LMFDB - Embedded newform 8008.2.a.q.1.7

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} - 53x^{3} + 252x^{2} - 69x - 26 $ x^9 - 3x^8 - 15x^7 + 45x^6 + 64x^5 - 201x^4 - 53x^3 + 252x^2 - 69x - 26
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.27011$ 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)

$f(q)$	$=$	$q+2.27011 q^{3} +1.08525 q^{5} +1.00000 q^{7} +2.15339 q^{9} +O(q^{10})$	$q+2.27011 q^{3} +1.08525 q^{5} +1.00000 q^{7} +2.15339 q^{9} +1.00000 q^{11} -1.00000 q^{13} +2.46363 q^{15} -2.01508 q^{17} -0.128221 q^{19} +2.27011 q^{21} +6.73579 q^{23} -3.82224 q^{25} -1.92189 q^{27} +2.70245 q^{29} +6.97612 q^{31} +2.27011 q^{33} +1.08525 q^{35} +7.85402 q^{37} -2.27011 q^{39} +6.28519 q^{41} +2.51164 q^{43} +2.33697 q^{45} -6.73579 q^{47} +1.00000 q^{49} -4.57445 q^{51} +6.71754 q^{53} +1.08525 q^{55} -0.291076 q^{57} -9.89513 q^{59} +2.42338 q^{61} +2.15339 q^{63} -1.08525 q^{65} -11.2747 q^{67} +15.2910 q^{69} -3.36875 q^{71} +9.69249 q^{73} -8.67689 q^{75} +1.00000 q^{77} +4.99948 q^{79} -10.8231 q^{81} +16.0328 q^{83} -2.18687 q^{85} +6.13486 q^{87} +4.67643 q^{89} -1.00000 q^{91} +15.8366 q^{93} -0.139152 q^{95} -8.61563 q^{97} +2.15339 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9	$ 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + q^{17} + 16 q^{19} + 3 q^{21} + 6 q^{23} + 11 q^{25} + 9 q^{27} - 2 q^{29} + 3 q^{31} + 3 q^{33} + 8 q^{35} - 4 q^{37} - 3 q^{39} + 20 q^{41} + 20 q^{43} + 8 q^{45} - 6 q^{47} + 9 q^{49} + 15 q^{51} + 15 q^{53} + 8 q^{55} + 16 q^{57} + 21 q^{59} + 12 q^{63} - 8 q^{65} + 18 q^{67} + 10 q^{69} + 12 q^{71} + 29 q^{73} + 12 q^{75} + 9 q^{77} - 6 q^{79} + 5 q^{81} + 25 q^{83} + 13 q^{85} + 17 q^{87} + 32 q^{89} - 9 q^{91} - 13 q^{93} + 38 q^{95} + 17 q^{97} + 12 q^{99}+O(q^{100}) $ 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 - 9 * q^15 + q^17 + 16 * q^19 + 3 * q^21 + 6 * q^23 + 11 * q^25 + 9 * q^27 - 2 * q^29 + 3 * q^31 + 3 * q^33 + 8 * q^35 - 4 * q^37 - 3 * q^39 + 20 * q^41 + 20 * q^43 + 8 * q^45 - 6 * q^47 + 9 * q^49 + 15 * q^51 + 15 * q^53 + 8 * q^55 + 16 * q^57 + 21 * q^59 + 12 * q^63 - 8 * q^65 + 18 * q^67 + 10 * q^69 + 12 * q^71 + 29 * q^73 + 12 * q^75 + 9 * q^77 - 6 * q^79 + 5 * q^81 + 25 * q^83 + 13 * q^85 + 17 * q^87 + 32 * q^89 - 9 * q^91 - 13 * q^93 + 38 * q^95 + 17 * 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
$3$	2.27011	1.31065	0.655324	−	0.755348i	$-0.272532\pi$
$3$	2.27011	1.31065	0.655324	+	0.755348i	$0.272532\pi$
$4$	0	0
$5$	1.08525	0.485338	0.242669	−	0.970109i	$-0.421977\pi$
$5$	1.08525	0.485338	0.242669	+	0.970109i	$0.421977\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	2.15339	0.717798
$10$	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
$15$	2.46363	0.636107
$16$	0	0
$17$	−2.01508	−0.488729	−0.244365	−	0.969683i	$-0.578579\pi$
$17$	−2.01508	−0.488729	−0.244365	+	0.969683i	$0.578579\pi$
$18$	0	0
$19$	−0.128221	−0.0294160	−0.0147080	−	0.999892i	$-0.504682\pi$
$19$	−0.128221	−0.0294160	−0.0147080	+	0.999892i	$0.504682\pi$
$20$	0	0
$21$	2.27011	0.495378
$22$	0	0
$23$	6.73579	1.40451	0.702255	−	0.711926i	$-0.252177\pi$
$23$	6.73579	1.40451	0.702255	+	0.711926i	$0.252177\pi$
$24$	0	0
$25$	−3.82224	−0.764447
$26$	0	0
$27$	−1.92189	−0.369868
$28$	0	0
$29$	2.70245	0.501833	0.250917	−	0.968009i	$-0.419268\pi$
$29$	2.70245	0.501833	0.250917	+	0.968009i	$0.419268\pi$
$30$	0	0
$31$	6.97612	1.25295	0.626474	−	0.779442i	$-0.284497\pi$
$31$	6.97612	1.25295	0.626474	+	0.779442i	$0.284497\pi$
$32$	0	0
$33$	2.27011	0.395175
$34$	0	0
$35$	1.08525	0.183441
$36$	0	0
$37$	7.85402	1.29119	0.645596	−	0.763679i	$-0.276609\pi$
$37$	7.85402	1.29119	0.645596	+	0.763679i	$0.276609\pi$
$38$	0	0
$39$	−2.27011	−0.363508
$40$	0	0
