LMFDB - Embedded newform 8280.2.a.bs.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.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:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.13955077.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 14x^{3} - x^{2} + 32x + 16 $ x^5 - 14x^3 - x^2 + 32x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.568386$ of defining polynomial
Character	$\chi$	$=$	8280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +4.73770 q^{7} +0.360532 q^{11} +5.26123 q^{13} -0.370852 q^{17} -4.60586 q^{19} +1.00000 q^{23} +1.00000 q^{25} -0.939238 q^{29} +9.66662 q^{31} +4.73770 q^{35} +3.26862 q^{37} -5.29977 q^{41} +1.25491 q^{47} +15.4458 q^{49} -10.9278 q^{53} +0.360532 q^{55} +9.66955 q^{59} +9.71441 q^{61} +5.26123 q^{65} -7.07001 q^{67} -11.3747 q^{71} +0.745086 q^{73} +1.70809 q^{77} +0.415709 q^{79} +9.26862 q^{83} -0.370852 q^{85} -12.6122 q^{89} +24.9261 q^{91} -4.60586 q^{95} +14.0404 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{5} - 2 q^{7} + q^{11} + 4 q^{13} - 4 q^{17} + 7 q^{19} + 5 q^{23} + 5 q^{25} - 4 q^{29} + 19 q^{31} - 2 q^{35} + 15 q^{37} - 25 q^{41} + 11 q^{47} + 25 q^{49} - 3 q^{53} + q^{55} + q^{59}+ \cdots + 25 q^{97}+O(q^{100}) $ 5 * q + 5 * q^5 - 2 * q^7 + q^11 + 4 * q^13 - 4 * q^17 + 7 * q^19 + 5 * q^23 + 5 * q^25 - 4 * q^29 + 19 * q^31 - 2 * q^35 + 15 * q^37 - 25 * q^41 + 11 * q^47 + 25 * q^49 - 3 * q^53 + q^55 + q^59 - 5 * q^61 + 4 * q^65 + 9 * q^67 - q^71 - q^73 - 18 * q^77 - 2 * q^79 + 45 * q^83 - 4 * q^85 - 6 * q^89 + 11 * q^91 + 7 * q^95 + 25 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.73770	1.79068	0.895341	−	0.445381i	$-0.146932\pi$
$7$	4.73770	1.79068	0.895341	+	0.445381i	$0.146932\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.360532	0.108704	0.0543522	−	0.998522i	$-0.482691\pi$
$11$	0.360532	0.108704	0.0543522	+	0.998522i	$0.482691\pi$
$12$	0	0
$13$	5.26123	1.45920	0.729601	−	0.683873i	$-0.239706\pi$
$13$	5.26123	1.45920	0.729601	+	0.683873i	$0.239706\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.370852	−0.0899449	−0.0449724	−	0.998988i	$-0.514320\pi$
$17$	−0.370852	−0.0899449	−0.0449724	+	0.998988i	$0.514320\pi$
$18$	0	0
$19$	−4.60586	−1.05666	−0.528328	−	0.849040i	$-0.677181\pi$
$19$	−4.60586	−1.05666	−0.528328	+	0.849040i	$0.677181\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.939238	−0.174412	−0.0872061	−	0.996190i	$-0.527794\pi$
$29$	−0.939238	−0.174412	−0.0872061	+	0.996190i	$0.527794\pi$
$30$	0	0
$31$	9.66662	1.73618	0.868088	−	0.496411i	$-0.165349\pi$
$31$	9.66662	1.73618	0.868088	+	0.496411i	$0.165349\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.73770	0.800817
$36$	0	0
$37$	3.26862	0.537357	0.268679	−	0.963230i	$-0.413413\pi$
$37$	3.26862	0.537357	0.268679	+	0.963230i	$0.413413\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.29977	−0.827685	−0.413843	−	0.910348i	$-0.635814\pi$
$41$	−5.29977	−0.827685	−0.413843	+	0.910348i	$0.635814\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.25491	0.183048	0.0915240	−	0.995803i	$-0.470826\pi$
$47$	1.25491	0.183048	0.0915240	+	0.995803i	$0.470826\pi$
$48$	0	0
$49$	15.4458	2.20654
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.9278	−1.50106	−0.750528	−	0.660839i	$-0.770201\pi$
$53$	−10.9278	−1.50106	−0.750528	+	0.660839i	$0.770201\pi$
$54$	0	0
$55$	0.360532	0.0486141
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.66955	1.25887	0.629434	−	0.777054i	$-0.283287\pi$
$59$	9.66955	1.25887	0.629434	+	0.777054i	$0.283287\pi$
$60$	0	0
$61$	9.71441	1.24380	0.621901	−	0.783096i	$-0.286361\pi$
$61$	9.71441	1.24380	0.621901	+	0.783096i	$0.286361\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.26123	0.652575
$66$	0	0
$67$	−7.07001	−0.863739	−0.431870	−	0.901936i	$-0.642146\pi$
$67$	−7.07001	−0.863739	−0.431870	+	0.901936i	$0.642146\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.3747	−1.34993	−0.674965	−	0.737850i	$-0.735841\pi$
$71$	−11.3747	−1.34993	−0.674965	+	0.737850i	$0.735841\pi$
$72$	0	0
$73$	0.745086	0.0872057	0.0436029	−	0.999049i	$-0.486116\pi$
$73$	0.745086	0.0872057	0.0436029	+	0.999049i	$0.486116\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.70809	0.194655
$78$	0	0
$79$	0.415709	0.0467709	0.0233854	−	0.999727i	$-0.492556\pi$
$79$	0.415709	0.0467709	0.0233854	+	0.999727i	$0.492556\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.26862	1.01736	0.508681	−	0.860955i	$-0.330133\pi$
