LMFDB - Embedded newform 8008.2.a.n.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:	$0$
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} - 3x^{8} - 15x^{7} + 45x^{6} + 64x^{5} - 201x^{4} - 63x^{3} + 282x^{2} + 3x - 116 $ x^9 - 3x^8 - 15x^7 + 45x^6 + 64x^5 - 201x^4 - 63x^3 + 282x^2 + 3x - 116
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-2.29224$ 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)

$f(q)$	$=$	$q+2.29224 q^{3} +3.91599 q^{5} -1.00000 q^{7} +2.25436 q^{9} +O(q^{10})$	$q+2.29224 q^{3} +3.91599 q^{5} -1.00000 q^{7} +2.25436 q^{9} +1.00000 q^{11} -1.00000 q^{13} +8.97640 q^{15} +5.29356 q^{17} +2.52623 q^{19} -2.29224 q^{21} -4.13217 q^{23} +10.3350 q^{25} -1.70918 q^{27} +6.65785 q^{29} -3.98607 q^{31} +2.29224 q^{33} -3.91599 q^{35} +2.57423 q^{37} -2.29224 q^{39} +4.72390 q^{41} -7.81748 q^{43} +8.82808 q^{45} +0.823244 q^{47} +1.00000 q^{49} +12.1341 q^{51} +0.474003 q^{53} +3.91599 q^{55} +5.79073 q^{57} +9.39011 q^{59} +6.43606 q^{61} -2.25436 q^{63} -3.91599 q^{65} -13.5133 q^{67} -9.47193 q^{69} +11.2332 q^{71} +10.9199 q^{73} +23.6903 q^{75} -1.00000 q^{77} +14.4865 q^{79} -10.6809 q^{81} -15.6479 q^{83} +20.7296 q^{85} +15.2614 q^{87} -16.0211 q^{89} +1.00000 q^{91} -9.13704 q^{93} +9.89271 q^{95} +1.43297 q^{97} +2.25436 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q - 3 * q^3 + q^5 - 9 * q^7 + 12 * q^9	$ 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} + 11 q^{15} + 7 q^{17} - 17 q^{19} + 3 q^{21} + 11 q^{23} + 18 q^{25} - 9 q^{27} + 9 q^{29} - 8 q^{31} - 3 q^{33} - q^{35} + 2 q^{37} + 3 q^{39} + 18 q^{41} + 7 q^{43} + 5 q^{45} + 15 q^{47} + 9 q^{49} - 7 q^{51} - 4 q^{53} + q^{55} + 22 q^{57} - 23 q^{59} + 12 q^{61} - 12 q^{63} - q^{65} - 16 q^{67} - 32 q^{69} - 6 q^{71} + 4 q^{73} - 14 q^{75} - 9 q^{77} + 21 q^{79} + 5 q^{81} - 16 q^{83} + 53 q^{85} + 41 q^{87} + 5 q^{89} + 9 q^{91} + 29 q^{93} + 19 q^{95} + 18 q^{97} + 12 q^{99}+O(q^{100}) $ 9 * q - 3 * q^3 + q^5 - 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 + 11 * q^15 + 7 * q^17 - 17 * q^19 + 3 * q^21 + 11 * q^23 + 18 * q^25 - 9 * q^27 + 9 * q^29 - 8 * q^31 - 3 * q^33 - q^35 + 2 * q^37 + 3 * q^39 + 18 * q^41 + 7 * q^43 + 5 * q^45 + 15 * q^47 + 9 * q^49 - 7 * q^51 - 4 * q^53 + q^55 + 22 * q^57 - 23 * q^59 + 12 * q^61 - 12 * q^63 - q^65 - 16 * q^67 - 32 * q^69 - 6 * q^71 + 4 * q^73 - 14 * q^75 - 9 * q^77 + 21 * q^79 + 5 * q^81 - 16 * q^83 + 53 * q^85 + 41 * q^87 + 5 * q^89 + 9 * q^91 + 29 * q^93 + 19 * q^95 + 18 * q^97 + 12 * 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
$3$	2.29224	1.32343	0.661713	−	0.749758i	$-0.269830\pi$
$3$	2.29224	1.32343	0.661713	+	0.749758i	$0.269830\pi$
$4$	0	0
$5$	3.91599	1.75129	0.875643	−	0.482959i	$-0.160438\pi$
$5$	3.91599	1.75129	0.875643	+	0.482959i	$0.160438\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.25436	0.751455
$10$	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
$15$	8.97640	2.31770
$16$	0	0
$17$	5.29356	1.28388	0.641939	−	0.766756i	$-0.278130\pi$
$17$	5.29356	1.28388	0.641939	+	0.766756i	$0.278130\pi$
$18$	0	0
$19$	2.52623	0.579557	0.289779	−	0.957094i	$-0.406418\pi$
$19$	2.52623	0.579557	0.289779	+	0.957094i	$0.406418\pi$
$20$	0	0
$21$	−2.29224	−0.500208
$22$	0	0
$23$	−4.13217	−0.861617	−0.430809	−	0.902443i	$-0.641772\pi$
$23$	−4.13217	−0.861617	−0.430809	+	0.902443i	$0.641772\pi$
$24$	0	0
$25$	10.3350	2.06700
$26$	0	0
$27$	−1.70918	−0.328931
$28$	0	0
$29$	6.65785	1.23633	0.618166	−	0.786048i	$-0.287876\pi$
$29$	6.65785	1.23633	0.618166	+	0.786048i	$0.287876\pi$
$30$	0	0
$31$	−3.98607	−0.715920	−0.357960	−	0.933737i	$-0.616528\pi$
$31$	−3.98607	−0.715920	−0.357960	+	0.933737i	$0.616528\pi$
$32$	0	0
$33$	2.29224	0.399028
$34$	0	0
$35$	−3.91599	−0.661924
$36$	0	0
$37$	2.57423	0.423201	0.211601	−	0.977356i	$-0.432132\pi$
$37$	2.57423	0.423201	0.211601	+	0.977356i	$0.432132\pi$
$38$	0	0
$39$	−2.29224	−0.367052
$40$	0	0
$41$	4.72390	0.737749	0.368874	−	0.929479i	$-0.379743\pi$
$41$	4.72390	0.737749	0.368874	+	0.929479i	$0.379743\pi$
$42$	0	0
$43$	−7.81748	−1.19215	−0.596077	−	0.802927i	$-0.703275\pi$
$43$	−7.81748	−1.19215	−0.596077	+	0.802927i	$0.703275\pi$
$44$	0	0
$45$	8.82808	1.31601
$46$	0	0
$47$	0.823244	0.120082	0.0600412	−	0.998196i	$-0.480877\pi$
$47$	0.823244	0.120082	0.0600412	+	0.998196i	$0.480877\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	12.1341	1.69912
$52$	0	0
$53$	0.474003	0.0651093	0.0325546	−	0.999470i	$-0.489636\pi$
$53$	0.474003	0.0651093	0.0325546	+	0.999470i	$0.489636\pi$
$54$	0	0
$55$	3.91599	0.528033
$56$	0	0
$57$	5.79073	0.767001
$58$	0	0
$59$	9.39011	1.22249	0.611244	−	0.791442i	$-0.290669\pi$
$59$	9.39011	1.22249	0.611244	+	0.791442i	$0.290669\pi$
