LMFDB - Embedded newform 8008.2.a.p.1.8

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			1.8
Root			$2.24883$ 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)

$f(q)$	$=$	$q+2.24883 q^{3} -0.903831 q^{5} -1.00000 q^{7} +2.05724 q^{9} +O(q^{10})$	$q+2.24883 q^{3} -0.903831 q^{5} -1.00000 q^{7} +2.05724 q^{9} +1.00000 q^{11} +1.00000 q^{13} -2.03256 q^{15} +6.24184 q^{17} -3.18001 q^{19} -2.24883 q^{21} -3.89934 q^{23} -4.18309 q^{25} -2.12010 q^{27} -7.45322 q^{29} -5.90291 q^{31} +2.24883 q^{33} +0.903831 q^{35} -5.53086 q^{37} +2.24883 q^{39} +1.74830 q^{41} +1.93088 q^{43} -1.85940 q^{45} +6.48238 q^{47} +1.00000 q^{49} +14.0368 q^{51} -5.50070 q^{53} -0.903831 q^{55} -7.15130 q^{57} +4.02967 q^{59} -9.49947 q^{61} -2.05724 q^{63} -0.903831 q^{65} +3.09877 q^{67} -8.76896 q^{69} +7.68736 q^{71} -7.41380 q^{73} -9.40707 q^{75} -1.00000 q^{77} -11.6456 q^{79} -10.9395 q^{81} -5.15710 q^{83} -5.64157 q^{85} -16.7610 q^{87} -6.64006 q^{89} -1.00000 q^{91} -13.2747 q^{93} +2.87419 q^{95} -7.18680 q^{97} +2.05724 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.24883	1.29836	0.649182	−	0.760633i	$-0.275111\pi$
$3$	2.24883	1.29836	0.649182	+	0.760633i	$0.275111\pi$
$4$	0	0
$5$	−0.903831	−0.404205	−0.202103	−	0.979364i	$-0.564777\pi$
$5$	−0.903831	−0.404205	−0.202103	+	0.979364i	$0.564777\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.05724	0.685748
$10$	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
$15$	−2.03256	−0.524806
$16$	0	0
$17$	6.24184	1.51387	0.756934	−	0.653491i	$-0.226696\pi$
$17$	6.24184	1.51387	0.756934	+	0.653491i	$0.226696\pi$
$18$	0	0
$19$	−3.18001	−0.729544	−0.364772	−	0.931097i	$-0.618853\pi$
$19$	−3.18001	−0.729544	−0.364772	+	0.931097i	$0.618853\pi$
$20$	0	0
$21$	−2.24883	−0.490735
$22$	0	0
$23$	−3.89934	−0.813068	−0.406534	−	0.913636i	$-0.633263\pi$
$23$	−3.89934	−0.813068	−0.406534	+	0.913636i	$0.633263\pi$
$24$	0	0
$25$	−4.18309	−0.836618
$26$	0	0
$27$	−2.12010	−0.408013
$28$	0	0
$29$	−7.45322	−1.38403	−0.692015	−	0.721884i	$-0.743277\pi$
$29$	−7.45322	−1.38403	−0.692015	+	0.721884i	$0.743277\pi$
$30$	0	0
$31$	−5.90291	−1.06019	−0.530097	−	0.847937i	$-0.677845\pi$
$31$	−5.90291	−1.06019	−0.530097	+	0.847937i	$0.677845\pi$
$32$	0	0
$33$	2.24883	0.391471
$34$	0	0
$35$	0.903831	0.152775
$36$	0	0
$37$	−5.53086	−0.909269	−0.454634	−	0.890678i	$-0.650230\pi$
$37$	−5.53086	−0.909269	−0.454634	+	0.890678i	$0.650230\pi$
$38$	0	0
$39$	2.24883	0.360101
$40$	0	0
$41$	1.74830	0.273038	0.136519	−	0.990637i	$-0.456408\pi$
$41$	1.74830	0.273038	0.136519	+	0.990637i	$0.456408\pi$
$42$	0	0
$43$	1.93088	0.294456	0.147228	−	0.989103i	$-0.452965\pi$
$43$	1.93088	0.294456	0.147228	+	0.989103i	$0.452965\pi$
$44$	0	0
$45$	−1.85940	−0.277183
$46$	0	0
$47$	6.48238	0.945552	0.472776	−	0.881183i	$-0.343252\pi$
$47$	6.48238	0.945552	0.472776	+	0.881183i	$0.343252\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	14.0368	1.96555
$52$	0	0
$53$	−5.50070	−0.755579	−0.377789	−	0.925892i	$-0.623316\pi$
$53$	−5.50070	−0.755579	−0.377789	+	0.925892i	$0.623316\pi$
$54$	0	0
$55$	−0.903831	−0.121873
$56$	0	0
$57$	−7.15130	−0.947213
$58$	0	0
$59$	4.02967	0.524618	0.262309	−	0.964984i	$-0.415516\pi$
$59$	4.02967	0.524618	0.262309	+	0.964984i	$0.415516\pi$
$60$	0	0
$61$	−9.49947	−1.21628	−0.608141	−	0.793829i	$-0.708084\pi$
$61$	−9.49947	−1.21628	−0.608141	+	0.793829i	$0.708084\pi$
$62$	0	0
$63$	−2.05724	−0.259188
$64$	0	0
$65$	−0.903831	−0.112106
$66$	0	0
$67$	3.09877	0.378575	0.189288	−	0.981922i	$-0.439382\pi$
$67$	3.09877	0.378575	0.189288	+	0.981922i	$0.439382\pi$
$68$	0	0
$69$	−8.76896	−1.05566
$70$	0	0
$71$	7.68736	0.912322	0.456161	−	0.889897i	$-0.349224\pi$
$71$	7.68736	0.912322	0.456161	+	0.889897i	$0.349224\pi$
$72$	0	0
$73$	−7.41380	−0.867720	−0.433860	−	0.900980i	$-0.642849\pi$
$73$	−7.41380	−0.867720	−0.433860	+	0.900980i	$0.642849\pi$
$74$	0	0
$75$	−9.40707	−1.08623
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−11.6456	−1.31023	−0.655116	−	0.755528i	$-0.727380\pi$
$79$	−11.6456	−1.31023	−0.655116	+	0.755528i	$0.727380\pi$
$80$	0	0
$81$	−10.9395	−1.21550
$82$	0	0
$83$	−5.15710	−0.566065	−0.283033	−	0.959110i	$-0.591340\pi$
$83$	−5.15710	−0.566065	−0.283033	+	0.959110i	$0.591340\pi$
$84$	0	0
$85$	−5.64157	−0.611914
$86$	0	0
$87$	−16.7610	−1.79697
$88$	0	0
$89$	−6.64006	−0.703845	−0.351923	−	0.936029i	$-0.614472\pi$
