LMFDB - Embedded newform 8008.2.a.x.1.11

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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 17 x^{10} + 79 x^{9} + 80 x^{8} - 536 x^{7} - 4 x^{6} + 1484 x^{5} - 682 x^{4} + \cdots - 64 $ x^12 - 4x^11 - 17x^10 + 79x^9 + 80x^8 - 536x^7 - 4x^6 + 1484x^5 - 682x^4 - 1431x^3 + 1069x^2 - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$2.98199$ 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)

$f(q)$	$=$	$q+2.98199 q^{3} +1.64834 q^{5} +1.00000 q^{7} +5.89225 q^{9} +O(q^{10})$	$q+2.98199 q^{3} +1.64834 q^{5} +1.00000 q^{7} +5.89225 q^{9} +1.00000 q^{11} +1.00000 q^{13} +4.91533 q^{15} +6.79730 q^{17} -4.90963 q^{19} +2.98199 q^{21} +3.96169 q^{23} -2.28297 q^{25} +8.62465 q^{27} -1.97906 q^{29} +9.49225 q^{31} +2.98199 q^{33} +1.64834 q^{35} -10.6949 q^{37} +2.98199 q^{39} -0.717475 q^{41} +6.18468 q^{43} +9.71244 q^{45} -12.0658 q^{47} +1.00000 q^{49} +20.2695 q^{51} +12.1807 q^{53} +1.64834 q^{55} -14.6404 q^{57} -2.55616 q^{59} +2.51108 q^{61} +5.89225 q^{63} +1.64834 q^{65} -11.2522 q^{67} +11.8137 q^{69} -6.36273 q^{71} +11.9803 q^{73} -6.80779 q^{75} +1.00000 q^{77} +12.8856 q^{79} +8.04186 q^{81} -4.62930 q^{83} +11.2043 q^{85} -5.90154 q^{87} -0.496767 q^{89} +1.00000 q^{91} +28.3058 q^{93} -8.09274 q^{95} -4.84522 q^{97} +5.89225 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9}+O(q^{10}) $ 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9	$ 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9} + 12 q^{11} + 12 q^{13} - 3 q^{15} + 16 q^{17} - 2 q^{19} + 4 q^{21} + 9 q^{23} + 14 q^{25} + 7 q^{27} + 15 q^{29} + 10 q^{31} + 4 q^{33} + 6 q^{35} + 18 q^{37} + 4 q^{39} + 24 q^{41} + 15 q^{45} + 5 q^{47} + 12 q^{49} + 4 q^{51} + 15 q^{53} + 6 q^{55} - 4 q^{57} + 15 q^{59} + 17 q^{61} + 14 q^{63} + 6 q^{65} - 7 q^{67} + 9 q^{71} + 32 q^{73} - 8 q^{75} + 12 q^{77} + 20 q^{79} - 4 q^{81} - 5 q^{83} + 25 q^{85} + 19 q^{87} + 16 q^{89} + 12 q^{91} + 21 q^{93} + 8 q^{95} + 10 q^{97} + 14 q^{99}+O(q^{100}) $ 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9 + 12 * q^11 + 12 * q^13 - 3 * q^15 + 16 * q^17 - 2 * q^19 + 4 * q^21 + 9 * q^23 + 14 * q^25 + 7 * q^27 + 15 * q^29 + 10 * q^31 + 4 * q^33 + 6 * q^35 + 18 * q^37 + 4 * q^39 + 24 * q^41 + 15 * q^45 + 5 * q^47 + 12 * q^49 + 4 * q^51 + 15 * q^53 + 6 * q^55 - 4 * q^57 + 15 * q^59 + 17 * q^61 + 14 * q^63 + 6 * q^65 - 7 * q^67 + 9 * q^71 + 32 * q^73 - 8 * q^75 + 12 * q^77 + 20 * q^79 - 4 * q^81 - 5 * q^83 + 25 * q^85 + 19 * q^87 + 16 * q^89 + 12 * q^91 + 21 * q^93 + 8 * q^95 + 10 * q^97 + 14 * 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
$3$	2.98199	1.72165	0.860826	−	0.508900i	$-0.169948\pi$
$3$	2.98199	1.72165	0.860826	+	0.508900i	$0.169948\pi$
$4$	0	0
$5$	1.64834	0.737161	0.368580	−	0.929596i	$-0.379844\pi$
$5$	1.64834	0.737161	0.368580	+	0.929596i	$0.379844\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	5.89225	1.96408
$10$	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
$15$	4.91533	1.26913
$16$	0	0
$17$	6.79730	1.64859	0.824294	−	0.566163i	$-0.191573\pi$
$17$	6.79730	1.64859	0.824294	+	0.566163i	$0.191573\pi$
$18$	0	0
$19$	−4.90963	−1.12635	−0.563173	−	0.826339i	$-0.690419\pi$
$19$	−4.90963	−1.12635	−0.563173	+	0.826339i	$0.690419\pi$
$20$	0	0
$21$	2.98199	0.650723
$22$	0	0
$23$	3.96169	0.826069	0.413035	−	0.910715i	$-0.364469\pi$
$23$	3.96169	0.826069	0.413035	+	0.910715i	$0.364469\pi$
$24$	0	0
$25$	−2.28297	−0.456594
$26$	0	0
$27$	8.62465	1.65982
$28$	0	0
$29$	−1.97906	−0.367503	−0.183751	−	0.982973i	$-0.558824\pi$
$29$	−1.97906	−0.367503	−0.183751	+	0.982973i	$0.558824\pi$
$30$	0	0
$31$	9.49225	1.70486	0.852430	−	0.522842i	$-0.175128\pi$
$31$	9.49225	1.70486	0.852430	+	0.522842i	$0.175128\pi$
$32$	0	0
$33$	2.98199	0.519097
$34$	0	0
$35$	1.64834	0.278621
$36$	0	0
$37$	−10.6949	−1.75824	−0.879119	−	0.476601i	$-0.841868\pi$
$37$	−10.6949	−1.75824	−0.879119	+	0.476601i	$0.841868\pi$
$38$	0	0
$39$	2.98199	0.477500
$40$	0	0
$41$	−0.717475	−0.112051	−0.0560254	−	0.998429i	$-0.517843\pi$
$41$	−0.717475	−0.112051	−0.0560254	+	0.998429i	$0.517843\pi$
$42$	0	0
$43$	6.18468	0.943155	0.471578	−	0.881825i	$-0.343685\pi$
$43$	6.18468	0.943155	0.471578	+	0.881825i	$0.343685\pi$
$44$	0	0
$45$	9.71244	1.44785
$46$	0	0
$47$	−12.0658	−1.75998	−0.879988	−	0.474996i	$-0.842450\pi$
$47$	−12.0658	−1.75998	−0.879988	+	0.474996i	$0.842450\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	20.2695	2.83829
$52$	0	0
$53$	12.1807	1.67314	0.836571	−	0.547858i	$-0.184557\pi$
$53$	12.1807	1.67314	0.836571	+	0.547858i	$0.184557\pi$
$54$	0	0
$55$	1.64834	0.222262
$56$	0	0
$57$	−14.6404	−1.93917
$58$	0	0
$59$	−2.55616	−0.332783	−0.166392	−	0.986060i	$-0.553212\pi$
