LMFDB - Embedded newform 8008.2.a.p.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 15x^{7} + 15x^{6} + 66x^{5} - 59x^{4} - 77x^{3} + 34x^{2} + 11x - 2 $ x^9 - x^8 - 15x^7 + 15x^6 + 66x^5 - 59x^4 - 77x^3 + 34x^2 + 11*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.141545$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.141545 q^{3} +0.343768 q^{5} -1.00000 q^{7} -2.97996 q^{9} +O(q^{10})$	$q+0.141545 q^{3} +0.343768 q^{5} -1.00000 q^{7} -2.97996 q^{9} +1.00000 q^{11} +1.00000 q^{13} +0.0486587 q^{15} -1.94296 q^{17} -0.383972 q^{19} -0.141545 q^{21} -1.66573 q^{23} -4.88182 q^{25} -0.846436 q^{27} +3.52781 q^{29} +8.92433 q^{31} +0.141545 q^{33} -0.343768 q^{35} +5.87381 q^{37} +0.141545 q^{39} +6.78021 q^{41} -1.87960 q^{43} -1.02442 q^{45} +2.51467 q^{47} +1.00000 q^{49} -0.275017 q^{51} +0.793466 q^{53} +0.343768 q^{55} -0.0543494 q^{57} -14.0287 q^{59} +6.36966 q^{61} +2.97996 q^{63} +0.343768 q^{65} -8.05936 q^{67} -0.235776 q^{69} +0.428251 q^{71} -12.1299 q^{73} -0.690999 q^{75} -1.00000 q^{77} +9.37219 q^{79} +8.82009 q^{81} -4.37797 q^{83} -0.667928 q^{85} +0.499346 q^{87} -4.95758 q^{89} -1.00000 q^{91} +1.26320 q^{93} -0.131997 q^{95} -13.5322 q^{97} -2.97996 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{3} - 4 q^{5} - 9 q^{7} + 4 q^{9}+O(q^{10}) $ 9 * q + q^3 - 4 * q^5 - 9 * q^7 + 4 * q^9	$ 9 q + q^{3} - 4 q^{5} - 9 q^{7} + 4 q^{9} + 9 q^{11} + 9 q^{13} - 9 q^{15} - 11 q^{17} + 10 q^{19} - q^{21} - 14 q^{23} - q^{25} - 5 q^{27} - 10 q^{29} + 5 q^{31} + q^{33} + 4 q^{35} - 16 q^{37} + q^{39} + 2 q^{41} + 4 q^{43} - 30 q^{45} + 9 q^{49} + 3 q^{51} - 23 q^{53} - 4 q^{55} + 14 q^{57} + 9 q^{59} - 14 q^{61} - 4 q^{63} - 4 q^{65} + 8 q^{67} - 26 q^{69} - 20 q^{71} - 23 q^{73} + 32 q^{75} - 9 q^{77} + 2 q^{79} - 11 q^{81} - 9 q^{83} - 3 q^{85} - 7 q^{87} - 6 q^{89} - 9 q^{91} - 19 q^{93} - 4 q^{95} - 3 q^{97} + 4 q^{99}+O(q^{100}) $ 9 * q + q^3 - 4 * q^5 - 9 * q^7 + 4 * q^9 + 9 * q^11 + 9 * q^13 - 9 * q^15 - 11 * q^17 + 10 * q^19 - q^21 - 14 * q^23 - q^25 - 5 * q^27 - 10 * q^29 + 5 * q^31 + q^33 + 4 * q^35 - 16 * q^37 + q^39 + 2 * q^41 + 4 * q^43 - 30 * q^45 + 9 * q^49 + 3 * q^51 - 23 * q^53 - 4 * q^55 + 14 * q^57 + 9 * q^59 - 14 * q^61 - 4 * q^63 - 4 * q^65 + 8 * q^67 - 26 * q^69 - 20 * q^71 - 23 * q^73 + 32 * q^75 - 9 * q^77 + 2 * q^79 - 11 * q^81 - 9 * q^83 - 3 * q^85 - 7 * q^87 - 6 * q^89 - 9 * q^91 - 19 * q^93 - 4 * q^95 - 3 * q^97 + 4 * q^99

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
$3$	0.141545	0.0817212	0.0408606	−	0.999165i	$-0.486990\pi$
$3$	0.141545	0.0817212	0.0408606	+	0.999165i	$0.486990\pi$
$4$	0	0
$5$	0.343768	0.153738	0.0768688	−	0.997041i	$-0.475508\pi$
$5$	0.343768	0.153738	0.0768688	+	0.997041i	$0.475508\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.97996	−0.993322
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0.0486587	0.0125636
$16$	0	0
$17$	−1.94296	−0.471238	−0.235619	−	0.971846i	$-0.575712\pi$
$17$	−1.94296	−0.471238	−0.235619	+	0.971846i	$0.575712\pi$
$18$	0	0
$19$	−0.383972	−0.0880892	−0.0440446	−	0.999030i	$-0.514024\pi$
$19$	−0.383972	−0.0880892	−0.0440446	+	0.999030i	$0.514024\pi$
$20$	0	0
$21$	−0.141545	−0.0308877
$22$	0	0
$23$	−1.66573	−0.347328	−0.173664	−	0.984805i	$-0.555561\pi$
$23$	−1.66573	−0.347328	−0.173664	+	0.984805i	$0.555561\pi$
$24$	0	0
$25$	−4.88182	−0.976365
$26$	0	0
$27$	−0.846436	−0.162897
$28$	0	0
$29$	3.52781	0.655098	0.327549	−	0.944834i	$-0.393777\pi$
$29$	3.52781	0.655098	0.327549	+	0.944834i	$0.393777\pi$
$30$	0	0
$31$	8.92433	1.60286	0.801429	−	0.598090i	$-0.204074\pi$
$31$	8.92433	1.60286	0.801429	+	0.598090i	$0.204074\pi$
$32$	0	0
$33$	0.141545	0.0246399
$34$	0	0
$35$	−0.343768	−0.0581074
$36$	0	0
$37$	5.87381	0.965649	0.482825	−	0.875717i	$-0.339611\pi$
$37$	5.87381	0.965649	0.482825	+	0.875717i	$0.339611\pi$
$38$	0	0
$39$	0.141545	0.0226654
$40$	0	0
$41$	6.78021	1.05889	0.529446	−	0.848344i	$-0.322400\pi$
$41$	6.78021	1.05889	0.529446	+	0.848344i	$0.322400\pi$
$42$	0	0
$43$	−1.87960	−0.286636	−0.143318	−	0.989677i	$-0.545777\pi$
$43$	−1.87960	−0.286636	−0.143318	+	0.989677i	$0.545777\pi$
$44$	0	0
$45$	−1.02442	−0.152711
$46$	0	0
$47$	2.51467	0.366803	0.183401	−	0.983038i	$-0.441289\pi$
$47$	2.51467	0.366803	0.183401	+	0.983038i	$0.441289\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.275017	−0.0385101
$52$	0	0
$53$	0.793466	0.108991	0.0544954	−	0.998514i	$-0.482645\pi$
$53$	0.793466	0.108991	0.0544954	+	0.998514i	$0.482645\pi$
$54$	0	0
$55$	0.343768	0.0463536
$56$	0	0
$57$	−0.0543494	−0.00719876
$58$	0	0
$59$	−14.0287	−1.82638	−0.913191	−	0.407531i	$-0.866390\pi$
$59$	−14.0287	−1.82638	−0.913191	+	0.407531i	$0.866390\pi$
$60$	0	0
$61$	6.36966	0.815551	0.407776	−	0.913082i	$-0.366305\pi$