$83$	9.26862	1.01736	0.508681	+	0.860955i	$0.330133\pi$
$84$	0	0
$85$	−0.370852	−0.0402246
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.6122	−1.33689	−0.668444	−	0.743763i	$-0.733039\pi$
$89$	−12.6122	−1.33689	−0.668444	+	0.743763i	$0.733039\pi$
$90$	0	0
$91$	24.9261	2.61297
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.60586	−0.472551
$96$	0	0
$97$	14.0404	1.42559	0.712793	−	0.701374i	$-0.247430\pi$
$97$	14.0404	1.42559	0.712793	+	0.701374i	$0.247430\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.7440	1.26808	0.634038	−	0.773302i	$-0.281396\pi$
$101$	12.7440	1.26808	0.634038	+	0.773302i	$0.281396\pi$
$102$	0	0
$103$	−3.31347	−0.326486	−0.163243	−	0.986586i	$-0.552195\pi$
$103$	−3.31347	−0.326486	−0.163243	+	0.986586i	$0.552195\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.13184	0.786135	0.393067	−	0.919510i	$-0.371414\pi$
$107$	8.13184	0.786135	0.393067	+	0.919510i	$0.371414\pi$
$108$	0	0
$109$	−5.79508	−0.555068	−0.277534	−	0.960716i	$-0.589517\pi$
$109$	−5.79508	−0.555068	−0.277534	+	0.960716i	$0.589517\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.60724	0.715629	0.357815	−	0.933793i	$-0.383522\pi$
$113$	7.60724	0.715629	0.357815	+	0.933793i	$0.383522\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.75699	−0.161063
$120$	0	0
$121$	−10.8700	−0.988183
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.13077	−0.366547	−0.183273	−	0.983062i	$-0.558669\pi$
$127$	−4.13077	−0.366547	−0.183273	+	0.983062i	$0.558669\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	17.7582	1.55154	0.775770	−	0.631016i	$-0.217362\pi$
$131$	17.7582	1.55154	0.775770	+	0.631016i	$0.217362\pi$
$132$	0	0
$133$	−21.8212	−1.89213
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.86954	−0.586905	−0.293452	−	0.955974i	$-0.594804\pi$
$137$	−6.86954	−0.586905	−0.293452	+	0.955974i	$0.594804\pi$
$138$	0	0
$139$	20.6635	1.75266	0.876330	−	0.481712i	$-0.159985\pi$
$139$	20.6635	1.75266	0.876330	+	0.481712i	$0.159985\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.89684	0.158622
$144$	0	0
$145$	−0.939238	−0.0779995
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.4763	1.34979	0.674896	−	0.737912i	$-0.264188\pi$
$149$	16.4763	1.34979	0.674896	+	0.737912i	$0.264188\pi$
$150$	0	0
$151$	−8.89224	−0.723640	−0.361820	−	0.932248i	$-0.617844\pi$
$151$	−8.89224	−0.723640	−0.361820	+	0.932248i	$0.617844\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	9.66662	0.776441
$156$	0	0
$157$	−6.62249	−0.528532	−0.264266	−	0.964450i	$-0.585130\pi$
$157$	−6.62249	−0.528532	−0.264266	+	0.964450i	$0.585130\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.73770	0.373383
$162$	0	0
$163$	−16.0779	−1.25932	−0.629658	−	0.776872i	$-0.716805\pi$
$163$	−16.0779	−1.25932	−0.629658	+	0.776872i	$0.716805\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.8763	−1.07378	−0.536891	−	0.843651i	$-0.680402\pi$
$167$	−13.8763	−1.07378	−0.536891	+	0.843651i	$0.680402\pi$
$168$	0	0
$169$	14.6805	1.12927
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.05518	0.156252	0.0781261	−	0.996943i	$-0.475106\pi$
$173$	2.05518	0.156252	0.0781261	+	0.996943i	$0.475106\pi$
$174$	0	0
$175$	4.73770	0.358136
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−17.6478	−1.31906	−0.659531	−	0.751678i	$-0.729245\pi$
$179$	−17.6478	−1.31906	−0.659531	+	0.751678i	$0.729245\pi$
$180$	0	0
$181$	2.85477	0.212193	0.106097	−	0.994356i	$-0.466165\pi$
$181$	2.85477	0.212193	0.106097	+	0.994356i	$0.466165\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.26862	0.240313
$186$	0	0
$187$	−0.133704	−0.00977741
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.13677	−0.0822540	−0.0411270	−	0.999154i	$-0.513095\pi$
$191$	−1.13677	−0.0822540	−0.0411270	+	0.999154i	$0.513095\pi$
$192$	0	0
$193$	−26.4694	−1.90531	−0.952654	−	0.304055i	$-0.901659\pi$
$193$	−26.4694	−1.90531	−0.952654	+	0.304055i	$0.901659\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.44424	−0.387886	−0.193943	−	0.981013i	$-0.562128\pi$
$197$	−5.44424	−0.387886	−0.193943	+	0.981013i	$0.562128\pi$
$198$	0	0
$199$	−18.6122	−1.31938	−0.659691	−	0.751537i	$-0.729313\pi$
$199$	−18.6122	−1.31938	−0.659691	+	0.751537i	$0.729313\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.44983	−0.312317
$204$	0	0
$205$	−5.29977	−0.370152
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.66056	−0.114863
$210$	0	0
$211$	6.10003	0.419944	0.209972	−	0.977707i	$-0.432663\pi$