$60$	0	0
$61$	6.43606	0.824053	0.412026	−	0.911172i	$-0.364821\pi$
$61$	6.43606	0.824053	0.412026	+	0.911172i	$0.364821\pi$
$62$	0	0
$63$	−2.25436	−0.284023
$64$	0	0
$65$	−3.91599	−0.485719
$66$	0	0
$67$	−13.5133	−1.65091	−0.825457	−	0.564465i	$-0.809083\pi$
$67$	−13.5133	−1.65091	−0.825457	+	0.564465i	$0.809083\pi$
$68$	0	0
$69$	−9.47193	−1.14029
$70$	0	0
$71$	11.2332	1.33313	0.666567	−	0.745445i	$-0.267763\pi$
$71$	11.2332	1.33313	0.666567	+	0.745445i	$0.267763\pi$
$72$	0	0
$73$	10.9199	1.27808	0.639041	−	0.769173i	$-0.279332\pi$
$73$	10.9199	1.27808	0.639041	+	0.769173i	$0.279332\pi$
$74$	0	0
$75$	23.6903	2.73552
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	14.4865	1.62986	0.814932	−	0.579557i	$-0.196774\pi$
$79$	14.4865	1.62986	0.814932	+	0.579557i	$0.196774\pi$
$80$	0	0
$81$	−10.6809	−1.18677
$82$	0	0
$83$	−15.6479	−1.71758	−0.858792	−	0.512324i	$-0.828785\pi$
$83$	−15.6479	−1.71758	−0.858792	+	0.512324i	$0.828785\pi$
$84$	0	0
$85$	20.7296	2.24844
$86$	0	0
$87$	15.2614	1.63619
$88$	0	0
$89$	−16.0211	−1.69824	−0.849119	−	0.528201i	$-0.822867\pi$
$89$	−16.0211	−1.69824	−0.849119	+	0.528201i	$0.822867\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−9.13704	−0.947467
$94$	0	0
$95$	9.89271	1.01497
$96$	0	0
$97$	1.43297	0.145496	0.0727481	−	0.997350i	$-0.476823\pi$
$97$	1.43297	0.145496	0.0727481	+	0.997350i	$0.476823\pi$
$98$	0	0
$99$	2.25436	0.226572
$100$	0	0
$101$	−9.90176	−0.985262	−0.492631	−	0.870238i	$-0.663965\pi$
$101$	−9.90176	−0.985262	−0.492631	+	0.870238i	$0.663965\pi$
$102$	0	0
$103$	−7.35242	−0.724456	−0.362228	−	0.932090i	$-0.617984\pi$
$103$	−7.35242	−0.724456	−0.362228	+	0.932090i	$0.617984\pi$
$104$	0	0
$105$	−8.97640	−0.876007
$106$	0	0
$107$	4.93360	0.476949	0.238475	−	0.971149i	$-0.423353\pi$
$107$	4.93360	0.476949	0.238475	+	0.971149i	$0.423353\pi$
$108$	0	0
$109$	−10.7579	−1.03042	−0.515209	−	0.857065i	$-0.672286\pi$
$109$	−10.7579	−1.03042	−0.515209	+	0.857065i	$0.672286\pi$
$110$	0	0
$111$	5.90076	0.560075
$112$	0	0
$113$	−4.60386	−0.433094	−0.216547	−	0.976272i	$-0.569480\pi$
$113$	−4.60386	−0.433094	−0.216547	+	0.976272i	$0.569480\pi$
$114$	0	0
$115$	−16.1816	−1.50894
$116$	0	0
$117$	−2.25436	−0.208416
$118$	0	0
$119$	−5.29356	−0.485260
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	10.8283	0.976355
$124$	0	0
$125$	20.8919	1.86863
$126$	0	0
$127$	6.56009	0.582114	0.291057	−	0.956706i	$-0.405993\pi$
$127$	6.56009	0.582114	0.291057	+	0.956706i	$0.405993\pi$
$128$	0	0
$129$	−17.9195	−1.57773
$130$	0	0
$131$	1.85601	0.162160	0.0810801	−	0.996708i	$-0.474163\pi$
$131$	1.85601	0.162160	0.0810801	+	0.996708i	$0.474163\pi$
$132$	0	0
$133$	−2.52623	−0.219052
$134$	0	0
$135$	−6.69313	−0.576053
$136$	0	0
$137$	−0.862991	−0.0737303	−0.0368651	−	0.999320i	$-0.511737\pi$
$137$	−0.862991	−0.0737303	−0.0368651	+	0.999320i	$0.511737\pi$
$138$	0	0
$139$	−7.44853	−0.631776	−0.315888	−	0.948797i	$-0.602302\pi$
$139$	−7.44853	−0.631776	−0.315888	+	0.948797i	$0.602302\pi$
$140$	0	0
$141$	1.88707	0.158920
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	26.0721	2.16517
$146$	0	0
$147$	2.29224	0.189061
$148$	0	0
$149$	8.28649	0.678856	0.339428	−	0.940632i	$-0.389767\pi$
$149$	8.28649	0.678856	0.339428	+	0.940632i	$0.389767\pi$
$150$	0	0
$151$	7.66565	0.623822	0.311911	−	0.950111i	$-0.399031\pi$
$151$	7.66565	0.623822	0.311911	+	0.950111i	$0.399031\pi$
$152$	0	0
$153$	11.9336	0.964775
$154$	0	0
$155$	−15.6094	−1.25378
$156$	0	0
$157$	23.6516	1.88760	0.943800	−	0.330518i	$-0.107223\pi$
$157$	23.6516	1.88760	0.943800	+	0.330518i	$0.107223\pi$
$158$	0	0
$159$	1.08653	0.0861672
$160$	0	0
$161$	4.13217	0.325661
$162$	0	0
$163$	−4.20451	−0.329323	−0.164662	−	0.986350i	$-0.552653\pi$
$163$	−4.20451	−0.329323	−0.164662	+	0.986350i	$0.552653\pi$
$164$	0	0
$165$	8.97640	0.698812
$166$	0	0
$167$	21.6773	1.67744	0.838721	−	0.544561i	$-0.183304\pi$
$167$	21.6773	1.67744	0.838721	+	0.544561i	$0.183304\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	5.69505	0.435511
$172$	0	0
$173$	−5.65944	−0.430280	−0.215140	−	0.976583i	$-0.569021\pi$
$173$	−5.65944	−0.430280	−0.215140	+	0.976583i	$0.569021\pi$
$174$	0	0
$175$	−10.3350	−0.781254
$176$	0	0
$177$	21.5244	1.61787
$178$	0	0
$179$	−9.09626	−0.679887	−0.339943	−	0.940446i	$-0.610408\pi$
$179$	−9.09626	−0.679887	−0.339943	+	0.940446i	$0.610408\pi$
$180$	0	0
$181$	−26.4678	−1.96733	−0.983667	−	0.179998i	$-0.942391\pi$
$181$	−26.4678	−1.96733	−0.983667	+	0.179998i	$0.942391\pi$
$182$	0	0
$183$	14.7530	1.09057
$184$	0	0
$185$	10.0807	0.741146
$186$	0	0
$187$	5.29356	0.387104
$188$	0	0
$189$	1.70918	0.124324
$190$	0	0
$191$	−15.7289	−1.13810	−0.569052	−	0.822301i	$-0.692690\pi$