$61$	6.36966	0.815551	0.407776	+	0.913082i	$0.366305\pi$
$62$	0	0
$63$	2.97996	0.375440
$64$	0	0
$65$	0.343768	0.0426392
$66$	0	0
$67$	−8.05936	−0.984607	−0.492304	−	0.870423i	$-0.663845\pi$
$67$	−8.05936	−0.984607	−0.492304	+	0.870423i	$0.663845\pi$
$68$	0	0
$69$	−0.235776	−0.0283841
$70$	0	0
$71$	0.428251	0.0508241	0.0254120	−	0.999677i	$-0.491910\pi$
$71$	0.428251	0.0508241	0.0254120	+	0.999677i	$0.491910\pi$
$72$	0	0
$73$	−12.1299	−1.41970	−0.709848	−	0.704354i	$-0.751237\pi$
$73$	−12.1299	−1.41970	−0.709848	+	0.704354i	$0.751237\pi$
$74$	0	0
$75$	−0.690999	−0.0797897
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	9.37219	1.05445	0.527227	−	0.849725i	$-0.323232\pi$
$79$	9.37219	1.05445	0.527227	+	0.849725i	$0.323232\pi$
$80$	0	0
$81$	8.82009	0.980010
$82$	0	0
$83$	−4.37797	−0.480544	−0.240272	−	0.970706i	$-0.577237\pi$
$83$	−4.37797	−0.480544	−0.240272	+	0.970706i	$0.577237\pi$
$84$	0	0
$85$	−0.667928	−0.0724470
$86$	0	0
$87$	0.499346	0.0535355
$88$	0	0
$89$	−4.95758	−0.525502	−0.262751	−	0.964864i	$-0.584630\pi$
$89$	−4.95758	−0.525502	−0.262751	+	0.964864i	$0.584630\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	1.26320	0.130987
$94$	0	0
$95$	−0.131997	−0.0135426
$96$	0	0
$97$	−13.5322	−1.37399	−0.686994	−	0.726664i	$-0.741070\pi$
$97$	−13.5322	−1.37399	−0.686994	+	0.726664i	$0.741070\pi$
$98$	0	0
$99$	−2.97996	−0.299498
$100$	0	0
$101$	−5.38219	−0.535548	−0.267774	−	0.963482i	$-0.586288\pi$
$101$	−5.38219	−0.535548	−0.267774	+	0.963482i	$0.586288\pi$
$102$	0	0
$103$	−12.3682	−1.21868	−0.609339	−	0.792910i	$-0.708565\pi$
$103$	−12.3682	−1.21868	−0.609339	+	0.792910i	$0.708565\pi$
$104$	0	0
$105$	−0.0486587	−0.00474861
$106$	0	0
$107$	−0.508008	−0.0491110	−0.0245555	−	0.999698i	$-0.507817\pi$
$107$	−0.508008	−0.0491110	−0.0245555	+	0.999698i	$0.507817\pi$
$108$	0	0
$109$	1.42923	0.136896	0.0684478	−	0.997655i	$-0.478195\pi$
$109$	1.42923	0.136896	0.0684478	+	0.997655i	$0.478195\pi$
$110$	0	0
$111$	0.831411	0.0789141
$112$	0	0
$113$	−9.14399	−0.860194	−0.430097	−	0.902783i	$-0.641521\pi$
$113$	−9.14399	−0.860194	−0.430097	+	0.902783i	$0.641521\pi$
$114$	0	0
$115$	−0.572623	−0.0533974
$116$	0	0
$117$	−2.97996	−0.275498
$118$	0	0
$119$	1.94296	0.178111
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.959707	0.0865339
$124$	0	0
$125$	−3.39705	−0.303842
$126$	0	0
$127$	−13.2998	−1.18017	−0.590083	−	0.807343i	$-0.700905\pi$
$127$	−13.2998	−1.18017	−0.590083	+	0.807343i	$0.700905\pi$
$128$	0	0
$129$	−0.266048	−0.0234242
$130$	0	0
$131$	17.2076	1.50344	0.751719	−	0.659483i	$-0.229225\pi$
$131$	17.2076	1.50344	0.751719	+	0.659483i	$0.229225\pi$
$132$	0	0
$133$	0.383972	0.0332946
$134$	0	0
$135$	−0.290978	−0.0250434
$136$	0	0
$137$	−16.9509	−1.44822	−0.724109	−	0.689686i	$-0.757749\pi$
$137$	−16.9509	−1.44822	−0.724109	+	0.689686i	$0.757749\pi$
$138$	0	0
$139$	−11.0975	−0.941275	−0.470638	−	0.882327i	$-0.655976\pi$
$139$	−11.0975	−0.941275	−0.470638	+	0.882327i	$0.655976\pi$
$140$	0	0
$141$	0.355940	0.0299756
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	1.21275	0.100713
$146$	0	0
$147$	0.141545	0.0116745
$148$	0	0
$149$	−10.3649	−0.849127	−0.424563	−	0.905398i	$-0.639572\pi$
$149$	−10.3649	−0.849127	−0.424563	+	0.905398i	$0.639572\pi$
$150$	0	0
$151$	6.99428	0.569187	0.284593	−	0.958648i	$-0.408141\pi$
$151$	6.99428	0.569187	0.284593	+	0.958648i	$0.408141\pi$
$152$	0	0
$153$	5.78996	0.468091
$154$	0	0
$155$	3.06790	0.246420
$156$	0	0
$157$	−5.89153	−0.470195	−0.235098	−	0.971972i	$-0.575541\pi$
$157$	−5.89153	−0.470195	−0.235098	+	0.971972i	$0.575541\pi$
$158$	0	0
$159$	0.112311	0.00890687
$160$	0	0
$161$	1.66573	0.131278
$162$	0	0
$163$	1.13057	0.0885533	0.0442767	−	0.999019i	$-0.485902\pi$
$163$	1.13057	0.0885533	0.0442767	+	0.999019i	$0.485902\pi$
$164$	0	0
$165$	0.0486587	0.00378808
$166$	0	0
$167$	−15.9501	−1.23425	−0.617127	−	0.786864i	$-0.711703\pi$
$167$	−15.9501	−1.23425	−0.617127	+	0.786864i	$0.711703\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.14422	0.0875009
$172$	0	0
$173$	−22.4356	−1.70574	−0.852872	−	0.522120i	$-0.825141\pi$
$173$	−22.4356	−1.70574	−0.852872	+	0.522120i	$0.825141\pi$
$174$	0	0
$175$	4.88182	0.369031
$176$	0	0
$177$	−1.98570	−0.149254
$178$	0	0
$179$	12.9674	0.969228	0.484614	−	0.874728i	$-0.338960\pi$
$179$	12.9674	0.969228	0.484614	+	0.874728i	$0.338960\pi$
$180$	0	0
$181$	−0.833575	−0.0619591	−0.0309796	−	0.999520i	$-0.509863\pi$
$181$	−0.833575	−0.0619591	−0.0309796	+	0.999520i	$0.509863\pi$
$182$	0	0
$183$	0.901595	0.0666478
$184$	0	0
$185$	2.01923	0.148457
$186$	0	0
$187$	−1.94296	−0.142084
$188$	0	0
$189$	0.846436	0.0615692
$190$	0	0
$191$	16.3507	1.18309	0.591547	−	0.806271i	$-0.298517\pi$
$191$	16.3507	1.18309	0.591547	+	0.806271i	$0.298517\pi$
$192$	0	0
