.320217	−	0.947344i	$-0.396244\pi$
$613$	15.8564	0.640434	0.320217	+	0.947344i	$0.396244\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−15.7322	−0.633355	−0.316678	−	0.948533i	$-0.602567\pi$
$617$	−15.7322	−0.633355	−0.316678	+	0.948533i	$0.602567\pi$
$618$	0	0
$619$	27.2490	1.09523	0.547615	−	0.836731i	$-0.315536\pi$
$619$	27.2490	1.09523	0.547615	+	0.836731i	$0.315536\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.287187	0.0115059
$624$	0	0
$625$	23.9282	0.957128
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.3137	0.451107
$630$	0	0
$631$	17.4510	0.694715	0.347357	−	0.937733i	$-0.387079\pi$
$631$	17.4510	0.694715	0.347357	+	0.937733i	$0.387079\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−67.7128	−2.68710
$636$	0	0
$637$	−5.92820	−0.234884
$638$	0	0
$639$	0	0
$640$	0	0
$641$	32.4254	1.28073	0.640363	−	0.768073i	$-0.278784\pi$
$641$	32.4254	1.28073	0.640363	+	0.768073i	$0.278784\pi$
$642$	0	0
$643$	6.69213	0.263912	0.131956	−	0.991256i	$-0.457874\pi$
$643$	6.69213	0.263912	0.131956	+	0.991256i	$0.457874\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18.1436	0.713298	0.356649	−	0.934238i	$-0.383919\pi$
$647$	18.1436	0.713298	0.356649	+	0.934238i	$0.383919\pi$
$648$	0	0
$649$	13.8564	0.543912
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−45.8096	−1.79267	−0.896335	−	0.443378i	$-0.853780\pi$
$653$	−45.8096	−1.79267	−0.896335	+	0.443378i	$0.853780\pi$
$654$	0	0
$655$	4.14110	0.161806
$656$	0	0
$657$	0	0
$658$	0	0
$659$	37.8564	1.47468	0.737338	−	0.675524i	$-0.236082\pi$
$659$	37.8564	1.47468	0.737338	+	0.675524i	$0.236082\pi$
$660$	0	0
$661$	15.8564	0.616743	0.308371	−	0.951266i	$-0.400216\pi$
$661$	15.8564	0.616743	0.308371	+	0.951266i	$0.400216\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−26.7685	−1.03804
$666$	0	0
$667$	53.5370	2.07296
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−26.9282	−1.03955
$672$	0	0
$673$	−28.9282	−1.11510	−0.557550	−	0.830143i	$-0.688259\pi$
$673$	−28.9282	−1.11510	−0.557550	+	0.830143i	$0.688259\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16.9706	0.652232	0.326116	−	0.945330i	$-0.394260\pi$
$677$	16.9706	0.652232	0.326116	+	0.945330i	$0.394260\pi$
$678$	0	0
$679$	6.21166	0.238382
$680$	0	0
$681$	0	0
$682$	0	0
$683$	17.1769	0.657256	0.328628	−	0.944459i	$-0.393414\pi$
$683$	17.1769	0.657256	0.328628	+	0.944459i	$0.393414\pi$
$684$	0	0
$685$	74.6410	2.85189
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5.65685	−0.215509
$690$	0	0
$691$	14.9743	0.569651	0.284825	−	0.958579i	$-0.408064\pi$
$691$	14.9743	0.569651	0.284825	+	0.958579i	$0.408064\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−13.8564	−0.525603
$696$	0	0
$697$	21.8564	0.827870
$698$	0	0
$699$	0	0
$700$	0	0
$701$	19.5959	0.740128	0.370064	−	0.929006i	$-0.379336\pi$
$701$	19.5959	0.740128	0.370064	+	0.929006i	$0.379336\pi$
$702$	0	0
$703$	−13.3843	−0.504797
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.14359	0.0806181
$708$	0	0
$709$	−37.7128	−1.41633	−0.708167	−	0.706045i	$-0.750478\pi$
$709$	−37.7128	−1.41633	−0.708167	+	0.706045i	$0.750478\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.17260	0.268616
$714$	0	0
$715$	21.1117	0.789532
$716$	0	0
$717$	0	0
$718$	0	0
$719$	3.21539	0.119914	0.0599569	−	0.998201i	$-0.480904\pi$
$719$	3.21539	0.119914	0.0599569	+	0.998201i	$0.480904\pi$
$720$	0	0
$721$	−17.5692	−0.654312
