LMFDB - Embedded newform 8008.2.a.x.1.9

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			1.9
Root			$1.75763$ 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)

$f(q)$	$=$	$q+1.75763 q^{3} -3.24204 q^{5} +1.00000 q^{7} +0.0892594 q^{9} +O(q^{10})$	$q+1.75763 q^{3} -3.24204 q^{5} +1.00000 q^{7} +0.0892594 q^{9} +1.00000 q^{11} +1.00000 q^{13} -5.69831 q^{15} +0.0754893 q^{17} -6.53631 q^{19} +1.75763 q^{21} -8.67581 q^{23} +5.51085 q^{25} -5.11600 q^{27} +3.22876 q^{29} +5.81883 q^{31} +1.75763 q^{33} -3.24204 q^{35} +1.83193 q^{37} +1.75763 q^{39} +9.76053 q^{41} -3.89313 q^{43} -0.289383 q^{45} -1.46368 q^{47} +1.00000 q^{49} +0.132682 q^{51} +8.85251 q^{53} -3.24204 q^{55} -11.4884 q^{57} +11.8022 q^{59} -0.744116 q^{61} +0.0892594 q^{63} -3.24204 q^{65} -10.6894 q^{67} -15.2489 q^{69} +7.68968 q^{71} +9.86228 q^{73} +9.68604 q^{75} +1.00000 q^{77} +10.6204 q^{79} -9.25981 q^{81} -3.12381 q^{83} -0.244740 q^{85} +5.67496 q^{87} +2.35214 q^{89} +1.00000 q^{91} +10.2273 q^{93} +21.1910 q^{95} -16.9322 q^{97} +0.0892594 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.75763	1.01477	0.507384	−	0.861720i	$-0.330613\pi$
$3$	1.75763	1.01477	0.507384	+	0.861720i	$0.330613\pi$
$4$	0	0
$5$	−3.24204	−1.44989	−0.724943	−	0.688809i	$-0.758134\pi$
$5$	−3.24204	−1.44989	−0.724943	+	0.688809i	$0.758134\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0.0892594	0.0297531
$10$	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
$15$	−5.69831	−1.47130
$16$	0	0
$17$	0.0754893	0.0183088	0.00915442	−	0.999958i	$-0.497086\pi$
$17$	0.0754893	0.0183088	0.00915442	+	0.999958i	$0.497086\pi$
$18$	0	0
$19$	−6.53631	−1.49953	−0.749766	−	0.661703i	$-0.769834\pi$
$19$	−6.53631	−1.49953	−0.749766	+	0.661703i	$0.769834\pi$
$20$	0	0
$21$	1.75763	0.383546
$22$	0	0
$23$	−8.67581	−1.80903	−0.904516	−	0.426441i	$-0.859767\pi$
$23$	−8.67581	−1.80903	−0.904516	+	0.426441i	$0.859767\pi$
$24$	0	0
$25$	5.51085	1.10217
$26$	0	0
$27$	−5.11600	−0.984575
$28$	0	0
$29$	3.22876	0.599566	0.299783	−	0.954007i	$-0.403086\pi$
$29$	3.22876	0.599566	0.299783	+	0.954007i	$0.403086\pi$
$30$	0	0
$31$	5.81883	1.04509	0.522546	−	0.852611i	$-0.324982\pi$
$31$	5.81883	1.04509	0.522546	+	0.852611i	$0.324982\pi$
$32$	0	0
$33$	1.75763	0.305964
$34$	0	0
$35$	−3.24204	−0.548006
$36$	0	0
$37$	1.83193	0.301168	0.150584	−	0.988597i	$-0.451885\pi$
$37$	1.83193	0.301168	0.150584	+	0.988597i	$0.451885\pi$
$38$	0	0
$39$	1.75763	0.281446
$40$	0	0
$41$	9.76053	1.52434	0.762169	−	0.647378i	$-0.224134\pi$
$41$	9.76053	1.52434	0.762169	+	0.647378i	$0.224134\pi$
$42$	0	0
$43$	−3.89313	−0.593697	−0.296848	−	0.954925i	$-0.595936\pi$
$43$	−3.89313	−0.593697	−0.296848	+	0.954925i	$0.595936\pi$
$44$	0	0
$45$	−0.289383	−0.0431387
$46$	0	0
$47$	−1.46368	−0.213500	−0.106750	−	0.994286i	$-0.534044\pi$
$47$	−1.46368	−0.213500	−0.106750	+	0.994286i	$0.534044\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.132682	0.0185792
$52$	0	0
$53$	8.85251	1.21599	0.607993	−	0.793942i	$-0.291975\pi$
$53$	8.85251	1.21599	0.607993	+	0.793942i	$0.291975\pi$
$54$	0	0
$55$	−3.24204	−0.437157
$56$	0	0
$57$	−11.4884	−1.52168
$58$	0	0
$59$	11.8022	1.53651	0.768257	−	0.640141i	$-0.221124\pi$
$59$	11.8022	1.53651	0.768257	+	0.640141i	$0.221124\pi$
$60$	0	0
$61$	−0.744116	−0.0952742	−0.0476371	−	0.998865i	$-0.515169\pi$
$61$	−0.744116	−0.0952742	−0.0476371	+	0.998865i	$0.515169\pi$
$62$	0	0
$63$	0.0892594	0.0112456
$64$	0	0
$65$	−3.24204	−0.402126
$66$	0	0
$67$	−10.6894	−1.30591	−0.652957	−	0.757395i	$-0.726472\pi$
$67$	−10.6894	−1.30591	−0.652957	+	0.757395i	$0.726472\pi$
$68$	0	0
$69$	−15.2489	−1.83575
$70$	0	0
$71$	7.68968	0.912597	0.456298	−	0.889827i	$-0.349175\pi$
$71$	7.68968	0.912597	0.456298	+	0.889827i	$0.349175\pi$
$72$	0	0
$73$	9.86228	1.15429	0.577146	−	0.816641i	$-0.304166\pi$
$73$	9.86228	1.15429	0.577146	+	0.816641i	$0.304166\pi$
$74$	0	0
$75$	9.68604	1.11845
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	10.6204	1.19489	0.597447	−	0.801908i	$-0.296182\pi$
$79$	10.6204	1.19489	0.597447	+	0.801908i	$0.296182\pi$
$80$	0	0
$81$	−9.25981	−1.02887
$82$	0	0
$83$	−3.12381	−0.342883	−0.171441	−	0.985194i	$-0.554842\pi$
$83$	−3.12381	−0.342883	−0.171441	+	0.985194i	$0.554842\pi$
$84$	0	0
$85$	−0.244740	−0.0265458
$86$	0	0
$87$	5.67496	0.608420
$88$	0	0
$89$	2.35214	0.249327	0.124663	−	0.992199i	$-0.460215\pi$