$41$	6.28519	0.981582	0.490791	−	0.871277i	$-0.336708\pi$
$41$	6.28519	0.981582	0.490791	+	0.871277i	$0.336708\pi$
$42$	0	0
$43$	2.51164	0.383021	0.191510	−	0.981491i	$-0.438661\pi$
$43$	2.51164	0.383021	0.191510	+	0.981491i	$0.438661\pi$
$44$	0	0
$45$	2.33697	0.348375
$46$	0	0
$47$	−6.73579	−0.982516	−0.491258	−	0.871014i	$-0.663463\pi$
$47$	−6.73579	−0.982516	−0.491258	+	0.871014i	$0.663463\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.57445	−0.640552
$52$	0	0
$53$	6.71754	0.922724	0.461362	−	0.887212i	$-0.347361\pi$
$53$	6.71754	0.922724	0.461362	+	0.887212i	$0.347361\pi$
$54$	0	0
$55$	1.08525	0.146335
$56$	0	0
$57$	−0.291076	−0.0385540
$58$	0	0
$59$	−9.89513	−1.28824	−0.644118	−	0.764926i	$-0.722775\pi$
$59$	−9.89513	−1.28824	−0.644118	+	0.764926i	$0.722775\pi$
$60$	0	0
$61$	2.42338	0.310282	0.155141	−	0.987892i	$-0.450417\pi$
$61$	2.42338	0.310282	0.155141	+	0.987892i	$0.450417\pi$
$62$	0	0
$63$	2.15339	0.271302
$64$	0	0
$65$	−1.08525	−0.134609
$66$	0	0
$67$	−11.2747	−1.37742	−0.688712	−	0.725035i	$-0.741824\pi$
$67$	−11.2747	−1.37742	−0.688712	+	0.725035i	$0.741824\pi$
$68$	0	0
$69$	15.2910	1.84082
$70$	0	0
$71$	−3.36875	−0.399797	−0.199899	−	0.979817i	$-0.564061\pi$
$71$	−3.36875	−0.399797	−0.199899	+	0.979817i	$0.564061\pi$
$72$	0	0
$73$	9.69249	1.13442	0.567210	−	0.823573i	$-0.308023\pi$
$73$	9.69249	1.13442	0.567210	+	0.823573i	$0.308023\pi$
$74$	0	0
$75$	−8.67689	−1.00192
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	4.99948	0.562486	0.281243	−	0.959637i	$-0.409253\pi$
$79$	4.99948	0.562486	0.281243	+	0.959637i	$0.409253\pi$
$80$	0	0
$81$	−10.8231	−1.20256
$82$	0	0
$83$	16.0328	1.75983	0.879915	−	0.475132i	$-0.157600\pi$
$83$	16.0328	1.75983	0.879915	+	0.475132i	$0.157600\pi$
$84$	0	0
$85$	−2.18687	−0.237199
$86$	0	0
$87$	6.13486	0.657726
$88$	0	0
$89$	4.67643	0.495700	0.247850	−	0.968798i	$-0.420276\pi$
$89$	4.67643	0.495700	0.247850	+	0.968798i	$0.420276\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	15.8366	1.64217
$94$	0	0
$95$	−0.139152	−0.0142767
$96$	0	0
$97$	−8.61563	−0.874785	−0.437392	−	0.899271i	$-0.644098\pi$
$97$	−8.61563	−0.874785	−0.437392	+	0.899271i	$0.644098\pi$
$98$	0	0
$99$	2.15339	0.216424
$100$	0	0
$101$	−5.36500	−0.533838	−0.266919	−	0.963719i	$-0.586006\pi$
$101$	−5.36500	−0.533838	−0.266919	+	0.963719i	$0.586006\pi$
$102$	0	0
$103$	5.69749	0.561391	0.280695	−	0.959797i	$-0.409435\pi$
$103$	5.69749	0.561391	0.280695	+	0.959797i	$0.409435\pi$
$104$	0	0
$105$	2.46363	0.240426
$106$	0	0
$107$	6.12636	0.592257	0.296129	−	0.955148i	$-0.404304\pi$
$107$	6.12636	0.592257	0.296129	+	0.955148i	$0.404304\pi$
$108$	0	0
$109$	4.89931	0.469269	0.234635	−	0.972084i	$-0.424611\pi$
$109$	4.89931	0.469269	0.234635	+	0.972084i	$0.424611\pi$
$110$	0	0
$111$	17.8295	1.69230
$112$	0	0
$113$	14.1187	1.32817	0.664087	−	0.747656i	$-0.268821\pi$
$113$	14.1187	1.32817	0.664087	+	0.747656i	$0.268821\pi$
$114$	0	0
$115$	7.31001	0.681662
$116$	0	0
$117$	−2.15339	−0.199081
$118$	0	0
$119$	−2.01508	−0.184722
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	14.2681	1.28651
$124$	0	0
$125$	−9.57432	−0.856353
$126$	0	0
$127$	−7.12479	−0.632223	−0.316112	−	0.948722i	$-0.602377\pi$
$127$	−7.12479	−0.632223	−0.316112	+	0.948722i	$0.602377\pi$
$128$	0	0
$129$	5.70169	0.502005
$130$	0	0
$131$	8.26930	0.722492	0.361246	−	0.932471i	$-0.382351\pi$
$131$	8.26930	0.722492	0.361246	+	0.932471i	$0.382351\pi$
$132$	0	0
$133$	−0.128221	−0.0111182
$134$	0	0
$135$	−2.08573	−0.179511
$136$	0	0
$137$	−9.57317	−0.817891	−0.408946	−	0.912559i	$-0.634103\pi$
$137$	−9.57317	−0.817891	−0.408946	+	0.912559i	$0.634103\pi$
$138$	0	0
$139$	12.7851	1.08441	0.542207	−	0.840245i	$-0.317589\pi$
$139$	12.7851	1.08441	0.542207	+	0.840245i	$0.317589\pi$