$191$	−15.7289	−1.13810	−0.569052	+	0.822301i	$0.692690\pi$
$192$	0	0
$193$	7.59521	0.546715	0.273358	−	0.961912i	$-0.411866\pi$
$193$	7.59521	0.546715	0.273358	+	0.961912i	$0.411866\pi$
$194$	0	0
$195$	−8.97640	−0.642813
$196$	0	0
$197$	11.6302	0.828615	0.414308	−	0.910137i	$-0.364024\pi$
$197$	11.6302	0.828615	0.414308	+	0.910137i	$0.364024\pi$
$198$	0	0
$199$	−2.61756	−0.185554	−0.0927769	−	0.995687i	$-0.529574\pi$
$199$	−2.61756	−0.185554	−0.0927769	+	0.995687i	$0.529574\pi$
$200$	0	0
$201$	−30.9758	−2.18486
$202$	0	0
$203$	−6.65785	−0.467289
$204$	0	0
$205$	18.4987	1.29201
$206$	0	0
$207$	−9.31542	−0.647466
$208$	0	0
$209$	2.52623	0.174743
$210$	0	0
$211$	10.4912	0.722244	0.361122	−	0.932519i	$-0.382394\pi$
$211$	10.4912	0.722244	0.361122	+	0.932519i	$0.382394\pi$
$212$	0	0
$213$	25.7492	1.76430
$214$	0	0
$215$	−30.6132	−2.08780
$216$	0	0
$217$	3.98607	0.270592
$218$	0	0
$219$	25.0311	1.69145
$220$	0	0
$221$	−5.29356	−0.356083
$222$	0	0
$223$	12.0020	0.803715	0.401857	−	0.915702i	$-0.368365\pi$
$223$	12.0020	0.803715	0.401857	+	0.915702i	$0.368365\pi$
$224$	0	0
$225$	23.2989	1.55326
$226$	0	0
$227$	−3.60882	−0.239526	−0.119763	−	0.992803i	$-0.538213\pi$
$227$	−3.60882	−0.239526	−0.119763	+	0.992803i	$0.538213\pi$
$228$	0	0
$229$	8.09460	0.534906	0.267453	−	0.963571i	$-0.413818\pi$
$229$	8.09460	0.534906	0.267453	+	0.963571i	$0.413818\pi$
$230$	0	0
$231$	−2.29224	−0.150818
$232$	0	0
$233$	5.11617	0.335171	0.167586	−	0.985858i	$-0.446403\pi$
$233$	5.11617	0.335171	0.167586	+	0.985858i	$0.446403\pi$
$234$	0	0
$235$	3.22382	0.210299
$236$	0	0
$237$	33.2066	2.15700
$238$	0	0
$239$	−5.77529	−0.373573	−0.186786	−	0.982401i	$-0.559807\pi$
$239$	−5.77529	−0.373573	−0.186786	+	0.982401i	$0.559807\pi$
$240$	0	0
$241$	−16.7524	−1.07912	−0.539559	−	0.841948i	$-0.681409\pi$
$241$	−16.7524	−1.07912	−0.539559	+	0.841948i	$0.681409\pi$
$242$	0	0
$243$	−19.3557	−1.24167
$244$	0	0
$245$	3.91599	0.250184
$246$	0	0
$247$	−2.52623	−0.160740
$248$	0	0
$249$	−35.8688	−2.27310
$250$	0	0
$251$	−14.7959	−0.933908	−0.466954	−	0.884282i	$-0.654649\pi$
$251$	−14.7959	−0.933908	−0.466954	+	0.884282i	$0.654649\pi$
$252$	0	0
$253$	−4.13217	−0.259787
$254$	0	0
$255$	47.5171	2.97564
$256$	0	0
$257$	17.3463	1.08203	0.541016	−	0.841012i	$-0.318040\pi$
$257$	17.3463	1.08203	0.541016	+	0.841012i	$0.318040\pi$
$258$	0	0
$259$	−2.57423	−0.159955
$260$	0	0
$261$	15.0092	0.929047
$262$	0	0
$263$	11.2871	0.695993	0.347996	−	0.937496i	$-0.386862\pi$
$263$	11.2871	0.695993	0.347996	+	0.937496i	$0.386862\pi$
$264$	0	0
$265$	1.85619	0.114025
$266$	0	0
$267$	−36.7243	−2.24749
$268$	0	0
$269$	−1.24078	−0.0756517	−0.0378259	−	0.999284i	$-0.512043\pi$
$269$	−1.24078	−0.0756517	−0.0378259	+	0.999284i	$0.512043\pi$
$270$	0	0
$271$	8.75377	0.531754	0.265877	−	0.964007i	$-0.414339\pi$
$271$	8.75377	0.531754	0.265877	+	0.964007i	$0.414339\pi$
$272$	0	0
$273$	2.29224	0.138733
$274$	0	0
$275$	10.3350	0.623225
$276$	0	0
$277$	−19.0520	−1.14472	−0.572361	−	0.820002i	$-0.693972\pi$
$277$	−19.0520	−1.14472	−0.572361	+	0.820002i	$0.693972\pi$
$278$	0	0
$279$	−8.98606	−0.537981
$280$	0	0
$281$	27.3417	1.63107	0.815534	−	0.578709i	$-0.196443\pi$
$281$	27.3417	1.63107	0.815534	+	0.578709i	$0.196443\pi$
$282$	0	0
$283$	25.9988	1.54547	0.772735	−	0.634729i	$-0.218888\pi$
$283$	25.9988	1.54547	0.772735	+	0.634729i	$0.218888\pi$
$284$	0	0
$285$	22.6765	1.34324
$286$	0	0
$287$	−4.72390	−0.278843
$288$	0	0
$289$	11.0218	0.648340
$290$	0	0
$291$	3.28471	0.192553
$292$	0	0
$293$	−20.2194	−1.18123	−0.590614	−	0.806954i	$-0.701114\pi$
$293$	−20.2194	−1.18123	−0.590614	+	0.806954i	$0.701114\pi$
$294$	0	0
$295$	36.7716	2.14093
$296$	0	0
$297$	−1.70918	−0.0991765
$298$	0	0
$299$	4.13217	0.238970
$300$	0	0
$301$	7.81748	0.450592
$302$	0	0
$303$	−22.6972	−1.30392
$304$	0	0
$305$	25.2036	1.44315
$306$	0	0
$307$	−10.4580	−0.596868	−0.298434	−	0.954430i	$-0.596464\pi$
$307$	−10.4580	−0.596868	−0.298434	+	0.954430i	$0.596464\pi$
$308$	0	0
$309$	−16.8535	−0.958763
$310$	0	0
$311$	−14.7905	−0.838693	−0.419347	−	0.907826i	$-0.637741\pi$
$311$	−14.7905	−0.838693	−0.419347	+	0.907826i	$0.637741\pi$
$312$	0	0
$313$	26.2194	1.48201	0.741005	−	0.671499i	$-0.234349\pi$
$313$	26.2194	1.48201	0.741005	+	0.671499i	$0.234349\pi$
$314$	0	0
$315$	−8.82808	−0.497406
$316$	0	0
$317$	−22.7090	−1.27546	−0.637732	−	0.770258i	$-0.720127\pi$
$317$	−22.7090	−1.27546	−0.637732	+	0.770258i	$0.720127\pi$
$318$	0	0
$319$	6.65785	0.372768
$320$	0	0
$321$	11.3090	0.631207
$322$	0	0
$323$	13.3728	0.744080
$324$	0	0
$325$	−10.3350	−0.573283
$326$	0	0
$327$	−24.6596	−1.36368
$328$	0	0
$329$	−0.823244	−0.0453869
$330$	0	0
$331$	−22.1276	−1.21624	−0.608121	−	0.793845i	$-0.708076\pi$