$722$	0	0
$723$	0	0
$724$	0	0
$725$	76.7193	2.84928
$726$	0	0
$727$	−2.07055	−0.0767925	−0.0383963	−	0.999263i	$-0.512225\pi$
$727$	−2.07055	−0.0767925	−0.0383963	+	0.999263i	$0.512225\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−32.0000	−1.18356
$732$	0	0
$733$	−37.7128	−1.39295	−0.696477	−	0.717579i	$-0.745250\pi$
$733$	−37.7128	−1.39295	−0.696477	+	0.717579i	$0.745250\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	36.5665	1.34694
$738$	0	0
$739$	39.6723	1.45937	0.729685	−	0.683784i	$-0.239667\pi$
$739$	39.6723	1.45937	0.729685	+	0.683784i	$0.239667\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	18.2487	0.669480	0.334740	−	0.942310i	$-0.391351\pi$
$743$	18.2487	0.669480	0.334740	+	0.942310i	$0.391351\pi$
$744$	0	0
$745$	−90.6410	−3.32083
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−7.17260	−0.262081
$750$	0	0
$751$	−52.0213	−1.89828	−0.949142	−	0.314848i	$-0.898046\pi$
$751$	−52.0213	−1.89828	−0.949142	+	0.314848i	$0.898046\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−63.7128	−2.31875
$756$	0	0
$757$	40.6410	1.47712	0.738561	−	0.674186i	$-0.235505\pi$
$757$	40.6410	1.47712	0.738561	+	0.674186i	$0.235505\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	41.9459	1.52054	0.760269	−	0.649608i	$-0.225067\pi$
$761$	41.9459	1.52054	0.760269	+	0.649608i	$0.225067\pi$
$762$	0	0
$763$	6.21166	0.224877
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.53590	0.0915660
$768$	0	0
$769$	−33.7128	−1.21572	−0.607858	−	0.794046i	$-0.707971\pi$
$769$	−33.7128	−1.21572	−0.607858	+	0.794046i	$0.707971\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−49.1185	−1.76667	−0.883335	−	0.468741i	$-0.844708\pi$
$773$	−49.1185	−1.76667	−0.883335	+	0.468741i	$0.844708\pi$
$774$	0	0
$775$	10.2784	0.369212
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−25.8564	−0.926402
$780$	0	0
$781$	−51.7128	−1.85043
$782$	0	0
$783$	0	0
$784$	0	0
$785$	65.4056	2.33442
$786$	0	0
$787$	−14.9743	−0.533778	−0.266889	−	0.963727i	$-0.585996\pi$
$787$	−14.9743	−0.533778	−0.266889	+	0.963727i	$0.585996\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−9.56922	−0.340242
$792$	0	0
$793$	−4.92820	−0.175006
$794$	0	0
$795$	0	0
$796$	0	0
$797$	46.3644	1.64231	0.821156	−	0.570703i	$-0.193329\pi$
$797$	46.3644	1.64231	0.821156	+	0.570703i	$0.193329\pi$
$798$	0	0
$799$	−53.5370	−1.89400
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−86.6410	−3.05750
$804$	0	0
$805$	27.7128	0.976748
$806$	0	0
$807$	0	0
$808$	0	0
$809$	28.8391	1.01393	0.506964	−	0.861967i	$-0.330768\pi$
$809$	28.8391	1.01393	0.506964	+	0.861967i	$0.330768\pi$
$810$	0	0
$811$	−44.7744	−1.57224	−0.786120	−	0.618074i	$-0.787913\pi$
$811$	−44.7744	−1.57224	−0.786120	+	0.618074i	$0.787913\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−49.8564	−1.74639
$816$	0	0
$817$	37.8564	1.32443
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−15.1774	−0.529695	−0.264848	−	0.964290i	$-0.585322\pi$
$821$	−15.1774	−0.529695	−0.264848	+	0.964290i	$0.585322\pi$
$822$	0	0
$823$	24.1432	0.841578	0.420789	−	0.907159i	$-0.361753\pi$
$823$	24.1432	0.841578	0.420789	+	0.907159i	$0.361753\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	39.0333	1.35732	0.678661	−	0.734452i	$-0.262561\pi$
$827$	39.0333	1.35732	0.678661	+	0.734452i	$0.262561\pi$
$828$	0	0
$829$	34.7846	1.20812	0.604060	−	0.796939i	$-0.293549\pi$