$140$	0	0
$141$	−15.2910	−1.28773
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	2.93283	0.243559
$146$	0	0
$147$	2.27011	0.187235
$148$	0	0
$149$	0.119981	0.00982920	0.00491460	−	0.999988i	$-0.498436\pi$
$149$	0.119981	0.00982920	0.00491460	+	0.999988i	$0.498436\pi$
$150$	0	0
$151$	14.1672	1.15291	0.576454	−	0.817129i	$-0.304436\pi$
$151$	14.1672	1.15291	0.576454	+	0.817129i	$0.304436\pi$
$152$	0	0
$153$	−4.33926	−0.350809
$154$	0	0
$155$	7.57083	0.608104
$156$	0	0
$157$	19.1014	1.52445	0.762227	−	0.647309i	$-0.224106\pi$
$157$	19.1014	1.52445	0.762227	+	0.647309i	$0.224106\pi$
$158$	0	0
$159$	15.2495	1.20937
$160$	0	0
$161$	6.73579	0.530855
$162$	0	0
$163$	−19.0722	−1.49385	−0.746925	−	0.664908i	$-0.768471\pi$
$163$	−19.0722	−1.49385	−0.746925	+	0.664908i	$0.768471\pi$
$164$	0	0
$165$	2.46363	0.191794
$166$	0	0
$167$	4.72125	0.365341	0.182670	−	0.983174i	$-0.441526\pi$
$167$	4.72125	0.365341	0.182670	+	0.983174i	$0.441526\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.276111	−0.0211147
$172$	0	0
$173$	−4.28187	−0.325544	−0.162772	−	0.986664i	$-0.552044\pi$
$173$	−4.28187	−0.325544	−0.162772	+	0.986664i	$0.552044\pi$
$174$	0	0
$175$	−3.82224	−0.288934
$176$	0	0
$177$	−22.4630	−1.68842
$178$	0	0
$179$	24.1141	1.80237	0.901187	−	0.433430i	$-0.142697\pi$
$179$	24.1141	1.80237	0.901187	+	0.433430i	$0.142697\pi$
$180$	0	0
$181$	14.2499	1.05919	0.529594	−	0.848251i	$-0.322344\pi$
$181$	14.2499	1.05919	0.529594	+	0.848251i	$0.322344\pi$
$182$	0	0
$183$	5.50133	0.406670
$184$	0	0
$185$	8.52356	0.626665
$186$	0	0
$187$	−2.01508	−0.147357
$188$	0	0
$189$	−1.92189	−0.139797
$190$	0	0
$191$	−24.7000	−1.78723	−0.893614	−	0.448836i	$-0.851839\pi$
$191$	−24.7000	−1.78723	−0.893614	+	0.448836i	$0.851839\pi$
$192$	0	0
$193$	−2.46278	−0.177275	−0.0886373	−	0.996064i	$-0.528251\pi$
$193$	−2.46278	−0.177275	−0.0886373	+	0.996064i	$0.528251\pi$
$194$	0	0
$195$	−2.46363	−0.176424
$196$	0	0
$197$	−24.2988	−1.73122	−0.865608	−	0.500722i	$-0.833068\pi$
$197$	−24.2988	−1.73122	−0.865608	+	0.500722i	$0.833068\pi$
$198$	0	0
$199$	−17.0959	−1.21190	−0.605949	−	0.795504i	$-0.707206\pi$
$199$	−17.0959	−1.21190	−0.605949	+	0.795504i	$0.707206\pi$
$200$	0	0
$201$	−25.5948	−1.80532
$202$	0	0
$203$	2.70245	0.189675
$204$	0	0
$205$	6.82100	0.476399
$206$	0	0
$207$	14.5048	1.00815
$208$	0	0
$209$	−0.128221	−0.00886925
$210$	0	0
$211$	−14.3165	−0.985587	−0.492794	−	0.870146i	$-0.664024\pi$
$211$	−14.3165	−0.985587	−0.492794	+	0.870146i	$0.664024\pi$
$212$	0	0
$213$	−7.64743	−0.523993
$214$	0	0
$215$	2.72575	0.185895
$216$	0	0
$217$	6.97612	0.473570
$218$	0	0
$219$	22.0030	1.48682
$220$	0	0
$221$	2.01508	0.135549
$222$	0	0
$223$	−1.54928	−0.103747	−0.0518737	−	0.998654i	$-0.516519\pi$
$223$	−1.54928	−0.103747	−0.0518737	+	0.998654i	$0.516519\pi$
$224$	0	0
$225$	−8.23078	−0.548718
$226$	0	0
$227$	−7.50691	−0.498251	−0.249125	−	0.968471i	$-0.580143\pi$
$227$	−7.50691	−0.498251	−0.249125	+	0.968471i	$0.580143\pi$
$228$	0	0
$229$	−5.67857	−0.375250	−0.187625	−	0.982241i	$-0.560079\pi$
$229$	−5.67857	−0.375250	−0.187625	+	0.982241i	$0.560079\pi$
$230$	0	0
$231$	2.27011	0.149362
$232$	0	0
$233$	22.7137	1.48802	0.744012	−	0.668166i	$-0.232920\pi$
$233$	22.7137	1.48802	0.744012	+	0.668166i	$0.232920\pi$
$234$	0	0
$235$	−7.31001	−0.476853
$236$	0	0
$237$	11.3494	0.737221
$238$	0	0
$239$	17.4855	1.13104	0.565521	−	0.824734i	$-0.308675\pi$
$239$	17.4855	1.13104	0.565521	+	0.824734i	$0.308675\pi$
$240$	0	0
$241$	16.6343	1.07151	0.535756	−	0.844373i	$-0.320027\pi$
$241$	16.6343	1.07151	0.535756	+	0.844373i	$0.320027\pi$
$242$	0	0
$243$	−18.8039	−1.20627
$244$	0	0
$245$	1.08525	0.0693340
$246$	0	0
$247$	0.128221	0.00815853
$248$	0	0
$249$	36.3962	2.30652