$331$	−22.1276	−1.21624	−0.608121	+	0.793845i	$0.708076\pi$
$332$	0	0
$333$	5.80326	0.318016
$334$	0	0
$335$	−52.9181	−2.89122
$336$	0	0
$337$	−21.5860	−1.17586	−0.587931	−	0.808911i	$-0.700057\pi$
$337$	−21.5860	−1.17586	−0.587931	+	0.808911i	$0.700057\pi$
$338$	0	0
$339$	−10.5531	−0.573168
$340$	0	0
$341$	−3.98607	−0.215858
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−37.0920	−1.99697
$346$	0	0
$347$	−11.7174	−0.629021	−0.314511	−	0.949254i	$-0.601840\pi$
$347$	−11.7174	−0.629021	−0.314511	+	0.949254i	$0.601840\pi$
$348$	0	0
$349$	−12.0852	−0.646906	−0.323453	−	0.946244i	$-0.604844\pi$
$349$	−12.0852	−0.646906	−0.323453	+	0.946244i	$0.604844\pi$
$350$	0	0
$351$	1.70918	0.0912291
$352$	0	0
$353$	−2.29821	−0.122321	−0.0611607	−	0.998128i	$-0.519480\pi$
$353$	−2.29821	−0.122321	−0.0611607	+	0.998128i	$0.519480\pi$
$354$	0	0
$355$	43.9891	2.33470
$356$	0	0
$357$	−12.1341	−0.642205
$358$	0	0
$359$	−10.3778	−0.547720	−0.273860	−	0.961770i	$-0.588301\pi$
$359$	−10.3778	−0.547720	−0.273860	+	0.961770i	$0.588301\pi$
$360$	0	0
$361$	−12.6182	−0.664113
$362$	0	0
$363$	2.29224	0.120311
$364$	0	0
$365$	42.7624	2.23829
$366$	0	0
$367$	9.53714	0.497835	0.248917	−	0.968525i	$-0.419925\pi$
$367$	9.53714	0.497835	0.248917	+	0.968525i	$0.419925\pi$
$368$	0	0
$369$	10.6494	0.554385
$370$	0	0
$371$	−0.474003	−0.0246090
$372$	0	0
$373$	31.9254	1.65304	0.826518	−	0.562910i	$-0.190318\pi$
$373$	31.9254	1.65304	0.826518	+	0.562910i	$0.190318\pi$
$374$	0	0
$375$	47.8892	2.47299
$376$	0	0
$377$	−6.65785	−0.342897
$378$	0	0
$379$	−9.17132	−0.471099	−0.235550	−	0.971862i	$-0.575689\pi$
$379$	−9.17132	−0.471099	−0.235550	+	0.971862i	$0.575689\pi$
$380$	0	0
$381$	15.0373	0.770384
$382$	0	0
$383$	3.89314	0.198930	0.0994651	−	0.995041i	$-0.468287\pi$
$383$	3.89314	0.198930	0.0994651	+	0.995041i	$0.468287\pi$
$384$	0	0
$385$	−3.91599	−0.199578
$386$	0	0
$387$	−17.6234	−0.895850
$388$	0	0
$389$	0.762615	0.0386661	0.0193331	−	0.999813i	$-0.493846\pi$
$389$	0.762615	0.0386661	0.0193331	+	0.999813i	$0.493846\pi$
$390$	0	0
$391$	−21.8739	−1.10621
$392$	0	0
$393$	4.25442	0.214607
$394$	0	0
$395$	56.7292	2.85436
$396$	0	0
$397$	−11.5906	−0.581715	−0.290858	−	0.956766i	$-0.593941\pi$
$397$	−11.5906	−0.581715	−0.290858	+	0.956766i	$0.593941\pi$
$398$	0	0
$399$	−5.79073	−0.289899
$400$	0	0
$401$	−3.05610	−0.152614	−0.0763071	−	0.997084i	$-0.524313\pi$
$401$	−3.05610	−0.152614	−0.0763071	+	0.997084i	$0.524313\pi$
$402$	0	0
$403$	3.98607	0.198560
$404$	0	0
$405$	−41.8265	−2.07837
$406$	0	0
$407$	2.57423	0.127600
$408$	0	0
$409$	−35.9091	−1.77559	−0.887795	−	0.460238i	$-0.847764\pi$
$409$	−35.9091	−1.77559	−0.887795	+	0.460238i	$0.847764\pi$
$410$	0	0
$411$	−1.97818	−0.0975765
$412$	0	0
$413$	−9.39011	−0.462057
$414$	0	0
$415$	−61.2773	−3.00798
$416$	0	0
$417$	−17.0738	−0.836108
$418$	0	0
$419$	−6.89573	−0.336878	−0.168439	−	0.985712i	$-0.553873\pi$
$419$	−6.89573	−0.336878	−0.168439	+	0.985712i	$0.553873\pi$
$420$	0	0
$421$	−20.0036	−0.974917	−0.487459	−	0.873146i	$-0.662076\pi$
$421$	−20.0036	−0.974917	−0.487459	+	0.873146i	$0.662076\pi$
$422$	0	0
$423$	1.85589	0.0902365
$424$	0	0
$425$	54.7090	2.65378
$426$	0	0
$427$	−6.43606	−0.311463
$428$	0	0
$429$	−2.29224	−0.110670
$430$	0	0
$431$	−37.6016	−1.81121	−0.905603	−	0.424126i	$-0.860581\pi$
$431$	−37.6016	−1.81121	−0.905603	+	0.424126i	$0.860581\pi$
$432$	0	0
$433$	5.69079	0.273482	0.136741	−	0.990607i	$-0.456337\pi$
$433$	5.69079	0.273482	0.136741	+	0.990607i	$0.456337\pi$
$434$	0	0
$435$	59.7635	2.86544
$436$	0	0
$437$	−10.4388	−0.499356
$438$	0	0
$439$	19.1665	0.914766	0.457383	−	0.889270i	$-0.348787\pi$
$439$	19.1665	0.914766	0.457383	+	0.889270i	$0.348787\pi$
$440$	0	0
$441$	2.25436	0.107351
$442$	0	0
$443$	−33.2787	−1.58112	−0.790559	−	0.612386i	$-0.790210\pi$
$443$	−33.2787	−1.58112	−0.790559	+	0.612386i	$0.790210\pi$
$444$	0	0
$445$	−62.7387	−2.97410
$446$	0	0
$447$	18.9946	0.898415
$448$	0	0
$449$	−38.2020	−1.80287	−0.901433	−	0.432919i	$-0.857483\pi$
$449$	−38.2020	−1.80287	−0.901433	+	0.432919i	$0.857483\pi$
$450$	0	0
$451$	4.72390	0.222440
$452$	0	0
$453$	17.5715	0.825582
$454$	0	0
$455$	3.91599	0.183585
$456$	0	0
$457$	30.5982	1.43132	0.715662	−	0.698447i	$-0.246125\pi$
$457$	30.5982	1.43132	0.715662	+	0.698447i	$0.246125\pi$
$458$	0	0
$459$	−9.04763	−0.422307
$460$	0	0
$461$	−18.4074	−0.857319	−0.428659	−	0.903466i	$-0.641014\pi$
$461$	−18.4074	−0.857319	−0.428659	+	0.903466i	$0.641014\pi$
$462$	0	0
$463$	12.8888	0.598993	0.299496	−	0.954097i	$-0.403181\pi$
$463$	12.8888	0.598993	0.299496	+	0.954097i	$0.403181\pi$
$464$	0	0
$465$	−35.7806	−1.65929
$466$	0	0
$467$	37.4568	1.73329	0.866647	−	0.498921i	$-0.166270\pi$
$467$	37.4568	1.73329	0.866647	+	0.498921i	$0.166270\pi$
