LMFDB - Embedded newform 8008.2.a.u.1.6

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 16x^{8} + 26x^{7} + 93x^{6} - 113x^{5} - 230x^{4} + 197x^{3} + 201x^{2} - 115x + 8 $ x^10 - 2x^9 - 16x^8 + 26x^7 + 93x^6 - 113x^5 - 230x^4 + 197x^3 + 201x^2 - 115*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.398142$ 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)

$f(q)$	$=$	$q+0.398142 q^{3} +3.53033 q^{5} -1.00000 q^{7} -2.84148 q^{9} +O(q^{10})$	$q+0.398142 q^{3} +3.53033 q^{5} -1.00000 q^{7} -2.84148 q^{9} -1.00000 q^{11} -1.00000 q^{13} +1.40557 q^{15} -1.07861 q^{17} +5.12685 q^{19} -0.398142 q^{21} -5.71619 q^{23} +7.46325 q^{25} -2.32574 q^{27} -1.17565 q^{29} +1.41204 q^{31} -0.398142 q^{33} -3.53033 q^{35} -7.80593 q^{37} -0.398142 q^{39} -5.27856 q^{41} -2.86769 q^{43} -10.0314 q^{45} -2.62898 q^{47} +1.00000 q^{49} -0.429439 q^{51} -6.09517 q^{53} -3.53033 q^{55} +2.04121 q^{57} +4.25733 q^{59} -1.98172 q^{61} +2.84148 q^{63} -3.53033 q^{65} +13.4087 q^{67} -2.27585 q^{69} -8.91852 q^{71} -10.4436 q^{73} +2.97143 q^{75} +1.00000 q^{77} +5.29521 q^{79} +7.59848 q^{81} +2.00236 q^{83} -3.80785 q^{85} -0.468076 q^{87} -5.89528 q^{89} +1.00000 q^{91} +0.562193 q^{93} +18.0995 q^{95} +10.8363 q^{97} +2.84148 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9	$ 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} - 3 q^{17} + 13 q^{19} - 2 q^{21} + 6 q^{25} + 14 q^{27} - 3 q^{29} - q^{31} - 2 q^{33} + 4 q^{35} - 5 q^{37} - 2 q^{39} + 4 q^{41} + 19 q^{43} - 11 q^{45} + q^{47} + 10 q^{49} + 15 q^{51} - 6 q^{53} + 4 q^{55} - 6 q^{57} + 4 q^{59} - 18 q^{61} - 6 q^{63} + 4 q^{65} + q^{67} - 11 q^{69} - 21 q^{71} - 10 q^{73} + 10 q^{77} + 3 q^{79} - 30 q^{81} + 6 q^{83} - 33 q^{85} - 9 q^{87} - 22 q^{89} + 10 q^{91} - 34 q^{93} - 3 q^{95} - 9 q^{97} - 6 q^{99}+O(q^{100}) $ 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 - 3 * q^17 + 13 * q^19 - 2 * q^21 + 6 * q^25 + 14 * q^27 - 3 * q^29 - q^31 - 2 * q^33 + 4 * q^35 - 5 * q^37 - 2 * q^39 + 4 * q^41 + 19 * q^43 - 11 * q^45 + q^47 + 10 * q^49 + 15 * q^51 - 6 * q^53 + 4 * q^55 - 6 * q^57 + 4 * q^59 - 18 * q^61 - 6 * q^63 + 4 * q^65 + q^67 - 11 * q^69 - 21 * q^71 - 10 * q^73 + 10 * q^77 + 3 * q^79 - 30 * q^81 + 6 * q^83 - 33 * q^85 - 9 * q^87 - 22 * q^89 + 10 * q^91 - 34 * q^93 - 3 * q^95 - 9 * q^97 - 6 * 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
$3$	0.398142	0.229867	0.114934	−	0.993373i	$-0.463334\pi$
$3$	0.398142	0.229867	0.114934	+	0.993373i	$0.463334\pi$
$4$	0	0
$5$	3.53033	1.57881	0.789406	−	0.613871i	$-0.210389\pi$
$5$	3.53033	1.57881	0.789406	+	0.613871i	$0.210389\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.84148	−0.947161
$10$	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
$15$	1.40557	0.362917
$16$	0	0
$17$	−1.07861	−0.261601	−0.130801	−	0.991409i	$-0.541755\pi$
$17$	−1.07861	−0.261601	−0.130801	+	0.991409i	$0.541755\pi$
$18$	0	0
$19$	5.12685	1.17618	0.588090	−	0.808796i	$-0.299880\pi$
$19$	5.12685	1.17618	0.588090	+	0.808796i	$0.299880\pi$
$20$	0	0
$21$	−0.398142	−0.0868816
$22$	0	0
$23$	−5.71619	−1.19191	−0.595954	−	0.803019i	$-0.703226\pi$
$23$	−5.71619	−1.19191	−0.595954	+	0.803019i	$0.703226\pi$
$24$	0	0
$25$	7.46325	1.49265
$26$	0	0
$27$	−2.32574	−0.447588
$28$	0	0
$29$	−1.17565	−0.218313	−0.109157	−	0.994025i	$-0.534815\pi$
$29$	−1.17565	−0.218313	−0.109157	+	0.994025i	$0.534815\pi$
$30$	0	0
$31$	1.41204	0.253611	0.126805	−	0.991928i	$-0.459528\pi$
$31$	1.41204	0.253611	0.126805	+	0.991928i	$0.459528\pi$
$32$	0	0
$33$	−0.398142	−0.0693076
$34$	0	0
$35$	−3.53033	−0.596735
$36$	0	0
$37$	−7.80593	−1.28329	−0.641643	−	0.767003i	$-0.721747\pi$
$37$	−7.80593	−1.28329	−0.641643	+	0.767003i	$0.721747\pi$
$38$	0	0
$39$	−0.398142	−0.0637537
$40$	0	0
$41$	−5.27856	−0.824372	−0.412186	−	0.911100i	$-0.635235\pi$
$41$	−5.27856	−0.824372	−0.412186	+	0.911100i	$0.635235\pi$
$42$	0	0
$43$	−2.86769	−0.437318	−0.218659	−	0.975801i	$-0.570168\pi$
$43$	−2.86769	−0.437318	−0.218659	+	0.975801i	$0.570168\pi$
$44$	0	0
$45$	−10.0314	−1.49539
$46$	0	0
$47$	−2.62898	−0.383476	−0.191738	−	0.981446i	$-0.561412\pi$
$47$	−2.62898	−0.383476	−0.191738	+	0.981446i	$0.561412\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.429439	−0.0601335
$52$	0	0
$53$	−6.09517	−0.837236	−0.418618	−	0.908163i	$-0.637485\pi$
$53$	−6.09517	−0.837236	−0.418618	+	0.908163i	$0.637485\pi$
$54$	0	0
$55$	−3.53033	−0.476030
$56$	0	0
$57$	2.04121	0.270365
$58$	0	0
$59$	4.25733	0.554257	0.277129	−	0.960833i	$-0.410617\pi$
$59$	4.25733	0.554257	0.277129	+	0.960833i	$0.410617\pi$
$60$	0	0
$61$	−1.98172	−0.253733	−0.126866	−	0.991920i	$-0.540492\pi$
$61$	−1.98172	−0.253733	−0.126866	+	0.991920i	$0.540492\pi$
$62$	0	0