$250$	0	0
$251$	5.08269	0.320817	0.160408	−	0.987051i	$-0.448719\pi$
$251$	5.08269	0.320817	0.160408	+	0.987051i	$0.448719\pi$
$252$	0	0
$253$	6.73579	0.423476
$254$	0	0
$255$	−4.96442	−0.310884
$256$	0	0
$257$	0.768872	0.0479609	0.0239805	−	0.999712i	$-0.492366\pi$
$257$	0.768872	0.0479609	0.0239805	+	0.999712i	$0.492366\pi$
$258$	0	0
$259$	7.85402	0.488025
$260$	0	0
$261$	5.81945	0.360215
$262$	0	0
$263$	−28.9224	−1.78343	−0.891717	−	0.452593i	$-0.850499\pi$
$263$	−28.9224	−1.78343	−0.891717	+	0.452593i	$0.850499\pi$
$264$	0	0
$265$	7.29020	0.447833
$266$	0	0
$267$	10.6160	0.649688
$268$	0	0
$269$	−27.9302	−1.70293	−0.851467	−	0.524408i	$-0.824287\pi$
$269$	−27.9302	−1.70293	−0.851467	+	0.524408i	$0.824287\pi$
$270$	0	0
$271$	0.585342	0.0355570	0.0177785	−	0.999842i	$-0.494341\pi$
$271$	0.585342	0.0355570	0.0177785	+	0.999842i	$0.494341\pi$
$272$	0	0
$273$	−2.27011	−0.137393
$274$	0	0
$275$	−3.82224	−0.230489
$276$	0	0
$277$	−23.7900	−1.42940	−0.714700	−	0.699431i	$-0.753437\pi$
$277$	−23.7900	−1.42940	−0.714700	+	0.699431i	$0.753437\pi$
$278$	0	0
$279$	15.0223	0.899364
$280$	0	0
$281$	7.59108	0.452846	0.226423	−	0.974029i	$-0.427297\pi$
$281$	7.59108	0.452846	0.226423	+	0.974029i	$0.427297\pi$
$282$	0	0
$283$	11.5567	0.686975	0.343487	−	0.939157i	$-0.388392\pi$
$283$	11.5567	0.686975	0.343487	+	0.939157i	$0.388392\pi$
$284$	0	0
$285$	−0.315890	−0.0187117
$286$	0	0
$287$	6.28519	0.371003
$288$	0	0
$289$	−12.9394	−0.761144
$290$	0	0
$291$	−19.5584	−1.14653
$292$	0	0
$293$	−13.5922	−0.794064	−0.397032	−	0.917805i	$-0.629960\pi$
$293$	−13.5922	−0.794064	−0.397032	+	0.917805i	$0.629960\pi$
$294$	0	0
$295$	−10.7387	−0.625230
$296$	0	0
$297$	−1.92189	−0.111519
$298$	0	0
$299$	−6.73579	−0.389541
$300$	0	0
$301$	2.51164	0.144768
$302$	0	0
$303$	−12.1791	−0.699673
$304$	0	0
$305$	2.62997	0.150592
$306$	0	0
$307$	21.0249	1.19996	0.599978	−	0.800016i	$-0.295176\pi$
$307$	21.0249	1.19996	0.599978	+	0.800016i	$0.295176\pi$
$308$	0	0
$309$	12.9339	0.735786
$310$	0	0
$311$	−19.5358	−1.10777	−0.553886	−	0.832593i	$-0.686856\pi$
$311$	−19.5358	−1.10777	−0.553886	+	0.832593i	$0.686856\pi$
$312$	0	0
$313$	10.3037	0.582399	0.291200	−	0.956662i	$-0.405946\pi$
$313$	10.3037	0.582399	0.291200	+	0.956662i	$0.405946\pi$
$314$	0	0
$315$	2.33697	0.131673
$316$	0	0
$317$	5.25662	0.295241	0.147621	−	0.989044i	$-0.452839\pi$
$317$	5.25662	0.295241	0.147621	+	0.989044i	$0.452839\pi$
$318$	0	0
$319$	2.70245	0.151308
$320$	0	0
$321$	13.9075	0.776241
$322$	0	0
$323$	0.258376	0.0143765
$324$	0	0
$325$	3.82224	0.212019
$326$	0	0
$327$	11.1220	0.615047
$328$	0	0
$329$	−6.73579	−0.371356
$330$	0	0
$331$	14.6756	0.806646	0.403323	−	0.915058i	$-0.367855\pi$
$331$	14.6756	0.806646	0.403323	+	0.915058i	$0.367855\pi$
$332$	0	0
$333$	16.9128	0.926815
$334$	0	0
$335$	−12.2359	−0.668517
$336$	0	0
$337$	−30.8411	−1.68002	−0.840012	−	0.542569i	$-0.817452\pi$
$337$	−30.8411	−1.68002	−0.840012	+	0.542569i	$0.817452\pi$
$338$	0	0
$339$	32.0509	1.74077
$340$	0	0
$341$	6.97612	0.377778
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	16.5945	0.893419
$346$	0	0
$347$	8.26828	0.443865	0.221932	−	0.975062i	$-0.428764\pi$
$347$	8.26828	0.443865	0.221932	+	0.975062i	$0.428764\pi$
$348$	0	0
$349$	−3.45200	−0.184781	−0.0923907	−	0.995723i	$-0.529451\pi$
$349$	−3.45200	−0.184781	−0.0923907	+	0.995723i	$0.529451\pi$
$350$	0	0
$351$	1.92189	0.102583
$352$	0	0
$353$	24.1450	1.28511	0.642555	−	0.766239i	$-0.277874\pi$
$353$	24.1450	1.28511	0.642555	+	0.766239i	$0.277874\pi$
$354$	0	0
$355$	−3.65593	−0.194037
$356$	0	0
$357$	−4.57445	−0.242106
$358$	0	0
$359$	−1.38432	−0.0730616	−0.0365308	−	0.999333i	$-0.511631\pi$
$359$	−1.38432	−0.0730616	−0.0365308	+	0.999333i	$0.511631\pi$
$360$	0	0