$59$	−2.55616	−0.332783	−0.166392	+	0.986060i	$0.553212\pi$
$60$	0	0
$61$	2.51108	0.321511	0.160756	−	0.986994i	$-0.448607\pi$
$61$	2.51108	0.321511	0.160756	+	0.986994i	$0.448607\pi$
$62$	0	0
$63$	5.89225	0.742354
$64$	0	0
$65$	1.64834	0.204452
$66$	0	0
$67$	−11.2522	−1.37468	−0.687338	−	0.726338i	$-0.741221\pi$
$67$	−11.2522	−1.37468	−0.687338	+	0.726338i	$0.741221\pi$
$68$	0	0
$69$	11.8137	1.42220
$70$	0	0
$71$	−6.36273	−0.755118	−0.377559	−	0.925986i	$-0.623236\pi$
$71$	−6.36273	−0.755118	−0.377559	+	0.925986i	$0.623236\pi$
$72$	0	0
$73$	11.9803	1.40218	0.701091	−	0.713071i	$-0.252697\pi$
$73$	11.9803	1.40218	0.701091	+	0.713071i	$0.252697\pi$
$74$	0	0
$75$	−6.80779	−0.786095
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	12.8856	1.44974	0.724872	−	0.688884i	$-0.241899\pi$
$79$	12.8856	1.44974	0.724872	+	0.688884i	$0.241899\pi$
$80$	0	0
$81$	8.04186	0.893540
$82$	0	0
$83$	−4.62930	−0.508131	−0.254066	−	0.967187i	$-0.581768\pi$
$83$	−4.62930	−0.508131	−0.254066	+	0.967187i	$0.581768\pi$
$84$	0	0
$85$	11.2043	1.21527
$86$	0	0
$87$	−5.90154	−0.632712
$88$	0	0
$89$	−0.496767	−0.0526572	−0.0263286	−	0.999653i	$-0.508382\pi$
$89$	−0.496767	−0.0526572	−0.0263286	+	0.999653i	$0.508382\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	28.3058	2.93517
$94$	0	0
$95$	−8.09274	−0.830298
$96$	0	0
$97$	−4.84522	−0.491958	−0.245979	−	0.969275i	$-0.579109\pi$
$97$	−4.84522	−0.491958	−0.245979	+	0.969275i	$0.579109\pi$
$98$	0	0
$99$	5.89225	0.592193
$100$	0	0
$101$	7.60638	0.756863	0.378432	−	0.925629i	$-0.376464\pi$
$101$	7.60638	0.756863	0.378432	+	0.925629i	$0.376464\pi$
$102$	0	0
$103$	−6.53211	−0.643628	−0.321814	−	0.946803i	$-0.604293\pi$
$103$	−6.53211	−0.643628	−0.321814	+	0.946803i	$0.604293\pi$
$104$	0	0
$105$	4.91533	0.479688
$106$	0	0
$107$	−15.1109	−1.46082	−0.730411	−	0.683008i	$-0.760671\pi$
$107$	−15.1109	−1.46082	−0.730411	+	0.683008i	$0.760671\pi$
$108$	0	0
$109$	−10.1957	−0.976572	−0.488286	−	0.872684i	$-0.662378\pi$
$109$	−10.1957	−0.976572	−0.488286	+	0.872684i	$0.662378\pi$
$110$	0	0
$111$	−31.8922	−3.02707
$112$	0	0
$113$	−4.16958	−0.392242	−0.196121	−	0.980580i	$-0.562834\pi$
$113$	−4.16958	−0.392242	−0.196121	+	0.980580i	$0.562834\pi$
$114$	0	0
$115$	6.53022	0.608946
$116$	0	0
$117$	5.89225	0.544739
$118$	0	0
$119$	6.79730	0.623107
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.13950	−0.192912
$124$	0	0
$125$	−12.0048	−1.07374
$126$	0	0
$127$	15.1320	1.34275	0.671373	−	0.741119i	$-0.265705\pi$
$127$	15.1320	1.34275	0.671373	+	0.741119i	$0.265705\pi$
$128$	0	0
$129$	18.4426	1.62378
$130$	0	0
$131$	4.43263	0.387280	0.193640	−	0.981073i	$-0.437971\pi$
$131$	4.43263	0.387280	0.193640	+	0.981073i	$0.437971\pi$
$132$	0	0
$133$	−4.90963	−0.425718
$134$	0	0
$135$	14.2164	1.22355
$136$	0	0
$137$	14.9249	1.27512	0.637562	−	0.770399i	$-0.279943\pi$
$137$	14.9249	1.27512	0.637562	+	0.770399i	$0.279943\pi$
$138$	0	0
$139$	−12.8272	−1.08799	−0.543997	−	0.839087i	$-0.683090\pi$
$139$	−12.8272	−1.08799	−0.543997	+	0.839087i	$0.683090\pi$
$140$	0	0
$141$	−35.9800	−3.03007
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−3.26217	−0.270909
$146$	0	0
$147$	2.98199	0.245950
$148$	0	0
$149$	−2.03309	−0.166557	−0.0832787	−	0.996526i	$-0.526539\pi$
$149$	−2.03309	−0.166557	−0.0832787	+	0.996526i	$0.526539\pi$
$150$	0	0
$151$	14.2589	1.16038	0.580188	−	0.814482i	$-0.302979\pi$
$151$	14.2589	1.16038	0.580188	+	0.814482i	$0.302979\pi$
$152$	0	0
$153$	40.0514	3.23796
$154$	0	0
$155$	15.6465	1.25676
$156$	0	0
$157$	−12.4126	−0.990630	−0.495315	−	0.868713i	$-0.664947\pi$
$157$	−12.4126	−0.990630	−0.495315	+	0.868713i	$0.664947\pi$
$158$	0	0
$159$	36.3226	2.88057
$160$	0	0
$161$	3.96169	0.312225
$162$	0	0
$163$	9.53508	0.746845	0.373423	−	0.927661i	$-0.378184\pi$
$163$	9.53508	0.746845	0.373423	+	0.927661i	$0.378184\pi$
$164$	0	0
$165$	4.91533	0.382658
$166$	0	0
$167$	7.08700	0.548408	0.274204	−	0.961672i	$-0.411586\pi$
$167$	7.08700	0.548408	0.274204	+	0.961672i	$0.411586\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−28.9287	−2.21224
$172$	0	0
$173$	−8.95669	−0.680965	−0.340482	−	0.940251i	$-0.610590\pi$
$173$	−8.95669	−0.680965	−0.340482	+	0.940251i	$0.610590\pi$
$174$	0	0
$175$	−2.28297	−0.172576
$176$	0	0
$177$	−7.62243	−0.572937
$178$	0	0
$179$	−16.3416	−1.22143	−0.610713	−	0.791852i	$-0.709117\pi$
$179$	−16.3416	−1.22143	−0.610713	+	0.791852i	$0.709117\pi$
$180$	0	0
$181$	−2.27522	−0.169116	−0.0845578	−	0.996419i	$-0.526948\pi$
$181$	−2.27522	−0.169116	−0.0845578	+	0.996419i	$0.526948\pi$
$182$	0	0
$183$	7.48802	0.553530
$184$	0	0
$185$	−17.6289	−1.29610
$186$	0	0
$187$	6.79730	0.497068
$188$	0	0
$189$	8.62465	0.627351
$190$	0	0
$191$	−24.9799	−1.80748	−0.903741	−	0.428081i	$-0.859190\pi$