$63$	2.84148	0.357993
$64$	0	0
$65$	−3.53033	−0.437884
$66$	0	0
$67$	13.4087	1.63814	0.819068	−	0.573696i	$-0.194491\pi$
$67$	13.4087	1.63814	0.819068	+	0.573696i	$0.194491\pi$
$68$	0	0
$69$	−2.27585	−0.273980
$70$	0	0
$71$	−8.91852	−1.05843	−0.529217	−	0.848487i	$-0.677514\pi$
$71$	−8.91852	−1.05843	−0.529217	+	0.848487i	$0.677514\pi$
$72$	0	0
$73$	−10.4436	−1.22233	−0.611164	−	0.791504i	$-0.709299\pi$
$73$	−10.4436	−1.22233	−0.611164	+	0.791504i	$0.709299\pi$
$74$	0	0
$75$	2.97143	0.343111
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	5.29521	0.595758	0.297879	−	0.954604i	$-0.403721\pi$
$79$	5.29521	0.595758	0.297879	+	0.954604i	$0.403721\pi$
$80$	0	0
$81$	7.59848	0.844275
$82$	0	0
$83$	2.00236	0.219787	0.109894	−	0.993943i	$-0.464949\pi$
$83$	2.00236	0.219787	0.109894	+	0.993943i	$0.464949\pi$
$84$	0	0
$85$	−3.80785	−0.413019
$86$	0	0
$87$	−0.468076	−0.0501830
$88$	0	0
$89$	−5.89528	−0.624899	−0.312449	−	0.949934i	$-0.601149\pi$
$89$	−5.89528	−0.624899	−0.312449	+	0.949934i	$0.601149\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0.562193	0.0582967
$94$	0	0
$95$	18.0995	1.85697
$96$	0	0
$97$	10.8363	1.10026	0.550129	−	0.835080i	$-0.314579\pi$
$97$	10.8363	1.10026	0.550129	+	0.835080i	$0.314579\pi$
$98$	0	0
$99$	2.84148	0.285580
$100$	0	0
$101$	−6.40496	−0.637317	−0.318659	−	0.947870i	$-0.603232\pi$
$101$	−6.40496	−0.637317	−0.318659	+	0.947870i	$0.603232\pi$
$102$	0	0
$103$	−2.99636	−0.295241	−0.147620	−	0.989044i	$-0.547161\pi$
$103$	−2.99636	−0.295241	−0.147620	+	0.989044i	$0.547161\pi$
$104$	0	0
$105$	−1.40557	−0.137170
$106$	0	0
$107$	−14.7546	−1.42638	−0.713189	−	0.700972i	$-0.752750\pi$
$107$	−14.7546	−1.42638	−0.713189	+	0.700972i	$0.752750\pi$
$108$	0	0
$109$	15.8020	1.51356	0.756780	−	0.653670i	$-0.226771\pi$
$109$	15.8020	1.51356	0.756780	+	0.653670i	$0.226771\pi$
$110$	0	0
$111$	−3.10786	−0.294985
$112$	0	0
$113$	−12.3822	−1.16482	−0.582408	−	0.812897i	$-0.697889\pi$
$113$	−12.3822	−1.16482	−0.582408	+	0.812897i	$0.697889\pi$
$114$	0	0
$115$	−20.1800	−1.88180
$116$	0	0
$117$	2.84148	0.262695
$118$	0	0
$119$	1.07861	0.0988760
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.10161	−0.189496
$124$	0	0
$125$	8.69608	0.777801
$126$	0	0
$127$	−14.1106	−1.25211	−0.626057	−	0.779777i	$-0.715332\pi$
$127$	−14.1106	−1.25211	−0.626057	+	0.779777i	$0.715332\pi$
$128$	0	0
$129$	−1.14175	−0.100525
$130$	0	0
$131$	−0.00489576	−0.000427744 0	−0.000213872	−	1.00000i	$-0.500068\pi$
$131$	−0.00489576	−0.000427744 0	−0.000213872		1.00000i	$0.500068\pi$
$132$	0	0
$133$	−5.12685	−0.444554
$134$	0	0
$135$	−8.21063	−0.706658
$136$	0	0
$137$	−15.7889	−1.34894	−0.674470	−	0.738303i	$-0.735628\pi$
$137$	−15.7889	−1.34894	−0.674470	+	0.738303i	$0.735628\pi$
$138$	0	0
$139$	10.7834	0.914634	0.457317	−	0.889304i	$-0.348811\pi$
$139$	10.7834	0.914634	0.457317	+	0.889304i	$0.348811\pi$
$140$	0	0
$141$	−1.04671	−0.0881486
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−4.15044	−0.344676
$146$	0	0
$147$	0.398142	0.0328382
$148$	0	0
$149$	−0.489805	−0.0401264	−0.0200632	−	0.999799i	$-0.506387\pi$
$149$	−0.489805	−0.0401264	−0.0200632	+	0.999799i	$0.506387\pi$
$150$	0	0
$151$	−15.9015	−1.29405	−0.647023	−	0.762470i	$-0.723986\pi$
$151$	−15.9015	−1.29405	−0.647023	+	0.762470i	$0.723986\pi$
$152$	0	0
$153$	3.06485	0.247778
$154$	0	0
$155$	4.98498	0.400404
$156$	0	0
$157$	−22.7567	−1.81619	−0.908093	−	0.418769i	$-0.862462\pi$
$157$	−22.7567	−1.81619	−0.908093	+	0.418769i	$0.862462\pi$
$158$	0	0
$159$	−2.42674	−0.192453
$160$	0	0
$161$	5.71619	0.450499
$162$	0	0
$163$	14.7578	1.15592	0.577959	−	0.816066i	$-0.303849\pi$
$163$	14.7578	1.15592	0.577959	+	0.816066i	$0.303849\pi$
$164$	0	0
$165$	−1.40557	−0.109424
$166$	0	0
$167$	−5.64625	−0.436920	−0.218460	−	0.975846i	$-0.570103\pi$
$167$	−5.64625	−0.436920	−0.218460	+	0.975846i	$0.570103\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−14.5679	−1.11403
$172$	0	0
$173$	−19.2055	−1.46017	−0.730084	−	0.683357i	$-0.760519\pi$
$173$	−19.2055	−1.46017	−0.730084	+	0.683357i	$0.760519\pi$
$174$	0	0
$175$	−7.46325	−0.564168
$176$	0	0
$177$	1.69502	0.127406
$178$	0	0
$179$	−14.8267	−1.10820	−0.554101	−	0.832450i	$-0.686938\pi$
$179$	−14.8267	−1.10820	−0.554101	+	0.832450i	$0.686938\pi$
$180$	0	0
$181$	−17.9259	−1.33242	−0.666210	−	0.745764i	$-0.732085\pi$
$181$	−17.9259	−1.33242	−0.666210	+	0.745764i	$0.732085\pi$
$182$	0	0
$183$	−0.789004	−0.0583249
$184$	0	0
$185$	−27.5575	−2.02607
$186$	0	0
$187$	1.07861	0.0788757
$188$	0	0
$189$	2.32574	0.169173
$190$	0	0
$191$	13.3986	0.969488	0.484744	−	0.874656i	$-0.338913\pi$
$191$	13.3986	0.969488	0.484744	+	0.874656i	$0.338913\pi$
$192$	0	0
$193$	13.2546	0.954088	0.477044	−	0.878879i	$-0.341708\pi$
