LMFDB - Embedded newform 4176.2.a.bu.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4176,2,Mod(1,4176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4176, 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("4176.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4176 = 2^{4} \cdot 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4176.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:	$33.3455278841$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.12489$ of defining polynomial
Character	$\chi$	$=$	4176.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.48486 q^{5} +O(q^{10})$	$q-1.48486 q^{5} +3.15516 q^{11} +6.76491 q^{13} -2.00000 q^{17} +1.03028 q^{19} +4.24977 q^{23} -2.79518 q^{25} -1.00000 q^{29} +1.87511 q^{31} -0.969724 q^{37} +7.52982 q^{41} -1.09461 q^{43} -9.34438 q^{47} -7.00000 q^{49} -5.73463 q^{53} -4.68498 q^{55} +8.24977 q^{59} -10.4995 q^{61} -10.0450 q^{65} +4.49954 q^{67} +10.1892 q^{71} +11.5298 q^{73} +13.0946 q^{79} +16.2498 q^{83} +2.96972 q^{85} -13.5904 q^{89} -1.52982 q^{95} -4.96972 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{5}+O(q^{10}) $ 3 * q - 4 * q^5	$ 3 q - 4 q^{5} + 2 q^{11} + 4 q^{13} - 6 q^{17} + 4 q^{19} - 4 q^{23} + 7 q^{25} - 3 q^{29} + 14 q^{31} - 2 q^{37} - 10 q^{41} + 6 q^{43} - 2 q^{47} - 21 q^{49} + 26 q^{55} + 8 q^{59} + 2 q^{61} + 2 q^{65} - 20 q^{67} + 12 q^{71} + 2 q^{73} + 30 q^{79} + 32 q^{83} + 8 q^{85} - 10 q^{89} + 28 q^{95} - 14 q^{97}+O(q^{100}) $ 3 * q - 4 * q^5 + 2 * q^11 + 4 * q^13 - 6 * q^17 + 4 * q^19 - 4 * q^23 + 7 * q^25 - 3 * q^29 + 14 * q^31 - 2 * q^37 - 10 * q^41 + 6 * q^43 - 2 * q^47 - 21 * q^49 + 26 * q^55 + 8 * q^59 + 2 * q^61 + 2 * q^65 - 20 * q^67 + 12 * q^71 + 2 * q^73 + 30 * q^79 + 32 * q^83 + 8 * q^85 - 10 * q^89 + 28 * q^95 - 14 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.48486	−0.664050	−0.332025	−	0.943271i	$-0.607732\pi$
$5$	−1.48486	−0.664050	−0.332025	+	0.943271i	$0.607732\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.15516	0.951317	0.475658	−	0.879630i	$-0.342210\pi$
$11$	3.15516	0.951317	0.475658	+	0.879630i	$0.342210\pi$
$12$	0	0
$13$	6.76491	1.87625	0.938124	−	0.346299i	$-0.112562\pi$
$13$	6.76491	1.87625	0.938124	+	0.346299i	$0.112562\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	1.03028	0.236362	0.118181	−	0.992992i	$-0.462294\pi$
$19$	1.03028	0.236362	0.118181	+	0.992992i	$0.462294\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.24977	0.886138	0.443069	−	0.896487i	$-0.353890\pi$
$23$	4.24977	0.886138	0.443069	+	0.896487i	$0.353890\pi$
$24$	0	0
$25$	−2.79518	−0.559037
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	1.87511	0.336781	0.168390	−	0.985720i	$-0.446143\pi$
$31$	1.87511	0.336781	0.168390	+	0.985720i	$0.446143\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.969724	−0.159422	−0.0797108	−	0.996818i	$-0.525400\pi$
$37$	−0.969724	−0.159422	−0.0797108	+	0.996818i	$0.525400\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.52982	1.17596	0.587980	−	0.808875i	$-0.299923\pi$
$41$	7.52982	1.17596	0.587980	+	0.808875i	$0.299923\pi$
$42$	0	0
$43$	−1.09461	−0.166926	−0.0834632	−	0.996511i	$-0.526598\pi$
$43$	−1.09461	−0.166926	−0.0834632	+	0.996511i	$0.526598\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.34438	−1.36302	−0.681509	−	0.731810i	$-0.738676\pi$
$47$	−9.34438	−1.36302	−0.681509	+	0.731810i	$0.738676\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.73463	−0.787712	−0.393856	−	0.919172i	$-0.628859\pi$
$53$	−5.73463	−0.787712	−0.393856	+	0.919172i	$0.628859\pi$
$54$	0	0
$55$	−4.68498	−0.631722
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.24977	1.07403	0.537014	−	0.843573i	$-0.319552\pi$
$59$	8.24977	1.07403	0.537014	+	0.843573i	$0.319552\pi$
$60$	0	0
$61$	−10.4995	−1.34433	−0.672164	−	0.740402i	$-0.734635\pi$
$61$	−10.4995	−1.34433	−0.672164	+	0.740402i	$0.734635\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−10.0450	−1.24592
$66$	0	0
$67$	4.49954	0.549707	0.274853	−	0.961486i	$-0.411371\pi$
$67$	4.49954	0.549707	0.274853	+	0.961486i	$0.411371\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.1892	1.20924	0.604619	−	0.796515i	$-0.293325\pi$
$71$	10.1892	1.20924	0.604619	+	0.796515i	$0.293325\pi$
$72$	0	0
$73$	11.5298	1.34946	0.674732	−	0.738063i	$-0.264259\pi$