$193$	18.1486	1.30636	0.653182	−	0.757201i	$-0.273434\pi$
$193$	18.1486	1.30636	0.653182	+	0.757201i	$0.273434\pi$
$194$	0	0
$195$	0.0486587	0.00348452
$196$	0	0
$197$	−22.4904	−1.60237	−0.801186	−	0.598416i	$-0.795797\pi$
$197$	−22.4904	−1.60237	−0.801186	+	0.598416i	$0.795797\pi$
$198$	0	0
$199$	13.1359	0.931182	0.465591	−	0.885000i	$-0.345842\pi$
$199$	13.1359	0.931182	0.465591	+	0.885000i	$0.345842\pi$
$200$	0	0
$201$	−1.14076	−0.0804633
$202$	0	0
$203$	−3.52781	−0.247604
$204$	0	0
$205$	2.33082	0.162791
$206$	0	0
$207$	4.96380	0.345008
$208$	0	0
$209$	−0.383972	−0.0265599
$210$	0	0
$211$	−18.0317	−1.24135	−0.620677	−	0.784066i	$-0.713142\pi$
$211$	−18.0317	−1.24135	−0.620677	+	0.784066i	$0.713142\pi$
$212$	0	0
$213$	0.0606170	0.00415341
$214$	0	0
$215$	−0.646145	−0.0440667
$216$	0	0
$217$	−8.92433	−0.605823
$218$	0	0
$219$	−1.71693	−0.116019
$220$	0	0
$221$	−1.94296	−0.130698
$222$	0	0
$223$	6.34803	0.425096	0.212548	−	0.977151i	$-0.431824\pi$
$223$	6.34803	0.425096	0.212548	+	0.977151i	$0.431824\pi$
$224$	0	0
$225$	14.5477	0.969844
$226$	0	0
$227$	−3.57190	−0.237075	−0.118538	−	0.992950i	$-0.537821\pi$
$227$	−3.57190	−0.237075	−0.118538	+	0.992950i	$0.537821\pi$
$228$	0	0
$229$	−6.43178	−0.425024	−0.212512	−	0.977158i	$-0.568164\pi$
$229$	−6.43178	−0.425024	−0.212512	+	0.977158i	$0.568164\pi$
$230$	0	0
$231$	−0.141545	−0.00931300
$232$	0	0
$233$	11.7243	0.768085	0.384043	−	0.923315i	$-0.374532\pi$
$233$	11.7243	0.768085	0.384043	+	0.923315i	$0.374532\pi$
$234$	0	0
$235$	0.864463	0.0563914
$236$	0	0
$237$	1.32659	0.0861713
$238$	0	0
$239$	−13.8255	−0.894299	−0.447150	−	0.894459i	$-0.647561\pi$
$239$	−13.8255	−0.894299	−0.447150	+	0.894459i	$0.647561\pi$
$240$	0	0
$241$	−14.5623	−0.938040	−0.469020	−	0.883187i	$-0.655393\pi$
$241$	−14.5623	−0.938040	−0.469020	+	0.883187i	$0.655393\pi$
$242$	0	0
$243$	3.78775	0.242984
$244$	0	0
$245$	0.343768	0.0219625
$246$	0	0
$247$	−0.383972	−0.0244315
$248$	0	0
$249$	−0.619681	−0.0392707
$250$	0	0
$251$	9.48632	0.598771	0.299386	−	0.954132i	$-0.403218\pi$
$251$	9.48632	0.598771	0.299386	+	0.954132i	$0.403218\pi$
$252$	0	0
$253$	−1.66573	−0.104723
$254$	0	0
$255$	−0.0945421	−0.00592046
$256$	0	0
$257$	14.0937	0.879143	0.439572	−	0.898208i	$-0.355130\pi$
$257$	14.0937	0.879143	0.439572	+	0.898208i	$0.355130\pi$
$258$	0	0
$259$	−5.87381	−0.364981
$260$	0	0
$261$	−10.5128	−0.650723
$262$	0	0
$263$	15.5786	0.960615	0.480307	−	0.877100i	$-0.340525\pi$
$263$	15.5786	0.960615	0.480307	+	0.877100i	$0.340525\pi$
$264$	0	0
$265$	0.272768	0.0167560
$266$	0	0
$267$	−0.701722	−0.0429447
$268$	0	0
$269$	−16.3016	−0.993924	−0.496962	−	0.867772i	$-0.665551\pi$
$269$	−16.3016	−0.993924	−0.496962	+	0.867772i	$0.665551\pi$
$270$	0	0
$271$	3.78877	0.230151	0.115076	−	0.993357i	$-0.463289\pi$
$271$	3.78877	0.230151	0.115076	+	0.993357i	$0.463289\pi$
$272$	0	0
$273$	−0.141545	−0.00856671
$274$	0	0
$275$	−4.88182	−0.294385
$276$	0	0
$277$	−19.7291	−1.18541	−0.592703	−	0.805421i	$-0.701939\pi$
$277$	−19.7291	−1.18541	−0.592703	+	0.805421i	$0.701939\pi$
$278$	0	0
$279$	−26.5942	−1.59215
$280$	0	0
$281$	−18.8178	−1.12257	−0.561286	−	0.827622i	$-0.689693\pi$
$281$	−18.8178	−1.12257	−0.561286	+	0.827622i	$0.689693\pi$
$282$	0	0
$283$	−6.57588	−0.390895	−0.195448	−	0.980714i	$-0.562616\pi$
$283$	−6.57588	−0.390895	−0.195448	+	0.980714i	$0.562616\pi$
$284$	0	0
$285$	−0.0186836	−0.00110672
$286$	0	0
$287$	−6.78021	−0.400223
$288$	0	0
$289$	−13.2249	−0.777935
$290$	0	0
$291$	−1.91542	−0.112284
$292$	0	0
$293$	8.44965	0.493634	0.246817	−	0.969062i	$-0.420615\pi$
$293$	8.44965	0.493634	0.246817	+	0.969062i	$0.420615\pi$
$294$	0	0
$295$	−4.82262	−0.280784
$296$	0	0
$297$	−0.846436	−0.0491152
$298$	0	0
$299$	−1.66573	−0.0963314
$300$	0	0
$301$	1.87960	0.108338
$302$	0	0
$303$	−0.761824	−0.0437656
$304$	0	0
$305$	2.18968	0.125381
$306$	0	0
$307$	−0.311354	−0.0177699	−0.00888496	−	0.999961i	$-0.502828\pi$
$307$	−0.311354	−0.0177699	−0.00888496	+	0.999961i	$0.502828\pi$
$308$	0	0
$309$	−1.75067	−0.0995919
$310$	0	0
$311$	−16.0877	−0.912249	−0.456125	−	0.889916i	$-0.650763\pi$
$311$	−16.0877	−0.912249	−0.456125	+	0.889916i	$0.650763\pi$
$312$	0	0
$313$	17.5991	0.994758	0.497379	−	0.867533i	$-0.334296\pi$
$313$	17.5991	0.994758	0.497379	+	0.867533i	$0.334296\pi$
$314$	0	0
$315$	1.02442	0.0577193
$316$	0	0
$317$	−19.0753	−1.07138	−0.535688	−	0.844416i	$-0.679948\pi$
$317$	−19.0753	−1.07138	−0.535688	+	0.844416i	$0.679948\pi$
$318$	0	0
$319$	3.52781	0.197520
$320$	0	0
$321$	−0.0719061	−0.00401341
$322$	0	0
$323$	0.746043	0.0415109
$324$	0	0
$325$	−4.88182	−0.270795
$326$	0	0
$327$	0.202301	0.0111873
$328$	0	0
$329$	−2.51467	−0.138638
$330$	0	0
$331$	−34.2735	−1.88384	−0.941922	−	0.335832i	$-0.890983\pi$
$331$	−34.2735	−1.88384	−0.941922	+	0.335832i	$0.890983\pi$