$193$	13.2546	0.954088	0.477044	+	0.878879i	$0.341708\pi$
$194$	0	0
$195$	−1.40557	−0.100655
$196$	0	0
$197$	−9.52971	−0.678964	−0.339482	−	0.940613i	$-0.610252\pi$
$197$	−9.52971	−0.678964	−0.339482	+	0.940613i	$0.610252\pi$
$198$	0	0
$199$	16.0569	1.13824	0.569122	−	0.822253i	$-0.307283\pi$
$199$	16.0569	1.13824	0.569122	+	0.822253i	$0.307283\pi$
$200$	0	0
$201$	5.33857	0.376554
$202$	0	0
$203$	1.17565	0.0825146
$204$	0	0
$205$	−18.6351	−1.30153
$206$	0	0
$207$	16.2424	1.12893
$208$	0	0
$209$	−5.12685	−0.354631
$210$	0	0
$211$	10.3841	0.714869	0.357434	−	0.933938i	$-0.383652\pi$
$211$	10.3841	0.714869	0.357434	+	0.933938i	$0.383652\pi$
$212$	0	0
$213$	−3.55083	−0.243299
$214$	0	0
$215$	−10.1239	−0.690443
$216$	0	0
$217$	−1.41204	−0.0958558
$218$	0	0
$219$	−4.15802	−0.280973
$220$	0	0
$221$	1.07861	0.0725551
$222$	0	0
$223$	2.73987	0.183475	0.0917375	−	0.995783i	$-0.470758\pi$
$223$	2.73987	0.183475	0.0917375	+	0.995783i	$0.470758\pi$
$224$	0	0
$225$	−21.2067	−1.41378
$226$	0	0
$227$	−11.9545	−0.793447	−0.396724	−	0.917938i	$-0.629853\pi$
$227$	−11.9545	−0.793447	−0.396724	+	0.917938i	$0.629853\pi$
$228$	0	0
$229$	−6.05664	−0.400234	−0.200117	−	0.979772i	$-0.564132\pi$
$229$	−6.05664	−0.400234	−0.200117	+	0.979772i	$0.564132\pi$
$230$	0	0
$231$	0.398142	0.0261958
$232$	0	0
$233$	6.31979	0.414023	0.207012	−	0.978338i	$-0.433626\pi$
$233$	6.31979	0.414023	0.207012	+	0.978338i	$0.433626\pi$
$234$	0	0
$235$	−9.28118	−0.605437
$236$	0	0
$237$	2.10824	0.136945
$238$	0	0
$239$	26.2496	1.69795	0.848973	−	0.528436i	$-0.177221\pi$
$239$	26.2496	1.69795	0.848973	+	0.528436i	$0.177221\pi$
$240$	0	0
$241$	−3.11783	−0.200837	−0.100419	−	0.994945i	$-0.532018\pi$
$241$	−3.11783	−0.200837	−0.100419	+	0.994945i	$0.532018\pi$
$242$	0	0
$243$	10.0025	0.641660
$244$	0	0
$245$	3.53033	0.225545
$246$	0	0
$247$	−5.12685	−0.326213
$248$	0	0
$249$	0.797221	0.0505218
$250$	0	0
$251$	−6.71557	−0.423883	−0.211942	−	0.977282i	$-0.567979\pi$
$251$	−6.71557	−0.423883	−0.211942	+	0.977282i	$0.567979\pi$
$252$	0	0
$253$	5.71619	0.359374
$254$	0	0
$255$	−1.51606	−0.0949396
$256$	0	0
$257$	−1.71946	−0.107257	−0.0536284	−	0.998561i	$-0.517079\pi$
$257$	−1.71946	−0.107257	−0.0536284	+	0.998561i	$0.517079\pi$
$258$	0	0
$259$	7.80593	0.485037
$260$	0	0
$261$	3.34060	0.206778
$262$	0	0
$263$	2.39328	0.147576	0.0737881	−	0.997274i	$-0.476491\pi$
$263$	2.39328	0.147576	0.0737881	+	0.997274i	$0.476491\pi$
$264$	0	0
$265$	−21.5180	−1.32184
$266$	0	0
$267$	−2.34716	−0.143644
$268$	0	0
$269$	−23.2462	−1.41734	−0.708672	−	0.705538i	$-0.750705\pi$
$269$	−23.2462	−1.41734	−0.708672	+	0.705538i	$0.750705\pi$
$270$	0	0
$271$	9.13004	0.554610	0.277305	−	0.960782i	$-0.410559\pi$
$271$	9.13004	0.554610	0.277305	+	0.960782i	$0.410559\pi$
$272$	0	0
$273$	0.398142	0.0240966
$274$	0	0
$275$	−7.46325	−0.450051
$276$	0	0
$277$	18.1440	1.09017	0.545084	−	0.838382i	$-0.316498\pi$
$277$	18.1440	1.09017	0.545084	+	0.838382i	$0.316498\pi$
$278$	0	0
$279$	−4.01230	−0.240210
$280$	0	0
$281$	11.9833	0.714865	0.357433	−	0.933939i	$-0.383652\pi$
$281$	11.9833	0.714865	0.357433	+	0.933939i	$0.383652\pi$
$282$	0	0
$283$	−14.4334	−0.857978	−0.428989	−	0.903310i	$-0.641130\pi$
$283$	−14.4334	−0.857978	−0.428989	+	0.903310i	$0.641130\pi$
$284$	0	0
$285$	7.20616	0.426856
$286$	0	0
$287$	5.27856	0.311583
$288$	0	0
$289$	−15.8366	−0.931565
$290$	0	0
$291$	4.31438	0.252913
$292$	0	0
$293$	−7.10073	−0.414829	−0.207414	−	0.978253i	$-0.566505\pi$
$293$	−7.10073	−0.414829	−0.207414	+	0.978253i	$0.566505\pi$
$294$	0	0
$295$	15.0298	0.875068
$296$	0	0
$297$	2.32574	0.134953
$298$	0	0
$299$	5.71619	0.330576
$300$	0	0
$301$	2.86769	0.165291
$302$	0	0
$303$	−2.55008	−0.146498
$304$	0	0
$305$	−6.99612	−0.400597
$306$	0	0
$307$	0.0952457	0.00543596	0.00271798	−	0.999996i	$-0.499135\pi$
$307$	0.0952457	0.00543596	0.00271798	+	0.999996i	$0.499135\pi$
$308$	0	0
$309$	−1.19298	−0.0678661
$310$	0	0
$311$	31.5533	1.78923	0.894613	−	0.446842i	$-0.147451\pi$
$311$	31.5533	1.78923	0.894613	+	0.446842i	$0.147451\pi$
$312$	0	0
$313$	8.06515	0.455869	0.227935	−	0.973676i	$-0.426803\pi$
$313$	8.06515	0.455869	0.227935	+	0.973676i	$0.426803\pi$
$314$	0	0
$315$	10.0314	0.565204
$316$	0	0
$317$	18.6840	1.04940	0.524700	−	0.851287i	$-0.324177\pi$
$317$	18.6840	1.04940	0.524700	+	0.851287i	$0.324177\pi$
$318$	0	0
$319$	1.17565	0.0658239
$320$	0	0
$321$	−5.87441	−0.327877
$322$	0	0
$323$	−5.52987	−0.307690
$324$	0	0
$325$	−7.46325	−0.413986
$326$	0	0
$327$	6.29145	0.347918
$328$	0	0
$329$	2.62898	0.144940
$330$	0	0
$331$	−11.2983	−0.621011	−0.310506	−	0.950572i	$-0.600498\pi$
$331$	−11.2983	−0.621011	−0.310506	+	0.950572i	$0.600498\pi$
$332$	0	0
