LMFDB - Embedded newform 8008.2.a.r.1.2

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} - 4x^{8} - 9x^{7} + 36x^{6} + 23x^{5} - 89x^{4} - 20x^{3} + 51x^{2} + 18x + 1 $ x^9 - 4x^8 - 9x^7 + 36x^6 + 23x^5 - 89x^4 - 20x^3 + 51x^2 + 18x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.28175$ 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)

$f(q)$	$=$	$q-1.28175 q^{3} +0.660545 q^{5} -1.00000 q^{7} -1.35712 q^{9} +O(q^{10})$	$q-1.28175 q^{3} +0.660545 q^{5} -1.00000 q^{7} -1.35712 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.846653 q^{15} +6.29996 q^{17} +6.89288 q^{19} +1.28175 q^{21} -0.507379 q^{23} -4.56368 q^{25} +5.58473 q^{27} -5.42153 q^{29} +6.19617 q^{31} +1.28175 q^{33} -0.660545 q^{35} +1.86702 q^{37} +1.28175 q^{39} +5.59719 q^{41} -6.02385 q^{43} -0.896437 q^{45} -4.97095 q^{47} +1.00000 q^{49} -8.07497 q^{51} -4.45633 q^{53} -0.660545 q^{55} -8.83494 q^{57} +5.22536 q^{59} +2.34510 q^{61} +1.35712 q^{63} -0.660545 q^{65} -9.97202 q^{67} +0.650333 q^{69} -7.59229 q^{71} -9.30560 q^{73} +5.84950 q^{75} +1.00000 q^{77} -14.6197 q^{79} -3.08688 q^{81} +4.83006 q^{83} +4.16141 q^{85} +6.94904 q^{87} -4.09889 q^{89} +1.00000 q^{91} -7.94194 q^{93} +4.55306 q^{95} +0.382180 q^{97} +1.35712 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9}+O(q^{10}) $ 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9	$ 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9} - 9 q^{11} - 9 q^{13} - 3 q^{15} - 7 q^{17} + 13 q^{19} - 5 q^{21} + 9 q^{23} - 2 q^{25} + 5 q^{27} + q^{29} + 10 q^{31} - 5 q^{33} - 3 q^{35} + 14 q^{37} - 5 q^{39} - 2 q^{41} + 5 q^{43} - 15 q^{45} + 13 q^{47} + 9 q^{49} - 3 q^{51} + 22 q^{53} - 3 q^{55} + 16 q^{57} + 43 q^{59} - 10 q^{61} - 8 q^{63} - 3 q^{65} + 26 q^{67} - 30 q^{69} + 18 q^{71} - 8 q^{73} + 28 q^{75} + 9 q^{77} - 9 q^{79} + 33 q^{81} + 4 q^{83} + 5 q^{85} + 33 q^{87} + 7 q^{89} + 9 q^{91} + 13 q^{93} + 7 q^{95} + 2 q^{97} - 8 q^{99}+O(q^{100}) $ 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9 - 9 * q^11 - 9 * q^13 - 3 * q^15 - 7 * q^17 + 13 * q^19 - 5 * q^21 + 9 * q^23 - 2 * q^25 + 5 * q^27 + q^29 + 10 * q^31 - 5 * q^33 - 3 * q^35 + 14 * q^37 - 5 * q^39 - 2 * q^41 + 5 * q^43 - 15 * q^45 + 13 * q^47 + 9 * q^49 - 3 * q^51 + 22 * q^53 - 3 * q^55 + 16 * q^57 + 43 * q^59 - 10 * q^61 - 8 * q^63 - 3 * q^65 + 26 * q^67 - 30 * q^69 + 18 * q^71 - 8 * q^73 + 28 * q^75 + 9 * q^77 - 9 * q^79 + 33 * q^81 + 4 * q^83 + 5 * q^85 + 33 * q^87 + 7 * q^89 + 9 * q^91 + 13 * q^93 + 7 * q^95 + 2 * q^97 - 8 * 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
$3$	−1.28175	−0.740019	−0.370009	−	0.929028i	$-0.620646\pi$
$3$	−1.28175	−0.740019	−0.370009	+	0.929028i	$0.620646\pi$
$4$	0	0
$5$	0.660545	0.295405	0.147702	−	0.989032i	$-0.452812\pi$
$5$	0.660545	0.295405	0.147702	+	0.989032i	$0.452812\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.35712	−0.452372
$10$	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
$15$	−0.846653	−0.218605
$16$	0	0
$17$	6.29996	1.52796	0.763982	−	0.645237i	$-0.223241\pi$
$17$	6.29996	1.52796	0.763982	+	0.645237i	$0.223241\pi$
$18$	0	0
$19$	6.89288	1.58133	0.790667	−	0.612246i	$-0.209734\pi$
$19$	6.89288	1.58133	0.790667	+	0.612246i	$0.209734\pi$
$20$	0	0
$21$	1.28175	0.279701
$22$	0	0
$23$	−0.507379	−0.105796	−0.0528979	−	0.998600i	$-0.516846\pi$
$23$	−0.507379	−0.105796	−0.0528979	+	0.998600i	$0.516846\pi$
$24$	0	0
$25$	−4.56368	−0.912736
$26$	0	0
$27$	5.58473	1.07478
$28$	0	0
$29$	−5.42153	−1.00675	−0.503376	−	0.864067i	$-0.667909\pi$
$29$	−5.42153	−1.00675	−0.503376	+	0.864067i	$0.667909\pi$
$30$	0	0
$31$	6.19617	1.11287	0.556433	−	0.830893i	$-0.312170\pi$
$31$	6.19617	1.11287	0.556433	+	0.830893i	$0.312170\pi$
$32$	0	0
$33$	1.28175	0.223124
$34$	0	0
$35$	−0.660545	−0.111652
$36$	0	0
$37$	1.86702	0.306936	0.153468	−	0.988154i	$-0.450956\pi$
$37$	1.86702	0.306936	0.153468	+	0.988154i	$0.450956\pi$
$38$	0	0
$39$	1.28175	0.205244
$40$	0	0
$41$	5.59719	0.874134	0.437067	−	0.899429i	$-0.356017\pi$
$41$	5.59719	0.874134	0.437067	+	0.899429i	$0.356017\pi$
$42$	0	0
$43$	−6.02385	−0.918628	−0.459314	−	0.888274i	$-0.651905\pi$
$43$	−6.02385	−0.918628	−0.459314	+	0.888274i	$0.651905\pi$
$44$	0	0
$45$	−0.896437	−0.133633
$46$	0	0
$47$	−4.97095	−0.725088	−0.362544	−	0.931967i	$-0.618092\pi$
$47$	−4.97095	−0.725088	−0.362544	+	0.931967i	$0.618092\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−8.07497	−1.13072
$52$	0	0
$53$	−4.45633	−0.612124	−0.306062	−	0.952012i	$-0.599011\pi$
$53$	−4.45633	−0.612124	−0.306062	+	0.952012i	$0.599011\pi$
$54$	0	0
$55$	−0.660545	−0.0890679
$56$	0	0
$57$	−8.83494	−1.17022
$58$	0	0
$59$	5.22536	0.680283	0.340142	−	0.940374i	$-0.389525\pi$
$59$	5.22536	0.680283	0.340142	+	0.940374i	$0.389525\pi$