$89$	2.35214	0.249327	0.124663	+	0.992199i	$0.460215\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	10.2273	1.06053
$94$	0	0
$95$	21.1910	2.17415
$96$	0	0
$97$	−16.9322	−1.71920	−0.859601	−	0.510966i	$-0.829288\pi$
$97$	−16.9322	−1.71920	−0.859601	+	0.510966i	$0.829288\pi$
$98$	0	0
$99$	0.0892594	0.00897091
$100$	0	0
$101$	−17.3213	−1.72353	−0.861764	−	0.507309i	$-0.830640\pi$
$101$	−17.3213	−1.72353	−0.861764	+	0.507309i	$0.830640\pi$
$102$	0	0
$103$	13.5088	1.33106	0.665530	−	0.746371i	$-0.268206\pi$
$103$	13.5088	1.33106	0.665530	+	0.746371i	$0.268206\pi$
$104$	0	0
$105$	−5.69831	−0.556098
$106$	0	0
$107$	15.2803	1.47720	0.738599	−	0.674145i	$-0.235488\pi$
$107$	15.2803	1.47720	0.738599	+	0.674145i	$0.235488\pi$
$108$	0	0
$109$	17.1537	1.64302	0.821511	−	0.570193i	$-0.193132\pi$
$109$	17.1537	1.64302	0.821511	+	0.570193i	$0.193132\pi$
$110$	0	0
$111$	3.21986	0.305615
$112$	0	0
$113$	19.7809	1.86084	0.930418	−	0.366501i	$-0.119445\pi$
$113$	19.7809	1.86084	0.930418	+	0.366501i	$0.119445\pi$
$114$	0	0
$115$	28.1274	2.62289
$116$	0	0
$117$	0.0892594	0.00825204
$118$	0	0
$119$	0.0754893	0.00692009
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	17.1554	1.54685
$124$	0	0
$125$	−1.65621	−0.148136
$126$	0	0
$127$	−8.10996	−0.719643	−0.359821	−	0.933021i	$-0.617162\pi$
$127$	−8.10996	−0.719643	−0.359821	+	0.933021i	$0.617162\pi$
$128$	0	0
$129$	−6.84268	−0.602464
$130$	0	0
$131$	−4.82740	−0.421772	−0.210886	−	0.977511i	$-0.567635\pi$
$131$	−4.82740	−0.421772	−0.210886	+	0.977511i	$0.567635\pi$
$132$	0	0
$133$	−6.53631	−0.566770
$134$	0	0
$135$	16.5863	1.42752
$136$	0	0
$137$	2.73129	0.233350	0.116675	−	0.993170i	$-0.462776\pi$
$137$	2.73129	0.233350	0.116675	+	0.993170i	$0.462776\pi$
$138$	0	0
$139$	3.80299	0.322565	0.161283	−	0.986908i	$-0.448437\pi$
$139$	3.80299	0.322565	0.161283	+	0.986908i	$0.448437\pi$
$140$	0	0
$141$	−2.57261	−0.216653
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−10.4678	−0.869302
$146$	0	0
$147$	1.75763	0.144967
$148$	0	0
$149$	−18.6555	−1.52832	−0.764160	−	0.645027i	$-0.776846\pi$
$149$	−18.6555	−1.52832	−0.764160	+	0.645027i	$0.776846\pi$
$150$	0	0
$151$	−22.5787	−1.83742	−0.918712	−	0.394927i	$-0.870770\pi$
$151$	−22.5787	−1.83742	−0.918712	+	0.394927i	$0.870770\pi$
$152$	0	0
$153$	0.00673813	0.000544746 0
$154$	0	0
$155$	−18.8649	−1.51526
$156$	0	0
$157$	18.9589	1.51308	0.756542	−	0.653945i	$-0.226887\pi$
$157$	18.9589	1.51308	0.756542	+	0.653945i	$0.226887\pi$
$158$	0	0
$159$	15.5594	1.23394
$160$	0	0
$161$	−8.67581	−0.683750
$162$	0	0
$163$	−8.19359	−0.641772	−0.320886	−	0.947118i	$-0.603981\pi$
$163$	−8.19359	−0.641772	−0.320886	+	0.947118i	$0.603981\pi$
$164$	0	0
$165$	−5.69831	−0.443613
$166$	0	0
$167$	0.673652	0.0521287	0.0260644	−	0.999660i	$-0.491703\pi$
$167$	0.673652	0.0521287	0.0260644	+	0.999660i	$0.491703\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.583427	−0.0446158
$172$	0	0
$173$	17.6596	1.34263	0.671316	−	0.741171i	$-0.265729\pi$
$173$	17.6596	1.34263	0.671316	+	0.741171i	$0.265729\pi$
$174$	0	0
$175$	5.51085	0.416581
$176$	0	0
$177$	20.7439	1.55921
$178$	0	0
$179$	−0.783353	−0.0585505	−0.0292753	−	0.999571i	$-0.509320\pi$
$179$	−0.783353	−0.0585505	−0.0292753	+	0.999571i	$0.509320\pi$
$180$	0	0
$181$	−6.49721	−0.482934	−0.241467	−	0.970409i	$-0.577629\pi$
$181$	−6.49721	−0.482934	−0.241467	+	0.970409i	$0.577629\pi$
$182$	0	0
$183$	−1.30788	−0.0966812
$184$	0	0
$185$	−5.93921	−0.436659
$186$	0	0
$187$	0.0754893	0.00552033
$188$	0	0
$189$	−5.11600	−0.372134
$190$	0	0
$191$	19.3294	1.39862	0.699311	−	0.714817i	$-0.253490\pi$
$191$	19.3294	1.39862	0.699311	+	0.714817i	$0.253490\pi$
$192$	0	0
$193$	−13.3860	−0.963542	−0.481771	−	0.876297i	$-0.660006\pi$
$193$	−13.3860	−0.963542	−0.481771	+	0.876297i	$0.660006\pi$
$194$	0	0
$195$	−5.69831	−0.408065
$196$	0	0
$197$	0.509594	0.0363071	0.0181535	−	0.999835i	$-0.494221\pi$
$197$	0.509594	0.0363071	0.0181535	+	0.999835i	$0.494221\pi$
$198$	0	0
$199$	16.2846	1.15439	0.577193	−	0.816608i	$-0.304148\pi$
$199$	16.2846	1.15439	0.577193	+	0.816608i	$0.304148\pi$
$200$	0	0
$201$	−18.7879	−1.32520
$202$	0	0
$203$	3.22876	0.226614
$204$	0	0
$205$	−31.6441	−2.21012
$206$	0	0
$207$	−0.774398	−0.0538244
$208$	0	0
$209$	−6.53631	−0.452126
$210$	0	0
$211$	−11.4003	−0.784828	−0.392414	−	0.919789i	$-0.628360\pi$
$211$	−11.4003	−0.784828	−0.392414	+	0.919789i	$0.628360\pi$
$212$	0	0
$213$	13.5156	0.926074
$214$	0	0
$215$	12.6217	0.860793
$216$	0	0