$89$	−6.64006	−0.703845	−0.351923	+	0.936029i	$0.614472\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−13.2747	−1.37652
$94$	0	0
$95$	2.87419	0.294886
$96$	0	0
$97$	−7.18680	−0.729709	−0.364855	−	0.931064i	$-0.618881\pi$
$97$	−7.18680	−0.729709	−0.364855	+	0.931064i	$0.618881\pi$
$98$	0	0
$99$	2.05724	0.206761
$100$	0	0
$101$	10.2582	1.02073	0.510366	−	0.859957i	$-0.329510\pi$
$101$	10.2582	1.02073	0.510366	+	0.859957i	$0.329510\pi$
$102$	0	0
$103$	−15.7974	−1.55657	−0.778284	−	0.627913i	$-0.783910\pi$
$103$	−15.7974	−1.55657	−0.778284	+	0.627913i	$0.783910\pi$
$104$	0	0
$105$	2.03256	0.198358
$106$	0	0
$107$	9.13768	0.883372	0.441686	−	0.897170i	$-0.354380\pi$
$107$	9.13768	0.883372	0.441686	+	0.897170i	$0.354380\pi$
$108$	0	0
$109$	10.8358	1.03788	0.518939	−	0.854811i	$-0.326327\pi$
$109$	10.8358	1.03788	0.518939	+	0.854811i	$0.326327\pi$
$110$	0	0
$111$	−12.4380	−1.18056
$112$	0	0
$113$	15.4913	1.45730	0.728648	−	0.684889i	$-0.240149\pi$
$113$	15.4913	1.45730	0.728648	+	0.684889i	$0.240149\pi$
$114$	0	0
$115$	3.52434	0.328647
$116$	0	0
$117$	2.05724	0.190192
$118$	0	0
$119$	−6.24184	−0.572188
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.93163	0.354503
$124$	0	0
$125$	8.29996	0.742371
$126$	0	0
$127$	−4.77207	−0.423452	−0.211726	−	0.977329i	$-0.567909\pi$
$127$	−4.77207	−0.423452	−0.211726	+	0.977329i	$0.567909\pi$
$128$	0	0
$129$	4.34222	0.382312
$130$	0	0
$131$	−12.3855	−1.08212	−0.541062	−	0.840982i	$-0.681978\pi$
$131$	−12.3855	−1.08212	−0.541062	+	0.840982i	$0.681978\pi$
$132$	0	0
$133$	3.18001	0.275742
$134$	0	0
$135$	1.91621	0.164921
$136$	0	0
$137$	−5.16125	−0.440955	−0.220478	−	0.975392i	$-0.570762\pi$
$137$	−5.16125	−0.440955	−0.220478	+	0.975392i	$0.570762\pi$
$138$	0	0
$139$	8.79830	0.746262	0.373131	−	0.927779i	$-0.378284\pi$
$139$	8.79830	0.746262	0.373131	+	0.927779i	$0.378284\pi$
$140$	0	0
$141$	14.5778	1.22767
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	6.73645	0.559432
$146$	0	0
$147$	2.24883	0.185481
$148$	0	0
$149$	12.2202	1.00112	0.500558	−	0.865703i	$-0.333128\pi$
$149$	12.2202	1.00112	0.500558	+	0.865703i	$0.333128\pi$
$150$	0	0
$151$	17.5452	1.42781	0.713903	−	0.700245i	$-0.246926\pi$
$151$	17.5452	1.42781	0.713903	+	0.700245i	$0.246926\pi$
$152$	0	0
$153$	12.8410	1.03813
$154$	0	0
$155$	5.33524	0.428536
$156$	0	0
$157$	7.40036	0.590613	0.295307	−	0.955403i	$-0.404578\pi$
$157$	7.40036	0.590613	0.295307	+	0.955403i	$0.404578\pi$
$158$	0	0
$159$	−12.3701	−0.981016
$160$	0	0
$161$	3.89934	0.307311
$162$	0	0
$163$	14.8288	1.16148	0.580741	−	0.814088i	$-0.302763\pi$
$163$	14.8288	1.16148	0.580741	+	0.814088i	$0.302763\pi$
$164$	0	0
$165$	−2.03256	−0.158235
$166$	0	0
$167$	−20.5373	−1.58923	−0.794613	−	0.607117i	$-0.792326\pi$
$167$	−20.5373	−1.58923	−0.794613	+	0.607117i	$0.792326\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.54205	−0.500283
$172$	0	0
$173$	−11.5283	−0.876480	−0.438240	−	0.898858i	$-0.644398\pi$
$173$	−11.5283	−0.876480	−0.438240	+	0.898858i	$0.644398\pi$
$174$	0	0
$175$	4.18309	0.316212
$176$	0	0
$177$	9.06204	0.681145
$178$	0	0
$179$	−16.0348	−1.19850	−0.599250	−	0.800562i	$-0.704534\pi$
$179$	−16.0348	−1.19850	−0.599250	+	0.800562i	$0.704534\pi$
$180$	0	0
$181$	−23.3798	−1.73781	−0.868903	−	0.494982i	$-0.835175\pi$
$181$	−23.3798	−1.73781	−0.868903	+	0.494982i	$0.835175\pi$
$182$	0	0
$183$	−21.3627	−1.57918
$184$	0	0
$185$	4.99897	0.367531
$186$	0	0
$187$	6.24184	0.456448
$188$	0	0
$189$	2.12010	0.154215
$190$	0	0
$191$	−9.17954	−0.664208	−0.332104	−	0.943243i	$-0.607758\pi$
$191$	−9.17954	−0.664208	−0.332104	+	0.943243i	$0.607758\pi$
$192$	0	0
$193$	6.87634	0.494970	0.247485	−	0.968892i	$-0.420396\pi$
$193$	6.87634	0.494970	0.247485	+	0.968892i	$0.420396\pi$
$194$	0	0
$195$	−2.03256	−0.145555
$196$	0	0
$197$	−20.9814	−1.49486	−0.747431	−	0.664339i	$-0.768713\pi$
$197$	−20.9814	−1.49486	−0.747431	+	0.664339i	$0.768713\pi$
$198$	0	0
$199$	4.74468	0.336341	0.168171	−	0.985758i	$-0.446214\pi$
$199$	4.74468	0.336341	0.168171	+	0.985758i	$0.446214\pi$
$200$	0	0
$201$	6.96862	0.491528
$202$	0	0
$203$	7.45322	0.523114
$204$	0	0
$205$	−1.58017	−0.110364
$206$	0	0
$207$	−8.02189	−0.557560
$208$	0	0
$209$	−3.18001	−0.219966
$210$	0	0
$211$	−18.1416	−1.24892	−0.624459	−	0.781058i	$-0.714680\pi$
$211$	−18.1416	−1.24892	−0.624459	+	0.781058i	$0.714680\pi$
$212$	0	0
$213$	17.2876	1.18453
$214$	0	0
$215$	−1.74519	−0.119021
$216$	0	0