$333$	22.1804	1.21548
$334$	0	0
$335$	47.3373	2.58631
$336$	0	0
$337$	−5.26007	−0.286534	−0.143267	−	0.989684i	$-0.545761\pi$
$337$	−5.26007	−0.286534	−0.143267	+	0.989684i	$0.545761\pi$
$338$	0	0
$339$	−4.92985	−0.267753
$340$	0	0
$341$	−1.41204	−0.0764665
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−8.03451	−0.432564
$346$	0	0
$347$	−0.564314	−0.0302939	−0.0151470	−	0.999885i	$-0.504822\pi$
$347$	−0.564314	−0.0302939	−0.0151470	+	0.999885i	$0.504822\pi$
$348$	0	0
$349$	11.8039	0.631849	0.315925	−	0.948784i	$-0.397685\pi$
$349$	11.8039	0.631849	0.315925	+	0.948784i	$0.397685\pi$
$350$	0	0
$351$	2.32574	0.124139
$352$	0	0
$353$	9.12891	0.485883	0.242942	−	0.970041i	$-0.421888\pi$
$353$	9.12891	0.485883	0.242942	+	0.970041i	$0.421888\pi$
$354$	0	0
$355$	−31.4853	−1.67107
$356$	0	0
$357$	0.429439	0.0227283
$358$	0	0
$359$	28.4115	1.49950	0.749750	−	0.661721i	$-0.230174\pi$
$359$	28.4115	1.49950	0.749750	+	0.661721i	$0.230174\pi$
$360$	0	0
$361$	7.28457	0.383398
$362$	0	0
$363$	0.398142	0.0208970
$364$	0	0
$365$	−36.8693	−1.92983
$366$	0	0
$367$	−29.4820	−1.53895	−0.769475	−	0.638677i	$-0.779482\pi$
$367$	−29.4820	−1.53895	−0.769475	+	0.638677i	$0.779482\pi$
$368$	0	0
$369$	14.9989	0.780813
$370$	0	0
$371$	6.09517	0.316445
$372$	0	0
$373$	−21.0819	−1.09158	−0.545790	−	0.837922i	$-0.683770\pi$
$373$	−21.0819	−1.09158	−0.545790	+	0.837922i	$0.683770\pi$
$374$	0	0
$375$	3.46227	0.178791
$376$	0	0
$377$	1.17565	0.0605492
$378$	0	0
$379$	−16.1750	−0.830852	−0.415426	−	0.909627i	$-0.636368\pi$
$379$	−16.1750	−0.830852	−0.415426	+	0.909627i	$0.636368\pi$
$380$	0	0
$381$	−5.61802	−0.287820
$382$	0	0
$383$	30.5596	1.56152	0.780761	−	0.624830i	$-0.214832\pi$
$383$	30.5596	1.56152	0.780761	+	0.624830i	$0.214832\pi$
$384$	0	0
$385$	3.53033	0.179922
$386$	0	0
$387$	8.14848	0.414211
$388$	0	0
$389$	−11.0373	−0.559612	−0.279806	−	0.960057i	$-0.590270\pi$
$389$	−11.0373	−0.559612	−0.279806	+	0.960057i	$0.590270\pi$
$390$	0	0
$391$	6.16553	0.311804
$392$	0	0
$393$	−0.00194921	−9.83244e−5 0
$394$	0	0
$395$	18.6938	0.940590
$396$	0	0
$397$	−12.1409	−0.609335	−0.304668	−	0.952459i	$-0.598545\pi$
$397$	−12.1409	−0.609335	−0.304668	+	0.952459i	$0.598545\pi$
$398$	0	0
$399$	−2.04121	−0.102188
$400$	0	0
$401$	−34.0372	−1.69974	−0.849869	−	0.526994i	$-0.823319\pi$
$401$	−34.0372	−1.69974	−0.849869	+	0.526994i	$0.823319\pi$
$402$	0	0
$403$	−1.41204	−0.0703389
$404$	0	0
$405$	26.8251	1.33295
$406$	0	0
$407$	7.80593	0.386925
$408$	0	0
$409$	19.7905	0.978575	0.489288	−	0.872122i	$-0.337257\pi$
$409$	19.7905	0.978575	0.489288	+	0.872122i	$0.337257\pi$
$410$	0	0
$411$	−6.28623	−0.310077
$412$	0	0
$413$	−4.25733	−0.209489
$414$	0	0
$415$	7.06898	0.347003
$416$	0	0
$417$	4.29331	0.210244
$418$	0	0
$419$	−27.9759	−1.36671	−0.683357	−	0.730085i	$-0.739481\pi$
$419$	−27.9759	−1.36671	−0.683357	+	0.730085i	$0.739481\pi$
$420$	0	0
$421$	22.9430	1.11817	0.559086	−	0.829110i	$-0.311152\pi$
$421$	22.9430	1.11817	0.559086	+	0.829110i	$0.311152\pi$
$422$	0	0
$423$	7.47021	0.363214
$424$	0	0
$425$	−8.04993	−0.390479
$426$	0	0
$427$	1.98172	0.0959020
$428$	0	0
$429$	0.398142	0.0192225
$430$	0	0
$431$	−12.6261	−0.608176	−0.304088	−	0.952644i	$-0.598352\pi$
$431$	−12.6261	−0.608176	−0.304088	+	0.952644i	$0.598352\pi$
$432$	0	0
$433$	−24.0580	−1.15616	−0.578078	−	0.815982i	$-0.696197\pi$
$433$	−24.0580	−1.15616	−0.578078	+	0.815982i	$0.696197\pi$
$434$	0	0
$435$	−1.65246	−0.0792296
$436$	0	0
$437$	−29.3060	−1.40190
$438$	0	0
$439$	−20.4145	−0.974329	−0.487165	−	0.873310i	$-0.661969\pi$
$439$	−20.4145	−0.974329	−0.487165	+	0.873310i	$0.661969\pi$
$440$	0	0
$441$	−2.84148	−0.135309
$442$	0	0
$443$	−12.9202	−0.613858	−0.306929	−	0.951732i	$-0.599301\pi$
$443$	−12.9202	−0.613858	−0.306929	+	0.951732i	$0.599301\pi$
$444$	0	0
$445$	−20.8123	−0.986598
$446$	0	0
$447$	−0.195012	−0.00922374
$448$	0	0
$449$	−24.2981	−1.14670	−0.573350	−	0.819311i	$-0.694356\pi$
$449$	−24.2981	−1.14670	−0.573350	+	0.819311i	$0.694356\pi$
$450$	0	0
$451$	5.27856	0.248558
$452$	0	0
$453$	−6.33105	−0.297459
$454$	0	0
$455$	3.53033	0.165505
$456$	0	0
$457$	34.4161	1.60992	0.804958	−	0.593331i	$-0.202188\pi$
$457$	34.4161	1.60992	0.804958	+	0.593331i	$0.202188\pi$
$458$	0	0
$459$	2.50856	0.117090
$460$	0	0
$461$	18.6720	0.869642	0.434821	−	0.900517i	$-0.356812\pi$
$461$	18.6720	0.869642	0.434821	+	0.900517i	$0.356812\pi$
$462$	0	0
$463$	28.4013	1.31992	0.659960	−	0.751301i	$-0.270573\pi$
$463$	28.4013	1.31992	0.659960	+	0.751301i	$0.270573\pi$
$464$	0	0
$465$	1.98473	0.0920396
$466$	0	0
$467$	−12.5181	−0.579269	−0.289635	−	0.957137i	$-0.593534\pi$
$467$	−12.5181	−0.579269	−0.289635	+	0.957137i	$0.593534\pi$
$468$	0	0
$469$	−13.4087	−0.619157
$470$	0	0