$361$	−18.9836	−0.999135
$362$	0	0
$363$	2.27011	0.119150
$364$	0	0
$365$	10.5188	0.550577
$366$	0	0
$367$	2.27635	0.118824	0.0594122	−	0.998234i	$-0.481077\pi$
$367$	2.27635	0.118824	0.0594122	+	0.998234i	$0.481077\pi$
$368$	0	0
$369$	13.5345	0.704577
$370$	0	0
$371$	6.71754	0.348757
$372$	0	0
$373$	26.7483	1.38497	0.692487	−	0.721430i	$-0.256515\pi$
$373$	26.7483	1.38497	0.692487	+	0.721430i	$0.256515\pi$
$374$	0	0
$375$	−21.7347	−1.12238
$376$	0	0
$377$	−2.70245	−0.139183
$378$	0	0
$379$	7.05389	0.362334	0.181167	−	0.983452i	$-0.442013\pi$
$379$	7.05389	0.362334	0.181167	+	0.983452i	$0.442013\pi$
$380$	0	0
$381$	−16.1741	−0.828622
$382$	0	0
$383$	11.2256	0.573599	0.286799	−	0.957991i	$-0.407409\pi$
$383$	11.2256	0.573599	0.286799	+	0.957991i	$0.407409\pi$
$384$	0	0
$385$	1.08525	0.0553094
$386$	0	0
$387$	5.40854	0.274932
$388$	0	0
$389$	−7.08291	−0.359118	−0.179559	−	0.983747i	$-0.557467\pi$
$389$	−7.08291	−0.359118	−0.179559	+	0.983747i	$0.557467\pi$
$390$	0	0
$391$	−13.5732	−0.686425
$392$	0	0
$393$	18.7722	0.946933
$394$	0	0
$395$	5.42568	0.272996
$396$	0	0
$397$	−7.98693	−0.400852	−0.200426	−	0.979709i	$-0.564233\pi$
$397$	−7.98693	−0.400852	−0.200426	+	0.979709i	$0.564233\pi$
$398$	0	0
$399$	−0.291076	−0.0145720
$400$	0	0
$401$	−21.6997	−1.08363	−0.541816	−	0.840497i	$-0.682263\pi$
$401$	−21.6997	−1.08363	−0.541816	+	0.840497i	$0.682263\pi$
$402$	0	0
$403$	−6.97612	−0.347505
$404$	0	0
$405$	−11.7457	−0.583650
$406$	0	0
$407$	7.85402	0.389309
$408$	0	0
$409$	2.92255	0.144511	0.0722553	−	0.997386i	$-0.476980\pi$
$409$	2.92255	0.144511	0.0722553	+	0.997386i	$0.476980\pi$
$410$	0	0
$411$	−21.7321	−1.07197
$412$	0	0
$413$	−9.89513	−0.486907
$414$	0	0
$415$	17.3996	0.854112
$416$	0	0
$417$	29.0235	1.42129
$418$	0	0
$419$	−19.4157	−0.948518	−0.474259	−	0.880385i	$-0.657284\pi$
$419$	−19.4157	−0.948518	−0.474259	+	0.880385i	$0.657284\pi$
$420$	0	0
$421$	−29.7711	−1.45095	−0.725477	−	0.688246i	$-0.758381\pi$
$421$	−29.7711	−1.45095	−0.725477	+	0.688246i	$0.758381\pi$
$422$	0	0
$423$	−14.5048	−0.705248
$424$	0	0
$425$	7.70212	0.373608
$426$	0	0
$427$	2.42338	0.117276
$428$	0	0
$429$	−2.27011	−0.109602
$430$	0	0
$431$	21.7679	1.04852	0.524261	−	0.851558i	$-0.324342\pi$
$431$	21.7679	1.04852	0.524261	+	0.851558i	$0.324342\pi$
$432$	0	0
$433$	−26.2082	−1.25949	−0.629744	−	0.776803i	$-0.716840\pi$
$433$	−26.2082	−1.25949	−0.629744	+	0.776803i	$0.716840\pi$
$434$	0	0
$435$	6.65785	0.319220
$436$	0	0
$437$	−0.863672	−0.0413151
$438$	0	0
$439$	−37.0713	−1.76932	−0.884659	−	0.466238i	$-0.845609\pi$
$439$	−37.0713	−1.76932	−0.884659	+	0.466238i	$0.845609\pi$
$440$	0	0
$441$	2.15339	0.102543
$442$	0	0
$443$	−36.6514	−1.74136	−0.870680	−	0.491850i	$-0.836321\pi$
$443$	−36.6514	−1.74136	−0.870680	+	0.491850i	$0.836321\pi$
$444$	0	0
$445$	5.07509	0.240582
$446$	0	0
$447$	0.272369	0.0128826
$448$	0	0
$449$	−34.7450	−1.63972	−0.819859	−	0.572566i	$-0.805948\pi$
$449$	−34.7450	−1.63972	−0.819859	+	0.572566i	$0.805948\pi$
$450$	0	0
$451$	6.28519	0.295958
$452$	0	0
$453$	32.1610	1.51106
$454$	0	0
$455$	−1.08525	−0.0508772
$456$	0	0
$457$	5.17955	0.242289	0.121145	−	0.992635i	$-0.461344\pi$
$457$	5.17955	0.242289	0.121145	+	0.992635i	$0.461344\pi$
$458$	0	0
$459$	3.87276	0.180765
$460$	0	0
$461$	28.6476	1.33425	0.667125	−	0.744945i	$-0.267525\pi$
$461$	28.6476	1.33425	0.667125	+	0.744945i	$0.267525\pi$
$462$	0	0
$463$	4.50924	0.209562	0.104781	−	0.994495i	$-0.466586\pi$
$463$	4.50924	0.209562	0.104781	+	0.994495i	$0.466586\pi$
$464$	0	0
$465$	17.1866	0.797010
$466$	0	0
$467$	17.7537	0.821543	0.410772	−	0.911738i	$-0.365259\pi$
$467$	17.7537	0.821543	0.410772	+	0.911738i	$0.365259\pi$
$468$	0	0
$469$	−11.2747	−0.520618
$470$	0	0
$471$	43.3622	1.99802