$191$	−24.9799	−1.80748	−0.903741	+	0.428081i	$0.859190\pi$
$192$	0	0
$193$	3.88818	0.279878	0.139939	−	0.990160i	$-0.455309\pi$
$193$	3.88818	0.279878	0.139939	+	0.990160i	$0.455309\pi$
$194$	0	0
$195$	4.91533	0.351994
$196$	0	0
$197$	2.48256	0.176875	0.0884376	−	0.996082i	$-0.471813\pi$
$197$	2.48256	0.176875	0.0884376	+	0.996082i	$0.471813\pi$
$198$	0	0
$199$	16.8704	1.19591	0.597957	−	0.801528i	$-0.295979\pi$
$199$	16.8704	1.19591	0.597957	+	0.801528i	$0.295979\pi$
$200$	0	0
$201$	−33.5539	−2.36671
$202$	0	0
$203$	−1.97906	−0.138903
$204$	0	0
$205$	−1.18264	−0.0825995
$206$	0	0
$207$	23.3433	1.62247
$208$	0	0
$209$	−4.90963	−0.339606
$210$	0	0
$211$	−15.8842	−1.09351	−0.546757	−	0.837291i	$-0.684138\pi$
$211$	−15.8842	−1.09351	−0.546757	+	0.837291i	$0.684138\pi$
$212$	0	0
$213$	−18.9736	−1.30005
$214$	0	0
$215$	10.1945	0.695257
$216$	0	0
$217$	9.49225	0.644376
$218$	0	0
$219$	35.7250	2.41407
$220$	0	0
$221$	6.79730	0.457236
$222$	0	0
$223$	28.9078	1.93581	0.967904	−	0.251319i	$-0.0808642\pi$
$223$	28.9078	1.93581	0.967904	+	0.251319i	$0.0808642\pi$
$224$	0	0
$225$	−13.4518	−0.896788
$226$	0	0
$227$	−14.4119	−0.956551	−0.478275	−	0.878210i	$-0.658738\pi$
$227$	−14.4119	−0.956551	−0.478275	+	0.878210i	$0.658738\pi$
$228$	0	0
$229$	−8.50439	−0.561986	−0.280993	−	0.959710i	$-0.590664\pi$
$229$	−8.50439	−0.561986	−0.280993	+	0.959710i	$0.590664\pi$
$230$	0	0
$231$	2.98199	0.196200
$232$	0	0
$233$	−10.0643	−0.659336	−0.329668	−	0.944097i	$-0.606937\pi$
$233$	−10.0643	−0.659336	−0.329668	+	0.944097i	$0.606937\pi$
$234$	0	0
$235$	−19.8885	−1.29739
$236$	0	0
$237$	38.4247	2.49595
$238$	0	0
$239$	−16.5664	−1.07159	−0.535797	−	0.844347i	$-0.679989\pi$
$239$	−16.5664	−1.07159	−0.535797	+	0.844347i	$0.679989\pi$
$240$	0	0
$241$	4.90286	0.315821	0.157910	−	0.987453i	$-0.449524\pi$
$241$	4.90286	0.315821	0.157910	+	0.987453i	$0.449524\pi$
$242$	0	0
$243$	−1.89323	−0.121451
$244$	0	0
$245$	1.64834	0.105309
$246$	0	0
$247$	−4.90963	−0.312392
$248$	0	0
$249$	−13.8045	−0.874825
$250$	0	0
$251$	−14.5421	−0.917890	−0.458945	−	0.888465i	$-0.651772\pi$
$251$	−14.5421	−0.917890	−0.458945	+	0.888465i	$0.651772\pi$
$252$	0	0
$253$	3.96169	0.249069
$254$	0	0
$255$	33.4110	2.09228
$256$	0	0
$257$	−12.9532	−0.807996	−0.403998	−	0.914760i	$-0.632380\pi$
$257$	−12.9532	−0.807996	−0.403998	+	0.914760i	$0.632380\pi$
$258$	0	0
$259$	−10.6949	−0.664552
$260$	0	0
$261$	−11.6611	−0.721806
$262$	0	0
$263$	22.4953	1.38712	0.693561	−	0.720398i	$-0.256041\pi$
$263$	22.4953	1.38712	0.693561	+	0.720398i	$0.256041\pi$
$264$	0	0
$265$	20.0779	1.23338
$266$	0	0
$267$	−1.48135	−0.0906574
$268$	0	0
$269$	7.84242	0.478161	0.239081	−	0.971000i	$-0.423154\pi$
$269$	7.84242	0.478161	0.239081	+	0.971000i	$0.423154\pi$
$270$	0	0
$271$	−25.0606	−1.52232	−0.761162	−	0.648562i	$-0.775371\pi$
$271$	−25.0606	−1.52232	−0.761162	+	0.648562i	$0.775371\pi$
$272$	0	0
$273$	2.98199	0.180478
$274$	0	0
$275$	−2.28297	−0.137668
$276$	0	0
$277$	17.1258	1.02899	0.514494	−	0.857494i	$-0.327980\pi$
$277$	17.1258	1.02899	0.514494	+	0.857494i	$0.327980\pi$
$278$	0	0
$279$	55.9307	3.34849
$280$	0	0
$281$	15.4172	0.919715	0.459857	−	0.887993i	$-0.347901\pi$
$281$	15.4172	0.919715	0.459857	+	0.887993i	$0.347901\pi$
$282$	0	0
$283$	13.9782	0.830915	0.415457	−	0.909613i	$-0.363622\pi$
$283$	13.9782	0.830915	0.415457	+	0.909613i	$0.363622\pi$
$284$	0	0
$285$	−24.1325	−1.42948
$286$	0	0
$287$	−0.717475	−0.0423512
$288$	0	0
$289$	29.2033	1.71784
$290$	0	0
$291$	−14.4484	−0.846980
$292$	0	0
$293$	16.5764	0.968403	0.484201	−	0.874957i	$-0.339110\pi$
$293$	16.5764	0.968403	0.484201	+	0.874957i	$0.339110\pi$
$294$	0	0
$295$	−4.21342	−0.245315
$296$	0	0
$297$	8.62465	0.500453
$298$	0	0
$299$	3.96169	0.229110
$300$	0	0
$301$	6.18468	0.356479
$302$	0	0
$303$	22.6821	1.30305
$304$	0	0
$305$	4.13912	0.237006
$306$	0	0
$307$	16.4896	0.941113	0.470556	−	0.882370i	$-0.344053\pi$
$307$	16.4896	0.941113	0.470556	+	0.882370i	$0.344053\pi$
$308$	0	0
$309$	−19.4787	−1.10810
$310$	0	0
$311$	11.3038	0.640977	0.320489	−	0.947252i	$-0.396153\pi$
$311$	11.3038	0.640977	0.320489	+	0.947252i	$0.396153\pi$
$312$	0	0
$313$	3.21508	0.181727	0.0908636	−	0.995863i	$-0.471037\pi$
$313$	3.21508	0.181727	0.0908636	+	0.995863i	$0.471037\pi$
$314$	0	0
$315$	9.71244	0.547234
$316$	0	0
$317$	−16.4394	−0.923328	−0.461664	−	0.887055i	$-0.652747\pi$
$317$	−16.4394	−0.923328	−0.461664	+	0.887055i	$0.652747\pi$
$318$	0	0
$319$	−1.97906	−0.110806
$320$	0	0
$321$	−45.0604	−2.51502
$322$	0	0
$323$	−33.3722	−1.85688
$324$	0	0
$325$	−2.28297	−0.126636
$326$	0	0
$327$	−30.4035	−1.68132
$328$	0	0
$329$	−12.0658	−0.665208
$330$	0	0
$331$	−11.6524	−0.640473	−0.320236	−	0.947338i	$-0.603762\pi$
$331$	−11.6524	−0.640473	−0.320236	+	0.947338i	$0.603762\pi$