$471$	−9.06041	−0.417481
$472$	0	0
$473$	2.86769	0.131856
$474$	0	0
$475$	38.2629	1.75562
$476$	0	0
$477$	17.3193	0.792997
$478$	0	0
$479$	−12.6426	−0.577657	−0.288829	−	0.957381i	$-0.593266\pi$
$479$	−12.6426	−0.577657	−0.288829	+	0.957381i	$0.593266\pi$
$480$	0	0
$481$	7.80593	0.355920
$482$	0	0
$483$	2.27585	0.103555
$484$	0	0
$485$	38.2557	1.73710
$486$	0	0
$487$	−7.51778	−0.340663	−0.170332	−	0.985387i	$-0.554484\pi$
$487$	−7.51778	−0.340663	−0.170332	+	0.985387i	$0.554484\pi$
$488$	0	0
$489$	5.87568	0.265708
$490$	0	0
$491$	32.9453	1.48680	0.743399	−	0.668848i	$-0.233212\pi$
$491$	32.9453	1.48680	0.743399	+	0.668848i	$0.233212\pi$
$492$	0	0
$493$	1.26807	0.0571110
$494$	0	0
$495$	10.0314	0.450877
$496$	0	0
$497$	8.91852	0.400050
$498$	0	0
$499$	17.0446	0.763021	0.381511	−	0.924364i	$-0.375404\pi$
$499$	17.0446	0.763021	0.381511	+	0.924364i	$0.375404\pi$
$500$	0	0
$501$	−2.24801	−0.100434
$502$	0	0
$503$	−4.96500	−0.221378	−0.110689	−	0.993855i	$-0.535306\pi$
$503$	−4.96500	−0.221378	−0.110689	+	0.993855i	$0.535306\pi$
$504$	0	0
$505$	−22.6116	−1.00620
$506$	0	0
$507$	0.398142	0.0176821
$508$	0	0
$509$	13.2498	0.587287	0.293643	−	0.955915i	$-0.405132\pi$
$509$	13.2498	0.587287	0.293643	+	0.955915i	$0.405132\pi$
$510$	0	0
$511$	10.4436	0.461997
$512$	0	0
$513$	−11.9237	−0.526444
$514$	0	0
$515$	−10.5782	−0.466130
$516$	0	0
$517$	2.62898	0.115622
$518$	0	0
$519$	−7.64651	−0.335645
$520$	0	0
$521$	27.8348	1.21947	0.609733	−	0.792607i	$-0.291277\pi$
$521$	27.8348	1.21947	0.609733	+	0.792607i	$0.291277\pi$
$522$	0	0
$523$	−14.2962	−0.625127	−0.312564	−	0.949897i	$-0.601188\pi$
$523$	−14.2962	−0.625127	−0.312564	+	0.949897i	$0.601188\pi$
$524$	0	0
$525$	−2.97143	−0.129684
$526$	0	0
$527$	−1.52304	−0.0663448
$528$	0	0
$529$	9.67478	0.420643
$530$	0	0
$531$	−12.0971	−0.524971
$532$	0	0
$533$	5.27856	0.228640
$534$	0	0
$535$	−52.0885	−2.25198
$536$	0	0
$537$	−5.90314	−0.254739
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	27.6088	1.18700	0.593498	−	0.804835i	$-0.297746\pi$
$541$	27.6088	1.18700	0.593498	+	0.804835i	$0.297746\pi$
$542$	0	0
$543$	−7.13704	−0.306280
$544$	0	0
$545$	55.7864	2.38963
$546$	0	0
$547$	9.11449	0.389707	0.194854	−	0.980832i	$-0.437577\pi$
$547$	9.11449	0.389707	0.194854	+	0.980832i	$0.437577\pi$
$548$	0	0
$549$	5.63102	0.240326
$550$	0	0
$551$	−6.02739	−0.256776
$552$	0	0
$553$	−5.29521	−0.225175
$554$	0	0
$555$	−10.9718	−0.465727
$556$	0	0
$557$	−31.3149	−1.32685	−0.663426	−	0.748242i	$-0.730898\pi$
$557$	−31.3149	−1.32685	−0.663426	+	0.748242i	$0.730898\pi$
$558$	0	0
$559$	2.86769	0.121290
$560$	0	0
$561$	0.429439	0.0181309
$562$	0	0
$563$	−9.73574	−0.410313	−0.205156	−	0.978729i	$-0.565770\pi$
$563$	−9.73574	−0.410313	−0.205156	+	0.978729i	$0.565770\pi$
$564$	0	0
$565$	−43.7131	−1.83903
$566$	0	0
$567$	−7.59848	−0.319106
$568$	0	0
$569$	31.8199	1.33396	0.666980	−	0.745075i	$-0.267586\pi$
$569$	31.8199	1.33396	0.666980	+	0.745075i	$0.267586\pi$
$570$	0	0
$571$	0.0548895	0.00229705	0.00114853	−	0.999999i	$-0.499634\pi$
$571$	0.0548895	0.00229705	0.00114853	+	0.999999i	$0.499634\pi$
$572$	0	0
$573$	5.33454	0.222854
$574$	0	0
$575$	−42.6613	−1.77910
$576$	0	0
$577$	−34.0582	−1.41786	−0.708931	−	0.705278i	$-0.750822\pi$
$577$	−34.0582	−1.41786	−0.708931	+	0.705278i	$0.750822\pi$
$578$	0	0
$579$	5.27722	0.219314
$580$	0	0
$581$	−2.00236	−0.0830717
$582$	0	0
$583$	6.09517	0.252436
$584$	0	0
$585$	10.0314	0.414747
$586$	0	0
$587$	27.5504	1.13713	0.568563	−	0.822639i	$-0.307499\pi$
$587$	27.5504	1.13713	0.568563	+	0.822639i	$0.307499\pi$
$588$	0	0
$589$	7.23933	0.298292
$590$	0	0
$591$	−3.79417	−0.156071
$592$	0	0
$593$	9.95074	0.408628	0.204314	−	0.978905i	$-0.434504\pi$
$593$	9.95074	0.408628	0.204314	+	0.978905i	$0.434504\pi$
$594$	0	0
$595$	3.80785	0.156107
$596$	0	0
$597$	6.39293	0.261645
$598$	0	0
$599$	−6.97062	−0.284812	−0.142406	−	0.989808i	$-0.545484\pi$
$599$	−6.97062	−0.284812	−0.142406	+	0.989808i	$0.545484\pi$
$600$	0	0
$601$	5.23766	0.213649	0.106824	−	0.994278i	$-0.465932\pi$
$601$	5.23766	0.213649	0.106824	+	0.994278i	$0.465932\pi$
$602$	0	0
$603$	−38.1007	−1.55158
$604$	0	0
$605$	3.53033	0.143528
$606$	0	0
$607$	46.8643	1.90216	0.951082	−	0.308938i	$-0.0999736\pi$
$607$	46.8643	1.90216	0.951082	+	0.308938i	$0.0999736\pi$
$608$	0	0
$609$	0.468076	0.0189674
$610$	0	0
$611$	2.62898	0.106357
$612$	0	0
$613$	−8.92271	−0.360385	−0.180192	−	0.983631i	$-0.557672\pi$
$613$	−8.92271	−0.360385	−0.180192	+	0.983631i	$0.557672\pi$
$614$	0	0
$615$	−7.41940	−0.299179
$616$	0	0
$617$	33.8268	1.36182	0.680909	−	0.732368i	$-0.261585\pi$
$617$	33.8268	1.36182	0.680909	+	0.732368i	$0.261585\pi$
$618$	0	0
$619$	−35.0850	−1.41018	−0.705092	−	0.709115i	$-0.749095\pi$