$332$	0	0
$333$	−17.5038	−0.959200
$334$	0	0
$335$	−2.77055	−0.151371
$336$	0	0
$337$	20.3206	1.10693	0.553466	−	0.832872i	$-0.313305\pi$
$337$	20.3206	1.10693	0.553466	+	0.832872i	$0.313305\pi$
$338$	0	0
$339$	−1.29429	−0.0702961
$340$	0	0
$341$	8.92433	0.483280
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−0.0810521	−0.00436370
$346$	0	0
$347$	−1.03772	−0.0557080	−0.0278540	−	0.999612i	$-0.508867\pi$
$347$	−1.03772	−0.0557080	−0.0278540	+	0.999612i	$0.508867\pi$
$348$	0	0
$349$	−13.8448	−0.741096	−0.370548	−	0.928813i	$-0.620830\pi$
$349$	−13.8448	−0.741096	−0.370548	+	0.928813i	$0.620830\pi$
$350$	0	0
$351$	−0.846436	−0.0451794
$352$	0	0
$353$	−16.0955	−0.856678	−0.428339	−	0.903618i	$-0.640901\pi$
$353$	−16.0955	−0.856678	−0.428339	+	0.903618i	$0.640901\pi$
$354$	0	0
$355$	0.147219	0.00781357
$356$	0	0
$357$	0.275017	0.0145555
$358$	0	0
$359$	3.89822	0.205740	0.102870	−	0.994695i	$-0.467197\pi$
$359$	3.89822	0.205740	0.102870	+	0.994695i	$0.467197\pi$
$360$	0	0
$361$	−18.8526	−0.992240
$362$	0	0
$363$	0.141545	0.00742920
$364$	0	0
$365$	−4.16987	−0.218261
$366$	0	0
$367$	35.9022	1.87408	0.937041	−	0.349220i	$-0.113553\pi$
$367$	35.9022	1.87408	0.937041	+	0.349220i	$0.113553\pi$
$368$	0	0
$369$	−20.2048	−1.05182
$370$	0	0
$371$	−0.793466	−0.0411947
$372$	0	0
$373$	−17.8176	−0.922563	−0.461281	−	0.887254i	$-0.652610\pi$
$373$	−17.8176	−0.922563	−0.461281	+	0.887254i	$0.652610\pi$
$374$	0	0
$375$	−0.480837	−0.0248303
$376$	0	0
$377$	3.52781	0.181692
$378$	0	0
$379$	0.508512	0.0261205	0.0130603	−	0.999915i	$-0.495843\pi$
$379$	0.508512	0.0261205	0.0130603	+	0.999915i	$0.495843\pi$
$380$	0	0
$381$	−1.88252	−0.0964446
$382$	0	0
$383$	31.2777	1.59822	0.799109	−	0.601187i	$-0.205305\pi$
$383$	31.2777	1.59822	0.799109	+	0.601187i	$0.205305\pi$
$384$	0	0
$385$	−0.343768	−0.0175200
$386$	0	0
$387$	5.60114	0.284722
$388$	0	0
$389$	−10.1369	−0.513963	−0.256981	−	0.966416i	$-0.582728\pi$
$389$	−10.1369	−0.513963	−0.256981	+	0.966416i	$0.582728\pi$
$390$	0	0
$391$	3.23644	0.163674
$392$	0	0
$393$	2.43566	0.122863
$394$	0	0
$395$	3.22186	0.162109
$396$	0	0
$397$	−2.86914	−0.143998	−0.0719989	−	0.997405i	$-0.522938\pi$
$397$	−2.86914	−0.143998	−0.0719989	+	0.997405i	$0.522938\pi$
$398$	0	0
$399$	0.0543494	0.00272087
$400$	0	0
$401$	24.7925	1.23808	0.619040	−	0.785359i	$-0.287522\pi$
$401$	24.7925	1.23808	0.619040	+	0.785359i	$0.287522\pi$
$402$	0	0
$403$	8.92433	0.444553
$404$	0	0
$405$	3.03206	0.150664
$406$	0	0
$407$	5.87381	0.291154
$408$	0	0
$409$	18.9450	0.936769	0.468384	−	0.883525i	$-0.344836\pi$
$409$	18.9450	0.936769	0.468384	+	0.883525i	$0.344836\pi$
$410$	0	0
$411$	−2.39933	−0.118350
$412$	0	0
$413$	14.0287	0.690308
$414$	0	0
$415$	−1.50500	−0.0738778
$416$	0	0
$417$	−1.57080	−0.0769222
$418$	0	0
$419$	1.32966	0.0649579	0.0324790	−	0.999472i	$-0.489660\pi$
$419$	1.32966	0.0649579	0.0324790	+	0.999472i	$0.489660\pi$
$420$	0	0
$421$	−8.02264	−0.390999	−0.195500	−	0.980704i	$-0.562633\pi$
$421$	−8.02264	−0.390999	−0.195500	+	0.980704i	$0.562633\pi$
$422$	0	0
$423$	−7.49363	−0.364353
$424$	0	0
$425$	9.48520	0.460100
$426$	0	0
$427$	−6.36966	−0.308249
$428$	0	0
$429$	0.141545	0.00683387
$430$	0	0
$431$	−9.68512	−0.466516	−0.233258	−	0.972415i	$-0.574939\pi$
$431$	−9.68512	−0.466516	−0.233258	+	0.972415i	$0.574939\pi$
$432$	0	0
$433$	−28.7157	−1.37999	−0.689994	−	0.723815i	$-0.742387\pi$
$433$	−28.7157	−1.37999	−0.689994	+	0.723815i	$0.742387\pi$
$434$	0	0
$435$	0.171659	0.00823042
$436$	0	0
$437$	0.639592	0.0305958
$438$	0	0
$439$	−8.65803	−0.413225	−0.206613	−	0.978423i	$-0.566244\pi$
$439$	−8.65803	−0.413225	−0.206613	+	0.978423i	$0.566244\pi$
$440$	0	0
$441$	−2.97996	−0.141903
$442$	0	0
$443$	−20.9580	−0.995745	−0.497872	−	0.867250i	$-0.665885\pi$
$443$	−20.9580	−0.995745	−0.497872	+	0.867250i	$0.665885\pi$
$444$	0	0
$445$	−1.70426	−0.0807895
$446$	0	0
$447$	−1.46710	−0.0693917
$448$	0	0
$449$	−33.5646	−1.58401	−0.792006	−	0.610513i	$-0.790963\pi$
$449$	−33.5646	−1.58401	−0.792006	+	0.610513i	$0.790963\pi$
$450$	0	0
$451$	6.78021	0.319268
$452$	0	0
$453$	0.990008	0.0465146
$454$	0	0
$455$	−0.343768	−0.0161161
$456$	0	0
$457$	−21.1902	−0.991235	−0.495617	−	0.868541i	$-0.665058\pi$
$457$	−21.1902	−0.991235	−0.495617	+	0.868541i	$0.665058\pi$
$458$	0	0
$459$	1.64459	0.0767631
$460$	0	0
$461$	36.6365	1.70633	0.853166	−	0.521640i	$-0.174680\pi$
$461$	36.6365	1.70633	0.853166	+	0.521640i	$0.174680\pi$
$462$	0	0
$463$	−17.3814	−0.807782	−0.403891	−	0.914807i	$-0.632342\pi$
$463$	−17.3814	−0.807782	−0.403891	+	0.914807i	$0.632342\pi$
$464$	0	0
$465$	0.434247	0.0201377
$466$	0	0
$467$	25.9327	1.20002	0.600012	−	0.799991i	$-0.295163\pi$
$467$	25.9327	1.20002	0.600012	+	0.799991i	$0.295163\pi$
$468$	0	0
$469$	8.05936	0.372147
$470$	0	0