$829$	34.7846	1.20812	0.604060	+	0.796939i	$0.293549\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−33.5350	−1.16192
$834$	0	0
$835$	−36.5665	−1.26544
$836$	0	0
$837$	0	0
$838$	0	0
$839$	15.3205	0.528923	0.264461	−	0.964396i	$-0.414806\pi$
$839$	15.3205	0.528923	0.264461	+	0.964396i	$0.414806\pi$
$840$	0	0
$841$	30.7128	1.05906
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.86370	0.132915
$846$	0	0
$847$	19.5216	0.670770
$848$	0	0
$849$	0	0
$850$	0	0
$851$	13.8564	0.474991
$852$	0	0
$853$	13.7128	0.469518	0.234759	−	0.972054i	$-0.424570\pi$
$853$	13.7128	0.469518	0.234759	+	0.972054i	$0.424570\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−14.9000	−0.508975	−0.254487	−	0.967076i	$-0.581907\pi$
$857$	−14.9000	−0.508975	−0.254487	+	0.967076i	$0.581907\pi$
$858$	0	0
$859$	33.9411	1.15806	0.579028	−	0.815308i	$-0.303432\pi$
$859$	33.9411	1.15806	0.579028	+	0.815308i	$0.303432\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	45.1769	1.53784	0.768920	−	0.639345i	$-0.220794\pi$
$863$	45.1769	1.53784	0.768920	+	0.639345i	$0.220794\pi$
$864$	0	0
$865$	24.0000	0.816024
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	6.69213	0.226754
$872$	0	0
$873$	0	0
$874$	0	0
$875$	19.7128	0.666415
$876$	0	0
$877$	26.0000	0.877958	0.438979	−	0.898497i	$-0.355340\pi$
$877$	26.0000	0.877958	0.438979	+	0.898497i	$0.355340\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	45.8096	1.54337	0.771683	−	0.636007i	$-0.219415\pi$
$881$	45.8096	1.54337	0.771683	+	0.636007i	$0.219415\pi$
$882$	0	0
$883$	17.5254	0.589776	0.294888	−	0.955532i	$-0.404718\pi$
$883$	17.5254	0.589776	0.294888	+	0.955532i	$0.404718\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	49.8564	1.67401	0.837007	−	0.547192i	$-0.184303\pi$
$887$	49.8564	1.67401	0.837007	+	0.547192i	$0.184303\pi$
$888$	0	0
$889$	−18.1436	−0.608517
$890$	0	0
$891$	0	0
$892$	0	0
$893$	63.3350	2.11943
$894$	0	0
$895$	68.9919	2.30614
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.00000	0.266815
$900$	0	0
$901$	−32.0000	−1.06607
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−88.0330	−2.92631
$906$	0	0
$907$	−32.9802	−1.09509	−0.547544	−	0.836777i	$-0.684437\pi$
$907$	−32.9802	−1.09509	−0.547544	+	0.836777i	$0.684437\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	39.7128	1.31574	0.657872	−	0.753130i	$-0.271457\pi$
$911$	39.7128	1.31574	0.657872	+	0.753130i	$0.271457\pi$
$912$	0	0
$913$	73.5692	2.43479
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.10961	0.0366424
$918$	0	0
$919$	−8.28221	−0.273205	−0.136602	−	0.990626i	$-0.543618\pi$
$919$	−8.28221	−0.273205	−0.136602	+	0.990626i	$0.543618\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−9.46410	−0.311515
$924$	0	0
$925$	19.8564	0.652875
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−14.6226	−0.479752	−0.239876	−	0.970804i	$-0.577107\pi$
$929$	−14.6226	−0.479752	−0.239876	+	0.970804i	$0.577107\pi$
$930$	0	0
$931$	39.6723	1.30021
$932$	0	0
$933$	0	0
$934$	0	0
$935$	119.426	3.90564
$936$	0	0
$937$	20.6410	0.674313	0.337156	−	0.941449i	$-0.390535\pi$
$937$	20.6410	0.674313	0.337156	+	0.941449i	$0.390535\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−34.7733	−1.13358	−0.566789	−	0.823863i	$-0.691815\pi$
$941$	−34.7733	−1.13358	−0.566789	+	0.823863i	$0.691815\pi$
$942$	0	0
$943$	26.7685	0.871703
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14.2487	0.463021	0.231510	−	0.972832i	$-0.425633\pi$