$619$	−35.0850	−1.41018	−0.705092	+	0.709115i	$0.749095\pi$
$620$	0	0
$621$	13.2943	0.533484
$622$	0	0
$623$	5.89528	0.236190
$624$	0	0
$625$	−6.61617	−0.264647
$626$	0	0
$627$	−2.04121	−0.0815181
$628$	0	0
$629$	8.41954	0.335709
$630$	0	0
$631$	−10.7092	−0.426326	−0.213163	−	0.977017i	$-0.568377\pi$
$631$	−10.7092	−0.426326	−0.213163	+	0.977017i	$0.568377\pi$
$632$	0	0
$633$	4.13433	0.164325
$634$	0	0
$635$	−49.8152	−1.97685
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	25.3418	1.00251
$640$	0	0
$641$	4.77153	0.188464	0.0942321	−	0.995550i	$-0.469960\pi$
$641$	4.77153	0.188464	0.0942321	+	0.995550i	$0.469960\pi$
$642$	0	0
$643$	−10.1320	−0.399568	−0.199784	−	0.979840i	$-0.564024\pi$
$643$	−10.1320	−0.399568	−0.199784	+	0.979840i	$0.564024\pi$
$644$	0	0
$645$	−4.03074	−0.158710
$646$	0	0
$647$	−2.26281	−0.0889602	−0.0444801	−	0.999010i	$-0.514163\pi$
$647$	−2.26281	−0.0889602	−0.0444801	+	0.999010i	$0.514163\pi$
$648$	0	0
$649$	−4.25733	−0.167115
$650$	0	0
$651$	−0.562193	−0.0220341
$652$	0	0
$653$	−35.7172	−1.39772	−0.698862	−	0.715257i	$-0.746310\pi$
$653$	−35.7172	−1.39772	−0.698862	+	0.715257i	$0.746310\pi$
$654$	0	0
$655$	−0.0172837	−0.000675328 0
$656$	0	0
$657$	29.6753	1.15774
$658$	0	0
$659$	−11.6107	−0.452288	−0.226144	−	0.974094i	$-0.572612\pi$
$659$	−11.6107	−0.452288	−0.226144	+	0.974094i	$0.572612\pi$
$660$	0	0
$661$	−18.6643	−0.725958	−0.362979	−	0.931797i	$-0.618240\pi$
$661$	−18.6643	−0.725958	−0.362979	+	0.931797i	$0.618240\pi$
$662$	0	0
$663$	0.429439	0.0166780
$664$	0	0
$665$	−18.0995	−0.701868
$666$	0	0
$667$	6.72025	0.260209
$668$	0	0
$669$	1.09085	0.0421749
$670$	0	0
$671$	1.98172	0.0765033
$672$	0	0
$673$	−47.3405	−1.82484	−0.912420	−	0.409255i	$-0.865789\pi$
$673$	−47.3405	−1.82484	−0.912420	+	0.409255i	$0.865789\pi$
$674$	0	0
$675$	−17.3576	−0.668093
$676$	0	0
$677$	−24.7467	−0.951094	−0.475547	−	0.879690i	$-0.657750\pi$
$677$	−24.7467	−0.951094	−0.475547	+	0.879690i	$0.657750\pi$
$678$	0	0
$679$	−10.8363	−0.415858
$680$	0	0
$681$	−4.75958	−0.182387
$682$	0	0
$683$	43.3261	1.65783	0.828914	−	0.559376i	$-0.188959\pi$
$683$	43.3261	1.65783	0.828914	+	0.559376i	$0.188959\pi$
$684$	0	0
$685$	−55.7402	−2.12972
$686$	0	0
$687$	−2.41140	−0.0920006
$688$	0	0
$689$	6.09517	0.232207
$690$	0	0
$691$	1.97337	0.0750707	0.0375353	−	0.999295i	$-0.488049\pi$
$691$	1.97337	0.0750707	0.0375353	+	0.999295i	$0.488049\pi$
$692$	0	0
$693$	−2.84148	−0.107939
$694$	0	0
$695$	38.0689	1.44404
$696$	0	0
$697$	5.69350	0.215657
$698$	0	0
$699$	2.51617	0.0951704
$700$	0	0
$701$	45.5082	1.71882	0.859410	−	0.511286i	$-0.170831\pi$
$701$	45.5082	1.71882	0.859410	+	0.511286i	$0.170831\pi$
$702$	0	0
$703$	−40.0198	−1.50938
$704$	0	0
$705$	−3.69522	−0.139170
$706$	0	0
$707$	6.40496	0.240883
$708$	0	0
$709$	−29.0351	−1.09043	−0.545217	−	0.838295i	$-0.683553\pi$
$709$	−29.0351	−1.09043	−0.545217	+	0.838295i	$0.683553\pi$
$710$	0	0
$711$	−15.0462	−0.564278
$712$	0	0
$713$	−8.07151	−0.302280
$714$	0	0
$715$	3.53033	0.132027
$716$	0	0
$717$	10.4511	0.390302
$718$	0	0
$719$	−4.14268	−0.154496	−0.0772480	−	0.997012i	$-0.524613\pi$
$719$	−4.14268	−0.154496	−0.0772480	+	0.997012i	$0.524613\pi$
$720$	0	0
$721$	2.99636	0.111590
$722$	0	0
$723$	−1.24134	−0.0461659
$724$	0	0
$725$	−8.77419	−0.325865
$726$	0	0
$727$	−50.0070	−1.85466	−0.927329	−	0.374247i	$-0.877901\pi$
$727$	−50.0070	−1.85466	−0.927329	+	0.374247i	$0.877901\pi$
$728$	0	0
$729$	−18.8130	−0.696779
$730$	0	0
$731$	3.09311	0.114403
$732$	0	0
$733$	8.24352	0.304481	0.152241	−	0.988343i	$-0.451351\pi$
$733$	8.24352	0.304481	0.152241	+	0.988343i	$0.451351\pi$
$734$	0	0
$735$	1.40557	0.0518453
$736$	0	0
$737$	−13.4087	−0.493917
$738$	0	0
$739$	−13.5649	−0.498993	−0.249496	−	0.968376i	$-0.580265\pi$
$739$	−13.5649	−0.498993	−0.249496	+	0.968376i	$0.580265\pi$
$740$	0	0
$741$	−2.04121	−0.0749858
$742$	0	0
$743$	42.7226	1.56734	0.783670	−	0.621177i	$-0.213345\pi$
$743$	42.7226	1.56734	0.783670	+	0.621177i	$0.213345\pi$
$744$	0	0
$745$	−1.72918	−0.0633521
$746$	0	0
$747$	−5.68966	−0.208174
$748$	0	0
$749$	14.7546	0.539120
$750$	0	0
$751$	−5.25317	−0.191691	−0.0958454	−	0.995396i	$-0.530555\pi$
$751$	−5.25317	−0.191691	−0.0958454	+	0.995396i	$0.530555\pi$
$752$	0	0
$753$	−2.67375	−0.0974369
$754$	0	0
$755$	−56.1376	−2.04306
$756$	0	0
$757$	9.55305	0.347211	0.173606	−	0.984815i	$-0.444458\pi$
$757$	9.55305	0.347211	0.173606	+	0.984815i	$0.444458\pi$
$758$	0	0
$759$	2.27585	0.0826082
$760$	0	0
$761$	17.8079	0.645536	0.322768	−	0.946478i	$-0.395387\pi$
$761$	17.8079	0.645536	0.322768	+	0.946478i	$0.395387\pi$
$762$	0	0
$763$	−15.8020	−0.572072
$764$	0	0
$765$	10.8199	0.391196
$766$	0	0
$767$	−4.25733	−0.153723