$217$	5.90291	0.400716
$218$	0	0
$219$	−16.6724	−1.12662
$220$	0	0
$221$	6.24184	0.419871
$222$	0	0
$223$	13.4791	0.902625	0.451313	−	0.892366i	$-0.350956\pi$
$223$	13.4791	0.902625	0.451313	+	0.892366i	$0.350956\pi$
$224$	0	0
$225$	−8.60564	−0.573709
$226$	0	0
$227$	8.81639	0.585164	0.292582	−	0.956240i	$-0.405485\pi$
$227$	8.81639	0.585164	0.292582	+	0.956240i	$0.405485\pi$
$228$	0	0
$229$	−7.45110	−0.492383	−0.246191	−	0.969221i	$-0.579179\pi$
$229$	−7.45110	−0.492383	−0.246191	+	0.969221i	$0.579179\pi$
$230$	0	0
$231$	−2.24883	−0.147962
$232$	0	0
$233$	−3.47202	−0.227460	−0.113730	−	0.993512i	$-0.536280\pi$
$233$	−3.47202	−0.227460	−0.113730	+	0.993512i	$0.536280\pi$
$234$	0	0
$235$	−5.85897	−0.382197
$236$	0	0
$237$	−26.1890	−1.70116
$238$	0	0
$239$	−17.2438	−1.11541	−0.557704	−	0.830040i	$-0.688318\pi$
$239$	−17.2438	−1.11541	−0.557704	+	0.830040i	$0.688318\pi$
$240$	0	0
$241$	−18.1758	−1.17080	−0.585402	−	0.810743i	$-0.699063\pi$
$241$	−18.1758	−1.17080	−0.585402	+	0.810743i	$0.699063\pi$
$242$	0	0
$243$	−18.2407	−1.17014
$244$	0	0
$245$	−0.903831	−0.0577436
$246$	0	0
$247$	−3.18001	−0.202339
$248$	0	0
$249$	−11.5975	−0.734959
$250$	0	0
$251$	25.5228	1.61099	0.805493	−	0.592605i	$-0.201900\pi$
$251$	25.5228	1.61099	0.805493	+	0.592605i	$0.201900\pi$
$252$	0	0
$253$	−3.89934	−0.245149
$254$	0	0
$255$	−12.6869	−0.794486
$256$	0	0
$257$	−28.7020	−1.79038	−0.895190	−	0.445685i	$-0.852960\pi$
$257$	−28.7020	−1.79038	−0.895190	+	0.445685i	$0.852960\pi$
$258$	0	0
$259$	5.53086	0.343671
$260$	0	0
$261$	−15.3331	−0.949095
$262$	0	0
$263$	−4.38449	−0.270359	−0.135180	−	0.990821i	$-0.543161\pi$
$263$	−4.38449	−0.270359	−0.135180	+	0.990821i	$0.543161\pi$
$264$	0	0
$265$	4.97170	0.305409
$266$	0	0
$267$	−14.9324	−0.913847
$268$	0	0
$269$	7.19130	0.438461	0.219231	−	0.975673i	$-0.429645\pi$
$269$	7.19130	0.438461	0.219231	+	0.975673i	$0.429645\pi$
$270$	0	0
$271$	17.1267	1.04037	0.520187	−	0.854053i	$-0.325862\pi$
$271$	17.1267	1.04037	0.520187	+	0.854053i	$0.325862\pi$
$272$	0	0
$273$	−2.24883	−0.136105
$274$	0	0
$275$	−4.18309	−0.252250
$276$	0	0
$277$	−2.61283	−0.156990	−0.0784950	−	0.996915i	$-0.525011\pi$
$277$	−2.61283	−0.156990	−0.0784950	+	0.996915i	$0.525011\pi$
$278$	0	0
$279$	−12.1437	−0.727027
$280$	0	0
$281$	−2.10599	−0.125633	−0.0628165	−	0.998025i	$-0.520008\pi$
$281$	−2.10599	−0.125633	−0.0628165	+	0.998025i	$0.520008\pi$
$282$	0	0
$283$	4.85673	0.288703	0.144351	−	0.989526i	$-0.453890\pi$
$283$	4.85673	0.288703	0.144351	+	0.989526i	$0.453890\pi$
$284$	0	0
$285$	6.46357	0.382869
$286$	0	0
$287$	−1.74830	−0.103199
$288$	0	0
$289$	21.9605	1.29180
$290$	0	0
$291$	−16.1619	−0.947428
$292$	0	0
$293$	−9.09597	−0.531392	−0.265696	−	0.964057i	$-0.585602\pi$
$293$	−9.09597	−0.531392	−0.265696	+	0.964057i	$0.585602\pi$
$294$	0	0
$295$	−3.64214	−0.212053
$296$	0	0
$297$	−2.12010	−0.123021
$298$	0	0
$299$	−3.89934	−0.225505
$300$	0	0
$301$	−1.93088	−0.111294
$302$	0	0
$303$	23.0690	1.32528
$304$	0	0
$305$	8.58591	0.491628
$306$	0	0
$307$	−31.2366	−1.78277	−0.891383	−	0.453251i	$-0.850264\pi$
$307$	−31.2366	−1.78277	−0.891383	+	0.453251i	$0.850264\pi$
$308$	0	0
$309$	−35.5258	−2.02099
$310$	0	0
$311$	−27.8100	−1.57696	−0.788481	−	0.615059i	$-0.789132\pi$
$311$	−27.8100	−1.57696	−0.788481	+	0.615059i	$0.789132\pi$
$312$	0	0
$313$	−7.79549	−0.440627	−0.220314	−	0.975429i	$-0.570708\pi$
$313$	−7.79549	−0.440627	−0.220314	+	0.975429i	$0.570708\pi$
$314$	0	0
$315$	1.85940	0.104765
$316$	0	0
$317$	24.4193	1.37152	0.685762	−	0.727826i	$-0.259469\pi$
$317$	24.4193	1.37152	0.685762	+	0.727826i	$0.259469\pi$
$318$	0	0
$319$	−7.45322	−0.417300
$320$	0	0
$321$	20.5491	1.14694
$322$	0	0
$323$	−19.8491	−1.10443
$324$	0	0
$325$	−4.18309	−0.232036
$326$	0	0
$327$	24.3678	1.34754
$328$	0	0
$329$	−6.48238	−0.357385
$330$	0	0
$331$	−7.27763	−0.400014	−0.200007	−	0.979794i	$-0.564097\pi$
$331$	−7.27763	−0.400014	−0.200007	+	0.979794i	$0.564097\pi$
$332$	0	0
$333$	−11.3783	−0.623529
$334$	0	0
$335$	−2.80077	−0.153022
$336$	0	0
$337$	14.6862	0.800007	0.400004	−	0.916514i	$-0.369009\pi$
$337$	14.6862	0.800007	0.400004	+	0.916514i	$0.369009\pi$
$338$	0	0
$339$	34.8372	1.89210
$340$	0	0
$341$	−5.90291	−0.319661
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	7.92565	0.426703
$346$	0	0
$347$	16.5963	0.890939	0.445469	−	0.895297i	$-0.353037\pi$
$347$	16.5963	0.890939	0.445469	+	0.895297i	$0.353037\pi$