$211$	6.10003	0.419944	0.209972	+	0.977707i	$0.432663\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	45.7975	3.10894
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.95114	−0.131248
$222$	0	0
$223$	−16.4136	−1.09913	−0.549567	−	0.835450i	$-0.685207\pi$
$223$	−16.4136	−1.09913	−0.549567	+	0.835450i	$0.685207\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.21171	−0.0804240	−0.0402120	−	0.999191i	$-0.512803\pi$
$227$	−1.21171	−0.0804240	−0.0402120	+	0.999191i	$0.512803\pi$
$228$	0	0
$229$	−22.8293	−1.50860	−0.754300	−	0.656529i	$-0.772024\pi$
$229$	−22.8293	−1.50860	−0.754300	+	0.656529i	$0.772024\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−21.9420	−1.43747	−0.718735	−	0.695284i	$-0.755278\pi$
$233$	−21.9420	−1.43747	−0.718735	+	0.695284i	$0.755278\pi$
$234$	0	0
$235$	1.25491	0.0818616
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.58274	0.555172	0.277586	−	0.960701i	$-0.410466\pi$
$239$	8.58274	0.555172	0.277586	+	0.960701i	$0.410466\pi$
$240$	0	0
$241$	15.3857	0.991079	0.495540	−	0.868585i	$-0.334970\pi$
$241$	15.3857	0.991079	0.495540	+	0.868585i	$0.334970\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	15.4458	0.986795
$246$	0	0
$247$	−24.2325	−1.54187
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.85477	0.558908	0.279454	−	0.960159i	$-0.409847\pi$
$251$	8.85477	0.558908	0.279454	+	0.960159i	$0.409847\pi$
$252$	0	0
$253$	0.360532	0.0226664
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.6008	1.40980	0.704899	−	0.709308i	$-0.250992\pi$
$257$	22.6008	1.40980	0.704899	+	0.709308i	$0.250992\pi$
$258$	0	0
$259$	15.4857	0.962236
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.73590	0.477016	0.238508	−	0.971141i	$-0.423342\pi$
$263$	7.73590	0.477016	0.238508	+	0.971141i	$0.423342\pi$
$264$	0	0
$265$	−10.9278	−0.671292
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.87154	0.236052	0.118026	−	0.993011i	$-0.462343\pi$
$269$	3.87154	0.236052	0.118026	+	0.993011i	$0.462343\pi$
$270$	0	0
$271$	4.24705	0.257990	0.128995	−	0.991645i	$-0.458825\pi$
$271$	4.24705	0.257990	0.128995	+	0.991645i	$0.458825\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.360532	0.0217409
$276$	0	0
$277$	−7.26754	−0.436664	−0.218332	−	0.975875i	$-0.570062\pi$
$277$	−7.26754	−0.436664	−0.218332	+	0.975875i	$0.570062\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	13.0131	0.776297	0.388148	−	0.921597i	$-0.373115\pi$
$281$	13.0131	0.776297	0.388148	+	0.921597i	$0.373115\pi$
$282$	0	0
$283$	17.8342	1.06013	0.530067	−	0.847956i	$-0.322167\pi$
$283$	17.8342	1.06013	0.530067	+	0.847956i	$0.322167\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−25.1087	−1.48212
$288$	0	0
$289$	−16.8625	−0.991910
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−7.32291	−0.427809	−0.213905	−	0.976855i	$-0.568618\pi$
$293$	−7.32291	−0.427809	−0.213905	+	0.976855i	$0.568618\pi$
$294$	0	0
$295$	9.66955	0.562983
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.26123	0.304265
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.71441	0.556245
$306$	0	0
$307$	−29.1127	−1.66155	−0.830775	−	0.556608i	$-0.812103\pi$
$307$	−29.1127	−1.66155	−0.830775	+	0.556608i	$0.812103\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−0.458912	−0.0260225	−0.0130113	−	0.999915i	$-0.504142\pi$
$311$	−0.458912	−0.0260225	−0.0130113	+	0.999915i	$0.504142\pi$
$312$	0	0
$313$	−17.8279	−1.00769	−0.503846	−	0.863794i	$-0.668082\pi$
$313$	−17.8279	−1.00769	−0.503846	+	0.863794i	$0.668082\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−23.8799	−1.34123	−0.670614	−	0.741807i	$-0.733969\pi$
$317$	−23.8799	−1.34123	−0.670614	+	0.741807i	$0.733969\pi$
$318$	0	0
$319$	−0.338625	−0.0189594
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.70809	0.0950407
$324$	0	0
$325$	5.26123	0.291840
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.94540	0.327781
$330$	0	0
$331$	30.0214	1.65013	0.825063	−	0.565041i	$-0.191140\pi$
$331$	30.0214	1.65013	0.825063	+	0.565041i	$0.191140\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7.07001	−0.386276
$336$	0	0
$337$	11.9440	0.650631	0.325316	−	0.945605i	$-0.394529\pi$
$337$	11.9440	0.650631	0.325316	+	0.945605i	$0.394529\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.48512	0.188730
$342$	0	0
$343$	40.0136	2.16053
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.873132	0.0468722	0.0234361	−	0.999725i	$-0.492539\pi$