$468$	0	0
$469$	13.5133	0.623987
$470$	0	0
$471$	54.2150	2.49810
$472$	0	0
$473$	−7.81748	−0.359448
$474$	0	0
$475$	26.1086	1.19795
$476$	0	0
$477$	1.06857	0.0489267
$478$	0	0
$479$	−17.5064	−0.799889	−0.399945	−	0.916539i	$-0.630971\pi$
$479$	−17.5064	−0.799889	−0.399945	+	0.916539i	$0.630971\pi$
$480$	0	0
$481$	−2.57423	−0.117375
$482$	0	0
$483$	9.47193	0.430988
$484$	0	0
$485$	5.61151	0.254806
$486$	0	0
$487$	42.3008	1.91683	0.958416	−	0.285375i	$-0.0921181\pi$
$487$	42.3008	1.91683	0.958416	+	0.285375i	$0.0921181\pi$
$488$	0	0
$489$	−9.63776	−0.435834
$490$	0	0
$491$	8.16601	0.368527	0.184263	−	0.982877i	$-0.441010\pi$
$491$	8.16601	0.368527	0.184263	+	0.982877i	$0.441010\pi$
$492$	0	0
$493$	35.2437	1.58730
$494$	0	0
$495$	8.82808	0.396793
$496$	0	0
$497$	−11.2332	−0.503878
$498$	0	0
$499$	−31.4136	−1.40627	−0.703134	−	0.711058i	$-0.748216\pi$
$499$	−31.4136	−1.40627	−0.703134	+	0.711058i	$0.748216\pi$
$500$	0	0
$501$	49.6897	2.21997
$502$	0	0
$503$	−16.1653	−0.720776	−0.360388	−	0.932802i	$-0.617356\pi$
$503$	−16.1653	−0.720776	−0.360388	+	0.932802i	$0.617356\pi$
$504$	0	0
$505$	−38.7753	−1.72548
$506$	0	0
$507$	2.29224	0.101802
$508$	0	0
$509$	29.7395	1.31818	0.659090	−	0.752064i	$-0.270941\pi$
$509$	29.7395	1.31818	0.659090	+	0.752064i	$0.270941\pi$
$510$	0	0
$511$	−10.9199	−0.483069
$512$	0	0
$513$	−4.31778	−0.190634
$514$	0	0
$515$	−28.7920	−1.26873
$516$	0	0
$517$	0.823244	0.0362062
$518$	0	0
$519$	−12.9728	−0.569443
$520$	0	0
$521$	13.6438	0.597748	0.298874	−	0.954293i	$-0.403389\pi$
$521$	13.6438	0.597748	0.298874	+	0.954293i	$0.403389\pi$
$522$	0	0
$523$	5.94985	0.260169	0.130084	−	0.991503i	$-0.458475\pi$
$523$	5.94985	0.260169	0.130084	+	0.991503i	$0.458475\pi$
$524$	0	0
$525$	−23.6903	−1.03393
$526$	0	0
$527$	−21.1005	−0.919153
$528$	0	0
$529$	−5.92517	−0.257616
$530$	0	0
$531$	21.1687	0.918645
$532$	0	0
$533$	−4.72390	−0.204615
$534$	0	0
$535$	19.3200	0.835275
$536$	0	0
$537$	−20.8508	−0.899779
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	7.45868	0.320674	0.160337	−	0.987062i	$-0.448742\pi$
$541$	7.45868	0.320674	0.160337	+	0.987062i	$0.448742\pi$
$542$	0	0
$543$	−60.6705	−2.60362
$544$	0	0
$545$	−42.1278	−1.80456
$546$	0	0
$547$	−33.2378	−1.42115	−0.710573	−	0.703623i	$-0.751564\pi$
$547$	−33.2378	−1.42115	−0.710573	+	0.703623i	$0.751564\pi$
$548$	0	0
$549$	14.5092	0.619238
$550$	0	0
$551$	16.8193	0.716525
$552$	0	0
$553$	−14.4865	−0.616031
$554$	0	0
$555$	23.1073	0.980852
$556$	0	0
$557$	9.73600	0.412527	0.206264	−	0.978496i	$-0.433870\pi$
$557$	9.73600	0.412527	0.206264	+	0.978496i	$0.433870\pi$
$558$	0	0
$559$	7.81748	0.330644
$560$	0	0
$561$	12.1341	0.512303
$562$	0	0
$563$	−26.1109	−1.10044	−0.550221	−	0.835019i	$-0.685457\pi$
$563$	−26.1109	−1.10044	−0.550221	+	0.835019i	$0.685457\pi$
$564$	0	0
$565$	−18.0287	−0.758472
$566$	0	0
$567$	10.6809	0.448557
$568$	0	0
$569$	4.95977	0.207924	0.103962	−	0.994581i	$-0.466848\pi$
$569$	4.95977	0.207924	0.103962	+	0.994581i	$0.466848\pi$
$570$	0	0
$571$	4.17043	0.174527	0.0872635	−	0.996185i	$-0.472188\pi$
$571$	4.17043	0.174527	0.0872635	+	0.996185i	$0.472188\pi$
$572$	0	0
$573$	−36.0544	−1.50620
$574$	0	0
$575$	−42.7060	−1.78096
$576$	0	0
$577$	−0.106554	−0.00443592	−0.00221796	−	0.999998i	$-0.500706\pi$
$577$	−0.106554	−0.00443592	−0.00221796	+	0.999998i	$0.500706\pi$
$578$	0	0
$579$	17.4100	0.723537
$580$	0	0
$581$	15.6479	0.649186
$582$	0	0
$583$	0.474003	0.0196312
$584$	0	0
$585$	−8.82808	−0.364996
$586$	0	0
$587$	21.2490	0.877041	0.438520	−	0.898721i	$-0.355503\pi$
$587$	21.2490	0.877041	0.438520	+	0.898721i	$0.355503\pi$
$588$	0	0
$589$	−10.0697	−0.414917
$590$	0	0
$591$	26.6591	1.09661
$592$	0	0
$593$	−14.0513	−0.577018	−0.288509	−	0.957477i	$-0.593160\pi$
$593$	−14.0513	−0.577018	−0.288509	+	0.957477i	$0.593160\pi$
$594$	0	0
$595$	−20.7296	−0.849829
$596$	0	0
$597$	−6.00007	−0.245567
$598$	0	0
$599$	−1.23424	−0.0504296	−0.0252148	−	0.999682i	$-0.508027\pi$
$599$	−1.23424	−0.0504296	−0.0252148	+	0.999682i	$0.508027\pi$
$600$	0	0
$601$	1.92576	0.0785536	0.0392768	−	0.999228i	$-0.487495\pi$
$601$	1.92576	0.0785536	0.0392768	+	0.999228i	$0.487495\pi$
$602$	0	0
$603$	−30.4639	−1.24059
$604$	0	0
$605$	3.91599	0.159208
$606$	0	0
$607$	−33.4630	−1.35822	−0.679110	−	0.734036i	$-0.737634\pi$
$607$	−33.4630	−1.35822	−0.679110	+	0.734036i	$0.737634\pi$
$608$	0	0
$609$	−15.2614	−0.618422
$610$	0	0
$611$	−0.823244	−0.0333049
$612$	0	0
$613$	−28.4200	−1.14787	−0.573937	−	0.818900i	$-0.694584\pi$
$613$	−28.4200	−1.14787	−0.573937	+	0.818900i	$0.694584\pi$
$614$	0	0
$615$	42.4036	1.70988
$616$	0	0
$617$	−43.9714	−1.77022	−0.885112	−	0.465379i	$-0.845918\pi$