$472$	0	0
$473$	2.51164	0.115485
$474$	0	0
$475$	0.490092	0.0224870
$476$	0	0
$477$	14.4655	0.662330
$478$	0	0
$479$	−31.8453	−1.45505	−0.727525	−	0.686081i	$-0.759329\pi$
$479$	−31.8453	−1.45505	−0.727525	+	0.686081i	$0.759329\pi$
$480$	0	0
$481$	−7.85402	−0.358112
$482$	0	0
$483$	15.2910	0.695764
$484$	0	0
$485$	−9.35010	−0.424566
$486$	0	0
$487$	29.3524	1.33008	0.665042	−	0.746806i	$-0.268414\pi$
$487$	29.3524	1.33008	0.665042	+	0.746806i	$0.268414\pi$
$488$	0	0
$489$	−43.2960	−1.95791
$490$	0	0
$491$	−3.40365	−0.153604	−0.0768022	−	0.997046i	$-0.524471\pi$
$491$	−3.40365	−0.153604	−0.0768022	+	0.997046i	$0.524471\pi$
$492$	0	0
$493$	−5.44566	−0.245260
$494$	0	0
$495$	2.33697	0.105039
$496$	0	0
$497$	−3.36875	−0.151109
$498$	0	0
$499$	16.2010	0.725257	0.362629	−	0.931934i	$-0.381879\pi$
$499$	16.2010	0.725257	0.362629	+	0.931934i	$0.381879\pi$
$500$	0	0
$501$	10.7177	0.478833
$502$	0	0
$503$	7.91042	0.352708	0.176354	−	0.984327i	$-0.443570\pi$
$503$	7.91042	0.352708	0.176354	+	0.984327i	$0.443570\pi$
$504$	0	0
$505$	−5.82236	−0.259092
$506$	0	0
$507$	2.27011	0.100819
$508$	0	0
$509$	−15.9294	−0.706058	−0.353029	−	0.935612i	$-0.614848\pi$
$509$	−15.9294	−0.706058	−0.353029	+	0.935612i	$0.614848\pi$
$510$	0	0
$511$	9.69249	0.428770
$512$	0	0
$513$	0.246427	0.0108800
$514$	0	0
$515$	6.18320	0.272464
$516$	0	0
$517$	−6.73579	−0.296240
$518$	0	0
$519$	−9.72030	−0.426674
$520$	0	0
$521$	−9.40310	−0.411957	−0.205979	−	0.978556i	$-0.566038\pi$
$521$	−9.40310	−0.411957	−0.205979	+	0.978556i	$0.566038\pi$
$522$	0	0
$523$	−34.3692	−1.50286	−0.751429	−	0.659814i	$-0.770635\pi$
$523$	−34.3692	−1.50286	−0.751429	+	0.659814i	$0.770635\pi$
$524$	0	0
$525$	−8.67689	−0.378690
$526$	0	0
$527$	−14.0575	−0.612353
$528$	0	0
$529$	22.3709	0.972648
$530$	0	0
$531$	−21.3081	−0.924693
$532$	0	0
$533$	−6.28519	−0.272242
$534$	0	0
$535$	6.64862	0.287445
$536$	0	0
$537$	54.7417	2.36228
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−16.2950	−0.700577	−0.350288	−	0.936642i	$-0.613916\pi$
$541$	−16.2950	−0.700577	−0.350288	+	0.936642i	$0.613916\pi$
$542$	0	0
$543$	32.3489	1.38822
$544$	0	0
$545$	5.31698	0.227754
$546$	0	0
$547$	15.6185	0.667798	0.333899	−	0.942609i	$-0.391636\pi$
$547$	15.6185	0.667798	0.333899	+	0.942609i	$0.391636\pi$
$548$	0	0
$549$	5.21849	0.222720
$550$	0	0
$551$	−0.346512	−0.0147619
$552$	0	0
$553$	4.99948	0.212600
$554$	0	0
$555$	19.3494	0.821337
$556$	0	0
$557$	11.9695	0.507162	0.253581	−	0.967314i	$-0.418392\pi$
$557$	11.9695	0.507162	0.253581	+	0.967314i	$0.418392\pi$
$558$	0	0
$559$	−2.51164	−0.106231
$560$	0	0
$561$	−4.57445	−0.193134
$562$	0	0
$563$	−23.2636	−0.980443	−0.490221	−	0.871598i	$-0.663084\pi$
$563$	−23.2636	−0.980443	−0.490221	+	0.871598i	$0.663084\pi$
$564$	0	0
$565$	15.3223	0.644613
$566$	0	0
$567$	−10.8231	−0.454526
$568$	0	0
$569$	27.9527	1.17184	0.585919	−	0.810369i	$-0.300733\pi$
$569$	27.9527	1.17184	0.585919	+	0.810369i	$0.300733\pi$
$570$	0	0
$571$	18.9453	0.792835	0.396418	−	0.918070i	$-0.370253\pi$
$571$	18.9453	0.792835	0.396418	+	0.918070i	$0.370253\pi$
$572$	0	0
$573$	−56.0717	−2.34243
$574$	0	0
$575$	−25.7458	−1.07367
$576$	0	0
$577$	11.9772	0.498618	0.249309	−	0.968424i	$-0.419796\pi$
$577$	11.9772	0.498618	0.249309	+	0.968424i	$0.419796\pi$
$578$	0	0
$579$	−5.59077	−0.232345
$580$	0	0
$581$	16.0328	0.665153
$582$	0	0
$583$	6.71754	0.278212
$584$	0	0
$585$	−2.33697	−0.0966217
$586$	0	0
$587$	12.7317	0.525493	0.262746	−	0.964865i	$-0.415372\pi$
$587$	12.7317	0.525493	0.262746	+	0.964865i	$0.415372\pi$
$588$	0	0
$589$	−0.894488	−0.0368567
$590$	0	0
$591$	−55.1609	−2.26901
$592$	0	0
$593$	30.9107	1.26935	0.634675	−	0.772779i	$-0.281134\pi$
$593$	30.9107	1.26935	0.634675	+	0.772779i	$0.281134\pi$