$332$	0	0
$333$	−63.0173	−3.45333
$334$	0	0
$335$	−18.5475	−1.01336
$336$	0	0
$337$	12.9291	0.704294	0.352147	−	0.935945i	$-0.385452\pi$
$337$	12.9291	0.704294	0.352147	+	0.935945i	$0.385452\pi$
$338$	0	0
$339$	−12.4336	−0.675303
$340$	0	0
$341$	9.49225	0.514034
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	19.4730	1.04839
$346$	0	0
$347$	10.3951	0.558036	0.279018	−	0.960286i	$-0.409991\pi$
$347$	10.3951	0.558036	0.279018	+	0.960286i	$0.409991\pi$
$348$	0	0
$349$	−20.1780	−1.08010	−0.540052	−	0.841632i	$-0.681595\pi$
$349$	−20.1780	−1.08010	−0.540052	+	0.841632i	$0.681595\pi$
$350$	0	0
$351$	8.62465	0.460350
$352$	0	0
$353$	−14.0778	−0.749288	−0.374644	−	0.927169i	$-0.622235\pi$
$353$	−14.0778	−0.749288	−0.374644	+	0.927169i	$0.622235\pi$
$354$	0	0
$355$	−10.4880	−0.556643
$356$	0	0
$357$	20.2695	1.07277
$358$	0	0
$359$	14.2994	0.754695	0.377347	−	0.926072i	$-0.376836\pi$
$359$	14.2994	0.754695	0.377347	+	0.926072i	$0.376836\pi$
$360$	0	0
$361$	5.10442	0.268654
$362$	0	0
$363$	2.98199	0.156514
$364$	0	0
$365$	19.7476	1.03363
$366$	0	0
$367$	−2.92040	−0.152443	−0.0762217	−	0.997091i	$-0.524286\pi$
$367$	−2.92040	−0.152443	−0.0762217	+	0.997091i	$0.524286\pi$
$368$	0	0
$369$	−4.22754	−0.220077
$370$	0	0
$371$	12.1807	0.632388
$372$	0	0
$373$	−26.2299	−1.35814	−0.679068	−	0.734076i	$-0.737616\pi$
$373$	−26.2299	−1.35814	−0.679068	+	0.734076i	$0.737616\pi$
$374$	0	0
$375$	−35.7982	−1.84861
$376$	0	0
$377$	−1.97906	−0.101927
$378$	0	0
$379$	10.2559	0.526808	0.263404	−	0.964686i	$-0.415155\pi$
$379$	10.2559	0.526808	0.263404	+	0.964686i	$0.415155\pi$
$380$	0	0
$381$	45.1234	2.31174
$382$	0	0
$383$	25.2208	1.28872	0.644360	−	0.764722i	$-0.277124\pi$
$383$	25.2208	1.28872	0.644360	+	0.764722i	$0.277124\pi$
$384$	0	0
$385$	1.64834	0.0840073
$386$	0	0
$387$	36.4417	1.85244
$388$	0	0
$389$	22.6695	1.14939	0.574694	−	0.818368i	$-0.305121\pi$
$389$	22.6695	1.14939	0.574694	+	0.818368i	$0.305121\pi$
$390$	0	0
$391$	26.9288	1.36185
$392$	0	0
$393$	13.2180	0.666762
$394$	0	0
$395$	21.2399	1.06869
$396$	0	0
$397$	2.28885	0.114874	0.0574371	−	0.998349i	$-0.481707\pi$
$397$	2.28885	0.114874	0.0574371	+	0.998349i	$0.481707\pi$
$398$	0	0
$399$	−14.6404	−0.732939
$400$	0	0
$401$	−22.3095	−1.11408	−0.557042	−	0.830484i	$-0.688064\pi$
$401$	−22.3095	−1.11408	−0.557042	+	0.830484i	$0.688064\pi$
$402$	0	0
$403$	9.49225	0.472843
$404$	0	0
$405$	13.2557	0.658683
$406$	0	0
$407$	−10.6949	−0.530129
$408$	0	0
$409$	−5.02468	−0.248455	−0.124227	−	0.992254i	$-0.539645\pi$
$409$	−5.02468	−0.248455	−0.124227	+	0.992254i	$0.539645\pi$
$410$	0	0
$411$	44.5060	2.19532
$412$	0	0
$413$	−2.55616	−0.125780
$414$	0	0
$415$	−7.63066	−0.374575
$416$	0	0
$417$	−38.2507	−1.87314
$418$	0	0
$419$	−12.9607	−0.633171	−0.316585	−	0.948564i	$-0.602536\pi$
$419$	−12.9607	−0.633171	−0.316585	+	0.948564i	$0.602536\pi$
$420$	0	0
$421$	0.269186	0.0131193	0.00655967	−	0.999978i	$-0.497912\pi$
$421$	0.269186	0.0131193	0.00655967	+	0.999978i	$0.497912\pi$
$422$	0	0
$423$	−71.0946	−3.45674
$424$	0	0
$425$	−15.5180	−0.752735
$426$	0	0
$427$	2.51108	0.121520
$428$	0	0
$429$	2.98199	0.143972
$430$	0	0
$431$	−9.57396	−0.461161	−0.230581	−	0.973053i	$-0.574063\pi$
$431$	−9.57396	−0.461161	−0.230581	+	0.973053i	$0.574063\pi$
$432$	0	0
$433$	7.60070	0.365266	0.182633	−	0.983181i	$-0.441538\pi$
$433$	7.60070	0.365266	0.182633	+	0.983181i	$0.441538\pi$
$434$	0	0
$435$	−9.72776	−0.466410
$436$	0	0
$437$	−19.4504	−0.930439
$438$	0	0
$439$	−2.10798	−0.100609	−0.0503043	−	0.998734i	$-0.516019\pi$
$439$	−2.10798	−0.100609	−0.0503043	+	0.998734i	$0.516019\pi$
$440$	0	0
$441$	5.89225	0.280583
$442$	0	0
$443$	−15.2085	−0.722575	−0.361288	−	0.932454i	$-0.617663\pi$
$443$	−15.2085	−0.722575	−0.361288	+	0.932454i	$0.617663\pi$
$444$	0	0
$445$	−0.818842	−0.0388168
$446$	0	0
$447$	−6.06265	−0.286754
$448$	0	0
$449$	31.5640	1.48960	0.744799	−	0.667289i	$-0.232545\pi$
$449$	31.5640	1.48960	0.744799	+	0.667289i	$0.232545\pi$
$450$	0	0
$451$	−0.717475	−0.0337846
$452$	0	0
$453$	42.5200	1.99776
$454$	0	0
$455$	1.64834	0.0772755
$456$	0	0
$457$	−16.4395	−0.769006	−0.384503	−	0.923124i	$-0.625627\pi$
$457$	−16.4395	−0.769006	−0.384503	+	0.923124i	$0.625627\pi$
$458$	0	0
$459$	58.6244	2.73635
$460$	0	0
$461$	41.9615	1.95434	0.977172	−	0.212449i	$-0.0681441\pi$
$461$	41.9615	1.95434	0.977172	+	0.212449i	$0.0681441\pi$
$462$	0	0
$463$	−29.1385	−1.35418	−0.677089	−	0.735901i	$-0.736759\pi$
$463$	−29.1385	−1.35418	−0.677089	+	0.735901i	$0.736759\pi$
$464$	0	0
$465$	46.6576	2.16369
$466$	0	0
$467$	−26.9261	−1.24599	−0.622995	−	0.782226i	$-0.714084\pi$
$467$	−26.9261	−1.24599	−0.622995	+	0.782226i	$0.714084\pi$
$468$	0	0
$469$	−11.2522	−0.519579
$470$	0	0