$217$	5.81883	0.395008
$218$	0	0
$219$	17.3342	1.17134
$220$	0	0
$221$	0.0754893	0.00507796
$222$	0	0
$223$	−13.1093	−0.877867	−0.438933	−	0.898520i	$-0.644644\pi$
$223$	−13.1093	−0.877867	−0.438933	+	0.898520i	$0.644644\pi$
$224$	0	0
$225$	0.491896	0.0327930
$226$	0	0
$227$	−10.1084	−0.670916	−0.335458	−	0.942055i	$-0.608891\pi$
$227$	−10.1084	−0.670916	−0.335458	+	0.942055i	$0.608891\pi$
$228$	0	0
$229$	5.22668	0.345388	0.172694	−	0.984975i	$-0.444753\pi$
$229$	5.22668	0.345388	0.172694	+	0.984975i	$0.444753\pi$
$230$	0	0
$231$	1.75763	0.115643
$232$	0	0
$233$	10.7762	0.705972	0.352986	−	0.935629i	$-0.385166\pi$
$233$	10.7762	0.705972	0.352986	+	0.935629i	$0.385166\pi$
$234$	0	0
$235$	4.74532	0.309551
$236$	0	0
$237$	18.6668	1.21254
$238$	0	0
$239$	27.3801	1.77107	0.885535	−	0.464573i	$-0.153792\pi$
$239$	27.3801	1.77107	0.885535	+	0.464573i	$0.153792\pi$
$240$	0	0
$241$	10.7988	0.695610	0.347805	−	0.937567i	$-0.386927\pi$
$241$	10.7988	0.695610	0.347805	+	0.937567i	$0.386927\pi$
$242$	0	0
$243$	−0.927306	−0.0594867
$244$	0	0
$245$	−3.24204	−0.207127
$246$	0	0
$247$	−6.53631	−0.415895
$248$	0	0
$249$	−5.49050	−0.347946
$250$	0	0
$251$	−16.1726	−1.02080	−0.510402	−	0.859936i	$-0.670503\pi$
$251$	−16.1726	−1.02080	−0.510402	+	0.859936i	$0.670503\pi$
$252$	0	0
$253$	−8.67581	−0.545443
$254$	0	0
$255$	−0.430162	−0.0269378
$256$	0	0
$257$	11.5195	0.718568	0.359284	−	0.933228i	$-0.383021\pi$
$257$	11.5195	0.718568	0.359284	+	0.933228i	$0.383021\pi$
$258$	0	0
$259$	1.83193	0.113831
$260$	0	0
$261$	0.288197	0.0178390
$262$	0	0
$263$	28.7721	1.77416	0.887080	−	0.461615i	$-0.152730\pi$
$263$	28.7721	1.77416	0.887080	+	0.461615i	$0.152730\pi$
$264$	0	0
$265$	−28.7002	−1.76304
$266$	0	0
$267$	4.13420	0.253009
$268$	0	0
$269$	16.4914	1.00550	0.502749	−	0.864433i	$-0.332322\pi$
$269$	16.4914	1.00550	0.502749	+	0.864433i	$0.332322\pi$
$270$	0	0
$271$	−15.0468	−0.914027	−0.457014	−	0.889460i	$-0.651081\pi$
$271$	−15.0468	−0.914027	−0.457014	+	0.889460i	$0.651081\pi$
$272$	0	0
$273$	1.75763	0.106377
$274$	0	0
$275$	5.51085	0.332317
$276$	0	0
$277$	−10.7234	−0.644304	−0.322152	−	0.946688i	$-0.604406\pi$
$277$	−10.7234	−0.644304	−0.322152	+	0.946688i	$0.604406\pi$
$278$	0	0
$279$	0.519385	0.0310948
$280$	0	0
$281$	−21.8225	−1.30182	−0.650910	−	0.759155i	$-0.725613\pi$
$281$	−21.8225	−1.30182	−0.650910	+	0.759155i	$0.725613\pi$
$282$	0	0
$283$	7.44645	0.442646	0.221323	−	0.975201i	$-0.428963\pi$
$283$	7.44645	0.442646	0.221323	+	0.975201i	$0.428963\pi$
$284$	0	0
$285$	37.2459	2.20626
$286$	0	0
$287$	9.76053	0.576146
$288$	0	0
$289$	−16.9943	−0.999665
$290$	0	0
$291$	−29.7605	−1.74459
$292$	0	0
$293$	30.0759	1.75705	0.878527	−	0.477693i	$-0.158527\pi$
$293$	30.0759	1.75705	0.878527	+	0.477693i	$0.158527\pi$
$294$	0	0
$295$	−38.2632	−2.22777
$296$	0	0
$297$	−5.11600	−0.296861
$298$	0	0
$299$	−8.67581	−0.501735
$300$	0	0
$301$	−3.89313	−0.224396
$302$	0	0
$303$	−30.4443	−1.74898
$304$	0	0
$305$	2.41246	0.138137
$306$	0	0
$307$	22.3098	1.27329	0.636644	−	0.771158i	$-0.280322\pi$
$307$	22.3098	1.27329	0.636644	+	0.771158i	$0.280322\pi$
$308$	0	0
$309$	23.7434	1.35072
$310$	0	0
$311$	10.7857	0.611599	0.305800	−	0.952096i	$-0.401076\pi$
$311$	10.7857	0.611599	0.305800	+	0.952096i	$0.401076\pi$
$312$	0	0
$313$	−18.6570	−1.05455	−0.527277	−	0.849693i	$-0.676787\pi$
$313$	−18.6570	−1.05455	−0.527277	+	0.849693i	$0.676787\pi$
$314$	0	0
$315$	−0.289383	−0.0163049
$316$	0	0
$317$	32.1366	1.80497	0.902485	−	0.430722i	$-0.141741\pi$
$317$	32.1366	1.80497	0.902485	+	0.430722i	$0.141741\pi$
$318$	0	0
$319$	3.22876	0.180776
$320$	0	0
$321$	26.8570	1.49901
$322$	0	0
$323$	−0.493422	−0.0274547
$324$	0	0
$325$	5.51085	0.305687
$326$	0	0
$327$	30.1498	1.66729
$328$	0	0
$329$	−1.46368	−0.0806953
$330$	0	0
$331$	−0.278064	−0.0152838	−0.00764188	−	0.999971i	$-0.502433\pi$
$331$	−0.278064	−0.0152838	−0.00764188	+	0.999971i	$0.502433\pi$
$332$	0	0
$333$	0.163517	0.00896069
$334$	0	0
$335$	34.6554	1.89343
$336$	0	0
$337$	30.3463	1.65307	0.826533	−	0.562888i	$-0.190310\pi$
$337$	30.3463	1.65307	0.826533	+	0.562888i	$0.190310\pi$
$338$	0	0
$339$	34.7676	1.88831
$340$	0	0
$341$	5.81883	0.315107
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	49.4375	2.66162
$346$	0	0
$347$	24.5838	1.31973	0.659865	−	0.751384i	$-0.270614\pi$
$347$	24.5838	1.31973	0.659865	+	0.751384i	$0.270614\pi$
$348$	0	0
$349$	16.1744	0.865798	0.432899	−	0.901443i	$-0.357491\pi$