$348$	0	0
$349$	−0.874980	−0.0468366	−0.0234183	−	0.999726i	$-0.507455\pi$
$349$	−0.874980	−0.0468366	−0.0234183	+	0.999726i	$0.507455\pi$
$350$	0	0
$351$	−2.12010	−0.113163
$352$	0	0
$353$	26.2033	1.39466	0.697331	−	0.716750i	$-0.254371\pi$
$353$	26.2033	1.39466	0.697331	+	0.716750i	$0.254371\pi$
$354$	0	0
$355$	−6.94807	−0.368765
$356$	0	0
$357$	−14.0368	−0.742909
$358$	0	0
$359$	−8.60660	−0.454239	−0.227120	−	0.973867i	$-0.572931\pi$
$359$	−8.60660	−0.454239	−0.227120	+	0.973867i	$0.572931\pi$
$360$	0	0
$361$	−8.88755	−0.467766
$362$	0	0
$363$	2.24883	0.118033
$364$	0	0
$365$	6.70082	0.350737
$366$	0	0
$367$	−3.27914	−0.171170	−0.0855848	−	0.996331i	$-0.527276\pi$
$367$	−3.27914	−0.171170	−0.0855848	+	0.996331i	$0.527276\pi$
$368$	0	0
$369$	3.59668	0.187235
$370$	0	0
$371$	5.50070	0.285582
$372$	0	0
$373$	−26.3670	−1.36523	−0.682615	−	0.730778i	$-0.739157\pi$
$373$	−26.3670	−1.36523	−0.682615	+	0.730778i	$0.739157\pi$
$374$	0	0
$375$	18.6652	0.963867
$376$	0	0
$377$	−7.45322	−0.383861
$378$	0	0
$379$	−20.3220	−1.04387	−0.521935	−	0.852985i	$-0.674790\pi$
$379$	−20.3220	−1.04387	−0.521935	+	0.852985i	$0.674790\pi$
$380$	0	0
$381$	−10.7316	−0.549795
$382$	0	0
$383$	3.69412	0.188761	0.0943804	−	0.995536i	$-0.469913\pi$
$383$	3.69412	0.188761	0.0943804	+	0.995536i	$0.469913\pi$
$384$	0	0
$385$	0.903831	0.0460635
$386$	0	0
$387$	3.97229	0.201923
$388$	0	0
$389$	−12.3981	−0.628608	−0.314304	−	0.949322i	$-0.601771\pi$
$389$	−12.3981	−0.628608	−0.314304	+	0.949322i	$0.601771\pi$
$390$	0	0
$391$	−24.3390	−1.23088
$392$	0	0
$393$	−27.8529	−1.40499
$394$	0	0
$395$	10.5257	0.529603
$396$	0	0
$397$	−13.5349	−0.679296	−0.339648	−	0.940553i	$-0.610308\pi$
$397$	−13.5349	−0.679296	−0.339648	+	0.940553i	$0.610308\pi$
$398$	0	0
$399$	7.15130	0.358013
$400$	0	0
$401$	−20.3061	−1.01404	−0.507020	−	0.861934i	$-0.669253\pi$
$401$	−20.3061	−1.01404	−0.507020	+	0.861934i	$0.669253\pi$
$402$	0	0
$403$	−5.90291	−0.294045
$404$	0	0
$405$	9.88744	0.491311
$406$	0	0
$407$	−5.53086	−0.274155
$408$	0	0
$409$	−15.9099	−0.786692	−0.393346	−	0.919391i	$-0.628682\pi$
$409$	−15.9099	−0.786692	−0.393346	+	0.919391i	$0.628682\pi$
$410$	0	0
$411$	−11.6068	−0.572520
$412$	0	0
$413$	−4.02967	−0.198287
$414$	0	0
$415$	4.66115	0.228807
$416$	0	0
$417$	19.7859	0.968920
$418$	0	0
$419$	−3.60713	−0.176220	−0.0881100	−	0.996111i	$-0.528083\pi$
$419$	−3.60713	−0.176220	−0.0881100	+	0.996111i	$0.528083\pi$
$420$	0	0
$421$	37.9364	1.84891	0.924455	−	0.381292i	$-0.124521\pi$
$421$	37.9364	1.84891	0.924455	+	0.381292i	$0.124521\pi$
$422$	0	0
$423$	13.3358	0.648411
$424$	0	0
$425$	−26.1102	−1.26653
$426$	0	0
$427$	9.49947	0.459711
$428$	0	0
$429$	2.24883	0.108575
$430$	0	0
$431$	13.6439	0.657205	0.328602	−	0.944468i	$-0.393422\pi$
$431$	13.6439	0.657205	0.328602	+	0.944468i	$0.393422\pi$
$432$	0	0
$433$	3.93517	0.189112	0.0945560	−	0.995520i	$-0.469857\pi$
$433$	3.93517	0.189112	0.0945560	+	0.995520i	$0.469857\pi$
$434$	0	0
$435$	15.1491	0.726346
$436$	0	0
$437$	12.3999	0.593169
$438$	0	0
$439$	−8.99749	−0.429427	−0.214713	−	0.976677i	$-0.568882\pi$
$439$	−8.99749	−0.429427	−0.214713	+	0.976677i	$0.568882\pi$
$440$	0	0
$441$	2.05724	0.0979640
$442$	0	0
$443$	11.7651	0.558979	0.279490	−	0.960149i	$-0.409835\pi$
$443$	11.7651	0.558979	0.279490	+	0.960149i	$0.409835\pi$
$444$	0	0
$445$	6.00149	0.284498
$446$	0	0
$447$	27.4811	1.29981
$448$	0	0
$449$	−7.69749	−0.363267	−0.181634	−	0.983366i	$-0.558138\pi$
$449$	−7.69749	−0.363267	−0.181634	+	0.983366i	$0.558138\pi$
$450$	0	0
$451$	1.74830	0.0823241
$452$	0	0
$453$	39.4561	1.85381
$454$	0	0
$455$	0.903831	0.0423722
$456$	0	0
$457$	34.1264	1.59636	0.798182	−	0.602417i	$-0.205795\pi$
$457$	34.1264	1.59636	0.798182	+	0.602417i	$0.205795\pi$
$458$	0	0
$459$	−13.2333	−0.617678
$460$	0	0
$461$	30.9635	1.44211	0.721057	−	0.692875i	$-0.243656\pi$
$461$	30.9635	1.44211	0.721057	+	0.692875i	$0.243656\pi$
$462$	0	0
$463$	−1.61862	−0.0752238	−0.0376119	−	0.999292i	$-0.511975\pi$
$463$	−1.61862	−0.0752238	−0.0376119	+	0.999292i	$0.511975\pi$
$464$	0	0
$465$	11.9980	0.556396
$466$	0	0
$467$	2.82037	0.130511	0.0652557	−	0.997869i	$-0.479214\pi$
$467$	2.82037	0.130511	0.0652557	+	0.997869i	$0.479214\pi$
$468$	0	0
$469$	−3.09877	−0.143088
$470$	0	0
$471$	16.6422	0.766831
$472$	0	0
$473$	1.93088	0.0887820
$474$	0	0
$475$	13.3023	0.610350
$476$	0	0
$477$	−11.3163	−0.518137
$478$	0	0