$768$	0	0
$769$	11.4396	0.412523	0.206261	−	0.978497i	$-0.433870\pi$
$769$	11.4396	0.412523	0.206261	+	0.978497i	$0.433870\pi$
$770$	0	0
$771$	−0.684587	−0.0246548
$772$	0	0
$773$	−47.5847	−1.71150	−0.855751	−	0.517387i	$-0.826905\pi$
$773$	−47.5847	−1.71150	−0.855751	+	0.517387i	$0.826905\pi$
$774$	0	0
$775$	10.5384	0.378552
$776$	0	0
$777$	3.10786	0.111494
$778$	0	0
$779$	−27.0624	−0.969610
$780$	0	0
$781$	8.91852	0.319130
$782$	0	0
$783$	2.73426	0.0977145
$784$	0	0
$785$	−80.3389	−2.86742
$786$	0	0
$787$	40.6405	1.44868	0.724339	−	0.689444i	$-0.242145\pi$
$787$	40.6405	1.44868	0.724339	+	0.689444i	$0.242145\pi$
$788$	0	0
$789$	0.952865	0.0339229
$790$	0	0
$791$	12.3822	0.440259
$792$	0	0
$793$	1.98172	0.0703728
$794$	0	0
$795$	−8.56720	−0.303847
$796$	0	0
$797$	31.9673	1.13234	0.566169	−	0.824289i	$-0.308425\pi$
$797$	31.9673	1.13234	0.566169	+	0.824289i	$0.308425\pi$
$798$	0	0
$799$	2.83564	0.100318
$800$	0	0
$801$	16.7514	0.591880
$802$	0	0
$803$	10.4436	0.368546
$804$	0	0
$805$	20.1800	0.711253
$806$	0	0
$807$	−9.25526	−0.325801
$808$	0	0
$809$	28.8693	1.01499	0.507494	−	0.861655i	$-0.330572\pi$
$809$	28.8693	1.01499	0.507494	+	0.861655i	$0.330572\pi$
$810$	0	0
$811$	−18.9186	−0.664322	−0.332161	−	0.943223i	$-0.607778\pi$
$811$	−18.9186	−0.664322	−0.332161	+	0.943223i	$0.607778\pi$
$812$	0	0
$813$	3.63505	0.127487
$814$	0	0
$815$	52.0998	1.82498
$816$	0	0
$817$	−14.7022	−0.514365
$818$	0	0
$819$	−2.84148	−0.0992895
$820$	0	0
$821$	−12.7775	−0.445938	−0.222969	−	0.974826i	$-0.571575\pi$
$821$	−12.7775	−0.445938	−0.222969	+	0.974826i	$0.571575\pi$
$822$	0	0
$823$	24.4311	0.851615	0.425808	−	0.904814i	$-0.359990\pi$
$823$	24.4311	0.851615	0.425808	+	0.904814i	$0.359990\pi$
$824$	0	0
$825$	−2.97143	−0.103452
$826$	0	0
$827$	41.7407	1.45147	0.725733	−	0.687976i	$-0.241501\pi$
$827$	41.7407	1.45147	0.725733	+	0.687976i	$0.241501\pi$
$828$	0	0
$829$	0.530225	0.0184155	0.00920774	−	0.999958i	$-0.497069\pi$
$829$	0.530225	0.0184155	0.00920774	+	0.999958i	$0.497069\pi$
$830$	0	0
$831$	7.22388	0.250594
$832$	0	0
$833$	−1.07861	−0.0373716
$834$	0	0
$835$	−19.9331	−0.689815
$836$	0	0
$837$	−3.28404	−0.113513
$838$	0	0
$839$	−26.0810	−0.900417	−0.450208	−	0.892924i	$-0.648650\pi$
$839$	−26.0810	−0.900417	−0.450208	+	0.892924i	$0.648650\pi$
$840$	0	0
$841$	−27.6178	−0.952339
$842$	0	0
$843$	4.77106	0.164324
$844$	0	0
$845$	3.53033	0.121447
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−5.74655	−0.197221
$850$	0	0
$851$	44.6201	1.52956
$852$	0	0
$853$	27.1746	0.930441	0.465221	−	0.885195i	$-0.345975\pi$
$853$	27.1746	0.930441	0.465221	+	0.885195i	$0.345975\pi$
$854$	0	0
$855$	−51.4294	−1.75885
$856$	0	0
$857$	−39.8600	−1.36159	−0.680796	−	0.732473i	$-0.738366\pi$
$857$	−39.8600	−1.36159	−0.680796	+	0.732473i	$0.738366\pi$
$858$	0	0
$859$	6.99532	0.238677	0.119339	−	0.992854i	$-0.461923\pi$
$859$	6.99532	0.238677	0.119339	+	0.992854i	$0.461923\pi$
$860$	0	0
$861$	2.10161	0.0716228
$862$	0	0
$863$	−34.9134	−1.18847	−0.594234	−	0.804292i	$-0.702545\pi$
$863$	−34.9134	−1.18847	−0.594234	+	0.804292i	$0.702545\pi$
$864$	0	0
$865$	−67.8018	−2.30533
$866$	0	0
$867$	−6.30521	−0.214136
$868$	0	0
$869$	−5.29521	−0.179628
$870$	0	0
$871$	−13.4087	−0.454337
$872$	0	0
$873$	−30.7911	−1.04212
$874$	0	0
$875$	−8.69608	−0.293981
$876$	0	0
$877$	31.5141	1.06416	0.532078	−	0.846695i	$-0.321411\pi$
$877$	31.5141	1.06416	0.532078	+	0.846695i	$0.321411\pi$
$878$	0	0
$879$	−2.82710	−0.0953556
$880$	0	0
$881$	−6.50910	−0.219297	−0.109649	−	0.993970i	$-0.534973\pi$
$881$	−6.50910	−0.219297	−0.109649	+	0.993970i	$0.534973\pi$
$882$	0	0
$883$	19.4828	0.655647	0.327824	−	0.944739i	$-0.393685\pi$
$883$	19.4828	0.655647	0.327824	+	0.944739i	$0.393685\pi$
$884$	0	0
$885$	5.98398	0.201149
$886$	0	0
$887$	5.79773	0.194669	0.0973344	−	0.995252i	$-0.468968\pi$
$887$	5.79773	0.194669	0.0973344	+	0.995252i	$0.468968\pi$
$888$	0	0
$889$	14.1106	0.473255
$890$	0	0
$891$	−7.59848	−0.254559
$892$	0	0
$893$	−13.4784	−0.451037
$894$	0	0
$895$	−52.3433	−1.74964
$896$	0	0
$897$	2.27585	0.0759885
$898$	0	0
$899$	−1.66007	−0.0553665
$900$	0	0
$901$	6.57430	0.219022
$902$	0	0
$903$	1.14175	0.0379949
$904$	0	0
$905$	−63.2843	−2.10364
$906$	0	0
$907$	−37.2381	−1.23647	−0.618236	−	0.785993i	$-0.712152\pi$
$907$	−37.2381	−1.23647	−0.618236	+	0.785993i	$0.712152\pi$
$908$	0	0
$909$	18.1996	0.603642
$910$	0	0
$911$	−19.7431	−0.654120	−0.327060	−	0.945004i	$-0.606058\pi$
$911$	−19.7431	−0.654120	−0.327060	+	0.945004i	$0.606058\pi$
$912$	0	0
$913$	−2.00236	−0.0662683
$914$	0	0
$915$	−2.78545	−0.0920840
$916$	0	0
$917$	0.00489576	0.000161672 0
$918$	0	0
$919$	18.6009	0.613587	0.306794	−	0.951776i	$-0.400744\pi$