$73$	11.5298	1.34946	0.674732	+	0.738063i	$0.264259\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.0946	1.47326	0.736629	−	0.676297i	$-0.236416\pi$
$79$	13.0946	1.47326	0.736629	+	0.676297i	$0.236416\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	16.2498	1.78364	0.891822	−	0.452386i	$-0.149427\pi$
$83$	16.2498	1.78364	0.891822	+	0.452386i	$0.149427\pi$
$84$	0	0
$85$	2.96972	0.322112
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−13.5904	−1.44058	−0.720288	−	0.693675i	$-0.755990\pi$
$89$	−13.5904	−1.44058	−0.720288	+	0.693675i	$0.755990\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.52982	−0.156956
$96$	0	0
$97$	−4.96972	−0.504599	−0.252300	−	0.967649i	$-0.581187\pi$
$97$	−4.96972	−0.504599	−0.252300	+	0.967649i	$0.581187\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.46927	0.942227	0.471114	−	0.882073i	$-0.343852\pi$
$101$	9.46927	0.942227	0.471114	+	0.882073i	$0.343852\pi$
$102$	0	0
$103$	−4.24977	−0.418742	−0.209371	−	0.977836i	$-0.567142\pi$
$103$	−4.24977	−0.418742	−0.209371	+	0.977836i	$0.567142\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.56009	−0.634188	−0.317094	−	0.948394i	$-0.602707\pi$
$107$	−6.56009	−0.634188	−0.317094	+	0.948394i	$0.602707\pi$
$108$	0	0
$109$	−12.9541	−1.24078	−0.620390	−	0.784293i	$-0.713026\pi$
$109$	−12.9541	−1.24078	−0.620390	+	0.784293i	$0.713026\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.0605522	0.00569627	0.00284814	−	0.999996i	$-0.499093\pi$
$113$	0.0605522	0.00569627	0.00284814	+	0.999996i	$0.499093\pi$
$114$	0	0
$115$	−6.31032	−0.588441
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.04496	−0.0949960
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.5748	1.03528
$126$	0	0
$127$	13.5298	1.20058	0.600289	−	0.799783i	$-0.295052\pi$
$127$	13.5298	1.20058	0.600289	+	0.799783i	$0.295052\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.03028	0.788979	0.394489	−	0.918900i	$-0.370921\pi$
$131$	9.03028	0.788979	0.394489	+	0.918900i	$0.370921\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−0.969724	−0.0828491	−0.0414246	−	0.999142i	$-0.513190\pi$
$137$	−0.969724	−0.0828491	−0.0414246	+	0.999142i	$0.513190\pi$
$138$	0	0
$139$	−16.7493	−1.42066	−0.710329	−	0.703870i	$-0.751454\pi$
$139$	−16.7493	−1.42066	−0.710329	+	0.703870i	$0.751454\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	21.3444	1.78491
$144$	0	0
$145$	1.48486	0.123311
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.2947	1.66261	0.831304	−	0.555817i	$-0.187595\pi$
$149$	20.2947	1.66261	0.831304	+	0.555817i	$0.187595\pi$
$150$	0	0
$151$	18.5601	1.51040	0.755200	−	0.655495i	$-0.227540\pi$
$151$	18.5601	1.51040	0.755200	+	0.655495i	$0.227540\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.78429	−0.223639
$156$	0	0
$157$	7.03028	0.561077	0.280539	−	0.959843i	$-0.409487\pi$
$157$	7.03028	0.561077	0.280539	+	0.959843i	$0.409487\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	24.9348	1.95304	0.976520	−	0.215426i	$-0.0691140\pi$
$163$	24.9348	1.95304	0.976520	+	0.215426i	$0.0691140\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.1892	0.788465	0.394233	−	0.919011i	$-0.371010\pi$
$167$	10.1892	0.788465	0.394233	+	0.919011i	$0.371010\pi$
$168$	0	0
$169$	32.7640	2.52031
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−0.0605522	−0.00460370	−0.00230185	−	0.999997i	$-0.500733\pi$
$173$	−0.0605522	−0.00460370	−0.00230185	+	0.999997i	$0.500733\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−14.1892	−1.06055	−0.530276	−	0.847825i	$-0.677912\pi$
$179$	−14.1892	−1.06055	−0.530276	+	0.847825i	$0.677912\pi$
$180$	0	0
$181$	16.9541	1.26019	0.630095	−	0.776518i	$-0.283016\pi$
$181$	16.9541	1.26019	0.630095	+	0.776518i	$0.283016\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.43991	0.105864
$186$	0	0
$187$	−6.31032	−0.461457
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.59037	0.549220	0.274610	−	0.961556i	$-0.411451\pi$
$191$	7.59037	0.549220	0.274610	+	0.961556i	$0.411451\pi$
$192$	0	0
$193$	−13.4693	−0.969539	−0.484769	−	0.874642i	$-0.661097\pi$
$193$	−13.4693	−0.969539	−0.484769	+	0.874642i	$0.661097\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.93945	0.565662	0.282831	−	0.959170i	$-0.408726\pi$
$197$	7.93945	0.565662	0.282831	+	0.959170i	$0.408726\pi$
$198$	0	0
$199$	26.5601	1.88280	0.941398	−	0.337299i	$-0.109513\pi$
$199$	26.5601	1.88280	0.941398	+	0.337299i	$0.109513\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−11.1807	−0.780897