$60$	0	0
$61$	2.34510	0.300259	0.150130	−	0.988666i	$-0.452031\pi$
$61$	2.34510	0.300259	0.150130	+	0.988666i	$0.452031\pi$
$62$	0	0
$63$	1.35712	0.170981
$64$	0	0
$65$	−0.660545	−0.0819305
$66$	0	0
$67$	−9.97202	−1.21828	−0.609138	−	0.793064i	$-0.708484\pi$
$67$	−9.97202	−1.21828	−0.609138	+	0.793064i	$0.708484\pi$
$68$	0	0
$69$	0.650333	0.0782909
$70$	0	0
$71$	−7.59229	−0.901039	−0.450520	−	0.892766i	$-0.648761\pi$
$71$	−7.59229	−0.901039	−0.450520	+	0.892766i	$0.648761\pi$
$72$	0	0
$73$	−9.30560	−1.08914	−0.544569	−	0.838716i	$-0.683307\pi$
$73$	−9.30560	−1.08914	−0.544569	+	0.838716i	$0.683307\pi$
$74$	0	0
$75$	5.84950	0.675442
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−14.6197	−1.64485	−0.822424	−	0.568875i	$-0.807379\pi$
$79$	−14.6197	−1.64485	−0.822424	+	0.568875i	$0.807379\pi$
$80$	0	0
$81$	−3.08688	−0.342987
$82$	0	0
$83$	4.83006	0.530168	0.265084	−	0.964225i	$-0.414600\pi$
$83$	4.83006	0.530168	0.265084	+	0.964225i	$0.414600\pi$
$84$	0	0
$85$	4.16141	0.451368
$86$	0	0
$87$	6.94904	0.745015
$88$	0	0
$89$	−4.09889	−0.434482	−0.217241	−	0.976118i	$-0.569706\pi$
$89$	−4.09889	−0.434482	−0.217241	+	0.976118i	$0.569706\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−7.94194	−0.823541
$94$	0	0
$95$	4.55306	0.467134
$96$	0	0
$97$	0.382180	0.0388045	0.0194023	−	0.999812i	$-0.493824\pi$
$97$	0.382180	0.0388045	0.0194023	+	0.999812i	$0.493824\pi$
$98$	0	0
$99$	1.35712	0.136395
$100$	0	0
$101$	−15.2785	−1.52027	−0.760134	−	0.649766i	$-0.774867\pi$
$101$	−15.2785	−1.52027	−0.760134	+	0.649766i	$0.774867\pi$
$102$	0	0
$103$	17.6834	1.74240	0.871199	−	0.490931i	$-0.163343\pi$
$103$	17.6834	1.74240	0.871199	+	0.490931i	$0.163343\pi$
$104$	0	0
$105$	0.846653	0.0826249
$106$	0	0
$107$	17.5425	1.69590	0.847948	−	0.530079i	$-0.177838\pi$
$107$	17.5425	1.69590	0.847948	+	0.530079i	$0.177838\pi$
$108$	0	0
$109$	2.82274	0.270369	0.135185	−	0.990820i	$-0.456837\pi$
$109$	2.82274	0.270369	0.135185	+	0.990820i	$0.456837\pi$
$110$	0	0
$111$	−2.39305	−0.227138
$112$	0	0
$113$	7.48584	0.704209	0.352104	−	0.935961i	$-0.385466\pi$
$113$	7.48584	0.704209	0.352104	+	0.935961i	$0.385466\pi$
$114$	0	0
$115$	−0.335147	−0.0312526
$116$	0	0
$117$	1.35712	0.125466
$118$	0	0
$119$	−6.29996	−0.577516
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−7.17420	−0.646876
$124$	0	0
$125$	−6.31724	−0.565031
$126$	0	0
$127$	−1.18214	−0.104898	−0.0524488	−	0.998624i	$-0.516703\pi$
$127$	−1.18214	−0.104898	−0.0524488	+	0.998624i	$0.516703\pi$
$128$	0	0
$129$	7.72107	0.679802
$130$	0	0
$131$	11.7670	1.02809	0.514044	−	0.857764i	$-0.328147\pi$
$131$	11.7670	1.02809	0.514044	+	0.857764i	$0.328147\pi$
$132$	0	0
$133$	−6.89288	−0.597688
$134$	0	0
$135$	3.68897	0.317496
$136$	0	0
$137$	−3.49941	−0.298975	−0.149487	−	0.988764i	$-0.547762\pi$
$137$	−3.49941	−0.298975	−0.149487	+	0.988764i	$0.547762\pi$
$138$	0	0
$139$	10.9964	0.932698	0.466349	−	0.884601i	$-0.345569\pi$
$139$	10.9964	0.932698	0.466349	+	0.884601i	$0.345569\pi$
$140$	0	0
$141$	6.37151	0.536578
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−3.58116	−0.297399
$146$	0	0
$147$	−1.28175	−0.105717
$148$	0	0
$149$	−18.0331	−1.47733	−0.738664	−	0.674073i	$-0.764543\pi$
$149$	−18.0331	−1.47733	−0.738664	+	0.674073i	$0.764543\pi$
$150$	0	0
$151$	15.4122	1.25422	0.627112	−	0.778929i	$-0.284237\pi$
$151$	15.4122	1.25422	0.627112	+	0.778929i	$0.284237\pi$
$152$	0	0
$153$	−8.54979	−0.691209
$154$	0	0
$155$	4.09285	0.328746
$156$	0	0
$157$	18.7967	1.50014	0.750071	−	0.661357i	$-0.230019\pi$
$157$	18.7967	1.50014	0.750071	+	0.661357i	$0.230019\pi$
$158$	0	0
$159$	5.71190	0.452983
$160$	0	0
$161$	0.507379	0.0399871
$162$	0	0
$163$	15.6795	1.22811	0.614057	−	0.789262i	$-0.289536\pi$
$163$	15.6795	1.22811	0.614057	+	0.789262i	$0.289536\pi$
$164$	0	0
$165$	0.846653	0.0659119
$166$	0	0
$167$	15.1457	1.17201	0.586005	−	0.810307i	$-0.300700\pi$
$167$	15.1457	1.17201	0.586005	+	0.810307i	$0.300700\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−9.35444	−0.715352
$172$	0	0
$173$	8.53874	0.649188	0.324594	−	0.945853i	$-0.394772\pi$
$173$	8.53874	0.649188	0.324594	+	0.945853i	$0.394772\pi$
$174$	0	0
$175$	4.56368	0.344982
$176$	0	0
$177$	−6.69760	−0.503422
$178$	0	0
$179$	6.20350	0.463671	0.231836	−	0.972755i	$-0.425527\pi$
$179$	6.20350	0.463671	0.231836	+	0.972755i	$0.425527\pi$
$180$	0	0
$181$	−15.1875	−1.12888	−0.564440	−	0.825474i	$-0.690908\pi$
$181$	−15.1875	−1.12888	−0.564440	+	0.825474i	$0.690908\pi$
$182$	0	0
$183$	−3.00583	−0.222198
$184$	0	0
$185$	1.23325	0.0906704
$186$	0	0
$187$	−6.29996	−0.460699
$188$	0	0
$189$	−5.58473	−0.406230
$190$	0	0
$191$	0.298302	0.0215843	0.0107922	−	0.999942i	$-0.496565\pi$
$191$	0.298302	0.0215843	0.0107922	+	0.999942i	$0.496565\pi$