$479$	12.8970	0.589279	0.294639	−	0.955609i	$-0.404800\pi$
$479$	12.8970	0.589279	0.294639	+	0.955609i	$0.404800\pi$
$480$	0	0
$481$	−5.53086	−0.252186
$482$	0	0
$483$	8.76896	0.399001
$484$	0	0
$485$	6.49565	0.294952
$486$	0	0
$487$	23.1931	1.05098	0.525490	−	0.850800i	$-0.323882\pi$
$487$	23.1931	1.05098	0.525490	+	0.850800i	$0.323882\pi$
$488$	0	0
$489$	33.3475	1.50803
$490$	0	0
$491$	−22.4553	−1.01339	−0.506696	−	0.862125i	$-0.669133\pi$
$491$	−22.4553	−1.01339	−0.506696	+	0.862125i	$0.669133\pi$
$492$	0	0
$493$	−46.5218	−2.09524
$494$	0	0
$495$	−1.85940	−0.0835738
$496$	0	0
$497$	−7.68736	−0.344825
$498$	0	0
$499$	6.82022	0.305315	0.152657	−	0.988279i	$-0.451217\pi$
$499$	6.82022	0.305315	0.152657	+	0.988279i	$0.451217\pi$
$500$	0	0
$501$	−46.1850	−2.06339
$502$	0	0
$503$	−10.2676	−0.457808	−0.228904	−	0.973449i	$-0.573514\pi$
$503$	−10.2676	−0.457808	−0.228904	+	0.973449i	$0.573514\pi$
$504$	0	0
$505$	−9.27171	−0.412586
$506$	0	0
$507$	2.24883	0.0998741
$508$	0	0
$509$	−24.1361	−1.06982	−0.534908	−	0.844911i	$-0.679654\pi$
$509$	−24.1361	−1.06982	−0.534908	+	0.844911i	$0.679654\pi$
$510$	0	0
$511$	7.41380	0.327967
$512$	0	0
$513$	6.74193	0.297664
$514$	0	0
$515$	14.2782	0.629173
$516$	0	0
$517$	6.48238	0.285095
$518$	0	0
$519$	−25.9252	−1.13799
$520$	0	0
$521$	28.7456	1.25937	0.629684	−	0.776851i	$-0.283184\pi$
$521$	28.7456	1.25937	0.629684	+	0.776851i	$0.283184\pi$
$522$	0	0
$523$	−1.90369	−0.0832427	−0.0416213	−	0.999133i	$-0.513252\pi$
$523$	−1.90369	−0.0832427	−0.0416213	+	0.999133i	$0.513252\pi$
$524$	0	0
$525$	9.40707	0.410558
$526$	0	0
$527$	−36.8450	−1.60500
$528$	0	0
$529$	−7.79515	−0.338920
$530$	0	0
$531$	8.29001	0.359756
$532$	0	0
$533$	1.74830	0.0757272
$534$	0	0
$535$	−8.25891	−0.357064
$536$	0	0
$537$	−36.0596	−1.55609
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−22.4335	−0.964491	−0.482245	−	0.876036i	$-0.660179\pi$
$541$	−22.4335	−0.964491	−0.482245	+	0.876036i	$0.660179\pi$
$542$	0	0
$543$	−52.5772	−2.25630
$544$	0	0
$545$	−9.79370	−0.419516
$546$	0	0
$547$	20.1066	0.859695	0.429847	−	0.902902i	$-0.358567\pi$
$547$	20.1066	0.859695	0.429847	+	0.902902i	$0.358567\pi$
$548$	0	0
$549$	−19.5427	−0.834063
$550$	0	0
$551$	23.7013	1.00971
$552$	0	0
$553$	11.6456	0.495221
$554$	0	0
$555$	11.2418	0.477189
$556$	0	0
$557$	−13.9569	−0.591371	−0.295686	−	0.955285i	$-0.595548\pi$
$557$	−13.9569	−0.591371	−0.295686	+	0.955285i	$0.595548\pi$
$558$	0	0
$559$	1.93088	0.0816675
$560$	0	0
$561$	14.0368	0.592636
$562$	0	0
$563$	38.2900	1.61373	0.806867	−	0.590734i	$-0.201162\pi$
$563$	38.2900	1.61373	0.806867	+	0.590734i	$0.201162\pi$
$564$	0	0
$565$	−14.0015	−0.589047
$566$	0	0
$567$	10.9395	0.459415
$568$	0	0
$569$	−38.8497	−1.62866	−0.814331	−	0.580400i	$-0.802896\pi$
$569$	−38.8497	−1.62866	−0.814331	+	0.580400i	$0.802896\pi$
$570$	0	0
$571$	−15.1303	−0.633185	−0.316592	−	0.948562i	$-0.602539\pi$
$571$	−15.1303	−0.633185	−0.316592	+	0.948562i	$0.602539\pi$
$572$	0	0
$573$	−20.6432	−0.862383
$574$	0	0
$575$	16.3113	0.680228
$576$	0	0
$577$	19.1643	0.797818	0.398909	−	0.916990i	$-0.369389\pi$
$577$	19.1643	0.797818	0.398909	+	0.916990i	$0.369389\pi$
$578$	0	0
$579$	15.4637	0.642651
$580$	0	0
$581$	5.15710	0.213953
$582$	0	0
$583$	−5.50070	−0.227816
$584$	0	0
$585$	−1.85940	−0.0768767
$586$	0	0
$587$	28.6383	1.18203	0.591014	−	0.806661i	$-0.298728\pi$
$587$	28.6383	1.18203	0.591014	+	0.806661i	$0.298728\pi$
$588$	0	0
$589$	18.7713	0.773459
$590$	0	0
$591$	−47.1836	−1.94087
$592$	0	0
$593$	−5.02360	−0.206295	−0.103147	−	0.994666i	$-0.532891\pi$
$593$	−5.02360	−0.206295	−0.103147	+	0.994666i	$0.532891\pi$
$594$	0	0
$595$	5.64157	0.231282
$596$	0	0
$597$	10.6700	0.436693
$598$	0	0
$599$	3.48066	0.142216	0.0711080	−	0.997469i	$-0.477347\pi$
$599$	3.48066	0.142216	0.0711080	+	0.997469i	$0.477347\pi$
$600$	0	0
$601$	19.5135	0.795974	0.397987	−	0.917391i	$-0.369709\pi$
$601$	19.5135	0.795974	0.397987	+	0.917391i	$0.369709\pi$
$602$	0	0
$603$	6.37493	0.259607
$604$	0	0
$605$	−0.903831	−0.0367459
$606$	0	0
$607$	−0.256535	−0.0104124	−0.00520622	−	0.999986i	$-0.501657\pi$
$607$	−0.256535	−0.0104124	−0.00520622	+	0.999986i	$0.501657\pi$
$608$	0	0
$609$	16.7610	0.679192
$610$	0	0
$611$	6.48238	0.262249
$612$	0	0
$613$	−6.79629	−0.274500	−0.137250	−	0.990536i	$-0.543826\pi$
$613$	−6.79629	−0.274500	−0.137250	+	0.990536i	$0.543826\pi$
$614$	0	0
$615$	−3.55353	−0.143292
$616$	0	0