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.25069	0.224855
$210$	0	0
$211$	0.435208	0.0299610	0.0149805	−	0.999888i	$-0.495231\pi$
$211$	0.435208	0.0299610	0.0149805	+	0.999888i	$0.495231\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.62534	0.110848
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−13.5298	−0.910114
$222$	0	0
$223$	8.12867	0.544336	0.272168	−	0.962250i	$-0.412259\pi$
$223$	8.12867	0.544336	0.272168	+	0.962250i	$0.412259\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−29.3700	−1.94935	−0.974676	−	0.223620i	$-0.928212\pi$
$227$	−29.3700	−1.94935	−0.974676	+	0.223620i	$0.928212\pi$
$228$	0	0
$229$	−9.46927	−0.625747	−0.312873	−	0.949795i	$-0.601292\pi$
$229$	−9.46927	−0.625747	−0.312873	+	0.949795i	$0.601292\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.20482	−0.537515	−0.268758	−	0.963208i	$-0.586613\pi$
$233$	−8.20482	−0.537515	−0.268758	+	0.963208i	$0.586613\pi$
$234$	0	0
$235$	13.8751	0.905113
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.75023	0.242582	0.121291	−	0.992617i	$-0.461297\pi$
$239$	3.75023	0.242582	0.121291	+	0.992617i	$0.461297\pi$
$240$	0	0
$241$	28.2947	1.82262	0.911312	−	0.411717i	$-0.135071\pi$
$241$	28.2947	1.82262	0.911312	+	0.411717i	$0.135071\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	10.3940	0.664050
$246$	0	0
$247$	6.96972	0.443473
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−12.1854	−0.769138	−0.384569	−	0.923096i	$-0.625650\pi$
$251$	−12.1854	−0.769138	−0.384569	+	0.923096i	$0.625650\pi$
$252$	0	0
$253$	13.4087	0.842999
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.98532	−0.310976	−0.155488	−	0.987838i	$-0.549695\pi$
$257$	−4.98532	−0.310976	−0.155488	+	0.987838i	$0.549695\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.37466	−0.146428	−0.0732138	−	0.997316i	$-0.523326\pi$
$263$	−2.37466	−0.146428	−0.0732138	+	0.997316i	$0.523326\pi$
$264$	0	0
$265$	8.51514	0.523081
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−28.9385	−1.76441	−0.882207	−	0.470862i	$-0.843943\pi$
$269$	−28.9385	−1.76441	−0.882207	+	0.470862i	$0.843943\pi$
$270$	0	0
$271$	16.1854	0.983195	0.491598	−	0.870822i	$-0.336413\pi$
$271$	16.1854	0.983195	0.491598	+	0.870822i	$0.336413\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.81926	−0.531821
$276$	0	0
$277$	25.0596	1.50569	0.752844	−	0.658199i	$-0.228682\pi$
$277$	25.0596	1.50569	0.752844	+	0.658199i	$0.228682\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	13.2351	0.789539	0.394770	−	0.918780i	$-0.370824\pi$
$281$	13.2351	0.789539	0.394770	+	0.918780i	$0.370824\pi$
$282$	0	0
$283$	−18.3103	−1.08844	−0.544218	−	0.838944i	$-0.683174\pi$
$283$	−18.3103	−1.08844	−0.544218	+	0.838944i	$0.683174\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.93945	0.463827	0.231914	−	0.972736i	$-0.425501\pi$
$293$	7.93945	0.463827	0.231914	+	0.972736i	$0.425501\pi$
$294$	0	0
$295$	−12.2498	−0.713209
$296$	0	0
$297$	0	0
$298$	0	0
$299$	28.7493	1.66262
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	15.5904	0.892702
$306$	0	0
$307$	−28.1542	−1.60685	−0.803424	−	0.595408i	$-0.796991\pi$
$307$	−28.1542	−1.60685	−0.803424	+	0.595408i	$0.796991\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.53073	−0.256914	−0.128457	−	0.991715i	$-0.541002\pi$
$311$	−4.53073	−0.256914	−0.128457	+	0.991715i	$0.541002\pi$
$312$	0	0
$313$	5.48486	0.310023	0.155011	−	0.987913i	$-0.450459\pi$
$313$	5.48486	0.310023	0.155011	+	0.987913i	$0.450459\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−20.4390	−1.14797	−0.573984	−	0.818867i	$-0.694603\pi$
$317$	−20.4390	−1.14797	−0.573984	+	0.818867i	$0.694603\pi$
$318$	0	0
$319$	−3.15516	−0.176655
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.06055	−0.114652
$324$	0	0
$325$	−18.9092	−1.04889
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−24.5639	−1.35015	−0.675076	−	0.737748i	$-0.735889\pi$
$331$	−24.5639	−1.35015	−0.675076	+	0.737748i	$0.735889\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6.68120	−0.365033
$336$	0	0
$337$	26.4995	1.44352	0.721761	−	0.692142i	$-0.243333\pi$
$337$	26.4995	1.44352	0.721761	+	0.692142i	$0.243333\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.91629	0.320385
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.8704	−1.12038	−0.560191	−	0.828363i	$-0.689272\pi$