$192$	0	0
$193$	−9.39678	−0.676395	−0.338198	−	0.941075i	$-0.609817\pi$
$193$	−9.39678	−0.676395	−0.338198	+	0.941075i	$0.609817\pi$
$194$	0	0
$195$	0.846653	0.0606301
$196$	0	0
$197$	−4.05100	−0.288622	−0.144311	−	0.989532i	$-0.546097\pi$
$197$	−4.05100	−0.288622	−0.144311	+	0.989532i	$0.546097\pi$
$198$	0	0
$199$	5.02018	0.355871	0.177936	−	0.984042i	$-0.443058\pi$
$199$	5.02018	0.355871	0.177936	+	0.984042i	$0.443058\pi$
$200$	0	0
$201$	12.7816	0.901547
$202$	0	0
$203$	5.42153	0.380517
$204$	0	0
$205$	3.69720	0.258223
$206$	0	0
$207$	0.688573	0.0478591
$208$	0	0
$209$	−6.89288	−0.476790
$210$	0	0
$211$	26.9762	1.85712	0.928560	−	0.371183i	$-0.121048\pi$
$211$	26.9762	1.85712	0.928560	+	0.371183i	$0.121048\pi$
$212$	0	0
$213$	9.73142	0.666786
$214$	0	0
$215$	−3.97902	−0.271367
$216$	0	0
$217$	−6.19617	−0.420623
$218$	0	0
$219$	11.9275	0.805982
$220$	0	0
$221$	−6.29996	−0.423781
$222$	0	0
$223$	−17.9065	−1.19911	−0.599554	−	0.800334i	$-0.704655\pi$
$223$	−17.9065	−1.19911	−0.599554	+	0.800334i	$0.704655\pi$
$224$	0	0
$225$	6.19345	0.412897
$226$	0	0
$227$	26.4956	1.75858	0.879288	−	0.476291i	$-0.158019\pi$
$227$	26.4956	1.75858	0.879288	+	0.476291i	$0.158019\pi$
$228$	0	0
$229$	−14.6058	−0.965180	−0.482590	−	0.875846i	$-0.660304\pi$
$229$	−14.6058	−0.965180	−0.482590	+	0.875846i	$0.660304\pi$
$230$	0	0
$231$	−1.28175	−0.0843329
$232$	0	0
$233$	23.4453	1.53595	0.767976	−	0.640478i	$-0.221264\pi$
$233$	23.4453	1.53595	0.767976	+	0.640478i	$0.221264\pi$
$234$	0	0
$235$	−3.28354	−0.214194
$236$	0	0
$237$	18.7388	1.21722
$238$	0	0
$239$	2.97841	0.192657	0.0963287	−	0.995350i	$-0.469290\pi$
$239$	2.97841	0.192657	0.0963287	+	0.995350i	$0.469290\pi$
$240$	0	0
$241$	25.2013	1.62336	0.811679	−	0.584104i	$-0.198554\pi$
$241$	25.2013	1.62336	0.811679	+	0.584104i	$0.198554\pi$
$242$	0	0
$243$	−12.7976	−0.820966
$244$	0	0
$245$	0.660545	0.0422007
$246$	0	0
$247$	−6.89288	−0.438583
$248$	0	0
$249$	−6.19093	−0.392334
$250$	0	0
$251$	3.82529	0.241450	0.120725	−	0.992686i	$-0.461478\pi$
$251$	3.82529	0.241450	0.120725	+	0.992686i	$0.461478\pi$
$252$	0	0
$253$	0.507379	0.0318987
$254$	0	0
$255$	−5.33388	−0.334021
$256$	0	0
$257$	−4.01899	−0.250698	−0.125349	−	0.992113i	$-0.540005\pi$
$257$	−4.01899	−0.250698	−0.125349	+	0.992113i	$0.540005\pi$
$258$	0	0
$259$	−1.86702	−0.116011
$260$	0	0
$261$	7.35765	0.455427
$262$	0	0
$263$	18.3334	1.13048	0.565241	−	0.824926i	$-0.308783\pi$
$263$	18.3334	1.13048	0.565241	+	0.824926i	$0.308783\pi$
$264$	0	0
$265$	−2.94361	−0.180824
$266$	0	0
$267$	5.25376	0.321525
$268$	0	0
$269$	26.3417	1.60608	0.803042	−	0.595922i	$-0.203213\pi$
$269$	26.3417	1.60608	0.803042	+	0.595922i	$0.203213\pi$
$270$	0	0
$271$	5.03816	0.306046	0.153023	−	0.988223i	$-0.451099\pi$
$271$	5.03816	0.306046	0.153023	+	0.988223i	$0.451099\pi$
$272$	0	0
$273$	−1.28175	−0.0775750
$274$	0	0
$275$	4.56368	0.275200
$276$	0	0
$277$	7.88700	0.473884	0.236942	−	0.971524i	$-0.423855\pi$
$277$	7.88700	0.473884	0.236942	+	0.971524i	$0.423855\pi$
$278$	0	0
$279$	−8.40893	−0.503430
$280$	0	0
$281$	−2.48143	−0.148030	−0.0740149	−	0.997257i	$-0.523581\pi$
$281$	−2.48143	−0.148030	−0.0740149	+	0.997257i	$0.523581\pi$
$282$	0	0
$283$	−25.1137	−1.49285	−0.746426	−	0.665468i	$-0.768232\pi$
$283$	−25.1137	−1.49285	−0.746426	+	0.665468i	$0.768232\pi$
$284$	0	0
$285$	−5.83588	−0.345688
$286$	0	0
$287$	−5.59719	−0.330392
$288$	0	0
$289$	22.6895	1.33468
$290$	0	0
$291$	−0.489860	−0.0287161
$292$	0	0
$293$	−6.89671	−0.402910	−0.201455	−	0.979498i	$-0.564567\pi$
$293$	−6.89671	−0.402910	−0.201455	+	0.979498i	$0.564567\pi$
$294$	0	0
$295$	3.45158	0.200959
$296$	0	0
$297$	−5.58473	−0.324059
$298$	0	0
$299$	0.507379	0.0293425
$300$	0	0
$301$	6.02385	0.347209
$302$	0	0
$303$	19.5832	1.12503
$304$	0	0
$305$	1.54904	0.0886980
$306$	0	0
$307$	−7.55328	−0.431088	−0.215544	−	0.976494i	$-0.569152\pi$
$307$	−7.55328	−0.431088	−0.215544	+	0.976494i	$0.569152\pi$
$308$	0	0
$309$	−22.6657	−1.28941
$310$	0	0
$311$	4.40817	0.249965	0.124982	−	0.992159i	$-0.460113\pi$
$311$	4.40817	0.249965	0.124982	+	0.992159i	$0.460113\pi$
$312$	0	0
$313$	−11.5196	−0.651129	−0.325564	−	0.945520i	$-0.605554\pi$
$313$	−11.5196	−0.651129	−0.325564	+	0.945520i	$0.605554\pi$
$314$	0	0
$315$	0.896437	0.0505085
$316$	0	0
$317$	2.45519	0.137897	0.0689486	−	0.997620i	$-0.478036\pi$
$317$	2.45519	0.137897	0.0689486	+	0.997620i	$0.478036\pi$
$318$	0	0
$319$	5.42153	0.303547
$320$	0	0
$321$	−22.4851	−1.25500
$322$	0	0
$323$	43.4249	2.41622
$324$	0	0
$325$	4.56368	0.253147
$326$	0	0
$327$	−3.61804	−0.200078
$328$	0	0
$329$	4.97095	0.274057
$330$	0	0
$331$	−32.7909	−1.80235	−0.901177	−	0.433452i	$-0.857295\pi$
$331$	−32.7909	−1.80235	−0.901177	+	0.433452i	$0.857295\pi$