$347$	0.873132	0.0468722	0.0234361	+	0.999725i	$0.492539\pi$
$348$	0	0
$349$	−18.8868	−1.01099	−0.505493	−	0.862831i	$-0.668689\pi$
$349$	−18.8868	−1.01099	−0.505493	+	0.862831i	$0.668689\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.43875	0.289475	0.144738	−	0.989470i	$-0.453766\pi$
$353$	5.43875	0.289475	0.144738	+	0.989470i	$0.453766\pi$
$354$	0	0
$355$	−11.3747	−0.603707
$356$	0	0
$357$	0	0
$358$	0	0
$359$	23.4229	1.23622	0.618108	−	0.786093i	$-0.287899\pi$
$359$	23.4229	1.23622	0.618108	+	0.786093i	$0.287899\pi$
$360$	0	0
$361$	2.21390	0.116521
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.745086	0.0389996
$366$	0	0
$367$	27.2014	1.41990	0.709951	−	0.704252i	$-0.248717\pi$
$367$	27.2014	1.41990	0.709951	+	0.704252i	$0.248717\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−51.7728	−2.68791
$372$	0	0
$373$	20.0431	1.03779	0.518897	−	0.854837i	$-0.326343\pi$
$373$	20.0431	1.03779	0.518897	+	0.854837i	$0.326343\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.94155	−0.254503
$378$	0	0
$379$	−12.8364	−0.659362	−0.329681	−	0.944092i	$-0.606941\pi$
$379$	−12.8364	−0.659362	−0.329681	+	0.944092i	$0.606941\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	30.5405	1.56055	0.780273	−	0.625439i	$-0.215080\pi$
$383$	30.5405	1.56055	0.780273	+	0.625439i	$0.215080\pi$
$384$	0	0
$385$	1.70809	0.0870524
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−32.3392	−1.63966	−0.819831	−	0.572605i	$-0.805933\pi$
$389$	−32.3392	−1.63966	−0.819831	+	0.572605i	$0.805933\pi$
$390$	0	0
$391$	−0.370852	−0.0187548
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.415709	0.0209166
$396$	0	0
$397$	28.9147	1.45119	0.725594	−	0.688123i	$-0.241565\pi$
$397$	28.9147	1.45119	0.725594	+	0.688123i	$0.241565\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−24.2862	−1.21279	−0.606397	−	0.795162i	$-0.707386\pi$
$401$	−24.2862	−1.21279	−0.606397	+	0.795162i	$0.707386\pi$
$402$	0	0
$403$	50.8583	2.53343
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.17844	0.0584131
$408$	0	0
$409$	31.8371	1.57424	0.787122	−	0.616797i	$-0.211570\pi$
$409$	31.8371	1.57424	0.787122	+	0.616797i	$0.211570\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	45.8114	2.25423
$414$	0	0
$415$	9.26862	0.454978
$416$	0	0
$417$	0	0
$418$	0	0
$419$	21.4880	1.04976	0.524879	−	0.851177i	$-0.324110\pi$
$419$	21.4880	1.04976	0.524879	+	0.851177i	$0.324110\pi$
$420$	0	0
$421$	−13.2930	−0.647859	−0.323930	−	0.946081i	$-0.605004\pi$
$421$	−13.2930	−0.647859	−0.323930	+	0.946081i	$0.605004\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.370852	−0.0179890
$426$	0	0
$427$	46.0239	2.22725
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0.534460	0.0257440	0.0128720	−	0.999917i	$-0.495903\pi$
$431$	0.534460	0.0257440	0.0128720	+	0.999917i	$0.495903\pi$
$432$	0	0
$433$	16.3377	0.785140	0.392570	−	0.919722i	$-0.371586\pi$
$433$	16.3377	0.785140	0.392570	+	0.919722i	$0.371586\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.60586	−0.220328
$438$	0	0
$439$	28.7573	1.37251	0.686255	−	0.727361i	$-0.259254\pi$
$439$	28.7573	1.37251	0.686255	+	0.727361i	$0.259254\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	15.6542	0.743751	0.371876	−	0.928283i	$-0.378715\pi$
$443$	15.6542	0.743751	0.371876	+	0.928283i	$0.378715\pi$
$444$	0	0
$445$	−12.6122	−0.597874
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−38.4302	−1.81363	−0.906817	−	0.421525i	$-0.861495\pi$
$449$	−38.4302	−1.81363	−0.906817	+	0.421525i	$0.861495\pi$
$450$	0	0
$451$	−1.91074	−0.0899730
$452$	0	0
$453$	0	0
$454$	0	0
$455$	24.9261	1.16855
$456$	0	0
$457$	−37.2199	−1.74107	−0.870536	−	0.492104i	$-0.836228\pi$
$457$	−37.2199	−1.74107	−0.870536	+	0.492104i	$0.836228\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.25491	0.151596	0.0757982	−	0.997123i	$-0.475850\pi$
$461$	3.25491	0.151596	0.0757982	+	0.997123i	$0.475850\pi$
$462$	0	0
$463$	29.9534	1.39205	0.696027	−	0.718016i	$-0.254950\pi$
$463$	29.9534	1.39205	0.696027	+	0.718016i	$0.254950\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	39.3302	1.81999	0.909993	−	0.414623i	$-0.136087\pi$
$467$	39.3302	1.81999	0.909993	+	0.414623i	$0.136087\pi$
$468$	0	0
$469$	−33.4956	−1.54668
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−4.60586	−0.211331
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8.18660	−0.374055	−0.187028	−	0.982355i	$-0.559885\pi$
$479$	−8.18660	−0.374055	−0.187028	+	0.982355i	$0.559885\pi$
$480$	0	0