$349$	16.1744	0.865798	0.432899	+	0.901443i	$0.357491\pi$
$350$	0	0
$351$	−5.11600	−0.273072
$352$	0	0
$353$	−16.2617	−0.865525	−0.432763	−	0.901508i	$-0.642461\pi$
$353$	−16.2617	−0.865525	−0.432763	+	0.901508i	$0.642461\pi$
$354$	0	0
$355$	−24.9303	−1.32316
$356$	0	0
$357$	0.132682	0.00702229
$358$	0	0
$359$	−32.4039	−1.71021	−0.855105	−	0.518454i	$-0.826508\pi$
$359$	−32.4039	−1.71021	−0.855105	+	0.518454i	$0.826508\pi$
$360$	0	0
$361$	23.7233	1.24860
$362$	0	0
$363$	1.75763	0.0922516
$364$	0	0
$365$	−31.9740	−1.67359
$366$	0	0
$367$	34.4781	1.79974	0.899871	−	0.436157i	$-0.143661\pi$
$367$	34.4781	1.79974	0.899871	+	0.436157i	$0.143661\pi$
$368$	0	0
$369$	0.871219	0.0453539
$370$	0	0
$371$	8.85251	0.459599
$372$	0	0
$373$	18.3982	0.952621	0.476310	−	0.879277i	$-0.341974\pi$
$373$	18.3982	0.952621	0.476310	+	0.879277i	$0.341974\pi$
$374$	0	0
$375$	−2.91101	−0.150324
$376$	0	0
$377$	3.22876	0.166290
$378$	0	0
$379$	−22.6672	−1.16433	−0.582167	−	0.813069i	$-0.697795\pi$
$379$	−22.6672	−1.16433	−0.582167	+	0.813069i	$0.697795\pi$
$380$	0	0
$381$	−14.2543	−0.730270
$382$	0	0
$383$	−9.06237	−0.463065	−0.231533	−	0.972827i	$-0.574374\pi$
$383$	−9.06237	−0.463065	−0.231533	+	0.972827i	$0.574374\pi$
$384$	0	0
$385$	−3.24204	−0.165230
$386$	0	0
$387$	−0.347498	−0.0176643
$388$	0	0
$389$	−22.9903	−1.16566	−0.582828	−	0.812595i	$-0.698054\pi$
$389$	−22.9903	−1.16566	−0.582828	+	0.812595i	$0.698054\pi$
$390$	0	0
$391$	−0.654931	−0.0331213
$392$	0	0
$393$	−8.48478	−0.428001
$394$	0	0
$395$	−34.4320	−1.73246
$396$	0	0
$397$	−32.3968	−1.62595	−0.812973	−	0.582301i	$-0.802153\pi$
$397$	−32.3968	−1.62595	−0.812973	+	0.582301i	$0.802153\pi$
$398$	0	0
$399$	−11.4884	−0.575140
$400$	0	0
$401$	8.49899	0.424419	0.212210	−	0.977224i	$-0.431934\pi$
$401$	8.49899	0.424419	0.212210	+	0.977224i	$0.431934\pi$
$402$	0	0
$403$	5.81883	0.289856
$404$	0	0
$405$	30.0207	1.49174
$406$	0	0
$407$	1.83193	0.0908056
$408$	0	0
$409$	−3.26862	−0.161623	−0.0808113	−	0.996729i	$-0.525751\pi$
$409$	−3.26862	−0.161623	−0.0808113	+	0.996729i	$0.525751\pi$
$410$	0	0
$411$	4.80060	0.236796
$412$	0	0
$413$	11.8022	0.580748
$414$	0	0
$415$	10.1275	0.497141
$416$	0	0
$417$	6.68425	0.327329
$418$	0	0
$419$	18.0690	0.882727	0.441364	−	0.897328i	$-0.354495\pi$
$419$	18.0690	0.882727	0.441364	+	0.897328i	$0.354495\pi$
$420$	0	0
$421$	37.1551	1.81083	0.905414	−	0.424531i	$-0.139561\pi$
$421$	37.1551	1.81083	0.905414	+	0.424531i	$0.139561\pi$
$422$	0	0
$423$	−0.130647	−0.00635229
$424$	0	0
$425$	0.416011	0.0201795
$426$	0	0
$427$	−0.744116	−0.0360103
$428$	0	0
$429$	1.75763	0.0848591
$430$	0	0
$431$	−23.8682	−1.14969	−0.574845	−	0.818262i	$-0.694938\pi$
$431$	−23.8682	−1.14969	−0.574845	+	0.818262i	$0.694938\pi$
$432$	0	0
$433$	19.6722	0.945383	0.472692	−	0.881228i	$-0.343282\pi$
$433$	19.6722	0.945383	0.472692	+	0.881228i	$0.343282\pi$
$434$	0	0
$435$	−18.3985	−0.882140
$436$	0	0
$437$	56.7078	2.71270
$438$	0	0
$439$	9.84411	0.469834	0.234917	−	0.972015i	$-0.424518\pi$
$439$	9.84411	0.469834	0.234917	+	0.972015i	$0.424518\pi$
$440$	0	0
$441$	0.0892594	0.00425045
$442$	0	0
$443$	7.77433	0.369370	0.184685	−	0.982798i	$-0.440874\pi$
$443$	7.77433	0.369370	0.184685	+	0.982798i	$0.440874\pi$
$444$	0	0
$445$	−7.62576	−0.361496
$446$	0	0
$447$	−32.7895	−1.55089
$448$	0	0
$449$	20.8542	0.984169	0.492084	−	0.870548i	$-0.336235\pi$
$449$	20.8542	0.984169	0.492084	+	0.870548i	$0.336235\pi$
$450$	0	0
$451$	9.76053	0.459605
$452$	0	0
$453$	−39.6849	−1.86456
$454$	0	0
$455$	−3.24204	−0.151989
$456$	0	0
$457$	11.3836	0.532505	0.266252	−	0.963903i	$-0.414215\pi$
$457$	11.3836	0.532505	0.266252	+	0.963903i	$0.414215\pi$
$458$	0	0
$459$	−0.386204	−0.0180264
$460$	0	0
$461$	1.52353	0.0709581	0.0354790	−	0.999370i	$-0.488704\pi$
$461$	1.52353	0.0709581	0.0354790	+	0.999370i	$0.488704\pi$
$462$	0	0
$463$	4.76974	0.221668	0.110834	−	0.993839i	$-0.464648\pi$
$463$	4.76974	0.221668	0.110834	+	0.993839i	$0.464648\pi$
$464$	0	0
$465$	−33.1575	−1.53764
$466$	0	0
$467$	11.5561	0.534753	0.267377	−	0.963592i	$-0.413843\pi$
$467$	11.5561	0.534753	0.267377	+	0.963592i	$0.413843\pi$
$468$	0	0
$469$	−10.6894	−0.493589
$470$	0	0
$471$	33.3227	1.53543
$472$	0	0
$473$	−3.89313	−0.179006
$474$	0	0
$475$	−36.0206	−1.65274
$476$	0	0
$477$	0.790170	0.0361794
$478$	0	0
$479$	24.2969	1.11016	0.555078	−	0.831798i	$-0.312689\pi$
$479$	24.2969	1.11016	0.555078	+	0.831798i	$0.312689\pi$