$332$	0	0
$333$	−2.53377	−0.138849
$334$	0	0
$335$	−6.58697	−0.359884
$336$	0	0
$337$	12.9439	0.705099	0.352550	−	0.935793i	$-0.385315\pi$
$337$	12.9439	0.705099	0.352550	+	0.935793i	$0.385315\pi$
$338$	0	0
$339$	−9.59498	−0.521128
$340$	0	0
$341$	−6.19617	−0.335541
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0.429574	0.0231275
$346$	0	0
$347$	23.6859	1.27153	0.635763	−	0.771884i	$-0.280686\pi$
$347$	23.6859	1.27153	0.635763	+	0.771884i	$0.280686\pi$
$348$	0	0
$349$	6.17150	0.330353	0.165176	−	0.986264i	$-0.447181\pi$
$349$	6.17150	0.330353	0.165176	+	0.986264i	$0.447181\pi$
$350$	0	0
$351$	−5.58473	−0.298091
$352$	0	0
$353$	6.00669	0.319704	0.159852	−	0.987141i	$-0.448898\pi$
$353$	6.00669	0.319704	0.159852	+	0.987141i	$0.448898\pi$
$354$	0	0
$355$	−5.01505	−0.266171
$356$	0	0
$357$	8.07497	0.427373
$358$	0	0
$359$	−19.2935	−1.01827	−0.509137	−	0.860685i	$-0.670035\pi$
$359$	−19.2935	−1.01827	−0.509137	+	0.860685i	$0.670035\pi$
$360$	0	0
$361$	28.5118	1.50062
$362$	0	0
$363$	−1.28175	−0.0672744
$364$	0	0
$365$	−6.14677	−0.321737
$366$	0	0
$367$	31.6341	1.65129	0.825643	−	0.564192i	$-0.190812\pi$
$367$	31.6341	1.65129	0.825643	+	0.564192i	$0.190812\pi$
$368$	0	0
$369$	−7.59605	−0.395434
$370$	0	0
$371$	4.45633	0.231361
$372$	0	0
$373$	1.42361	0.0737117	0.0368559	−	0.999321i	$-0.488266\pi$
$373$	1.42361	0.0737117	0.0368559	+	0.999321i	$0.488266\pi$
$374$	0	0
$375$	8.09712	0.418134
$376$	0	0
$377$	5.42153	0.279223
$378$	0	0
$379$	19.4295	0.998025	0.499013	−	0.866595i	$-0.333696\pi$
$379$	19.4295	0.998025	0.499013	+	0.866595i	$0.333696\pi$
$380$	0	0
$381$	1.51520	0.0776262
$382$	0	0
$383$	−6.05814	−0.309556	−0.154778	−	0.987949i	$-0.549466\pi$
$383$	−6.05814	−0.309556	−0.154778	+	0.987949i	$0.549466\pi$
$384$	0	0
$385$	0.660545	0.0336645
$386$	0	0
$387$	8.17507	0.415562
$388$	0	0
$389$	−37.9160	−1.92242	−0.961208	−	0.275826i	$-0.911049\pi$
$389$	−37.9160	−1.92242	−0.961208	+	0.275826i	$0.911049\pi$
$390$	0	0
$391$	−3.19647	−0.161652
$392$	0	0
$393$	−15.0824	−0.760804
$394$	0	0
$395$	−9.65699	−0.485896
$396$	0	0
$397$	27.4681	1.37858	0.689291	−	0.724484i	$-0.257922\pi$
$397$	27.4681	1.37858	0.689291	+	0.724484i	$0.257922\pi$
$398$	0	0
$399$	8.83494	0.442300
$400$	0	0
$401$	6.45125	0.322160	0.161080	−	0.986941i	$-0.448502\pi$
$401$	6.45125	0.322160	0.161080	+	0.986941i	$0.448502\pi$
$402$	0	0
$403$	−6.19617	−0.308653
$404$	0	0
$405$	−2.03902	−0.101320
$406$	0	0
$407$	−1.86702	−0.0925447
$408$	0	0
$409$	23.4581	1.15993	0.579964	−	0.814642i	$-0.303066\pi$
$409$	23.4581	1.15993	0.579964	+	0.814642i	$0.303066\pi$
$410$	0	0
$411$	4.48537	0.221247
$412$	0	0
$413$	−5.22536	−0.257123
$414$	0	0
$415$	3.19047	0.156614
$416$	0	0
$417$	−14.0946	−0.690214
$418$	0	0
$419$	17.7615	0.867707	0.433854	−	0.900983i	$-0.357154\pi$
$419$	17.7615	0.867707	0.433854	+	0.900983i	$0.357154\pi$
$420$	0	0
$421$	0.418916	0.0204167	0.0102084	−	0.999948i	$-0.496751\pi$
$421$	0.418916	0.0204167	0.0102084	+	0.999948i	$0.496751\pi$
$422$	0	0
$423$	6.74616	0.328010
$424$	0	0
$425$	−28.7510	−1.39463
$426$	0	0
$427$	−2.34510	−0.113487
$428$	0	0
$429$	−1.28175	−0.0618835
$430$	0	0
$431$	−27.8435	−1.34117	−0.670587	−	0.741831i	$-0.733958\pi$
$431$	−27.8435	−1.34117	−0.670587	+	0.741831i	$0.733958\pi$
$432$	0	0
$433$	−18.5267	−0.890338	−0.445169	−	0.895447i	$-0.646856\pi$
$433$	−18.5267	−0.890338	−0.445169	+	0.895447i	$0.646856\pi$
$434$	0	0
$435$	4.59015	0.220081
$436$	0	0
$437$	−3.49730	−0.167299
$438$	0	0
$439$	−5.73042	−0.273498	−0.136749	−	0.990606i	$-0.543665\pi$
$439$	−5.73042	−0.273498	−0.136749	+	0.990606i	$0.543665\pi$
$440$	0	0
$441$	−1.35712	−0.0646246
$442$	0	0
$443$	−38.5522	−1.83167	−0.915836	−	0.401553i	$-0.868470\pi$
$443$	−38.5522	−1.83167	−0.915836	+	0.401553i	$0.868470\pi$
$444$	0	0
$445$	−2.70750	−0.128348
$446$	0	0
$447$	23.1139	1.09325
$448$	0	0
$449$	−27.8436	−1.31402	−0.657010	−	0.753881i	$-0.728179\pi$
$449$	−27.8436	−1.31402	−0.657010	+	0.753881i	$0.728179\pi$
$450$	0	0
$451$	−5.59719	−0.263561
$452$	0	0
$453$	−19.7545	−0.928149
$454$	0	0
$455$	0.660545	0.0309668
$456$	0	0
$457$	25.5091	1.19326	0.596632	−	0.802515i	$-0.296505\pi$
$457$	25.5091	1.19326	0.596632	+	0.802515i	$0.296505\pi$
$458$	0	0
$459$	35.1836	1.64223
$460$	0	0
$461$	−0.618587	−0.0288105	−0.0144052	−	0.999896i	$-0.504585\pi$
$461$	−0.618587	−0.0288105	−0.0144052	+	0.999896i	$0.504585\pi$
$462$	0	0
$463$	34.3772	1.59764	0.798822	−	0.601568i	$-0.205457\pi$
$463$	34.3772	1.59764	0.798822	+	0.601568i	$0.205457\pi$
$464$	0	0
$465$	−5.24601	−0.243278
$466$	0	0
$467$	−34.7955	−1.61015	−0.805073	−	0.593176i	$-0.797874\pi$
$467$	−34.7955	−1.61015	−0.805073	+	0.593176i	$0.797874\pi$
$468$	0	0
$469$	9.97202	0.460465
$470$	0	0