$617$	−43.9714	−1.77022	−0.885112	+	0.465379i	$0.845918\pi$
$618$	0	0
$619$	−4.17708	−0.167891	−0.0839456	−	0.996470i	$-0.526752\pi$
$619$	−4.17708	−0.167891	−0.0839456	+	0.996470i	$0.526752\pi$
$620$	0	0
$621$	7.06261	0.283413
$622$	0	0
$623$	16.0211	0.641874
$624$	0	0
$625$	30.1374	1.20550
$626$	0	0
$627$	5.79073	0.231259
$628$	0	0
$629$	13.6269	0.543338
$630$	0	0
$631$	−24.9393	−0.992816	−0.496408	−	0.868089i	$-0.665348\pi$
$631$	−24.9393	−0.992816	−0.496408	+	0.868089i	$0.665348\pi$
$632$	0	0
$633$	24.0483	0.955836
$634$	0	0
$635$	25.6893	1.01945
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	25.3237	1.00179
$640$	0	0
$641$	−18.0807	−0.714146	−0.357073	−	0.934077i	$-0.616225\pi$
$641$	−18.0807	−0.714146	−0.357073	+	0.934077i	$0.616225\pi$
$642$	0	0
$643$	29.3811	1.15868	0.579338	−	0.815087i	$-0.303311\pi$
$643$	29.3811	1.15868	0.579338	+	0.815087i	$0.303311\pi$
$644$	0	0
$645$	−70.1728	−2.76305
$646$	0	0
$647$	−22.0899	−0.868443	−0.434222	−	0.900806i	$-0.642977\pi$
$647$	−22.0899	−0.868443	−0.434222	+	0.900806i	$0.642977\pi$
$648$	0	0
$649$	9.39011	0.368594
$650$	0	0
$651$	9.13704	0.358109
$652$	0	0
$653$	−48.1624	−1.88474	−0.942369	−	0.334574i	$-0.891408\pi$
$653$	−48.1624	−1.88474	−0.942369	+	0.334574i	$0.891408\pi$
$654$	0	0
$655$	7.26812	0.283989
$656$	0	0
$657$	24.6175	0.960420
$658$	0	0
$659$	−27.4995	−1.07123	−0.535615	−	0.844462i	$-0.679920\pi$
$659$	−27.4995	−1.07123	−0.535615	+	0.844462i	$0.679920\pi$
$660$	0	0
$661$	38.4814	1.49675	0.748376	−	0.663275i	$-0.230834\pi$
$661$	38.4814	1.49675	0.748376	+	0.663275i	$0.230834\pi$
$662$	0	0
$663$	−12.1341	−0.471250
$664$	0	0
$665$	−9.89271	−0.383623
$666$	0	0
$667$	−27.5114	−1.06524
$668$	0	0
$669$	27.5115	1.06366
$670$	0	0
$671$	6.43606	0.248461
$672$	0	0
$673$	−13.7001	−0.528101	−0.264051	−	0.964509i	$-0.585059\pi$
$673$	−13.7001	−0.528101	−0.264051	+	0.964509i	$0.585059\pi$
$674$	0	0
$675$	−17.6644	−0.679902
$676$	0	0
$677$	42.1888	1.62145	0.810723	−	0.585430i	$-0.199074\pi$
$677$	42.1888	1.62145	0.810723	+	0.585430i	$0.199074\pi$
$678$	0	0
$679$	−1.43297	−0.0549924
$680$	0	0
$681$	−8.27228	−0.316994
$682$	0	0
$683$	38.4227	1.47020	0.735102	−	0.677956i	$-0.237134\pi$
$683$	38.4227	1.47020	0.735102	+	0.677956i	$0.237134\pi$
$684$	0	0
$685$	−3.37947	−0.129123
$686$	0	0
$687$	18.5548	0.707908
$688$	0	0
$689$	−0.474003	−0.0180581
$690$	0	0
$691$	40.1544	1.52755	0.763773	−	0.645485i	$-0.223345\pi$
$691$	40.1544	1.52755	0.763773	+	0.645485i	$0.223345\pi$
$692$	0	0
$693$	−2.25436	−0.0856362
$694$	0	0
$695$	−29.1684	−1.10642
$696$	0	0
$697$	25.0062	0.947178
$698$	0	0
$699$	11.7275	0.443574
$700$	0	0
$701$	13.1405	0.496308	0.248154	−	0.968721i	$-0.420176\pi$
$701$	13.1405	0.496308	0.248154	+	0.968721i	$0.420176\pi$
$702$	0	0
$703$	6.50311	0.245269
$704$	0	0
$705$	7.38977	0.278315
$706$	0	0
$707$	9.90176	0.372394
$708$	0	0
$709$	47.8981	1.79885	0.899425	−	0.437076i	$-0.143986\pi$
$709$	47.8981	1.79885	0.899425	+	0.437076i	$0.143986\pi$
$710$	0	0
$711$	32.6580	1.22477
$712$	0	0
$713$	16.4711	0.616849
$714$	0	0
$715$	−3.91599	−0.146450
$716$	0	0
$717$	−13.2384	−0.494395
$718$	0	0
$719$	−36.2505	−1.35192	−0.675959	−	0.736940i	$-0.736270\pi$
$719$	−36.2505	−1.35192	−0.675959	+	0.736940i	$0.736270\pi$
$720$	0	0
$721$	7.35242	0.273818
$722$	0	0
$723$	−38.4005	−1.42813
$724$	0	0
$725$	68.8089	2.55550
$726$	0	0
$727$	4.69897	0.174275	0.0871375	−	0.996196i	$-0.472228\pi$
$727$	4.69897	0.174275	0.0871375	+	0.996196i	$0.472228\pi$
$728$	0	0
$729$	−12.3252	−0.456488
$730$	0	0
$731$	−41.3823	−1.53058
$732$	0	0
$733$	21.8058	0.805414	0.402707	−	0.915329i	$-0.368069\pi$
$733$	21.8058	0.805414	0.402707	+	0.915329i	$0.368069\pi$
$734$	0	0
$735$	8.97640	0.331099
$736$	0	0
$737$	−13.5133	−0.497769
$738$	0	0
$739$	33.5000	1.23232	0.616158	−	0.787623i	$-0.288688\pi$
$739$	33.5000	1.23232	0.616158	+	0.787623i	$0.288688\pi$
$740$	0	0
$741$	−5.79073	−0.212728
$742$	0	0
$743$	53.3020	1.95546	0.977731	−	0.209863i	$-0.0673018\pi$
$743$	53.3020	1.95546	0.977731	+	0.209863i	$0.0673018\pi$
$744$	0	0
$745$	32.4499	1.18887
$746$	0	0
$747$	−35.2762	−1.29069
$748$	0	0
$749$	−4.93360	−0.180270
$750$	0	0
$751$	7.22369	0.263596	0.131798	−	0.991277i	$-0.457925\pi$
$751$	7.22369	0.263596	0.131798	+	0.991277i	$0.457925\pi$
$752$	0	0
$753$	−33.9157	−1.23596
$754$	0	0
$755$	30.0187	1.09249
$756$	0	0
$757$	−4.63587	−0.168493	−0.0842467	−	0.996445i	$-0.526848\pi$
$757$	−4.63587	−0.168493	−0.0842467	+	0.996445i	$0.526848\pi$
$758$	0	0
$759$	−9.47193	−0.343809
$760$	0	0
$761$	−34.3969	−1.24688	−0.623442	−	0.781869i	$-0.714266\pi$
$761$	−34.3969	−1.24688	−0.623442	+	0.781869i	$0.714266\pi$
$762$	0	0
$763$	10.7579	0.389461
$764$	0	0
$765$	46.7320	1.68960
$766$	0	0
$767$	−9.39011	−0.339057