$347$	−20.8704	−1.12038	−0.560191	+	0.828363i	$0.689272\pi$
$348$	0	0
$349$	−3.26445	−0.174742	−0.0873710	−	0.996176i	$-0.527847\pi$
$349$	−3.26445	−0.174742	−0.0873710	+	0.996176i	$0.527847\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	30.9991	1.64991	0.824957	−	0.565195i	$-0.191199\pi$
$353$	30.9991	1.64991	0.824957	+	0.565195i	$0.191199\pi$
$354$	0	0
$355$	−15.1296	−0.802995
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.03406	−0.160131	−0.0800657	−	0.996790i	$-0.525513\pi$
$359$	−3.03406	−0.160131	−0.0800657	+	0.996790i	$0.525513\pi$
$360$	0	0
$361$	−17.9385	−0.944133
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−17.1202	−0.896112
$366$	0	0
$367$	16.9092	0.882652	0.441326	−	0.897347i	$-0.354508\pi$
$367$	16.9092	0.882652	0.441326	+	0.897347i	$0.354508\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	12.5757	0.651145	0.325572	−	0.945517i	$-0.394443\pi$
$373$	12.5757	0.651145	0.325572	+	0.945517i	$0.394443\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.76491	−0.348411
$378$	0	0
$379$	13.4087	0.688759	0.344380	−	0.938830i	$-0.388089\pi$
$379$	13.4087	0.688759	0.344380	+	0.938830i	$0.388089\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	18.0606	0.922851	0.461426	−	0.887179i	$-0.347338\pi$
$383$	18.0606	0.922851	0.461426	+	0.887179i	$0.347338\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−0.0293595	−0.00148858	−0.000744292	−	1.00000i	$-0.500237\pi$
$389$	−0.0293595	−0.00148858	−0.000744292		1.00000i	$0.500237\pi$
$390$	0	0
$391$	−8.49954	−0.429840
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−19.4437	−0.978318
$396$	0	0
$397$	−26.1055	−1.31020	−0.655099	−	0.755543i	$-0.727373\pi$
$397$	−26.1055	−1.31020	−0.655099	+	0.755543i	$0.727373\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.2645	−0.962021	−0.481010	−	0.876715i	$-0.659730\pi$
$401$	−19.2645	−0.962021	−0.481010	+	0.876715i	$0.659730\pi$
$402$	0	0
$403$	12.6850	0.631884
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.05964	−0.151661
$408$	0	0
$409$	19.4986	0.964145	0.482072	−	0.876131i	$-0.339884\pi$
$409$	19.4986	0.964145	0.482072	+	0.876131i	$0.339884\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−24.1287	−1.18443
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−33.2489	−1.62431	−0.812156	−	0.583440i	$-0.801706\pi$
$419$	−33.2489	−1.62431	−0.812156	+	0.583440i	$0.801706\pi$
$420$	0	0
$421$	3.93945	0.191997	0.0959985	−	0.995381i	$-0.469396\pi$
$421$	3.93945	0.191997	0.0959985	+	0.995381i	$0.469396\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.59037	0.271173
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.3700	0.836681	0.418341	−	0.908290i	$-0.362612\pi$
$431$	17.3700	0.836681	0.418341	+	0.908290i	$0.362612\pi$
$432$	0	0
$433$	−15.5298	−0.746315	−0.373158	−	0.927768i	$-0.621725\pi$
$433$	−15.5298	−0.746315	−0.373158	+	0.927768i	$0.621725\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.37844	0.209449
$438$	0	0
$439$	−8.87042	−0.423362	−0.211681	−	0.977339i	$-0.567894\pi$
$439$	−8.87042	−0.423362	−0.211681	+	0.977339i	$0.567894\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.09083	−0.146850	−0.0734248	−	0.997301i	$-0.523393\pi$
$443$	−3.09083	−0.146850	−0.0734248	+	0.997301i	$0.523393\pi$
$444$	0	0
$445$	20.1798	0.956615
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−16.9385	−0.799379	−0.399689	−	0.916651i	$-0.630882\pi$
$449$	−16.9385	−0.799379	−0.399689	+	0.916651i	$0.630882\pi$
$450$	0	0
$451$	23.7578	1.11871
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.9991	0.514515	0.257258	−	0.966343i	$-0.417181\pi$
$457$	10.9991	0.514515	0.257258	+	0.966343i	$0.417181\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	23.1202	1.07681	0.538407	−	0.842685i	$-0.319026\pi$
$461$	23.1202	1.07681	0.538407	+	0.842685i	$0.319026\pi$
$462$	0	0
$463$	0.870417	0.0404517	0.0202259	−	0.999795i	$-0.493561\pi$
$463$	0.870417	0.0404517	0.0202259	+	0.999795i	$0.493561\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.2148	0.657782	0.328891	−	0.944368i	$-0.393325\pi$
$467$	14.2148	0.657782	0.328891	+	0.944368i	$0.393325\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.45367	−0.158800
$474$	0	0
$475$	−2.87981	−0.132135
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.56479	−0.162879	−0.0814397	−	0.996678i	$-0.525952\pi$