$471$	−24.0927	−1.11013
$472$	0	0
$473$	6.02385	0.276977
$474$	0	0
$475$	−31.4569	−1.44334
$476$	0	0
$477$	6.04776	0.276908
$478$	0	0
$479$	14.1374	0.645953	0.322977	−	0.946407i	$-0.395317\pi$
$479$	14.1374	0.645953	0.322977	+	0.946407i	$0.395317\pi$
$480$	0	0
$481$	−1.86702	−0.0851288
$482$	0	0
$483$	−0.650333	−0.0295912
$484$	0	0
$485$	0.252447	0.0114630
$486$	0	0
$487$	−12.8707	−0.583226	−0.291613	−	0.956536i	$-0.594192\pi$
$487$	−12.8707	−0.583226	−0.291613	+	0.956536i	$0.594192\pi$
$488$	0	0
$489$	−20.0972	−0.908827
$490$	0	0
$491$	34.5735	1.56028	0.780141	−	0.625604i	$-0.215147\pi$
$491$	34.5735	1.56028	0.780141	+	0.625604i	$0.215147\pi$
$492$	0	0
$493$	−34.1554	−1.53828
$494$	0	0
$495$	0.896437	0.0402919
$496$	0	0
$497$	7.59229	0.340561
$498$	0	0
$499$	−5.03551	−0.225420	−0.112710	−	0.993628i	$-0.535953\pi$
$499$	−5.03551	−0.225420	−0.112710	+	0.993628i	$0.535953\pi$
$500$	0	0
$501$	−19.4130	−0.867309
$502$	0	0
$503$	40.1200	1.78886	0.894431	−	0.447207i	$-0.147581\pi$
$503$	40.1200	1.78886	0.894431	+	0.447207i	$0.147581\pi$
$504$	0	0
$505$	−10.0921	−0.449094
$506$	0	0
$507$	−1.28175	−0.0569245
$508$	0	0
$509$	−8.02963	−0.355907	−0.177954	−	0.984039i	$-0.556948\pi$
$509$	−8.02963	−0.355907	−0.177954	+	0.984039i	$0.556948\pi$
$510$	0	0
$511$	9.30560	0.411655
$512$	0	0
$513$	38.4949	1.69959
$514$	0	0
$515$	11.6807	0.514712
$516$	0	0
$517$	4.97095	0.218622
$518$	0	0
$519$	−10.9445	−0.480411
$520$	0	0
$521$	5.36291	0.234953	0.117477	−	0.993076i	$-0.462519\pi$
$521$	5.36291	0.234953	0.117477	+	0.993076i	$0.462519\pi$
$522$	0	0
$523$	7.64386	0.334243	0.167121	−	0.985936i	$-0.446553\pi$
$523$	7.64386	0.334243	0.167121	+	0.985936i	$0.446553\pi$
$524$	0	0
$525$	−5.84950	−0.255293
$526$	0	0
$527$	39.0356	1.70042
$528$	0	0
$529$	−22.7426	−0.988807
$530$	0	0
$531$	−7.09142	−0.307741
$532$	0	0
$533$	−5.59719	−0.242441
$534$	0	0
$535$	11.5876	0.500976
$536$	0	0
$537$	−7.95133	−0.343125
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	30.6125	1.31613	0.658066	−	0.752960i	$-0.271375\pi$
$541$	30.6125	1.31613	0.658066	+	0.752960i	$0.271375\pi$
$542$	0	0
$543$	19.4666	0.835393
$544$	0	0
$545$	1.86454	0.0798683
$546$	0	0
$547$	−32.4164	−1.38602	−0.693012	−	0.720927i	$-0.743717\pi$
$547$	−32.4164	−1.38602	−0.693012	+	0.720927i	$0.743717\pi$
$548$	0	0
$549$	−3.18258	−0.135829
$550$	0	0
$551$	−37.3699	−1.59201
$552$	0	0
$553$	14.6197	0.621694
$554$	0	0
$555$	−1.58072	−0.0670978
$556$	0	0
$557$	44.7730	1.89709	0.948547	−	0.316638i	$-0.102554\pi$
$557$	44.7730	1.89709	0.948547	+	0.316638i	$0.102554\pi$
$558$	0	0
$559$	6.02385	0.254782
$560$	0	0
$561$	8.07497	0.340926
$562$	0	0
$563$	3.43465	0.144753	0.0723767	−	0.997377i	$-0.476942\pi$
$563$	3.43465	0.144753	0.0723767	+	0.997377i	$0.476942\pi$
$564$	0	0
$565$	4.94474	0.208027
$566$	0	0
$567$	3.08688	0.129637
$568$	0	0
$569$	−30.0636	−1.26033	−0.630165	−	0.776461i	$-0.717013\pi$
$569$	−30.0636	−1.26033	−0.630165	+	0.776461i	$0.717013\pi$
$570$	0	0
$571$	23.9050	1.00039	0.500196	−	0.865912i	$-0.333261\pi$
$571$	23.9050	1.00039	0.500196	+	0.865912i	$0.333261\pi$
$572$	0	0
$573$	−0.382348	−0.0159728
$574$	0	0
$575$	2.31552	0.0965637
$576$	0	0
$577$	−23.6430	−0.984270	−0.492135	−	0.870519i	$-0.663783\pi$
$577$	−23.6430	−0.984270	−0.492135	+	0.870519i	$0.663783\pi$
$578$	0	0
$579$	12.0443	0.500545
$580$	0	0
$581$	−4.83006	−0.200385
$582$	0	0
$583$	4.45633	0.184562
$584$	0	0
$585$	0.896437	0.0370631
$586$	0	0
$587$	30.4453	1.25661	0.628306	−	0.777967i	$-0.283749\pi$
$587$	30.4453	1.25661	0.628306	+	0.777967i	$0.283749\pi$
$588$	0	0
$589$	42.7094	1.75981
$590$	0	0
$591$	5.19237	0.213586
$592$	0	0
$593$	−8.60544	−0.353383	−0.176692	−	0.984266i	$-0.556540\pi$
$593$	−8.60544	−0.353383	−0.176692	+	0.984266i	$0.556540\pi$
$594$	0	0
$595$	−4.16141	−0.170601
$596$	0	0
$597$	−6.43462	−0.263351
$598$	0	0
$599$	32.4199	1.32464	0.662321	−	0.749220i	$-0.269571\pi$
$599$	32.4199	1.32464	0.662321	+	0.749220i	$0.269571\pi$
$600$	0	0
$601$	−28.2436	−1.15208	−0.576040	−	0.817421i	$-0.695403\pi$
$601$	−28.2436	−1.15208	−0.576040	+	0.817421i	$0.695403\pi$
$602$	0	0
$603$	13.5332	0.551115
$604$	0	0
$605$	0.660545	0.0268550
$606$	0	0
$607$	3.09550	0.125642	0.0628212	−	0.998025i	$-0.479990\pi$
$607$	3.09550	0.125642	0.0628212	+	0.998025i	$0.479990\pi$
$608$	0	0
$609$	−6.94904	−0.281589
$610$	0	0
$611$	4.97095	0.201103
$612$	0	0
$613$	−15.5707	−0.628895	−0.314447	−	0.949275i	$-0.601819\pi$
$613$	−15.5707	−0.628895	−0.314447	+	0.949275i	$0.601819\pi$
$614$	0	0
$615$	−4.73888	−0.191090
$616$	0	0
$617$	−21.4398	−0.863135	−0.431568	−	0.902081i	$-0.642039\pi$
$617$	−21.4398	−0.863135	−0.431568	+	0.902081i	$0.642039\pi$
$618$	0	0
$619$	44.0972	1.77242	0.886208	−	0.463288i	$-0.153330\pi$