$471$	−0.833919	−0.0384250
$472$	0	0
$473$	−1.87960	−0.0864240
$474$	0	0
$475$	1.87448	0.0860072
$476$	0	0
$477$	−2.36450	−0.108263
$478$	0	0
$479$	−21.2850	−0.972537	−0.486268	−	0.873810i	$-0.661642\pi$
$479$	−21.2850	−0.972537	−0.486268	+	0.873810i	$0.661642\pi$
$480$	0	0
$481$	5.87381	0.267823
$482$	0	0
$483$	0.235776	0.0107282
$484$	0	0
$485$	−4.65194	−0.211234
$486$	0	0
$487$	29.4443	1.33425	0.667123	−	0.744947i	$-0.267525\pi$
$487$	29.4443	1.33425	0.667123	+	0.744947i	$0.267525\pi$
$488$	0	0
$489$	0.160027	0.00723669
$490$	0	0
$491$	23.7655	1.07252	0.536261	−	0.844053i	$-0.319836\pi$
$491$	23.7655	1.07252	0.536261	+	0.844053i	$0.319836\pi$
$492$	0	0
$493$	−6.85441	−0.308707
$494$	0	0
$495$	−1.02442	−0.0460441
$496$	0	0
$497$	−0.428251	−0.0192097
$498$	0	0
$499$	39.2462	1.75690	0.878451	−	0.477833i	$-0.158578\pi$
$499$	39.2462	1.75690	0.878451	+	0.477833i	$0.158578\pi$
$500$	0	0
$501$	−2.25766	−0.100865
$502$	0	0
$503$	−14.9003	−0.664370	−0.332185	−	0.943214i	$-0.607786\pi$
$503$	−14.9003	−0.664370	−0.332185	+	0.943214i	$0.607786\pi$
$504$	0	0
$505$	−1.85022	−0.0823339
$506$	0	0
$507$	0.141545	0.00628625
$508$	0	0
$509$	15.3793	0.681677	0.340838	−	0.940122i	$-0.389289\pi$
$509$	15.3793	0.681677	0.340838	+	0.940122i	$0.389289\pi$
$510$	0	0
$511$	12.1299	0.536595
$512$	0	0
$513$	0.325008	0.0143494
$514$	0	0
$515$	−4.25180	−0.187357
$516$	0	0
$517$	2.51467	0.110595
$518$	0	0
$519$	−3.17565	−0.139396
$520$	0	0
$521$	−44.8745	−1.96599	−0.982994	−	0.183636i	$-0.941213\pi$
$521$	−44.8745	−1.96599	−0.982994	+	0.183636i	$0.941213\pi$
$522$	0	0
$523$	35.0853	1.53417	0.767086	−	0.641544i	$-0.221706\pi$
$523$	35.0853	1.53417	0.767086	+	0.641544i	$0.221706\pi$
$524$	0	0
$525$	0.690999	0.0301577
$526$	0	0
$527$	−17.3396	−0.755327
$528$	0	0
$529$	−20.2254	−0.879363
$530$	0	0
$531$	41.8051	1.81419
$532$	0	0
$533$	6.78021	0.293684
$534$	0	0
$535$	−0.174637	−0.00755021
$536$	0	0
$537$	1.83547	0.0792065
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	5.41564	0.232836	0.116418	−	0.993200i	$-0.462859\pi$
$541$	5.41564	0.232836	0.116418	+	0.993200i	$0.462859\pi$
$542$	0	0
$543$	−0.117989	−0.00506338
$544$	0	0
$545$	0.491324	0.0210460
$546$	0	0
$547$	8.15207	0.348557	0.174279	−	0.984696i	$-0.444241\pi$
$547$	8.15207	0.348557	0.174279	+	0.984696i	$0.444241\pi$
$548$	0	0
$549$	−18.9814	−0.810104
$550$	0	0
$551$	−1.35458	−0.0577071
$552$	0	0
$553$	−9.37219	−0.398546
$554$	0	0
$555$	0.285812	0.0121321
$556$	0	0
$557$	41.6501	1.76477	0.882385	−	0.470528i	$-0.155937\pi$
$557$	41.6501	1.76477	0.882385	+	0.470528i	$0.155937\pi$
$558$	0	0
$559$	−1.87960	−0.0794985
$560$	0	0
$561$	−0.275017	−0.0116112
$562$	0	0
$563$	−24.1701	−1.01865	−0.509324	−	0.860575i	$-0.670105\pi$
$563$	−24.1701	−1.01865	−0.509324	+	0.860575i	$0.670105\pi$
$564$	0	0
$565$	−3.14341	−0.132244
$566$	0	0
$567$	−8.82009	−0.370409
$568$	0	0
$569$	11.0116	0.461631	0.230815	−	0.972998i	$-0.425861\pi$
$569$	11.0116	0.461631	0.230815	+	0.972998i	$0.425861\pi$
$570$	0	0
$571$	−16.2224	−0.678885	−0.339443	−	0.940627i	$-0.610238\pi$
$571$	−16.2224	−0.678885	−0.339443	+	0.940627i	$0.610238\pi$
$572$	0	0
$573$	2.31436	0.0966839
$574$	0	0
$575$	8.13178	0.339119
$576$	0	0
$577$	9.16422	0.381512	0.190756	−	0.981638i	$-0.438906\pi$
$577$	9.16422	0.381512	0.190756	+	0.981638i	$0.438906\pi$
$578$	0	0
$579$	2.56885	0.106758
$580$	0	0
$581$	4.37797	0.181629
$582$	0	0
$583$	0.793466	0.0328620
$584$	0	0
$585$	−1.02442	−0.0423544
$586$	0	0
$587$	−28.4842	−1.17567	−0.587834	−	0.808982i	$-0.700019\pi$
$587$	−28.4842	−1.17567	−0.587834	+	0.808982i	$0.700019\pi$
$588$	0	0
$589$	−3.42669	−0.141194
$590$	0	0
$591$	−3.18341	−0.130948
$592$	0	0
$593$	11.2059	0.460172	0.230086	−	0.973170i	$-0.426099\pi$
$593$	11.2059	0.460172	0.230086	+	0.973170i	$0.426099\pi$
$594$	0	0
$595$	0.667928	0.0273824
$596$	0	0
$597$	1.85933	0.0760974
$598$	0	0
$599$	13.8800	0.567123	0.283561	−	0.958954i	$-0.408484\pi$
$599$	13.8800	0.567123	0.283561	+	0.958954i	$0.408484\pi$
$600$	0	0
$601$	11.7232	0.478200	0.239100	−	0.970995i	$-0.423148\pi$
$601$	11.7232	0.478200	0.239100	+	0.970995i	$0.423148\pi$
$602$	0	0
$603$	24.0166	0.978032
$604$	0	0
$605$	0.343768	0.0139762
$606$	0	0
$607$	−18.3454	−0.744618	−0.372309	−	0.928109i	$-0.621434\pi$
$607$	−18.3454	−0.744618	−0.372309	+	0.928109i	$0.621434\pi$
$608$	0	0
$609$	−0.499346	−0.0202345
$610$	0	0
$611$	2.51467	0.101733
$612$	0	0
$613$	−10.2327	−0.413296	−0.206648	−	0.978415i	$-0.566255\pi$
$613$	−10.2327	−0.413296	−0.206648	+	0.978415i	$0.566255\pi$
$614$	0	0
$615$	0.329917	0.0133035
$616$	0	0
$617$	2.92459	0.117739	0.0588697	−	0.998266i	$-0.481250\pi$
$617$	2.92459	0.117739	0.0588697	+	0.998266i	$0.481250\pi$
$618$	0	0
$619$	38.5588	1.54981	0.774904	−	0.632079i	$-0.217798\pi$