$768$	0	0
$769$	9.55462	0.344548	0.172274	−	0.985049i	$-0.444888\pi$
$769$	9.55462	0.344548	0.172274	+	0.985049i	$0.444888\pi$
$770$	0	0
$771$	39.7619	1.43199
$772$	0	0
$773$	−11.9002	−0.428019	−0.214010	−	0.976832i	$-0.568652\pi$
$773$	−11.9002	−0.428019	−0.214010	+	0.976832i	$0.568652\pi$
$774$	0	0
$775$	−41.1961	−1.47981
$776$	0	0
$777$	−5.90076	−0.211688
$778$	0	0
$779$	11.9337	0.427568
$780$	0	0
$781$	11.2332	0.401955
$782$	0	0
$783$	−11.3794	−0.406668
$784$	0	0
$785$	92.6194	3.30573
$786$	0	0
$787$	−22.1305	−0.788868	−0.394434	−	0.918924i	$-0.629059\pi$
$787$	−22.1305	−0.788868	−0.394434	+	0.918924i	$0.629059\pi$
$788$	0	0
$789$	25.8728	0.921095
$790$	0	0
$791$	4.60386	0.163694
$792$	0	0
$793$	−6.43606	−0.228551
$794$	0	0
$795$	4.25484	0.150903
$796$	0	0
$797$	−6.89964	−0.244398	−0.122199	−	0.992506i	$-0.538995\pi$
$797$	−6.89964	−0.244398	−0.122199	+	0.992506i	$0.538995\pi$
$798$	0	0
$799$	4.35789	0.154171
$800$	0	0
$801$	−36.1175	−1.27615
$802$	0	0
$803$	10.9199	0.385356
$804$	0	0
$805$	16.1816	0.570325
$806$	0	0
$807$	−2.84417	−0.100119
$808$	0	0
$809$	7.05115	0.247905	0.123953	−	0.992288i	$-0.460443\pi$
$809$	7.05115	0.247905	0.123953	+	0.992288i	$0.460443\pi$
$810$	0	0
$811$	45.5752	1.60036	0.800182	−	0.599758i	$-0.204736\pi$
$811$	45.5752	1.60036	0.800182	+	0.599758i	$0.204736\pi$
$812$	0	0
$813$	20.0657	0.703736
$814$	0	0
$815$	−16.4649	−0.576739
$816$	0	0
$817$	−19.7488	−0.690922
$818$	0	0
$819$	2.25436	0.0787739
$820$	0	0
$821$	−2.44063	−0.0851785	−0.0425893	−	0.999093i	$-0.513561\pi$
$821$	−2.44063	−0.0851785	−0.0425893	+	0.999093i	$0.513561\pi$
$822$	0	0
$823$	−22.5556	−0.786238	−0.393119	−	0.919488i	$-0.628604\pi$
$823$	−22.5556	−0.786238	−0.393119	+	0.919488i	$0.628604\pi$
$824$	0	0
$825$	23.6903	0.824791
$826$	0	0
$827$	−55.2486	−1.92118	−0.960592	−	0.277963i	$-0.910341\pi$
$827$	−55.2486	−1.92118	−0.960592	+	0.277963i	$0.910341\pi$
$828$	0	0
$829$	−0.745883	−0.0259056	−0.0129528	−	0.999916i	$-0.504123\pi$
$829$	−0.745883	−0.0259056	−0.0129528	+	0.999916i	$0.504123\pi$
$830$	0	0
$831$	−43.6717	−1.51495
$832$	0	0
$833$	5.29356	0.183411
$834$	0	0
$835$	84.8883	2.93768
$836$	0	0
$837$	6.81290	0.235488
$838$	0	0
$839$	−28.0788	−0.969387	−0.484694	−	0.874684i	$-0.661069\pi$
$839$	−28.0788	−0.969387	−0.484694	+	0.874684i	$0.661069\pi$
$840$	0	0
$841$	15.3269	0.528515
$842$	0	0
$843$	62.6737	2.15860
$844$	0	0
$845$	3.91599	0.134714
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	59.5955	2.04531
$850$	0	0
$851$	−10.6372	−0.364637
$852$	0	0
$853$	47.7511	1.63497	0.817483	−	0.575952i	$-0.195369\pi$
$853$	47.7511	1.63497	0.817483	+	0.575952i	$0.195369\pi$
$854$	0	0
$855$	22.3018	0.762704
$856$	0	0
$857$	−28.9286	−0.988182	−0.494091	−	0.869410i	$-0.664499\pi$
$857$	−28.9286	−0.988182	−0.494091	+	0.869410i	$0.664499\pi$
$858$	0	0
$859$	−11.2251	−0.382996	−0.191498	−	0.981493i	$-0.561335\pi$
$859$	−11.2251	−0.382996	−0.191498	+	0.981493i	$0.561335\pi$
$860$	0	0
$861$	−10.8283	−0.369028
$862$	0	0
$863$	−15.7525	−0.536222	−0.268111	−	0.963388i	$-0.586399\pi$
$863$	−15.7525	−0.536222	−0.268111	+	0.963388i	$0.586399\pi$
$864$	0	0
$865$	−22.1624	−0.753543
$866$	0	0
$867$	25.2646	0.858030
$868$	0	0
$869$	14.4865	0.491422
$870$	0	0
$871$	13.5133	0.457881
$872$	0	0
$873$	3.23044	0.109334
$874$	0	0
$875$	−20.8919	−0.706275
$876$	0	0
$877$	31.1425	1.05161	0.525805	−	0.850605i	$-0.323764\pi$
$877$	31.1425	1.05161	0.525805	+	0.850605i	$0.323764\pi$
$878$	0	0
$879$	−46.3476	−1.56327
$880$	0	0
$881$	−37.5111	−1.26378	−0.631891	−	0.775057i	$-0.717721\pi$
$881$	−37.5111	−1.26378	−0.631891	+	0.775057i	$0.717721\pi$
$882$	0	0
$883$	−28.7365	−0.967061	−0.483531	−	0.875327i	$-0.660646\pi$
$883$	−28.7365	−0.967061	−0.483531	+	0.875327i	$0.660646\pi$
$884$	0	0
$885$	84.2894	2.83336
$886$	0	0
$887$	−42.5333	−1.42813	−0.714065	−	0.700079i	$-0.753148\pi$
$887$	−42.5333	−1.42813	−0.714065	+	0.700079i	$0.753148\pi$
$888$	0	0
$889$	−6.56009	−0.220018
$890$	0	0
$891$	−10.6809	−0.357825
$892$	0	0
$893$	2.07971	0.0695947
$894$	0	0
$895$	−35.6209	−1.19068
$896$	0	0
$897$	9.47193	0.316258
$898$	0	0
$899$	−26.5387	−0.885114
$900$	0	0
$901$	2.50916	0.0835923
$902$	0	0
$903$	17.9195	0.596325
$904$	0	0
$905$	−103.648	−3.44536
$906$	0	0
$907$	14.5125	0.481881	0.240940	−	0.970540i	$-0.422544\pi$
$907$	14.5125	0.481881	0.240940	+	0.970540i	$0.422544\pi$
$908$	0	0
$909$	−22.3222	−0.740380
$910$	0	0
$911$	−12.1469	−0.402446	−0.201223	−	0.979545i	$-0.564492\pi$
$911$	−12.1469	−0.402446	−0.201223	+	0.979545i	$0.564492\pi$
$912$	0	0
$913$	−15.6479	−0.517871
$914$	0	0
$915$	57.7726	1.90990
$916$	0	0
$917$	−1.85601	−0.0612908
$918$	0	0
$919$	3.64879	0.120363	0.0601813	−	0.998187i	$-0.480832\pi$