$480$	0	0
$481$	1.83193	0.0835290
$482$	0	0
$483$	−15.2489	−0.693847
$484$	0	0
$485$	54.8949	2.49265
$486$	0	0
$487$	20.6886	0.937489	0.468744	−	0.883334i	$-0.344707\pi$
$487$	20.6886	0.937489	0.468744	+	0.883334i	$0.344707\pi$
$488$	0	0
$489$	−14.4013	−0.651249
$490$	0	0
$491$	−28.9083	−1.30461	−0.652307	−	0.757955i	$-0.726199\pi$
$491$	−28.9083	−1.30461	−0.652307	+	0.757955i	$0.726199\pi$
$492$	0	0
$493$	0.243737	0.0109774
$494$	0	0
$495$	−0.289383	−0.0130068
$496$	0	0
$497$	7.68968	0.344929
$498$	0	0
$499$	−19.2643	−0.862387	−0.431194	−	0.902259i	$-0.641907\pi$
$499$	−19.2643	−0.862387	−0.431194	+	0.902259i	$0.641907\pi$
$500$	0	0
$501$	1.18403	0.0528986
$502$	0	0
$503$	7.73844	0.345040	0.172520	−	0.985006i	$-0.444809\pi$
$503$	7.73844	0.345040	0.172520	+	0.985006i	$0.444809\pi$
$504$	0	0
$505$	56.1563	2.49892
$506$	0	0
$507$	1.75763	0.0780590
$508$	0	0
$509$	21.3765	0.947497	0.473748	−	0.880660i	$-0.342901\pi$
$509$	21.3765	0.947497	0.473748	+	0.880660i	$0.342901\pi$
$510$	0	0
$511$	9.86228	0.436282
$512$	0	0
$513$	33.4398	1.47640
$514$	0	0
$515$	−43.7961	−1.92989
$516$	0	0
$517$	−1.46368	−0.0643726
$518$	0	0
$519$	31.0390	1.36246
$520$	0	0
$521$	2.83720	0.124300	0.0621500	−	0.998067i	$-0.480204\pi$
$521$	2.83720	0.124300	0.0621500	+	0.998067i	$0.480204\pi$
$522$	0	0
$523$	−2.47295	−0.108134	−0.0540672	−	0.998537i	$-0.517219\pi$
$523$	−2.47295	−0.108134	−0.0540672	+	0.998537i	$0.517219\pi$
$524$	0	0
$525$	9.68604	0.422733
$526$	0	0
$527$	0.439259	0.0191344
$528$	0	0
$529$	52.2697	2.27259
$530$	0	0
$531$	1.05346	0.0457161
$532$	0	0
$533$	9.76053	0.422776
$534$	0	0
$535$	−49.5393	−2.14177
$536$	0	0
$537$	−1.37684	−0.0594152
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	14.5729	0.626540	0.313270	−	0.949664i	$-0.398576\pi$
$541$	14.5729	0.626540	0.313270	+	0.949664i	$0.398576\pi$
$542$	0	0
$543$	−11.4197	−0.490065
$544$	0	0
$545$	−55.6129	−2.38220
$546$	0	0
$547$	−22.4275	−0.958931	−0.479466	−	0.877561i	$-0.659169\pi$
$547$	−22.4275	−0.958931	−0.479466	+	0.877561i	$0.659169\pi$
$548$	0	0
$549$	−0.0664193	−0.00283471
$550$	0	0
$551$	−21.1042	−0.899068
$552$	0	0
$553$	10.6204	0.451627
$554$	0	0
$555$	−10.4389	−0.443108
$556$	0	0
$557$	−11.4660	−0.485832	−0.242916	−	0.970047i	$-0.578104\pi$
$557$	−11.4660	−0.485832	−0.242916	+	0.970047i	$0.578104\pi$
$558$	0	0
$559$	−3.89313	−0.164662
$560$	0	0
$561$	0.132682	0.00560185
$562$	0	0
$563$	23.2863	0.981401	0.490700	−	0.871328i	$-0.336741\pi$
$563$	23.2863	0.981401	0.490700	+	0.871328i	$0.336741\pi$
$564$	0	0
$565$	−64.1307	−2.69800
$566$	0	0
$567$	−9.25981	−0.388876
$568$	0	0
$569$	−14.3211	−0.600371	−0.300186	−	0.953881i	$-0.597049\pi$
$569$	−14.3211	−0.600371	−0.300186	+	0.953881i	$0.597049\pi$
$570$	0	0
$571$	−15.7368	−0.658564	−0.329282	−	0.944232i	$-0.606807\pi$
$571$	−15.7368	−0.658564	−0.329282	+	0.944232i	$0.606807\pi$
$572$	0	0
$573$	33.9738	1.41928
$574$	0	0
$575$	−47.8111	−1.99386
$576$	0	0
$577$	−13.0408	−0.542895	−0.271448	−	0.962453i	$-0.587502\pi$
$577$	−13.0408	−0.542895	−0.271448	+	0.962453i	$0.587502\pi$
$578$	0	0
$579$	−23.5275	−0.977771
$580$	0	0
$581$	−3.12381	−0.129597
$582$	0	0
$583$	8.85251	0.366634
$584$	0	0
$585$	−0.289383	−0.0119645
$586$	0	0
$587$	18.7549	0.774097	0.387048	−	0.922059i	$-0.373495\pi$
$587$	18.7549	0.774097	0.387048	+	0.922059i	$0.373495\pi$
$588$	0	0
$589$	−38.0336	−1.56715
$590$	0	0
$591$	0.895678	0.0368433
$592$	0	0
$593$	−38.4021	−1.57698	−0.788492	−	0.615045i	$-0.789138\pi$
$593$	−38.4021	−1.57698	−0.788492	+	0.615045i	$0.789138\pi$
$594$	0	0
$595$	−0.244740	−0.0100334
$596$	0	0
$597$	28.6223	1.17143
$598$	0	0
$599$	33.4888	1.36831	0.684157	−	0.729335i	$-0.260170\pi$
$599$	33.4888	1.36831	0.684157	+	0.729335i	$0.260170\pi$
$600$	0	0
$601$	23.7671	0.969481	0.484741	−	0.874658i	$-0.338914\pi$
$601$	23.7671	0.969481	0.484741	+	0.874658i	$0.338914\pi$
$602$	0	0
$603$	−0.954126	−0.0388550
$604$	0	0
$605$	−3.24204	−0.131808
$606$	0	0
$607$	35.8829	1.45644	0.728221	−	0.685343i	$-0.240348\pi$
$607$	35.8829	1.45644	0.728221	+	0.685343i	$0.240348\pi$
$608$	0	0
$609$	5.67496	0.229961
$610$	0	0
$611$	−1.46368	−0.0592142
$612$	0	0
$613$	−29.5774	−1.19462	−0.597309	−	0.802011i	$-0.703763\pi$
$613$	−29.5774	−1.19462	−0.597309	+	0.802011i	$0.703763\pi$
$614$	0	0
$615$	−55.6186	−2.24276
$616$	0	0
$617$	−40.5472	−1.63237	−0.816185	−	0.577790i	$-0.803915\pi$
$617$	−40.5472	−1.63237	−0.816185	+	0.577790i	$0.803915\pi$