$471$	−37.0141	−1.70552
$472$	0	0
$473$	6.18468	0.284372
$474$	0	0
$475$	11.2085	0.514282
$476$	0	0
$477$	71.7715	3.28619
$478$	0	0
$479$	7.72016	0.352743	0.176372	−	0.984324i	$-0.443564\pi$
$479$	7.72016	0.352743	0.176372	+	0.984324i	$0.443564\pi$
$480$	0	0
$481$	−10.6949	−0.487648
$482$	0	0
$483$	11.8137	0.537542
$484$	0	0
$485$	−7.98658	−0.362652
$486$	0	0
$487$	−39.8891	−1.80755	−0.903774	−	0.428009i	$-0.859215\pi$
$487$	−39.8891	−1.80755	−0.903774	+	0.428009i	$0.859215\pi$
$488$	0	0
$489$	28.4335	1.28581
$490$	0	0
$491$	14.3384	0.647081	0.323541	−	0.946214i	$-0.395127\pi$
$491$	14.3384	0.647081	0.323541	+	0.946214i	$0.395127\pi$
$492$	0	0
$493$	−13.4523	−0.605860
$494$	0	0
$495$	9.71244	0.436542
$496$	0	0
$497$	−6.36273	−0.285408
$498$	0	0
$499$	23.5498	1.05423	0.527117	−	0.849793i	$-0.323273\pi$
$499$	23.5498	1.05423	0.527117	+	0.849793i	$0.323273\pi$
$500$	0	0
$501$	21.1333	0.944168
$502$	0	0
$503$	1.98459	0.0884885	0.0442443	−	0.999021i	$-0.485912\pi$
$503$	1.98459	0.0884885	0.0442443	+	0.999021i	$0.485912\pi$
$504$	0	0
$505$	12.5379	0.557930
$506$	0	0
$507$	2.98199	0.132435
$508$	0	0
$509$	−16.2567	−0.720565	−0.360283	−	0.932843i	$-0.617320\pi$
$509$	−16.2567	−0.720565	−0.360283	+	0.932843i	$0.617320\pi$
$510$	0	0
$511$	11.9803	0.529975
$512$	0	0
$513$	−42.3438	−1.86953
$514$	0	0
$515$	−10.7671	−0.474457
$516$	0	0
$517$	−12.0658	−0.530653
$518$	0	0
$519$	−26.7087	−1.17238
$520$	0	0
$521$	30.1339	1.32019	0.660095	−	0.751182i	$-0.270516\pi$
$521$	30.1339	1.32019	0.660095	+	0.751182i	$0.270516\pi$
$522$	0	0
$523$	−8.95305	−0.391490	−0.195745	−	0.980655i	$-0.562712\pi$
$523$	−8.95305	−0.391490	−0.195745	+	0.980655i	$0.562712\pi$
$524$	0	0
$525$	−6.80779	−0.297116
$526$	0	0
$527$	64.5217	2.81061
$528$	0	0
$529$	−7.30503	−0.317610
$530$	0	0
$531$	−15.0615	−0.653615
$532$	0	0
$533$	−0.717475	−0.0310773
$534$	0	0
$535$	−24.9078	−1.07686
$536$	0	0
$537$	−48.7304	−2.10287
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−5.25125	−0.225769	−0.112884	−	0.993608i	$-0.536009\pi$
$541$	−5.25125	−0.225769	−0.112884	+	0.993608i	$0.536009\pi$
$542$	0	0
$543$	−6.78467	−0.291158
$544$	0	0
$545$	−16.8060	−0.719891
$546$	0	0
$547$	−17.4164	−0.744673	−0.372337	−	0.928098i	$-0.621443\pi$
$547$	−17.4164	−0.744673	−0.372337	+	0.928098i	$0.621443\pi$
$548$	0	0
$549$	14.7959	0.631475
$550$	0	0
$551$	9.71646	0.413935
$552$	0	0
$553$	12.8856	0.547952
$554$	0	0
$555$	−52.5693	−2.23144
$556$	0	0
$557$	39.7871	1.68583	0.842916	−	0.538045i	$-0.180837\pi$
$557$	39.7871	1.68583	0.842916	+	0.538045i	$0.180837\pi$
$558$	0	0
$559$	6.18468	0.261584
$560$	0	0
$561$	20.2695	0.855777
$562$	0	0
$563$	25.9935	1.09550	0.547749	−	0.836643i	$-0.315485\pi$
$563$	25.9935	1.09550	0.547749	+	0.836643i	$0.315485\pi$
$564$	0	0
$565$	−6.87290	−0.289145
$566$	0	0
$567$	8.04186	0.337726
$568$	0	0
$569$	−7.56934	−0.317323	−0.158662	−	0.987333i	$-0.550718\pi$
$569$	−7.56934	−0.317323	−0.158662	+	0.987333i	$0.550718\pi$
$570$	0	0
$571$	−27.7348	−1.16066	−0.580332	−	0.814380i	$-0.697077\pi$
$571$	−27.7348	−1.16066	−0.580332	+	0.814380i	$0.697077\pi$
$572$	0	0
$573$	−74.4897	−3.11185
$574$	0	0
$575$	−9.04441	−0.377178
$576$	0	0
$577$	−22.9015	−0.953403	−0.476701	−	0.879065i	$-0.658168\pi$
$577$	−22.9015	−0.953403	−0.476701	+	0.879065i	$0.658168\pi$
$578$	0	0
$579$	11.5945	0.481852
$580$	0	0
$581$	−4.62930	−0.192056
$582$	0	0
$583$	12.1807	0.504471
$584$	0	0
$585$	9.71244	0.401560
$586$	0	0
$587$	−40.2821	−1.66262	−0.831311	−	0.555808i	$-0.812409\pi$
$587$	−40.2821	−1.66262	−0.831311	+	0.555808i	$0.812409\pi$
$588$	0	0
$589$	−46.6034	−1.92026
$590$	0	0
$591$	7.40297	0.304517
$592$	0	0
$593$	−34.9963	−1.43712	−0.718562	−	0.695463i	$-0.755199\pi$
$593$	−34.9963	−1.43712	−0.718562	+	0.695463i	$0.755199\pi$
$594$	0	0
$595$	11.2043	0.459330
$596$	0	0
$597$	50.3074	2.05895
$598$	0	0
$599$	9.42970	0.385287	0.192644	−	0.981269i	$-0.438294\pi$
$599$	9.42970	0.385287	0.192644	+	0.981269i	$0.438294\pi$
$600$	0	0
$601$	18.2613	0.744892	0.372446	−	0.928054i	$-0.378519\pi$
$601$	18.2613	0.744892	0.372446	+	0.928054i	$0.378519\pi$
$602$	0	0
$603$	−66.3008	−2.69998
$604$	0	0
$605$	1.64834	0.0670146
$606$	0	0
$607$	−3.35809	−0.136301	−0.0681503	−	0.997675i	$-0.521710\pi$
$607$	−3.35809	−0.136301	−0.0681503	+	0.997675i	$0.521710\pi$
$608$	0	0
$609$	−5.90154	−0.239143
$610$	0	0
$611$	−12.0658	−0.488130
$612$	0	0
$613$	14.6981	0.593650	0.296825	−	0.954932i	$-0.404072\pi$
$613$	14.6981	0.593650	0.296825	+	0.954932i	$0.404072\pi$
$614$	0	0
$615$	−3.52663	−0.142207
$616$	0	0
$617$	22.2780	0.896877	0.448439	−	0.893814i	$-0.351980\pi$
$617$	22.2780	0.896877	0.448439	+	0.893814i	$0.351980\pi$
$618$	0	0
$619$	−45.8421	−1.84255	−0.921274	−	0.388914i	$-0.872850\pi$