$617$	2.84201	0.114415	0.0572075	−	0.998362i	$-0.481780\pi$
$617$	2.84201	0.114415	0.0572075	+	0.998362i	$0.481780\pi$
$618$	0	0
$619$	27.4698	1.10410	0.552052	−	0.833809i	$-0.313845\pi$
$619$	27.4698	1.10410	0.552052	+	0.833809i	$0.313845\pi$
$620$	0	0
$621$	8.26699	0.331743
$622$	0	0
$623$	6.64006	0.266029
$624$	0	0
$625$	13.4137	0.536548
$626$	0	0
$627$	−7.15130	−0.285596
$628$	0	0
$629$	−34.5228	−1.37651
$630$	0	0
$631$	28.1742	1.12160	0.560799	−	0.827952i	$-0.310494\pi$
$631$	28.1742	1.12160	0.560799	+	0.827952i	$0.310494\pi$
$632$	0	0
$633$	−40.7974	−1.62155
$634$	0	0
$635$	4.31314	0.171162
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	15.8148	0.625623
$640$	0	0
$641$	−49.2209	−1.94411	−0.972054	−	0.234758i	$-0.924570\pi$
$641$	−49.2209	−1.94411	−0.972054	+	0.234758i	$0.924570\pi$
$642$	0	0
$643$	30.3937	1.19861	0.599305	−	0.800521i	$-0.295444\pi$
$643$	30.3937	1.19861	0.599305	+	0.800521i	$0.295444\pi$
$644$	0	0
$645$	−3.92464	−0.154532
$646$	0	0
$647$	6.68020	0.262626	0.131313	−	0.991341i	$-0.458081\pi$
$647$	6.68020	0.262626	0.131313	+	0.991341i	$0.458081\pi$
$648$	0	0
$649$	4.02967	0.158178
$650$	0	0
$651$	13.2747	0.520275
$652$	0	0
$653$	41.4104	1.62052	0.810258	−	0.586073i	$-0.199327\pi$
$653$	41.4104	1.62052	0.810258	+	0.586073i	$0.199327\pi$
$654$	0	0
$655$	11.1944	0.437401
$656$	0	0
$657$	−15.2520	−0.595037
$658$	0	0
$659$	−50.9177	−1.98347	−0.991736	−	0.128292i	$-0.959051\pi$
$659$	−50.9177	−1.98347	−0.991736	+	0.128292i	$0.959051\pi$
$660$	0	0
$661$	11.9014	0.462912	0.231456	−	0.972845i	$-0.425651\pi$
$661$	11.9014	0.462912	0.231456	+	0.972845i	$0.425651\pi$
$662$	0	0
$663$	14.0368	0.545146
$664$	0	0
$665$	−2.87419	−0.111456
$666$	0	0
$667$	29.0627	1.12531
$668$	0	0
$669$	30.3122	1.17194
$670$	0	0
$671$	−9.49947	−0.366723
$672$	0	0
$673$	−2.53354	−0.0976609	−0.0488304	−	0.998807i	$-0.515549\pi$
$673$	−2.53354	−0.0976609	−0.0488304	+	0.998807i	$0.515549\pi$
$674$	0	0
$675$	8.86857	0.341351
$676$	0	0
$677$	33.1578	1.27436	0.637178	−	0.770716i	$-0.280101\pi$
$677$	33.1578	1.27436	0.637178	+	0.770716i	$0.280101\pi$
$678$	0	0
$679$	7.18680	0.275804
$680$	0	0
$681$	19.8266	0.759756
$682$	0	0
$683$	25.0003	0.956610	0.478305	−	0.878194i	$-0.341251\pi$
$683$	25.0003	0.956610	0.478305	+	0.878194i	$0.341251\pi$
$684$	0	0
$685$	4.66489	0.178236
$686$	0	0
$687$	−16.7563	−0.639292
$688$	0	0
$689$	−5.50070	−0.209560
$690$	0	0
$691$	24.4918	0.931712	0.465856	−	0.884861i	$-0.345747\pi$
$691$	24.4918	0.931712	0.465856	+	0.884861i	$0.345747\pi$
$692$	0	0
$693$	−2.05724	−0.0781482
$694$	0	0
$695$	−7.95218	−0.301643
$696$	0	0
$697$	10.9126	0.413344
$698$	0	0
$699$	−7.80799	−0.295325
$700$	0	0
$701$	−48.0349	−1.81425	−0.907127	−	0.420857i	$-0.861729\pi$
$701$	−48.0349	−1.81425	−0.907127	+	0.420857i	$0.861729\pi$
$702$	0	0
$703$	17.5882	0.663351
$704$	0	0
$705$	−13.1758	−0.496231
$706$	0	0
$707$	−10.2582	−0.385801
$708$	0	0
$709$	−17.2347	−0.647261	−0.323631	−	0.946184i	$-0.604904\pi$
$709$	−17.2347	−0.647261	−0.323631	+	0.946184i	$0.604904\pi$
$710$	0	0
$711$	−23.9578	−0.898489
$712$	0	0
$713$	23.0175	0.862011
$714$	0	0
$715$	−0.903831	−0.0338014
$716$	0	0
$717$	−38.7784	−1.44821
$718$	0	0
$719$	42.7531	1.59442	0.797212	−	0.603700i	$-0.206307\pi$
$719$	42.7531	1.59442	0.797212	+	0.603700i	$0.206307\pi$
$720$	0	0
$721$	15.7974	0.588327
$722$	0	0
$723$	−40.8743	−1.52013
$724$	0	0
$725$	31.1775	1.15790
$726$	0	0
$727$	42.0677	1.56021	0.780103	−	0.625652i	$-0.215167\pi$
$727$	42.0677	1.56021	0.780103	+	0.625652i	$0.215167\pi$
$728$	0	0
$729$	−8.20194	−0.303776
$730$	0	0
$731$	12.0522	0.445768
$732$	0	0
$733$	−36.9129	−1.36341	−0.681705	−	0.731627i	$-0.738761\pi$
$733$	−36.9129	−1.36341	−0.681705	+	0.731627i	$0.738761\pi$
$734$	0	0
$735$	−2.03256	−0.0749722
$736$	0	0
$737$	3.09877	0.114145
$738$	0	0
$739$	9.60434	0.353302	0.176651	−	0.984274i	$-0.443474\pi$
$739$	9.60434	0.353302	0.176651	+	0.984274i	$0.443474\pi$
$740$	0	0
$741$	−7.15130	−0.262710
$742$	0	0
$743$	11.8614	0.435153	0.217577	−	0.976043i	$-0.430185\pi$
$743$	11.8614	0.435153	0.217577	+	0.976043i	$0.430185\pi$
$744$	0	0
$745$	−11.0450	−0.404657
$746$	0	0
$747$	−10.6094	−0.388178
$748$	0	0
$749$	−9.13768	−0.333883
$750$	0	0
$751$	34.2351	1.24925	0.624627	−	0.780923i	$-0.285251\pi$
$751$	34.2351	1.24925	0.624627	+	0.780923i	$0.285251\pi$
$752$	0	0
$753$	57.3965	2.09165
$754$	0	0
$755$	−15.8579	−0.577127
$756$	0	0