$618$	0	0
$619$	4.13866	0.166347	0.0831735	−	0.996535i	$-0.473494\pi$
$619$	4.13866	0.166347	0.0831735	+	0.996535i	$0.473494\pi$
$620$	0	0
$621$	44.3855	1.78113
$622$	0	0
$623$	2.35214	0.0942367
$624$	0	0
$625$	−22.1848	−0.887390
$626$	0	0
$627$	−11.4884	−0.458803
$628$	0	0
$629$	0.138291	0.00551404
$630$	0	0
$631$	−18.1415	−0.722203	−0.361102	−	0.932526i	$-0.617599\pi$
$631$	−18.1415	−0.722203	−0.361102	+	0.932526i	$0.617599\pi$
$632$	0	0
$633$	−20.0375	−0.796418
$634$	0	0
$635$	26.2929	1.04340
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	0.686376	0.0271526
$640$	0	0
$641$	−8.31231	−0.328317	−0.164158	−	0.986434i	$-0.552491\pi$
$641$	−8.31231	−0.328317	−0.164158	+	0.986434i	$0.552491\pi$
$642$	0	0
$643$	−35.2533	−1.39025	−0.695127	−	0.718887i	$-0.744652\pi$
$643$	−35.2533	−1.39025	−0.695127	+	0.718887i	$0.744652\pi$
$644$	0	0
$645$	22.1843	0.873505
$646$	0	0
$647$	19.1678	0.753563	0.376781	−	0.926302i	$-0.377031\pi$
$647$	19.1678	0.753563	0.376781	+	0.926302i	$0.377031\pi$
$648$	0	0
$649$	11.8022	0.463277
$650$	0	0
$651$	10.2273	0.400841
$652$	0	0
$653$	13.6834	0.535471	0.267736	−	0.963492i	$-0.413725\pi$
$653$	13.6834	0.535471	0.267736	+	0.963492i	$0.413725\pi$
$654$	0	0
$655$	15.6507	0.611522
$656$	0	0
$657$	0.880301	0.0343438
$658$	0	0
$659$	6.23166	0.242751	0.121375	−	0.992607i	$-0.461270\pi$
$659$	6.23166	0.242751	0.121375	+	0.992607i	$0.461270\pi$
$660$	0	0
$661$	4.98032	0.193712	0.0968559	−	0.995298i	$-0.469121\pi$
$661$	4.98032	0.193712	0.0968559	+	0.995298i	$0.469121\pi$
$662$	0	0
$663$	0.132682	0.00515295
$664$	0	0
$665$	21.1910	0.821752
$666$	0	0
$667$	−28.0121	−1.08463
$668$	0	0
$669$	−23.0414	−0.890831
$670$	0	0
$671$	−0.744116	−0.0287263
$672$	0	0
$673$	−3.47082	−0.133790	−0.0668951	−	0.997760i	$-0.521309\pi$
$673$	−3.47082	−0.133790	−0.0668951	+	0.997760i	$0.521309\pi$
$674$	0	0
$675$	−28.1935	−1.08517
$676$	0	0
$677$	27.0278	1.03876	0.519382	−	0.854542i	$-0.326162\pi$
$677$	27.0278	1.03876	0.519382	+	0.854542i	$0.326162\pi$
$678$	0	0
$679$	−16.9322	−0.649797
$680$	0	0
$681$	−17.7668	−0.680824
$682$	0	0
$683$	−26.2660	−1.00504	−0.502521	−	0.864565i	$-0.667594\pi$
$683$	−26.2660	−1.00504	−0.502521	+	0.864565i	$0.667594\pi$
$684$	0	0
$685$	−8.85497	−0.338331
$686$	0	0
$687$	9.18656	0.350489
$688$	0	0
$689$	8.85251	0.337254
$690$	0	0
$691$	−42.8341	−1.62949	−0.814743	−	0.579822i	$-0.803122\pi$
$691$	−42.8341	−1.62949	−0.814743	+	0.579822i	$0.803122\pi$
$692$	0	0
$693$	0.0892594	0.00339068
$694$	0	0
$695$	−12.3295	−0.467683
$696$	0	0
$697$	0.736816	0.0279089
$698$	0	0
$699$	18.9405	0.716397
$700$	0	0
$701$	−32.4378	−1.22516	−0.612579	−	0.790409i	$-0.709868\pi$
$701$	−32.4378	−1.22516	−0.612579	+	0.790409i	$0.709868\pi$
$702$	0	0
$703$	−11.9741	−0.451611
$704$	0	0
$705$	8.34051	0.314122
$706$	0	0
$707$	−17.3213	−0.651433
$708$	0	0
$709$	13.5956	0.510593	0.255296	−	0.966863i	$-0.417827\pi$
$709$	13.5956	0.510593	0.255296	+	0.966863i	$0.417827\pi$
$710$	0	0
$711$	0.947975	0.0355518
$712$	0	0
$713$	−50.4830	−1.89060
$714$	0	0
$715$	−3.24204	−0.121246
$716$	0	0
$717$	48.1240	1.79722
$718$	0	0
$719$	3.14514	0.117294	0.0586469	−	0.998279i	$-0.481321\pi$
$719$	3.14514	0.117294	0.0586469	+	0.998279i	$0.481321\pi$
$720$	0	0
$721$	13.5088	0.503093
$722$	0	0
$723$	18.9802	0.705883
$724$	0	0
$725$	17.7932	0.660824
$726$	0	0
$727$	−18.2267	−0.675991	−0.337996	−	0.941148i	$-0.609749\pi$
$727$	−18.2267	−0.675991	−0.337996	+	0.941148i	$0.609749\pi$
$728$	0	0
$729$	26.1496	0.968503
$730$	0	0
$731$	−0.293890	−0.0108699
$732$	0	0
$733$	2.06580	0.0763022	0.0381511	−	0.999272i	$-0.487853\pi$
$733$	2.06580	0.0763022	0.0381511	+	0.999272i	$0.487853\pi$
$734$	0	0
$735$	−5.69831	−0.210185
$736$	0	0
$737$	−10.6894	−0.393748
$738$	0	0
$739$	1.79633	0.0660791	0.0330395	−	0.999454i	$-0.489481\pi$
$739$	1.79633	0.0660791	0.0330395	+	0.999454i	$0.489481\pi$
$740$	0	0
$741$	−11.4884	−0.422037
$742$	0	0
$743$	−30.3480	−1.11336	−0.556679	−	0.830728i	$-0.687925\pi$
$743$	−30.3480	−1.11336	−0.556679	+	0.830728i	$0.687925\pi$
$744$	0	0
$745$	60.4820	2.21589
$746$	0	0
$747$	−0.278829	−0.0102018
$748$	0	0
$749$	15.2803	0.558328
$750$	0	0
$751$	−7.40115	−0.270072	−0.135036	−	0.990841i	$-0.543115\pi$
$751$	−7.40115	−0.270072	−0.135036	+	0.990841i	$0.543115\pi$
$752$	0	0
$753$	−28.4254	−1.03588
$754$	0	0
$755$	73.2010	2.66406
$756$	0	0
$757$	−46.6920	−1.69705	−0.848525	−	0.529155i	$-0.822509\pi$