$619$	44.0972	1.77242	0.886208	+	0.463288i	$0.153330\pi$
$620$	0	0
$621$	−2.83358	−0.113708
$622$	0	0
$623$	4.09889	0.164219
$624$	0	0
$625$	18.6456	0.745823
$626$	0	0
$627$	8.83494	0.352834
$628$	0	0
$629$	11.7621	0.468988
$630$	0	0
$631$	30.6780	1.22127	0.610636	−	0.791911i	$-0.290914\pi$
$631$	30.6780	1.22127	0.610636	+	0.791911i	$0.290914\pi$
$632$	0	0
$633$	−34.5768	−1.37430
$634$	0	0
$635$	−0.780855	−0.0309873
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	10.3036	0.407605
$640$	0	0
$641$	4.35225	0.171904	0.0859518	−	0.996299i	$-0.472607\pi$
$641$	4.35225	0.171904	0.0859518	+	0.996299i	$0.472607\pi$
$642$	0	0
$643$	36.9544	1.45734	0.728670	−	0.684865i	$-0.240139\pi$
$643$	36.9544	1.45734	0.728670	+	0.684865i	$0.240139\pi$
$644$	0	0
$645$	5.10011	0.200817
$646$	0	0
$647$	23.5747	0.926819	0.463409	−	0.886144i	$-0.346626\pi$
$647$	23.5747	0.926819	0.463409	+	0.886144i	$0.346626\pi$
$648$	0	0
$649$	−5.22536	−0.205113
$650$	0	0
$651$	7.94194	0.311269
$652$	0	0
$653$	4.88399	0.191125	0.0955626	−	0.995423i	$-0.469535\pi$
$653$	4.88399	0.191125	0.0955626	+	0.995423i	$0.469535\pi$
$654$	0	0
$655$	7.77264	0.303702
$656$	0	0
$657$	12.6288	0.492696
$658$	0	0
$659$	9.28998	0.361886	0.180943	−	0.983494i	$-0.442085\pi$
$659$	9.28998	0.361886	0.180943	+	0.983494i	$0.442085\pi$
$660$	0	0
$661$	33.9853	1.32187	0.660937	−	0.750441i	$-0.270159\pi$
$661$	33.9853	1.32187	0.660937	+	0.750441i	$0.270159\pi$
$662$	0	0
$663$	8.07497	0.313606
$664$	0	0
$665$	−4.55306	−0.176560
$666$	0	0
$667$	2.75077	0.106510
$668$	0	0
$669$	22.9517	0.887363
$670$	0	0
$671$	−2.34510	−0.0905316
$672$	0	0
$673$	−26.9272	−1.03797	−0.518983	−	0.854785i	$-0.673689\pi$
$673$	−26.9272	−1.03797	−0.518983	+	0.854785i	$0.673689\pi$
$674$	0	0
$675$	−25.4869	−0.980993
$676$	0	0
$677$	13.1106	0.503882	0.251941	−	0.967743i	$-0.418931\pi$
$677$	13.1106	0.503882	0.251941	+	0.967743i	$0.418931\pi$
$678$	0	0
$679$	−0.382180	−0.0146667
$680$	0	0
$681$	−33.9607	−1.30138
$682$	0	0
$683$	16.6622	0.637562	0.318781	−	0.947828i	$-0.396727\pi$
$683$	16.6622	0.637562	0.318781	+	0.947828i	$0.396727\pi$
$684$	0	0
$685$	−2.31152	−0.0883186
$686$	0	0
$687$	18.7210	0.714251
$688$	0	0
$689$	4.45633	0.169773
$690$	0	0
$691$	39.2814	1.49433	0.747167	−	0.664636i	$-0.231413\pi$
$691$	39.2814	1.49433	0.747167	+	0.664636i	$0.231413\pi$
$692$	0	0
$693$	−1.35712	−0.0515526
$694$	0	0
$695$	7.26359	0.275523
$696$	0	0
$697$	35.2621	1.33565
$698$	0	0
$699$	−30.0510	−1.13663
$700$	0	0
$701$	46.4324	1.75373	0.876864	−	0.480739i	$-0.159632\pi$
$701$	46.4324	1.75373	0.876864	+	0.480739i	$0.159632\pi$
$702$	0	0
$703$	12.8691	0.485369
$704$	0	0
$705$	4.20867	0.158508
$706$	0	0
$707$	15.2785	0.574608
$708$	0	0
$709$	26.1815	0.983267	0.491633	−	0.870802i	$-0.336400\pi$
$709$	26.1815	0.983267	0.491633	+	0.870802i	$0.336400\pi$
$710$	0	0
$711$	19.8407	0.744084
$712$	0	0
$713$	−3.14381	−0.117737
$714$	0	0
$715$	0.660545	0.0247030
$716$	0	0
$717$	−3.81758	−0.142570
$718$	0	0
$719$	−28.3743	−1.05818	−0.529090	−	0.848565i	$-0.677467\pi$
$719$	−28.3743	−1.05818	−0.529090	+	0.848565i	$0.677467\pi$
$720$	0	0
$721$	−17.6834	−0.658564
$722$	0	0
$723$	−32.3017	−1.20131
$724$	0	0
$725$	24.7421	0.918899
$726$	0	0
$727$	22.3039	0.827205	0.413603	−	0.910457i	$-0.364270\pi$
$727$	22.3039	0.827205	0.413603	+	0.910457i	$0.364270\pi$
$728$	0	0
$729$	25.6640	0.950517
$730$	0	0
$731$	−37.9500	−1.40363
$732$	0	0
$733$	−3.81742	−0.141000	−0.0704998	−	0.997512i	$-0.522459\pi$
$733$	−3.81742	−0.141000	−0.0704998	+	0.997512i	$0.522459\pi$
$734$	0	0
$735$	−0.846653	−0.0312293
$736$	0	0
$737$	9.97202	0.367324
$738$	0	0
$739$	−11.0759	−0.407433	−0.203716	−	0.979030i	$-0.565302\pi$
$739$	−11.0759	−0.407433	−0.203716	+	0.979030i	$0.565302\pi$
$740$	0	0
$741$	8.83494	0.324560
$742$	0	0
$743$	−51.5784	−1.89223	−0.946114	−	0.323833i	$-0.895029\pi$
$743$	−51.5784	−1.89223	−0.946114	+	0.323833i	$0.895029\pi$
$744$	0	0
$745$	−11.9117	−0.436410
$746$	0	0
$747$	−6.55496	−0.239833
$748$	0	0
$749$	−17.5425	−0.640989
$750$	0	0
$751$	7.86277	0.286917	0.143458	−	0.989656i	$-0.454178\pi$
$751$	7.86277	0.286917	0.143458	+	0.989656i	$0.454178\pi$
$752$	0	0
$753$	−4.90306	−0.178677
$754$	0	0
$755$	10.1804	0.370504
$756$	0	0
$757$	11.4839	0.417389	0.208695	−	0.977981i	$-0.433079\pi$
$757$	11.4839	0.417389	0.208695	+	0.977981i	$0.433079\pi$
$758$	0	0
$759$	−0.650333	−0.0236056
$760$	0	0
$761$	22.3153	0.808930	0.404465	−	0.914554i	$-0.367458\pi$
$761$	22.3153	0.808930	0.404465	+	0.914554i	$0.367458\pi$
$762$	0	0
$763$	−2.82274	−0.102190
$764$	0	0
$765$	−5.64752	−0.204186
$766$	0	0
$767$	−5.22536	−0.188677
$768$	0	0
$769$	2.35022	0.0847511	0.0423756	−	0.999102i	$-0.486507\pi$
$769$	2.35022	0.0847511	0.0423756	+	0.999102i	$0.486507\pi$