$479$	−3.56479	−0.162879	−0.0814397	+	0.996678i	$0.525952\pi$
$480$	0	0
$481$	−6.56009	−0.299115
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.37935	0.335079
$486$	0	0
$487$	8.49954	0.385151	0.192575	−	0.981282i	$-0.438316\pi$
$487$	8.49954	0.385151	0.192575	+	0.981282i	$0.438316\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−0.595068	−0.0268550	−0.0134275	−	0.999910i	$-0.504274\pi$
$491$	−0.595068	−0.0268550	−0.0134275	+	0.999910i	$0.504274\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−9.24885	−0.414036	−0.207018	−	0.978337i	$-0.566376\pi$
$499$	−9.24885	−0.414036	−0.207018	+	0.978337i	$0.566376\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0.314104	0.0140052	0.00700260	−	0.999975i	$-0.497771\pi$
$503$	0.314104	0.0140052	0.00700260	+	0.999975i	$0.497771\pi$
$504$	0	0
$505$	−14.0606	−0.625686
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.11399	−0.0493766	−0.0246883	−	0.999695i	$-0.507859\pi$
$509$	−1.11399	−0.0493766	−0.0246883	+	0.999695i	$0.507859\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.31032	0.278066
$516$	0	0
$517$	−29.4830	−1.29666
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−11.1358	−0.487868	−0.243934	−	0.969792i	$-0.578438\pi$
$521$	−11.1358	−0.487868	−0.243934	+	0.969792i	$0.578438\pi$
$522$	0	0
$523$	4.49954	0.196751	0.0983756	−	0.995149i	$-0.468635\pi$
$523$	4.49954	0.196751	0.0983756	+	0.995149i	$0.468635\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.75023	−0.163363
$528$	0	0
$529$	−4.93945	−0.214759
$530$	0	0
$531$	0	0
$532$	0	0
$533$	50.9385	2.20639
$534$	0	0
$535$	9.74083	0.421133
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−22.0861	−0.951317
$540$	0	0
$541$	19.6509	0.844859	0.422430	−	0.906396i	$-0.361177\pi$
$541$	19.6509	0.844859	0.422430	+	0.906396i	$0.361177\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	19.2351	0.823941
$546$	0	0
$547$	1.93945	0.0829248	0.0414624	−	0.999140i	$-0.486798\pi$
$547$	1.93945	0.0829248	0.0414624	+	0.999140i	$0.486798\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.03028	−0.0438912
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	38.0587	1.61260	0.806300	−	0.591507i	$-0.201467\pi$
$557$	38.0587	1.61260	0.806300	+	0.591507i	$0.201467\pi$
$558$	0	0
$559$	−7.40493	−0.313195
$560$	0	0
$561$	0	0
$562$	0	0
$563$	45.1845	1.90430	0.952150	−	0.305630i	$-0.0988672\pi$
$563$	45.1845	1.90430	0.952150	+	0.305630i	$0.0988672\pi$
$564$	0	0
$565$	−0.0899116	−0.00378261
$566$	0	0
$567$	0	0
$568$	0	0
$569$	31.4986	1.32049	0.660246	−	0.751050i	$-0.270452\pi$
$569$	31.4986	1.32049	0.660246	+	0.751050i	$0.270452\pi$
$570$	0	0
$571$	−16.6206	−0.695552	−0.347776	−	0.937578i	$-0.613063\pi$
$571$	−16.6206	−0.695552	−0.347776	+	0.937578i	$0.613063\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−11.8789	−0.495384
$576$	0	0
$577$	−3.43991	−0.143205	−0.0716026	−	0.997433i	$-0.522811\pi$
$577$	−3.43991	−0.143205	−0.0716026	+	0.997433i	$0.522811\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−18.0937	−0.749364
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.93097	0.286072	0.143036	−	0.989718i	$-0.454314\pi$
$587$	6.93097	0.286072	0.143036	+	0.989718i	$0.454314\pi$
$588$	0	0
$589$	1.93189	0.0796020
$590$	0	0
$591$	0	0
$592$	0	0
$593$	31.6041	1.29783	0.648913	−	0.760863i	$-0.275224\pi$
$593$	31.6041	1.29783	0.648913	+	0.760863i	$0.275224\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−31.6547	−1.29338	−0.646688	−	0.762755i	$-0.723846\pi$
$599$	−31.6547	−1.29338	−0.646688	+	0.762755i	$0.723846\pi$
$600$	0	0
$601$	19.0303	0.776261	0.388131	−	0.921604i	$-0.373121\pi$
$601$	19.0303	0.776261	0.388131	+	0.921604i	$0.373121\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.55162	0.0630821
$606$	0	0
$607$	−24.3141	−0.986879	−0.493440	−	0.869780i	$-0.664261\pi$
$607$	−24.3141	−0.986879	−0.493440	+	0.869780i	$0.664261\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−63.2139	−2.55736
$612$	0	0
$613$	−8.07615	−0.326193	−0.163096	−	0.986610i	$-0.552148\pi$
$613$	−8.07615	−0.326193	−0.163096	+	0.986610i	$0.552148\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16.9697	−0.683175	−0.341588	−	0.939850i	$-0.610965\pi$
$617$	−16.9697	−0.683175	−0.341588	+	0.939850i	$0.610965\pi$
$618$	0	0
$619$	23.3737	0.939470	0.469735	−	0.882807i	$-0.344349\pi$
$619$	23.3737	0.939470	0.469735	+	0.882807i	$0.344349\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−3.21102	−0.128441