$481$	17.1969	0.784113
$482$	0	0
$483$	0	0
$484$	0	0
$485$	14.0404	0.637542
$486$	0	0
$487$	−31.9963	−1.44989	−0.724946	−	0.688806i	$-0.758135\pi$
$487$	−31.9963	−1.44989	−0.724946	+	0.688806i	$0.758135\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.12584	0.321585	0.160792	−	0.986988i	$-0.448595\pi$
$491$	7.12584	0.321585	0.160792	+	0.986988i	$0.448595\pi$
$492$	0	0
$493$	0.348319	0.0156875
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−53.8899	−2.41729
$498$	0	0
$499$	23.4219	1.04851	0.524254	−	0.851562i	$-0.324344\pi$
$499$	23.4219	1.04851	0.524254	+	0.851562i	$0.324344\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.94170	0.175752	0.0878758	−	0.996131i	$-0.471992\pi$
$503$	3.94170	0.175752	0.0878758	+	0.996131i	$0.471992\pi$
$504$	0	0
$505$	12.7440	0.567101
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−38.5237	−1.70753	−0.853766	−	0.520656i	$-0.825687\pi$
$509$	−38.5237	−1.70753	−0.853766	+	0.520656i	$0.825687\pi$
$510$	0	0
$511$	3.52999	0.156158
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.31347	−0.146009
$516$	0	0
$517$	0.452436	0.0198981
$518$	0	0
$519$	0	0
$520$	0	0
$521$	39.2428	1.71926	0.859630	−	0.510917i	$-0.170694\pi$
$521$	39.2428	1.71926	0.859630	+	0.510917i	$0.170694\pi$
$522$	0	0
$523$	−39.0190	−1.70618	−0.853090	−	0.521763i	$-0.825274\pi$
$523$	−39.0190	−1.70618	−0.853090	+	0.521763i	$0.825274\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.58489	−0.156160
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−27.8833	−1.20776
$534$	0	0
$535$	8.13184	0.351570
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.56870	0.239861
$540$	0	0
$541$	2.54061	0.109230	0.0546148	−	0.998508i	$-0.482607\pi$
$541$	2.54061	0.109230	0.0546148	+	0.998508i	$0.482607\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.79508	−0.248234
$546$	0	0
$547$	6.78776	0.290224	0.145112	−	0.989415i	$-0.453646\pi$
$547$	6.78776	0.290224	0.145112	+	0.989415i	$0.453646\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.32600	0.184294
$552$	0	0
$553$	1.96950	0.0837517
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.7086	1.04694	0.523468	−	0.852045i	$-0.324638\pi$
$557$	24.7086	1.04694	0.523468	+	0.852045i	$0.324638\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	28.1423	1.18606	0.593029	−	0.805181i	$-0.297932\pi$
$563$	28.1423	1.18606	0.593029	+	0.805181i	$0.297932\pi$
$564$	0	0
$565$	7.60724	0.320039
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−41.1938	−1.72694	−0.863468	−	0.504404i	$-0.831712\pi$
$569$	−41.1938	−1.72694	−0.863468	+	0.504404i	$0.831712\pi$
$570$	0	0
$571$	32.4687	1.35877	0.679387	−	0.733780i	$-0.262246\pi$
$571$	32.4687	1.35877	0.679387	+	0.733780i	$0.262246\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	6.61770	0.275498	0.137749	−	0.990467i	$-0.456013\pi$
$577$	6.61770	0.275498	0.137749	+	0.990467i	$0.456013\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	43.9119	1.82177
$582$	0	0
$583$	−3.93984	−0.163171
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.18008	0.337628	0.168814	−	0.985648i	$-0.446006\pi$
$587$	8.18008	0.337628	0.168814	+	0.985648i	$0.446006\pi$
$588$	0	0
$589$	−44.5230	−1.83454
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−29.7489	−1.22164	−0.610821	−	0.791768i	$-0.709161\pi$
$593$	−29.7489	−1.22164	−0.610821	+	0.791768i	$0.709161\pi$
$594$	0	0
$595$	−1.75699	−0.0720294
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−39.4069	−1.61012	−0.805061	−	0.593192i	$-0.797868\pi$
$599$	−39.4069	−1.61012	−0.805061	+	0.593192i	$0.797868\pi$
$600$	0	0
$601$	−34.3183	−1.39987	−0.699937	−	0.714205i	$-0.746788\pi$
$601$	−34.3183	−1.39987	−0.699937	+	0.714205i	$0.746788\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10.8700	−0.441929
$606$	0	0
$607$	−11.7570	−0.477200	−0.238600	−	0.971118i	$-0.576688\pi$
$607$	−11.7570	−0.477200	−0.238600	+	0.971118i	$0.576688\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.60239	0.267104
$612$	0	0
$613$	5.86862	0.237031	0.118516	−	0.992952i	$-0.462186\pi$
$613$	5.86862	0.237031	0.118516	+	0.992952i	$0.462186\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.45842	0.340523	0.170262	−	0.985399i	$-0.445539\pi$
$617$	8.45842	0.340523	0.170262	+	0.985399i	$0.445539\pi$
$618$	0	0
$619$	0.851047	0.0342065	0.0171032	−	0.999854i	$-0.494556\pi$
$619$	0.851047	0.0342065	0.0171032	+	0.999854i	$0.494556\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−59.7527	−2.39394