$619$	38.5588	1.54981	0.774904	+	0.632079i	$0.217798\pi$
$620$	0	0
$621$	1.40993	0.0565786
$622$	0	0
$623$	4.95758	0.198621
$624$	0	0
$625$	23.2413	0.929653
$626$	0	0
$627$	−0.0543494	−0.00217051
$628$	0	0
$629$	−11.4126	−0.455050
$630$	0	0
$631$	31.8716	1.26879	0.634395	−	0.773009i	$-0.281249\pi$
$631$	31.8716	1.26879	0.634395	+	0.773009i	$0.281249\pi$
$632$	0	0
$633$	−2.55230	−0.101445
$634$	0	0
$635$	−4.57204	−0.181436
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−1.27617	−0.0504846
$640$	0	0
$641$	40.7019	1.60763	0.803815	−	0.594879i	$-0.202800\pi$
$641$	40.7019	1.60763	0.803815	+	0.594879i	$0.202800\pi$
$642$	0	0
$643$	−28.5035	−1.12407	−0.562033	−	0.827114i	$-0.689981\pi$
$643$	−28.5035	−1.12407	−0.562033	+	0.827114i	$0.689981\pi$
$644$	0	0
$645$	−0.0914589	−0.00360119
$646$	0	0
$647$	−38.3960	−1.50950	−0.754751	−	0.656012i	$-0.772242\pi$
$647$	−38.3960	−1.50950	−0.754751	+	0.656012i	$0.772242\pi$
$648$	0	0
$649$	−14.0287	−0.550675
$650$	0	0
$651$	−1.26320	−0.0495086
$652$	0	0
$653$	−25.3523	−0.992111	−0.496055	−	0.868291i	$-0.665219\pi$
$653$	−25.3523	−0.992111	−0.496055	+	0.868291i	$0.665219\pi$
$654$	0	0
$655$	5.91543	0.231135
$656$	0	0
$657$	36.1467	1.41022
$658$	0	0
$659$	−3.42625	−0.133468	−0.0667339	−	0.997771i	$-0.521258\pi$
$659$	−3.42625	−0.133468	−0.0667339	+	0.997771i	$0.521258\pi$
$660$	0	0
$661$	−21.9669	−0.854413	−0.427207	−	0.904154i	$-0.640502\pi$
$661$	−21.9669	−0.854413	−0.427207	+	0.904154i	$0.640502\pi$
$662$	0	0
$663$	−0.275017	−0.0106808
$664$	0	0
$665$	0.131997	0.00511863
$666$	0	0
$667$	−5.87637	−0.227534
$668$	0	0
$669$	0.898534	0.0347393
$670$	0	0
$671$	6.36966	0.245898
$672$	0	0
$673$	−23.5578	−0.908087	−0.454044	−	0.890979i	$-0.650019\pi$
$673$	−23.5578	−0.908087	−0.454044	+	0.890979i	$0.650019\pi$
$674$	0	0
$675$	4.13215	0.159047
$676$	0	0
$677$	−45.9949	−1.76773	−0.883863	−	0.467745i	$-0.845066\pi$
$677$	−45.9949	−1.76773	−0.883863	+	0.467745i	$0.845066\pi$
$678$	0	0
$679$	13.5322	0.519318
$680$	0	0
$681$	−0.505586	−0.0193741
$682$	0	0
$683$	−5.43705	−0.208043	−0.104021	−	0.994575i	$-0.533171\pi$
$683$	−5.43705	−0.208043	−0.104021	+	0.994575i	$0.533171\pi$
$684$	0	0
$685$	−5.82719	−0.222646
$686$	0	0
$687$	−0.910388	−0.0347335
$688$	0	0
$689$	0.793466	0.0302286
$690$	0	0
$691$	26.9640	1.02576	0.512880	−	0.858460i	$-0.328579\pi$
$691$	26.9640	1.02576	0.512880	+	0.858460i	$0.328579\pi$
$692$	0	0
$693$	2.97996	0.113200
$694$	0	0
$695$	−3.81495	−0.144709
$696$	0	0
$697$	−13.1737	−0.498989
$698$	0	0
$699$	1.65952	0.0627689
$700$	0	0
$701$	−43.3979	−1.63911	−0.819557	−	0.572997i	$-0.805781\pi$
$701$	−43.3979	−1.63911	−0.819557	+	0.572997i	$0.805781\pi$
$702$	0	0
$703$	−2.25538	−0.0850632
$704$	0	0
$705$	0.122361	0.00460837
$706$	0	0
$707$	5.38219	0.202418
$708$	0	0
$709$	48.5676	1.82399	0.911997	−	0.410196i	$-0.134540\pi$
$709$	48.5676	1.82399	0.911997	+	0.410196i	$0.134540\pi$
$710$	0	0
$711$	−27.9288	−1.04741
$712$	0	0
$713$	−14.8655	−0.556717
$714$	0	0
$715$	0.343768	0.0128562
$716$	0	0
$717$	−1.95694	−0.0730833
$718$	0	0
$719$	−40.5701	−1.51301	−0.756504	−	0.653989i	$-0.773094\pi$
$719$	−40.5701	−1.51301	−0.756504	+	0.653989i	$0.773094\pi$
$720$	0	0
$721$	12.3682	0.460617
$722$	0	0
$723$	−2.06123	−0.0766578
$724$	0	0
$725$	−17.2222	−0.639615
$726$	0	0
$727$	15.9922	0.593117	0.296559	−	0.955015i	$-0.404161\pi$
$727$	15.9922	0.593117	0.296559	+	0.955015i	$0.404161\pi$
$728$	0	0
$729$	−25.9241	−0.960153
$730$	0	0
$731$	3.65199	0.135074
$732$	0	0
$733$	−17.4403	−0.644172	−0.322086	−	0.946710i	$-0.604384\pi$
$733$	−17.4403	−0.644172	−0.322086	+	0.946710i	$0.604384\pi$
$734$	0	0
$735$	0.0486587	0.00179480
$736$	0	0
$737$	−8.05936	−0.296870
$738$	0	0
$739$	−26.7320	−0.983352	−0.491676	−	0.870778i	$-0.663616\pi$
$739$	−26.7320	−0.983352	−0.491676	+	0.870778i	$0.663616\pi$
$740$	0	0
$741$	−0.0543494	−0.00199658
$742$	0	0
$743$	8.12671	0.298140	0.149070	−	0.988827i	$-0.452372\pi$
$743$	8.12671	0.298140	0.149070	+	0.988827i	$0.452372\pi$
$744$	0	0
$745$	−3.56312	−0.130543
$746$	0	0
$747$	13.0462	0.477335
$748$	0	0
$749$	0.508008	0.0185622
$750$	0	0
$751$	5.68036	0.207279	0.103640	−	0.994615i	$-0.466951\pi$
$751$	5.68036	0.207279	0.103640	+	0.994615i	$0.466951\pi$
$752$	0	0
$753$	1.34274	0.0489323
$754$	0	0
$755$	2.40441	0.0875054
$756$	0	0
$757$	−30.9880	−1.12628	−0.563138	−	0.826363i	$-0.690406\pi$
$757$	−30.9880	−1.12628	−0.563138	+	0.826363i	$0.690406\pi$
$758$	0	0
$759$	−0.235776	−0.00855812
$760$	0	0
$761$	−10.4708	−0.379566	−0.189783	−	0.981826i	$-0.560778\pi$
$761$	−10.4708	−0.379566	−0.189783	+	0.981826i	$0.560778\pi$
$762$	0	0
$763$	−1.42923	−0.0517417
$764$	0	0
$765$	1.99040	0.0719632
$766$	0	0
$767$	−14.0287	−0.506547
$768$	0	0
$769$	−42.0177	−1.51520	−0.757599	−	0.652720i	$-0.773628\pi$