$594$	0	0
$595$	−2.18687	−0.0896527
$596$	0	0
$597$	−38.8096	−1.58837
$598$	0	0
$599$	41.4811	1.69487	0.847437	−	0.530897i	$-0.178145\pi$
$599$	41.4811	1.69487	0.847437	+	0.530897i	$0.178145\pi$
$600$	0	0
$601$	8.55768	0.349075	0.174538	−	0.984651i	$-0.444157\pi$
$601$	8.55768	0.349075	0.174538	+	0.984651i	$0.444157\pi$
$602$	0	0
$603$	−24.2789	−0.988713
$604$	0	0
$605$	1.08525	0.0441216
$606$	0	0
$607$	−27.2598	−1.10644	−0.553221	−	0.833035i	$-0.686601\pi$
$607$	−27.2598	−1.10644	−0.553221	+	0.833035i	$0.686601\pi$
$608$	0	0
$609$	6.13486	0.248597
$610$	0	0
$611$	6.73579	0.272501
$612$	0	0
$613$	−40.6640	−1.64240	−0.821202	−	0.570637i	$-0.806696\pi$
$613$	−40.6640	−1.64240	−0.821202	+	0.570637i	$0.806696\pi$
$614$	0	0
$615$	15.4844	0.624391
$616$	0	0
$617$	6.61939	0.266487	0.133243	−	0.991083i	$-0.457461\pi$
$617$	6.61939	0.266487	0.133243	+	0.991083i	$0.457461\pi$
$618$	0	0
$619$	12.1172	0.487030	0.243515	−	0.969897i	$-0.421699\pi$
$619$	12.1172	0.487030	0.243515	+	0.969897i	$0.421699\pi$
$620$	0	0
$621$	−12.9454	−0.519483
$622$	0	0
$623$	4.67643	0.187357
$624$	0	0
$625$	8.72066	0.348826
$626$	0	0
$627$	−0.291076	−0.0116245
$628$	0	0
$629$	−15.8265	−0.631043
$630$	0	0
$631$	−4.61809	−0.183843	−0.0919217	−	0.995766i	$-0.529301\pi$
$631$	−4.61809	−0.183843	−0.0919217	+	0.995766i	$0.529301\pi$
$632$	0	0
$633$	−32.5000	−1.29176
$634$	0	0
$635$	−7.73217	−0.306842
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−7.25425	−0.286974
$640$	0	0
$641$	−32.7737	−1.29448	−0.647241	−	0.762286i	$-0.724077\pi$
$641$	−32.7737	−1.29448	−0.647241	+	0.762286i	$0.724077\pi$
$642$	0	0
$643$	9.83006	0.387660	0.193830	−	0.981035i	$-0.437909\pi$
$643$	9.83006	0.387660	0.193830	+	0.981035i	$0.437909\pi$
$644$	0	0
$645$	6.18775	0.243642
$646$	0	0
$647$	44.3598	1.74396	0.871982	−	0.489539i	$-0.162835\pi$
$647$	44.3598	1.74396	0.871982	+	0.489539i	$0.162835\pi$
$648$	0	0
$649$	−9.89513	−0.388418
$650$	0	0
$651$	15.8366	0.620684
$652$	0	0
$653$	38.5389	1.50814	0.754071	−	0.656792i	$-0.228087\pi$
$653$	38.5389	1.50814	0.754071	+	0.656792i	$0.228087\pi$
$654$	0	0
$655$	8.97425	0.350653
$656$	0	0
$657$	20.8717	0.814284
$658$	0	0
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405626\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405626\pi$
$660$	0	0
$661$	20.1844	0.785084	0.392542	−	0.919734i	$-0.371596\pi$
$661$	20.1844	0.785084	0.392542	+	0.919734i	$0.371596\pi$
$662$	0	0
$663$	4.57445	0.177657
$664$	0	0
$665$	−0.139152	−0.00539608
$666$	0	0
$667$	18.2032	0.704829
$668$	0	0
$669$	−3.51703	−0.135976
$670$	0	0
$671$	2.42338	0.0935535
$672$	0	0
$673$	−30.3903	−1.17146	−0.585729	−	0.810507i	$-0.699192\pi$
$673$	−30.3903	−1.17146	−0.585729	+	0.810507i	$0.699192\pi$
$674$	0	0
$675$	7.34591	0.282744
$676$	0	0
$677$	−20.1810	−0.775619	−0.387810	−	0.921740i	$-0.626768\pi$
$677$	−20.1810	−0.775619	−0.387810	+	0.921740i	$0.626768\pi$
$678$	0	0
$679$	−8.61563	−0.330638
$680$	0	0
$681$	−17.0415	−0.653031
$682$	0	0
$683$	4.22442	0.161643	0.0808214	−	0.996729i	$-0.474246\pi$
$683$	4.22442	0.161643	0.0808214	+	0.996729i	$0.474246\pi$
$684$	0	0
$685$	−10.3893	−0.396954
$686$	0	0
$687$	−12.8910	−0.491821
$688$	0	0
$689$	−6.71754	−0.255918
$690$	0	0
$691$	−16.0635	−0.611085	−0.305543	−	0.952178i	$-0.598838\pi$
$691$	−16.0635	−0.611085	−0.305543	+	0.952178i	$0.598838\pi$
$692$	0	0
$693$	2.15339	0.0818007
$694$	0	0
$695$	13.8750	0.526308
$696$	0	0
$697$	−12.6652	−0.479728
$698$	0	0
$699$	51.5626	1.95028
$700$	0	0
$701$	−39.1481	−1.47860	−0.739302	−	0.673374i	$-0.764844\pi$
$701$	−39.1481	−1.47860	−0.739302	+	0.673374i	$0.764844\pi$
$702$	0	0
$703$	−1.00705	−0.0379817
$704$	0	0
$705$	−16.5945	−0.624986
$706$	0	0
$707$	−5.36500	−0.201772
$708$	0	0