$619$	−45.8421	−1.84255	−0.921274	+	0.388914i	$0.872850\pi$
$620$	0	0
$621$	34.1682	1.37112
$622$	0	0
$623$	−0.496767	−0.0199026
$624$	0	0
$625$	−8.37320	−0.334928
$626$	0	0
$627$	−14.6404	−0.584683
$628$	0	0
$629$	−72.6968	−2.89861
$630$	0	0
$631$	16.0635	0.639477	0.319739	−	0.947506i	$-0.396405\pi$
$631$	16.0635	0.639477	0.319739	+	0.947506i	$0.396405\pi$
$632$	0	0
$633$	−47.3665	−1.88265
$634$	0	0
$635$	24.9427	0.989820
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−37.4908	−1.48311
$640$	0	0
$641$	3.22860	0.127522	0.0637610	−	0.997965i	$-0.479690\pi$
$641$	3.22860	0.127522	0.0637610	+	0.997965i	$0.479690\pi$
$642$	0	0
$643$	−10.7330	−0.423267	−0.211634	−	0.977349i	$-0.567878\pi$
$643$	−10.7330	−0.423267	−0.211634	+	0.977349i	$0.567878\pi$
$644$	0	0
$645$	30.3998	1.19699
$646$	0	0
$647$	−0.544527	−0.0214076	−0.0107038	−	0.999943i	$-0.503407\pi$
$647$	−0.544527	−0.0214076	−0.0107038	+	0.999943i	$0.503407\pi$
$648$	0	0
$649$	−2.55616	−0.100338
$650$	0	0
$651$	28.3058	1.10939
$652$	0	0
$653$	11.0985	0.434318	0.217159	−	0.976136i	$-0.430321\pi$
$653$	11.0985	0.434318	0.217159	+	0.976136i	$0.430321\pi$
$654$	0	0
$655$	7.30648	0.285488
$656$	0	0
$657$	70.5907	2.75400
$658$	0	0
$659$	−24.3204	−0.947387	−0.473694	−	0.880690i	$-0.657080\pi$
$659$	−24.3204	−0.947387	−0.473694	+	0.880690i	$0.657080\pi$
$660$	0	0
$661$	31.8438	1.23858	0.619289	−	0.785163i	$-0.287421\pi$
$661$	31.8438	1.23858	0.619289	+	0.785163i	$0.287421\pi$
$662$	0	0
$663$	20.2695	0.787201
$664$	0	0
$665$	−8.09274	−0.313823
$666$	0	0
$667$	−7.84043	−0.303583
$668$	0	0
$669$	86.2027	3.33279
$670$	0	0
$671$	2.51108	0.0969393
$672$	0	0
$673$	−26.7813	−1.03234	−0.516171	−	0.856486i	$-0.672643\pi$
$673$	−26.7813	−1.03234	−0.516171	+	0.856486i	$0.672643\pi$
$674$	0	0
$675$	−19.6898	−0.757862
$676$	0	0
$677$	−34.7853	−1.33691	−0.668453	−	0.743754i	$-0.733043\pi$
$677$	−34.7853	−1.33691	−0.668453	+	0.743754i	$0.733043\pi$
$678$	0	0
$679$	−4.84522	−0.185943
$680$	0	0
$681$	−42.9761	−1.64685
$682$	0	0
$683$	43.5369	1.66589	0.832946	−	0.553354i	$-0.186652\pi$
$683$	43.5369	1.66589	0.832946	+	0.553354i	$0.186652\pi$
$684$	0	0
$685$	24.6014	0.939971
$686$	0	0
$687$	−25.3600	−0.967544
$688$	0	0
$689$	12.1807	0.464046
$690$	0	0
$691$	−36.6571	−1.39450	−0.697252	−	0.716826i	$-0.745594\pi$
$691$	−36.6571	−1.39450	−0.697252	+	0.716826i	$0.745594\pi$
$692$	0	0
$693$	5.89225	0.223828
$694$	0	0
$695$	−21.1437	−0.802026
$696$	0	0
$697$	−4.87689	−0.184726
$698$	0	0
$699$	−30.0117	−1.13515
$700$	0	0
$701$	50.3757	1.90267	0.951333	−	0.308166i	$-0.0997151\pi$
$701$	50.3757	1.90267	0.951333	+	0.308166i	$0.0997151\pi$
$702$	0	0
$703$	52.5082	1.98038
$704$	0	0
$705$	−59.3074	−2.23365
$706$	0	0
$707$	7.60638	0.286067
$708$	0	0
$709$	19.5034	0.732464	0.366232	−	0.930523i	$-0.380648\pi$
$709$	19.5034	0.732464	0.366232	+	0.930523i	$0.380648\pi$
$710$	0	0
$711$	75.9252	2.84742
$712$	0	0
$713$	37.6053	1.40833
$714$	0	0
$715$	1.64834	0.0616445
$716$	0	0
$717$	−49.4009	−1.84491
$718$	0	0
$719$	−41.2601	−1.53874	−0.769371	−	0.638803i	$-0.779430\pi$
$719$	−41.2601	−1.53874	−0.769371	+	0.638803i	$0.779430\pi$
$720$	0	0
$721$	−6.53211	−0.243268
$722$	0	0
$723$	14.6203	0.543733
$724$	0	0
$725$	4.51814	0.167799
$726$	0	0
$727$	19.3953	0.719332	0.359666	−	0.933081i	$-0.382891\pi$
$727$	19.3953	0.719332	0.359666	+	0.933081i	$0.382891\pi$
$728$	0	0
$729$	−29.7712	−1.10264
$730$	0	0
$731$	42.0391	1.55487
$732$	0	0
$733$	−5.64796	−0.208612	−0.104306	−	0.994545i	$-0.533262\pi$
$733$	−5.64796	−0.208612	−0.104306	+	0.994545i	$0.533262\pi$
$734$	0	0
$735$	4.91533	0.181305
$736$	0	0
$737$	−11.2522	−0.414480
$738$	0	0
$739$	−10.4507	−0.384436	−0.192218	−	0.981352i	$-0.561568\pi$
$739$	−10.4507	−0.384436	−0.192218	+	0.981352i	$0.561568\pi$
$740$	0	0
$741$	−14.6404	−0.537830
$742$	0	0
$743$	−14.7272	−0.540288	−0.270144	−	0.962820i	$-0.587071\pi$
$743$	−14.7272	−0.540288	−0.270144	+	0.962820i	$0.587071\pi$
$744$	0	0
$745$	−3.35123	−0.122780
$746$	0	0
$747$	−27.2770	−0.998013
$748$	0	0
$749$	−15.1109	−0.552139
$750$	0	0
$751$	24.4163	0.890965	0.445482	−	0.895291i	$-0.353032\pi$
$751$	24.4163	0.890965	0.445482	+	0.895291i	$0.353032\pi$
$752$	0	0
$753$	−43.3644	−1.58029
$754$	0	0
$755$	23.5036	0.855384
$756$	0	0
$757$	50.1914	1.82424	0.912118	−	0.409928i	$-0.134446\pi$
$757$	50.1914	1.82424	0.912118	+	0.409928i	$0.134446\pi$
$758$	0	0
$759$	11.8137	0.428810
$760$	0	0
$761$	−0.526020	−0.0190682	−0.00953411	−	0.999955i	$-0.503035\pi$
$761$	−0.526020	−0.0190682	−0.00953411	+	0.999955i	$0.503035\pi$
$762$	0	0
$763$	−10.1957	−0.369110
$764$	0	0
$765$	66.0184	2.38690
$766$	0	0
$767$	−2.55616	−0.0922975
$768$	0	0
$769$	40.6516	1.46593	0.732967	−	0.680264i	$-0.238135\pi$
$769$	40.6516	1.46593	0.732967	+	0.680264i	$0.238135\pi$