$919$	18.6009	0.613587	0.306794	+	0.951776i	$0.400744\pi$
$920$	0	0
$921$	0.0379213	0.00124955
$922$	0	0
$923$	8.91852	0.293557
$924$	0	0
$925$	−58.2576	−1.91550
$926$	0	0
$927$	8.51412	0.279640
$928$	0	0
$929$	14.6458	0.480513	0.240256	−	0.970709i	$-0.422768\pi$
$929$	14.6458	0.480513	0.240256	+	0.970709i	$0.422768\pi$
$930$	0	0
$931$	5.12685	0.168026
$932$	0	0
$933$	12.5627	0.411284
$934$	0	0
$935$	3.80785	0.124530
$936$	0	0
$937$	8.68070	0.283586	0.141793	−	0.989896i	$-0.454713\pi$
$937$	8.68070	0.283586	0.141793	+	0.989896i	$0.454713\pi$
$938$	0	0
$939$	3.21107	0.104789
$940$	0	0
$941$	−14.5399	−0.473987	−0.236994	−	0.971511i	$-0.576162\pi$
$941$	−14.5399	−0.473987	−0.236994	+	0.971511i	$0.576162\pi$
$942$	0	0
$943$	30.1732	0.982575
$944$	0	0
$945$	8.21063	0.267092
$946$	0	0
$947$	−21.3164	−0.692690	−0.346345	−	0.938107i	$-0.612577\pi$
$947$	−21.3164	−0.692690	−0.346345	+	0.938107i	$0.612577\pi$
$948$	0	0
$949$	10.4436	0.339013
$950$	0	0
$951$	7.43889	0.241223
$952$	0	0
$953$	16.6026	0.537811	0.268906	−	0.963167i	$-0.413338\pi$
$953$	16.6026	0.537811	0.268906	+	0.963167i	$0.413338\pi$
$954$	0	0
$955$	47.3015	1.53064
$956$	0	0
$957$	0.468076	0.0151308
$958$	0	0
$959$	15.7889	0.509851
$960$	0	0
$961$	−29.0061	−0.935682
$962$	0	0
$963$	41.9249	1.35101
$964$	0	0
$965$	46.7932	1.50633
$966$	0	0
$967$	−29.3169	−0.942768	−0.471384	−	0.881928i	$-0.656245\pi$
$967$	−29.3169	−0.942768	−0.471384	+	0.881928i	$0.656245\pi$
$968$	0	0
$969$	−2.20167	−0.0707278
$970$	0	0
$971$	50.6171	1.62438	0.812189	−	0.583394i	$-0.198276\pi$
$971$	50.6171	1.62438	0.812189	+	0.583394i	$0.198276\pi$
$972$	0	0
$973$	−10.7834	−0.345699
$974$	0	0
$975$	−2.97143	−0.0951619
$976$	0	0
$977$	−28.4386	−0.909830	−0.454915	−	0.890535i	$-0.650330\pi$
$977$	−28.4386	−0.909830	−0.454915	+	0.890535i	$0.650330\pi$
$978$	0	0
$979$	5.89528	0.188414
$980$	0	0
$981$	−44.9012	−1.43358
$982$	0	0
$983$	53.0478	1.69196	0.845980	−	0.533214i	$-0.179016\pi$
$983$	53.0478	1.69196	0.845980	+	0.533214i	$0.179016\pi$
$984$	0	0
$985$	−33.6430	−1.07196
$986$	0	0
$987$	1.04671	0.0333170
$988$	0	0
$989$	16.3922	0.521243
$990$	0	0
$991$	37.2209	1.18236	0.591181	−	0.806539i	$-0.298662\pi$
$991$	37.2209	1.18236	0.591181	+	0.806539i	$0.298662\pi$
$992$	0	0
$993$	−4.49833	−0.142750
$994$	0	0
$995$	56.6862	1.79707
$996$	0	0
$997$	−19.5528	−0.619245	−0.309622	−	0.950860i	$-0.600203\pi$
$997$	−19.5528	−0.619245	−0.309622	+	0.950860i	$0.600203\pi$
$998$	0	0
$999$	18.1545	0.574384

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.u.1.6	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.u.1.6	✓	10	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)

\(f(q)\)	\(=\)	\(q+0.398142 q^{3} +3.53033 q^{5} -1.00000 q^{7} -2.84148 q^{9} +O(q^{10})\)	\(q+0.398142 q^{3} +3.53033 q^{5} -1.00000 q^{7} -2.84148 q^{9} -1.00000 q^{11} -1.00000 q^{13} +1.40557 q^{15} -1.07861 q^{17} +5.12685 q^{19} -0.398142 q^{21} -5.71619 q^{23} +7.46325 q^{25} -2.32574 q^{27} -1.17565 q^{29} +1.41204 q^{31} -0.398142 q^{33} -3.53033 q^{35} -7.80593 q^{37} -0.398142 q^{39} -5.27856 q^{41} -2.86769 q^{43} -10.0314 q^{45} -2.62898 q^{47} +1.00000 q^{49} -0.429439 q^{51} -6.09517 q^{53} -3.53033 q^{55} +2.04121 q^{57} +4.25733 q^{59} -1.98172 q^{61} +2.84148 q^{63} -3.53033 q^{65} +13.4087 q^{67} -2.27585 q^{69} -8.91852 q^{71} -10.4436 q^{73} +2.97143 q^{75} +1.00000 q^{77} +5.29521 q^{79} +7.59848 q^{81} +2.00236 q^{83} -3.80785 q^{85} -0.468076 q^{87} -5.89528 q^{89} +1.00000 q^{91} +0.562193 q^{93} +18.0995 q^{95} +10.8363 q^{97} +2.84148 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9	\( 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} - 3 q^{17} + 13 q^{19} - 2 q^{21} + 6 q^{25} + 14 q^{27} - 3 q^{29} - q^{31} - 2 q^{33} + 4 q^{35} - 5 q^{37} - 2 q^{39} + 4 q^{41} + 19 q^{43} - 11 q^{45} + q^{47} + 10 q^{49} + 15 q^{51} - 6 q^{53} + 4 q^{55} - 6 q^{57} + 4 q^{59} - 18 q^{61} - 6 q^{63} + 4 q^{65} + q^{67} - 11 q^{69} - 21 q^{71} - 10 q^{73} + 10 q^{77} + 3 q^{79} - 30 q^{81} + 6 q^{83} - 33 q^{85} - 9 q^{87} - 22 q^{89} + 10 q^{91} - 34 q^{93} - 3 q^{95} - 9 q^{97} - 6 q^{99}+O(q^{100}) \) 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 - 3 * q^17 + 13 * q^19 - 2 * q^21 + 6 * q^25 + 14 * q^27 - 3 * q^29 - q^31 - 2 * q^33 + 4 * q^35 - 5 * q^37 - 2 * q^39 + 4 * q^41 + 19 * q^43 - 11 * q^45 + q^47 + 10 * q^49 + 15 * q^51 - 6 * q^53 + 4 * q^55 - 6 * q^57 + 4 * q^59 - 18 * q^61 - 6 * q^63 + 4 * q^65 + q^67 - 11 * q^69 - 21 * q^71 - 10 * q^73 + 10 * q^77 + 3 * q^79 - 30 * q^81 + 6 * q^83 - 33 * q^85 - 9 * q^87 - 22 * q^89 + 10 * q^91 - 34 * q^93 - 3 * q^95 - 9 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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