$757$	1.55940	0.0566774	0.0283387	−	0.999598i	$-0.490978\pi$
$757$	1.55940	0.0566774	0.0283387	+	0.999598i	$0.490978\pi$
$758$	0	0
$759$	−8.76896	−0.318293
$760$	0	0
$761$	−2.70245	−0.0979636	−0.0489818	−	0.998800i	$-0.515598\pi$
$761$	−2.70245	−0.0979636	−0.0489818	+	0.998800i	$0.515598\pi$
$762$	0	0
$763$	−10.8358	−0.392281
$764$	0	0
$765$	−11.6061	−0.419619
$766$	0	0
$767$	4.02967	0.145503
$768$	0	0
$769$	−25.1828	−0.908116	−0.454058	−	0.890972i	$-0.650024\pi$
$769$	−25.1828	−0.908116	−0.454058	+	0.890972i	$0.650024\pi$
$770$	0	0
$771$	−64.5459	−2.32456
$772$	0	0
$773$	−29.9513	−1.07727	−0.538637	−	0.842538i	$-0.681061\pi$
$773$	−29.9513	−1.07727	−0.538637	+	0.842538i	$0.681061\pi$
$774$	0	0
$775$	24.6924	0.886978
$776$	0	0
$777$	12.4380	0.446210
$778$	0	0
$779$	−5.55960	−0.199193
$780$	0	0
$781$	7.68736	0.275075
$782$	0	0
$783$	15.8016	0.564702
$784$	0	0
$785$	−6.68867	−0.238729
$786$	0	0
$787$	16.2431	0.579002	0.289501	−	0.957178i	$-0.406511\pi$
$787$	16.2431	0.579002	0.289501	+	0.957178i	$0.406511\pi$
$788$	0	0
$789$	−9.85997	−0.351024
$790$	0	0
$791$	−15.4913	−0.550806
$792$	0	0
$793$	−9.49947	−0.337336
$794$	0	0
$795$	11.1805	0.396532
$796$	0	0
$797$	−42.5277	−1.50641	−0.753204	−	0.657787i	$-0.771493\pi$
$797$	−42.5277	−1.50641	−0.753204	+	0.657787i	$0.771493\pi$
$798$	0	0
$799$	40.4620	1.43144
$800$	0	0
$801$	−13.6602	−0.482661
$802$	0	0
$803$	−7.41380	−0.261627
$804$	0	0
$805$	−3.52434	−0.124217
$806$	0	0
$807$	16.1720	0.569282
$808$	0	0
$809$	51.1758	1.79925	0.899623	−	0.436666i	$-0.143841\pi$
$809$	51.1758	1.79925	0.899623	+	0.436666i	$0.143841\pi$
$810$	0	0
$811$	−26.9466	−0.946222	−0.473111	−	0.881003i	$-0.656869\pi$
$811$	−26.9466	−0.946222	−0.473111	+	0.881003i	$0.656869\pi$
$812$	0	0
$813$	38.5151	1.35078
$814$	0	0
$815$	−13.4027	−0.469478
$816$	0	0
$817$	−6.14021	−0.214819
$818$	0	0
$819$	−2.05724	−0.0718859
$820$	0	0
$821$	−20.0890	−0.701110	−0.350555	−	0.936542i	$-0.614007\pi$
$821$	−20.0890	−0.701110	−0.350555	+	0.936542i	$0.614007\pi$
$822$	0	0
$823$	−11.6383	−0.405686	−0.202843	−	0.979211i	$-0.565018\pi$
$823$	−11.6383	−0.405686	−0.202843	+	0.979211i	$0.565018\pi$
$824$	0	0
$825$	−9.40707	−0.327512
$826$	0	0
$827$	13.2406	0.460422	0.230211	−	0.973141i	$-0.426058\pi$
$827$	13.2406	0.460422	0.230211	+	0.973141i	$0.426058\pi$
$828$	0	0
$829$	10.4082	0.361493	0.180746	−	0.983530i	$-0.442149\pi$
$829$	10.4082	0.361493	0.180746	+	0.983530i	$0.442149\pi$
$830$	0	0
$831$	−5.87583	−0.203830
$832$	0	0
$833$	6.24184	0.216267
$834$	0	0
$835$	18.5623	0.642374
$836$	0	0
$837$	12.5148	0.432574
$838$	0	0
$839$	−36.0596	−1.24492	−0.622458	−	0.782653i	$-0.713866\pi$
$839$	−36.0596	−1.24492	−0.622458	+	0.782653i	$0.713866\pi$
$840$	0	0
$841$	26.5506	0.915536
$842$	0	0
$843$	−4.73602	−0.163117
$844$	0	0
$845$	−0.903831	−0.0310927
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	10.9220	0.374841
$850$	0	0
$851$	21.5667	0.739298
$852$	0	0
$853$	28.0126	0.959133	0.479567	−	0.877506i	$-0.340794\pi$
$853$	28.0126	0.959133	0.479567	+	0.877506i	$0.340794\pi$
$854$	0	0
$855$	5.91291	0.202217
$856$	0	0
$857$	−15.5237	−0.530280	−0.265140	−	0.964210i	$-0.585418\pi$
$857$	−15.5237	−0.530280	−0.265140	+	0.964210i	$0.585418\pi$
$858$	0	0
$859$	7.70075	0.262746	0.131373	−	0.991333i	$-0.458061\pi$
$859$	7.70075	0.262746	0.131373	+	0.991333i	$0.458061\pi$
$860$	0	0
$861$	−3.93163	−0.133990
$862$	0	0
$863$	−32.7843	−1.11599	−0.557995	−	0.829844i	$-0.688429\pi$
$863$	−32.7843	−1.11599	−0.557995	+	0.829844i	$0.688429\pi$
$864$	0	0
$865$	10.4196	0.354278
$866$	0	0
$867$	49.3856	1.67722
$868$	0	0
$869$	−11.6456	−0.395050
$870$	0	0
$871$	3.09877	0.104998
$872$	0	0
$873$	−14.7850	−0.500397
$874$	0	0
$875$	−8.29996	−0.280590
$876$	0	0
$877$	38.1649	1.28874	0.644369	−	0.764715i	$-0.277120\pi$
$877$	38.1649	1.28874	0.644369	+	0.764715i	$0.277120\pi$
$878$	0	0
$879$	−20.4553	−0.689940
$880$	0	0
$881$	38.2191	1.28763	0.643817	−	0.765179i	$-0.277350\pi$
$881$	38.2191	1.28763	0.643817	+	0.765179i	$0.277350\pi$
$882$	0	0
$883$	5.52445	0.185912	0.0929562	−	0.995670i	$-0.470368\pi$
$883$	5.52445	0.185912	0.0929562	+	0.995670i	$0.470368\pi$
$884$	0	0
$885$	−8.19055	−0.275322
$886$	0	0
$887$	−21.7944	−0.731785	−0.365892	−	0.930657i	$-0.619236\pi$
$887$	−21.7944	−0.731785	−0.365892	+	0.930657i	$0.619236\pi$
$888$	0	0
$889$	4.77207	0.160050
$890$	0	0
$891$	−10.9395	−0.366486
$892$	0	0
$893$	−20.6140	−0.689822