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.21217	−0.0483325
$630$	0	0
$631$	19.7131	0.784768	0.392384	−	0.919802i	$-0.371650\pi$
$631$	19.7131	0.784768	0.392384	+	0.919802i	$0.371650\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.13077	−0.163925
$636$	0	0
$637$	81.2638	3.21979
$638$	0	0
$639$	0	0
$640$	0	0
$641$	34.3593	1.35711	0.678555	−	0.734550i	$-0.262607\pi$
$641$	34.3593	1.35711	0.678555	+	0.734550i	$0.262607\pi$
$642$	0	0
$643$	−24.2817	−0.957578	−0.478789	−	0.877930i	$-0.658924\pi$
$643$	−24.2817	−0.957578	−0.478789	+	0.877930i	$0.658924\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	30.1465	1.18518	0.592590	−	0.805504i	$-0.298105\pi$
$647$	30.1465	1.18518	0.592590	+	0.805504i	$0.298105\pi$
$648$	0	0
$649$	3.48618	0.136845
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.17298	0.280700	0.140350	−	0.990102i	$-0.455177\pi$
$653$	7.17298	0.280700	0.140350	+	0.990102i	$0.455177\pi$
$654$	0	0
$655$	17.7582	0.693870
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.8521	0.500645	0.250323	−	0.968162i	$-0.419463\pi$
$659$	12.8521	0.500645	0.250323	+	0.968162i	$0.419463\pi$
$660$	0	0
$661$	3.72426	0.144857	0.0724285	−	0.997374i	$-0.476925\pi$
$661$	3.72426	0.144857	0.0724285	+	0.997374i	$0.476925\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−21.8212	−0.846188
$666$	0	0
$667$	−0.939238	−0.0363675
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.50235	0.135207
$672$	0	0
$673$	13.5204	0.521175	0.260587	−	0.965450i	$-0.416084\pi$
$673$	13.5204	0.521175	0.260587	+	0.965450i	$0.416084\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17.8534	0.686161	0.343081	−	0.939306i	$-0.388530\pi$
$677$	17.8534	0.686161	0.343081	+	0.939306i	$0.388530\pi$
$678$	0	0
$679$	66.5192	2.55277
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−15.8200	−0.605337	−0.302669	−	0.953096i	$-0.597878\pi$
$683$	−15.8200	−0.605337	−0.302669	+	0.953096i	$0.597878\pi$
$684$	0	0
$685$	−6.86954	−0.262472
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−57.4939	−2.19034
$690$	0	0
$691$	9.45476	0.359676	0.179838	−	0.983696i	$-0.442443\pi$
$691$	9.45476	0.359676	0.179838	+	0.983696i	$0.442443\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	20.6635	0.783813
$696$	0	0
$697$	1.96543	0.0744460
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−29.0096	−1.09568	−0.547838	−	0.836584i	$-0.684549\pi$
$701$	−29.0096	−1.09568	−0.547838	+	0.836584i	$0.684549\pi$
$702$	0	0
$703$	−15.0548	−0.567801
$704$	0	0
$705$	0	0
$706$	0	0
$707$	60.3773	2.27072
$708$	0	0
$709$	−8.22423	−0.308868	−0.154434	−	0.988003i	$-0.549355\pi$
$709$	−8.22423	−0.308868	−0.154434	+	0.988003i	$0.549355\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	9.66662	0.362018
$714$	0	0
$715$	1.89684	0.0709378
$716$	0	0
$717$	0	0
$718$	0	0
$719$	12.5468	0.467917	0.233958	−	0.972247i	$-0.424832\pi$
$719$	12.5468	0.467917	0.233958	+	0.972247i	$0.424832\pi$
$720$	0	0
$721$	−15.6982	−0.584633
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.939238	−0.0348824
$726$	0	0
$727$	−29.1952	−1.08279	−0.541395	−	0.840768i	$-0.682104\pi$
$727$	−29.1952	−1.08279	−0.541395	+	0.840768i	$0.682104\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−36.0414	−1.33122	−0.665611	−	0.746299i	$-0.731829\pi$
$733$	−36.0414	−1.33122	−0.665611	+	0.746299i	$0.731829\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.54896	−0.0938923
$738$	0	0
$739$	0.728002	0.0267800	0.0133900	−	0.999910i	$-0.495738\pi$
$739$	0.728002	0.0267800	0.0133900	+	0.999910i	$0.495738\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	32.8880	1.20655	0.603273	−	0.797535i	$-0.293863\pi$
$743$	32.8880	1.20655	0.603273	+	0.797535i	$0.293863\pi$
$744$	0	0
$745$	16.4763	0.603646
$746$	0	0
$747$	0	0
$748$	0	0
$749$	38.5262	1.40772
$750$	0	0
$751$	14.3337	0.523044	0.261522	−	0.965197i	$-0.415776\pi$
$751$	14.3337	0.523044	0.261522	+	0.965197i	$0.415776\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−8.89224	−0.323622
$756$	0	0
$757$	−0.0587519	−0.00213537	−0.00106769	−	0.999999i	$-0.500340\pi$
$757$	−0.0587519	−0.00213537	−0.00106769	+	0.999999i	$0.500340\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−45.7135	−1.65711	−0.828556	−	0.559906i	$-0.810837\pi$
$761$	−45.7135	−1.65711	−0.828556	+	0.559906i	$0.810837\pi$
$762$	0	0
$763$	−27.4553	−0.993950
$764$	0	0
$765$	0	0
$766$	0	0
$767$	50.8737	1.83694
$768$	0	0
$769$	−39.3949	−1.42061	−0.710307	−	0.703892i	$-0.751444\pi$