$757$	−46.6920	−1.69705	−0.848525	+	0.529155i	$0.822509\pi$
$758$	0	0
$759$	−15.2489	−0.553498
$760$	0	0
$761$	−3.34331	−0.121195	−0.0605975	−	0.998162i	$-0.519301\pi$
$761$	−3.34331	−0.121195	−0.0605975	+	0.998162i	$0.519301\pi$
$762$	0	0
$763$	17.1537	0.621004
$764$	0	0
$765$	−0.0218453	−0.000789820 0
$766$	0	0
$767$	11.8022	0.426153
$768$	0	0
$769$	−17.1711	−0.619205	−0.309603	−	0.950866i	$-0.600196\pi$
$769$	−17.1711	−0.619205	−0.309603	+	0.950866i	$0.600196\pi$
$770$	0	0
$771$	20.2470	0.729179
$772$	0	0
$773$	−16.0093	−0.575816	−0.287908	−	0.957658i	$-0.592960\pi$
$773$	−16.0093	−0.575816	−0.287908	+	0.957658i	$0.592960\pi$
$774$	0	0
$775$	32.0667	1.15187
$776$	0	0
$777$	3.21986	0.115512
$778$	0	0
$779$	−63.7978	−2.28580
$780$	0	0
$781$	7.68968	0.275158
$782$	0	0
$783$	−16.5183	−0.590317
$784$	0	0
$785$	−61.4656	−2.19380
$786$	0	0
$787$	6.08461	0.216893	0.108446	−	0.994102i	$-0.465412\pi$
$787$	6.08461	0.216893	0.108446	+	0.994102i	$0.465412\pi$
$788$	0	0
$789$	50.5706	1.80036
$790$	0	0
$791$	19.7809	0.703330
$792$	0	0
$793$	−0.744116	−0.0264243
$794$	0	0
$795$	−50.4444	−1.78908
$796$	0	0
$797$	4.91516	0.174104	0.0870520	−	0.996204i	$-0.472255\pi$
$797$	4.91516	0.174104	0.0870520	+	0.996204i	$0.472255\pi$
$798$	0	0
$799$	−0.110492	−0.00390894
$800$	0	0
$801$	0.209951	0.00741825
$802$	0	0
$803$	9.86228	0.348032
$804$	0	0
$805$	28.1274	0.991359
$806$	0	0
$807$	28.9857	1.02035
$808$	0	0
$809$	−29.3258	−1.03104	−0.515520	−	0.856877i	$-0.672401\pi$
$809$	−29.3258	−1.03104	−0.515520	+	0.856877i	$0.672401\pi$
$810$	0	0
$811$	30.3747	1.06660	0.533300	−	0.845926i	$-0.320952\pi$
$811$	30.3747	1.06660	0.533300	+	0.845926i	$0.320952\pi$
$812$	0	0
$813$	−26.4467	−0.927525
$814$	0	0
$815$	26.5640	0.930496
$816$	0	0
$817$	25.4467	0.890267
$818$	0	0
$819$	0.0892594	0.00311898
$820$	0	0
$821$	−1.15582	−0.0403383	−0.0201692	−	0.999797i	$-0.506420\pi$
$821$	−1.15582	−0.0403383	−0.0201692	+	0.999797i	$0.506420\pi$
$822$	0	0
$823$	45.3407	1.58048	0.790239	−	0.612799i	$-0.209956\pi$
$823$	45.3407	1.58048	0.790239	+	0.612799i	$0.209956\pi$
$824$	0	0
$825$	9.68604	0.337225
$826$	0	0
$827$	−47.2442	−1.64284	−0.821420	−	0.570323i	$-0.806818\pi$
$827$	−47.2442	−1.64284	−0.821420	+	0.570323i	$0.806818\pi$
$828$	0	0
$829$	10.0719	0.349812	0.174906	−	0.984585i	$-0.444038\pi$
$829$	10.0719	0.349812	0.174906	+	0.984585i	$0.444038\pi$
$830$	0	0
$831$	−18.8477	−0.653819
$832$	0	0
$833$	0.0754893	0.00261555
$834$	0	0
$835$	−2.18401	−0.0755808
$836$	0	0
$837$	−29.7691	−1.02897
$838$	0	0
$839$	46.2194	1.59567	0.797835	−	0.602875i	$-0.205978\pi$
$839$	46.2194	1.59567	0.797835	+	0.602875i	$0.205978\pi$
$840$	0	0
$841$	−18.5751	−0.640521
$842$	0	0
$843$	−38.3558	−1.32105
$844$	0	0
$845$	−3.24204	−0.111530
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	13.0881	0.449183
$850$	0	0
$851$	−15.8935	−0.544822
$852$	0	0
$853$	−17.2108	−0.589285	−0.294642	−	0.955608i	$-0.595201\pi$
$853$	−17.2108	−0.589285	−0.294642	+	0.955608i	$0.595201\pi$
$854$	0	0
$855$	1.89150	0.0646878
$856$	0	0
$857$	−35.7450	−1.22103	−0.610514	−	0.792006i	$-0.709037\pi$
$857$	−35.7450	−1.22103	−0.610514	+	0.792006i	$0.709037\pi$
$858$	0	0
$859$	−31.9760	−1.09101	−0.545504	−	0.838108i	$-0.683662\pi$
$859$	−31.9760	−1.09101	−0.545504	+	0.838108i	$0.683662\pi$
$860$	0	0
$861$	17.1554	0.584654
$862$	0	0
$863$	−32.3699	−1.10188	−0.550942	−	0.834543i	$-0.685731\pi$
$863$	−32.3699	−1.10188	−0.550942	+	0.834543i	$0.685731\pi$
$864$	0	0
$865$	−57.2531	−1.94667
$866$	0	0
$867$	−29.8697	−1.01443
$868$	0	0
$869$	10.6204	0.360274
$870$	0	0
$871$	−10.6894	−0.362195
$872$	0	0
$873$	−1.51136	−0.0511517
$874$	0	0
$875$	−1.65621	−0.0559902
$876$	0	0
$877$	−14.7473	−0.497980	−0.248990	−	0.968506i	$-0.580099\pi$
$877$	−14.7473	−0.497980	−0.248990	+	0.968506i	$0.580099\pi$
$878$	0	0
$879$	52.8623	1.78300
$880$	0	0
$881$	32.7221	1.10244	0.551218	−	0.834361i	$-0.314163\pi$
$881$	32.7221	1.10244	0.551218	+	0.834361i	$0.314163\pi$
$882$	0	0
$883$	−56.9772	−1.91743	−0.958717	−	0.284361i	$-0.908218\pi$
$883$	−56.9772	−1.91743	−0.958717	+	0.284361i	$0.908218\pi$
$884$	0	0
$885$	−67.2526	−2.26067
$886$	0	0
$887$	−11.6141	−0.389962	−0.194981	−	0.980807i	$-0.562465\pi$
$887$	−11.6141	−0.389962	−0.194981	+	0.980807i	$0.562465\pi$
$888$	0	0
$889$	−8.10996	−0.271999
$890$	0	0
$891$	−9.25981	−0.310215
$892$	0	0
$893$	9.56707	0.320150
$894$	0	0
$895$	2.53967	0.0848917
$896$	0	0
$897$	−15.2489	−0.509144