$770$	0	0
$771$	−38.6262	−1.39109
$772$	0	0
$773$	−47.5630	−1.71072	−0.855361	−	0.518032i	$-0.826665\pi$
$773$	−47.5630	−1.71072	−0.855361	+	0.518032i	$0.826665\pi$
$774$	0	0
$775$	−21.6705	−0.778428
$776$	0	0
$777$	−31.8922	−1.14413
$778$	0	0
$779$	3.52253	0.126208
$780$	0	0
$781$	−6.36273	−0.227677
$782$	0	0
$783$	−17.0687	−0.609987
$784$	0	0
$785$	−20.4601	−0.730254
$786$	0	0
$787$	−39.7722	−1.41772	−0.708862	−	0.705347i	$-0.750791\pi$
$787$	−39.7722	−1.41772	−0.708862	+	0.705347i	$0.750791\pi$
$788$	0	0
$789$	67.0809	2.38814
$790$	0	0
$791$	−4.16958	−0.148253
$792$	0	0
$793$	2.51108	0.0891712
$794$	0	0
$795$	59.8720	2.12344
$796$	0	0
$797$	36.3815	1.28870	0.644349	−	0.764732i	$-0.277128\pi$
$797$	36.3815	1.28870	0.644349	+	0.764732i	$0.277128\pi$
$798$	0	0
$799$	−82.0148	−2.90147
$800$	0	0
$801$	−2.92708	−0.103423
$802$	0	0
$803$	11.9803	0.422774
$804$	0	0
$805$	6.53022	0.230160
$806$	0	0
$807$	23.3860	0.823227
$808$	0	0
$809$	−19.3673	−0.680918	−0.340459	−	0.940259i	$-0.610582\pi$
$809$	−19.3673	−0.680918	−0.340459	+	0.940259i	$0.610582\pi$
$810$	0	0
$811$	7.76478	0.272658	0.136329	−	0.990664i	$-0.456470\pi$
$811$	7.76478	0.272658	0.136329	+	0.990664i	$0.456470\pi$
$812$	0	0
$813$	−74.7305	−2.62091
$814$	0	0
$815$	15.7171	0.550545
$816$	0	0
$817$	−30.3645	−1.06232
$818$	0	0
$819$	5.89225	0.205892
$820$	0	0
$821$	−11.4627	−0.400052	−0.200026	−	0.979791i	$-0.564103\pi$
$821$	−11.4627	−0.400052	−0.200026	+	0.979791i	$0.564103\pi$
$822$	0	0
$823$	−15.3045	−0.533480	−0.266740	−	0.963768i	$-0.585947\pi$
$823$	−15.3045	−0.533480	−0.266740	+	0.963768i	$0.585947\pi$
$824$	0	0
$825$	−6.80779	−0.237017
$826$	0	0
$827$	19.3272	0.672071	0.336036	−	0.941849i	$-0.390914\pi$
$827$	19.3272	0.672071	0.336036	+	0.941849i	$0.390914\pi$
$828$	0	0
$829$	8.37852	0.290998	0.145499	−	0.989358i	$-0.453521\pi$
$829$	8.37852	0.290998	0.145499	+	0.989358i	$0.453521\pi$
$830$	0	0
$831$	51.0688	1.77156
$832$	0	0
$833$	6.79730	0.235512
$834$	0	0
$835$	11.6818	0.404265
$836$	0	0
$837$	81.8674	2.82975
$838$	0	0
$839$	23.5821	0.814145	0.407073	−	0.913396i	$-0.366550\pi$
$839$	23.5821	0.814145	0.407073	+	0.913396i	$0.366550\pi$
$840$	0	0
$841$	−25.0833	−0.864942
$842$	0	0
$843$	45.9740	1.58343
$844$	0	0
$845$	1.64834	0.0567047
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	41.6827	1.43055
$850$	0	0
$851$	−42.3701	−1.45243
$852$	0	0
$853$	−54.9459	−1.88131	−0.940656	−	0.339362i	$-0.889789\pi$
$853$	−54.9459	−1.88131	−0.940656	+	0.339362i	$0.889789\pi$
$854$	0	0
$855$	−47.6845	−1.63077
$856$	0	0
$857$	−8.66416	−0.295962	−0.147981	−	0.988990i	$-0.547277\pi$
$857$	−8.66416	−0.295962	−0.147981	+	0.988990i	$0.547277\pi$
$858$	0	0
$859$	−8.85864	−0.302253	−0.151126	−	0.988514i	$-0.548290\pi$
$859$	−8.85864	−0.302253	−0.151126	+	0.988514i	$0.548290\pi$
$860$	0	0
$861$	−2.13950	−0.0729140
$862$	0	0
$863$	−27.5872	−0.939078	−0.469539	−	0.882912i	$-0.655580\pi$
$863$	−27.5872	−0.939078	−0.469539	+	0.882912i	$0.655580\pi$
$864$	0	0
$865$	−14.7637	−0.501981
$866$	0	0
$867$	87.0838	2.95752
$868$	0	0
$869$	12.8856	0.437114
$870$	0	0
$871$	−11.2522	−0.381267
$872$	0	0
$873$	−28.5493	−0.966246
$874$	0	0
$875$	−12.0048	−0.405837
$876$	0	0
$877$	−3.59159	−0.121279	−0.0606397	−	0.998160i	$-0.519314\pi$
$877$	−3.59159	−0.121279	−0.0606397	+	0.998160i	$0.519314\pi$
$878$	0	0
$879$	49.4306	1.66725
$880$	0	0
$881$	−19.1283	−0.644447	−0.322224	−	0.946664i	$-0.604430\pi$
$881$	−19.1283	−0.644447	−0.322224	+	0.946664i	$0.604430\pi$
$882$	0	0
$883$	51.6616	1.73855	0.869276	−	0.494328i	$-0.164586\pi$
$883$	51.6616	1.73855	0.869276	+	0.494328i	$0.164586\pi$
$884$	0	0
$885$	−12.5644	−0.422347
$886$	0	0
$887$	37.9610	1.27461	0.637303	−	0.770614i	$-0.280050\pi$
$887$	37.9610	1.27461	0.637303	+	0.770614i	$0.280050\pi$
$888$	0	0
$889$	15.1320	0.507511
$890$	0	0
$891$	8.04186	0.269413
$892$	0	0
$893$	59.2385	1.98234
$894$	0	0
$895$	−26.9365	−0.900388
$896$	0	0
$897$	11.8137	0.394448
$898$	0	0
$899$	−18.7858	−0.626540
$900$	0	0
$901$	82.7956	2.75832
$902$	0	0
$903$	18.4426	0.613733
$904$	0	0
$905$	−3.75033	−0.124665
$906$	0	0
$907$	−31.2698	−1.03830	−0.519148	−	0.854684i	$-0.673751\pi$
$907$	−31.2698	−1.03830	−0.519148	+	0.854684i	$0.673751\pi$
$908$	0	0
$909$	44.8187	1.48654
$910$	0	0
$911$	−31.6094	−1.04727	−0.523633	−	0.851944i	$-0.675424\pi$
$911$	−31.6094	−1.04727	−0.523633	+	0.851944i	$0.675424\pi$
$912$	0	0
$913$	−4.62930	−0.153207
$914$	0	0
$915$	12.3428	0.408041
$916$	0	0
$917$	4.43263	0.146378
$918$	0	0
$919$	−5.33101	−0.175854	−0.0879269	−	0.996127i	$-0.528024\pi$
$919$	−5.33101	−0.175854	−0.0879269	+	0.996127i	$0.528024\pi$
$920$	0	0
$921$	49.1719	1.62027
$922$	0	0
$923$	−6.36273	−0.209432
$924$	0	0
$925$	24.4162	0.802801
$926$	0	0