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.93945	0.0773308
$630$	0	0
$631$	3.75023	0.149294	0.0746471	−	0.997210i	$-0.476217\pi$
$631$	3.75023	0.149294	0.0746471	+	0.997210i	$0.476217\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−20.0899	−0.797244
$636$	0	0
$637$	−47.3544	−1.87625
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−29.5592	−1.16752	−0.583759	−	0.811927i	$-0.698419\pi$
$641$	−29.5592	−1.16752	−0.583759	+	0.811927i	$0.698419\pi$
$642$	0	0
$643$	−44.4995	−1.75489	−0.877445	−	0.479677i	$-0.840754\pi$
$643$	−44.4995	−1.75489	−0.877445	+	0.479677i	$0.840754\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.80986	−0.267723	−0.133862	−	0.991000i	$-0.542738\pi$
$647$	−6.80986	−0.267723	−0.133862	+	0.991000i	$0.542738\pi$
$648$	0	0
$649$	26.0294	1.02174
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−11.1514	−0.436387	−0.218194	−	0.975905i	$-0.570016\pi$
$653$	−11.1514	−0.436387	−0.218194	+	0.975905i	$0.570016\pi$
$654$	0	0
$655$	−13.4087	−0.523922
$656$	0	0
$657$	0	0
$658$	0	0
$659$	35.2526	1.37325	0.686624	−	0.727013i	$-0.259092\pi$
$659$	35.2526	1.37325	0.686624	+	0.727013i	$0.259092\pi$
$660$	0	0
$661$	38.9991	1.51689	0.758444	−	0.651738i	$-0.225960\pi$
$661$	38.9991	1.51689	0.758444	+	0.651738i	$0.225960\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.24977	−0.164552
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−33.1277	−1.27888
$672$	0	0
$673$	−10.1443	−0.391033	−0.195516	−	0.980700i	$-0.562638\pi$
$673$	−10.1443	−0.391033	−0.195516	+	0.980700i	$0.562638\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−47.4986	−1.82552	−0.912760	−	0.408496i	$-0.866053\pi$
$677$	−47.4986	−1.82552	−0.912760	+	0.408496i	$0.866053\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.43991	−0.0550965	−0.0275482	−	0.999620i	$-0.508770\pi$
$683$	−1.43991	−0.0550965	−0.0275482	+	0.999620i	$0.508770\pi$
$684$	0	0
$685$	1.43991	0.0550160
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−38.7943	−1.47794
$690$	0	0
$691$	−26.9385	−1.02479	−0.512395	−	0.858750i	$-0.671242\pi$
$691$	−26.9385	−1.02479	−0.512395	+	0.858750i	$0.671242\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	24.8704	0.943389
$696$	0	0
$697$	−15.0596	−0.570424
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−17.3250	−0.654356	−0.327178	−	0.944963i	$-0.606098\pi$
$701$	−17.3250	−0.654356	−0.327178	+	0.944963i	$0.606098\pi$
$702$	0	0
$703$	−0.999083	−0.0376811
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0.0837106	0.00314382	0.00157191	−	0.999999i	$-0.499500\pi$
$709$	0.0837106	0.00314382	0.00157191	+	0.999999i	$0.499500\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.96881	0.298434
$714$	0	0
$715$	−31.6935	−1.18527
$716$	0	0
$717$	0	0
$718$	0	0
$719$	19.4305	0.724636	0.362318	−	0.932055i	$-0.381985\pi$
$719$	19.4305	0.724636	0.362318	+	0.932055i	$0.381985\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.79518	0.103811
$726$	0	0
$727$	29.2876	1.08622	0.543109	−	0.839662i	$-0.317247\pi$
$727$	29.2876	1.08622	0.543109	+	0.839662i	$0.317247\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.18922	0.0809712
$732$	0	0
$733$	21.7115	0.801932	0.400966	−	0.916093i	$-0.368675\pi$
$733$	21.7115	0.801932	0.400966	+	0.916093i	$0.368675\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.1968	0.522945
$738$	0	0
$739$	−6.37466	−0.234496	−0.117248	−	0.993103i	$-0.537407\pi$
$739$	−6.37466	−0.234496	−0.117248	+	0.993103i	$0.537407\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−27.4693	−1.00775	−0.503875	−	0.863777i	$-0.668093\pi$
$743$	−27.4693	−1.00775	−0.503875	+	0.863777i	$0.668093\pi$
$744$	0	0
$745$	−30.1349	−1.10406
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	49.0890	1.79128	0.895641	−	0.444777i	$-0.146717\pi$
$751$	49.0890	1.79128	0.895641	+	0.444777i	$0.146717\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−27.5592	−1.00298
$756$	0	0
$757$	−35.2413	−1.28087	−0.640433	−	0.768014i	$-0.721245\pi$
$757$	−35.2413	−1.28087	−0.640433	+	0.768014i	$0.721245\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	25.0596	0.908411	0.454206	−	0.890897i	$-0.349923\pi$
$761$	25.0596	0.908411	0.454206	+	0.890897i	$0.349923\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	55.8089	2.01514
$768$	0	0
$769$	−29.6803	−1.07030	−0.535149	−	0.844758i	$-0.679745\pi$