$769$	−42.0177	−1.51520	−0.757599	+	0.652720i	$0.773628\pi$
$770$	0	0
$771$	1.99490	0.0718447
$772$	0	0
$773$	25.6652	0.923113	0.461557	−	0.887111i	$-0.347291\pi$
$773$	25.6652	0.923113	0.461557	+	0.887111i	$0.347291\pi$
$774$	0	0
$775$	−43.5670	−1.56497
$776$	0	0
$777$	−0.831411	−0.0298267
$778$	0	0
$779$	−2.60341	−0.0932768
$780$	0	0
$781$	0.428251	0.0153240
$782$	0	0
$783$	−2.98607	−0.106713
$784$	0	0
$785$	−2.02532	−0.0722868
$786$	0	0
$787$	39.4566	1.40648	0.703238	−	0.710955i	$-0.251737\pi$
$787$	39.4566	1.40648	0.703238	+	0.710955i	$0.251737\pi$
$788$	0	0
$789$	2.20507	0.0785026
$790$	0	0
$791$	9.14399	0.325123
$792$	0	0
$793$	6.36966	0.226193
$794$	0	0
$795$	0.0386090	0.00136932
$796$	0	0
$797$	0.454799	0.0161098	0.00805490	−	0.999968i	$-0.497436\pi$
$797$	0.454799	0.0161098	0.00805490	+	0.999968i	$0.497436\pi$
$798$	0	0
$799$	−4.88591	−0.172851
$800$	0	0
$801$	14.7734	0.521993
$802$	0	0
$803$	−12.1299	−0.428055
$804$	0	0
$805$	0.572623	0.0201823
$806$	0	0
$807$	−2.30741	−0.0812247
$808$	0	0
$809$	28.6757	1.00818	0.504092	−	0.863650i	$-0.331827\pi$
$809$	28.6757	1.00818	0.504092	+	0.863650i	$0.331827\pi$
$810$	0	0
$811$	24.3682	0.855682	0.427841	−	0.903854i	$-0.359274\pi$
$811$	24.3682	0.855682	0.427841	+	0.903854i	$0.359274\pi$
$812$	0	0
$813$	0.536282	0.0188082
$814$	0	0
$815$	0.388655	0.0136140
$816$	0	0
$817$	0.721712	0.0252495
$818$	0	0
$819$	2.97996	0.104128
$820$	0	0
$821$	38.7715	1.35313	0.676567	−	0.736381i	$-0.263467\pi$
$821$	38.7715	1.35313	0.676567	+	0.736381i	$0.263467\pi$
$822$	0	0
$823$	7.94497	0.276944	0.138472	−	0.990366i	$-0.455781\pi$
$823$	7.94497	0.276944	0.138472	+	0.990366i	$0.455781\pi$
$824$	0	0
$825$	−0.690999	−0.0240575
$826$	0	0
$827$	17.1512	0.596406	0.298203	−	0.954502i	$-0.403613\pi$
$827$	17.1512	0.596406	0.298203	+	0.954502i	$0.403613\pi$
$828$	0	0
$829$	14.6830	0.509961	0.254981	−	0.966946i	$-0.417931\pi$
$829$	14.6830	0.509961	0.254981	+	0.966946i	$0.417931\pi$
$830$	0	0
$831$	−2.79256	−0.0968729
$832$	0	0
$833$	−1.94296	−0.0673197
$834$	0	0
$835$	−5.48312	−0.189751
$836$	0	0
$837$	−7.55388	−0.261100
$838$	0	0
$839$	5.17883	0.178793	0.0893966	−	0.995996i	$-0.471506\pi$
$839$	5.17883	0.178793	0.0893966	+	0.995996i	$0.471506\pi$
$840$	0	0
$841$	−16.5545	−0.570846
$842$	0	0
$843$	−2.66357	−0.0917381
$844$	0	0
$845$	0.343768	0.0118260
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−0.930785	−0.0319444
$850$	0	0
$851$	−9.78417	−0.335397
$852$	0	0
$853$	3.44234	0.117864	0.0589318	−	0.998262i	$-0.481231\pi$
$853$	3.44234	0.117864	0.0589318	+	0.998262i	$0.481231\pi$
$854$	0	0
$855$	0.393347	0.0134522
$856$	0	0
$857$	−16.5475	−0.565252	−0.282626	−	0.959230i	$-0.591206\pi$
$857$	−16.5475	−0.565252	−0.282626	+	0.959230i	$0.591206\pi$
$858$	0	0
$859$	−37.9475	−1.29475	−0.647376	−	0.762171i	$-0.724133\pi$
$859$	−37.9475	−1.29475	−0.647376	+	0.762171i	$0.724133\pi$
$860$	0	0
$861$	−0.959707	−0.0327067
$862$	0	0
$863$	21.1359	0.719476	0.359738	−	0.933053i	$-0.382866\pi$
$863$	21.1359	0.719476	0.359738	+	0.933053i	$0.382866\pi$
$864$	0	0
$865$	−7.71262	−0.262237
$866$	0	0
$867$	−1.87192	−0.0635738
$868$	0	0
$869$	9.37219	0.317930
$870$	0	0
$871$	−8.05936	−0.273081
$872$	0	0
$873$	40.3255	1.36481
$874$	0	0
$875$	3.39705	0.114841
$876$	0	0
$877$	−31.7514	−1.07217	−0.536085	−	0.844164i	$-0.680097\pi$
$877$	−31.7514	−1.07217	−0.536085	+	0.844164i	$0.680097\pi$
$878$	0	0
$879$	1.19601	0.0403404
$880$	0	0
$881$	−26.7648	−0.901729	−0.450864	−	0.892593i	$-0.648884\pi$
$881$	−26.7648	−0.901729	−0.450864	+	0.892593i	$0.648884\pi$
$882$	0	0
$883$	−40.5131	−1.36338	−0.681688	−	0.731643i	$-0.738754\pi$
$883$	−40.5131	−1.36338	−0.681688	+	0.731643i	$0.738754\pi$
$884$	0	0
$885$	−0.682619	−0.0229460
$886$	0	0
$887$	27.5458	0.924897	0.462448	−	0.886646i	$-0.346971\pi$
$887$	27.5458	0.924897	0.462448	+	0.886646i	$0.346971\pi$
$888$	0	0
$889$	13.2998	0.446061
$890$	0	0
$891$	8.82009	0.295484
$892$	0	0
$893$	−0.965563	−0.0323113
$894$	0	0
$895$	4.45777	0.149007
$896$	0	0
$897$	−0.235776	−0.00787232
$898$	0	0
$899$	31.4834	1.05003
$900$	0	0
$901$	−1.54167	−0.0513606
$902$	0	0
$903$	0.266048	0.00885353
$904$	0	0
$905$	−0.286556	−0.00952545
$906$	0	0
$907$	21.6514	0.718925	0.359462	−	0.933160i	$-0.382960\pi$
$907$	21.6514	0.718925	0.359462	+	0.933160i	$0.382960\pi$
$908$	0	0
$909$	16.0387	0.531971
$910$	0	0
$911$	−37.6600	−1.24773	−0.623866	−	0.781532i	$-0.714439\pi$
$911$	−37.6600	−1.24773	−0.623866	+	0.781532i	$0.714439\pi$
$912$	0	0
$913$	−4.37797	−0.144890
$914$	0	0
$915$	0.309940	0.0102463
$916$	0	0
$917$	−17.2076	−0.568246
$918$	0	0
$919$	−45.8544	−1.51260	−0.756298	−	0.654227i	$-0.772994\pi$
$919$	−45.8544	−1.51260	−0.756298	+	0.654227i	$0.772994\pi$
$920$	0	0
$921$	−0.0440707	−0.00145218
$922$	0	0
$923$	0.428251	0.0140961