$898$	0	0
$899$	18.7876	0.626601
$900$	0	0
$901$	0.668270	0.0222633
$902$	0	0
$903$	−6.84268	−0.227710
$904$	0	0
$905$	21.0642	0.700199
$906$	0	0
$907$	−41.3541	−1.37314	−0.686570	−	0.727064i	$-0.740884\pi$
$907$	−41.3541	−1.37314	−0.686570	+	0.727064i	$0.740884\pi$
$908$	0	0
$909$	−1.54608	−0.0512804
$910$	0	0
$911$	40.2146	1.33237	0.666184	−	0.745787i	$-0.267927\pi$
$911$	40.2146	1.33237	0.666184	+	0.745787i	$0.267927\pi$
$912$	0	0
$913$	−3.12381	−0.103383
$914$	0	0
$915$	4.24020	0.140177
$916$	0	0
$917$	−4.82740	−0.159415
$918$	0	0
$919$	−3.68843	−0.121670	−0.0608350	−	0.998148i	$-0.519376\pi$
$919$	−3.68843	−0.121670	−0.0608350	+	0.998148i	$0.519376\pi$
$920$	0	0
$921$	39.2124	1.29209
$922$	0	0
$923$	7.68968	0.253109
$924$	0	0
$925$	10.0955	0.331939
$926$	0	0
$927$	1.20579	0.0396032
$928$	0	0
$929$	13.8268	0.453644	0.226822	−	0.973936i	$-0.427166\pi$
$929$	13.8268	0.453644	0.226822	+	0.973936i	$0.427166\pi$
$930$	0	0
$931$	−6.53631	−0.214219
$932$	0	0
$933$	18.9572	0.620631
$934$	0	0
$935$	−0.244740	−0.00800385
$936$	0	0
$937$	4.63089	0.151285	0.0756423	−	0.997135i	$-0.475899\pi$
$937$	4.63089	0.151285	0.0756423	+	0.997135i	$0.475899\pi$
$938$	0	0
$939$	−32.7920	−1.07013
$940$	0	0
$941$	−53.8959	−1.75695	−0.878477	−	0.477784i	$-0.841440\pi$
$941$	−53.8959	−1.75695	−0.878477	+	0.477784i	$0.841440\pi$
$942$	0	0
$943$	−84.6805	−2.75758
$944$	0	0
$945$	16.5863	0.539553
$946$	0	0
$947$	43.2368	1.40501	0.702503	−	0.711681i	$-0.252066\pi$
$947$	43.2368	1.40501	0.702503	+	0.711681i	$0.252066\pi$
$948$	0	0
$949$	9.86228	0.320143
$950$	0	0
$951$	56.4841	1.83162
$952$	0	0
$953$	60.4043	1.95669	0.978344	−	0.206984i	$-0.0663647\pi$
$953$	60.4043	1.95669	0.978344	+	0.206984i	$0.0663647\pi$
$954$	0	0
$955$	−62.6666	−2.02784
$956$	0	0
$957$	5.67496	0.183445
$958$	0	0
$959$	2.73129	0.0881980
$960$	0	0
$961$	2.85873	0.0922170
$962$	0	0
$963$	1.36391	0.0439513
$964$	0	0
$965$	43.3979	1.39703
$966$	0	0
$967$	−44.1489	−1.41973	−0.709866	−	0.704337i	$-0.751245\pi$
$967$	−44.1489	−1.41973	−0.709866	+	0.704337i	$0.751245\pi$
$968$	0	0
$969$	−0.867252	−0.0278601
$970$	0	0
$971$	8.39621	0.269447	0.134724	−	0.990883i	$-0.456985\pi$
$971$	8.39621	0.269447	0.134724	+	0.990883i	$0.456985\pi$
$972$	0	0
$973$	3.80299	0.121918
$974$	0	0
$975$	9.68604	0.310201
$976$	0	0
$977$	48.7280	1.55895	0.779473	−	0.626436i	$-0.215487\pi$
$977$	48.7280	1.55895	0.779473	+	0.626436i	$0.215487\pi$
$978$	0	0
$979$	2.35214	0.0751748
$980$	0	0
$981$	1.53112	0.0488851
$982$	0	0
$983$	−0.416637	−0.0132887	−0.00664433	−	0.999978i	$-0.502115\pi$
$983$	−0.416637	−0.0132887	−0.00664433	+	0.999978i	$0.502115\pi$
$984$	0	0
$985$	−1.65213	−0.0526412
$986$	0	0
$987$	−2.57261	−0.0818870
$988$	0	0
$989$	33.7760	1.07402
$990$	0	0
$991$	−50.6837	−1.61002	−0.805010	−	0.593261i	$-0.797840\pi$
$991$	−50.6837	−1.61002	−0.805010	+	0.593261i	$0.797840\pi$
$992$	0	0
$993$	−0.488733	−0.0155095
$994$	0	0
$995$	−52.7955	−1.67373
$996$	0	0
$997$	8.03497	0.254470	0.127235	−	0.991873i	$-0.459390\pi$
$997$	8.03497	0.254470	0.127235	+	0.991873i	$0.459390\pi$
$998$	0	0
$999$	−9.37217	−0.296522

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.x.1.9	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.x.1.9	✓	12	1.1	even	1	trivial

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Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.75763 q^{3} -3.24204 q^{5} +1.00000 q^{7} +0.0892594 q^{9} +O(q^{10})\)	\(q+1.75763 q^{3} -3.24204 q^{5} +1.00000 q^{7} +0.0892594 q^{9} +1.00000 q^{11} +1.00000 q^{13} -5.69831 q^{15} +0.0754893 q^{17} -6.53631 q^{19} +1.75763 q^{21} -8.67581 q^{23} +5.51085 q^{25} -5.11600 q^{27} +3.22876 q^{29} +5.81883 q^{31} +1.75763 q^{33} -3.24204 q^{35} +1.83193 q^{37} +1.75763 q^{39} +9.76053 q^{41} -3.89313 q^{43} -0.289383 q^{45} -1.46368 q^{47} +1.00000 q^{49} +0.132682 q^{51} +8.85251 q^{53} -3.24204 q^{55} -11.4884 q^{57} +11.8022 q^{59} -0.744116 q^{61} +0.0892594 q^{63} -3.24204 q^{65} -10.6894 q^{67} -15.2489 q^{69} +7.68968 q^{71} +9.86228 q^{73} +9.68604 q^{75} +1.00000 q^{77} +10.6204 q^{79} -9.25981 q^{81} -3.12381 q^{83} -0.244740 q^{85} +5.67496 q^{87} +2.35214 q^{89} +1.00000 q^{91} +10.2273 q^{93} +21.1910 q^{95} -16.9322 q^{97} +0.0892594 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9}+O(q^{10}) \) 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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