$769$	−29.6803	−1.07030	−0.535149	+	0.844758i	$0.679745\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−23.5298	−0.846309	−0.423154	−	0.906058i	$-0.639077\pi$
$773$	−23.5298	−0.846309	−0.423154	+	0.906058i	$0.639077\pi$
$774$	0	0
$775$	−5.24129	−0.188273
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.75779	0.277952
$780$	0	0
$781$	32.1486	1.15037
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−10.4390	−0.372584
$786$	0	0
$787$	41.7484	1.48817	0.744085	−	0.668085i	$-0.232886\pi$
$787$	41.7484	1.48817	0.744085	+	0.668085i	$0.232886\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−71.0284	−2.52229
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5.50046	−0.194836	−0.0974181	−	0.995244i	$-0.531058\pi$
$797$	−5.50046	−0.194836	−0.0974181	+	0.995244i	$0.531058\pi$
$798$	0	0
$799$	18.6888	0.661161
$800$	0	0
$801$	0	0
$802$	0	0
$803$	36.3784	1.28377
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−40.1505	−1.41162	−0.705808	−	0.708404i	$-0.749416\pi$
$809$	−40.1505	−1.41162	−0.705808	+	0.708404i	$0.749416\pi$
$810$	0	0
$811$	−14.0606	−0.493733	−0.246866	−	0.969050i	$-0.579401\pi$
$811$	−14.0606	−0.493733	−0.246866	+	0.969050i	$0.579401\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−37.0247	−1.29692
$816$	0	0
$817$	−1.12775	−0.0394550
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30.1131	1.05095	0.525477	−	0.850808i	$-0.323887\pi$
$821$	30.1131	1.05095	0.525477	+	0.850808i	$0.323887\pi$
$822$	0	0
$823$	−48.5895	−1.69372	−0.846861	−	0.531814i	$-0.821510\pi$
$823$	−48.5895	−1.69372	−0.846861	+	0.531814i	$0.821510\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31.0029	1.07808	0.539038	−	0.842282i	$-0.318788\pi$
$827$	31.0029	1.07808	0.539038	+	0.842282i	$0.318788\pi$
$828$	0	0
$829$	−46.6206	−1.61920	−0.809601	−	0.586981i	$-0.800316\pi$
$829$	−46.6206	−1.61920	−0.809601	+	0.586981i	$0.800316\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	14.0000	0.485071
$834$	0	0
$835$	−15.1296	−0.523581
$836$	0	0
$837$	0	0
$838$	0	0
$839$	4.09553	0.141393	0.0706966	−	0.997498i	$-0.477478\pi$
$839$	4.09553	0.141393	0.0706966	+	0.997498i	$0.477478\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−48.6500	−1.67361
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.12110	−0.141270
$852$	0	0
$853$	−28.0606	−0.960775	−0.480388	−	0.877056i	$-0.659504\pi$
$853$	−28.0606	−0.960775	−0.480388	+	0.877056i	$0.659504\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−9.60597	−0.328134	−0.164067	−	0.986449i	$-0.552461\pi$
$857$	−9.60597	−0.328134	−0.164067	+	0.986449i	$0.552461\pi$
$858$	0	0
$859$	−8.96594	−0.305914	−0.152957	−	0.988233i	$-0.548880\pi$
$859$	−8.96594	−0.305914	−0.152957	+	0.988233i	$0.548880\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−45.4911	−1.54853	−0.774267	−	0.632859i	$-0.781881\pi$
$863$	−45.4911	−1.54853	−0.774267	+	0.632859i	$0.781881\pi$
$864$	0	0
$865$	0.0899116	0.00305709
$866$	0	0
$867$	0	0
$868$	0	0
$869$	41.3156	1.40154
$870$	0	0
$871$	30.4390	1.03139
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−34.2654	−1.15706	−0.578530	−	0.815661i	$-0.696373\pi$
$877$	−34.2654	−1.15706	−0.578530	+	0.815661i	$0.696373\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	22.4995	0.758029	0.379014	−	0.925391i	$-0.376263\pi$
$881$	22.4995	0.758029	0.379014	+	0.925391i	$0.376263\pi$
$882$	0	0
$883$	−31.5592	−1.06205	−0.531025	−	0.847356i	$-0.678193\pi$
$883$	−31.5592	−1.06205	−0.531025	+	0.847356i	$0.678193\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	17.6841	0.593773	0.296886	−	0.954913i	$-0.404052\pi$
$887$	17.6841	0.593773	0.296886	+	0.954913i	$0.404052\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9.62729	−0.322165
$894$	0	0
$895$	21.0690	0.704260
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.87511	−0.0625386
$900$	0	0
$901$	11.4693	0.382097
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−25.1745	−0.836830
$906$	0	0
$907$	−1.03028	−0.0342098	−0.0171049	−	0.999854i	$-0.505445\pi$
$907$	−1.03028	−0.0342098	−0.0171049	+	0.999854i	$0.505445\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−21.3056	−0.705887	−0.352943	−	0.935645i	$-0.614819\pi$
$911$	−21.3056	−0.705887	−0.352943	+	0.935645i	$0.614819\pi$
$912$	0	0
$913$	51.2707	1.69681
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−19.1883	−0.632964	−0.316482	−	0.948599i	$-0.602502\pi$