$894$	0	0
$895$	14.4928	0.484440
$896$	0	0
$897$	−8.76896	−0.292787
$898$	0	0
$899$	43.9957	1.46734
$900$	0	0
$901$	−34.3345	−1.14385
$902$	0	0
$903$	−4.34222	−0.144500
$904$	0	0
$905$	21.1314	0.702431
$906$	0	0
$907$	19.7630	0.656218	0.328109	−	0.944640i	$-0.393589\pi$
$907$	19.7630	0.656218	0.328109	+	0.944640i	$0.393589\pi$
$908$	0	0
$909$	21.1037	0.699965
$910$	0	0
$911$	−38.6305	−1.27988	−0.639942	−	0.768423i	$-0.721042\pi$
$911$	−38.6305	−1.27988	−0.639942	+	0.768423i	$0.721042\pi$
$912$	0	0
$913$	−5.15710	−0.170675
$914$	0	0
$915$	19.3083	0.638312
$916$	0	0
$917$	12.3855	0.409005
$918$	0	0
$919$	−22.5736	−0.744635	−0.372318	−	0.928105i	$-0.621437\pi$
$919$	−22.5736	−0.744635	−0.372318	+	0.928105i	$0.621437\pi$
$920$	0	0
$921$	−70.2458	−2.31468
$922$	0	0
$923$	7.68736	0.253033
$924$	0	0
$925$	23.1361	0.760710
$926$	0	0
$927$	−32.4992	−1.06741
$928$	0	0
$929$	46.7458	1.53368	0.766840	−	0.641839i	$-0.221828\pi$
$929$	46.7458	1.53368	0.766840	+	0.641839i	$0.221828\pi$
$930$	0	0
$931$	−3.18001	−0.104221
$932$	0	0
$933$	−62.5401	−2.04747
$934$	0	0
$935$	−5.64157	−0.184499
$936$	0	0
$937$	24.3401	0.795157	0.397578	−	0.917568i	$-0.369851\pi$
$937$	24.3401	0.795157	0.397578	+	0.917568i	$0.369851\pi$
$938$	0	0
$939$	−17.5307	−0.572094
$940$	0	0
$941$	17.9933	0.586566	0.293283	−	0.956026i	$-0.405252\pi$
$941$	17.9933	0.586566	0.293283	+	0.956026i	$0.405252\pi$
$942$	0	0
$943$	−6.81721	−0.221999
$944$	0	0
$945$	−1.91621	−0.0623343
$946$	0	0
$947$	19.3456	0.628649	0.314324	−	0.949316i	$-0.398222\pi$
$947$	19.3456	0.628649	0.314324	+	0.949316i	$0.398222\pi$
$948$	0	0
$949$	−7.41380	−0.240662
$950$	0	0
$951$	54.9149	1.78074
$952$	0	0
$953$	27.3362	0.885505	0.442753	−	0.896644i	$-0.354002\pi$
$953$	27.3362	0.885505	0.442753	+	0.896644i	$0.354002\pi$
$954$	0	0
$955$	8.29675	0.268476
$956$	0	0
$957$	−16.7610	−0.541808
$958$	0	0
$959$	5.16125	0.166665
$960$	0	0
$961$	3.84440	0.124013
$962$	0	0
$963$	18.7984	0.605771
$964$	0	0
$965$	−6.21505	−0.200069
$966$	0	0
$967$	−45.4186	−1.46056	−0.730282	−	0.683145i	$-0.760612\pi$
$967$	−45.4186	−1.46056	−0.730282	+	0.683145i	$0.760612\pi$
$968$	0	0
$969$	−44.6373	−1.43396
$970$	0	0
$971$	60.0810	1.92809	0.964045	−	0.265739i	$-0.0856160\pi$
$971$	60.0810	1.92809	0.964045	+	0.265739i	$0.0856160\pi$
$972$	0	0
$973$	−8.79830	−0.282061
$974$	0	0
$975$	−9.40707	−0.301267
$976$	0	0
$977$	42.5380	1.36091	0.680456	−	0.732789i	$-0.261782\pi$
$977$	42.5380	1.36091	0.680456	+	0.732789i	$0.261782\pi$
$978$	0	0
$979$	−6.64006	−0.212217
$980$	0	0
$981$	22.2918	0.711723
$982$	0	0
$983$	18.0985	0.577254	0.288627	−	0.957442i	$-0.406801\pi$
$983$	18.0985	0.577254	0.288627	+	0.957442i	$0.406801\pi$
$984$	0	0
$985$	18.9636	0.604231
$986$	0	0
$987$	−14.5778	−0.464016
$988$	0	0
$989$	−7.52916	−0.239413
$990$	0	0
$991$	39.7987	1.26425	0.632123	−	0.774868i	$-0.282184\pi$
$991$	39.7987	1.26425	0.632123	+	0.774868i	$0.282184\pi$
$992$	0	0
$993$	−16.3662	−0.519364
$994$	0	0
$995$	−4.28838	−0.135951
$996$	0	0
$997$	−62.5265	−1.98023	−0.990116	−	0.140248i	$-0.955210\pi$
$997$	−62.5265	−1.98023	−0.990116	+	0.140248i	$0.955210\pi$
$998$	0	0
$999$	11.7260	0.370994

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.p.1.8	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.p.1.8	✓	9	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+2.24883 q^{3} -0.903831 q^{5} -1.00000 q^{7} +2.05724 q^{9} +O(q^{10})\)	\(q+2.24883 q^{3} -0.903831 q^{5} -1.00000 q^{7} +2.05724 q^{9} +1.00000 q^{11} +1.00000 q^{13} -2.03256 q^{15} +6.24184 q^{17} -3.18001 q^{19} -2.24883 q^{21} -3.89934 q^{23} -4.18309 q^{25} -2.12010 q^{27} -7.45322 q^{29} -5.90291 q^{31} +2.24883 q^{33} +0.903831 q^{35} -5.53086 q^{37} +2.24883 q^{39} +1.74830 q^{41} +1.93088 q^{43} -1.85940 q^{45} +6.48238 q^{47} +1.00000 q^{49} +14.0368 q^{51} -5.50070 q^{53} -0.903831 q^{55} -7.15130 q^{57} +4.02967 q^{59} -9.49947 q^{61} -2.05724 q^{63} -0.903831 q^{65} +3.09877 q^{67} -8.76896 q^{69} +7.68736 q^{71} -7.41380 q^{73} -9.40707 q^{75} -1.00000 q^{77} -11.6456 q^{79} -10.9395 q^{81} -5.15710 q^{83} -5.64157 q^{85} -16.7610 q^{87} -6.64006 q^{89} -1.00000 q^{91} -13.2747 q^{93} +2.87419 q^{95} -7.18680 q^{97} +2.05724 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{3} - 4 q^{5} - 9 q^{7} + 4 q^{9}+O(q^{10}) \) 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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