$919$	3.64879	0.120363	0.0601813	+	0.998187i	$0.480832\pi$
$920$	0	0
$921$	−23.9722	−0.789911
$922$	0	0
$923$	−11.2332	−0.369745
$924$	0	0
$925$	26.6047	0.874758
$926$	0	0
$927$	−16.5750	−0.544395
$928$	0	0
$929$	14.4675	0.474662	0.237331	−	0.971429i	$-0.423727\pi$
$929$	14.4675	0.474662	0.237331	+	0.971429i	$0.423727\pi$
$930$	0	0
$931$	2.52623	0.0827939
$932$	0	0
$933$	−33.9034	−1.10995
$934$	0	0
$935$	20.7296	0.677929
$936$	0	0
$937$	25.0598	0.818669	0.409334	−	0.912384i	$-0.365761\pi$
$937$	25.0598	0.818669	0.409334	+	0.912384i	$0.365761\pi$
$938$	0	0
$939$	60.1013	1.96133
$940$	0	0
$941$	5.56445	0.181396	0.0906979	−	0.995878i	$-0.471090\pi$
$941$	5.56445	0.181396	0.0906979	+	0.995878i	$0.471090\pi$
$942$	0	0
$943$	−19.5199	−0.635657
$944$	0	0
$945$	6.69313	0.217727
$946$	0	0
$947$	4.20309	0.136582	0.0682911	−	0.997665i	$-0.478245\pi$
$947$	4.20309	0.136582	0.0682911	+	0.997665i	$0.478245\pi$
$948$	0	0
$949$	−10.9199	−0.354476
$950$	0	0
$951$	−52.0545	−1.68798
$952$	0	0
$953$	−7.69251	−0.249185	−0.124592	−	0.992208i	$-0.539762\pi$
$953$	−7.69251	−0.249185	−0.124592	+	0.992208i	$0.539762\pi$
$954$	0	0
$955$	−61.5943	−1.99315
$956$	0	0
$957$	15.2614	0.493330
$958$	0	0
$959$	0.862991	0.0278674
$960$	0	0
$961$	−15.1112	−0.487459
$962$	0	0
$963$	11.1221	0.358406
$964$	0	0
$965$	29.7428	0.957455
$966$	0	0
$967$	27.3065	0.878119	0.439060	−	0.898458i	$-0.355312\pi$
$967$	27.3065	0.878119	0.439060	+	0.898458i	$0.355312\pi$
$968$	0	0
$969$	30.6536	0.984735
$970$	0	0
$971$	1.66328	0.0533771	0.0266886	−	0.999644i	$-0.491504\pi$
$971$	1.66328	0.0533771	0.0266886	+	0.999644i	$0.491504\pi$
$972$	0	0
$973$	7.44853	0.238789
$974$	0	0
$975$	−23.6903	−0.758698
$976$	0	0
$977$	52.0088	1.66391	0.831954	−	0.554845i	$-0.187222\pi$
$977$	52.0088	1.66391	0.831954	+	0.554845i	$0.187222\pi$
$978$	0	0
$979$	−16.0211	−0.512038
$980$	0	0
$981$	−24.2522	−0.774312
$982$	0	0
$983$	48.8967	1.55956	0.779781	−	0.626053i	$-0.215331\pi$
$983$	48.8967	1.55956	0.779781	+	0.626053i	$0.215331\pi$
$984$	0	0
$985$	45.5437	1.45114
$986$	0	0
$987$	−1.88707	−0.0600662
$988$	0	0
$989$	32.3032	1.02718
$990$	0	0
$991$	30.2429	0.960696	0.480348	−	0.877078i	$-0.340510\pi$
$991$	30.2429	0.960696	0.480348	+	0.877078i	$0.340510\pi$
$992$	0	0
$993$	−50.7217	−1.60960
$994$	0	0
$995$	−10.2503	−0.324958
$996$	0	0
$997$	41.1103	1.30198	0.650988	−	0.759088i	$-0.274355\pi$
$997$	41.1103	1.30198	0.650988	+	0.759088i	$0.274355\pi$
$998$	0	0
$999$	−4.39982	−0.139204

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.29224 q^{3} +3.91599 q^{5} -1.00000 q^{7} +2.25436 q^{9} +O(q^{10})\)	\(q+2.29224 q^{3} +3.91599 q^{5} -1.00000 q^{7} +2.25436 q^{9} +1.00000 q^{11} -1.00000 q^{13} +8.97640 q^{15} +5.29356 q^{17} +2.52623 q^{19} -2.29224 q^{21} -4.13217 q^{23} +10.3350 q^{25} -1.70918 q^{27} +6.65785 q^{29} -3.98607 q^{31} +2.29224 q^{33} -3.91599 q^{35} +2.57423 q^{37} -2.29224 q^{39} +4.72390 q^{41} -7.81748 q^{43} +8.82808 q^{45} +0.823244 q^{47} +1.00000 q^{49} +12.1341 q^{51} +0.474003 q^{53} +3.91599 q^{55} +5.79073 q^{57} +9.39011 q^{59} +6.43606 q^{61} -2.25436 q^{63} -3.91599 q^{65} -13.5133 q^{67} -9.47193 q^{69} +11.2332 q^{71} +10.9199 q^{73} +23.6903 q^{75} -1.00000 q^{77} +14.4865 q^{79} -10.6809 q^{81} -15.6479 q^{83} +20.7296 q^{85} +15.2614 q^{87} -16.0211 q^{89} +1.00000 q^{91} -9.13704 q^{93} +9.89271 q^{95} +1.43297 q^{97} +2.25436 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q - 3 * q^3 + q^5 - 9 * q^7 + 12 * q^9	\( 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} + 11 q^{15} + 7 q^{17} - 17 q^{19} + 3 q^{21} + 11 q^{23} + 18 q^{25} - 9 q^{27} + 9 q^{29} - 8 q^{31} - 3 q^{33} - q^{35} + 2 q^{37} + 3 q^{39} + 18 q^{41} + 7 q^{43} + 5 q^{45} + 15 q^{47} + 9 q^{49} - 7 q^{51} - 4 q^{53} + q^{55} + 22 q^{57} - 23 q^{59} + 12 q^{61} - 12 q^{63} - q^{65} - 16 q^{67} - 32 q^{69} - 6 q^{71} + 4 q^{73} - 14 q^{75} - 9 q^{77} + 21 q^{79} + 5 q^{81} - 16 q^{83} + 53 q^{85} + 41 q^{87} + 5 q^{89} + 9 q^{91} + 29 q^{93} + 19 q^{95} + 18 q^{97} + 12 q^{99}+O(q^{100}) \) 9 * q - 3 * q^3 + q^5 - 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 + 11 * q^15 + 7 * q^17 - 17 * q^19 + 3 * q^21 + 11 * q^23 + 18 * q^25 - 9 * q^27 + 9 * q^29 - 8 * q^31 - 3 * q^33 - q^35 + 2 * q^37 + 3 * q^39 + 18 * q^41 + 7 * q^43 + 5 * q^45 + 15 * q^47 + 9 * q^49 - 7 * q^51 - 4 * q^53 + q^55 + 22 * q^57 - 23 * q^59 + 12 * q^61 - 12 * q^63 - q^65 - 16 * q^67 - 32 * q^69 - 6 * q^71 + 4 * q^73 - 14 * q^75 - 9 * q^77 + 21 * q^79 + 5 * q^81 - 16 * q^83 + 53 * q^85 + 41 * q^87 + 5 * q^89 + 9 * q^91 + 29 * q^93 + 19 * q^95 + 18 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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