$919$	−19.1883	−0.632964	−0.316482	+	0.948599i	$0.602502\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	68.9291	2.26883
$924$	0	0
$925$	2.71056	0.0891226
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.18166	0.137196	0.0685979	−	0.997644i	$-0.478147\pi$
$929$	4.18166	0.137196	0.0685979	+	0.997644i	$0.478147\pi$
$930$	0	0
$931$	−7.21193	−0.236362
$932$	0	0
$933$	0	0
$934$	0	0
$935$	9.36996	0.306430
$936$	0	0
$937$	−45.1807	−1.47599	−0.737995	−	0.674806i	$-0.764227\pi$
$937$	−45.1807	−1.47599	−0.737995	+	0.674806i	$0.764227\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	17.0752	0.556637	0.278318	−	0.960489i	$-0.410223\pi$
$941$	17.0752	0.556637	0.278318	+	0.960489i	$0.410223\pi$
$942$	0	0
$943$	32.0000	1.04206
$944$	0	0
$945$	0	0
$946$	0	0
$947$	50.3359	1.63570	0.817849	−	0.575434i	$-0.195167\pi$
$947$	50.3359	1.63570	0.817849	+	0.575434i	$0.195167\pi$
$948$	0	0
$949$	77.9982	2.53193
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−8.94657	−0.289808	−0.144904	−	0.989446i	$-0.546287\pi$
$953$	−8.94657	−0.289808	−0.144904	+	0.989446i	$0.546287\pi$
$954$	0	0
$955$	−11.2707	−0.364710
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.4839	−0.886579
$962$	0	0
$963$	0	0
$964$	0	0
$965$	20.0000	0.643823
$966$	0	0
$967$	−44.5327	−1.43207	−0.716037	−	0.698062i	$-0.754046\pi$
$967$	−44.5327	−1.43207	−0.716037	+	0.698062i	$0.754046\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−42.0294	−1.34879	−0.674393	−	0.738372i	$-0.735595\pi$
$971$	−42.0294	−1.34879	−0.674393	+	0.738372i	$0.735595\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15.3931	0.492469	0.246235	−	0.969210i	$-0.420807\pi$
$977$	15.3931	0.492469	0.246235	+	0.969210i	$0.420807\pi$
$978$	0	0
$979$	−42.8798	−1.37044
$980$	0	0
$981$	0	0
$982$	0	0
$983$	33.7153	1.07535	0.537675	−	0.843152i	$-0.319303\pi$
$983$	33.7153	1.07535	0.537675	+	0.843152i	$0.319303\pi$
$984$	0	0
$985$	−11.7890	−0.375628
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.65184	−0.147920
$990$	0	0
$991$	23.3094	0.740448	0.370224	−	0.928943i	$-0.379281\pi$
$991$	23.3094	0.740448	0.370224	+	0.928943i	$0.379281\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−39.4381	−1.25027
$996$	0	0
$997$	−4.02936	−0.127611	−0.0638055	−	0.997962i	$-0.520324\pi$
$997$	−4.02936	−0.127611	−0.0638055	+	0.997962i	$0.520324\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4176.2.a.bu.1.2		3
3.2	odd	2		464.2.a.j.1.3		3
4.3	odd	2		2088.2.a.s.1.2		3
12.11	even	2		232.2.a.d.1.1	✓	3
24.5	odd	2		1856.2.a.y.1.1		3
24.11	even	2		1856.2.a.x.1.3		3
60.59	even	2		5800.2.a.p.1.3		3
348.347	even	2		6728.2.a.j.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.2.a.d.1.1	✓	3	12.11	even	2
464.2.a.j.1.3		3	3.2	odd	2
1856.2.a.x.1.3		3	24.11	even	2
1856.2.a.y.1.1		3	24.5	odd	2
2088.2.a.s.1.2		3	4.3	odd	2
4176.2.a.bu.1.2		3	1.1	even	1	trivial
5800.2.a.p.1.3		3	60.59	even	2
6728.2.a.j.1.3		3	348.347	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4176 = 2^{4} \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4176.a (trivial)

\(f(q)\)	\(=\)	\(q-1.48486 q^{5} +O(q^{10})\)	\(q-1.48486 q^{5} +3.15516 q^{11} +6.76491 q^{13} -2.00000 q^{17} +1.03028 q^{19} +4.24977 q^{23} -2.79518 q^{25} -1.00000 q^{29} +1.87511 q^{31} -0.969724 q^{37} +7.52982 q^{41} -1.09461 q^{43} -9.34438 q^{47} -7.00000 q^{49} -5.73463 q^{53} -4.68498 q^{55} +8.24977 q^{59} -10.4995 q^{61} -10.0450 q^{65} +4.49954 q^{67} +10.1892 q^{71} +11.5298 q^{73} +13.0946 q^{79} +16.2498 q^{83} +2.96972 q^{85} -13.5904 q^{89} -1.52982 q^{95} -4.96972 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{5}+O(q^{10}) \) 3 * q - 4 * q^5	\( 3 q - 4 q^{5} + 2 q^{11} + 4 q^{13} - 6 q^{17} + 4 q^{19} - 4 q^{23} + 7 q^{25} - 3 q^{29} + 14 q^{31} - 2 q^{37} - 10 q^{41} + 6 q^{43} - 2 q^{47} - 21 q^{49} + 26 q^{55} + 8 q^{59} + 2 q^{61} + 2 q^{65} - 20 q^{67} + 12 q^{71} + 2 q^{73} + 30 q^{79} + 32 q^{83} + 8 q^{85} - 10 q^{89} + 28 q^{95} - 14 q^{97}+O(q^{100}) \) 3 * q - 4 * q^5 + 2 * q^11 + 4 * q^13 - 6 * q^17 + 4 * q^19 - 4 * q^23 + 7 * q^25 - 3 * q^29 + 14 * q^31 - 2 * q^37 - 10 * q^41 + 6 * q^43 - 2 * q^47 - 21 * q^49 + 26 * q^55 + 8 * q^59 + 2 * q^61 + 2 * q^65 - 20 * q^67 + 12 * q^71 + 2 * q^73 + 30 * q^79 + 32 * q^83 + 8 * q^85 - 10 * q^89 + 28 * q^95 - 14 * q^97

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