$770$	0	0
$771$	5.15134	0.185521
$772$	0	0
$773$	6.93422	0.249406	0.124703	−	0.992194i	$-0.460202\pi$
$773$	6.93422	0.249406	0.124703	+	0.992194i	$0.460202\pi$
$774$	0	0
$775$	−28.2773	−1.01575
$776$	0	0
$777$	2.39305	0.0858503
$778$	0	0
$779$	38.5808	1.38230
$780$	0	0
$781$	7.59229	0.271674
$782$	0	0
$783$	−30.2778	−1.08204
$784$	0	0
$785$	12.4161	0.443149
$786$	0	0
$787$	14.5496	0.518638	0.259319	−	0.965792i	$-0.416502\pi$
$787$	14.5496	0.518638	0.259319	+	0.965792i	$0.416502\pi$
$788$	0	0
$789$	−23.4988	−0.836578
$790$	0	0
$791$	−7.48584	−0.266166
$792$	0	0
$793$	−2.34510	−0.0832770
$794$	0	0
$795$	3.77297	0.133813
$796$	0	0
$797$	21.9450	0.777331	0.388665	−	0.921379i	$-0.372936\pi$
$797$	21.9450	0.777331	0.388665	+	0.921379i	$0.372936\pi$
$798$	0	0
$799$	−31.3168	−1.10791
$800$	0	0
$801$	5.56268	0.196548
$802$	0	0
$803$	9.30560	0.328387
$804$	0	0
$805$	0.335147	0.0118124
$806$	0	0
$807$	−33.7635	−1.18853
$808$	0	0
$809$	34.5267	1.21389	0.606947	−	0.794742i	$-0.292394\pi$
$809$	34.5267	1.21389	0.606947	+	0.794742i	$0.292394\pi$
$810$	0	0
$811$	−31.9101	−1.12052	−0.560258	−	0.828318i	$-0.689298\pi$
$811$	−31.9101	−1.12052	−0.560258	+	0.828318i	$0.689298\pi$
$812$	0	0
$813$	−6.45766	−0.226480
$814$	0	0
$815$	10.3570	0.362791
$816$	0	0
$817$	−41.5217	−1.45266
$818$	0	0
$819$	−1.35712	−0.0474215
$820$	0	0
$821$	7.60265	0.265334	0.132667	−	0.991161i	$-0.457646\pi$
$821$	7.60265	0.265334	0.132667	+	0.991161i	$0.457646\pi$
$822$	0	0
$823$	−37.5716	−1.30966	−0.654832	−	0.755774i	$-0.727261\pi$
$823$	−37.5716	−1.30966	−0.654832	+	0.755774i	$0.727261\pi$
$824$	0	0
$825$	−5.84950	−0.203653
$826$	0	0
$827$	13.6982	0.476333	0.238166	−	0.971224i	$-0.423454\pi$
$827$	13.6982	0.476333	0.238166	+	0.971224i	$0.423454\pi$
$828$	0	0
$829$	3.19177	0.110855	0.0554274	−	0.998463i	$-0.482348\pi$
$829$	3.19177	0.110855	0.0554274	+	0.998463i	$0.482348\pi$
$830$	0	0
$831$	−10.1092	−0.350683
$832$	0	0
$833$	6.29996	0.218281
$834$	0	0
$835$	10.0044	0.346217
$836$	0	0
$837$	34.6040	1.19609
$838$	0	0
$839$	−48.7608	−1.68341	−0.841706	−	0.539937i	$-0.818448\pi$
$839$	−48.7608	−1.68341	−0.841706	+	0.539937i	$0.818448\pi$
$840$	0	0
$841$	0.392946	0.0135499
$842$	0	0
$843$	3.18058	0.109545
$844$	0	0
$845$	0.660545	0.0227234
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	32.1894	1.10474
$850$	0	0
$851$	−0.947287	−0.0324726
$852$	0	0
$853$	−17.9767	−0.615511	−0.307755	−	0.951466i	$-0.599578\pi$
$853$	−17.9767	−0.615511	−0.307755	+	0.951466i	$0.599578\pi$
$854$	0	0
$855$	−6.17903	−0.211318
$856$	0	0
$857$	16.7427	0.571920	0.285960	−	0.958242i	$-0.407688\pi$
$857$	16.7427	0.571920	0.285960	+	0.958242i	$0.407688\pi$
$858$	0	0
$859$	−50.5339	−1.72419	−0.862097	−	0.506743i	$-0.830849\pi$
$859$	−50.5339	−1.72419	−0.862097	+	0.506743i	$0.830849\pi$
$860$	0	0
$861$	7.17420	0.244496
$862$	0	0
$863$	25.5128	0.868466	0.434233	−	0.900801i	$-0.357019\pi$
$863$	25.5128	0.868466	0.434233	+	0.900801i	$0.357019\pi$
$864$	0	0
$865$	5.64022	0.191773
$866$	0	0
$867$	−29.0823	−0.987685
$868$	0	0
$869$	14.6197	0.495940
$870$	0	0
$871$	9.97202	0.337889
$872$	0	0
$873$	−0.518664	−0.0175541
$874$	0	0
$875$	6.31724	0.213562
$876$	0	0
$877$	−18.6839	−0.630909	−0.315455	−	0.948941i	$-0.602157\pi$
$877$	−18.6839	−0.630909	−0.315455	+	0.948941i	$0.602157\pi$
$878$	0	0
$879$	8.83986	0.298161
$880$	0	0
$881$	35.9689	1.21182	0.605912	−	0.795532i	$-0.292808\pi$
$881$	35.9689	1.21182	0.605912	+	0.795532i	$0.292808\pi$
$882$	0	0
$883$	23.7215	0.798291	0.399146	−	0.916888i	$-0.369307\pi$
$883$	23.7215	0.798291	0.399146	+	0.916888i	$0.369307\pi$
$884$	0	0
$885$	−4.42407	−0.148713
$886$	0	0
$887$	−57.4991	−1.93063	−0.965316	−	0.261085i	$-0.915920\pi$
$887$	−57.4991	−1.93063	−0.965316	+	0.261085i	$0.915920\pi$
$888$	0	0
$889$	1.18214	0.0396476
$890$	0	0
$891$	3.08688	0.103414
$892$	0	0
$893$	−34.2642	−1.14661
$894$	0	0
$895$	4.09769	0.136971
$896$	0	0
$897$	−0.650333	−0.0217140
$898$	0	0
$899$	−33.5927	−1.12038
$900$	0	0
$901$	−28.0747	−0.935304
$902$	0	0
$903$	−7.72107	−0.256941
$904$	0	0
$905$	−10.0320	−0.333477
$906$	0	0
$907$	33.5017	1.11241	0.556203	−	0.831046i	$-0.312258\pi$
$907$	33.5017	1.11241	0.556203	+	0.831046i	$0.312258\pi$
$908$	0	0
$909$	20.7347	0.687728
$910$	0	0
$911$	10.6127	0.351615	0.175808	−	0.984425i	$-0.443746\pi$
$911$	10.6127	0.351615	0.175808	+	0.984425i	$0.443746\pi$
$912$	0	0
$913$	−4.83006	−0.159852
$914$	0	0
$915$	−1.98549	−0.0656382
$916$	0	0
$917$	−11.7670	−0.388581
$918$	0	0
$919$	−37.8950	−1.25004	−0.625020	−	0.780609i	$-0.714909\pi$
$919$	−37.8950	−1.25004	−0.625020	+	0.780609i	$0.714909\pi$
$920$	0	0
$921$	9.68141	0.319013
$922$	0	0
$923$	7.59229	0.249903
$924$	0	0
$925$	−8.52048	−0.280152
$926$	0	0
$927$	−23.9985	−0.788213