$927$	−38.4888	−1.26414
$928$	0	0
$929$	−21.9623	−0.720558	−0.360279	−	0.932845i	$-0.617319\pi$
$929$	−21.9623	−0.720558	−0.360279	+	0.932845i	$0.617319\pi$
$930$	0	0
$931$	−4.90963	−0.160906
$932$	0	0
$933$	33.7077	1.10354
$934$	0	0
$935$	11.2043	0.366419
$936$	0	0
$937$	−57.0726	−1.86448	−0.932241	−	0.361839i	$-0.882149\pi$
$937$	−57.0726	−1.86448	−0.932241	+	0.361839i	$0.882149\pi$
$938$	0	0
$939$	9.58733	0.312871
$940$	0	0
$941$	−4.57465	−0.149129	−0.0745646	−	0.997216i	$-0.523757\pi$
$941$	−4.57465	−0.149129	−0.0745646	+	0.997216i	$0.523757\pi$
$942$	0	0
$943$	−2.84241	−0.0925617
$944$	0	0
$945$	14.2164	0.462459
$946$	0	0
$947$	−3.10700	−0.100964	−0.0504820	−	0.998725i	$-0.516076\pi$
$947$	−3.10700	−0.100964	−0.0504820	+	0.998725i	$0.516076\pi$
$948$	0	0
$949$	11.9803	0.388895
$950$	0	0
$951$	−49.0220	−1.58965
$952$	0	0
$953$	−8.42656	−0.272963	−0.136482	−	0.990643i	$-0.543579\pi$
$953$	−8.42656	−0.272963	−0.136482	+	0.990643i	$0.543579\pi$
$954$	0	0
$955$	−41.1754	−1.33240
$956$	0	0
$957$	−5.90154	−0.190770
$958$	0	0
$959$	14.9249	0.481951
$960$	0	0
$961$	59.1029	1.90654
$962$	0	0
$963$	−89.0369	−2.86917
$964$	0	0
$965$	6.40906	0.206315
$966$	0	0
$967$	3.24960	0.104500	0.0522500	−	0.998634i	$-0.483361\pi$
$967$	3.24960	0.104500	0.0522500	+	0.998634i	$0.483361\pi$
$968$	0	0
$969$	−99.5155	−3.19690
$970$	0	0
$971$	−22.6511	−0.726909	−0.363455	−	0.931612i	$-0.618403\pi$
$971$	−22.6511	−0.726909	−0.363455	+	0.931612i	$0.618403\pi$
$972$	0	0
$973$	−12.8272	−0.411223
$974$	0	0
$975$	−6.80779	−0.218024
$976$	0	0
$977$	13.9652	0.446787	0.223394	−	0.974728i	$-0.428286\pi$
$977$	13.9652	0.446787	0.223394	+	0.974728i	$0.428286\pi$
$978$	0	0
$979$	−0.496767	−0.0158767
$980$	0	0
$981$	−60.0757	−1.91807
$982$	0	0
$983$	−37.9390	−1.21006	−0.605032	−	0.796201i	$-0.706840\pi$
$983$	−37.9390	−1.21006	−0.605032	+	0.796201i	$0.706840\pi$
$984$	0	0
$985$	4.09211	0.130385
$986$	0	0
$987$	−35.9800	−1.14526
$988$	0	0
$989$	24.5018	0.779111
$990$	0	0
$991$	17.6116	0.559452	0.279726	−	0.960080i	$-0.409756\pi$
$991$	17.6116	0.559452	0.279726	+	0.960080i	$0.409756\pi$
$992$	0	0
$993$	−34.7472	−1.10267
$994$	0	0
$995$	27.8082	0.881581
$996$	0	0
$997$	−16.6683	−0.527889	−0.263944	−	0.964538i	$-0.585024\pi$
$997$	−16.6683	−0.527889	−0.263944	+	0.964538i	$0.585024\pi$
$998$	0	0
$999$	−92.2402	−2.91835

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.x.1.11	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.x.1.11	✓	12	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+2.98199 q^{3} +1.64834 q^{5} +1.00000 q^{7} +5.89225 q^{9} +O(q^{10})\)	\(q+2.98199 q^{3} +1.64834 q^{5} +1.00000 q^{7} +5.89225 q^{9} +1.00000 q^{11} +1.00000 q^{13} +4.91533 q^{15} +6.79730 q^{17} -4.90963 q^{19} +2.98199 q^{21} +3.96169 q^{23} -2.28297 q^{25} +8.62465 q^{27} -1.97906 q^{29} +9.49225 q^{31} +2.98199 q^{33} +1.64834 q^{35} -10.6949 q^{37} +2.98199 q^{39} -0.717475 q^{41} +6.18468 q^{43} +9.71244 q^{45} -12.0658 q^{47} +1.00000 q^{49} +20.2695 q^{51} +12.1807 q^{53} +1.64834 q^{55} -14.6404 q^{57} -2.55616 q^{59} +2.51108 q^{61} +5.89225 q^{63} +1.64834 q^{65} -11.2522 q^{67} +11.8137 q^{69} -6.36273 q^{71} +11.9803 q^{73} -6.80779 q^{75} +1.00000 q^{77} +12.8856 q^{79} +8.04186 q^{81} -4.62930 q^{83} +11.2043 q^{85} -5.90154 q^{87} -0.496767 q^{89} +1.00000 q^{91} +28.3058 q^{93} -8.09274 q^{95} -4.84522 q^{97} +5.89225 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9}+O(q^{10}) \) 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9	\( 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9} + 12 q^{11} + 12 q^{13} - 3 q^{15} + 16 q^{17} - 2 q^{19} + 4 q^{21} + 9 q^{23} + 14 q^{25} + 7 q^{27} + 15 q^{29} + 10 q^{31} + 4 q^{33} + 6 q^{35} + 18 q^{37} + 4 q^{39} + 24 q^{41} + 15 q^{45} + 5 q^{47} + 12 q^{49} + 4 q^{51} + 15 q^{53} + 6 q^{55} - 4 q^{57} + 15 q^{59} + 17 q^{61} + 14 q^{63} + 6 q^{65} - 7 q^{67} + 9 q^{71} + 32 q^{73} - 8 q^{75} + 12 q^{77} + 20 q^{79} - 4 q^{81} - 5 q^{83} + 25 q^{85} + 19 q^{87} + 16 q^{89} + 12 q^{91} + 21 q^{93} + 8 q^{95} + 10 q^{97} + 14 q^{99}+O(q^{100}) \) 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9 + 12 * q^11 + 12 * q^13 - 3 * q^15 + 16 * q^17 - 2 * q^19 + 4 * q^21 + 9 * q^23 + 14 * q^25 + 7 * q^27 + 15 * q^29 + 10 * q^31 + 4 * q^33 + 6 * q^35 + 18 * q^37 + 4 * q^39 + 24 * q^41 + 15 * q^45 + 5 * q^47 + 12 * q^49 + 4 * q^51 + 15 * q^53 + 6 * q^55 - 4 * q^57 + 15 * q^59 + 17 * q^61 + 14 * q^63 + 6 * q^65 - 7 * q^67 + 9 * q^71 + 32 * q^73 - 8 * q^75 + 12 * q^77 + 20 * q^79 - 4 * q^81 - 5 * q^83 + 25 * q^85 + 19 * q^87 + 16 * q^89 + 12 * q^91 + 21 * q^93 + 8 * q^95 + 10 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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