$924$	0	0
$925$	−28.6749	−0.942826
$926$	0	0
$927$	36.8569	1.21054
$928$	0	0
$929$	3.81073	0.125026	0.0625130	−	0.998044i	$-0.480089\pi$
$929$	3.81073	0.125026	0.0625130	+	0.998044i	$0.480089\pi$
$930$	0	0
$931$	−0.383972	−0.0125842
$932$	0	0
$933$	−2.27714	−0.0745501
$934$	0	0
$935$	−0.667928	−0.0218436
$936$	0	0
$937$	30.5064	0.996600	0.498300	−	0.867005i	$-0.333958\pi$
$937$	30.5064	0.996600	0.498300	+	0.867005i	$0.333958\pi$
$938$	0	0
$939$	2.49107	0.0812929
$940$	0	0
$941$	36.0054	1.17374	0.586871	−	0.809681i	$-0.300360\pi$
$941$	36.0054	1.17374	0.586871	+	0.809681i	$0.300360\pi$
$942$	0	0
$943$	−11.2940	−0.367782
$944$	0	0
$945$	0.290978	0.00946550
$946$	0	0
$947$	2.75429	0.0895023	0.0447512	−	0.998998i	$-0.485751\pi$
$947$	2.75429	0.0895023	0.0447512	+	0.998998i	$0.485751\pi$
$948$	0	0
$949$	−12.1299	−0.393753
$950$	0	0
$951$	−2.70002	−0.0875543
$952$	0	0
$953$	−56.0079	−1.81427	−0.907137	−	0.420836i	$-0.861737\pi$
$953$	−56.0079	−1.81427	−0.907137	+	0.420836i	$0.861737\pi$
$954$	0	0
$955$	5.62084	0.181886
$956$	0	0
$957$	0.499346	0.0161415
$958$	0	0
$959$	16.9509	0.547375
$960$	0	0
$961$	48.6437	1.56915
$962$	0	0
$963$	1.51385	0.0487830
$964$	0	0
$965$	6.23890	0.200837
$966$	0	0
$967$	−46.0658	−1.48138	−0.740688	−	0.671849i	$-0.765500\pi$
$967$	−46.0658	−1.48138	−0.740688	+	0.671849i	$0.765500\pi$
$968$	0	0
$969$	0.105599	0.00339233
$970$	0	0
$971$	−20.6368	−0.662268	−0.331134	−	0.943584i	$-0.607431\pi$
$971$	−20.6368	−0.662268	−0.331134	+	0.943584i	$0.607431\pi$
$972$	0	0
$973$	11.0975	0.355769
$974$	0	0
$975$	−0.690999	−0.0221297
$976$	0	0
$977$	6.08573	0.194700	0.0973498	−	0.995250i	$-0.468963\pi$
$977$	6.08573	0.194700	0.0973498	+	0.995250i	$0.468963\pi$
$978$	0	0
$979$	−4.95758	−0.158445
$980$	0	0
$981$	−4.25906	−0.135981
$982$	0	0
$983$	4.52353	0.144278	0.0721391	−	0.997395i	$-0.477017\pi$
$983$	4.52353	0.144278	0.0721391	+	0.997395i	$0.477017\pi$
$984$	0	0
$985$	−7.73146	−0.246345
$986$	0	0
$987$	−0.355940	−0.0113297
$988$	0	0
$989$	3.13089	0.0995567
$990$	0	0
$991$	−12.0636	−0.383212	−0.191606	−	0.981472i	$-0.561370\pi$
$991$	−12.0636	−0.383212	−0.191606	+	0.981472i	$0.561370\pi$
$992$	0	0
$993$	−4.85126	−0.153950
$994$	0	0
$995$	4.51572	0.143158
$996$	0	0
$997$	10.8318	0.343045	0.171523	−	0.985180i	$-0.445131\pi$
$997$	10.8318	0.343045	0.171523	+	0.985180i	$0.445131\pi$
$998$	0	0
$999$	−4.97181	−0.157301

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.p.1.5	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.p.1.5	✓	9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+0.141545 q^{3} +0.343768 q^{5} -1.00000 q^{7} -2.97996 q^{9} +O(q^{10})\)	\(q+0.141545 q^{3} +0.343768 q^{5} -1.00000 q^{7} -2.97996 q^{9} +1.00000 q^{11} +1.00000 q^{13} +0.0486587 q^{15} -1.94296 q^{17} -0.383972 q^{19} -0.141545 q^{21} -1.66573 q^{23} -4.88182 q^{25} -0.846436 q^{27} +3.52781 q^{29} +8.92433 q^{31} +0.141545 q^{33} -0.343768 q^{35} +5.87381 q^{37} +0.141545 q^{39} +6.78021 q^{41} -1.87960 q^{43} -1.02442 q^{45} +2.51467 q^{47} +1.00000 q^{49} -0.275017 q^{51} +0.793466 q^{53} +0.343768 q^{55} -0.0543494 q^{57} -14.0287 q^{59} +6.36966 q^{61} +2.97996 q^{63} +0.343768 q^{65} -8.05936 q^{67} -0.235776 q^{69} +0.428251 q^{71} -12.1299 q^{73} -0.690999 q^{75} -1.00000 q^{77} +9.37219 q^{79} +8.82009 q^{81} -4.37797 q^{83} -0.667928 q^{85} +0.499346 q^{87} -4.95758 q^{89} -1.00000 q^{91} +1.26320 q^{93} -0.131997 q^{95} -13.5322 q^{97} -2.97996 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{3} - 4 q^{5} - 9 q^{7} + 4 q^{9}+O(q^{10}) \) 9 * q + q^3 - 4 * q^5 - 9 * q^7 + 4 * q^9	\( 9 q + q^{3} - 4 q^{5} - 9 q^{7} + 4 q^{9} + 9 q^{11} + 9 q^{13} - 9 q^{15} - 11 q^{17} + 10 q^{19} - q^{21} - 14 q^{23} - q^{25} - 5 q^{27} - 10 q^{29} + 5 q^{31} + q^{33} + 4 q^{35} - 16 q^{37} + q^{39} + 2 q^{41} + 4 q^{43} - 30 q^{45} + 9 q^{49} + 3 q^{51} - 23 q^{53} - 4 q^{55} + 14 q^{57} + 9 q^{59} - 14 q^{61} - 4 q^{63} - 4 q^{65} + 8 q^{67} - 26 q^{69} - 20 q^{71} - 23 q^{73} + 32 q^{75} - 9 q^{77} + 2 q^{79} - 11 q^{81} - 9 q^{83} - 3 q^{85} - 7 q^{87} - 6 q^{89} - 9 q^{91} - 19 q^{93} - 4 q^{95} - 3 q^{97} + 4 q^{99}+O(q^{100}) \) 9 * q + q^3 - 4 * q^5 - 9 * q^7 + 4 * q^9 + 9 * q^11 + 9 * q^13 - 9 * q^15 - 11 * q^17 + 10 * q^19 - q^21 - 14 * q^23 - q^25 - 5 * q^27 - 10 * q^29 + 5 * q^31 + q^33 + 4 * q^35 - 16 * q^37 + q^39 + 2 * q^41 + 4 * q^43 - 30 * q^45 + 9 * q^49 + 3 * q^51 - 23 * q^53 - 4 * q^55 + 14 * q^57 + 9 * q^59 - 14 * q^61 - 4 * q^63 - 4 * q^65 + 8 * q^67 - 26 * q^69 - 20 * q^71 - 23 * q^73 + 32 * q^75 - 9 * q^77 + 2 * q^79 - 11 * q^81 - 9 * q^83 - 3 * q^85 - 7 * q^87 - 6 * q^89 - 9 * q^91 - 19 * q^93 - 4 * q^95 - 3 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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