$947$	14.2487	0.463021	0.231510	+	0.972832i	$0.425633\pi$
$948$	0	0
$949$	−15.8564	−0.514721
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−22.0726	−0.715002	−0.357501	−	0.933913i	$-0.616371\pi$
$953$	−22.0726	−0.715002	−0.357501	+	0.933913i	$0.616371\pi$
$954$	0	0
$955$	−77.2741	−2.50053
$956$	0	0
$957$	0	0
$958$	0	0
$959$	20.0000	0.645834
$960$	0	0
$961$	−29.9282	−0.965426
$962$	0	0
$963$	0	0
$964$	0	0
$965$	38.6370	1.24377
$966$	0	0
$967$	−10.2784	−0.330532	−0.165266	−	0.986249i	$-0.552848\pi$
$967$	−10.2784	−0.330532	−0.165266	+	0.986249i	$0.552848\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−11.7128	−0.375882	−0.187941	−	0.982180i	$-0.560181\pi$
$971$	−11.7128	−0.375882	−0.187941	+	0.982180i	$0.560181\pi$
$972$	0	0
$973$	−3.71281	−0.119027
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−50.2281	−1.60694	−0.803470	−	0.595345i	$-0.797015\pi$
$977$	−50.2281	−1.60694	−0.803470	+	0.595345i	$0.797015\pi$
$978$	0	0
$979$	1.51575	0.0484436
$980$	0	0
$981$	0	0
$982$	0	0
$983$	11.6077	0.370228	0.185114	−	0.982717i	$-0.440735\pi$
$983$	11.6077	0.370228	0.185114	+	0.982717i	$0.440735\pi$
$984$	0	0
$985$	−44.7846	−1.42696
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−39.1918	−1.24623
$990$	0	0
$991$	−3.58630	−0.113923	−0.0569613	−	0.998376i	$-0.518141\pi$
$991$	−3.58630	−0.113923	−0.0569613	+	0.998376i	$0.518141\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	51.7128	1.63941
$996$	0	0
$997$	−41.7128	−1.32106	−0.660529	−	0.750801i	$-0.729668\pi$
$997$	−41.7128	−1.32106	−0.660529	+	0.750801i	$0.729668\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7488.2.a.dc.1.4		4
3.2	odd	2		7488.2.a.db.1.1		4
4.3	odd	2		7488.2.a.db.1.4		4
8.3	odd	2		3744.2.a.bd.1.1	yes	4
8.5	even	2		3744.2.a.bc.1.1	✓	4
12.11	even	2	inner	7488.2.a.dc.1.1		4
24.5	odd	2		3744.2.a.bd.1.4	yes	4
24.11	even	2		3744.2.a.bc.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3744.2.a.bc.1.1	✓	4	8.5	even	2
3744.2.a.bc.1.4	yes	4	24.11	even	2
3744.2.a.bd.1.1	yes	4	8.3	odd	2
3744.2.a.bd.1.4	yes	4	24.5	odd	2
7488.2.a.db.1.1		4	3.2	odd	2
7488.2.a.db.1.4		4	4.3	odd	2
7488.2.a.dc.1.1		4	12.11	even	2	inner
7488.2.a.dc.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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7488 = 2^{6} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7488.a (trivial)

\(f(q)\)	\(=\)	\(q+3.86370 q^{5} +1.03528 q^{7} +5.46410 q^{11} +1.00000 q^{13} +5.65685 q^{17} -6.69213 q^{19} +6.92820 q^{23} +9.92820 q^{25} +7.72741 q^{29} +1.03528 q^{31} +4.00000 q^{35} +2.00000 q^{37} +3.86370 q^{41} -5.65685 q^{43} -9.46410 q^{47} -5.92820 q^{49} -5.65685 q^{53} +21.1117 q^{55} +2.53590 q^{59} -4.92820 q^{61} +3.86370 q^{65} +6.69213 q^{67} -9.46410 q^{71} -15.8564 q^{73} +5.65685 q^{77} +13.4641 q^{83} +21.8564 q^{85} +0.277401 q^{89} +1.03528 q^{91} -25.8564 q^{95} +6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{11} + 4 q^{13} + 12 q^{25} + 16 q^{35} + 8 q^{37} - 24 q^{47} + 4 q^{49} + 24 q^{59} + 8 q^{61} - 24 q^{71} - 8 q^{73} + 40 q^{83} + 32 q^{85} - 48 q^{95} + 24 q^{97}+O(q^{100}) \) 4 * q + 8 * q^11 + 4 * q^13 + 12 * q^25 + 16 * q^35 + 8 * q^37 - 24 * q^47 + 4 * q^49 + 24 * q^59 + 8 * q^61 - 24 * q^71 - 8 * q^73 + 40 * q^83 + 32 * q^85 - 48 * q^95 + 24 * q^97

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