$928$	0	0
$929$	−19.8334	−0.650713	−0.325357	−	0.945591i	$-0.605484\pi$
$929$	−19.8334	−0.650713	−0.325357	+	0.945591i	$0.605484\pi$
$930$	0	0
$931$	6.89288	0.225905
$932$	0	0
$933$	−5.65017	−0.184978
$934$	0	0
$935$	−4.16141	−0.136093
$936$	0	0
$937$	−45.9474	−1.50104	−0.750518	−	0.660849i	$-0.770196\pi$
$937$	−45.9474	−1.50104	−0.750518	+	0.660849i	$0.770196\pi$
$938$	0	0
$939$	14.7653	0.481847
$940$	0	0
$941$	58.4068	1.90401	0.952004	−	0.306087i	$-0.0990197\pi$
$941$	58.4068	1.90401	0.952004	+	0.306087i	$0.0990197\pi$
$942$	0	0
$943$	−2.83990	−0.0924798
$944$	0	0
$945$	−3.68897	−0.120002
$946$	0	0
$947$	4.56086	0.148208	0.0741041	−	0.997251i	$-0.476390\pi$
$947$	4.56086	0.148208	0.0741041	+	0.997251i	$0.476390\pi$
$948$	0	0
$949$	9.30560	0.302073
$950$	0	0
$951$	−3.14694	−0.102046
$952$	0	0
$953$	−53.7302	−1.74049	−0.870246	−	0.492617i	$-0.836040\pi$
$953$	−53.7302	−1.74049	−0.870246	+	0.492617i	$0.836040\pi$
$954$	0	0
$955$	0.197042	0.00637612
$956$	0	0
$957$	−6.94904	−0.224631
$958$	0	0
$959$	3.49941	0.113002
$960$	0	0
$961$	7.39253	0.238469
$962$	0	0
$963$	−23.8072	−0.767177
$964$	0	0
$965$	−6.20700	−0.199810
$966$	0	0
$967$	−5.27405	−0.169602	−0.0848010	−	0.996398i	$-0.527025\pi$
$967$	−5.27405	−0.169602	−0.0848010	+	0.996398i	$0.527025\pi$
$968$	0	0
$969$	−55.6598	−1.78805
$970$	0	0
$971$	48.5071	1.55667	0.778334	−	0.627850i	$-0.216065\pi$
$971$	48.5071	1.55667	0.778334	+	0.627850i	$0.216065\pi$
$972$	0	0
$973$	−10.9964	−0.352527
$974$	0	0
$975$	−5.84950	−0.187334
$976$	0	0
$977$	−14.6154	−0.467589	−0.233795	−	0.972286i	$-0.575114\pi$
$977$	−14.6154	−0.467589	−0.233795	+	0.972286i	$0.575114\pi$
$978$	0	0
$979$	4.09889	0.131001
$980$	0	0
$981$	−3.83078	−0.122308
$982$	0	0
$983$	43.9474	1.40171	0.700853	−	0.713306i	$-0.252803\pi$
$983$	43.9474	1.40171	0.700853	+	0.713306i	$0.252803\pi$
$984$	0	0
$985$	−2.67587	−0.0852603
$986$	0	0
$987$	−6.37151	−0.202808
$988$	0	0
$989$	3.05638	0.0971871
$990$	0	0
$991$	−34.7200	−1.10292	−0.551458	−	0.834203i	$-0.685928\pi$
$991$	−34.7200	−1.10292	−0.551458	+	0.834203i	$0.685928\pi$
$992$	0	0
$993$	42.0298	1.33378
$994$	0	0
$995$	3.31606	0.105126
$996$	0	0
$997$	40.7159	1.28949	0.644743	−	0.764399i	$-0.276964\pi$
$997$	40.7159	1.28949	0.644743	+	0.764399i	$0.276964\pi$
$998$	0	0
$999$	10.4268	0.329890

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.r.1.2	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.r.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-1.28175 q^{3} +0.660545 q^{5} -1.00000 q^{7} -1.35712 q^{9} +O(q^{10})\)	\(q-1.28175 q^{3} +0.660545 q^{5} -1.00000 q^{7} -1.35712 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.846653 q^{15} +6.29996 q^{17} +6.89288 q^{19} +1.28175 q^{21} -0.507379 q^{23} -4.56368 q^{25} +5.58473 q^{27} -5.42153 q^{29} +6.19617 q^{31} +1.28175 q^{33} -0.660545 q^{35} +1.86702 q^{37} +1.28175 q^{39} +5.59719 q^{41} -6.02385 q^{43} -0.896437 q^{45} -4.97095 q^{47} +1.00000 q^{49} -8.07497 q^{51} -4.45633 q^{53} -0.660545 q^{55} -8.83494 q^{57} +5.22536 q^{59} +2.34510 q^{61} +1.35712 q^{63} -0.660545 q^{65} -9.97202 q^{67} +0.650333 q^{69} -7.59229 q^{71} -9.30560 q^{73} +5.84950 q^{75} +1.00000 q^{77} -14.6197 q^{79} -3.08688 q^{81} +4.83006 q^{83} +4.16141 q^{85} +6.94904 q^{87} -4.09889 q^{89} +1.00000 q^{91} -7.94194 q^{93} +4.55306 q^{95} +0.382180 q^{97} +1.35712 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9}+O(q^{10}) \) 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9	\( 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9} - 9 q^{11} - 9 q^{13} - 3 q^{15} - 7 q^{17} + 13 q^{19} - 5 q^{21} + 9 q^{23} - 2 q^{25} + 5 q^{27} + q^{29} + 10 q^{31} - 5 q^{33} - 3 q^{35} + 14 q^{37} - 5 q^{39} - 2 q^{41} + 5 q^{43} - 15 q^{45} + 13 q^{47} + 9 q^{49} - 3 q^{51} + 22 q^{53} - 3 q^{55} + 16 q^{57} + 43 q^{59} - 10 q^{61} - 8 q^{63} - 3 q^{65} + 26 q^{67} - 30 q^{69} + 18 q^{71} - 8 q^{73} + 28 q^{75} + 9 q^{77} - 9 q^{79} + 33 q^{81} + 4 q^{83} + 5 q^{85} + 33 q^{87} + 7 q^{89} + 9 q^{91} + 13 q^{93} + 7 q^{95} + 2 q^{97} - 8 q^{99}+O(q^{100}) \) 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9 - 9 * q^11 - 9 * q^13 - 3 * q^15 - 7 * q^17 + 13 * q^19 - 5 * q^21 + 9 * q^23 - 2 * q^25 + 5 * q^27 + q^29 + 10 * q^31 - 5 * q^33 - 3 * q^35 + 14 * q^37 - 5 * q^39 - 2 * q^41 + 5 * q^43 - 15 * q^45 + 13 * q^47 + 9 * q^49 - 3 * q^51 + 22 * q^53 - 3 * q^55 + 16 * q^57 + 43 * q^59 - 10 * q^61 - 8 * q^63 - 3 * q^65 + 26 * q^67 - 30 * q^69 + 18 * q^71 - 8 * q^73 + 28 * q^75 + 9 * q^77 - 9 * q^79 + 33 * q^81 + 4 * q^83 + 5 * q^85 + 33 * q^87 + 7 * q^89 + 9 * q^91 + 13 * q^93 + 7 * q^95 + 2 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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