$769$	−39.3949	−1.42061	−0.710307	+	0.703892i	$0.751444\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−21.1799	−0.761788	−0.380894	−	0.924619i	$-0.624384\pi$
$773$	−21.1799	−0.761788	−0.380894	+	0.924619i	$0.624384\pi$
$774$	0	0
$775$	9.66662	0.347235
$776$	0	0
$777$	0	0
$778$	0	0
$779$	24.4100	0.874578
$780$	0	0
$781$	−4.10094	−0.146743
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.62249	−0.236367
$786$	0	0
$787$	21.5154	0.766941	0.383470	−	0.923553i	$-0.374729\pi$
$787$	21.5154	0.766941	0.383470	+	0.923553i	$0.374729\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	36.0408	1.28146
$792$	0	0
$793$	51.1097	1.81496
$794$	0	0
$795$	0	0
$796$	0	0
$797$	54.7227	1.93838	0.969189	−	0.246320i	$-0.0792215\pi$
$797$	54.7227	1.93838	0.969189	+	0.246320i	$0.0792215\pi$
$798$	0	0
$799$	−0.465388	−0.0164642
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0.268627	0.00947965
$804$	0	0
$805$	4.73770	0.166982
$806$	0	0
$807$	0	0
$808$	0	0
$809$	43.5554	1.53133	0.765663	−	0.643242i	$-0.222411\pi$
$809$	43.5554	1.53133	0.765663	+	0.643242i	$0.222411\pi$
$810$	0	0
$811$	−13.1837	−0.462944	−0.231472	−	0.972842i	$-0.574354\pi$
$811$	−13.1837	−0.462944	−0.231472	+	0.972842i	$0.574354\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−16.0779	−0.563183
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.63328	0.126803	0.0634013	−	0.997988i	$-0.479805\pi$
$821$	3.63328	0.126803	0.0634013	+	0.997988i	$0.479805\pi$
$822$	0	0
$823$	−39.5044	−1.37704	−0.688518	−	0.725220i	$-0.741738\pi$
$823$	−39.5044	−1.37704	−0.688518	+	0.725220i	$0.741738\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−48.3819	−1.68240	−0.841202	−	0.540721i	$-0.818152\pi$
$827$	−48.3819	−1.68240	−0.841202	+	0.540721i	$0.818152\pi$
$828$	0	0
$829$	−44.5864	−1.54855	−0.774275	−	0.632850i	$-0.781885\pi$
$829$	−44.5864	−1.54855	−0.774275	+	0.632850i	$0.781885\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−5.72810	−0.198467
$834$	0	0
$835$	−13.8763	−0.480210
$836$	0	0
$837$	0	0
$838$	0	0
$839$	29.2324	1.00921	0.504606	−	0.863350i	$-0.331638\pi$
$839$	29.2324	1.00921	0.504606	+	0.863350i	$0.331638\pi$
$840$	0	0
$841$	−28.1178	−0.969580
$842$	0	0
$843$	0	0
$844$	0	0
$845$	14.6805	0.505025
$846$	0	0
$847$	−51.4989	−1.76952
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.26862	0.112047
$852$	0	0
$853$	50.5599	1.73114	0.865569	−	0.500790i	$-0.166957\pi$
$853$	50.5599	1.73114	0.865569	+	0.500790i	$0.166957\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21.9568	0.750030	0.375015	−	0.927019i	$-0.377638\pi$
$857$	21.9568	0.750030	0.375015	+	0.927019i	$0.377638\pi$
$858$	0	0
$859$	13.7671	0.469726	0.234863	−	0.972029i	$-0.424536\pi$
$859$	13.7671	0.469726	0.234863	+	0.972029i	$0.424536\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26.5748	0.904618	0.452309	−	0.891861i	$-0.350600\pi$
$863$	26.5748	0.904618	0.452309	+	0.891861i	$0.350600\pi$
$864$	0	0
$865$	2.05518	0.0698781
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0.149876	0.00508420
$870$	0	0
$871$	−37.1969	−1.26037
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.73770	0.160163
$876$	0	0
$877$	45.9414	1.55133	0.775665	−	0.631145i	$-0.217415\pi$
$877$	45.9414	1.55133	0.775665	+	0.631145i	$0.217415\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	23.9054	0.805392	0.402696	−	0.915334i	$-0.368073\pi$
$881$	23.9054	0.805392	0.402696	+	0.915334i	$0.368073\pi$
$882$	0	0
$883$	52.4002	1.76341	0.881703	−	0.471804i	$-0.156397\pi$
$883$	52.4002	1.76341	0.881703	+	0.471804i	$0.156397\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.7608	1.10000	0.550000	−	0.835165i	$-0.314628\pi$
$887$	32.7608	1.10000	0.550000	+	0.835165i	$0.314628\pi$
$888$	0	0
$889$	−19.5703	−0.656368
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.77995	−0.193419
$894$	0	0
$895$	−17.6478	−0.589902
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−9.07926	−0.302810
$900$	0	0
$901$	4.05262	0.135012
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.85477	0.0948957
$906$	0	0
$907$	20.9426	0.695388	0.347694	−	0.937608i	$-0.386965\pi$
$907$	20.9426	0.695388	0.347694	+	0.937608i	$0.386965\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−40.6418	−1.34652	−0.673262	−	0.739404i	$-0.735107\pi$
$911$	−40.6418	−1.34652	−0.673262	+	0.739404i	$0.735107\pi$
$912$	0	0
$913$	3.34163	0.110592
$914$	0	0
$915$	0	0
$916$	0	0
$917$	84.1330	2.77831
$918$	0	0
$919$	−50.3241	−1.66004	−0.830019	−	0.557735i	$-0.811670\pi$
$919$	−50.3241	−1.66004	−0.830019	+	0.557735i	$0.811670\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−59.8449	−1.96982
$924$	0	0
$925$	3.26862	0.107471
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−24.1954	−0.793825	−0.396912	−	0.917856i	$-0.629918\pi$
$929$	−24.1954	−0.793825	−0.396912	+	0.917856i	$0.629918\pi$
$930$	0	0
$931$	−71.1411	−2.33155
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−0.133704	−0.00437259
$936$	0	0
$937$	−37.2988	−1.21850	−0.609250	−	0.792978i	$-0.708529\pi$
$937$	−37.2988	−1.21850	−0.609250	+	0.792978i	$0.708529\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−16.6467	−0.542667	−0.271334	−	0.962485i	$-0.587465\pi$
$941$	−16.6467	−0.542667	−0.271334	+	0.962485i	$0.587465\pi$
$942$	0	0
$943$	−5.29977	−0.172584
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.1868	0.980938	0.490469	−	0.871459i	$-0.336825\pi$
$947$	30.1868	0.980938	0.490469	+	0.871459i	$0.336825\pi$
$948$	0	0
$949$	3.92007	0.127251
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−34.6992	−1.12402	−0.562008	−	0.827132i	$-0.689971\pi$
$953$	−34.6992	−1.12402	−0.562008	+	0.827132i	$0.689971\pi$
$954$	0	0
$955$	−1.13677	−0.0367851
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−32.5458	−1.05096
$960$	0	0
$961$	62.4435	2.01431
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−26.4694	−0.852080
$966$	0	0
$967$	−11.4788	−0.369133	−0.184566	−	0.982820i	$-0.559088\pi$
$967$	−11.4788	−0.369133	−0.184566	+	0.982820i	$0.559088\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	46.8441	1.50330	0.751650	−	0.659563i	$-0.229258\pi$
$971$	46.8441	1.50330	0.751650	+	0.659563i	$0.229258\pi$
$972$	0	0
$973$	97.8977	3.13846
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−39.3737	−1.25968	−0.629838	−	0.776727i	$-0.716879\pi$
$977$	−39.3737	−1.25968	−0.629838	+	0.776727i	$0.716879\pi$
$978$	0	0
$979$	−4.54709	−0.145326
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−42.3505	−1.35077	−0.675385	−	0.737465i	$-0.736023\pi$
$983$	−42.3505	−1.35077	−0.675385	+	0.737465i	$0.736023\pi$
$984$	0	0
$985$	−5.44424	−0.173468
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−2.00104	−0.0635652	−0.0317826	−	0.999495i	$-0.510118\pi$
$991$	−2.00104	−0.0635652	−0.0317826	+	0.999495i	$0.510118\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−18.6122	−0.590045
$996$	0	0
$997$	1.97936	0.0626869	0.0313435	−	0.999509i	$-0.490021\pi$
$997$	1.97936	0.0626869	0.0313435	+	0.999509i	$0.490021\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bs.1.5		5
3.2	odd	2		920.2.a.j.1.3	✓	5
12.11	even	2		1840.2.a.v.1.3		5
15.2	even	4		4600.2.e.u.4049.6		10
15.8	even	4		4600.2.e.u.4049.5		10
15.14	odd	2		4600.2.a.be.1.3		5
24.5	odd	2		7360.2.a.co.1.3		5
24.11	even	2		7360.2.a.cp.1.3		5
60.59	even	2		9200.2.a.cu.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.j.1.3	✓	5	3.2	odd	2
1840.2.a.v.1.3		5	12.11	even	2
4600.2.a.be.1.3		5	15.14	odd	2
4600.2.e.u.4049.5		10	15.8	even	4
4600.2.e.u.4049.6		10	15.2	even	4
7360.2.a.co.1.3		5	24.5	odd	2
7360.2.a.cp.1.3		5	24.11	even	2
8280.2.a.bs.1.5		5	1.1	even	1	trivial
9200.2.a.cu.1.3		5	60.59	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:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +4.73770 q^{7} +0.360532 q^{11} +5.26123 q^{13} -0.370852 q^{17} -4.60586 q^{19} +1.00000 q^{23} +1.00000 q^{25} -0.939238 q^{29} +9.66662 q^{31} +4.73770 q^{35} +3.26862 q^{37} -5.29977 q^{41} +1.25491 q^{47} +15.4458 q^{49} -10.9278 q^{53} +0.360532 q^{55} +9.66955 q^{59} +9.71441 q^{61} +5.26123 q^{65} -7.07001 q^{67} -11.3747 q^{71} +0.745086 q^{73} +1.70809 q^{77} +0.415709 q^{79} +9.26862 q^{83} -0.370852 q^{85} -12.6122 q^{89} +24.9261 q^{91} -4.60586 q^{95} +14.0404 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{5} - 2 q^{7} + q^{11} + 4 q^{13} - 4 q^{17} + 7 q^{19} + 5 q^{23} + 5 q^{25} - 4 q^{29} + 19 q^{31} - 2 q^{35} + 15 q^{37} - 25 q^{41} + 11 q^{47} + 25 q^{49} - 3 q^{53} + q^{55} + q^{59}+ \cdots + 25 q^{97}+O(q^{100}) \) 5 * q + 5 * q^5 - 2 * q^7 + q^11 + 4 * q^13 - 4 * q^17 + 7 * q^19 + 5 * q^23 + 5 * q^25 - 4 * q^29 + 19 * q^31 - 2 * q^35 + 15 * q^37 - 25 * q^41 + 11 * q^47 + 25 * q^49 - 3 * q^53 + q^55 + q^59 - 5 * q^61 + 4 * q^65 + 9 * q^67 - q^71 - q^73 - 18 * q^77 - 2 * q^79 + 45 * q^83 - 4 * q^85 - 6 * q^89 + 11 * q^91 + 7 * q^95 + 25 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more