$709$	−40.7122	−1.52898	−0.764490	−	0.644635i	$-0.777009\pi$
$709$	−40.7122	−1.52898	−0.764490	+	0.644635i	$0.777009\pi$
$710$	0	0
$711$	10.7659	0.403751
$712$	0	0
$713$	46.9897	1.75978
$714$	0	0
$715$	−1.08525	−0.0405860
$716$	0	0
$717$	39.6940	1.48240
$718$	0	0
$719$	45.8724	1.71075	0.855375	−	0.518009i	$-0.173327\pi$
$719$	45.8724	1.71075	0.855375	+	0.518009i	$0.173327\pi$
$720$	0	0
$721$	5.69749	0.212186
$722$	0	0
$723$	37.7618	1.40438
$724$	0	0
$725$	−10.3294	−0.383625
$726$	0	0
$727$	−46.4950	−1.72441	−0.862203	−	0.506563i	$-0.830916\pi$
$727$	−46.4950	−1.72441	−0.862203	+	0.506563i	$0.830916\pi$
$728$	0	0
$729$	−10.2177	−0.378432
$730$	0	0
$731$	−5.06115	−0.187193
$732$	0	0
$733$	−9.17112	−0.338743	−0.169372	−	0.985552i	$-0.554174\pi$
$733$	−9.17112	−0.338743	−0.169372	+	0.985552i	$0.554174\pi$
$734$	0	0
$735$	2.46363	0.0908725
$736$	0	0
$737$	−11.2747	−0.415309
$738$	0	0
$739$	−26.6871	−0.981702	−0.490851	−	0.871244i	$-0.663314\pi$
$739$	−26.6871	−0.981702	−0.490851	+	0.871244i	$0.663314\pi$
$740$	0	0
$741$	0.291076	0.0106930
$742$	0	0
$743$	−2.24301	−0.0822880	−0.0411440	−	0.999153i	$-0.513100\pi$
$743$	−2.24301	−0.0822880	−0.0411440	+	0.999153i	$0.513100\pi$
$744$	0	0
$745$	0.130209	0.00477048
$746$	0	0
$747$	34.5249	1.26320
$748$	0	0
$749$	6.12636	0.223852
$750$	0	0
$751$	−47.0284	−1.71609	−0.858046	−	0.513573i	$-0.828322\pi$
$751$	−47.0284	−1.71609	−0.858046	+	0.513573i	$0.828322\pi$
$752$	0	0
$753$	11.5383	0.420478
$754$	0	0
$755$	15.3749	0.559550
$756$	0	0
$757$	−12.7686	−0.464081	−0.232040	−	0.972706i	$-0.574540\pi$
$757$	−12.7686	−0.464081	−0.232040	+	0.972706i	$0.574540\pi$
$758$	0	0
$759$	15.2910	0.555028
$760$	0	0
$761$	−22.0764	−0.800269	−0.400134	−	0.916456i	$-0.631037\pi$
$761$	−22.0764	−0.800269	−0.400134	+	0.916456i	$0.631037\pi$
$762$	0	0
$763$	4.89931	0.177367
$764$	0	0
$765$	−4.70918	−0.170261
$766$	0	0
$767$	9.89513	0.357292
$768$	0	0
$769$	34.9816	1.26147	0.630735	−	0.775999i	$-0.282754\pi$
$769$	34.9816	1.26147	0.630735	+	0.775999i	$0.282754\pi$
$770$	0	0
$771$	1.74542	0.0628599
$772$	0	0
$773$	−5.00734	−0.180101	−0.0900507	−	0.995937i	$-0.528703\pi$
$773$	−5.00734	−0.180101	−0.0900507	+	0.995937i	$0.528703\pi$
$774$	0	0
$775$	−26.6644	−0.957813
$776$	0	0
$777$	17.8295	0.639629
$778$	0	0
$779$	−0.805895	−0.0288742
$780$	0	0
$781$	−3.36875	−0.120543
$782$	0	0
$783$	−5.19381	−0.185612
$784$	0	0
$785$	20.7297	0.739876
$786$	0	0
$787$	9.29490	0.331327	0.165664	−	0.986182i	$-0.447023\pi$
$787$	9.29490	0.331327	0.165664	+	0.986182i	$0.447023\pi$
$788$	0	0
$789$	−65.6571	−2.33745
$790$	0	0
$791$	14.1187	0.502002
$792$	0	0
$793$	−2.42338	−0.0860567
$794$	0	0
$795$	16.5495	0.586952
$796$	0	0
$797$	−32.2488	−1.14231	−0.571156	−	0.820841i	$-0.693505\pi$
$797$	−32.2488	−1.14231	−0.571156	+	0.820841i	$0.693505\pi$
$798$	0	0
$799$	13.5732	0.480184
$800$	0	0
$801$	10.0702	0.355813
$802$	0	0
$803$	9.69249	0.342040
$804$	0	0
$805$	7.31001	0.257644
$806$	0	0
$807$	−63.4046	−2.23195
$808$	0	0
$809$	7.52955	0.264725	0.132362	−	0.991201i	$-0.457744\pi$
$809$	7.52955	0.264725	0.132362	+	0.991201i	$0.457744\pi$
$810$	0	0
$811$	38.7800	1.36175	0.680874	−	0.732400i	$-0.261600\pi$
$811$	38.7800	1.36175	0.680874	+	0.732400i	$0.261600\pi$
$812$	0	0
$813$	1.32879	0.0466027
$814$	0	0
$815$	−20.6981	−0.725023
$816$	0	0
$817$	−0.322045	−0.0112669
$818$	0	0
$819$	−2.15339	−0.0752457
$820$	0	0
$821$	30.4309	1.06205	0.531023	−	0.847357i	$-0.321808\pi$
$821$	30.4309	1.06205	0.531023	+	0.847357i	$0.321808\pi$
$822$	0	0
$823$	−38.9114	−1.35637	−0.678183	−	0.734893i	$-0.737232\pi$
$823$	−38.9114	−1.35637	−0.678183	+	0.734893i	$0.737232\pi$
$824$	0	0
$825$	−8.67689	−0.302091
$826$	0	0
$827$	−29.1921	−1.01511	−0.507554	−	0.861620i	$-0.669450\pi$
$827$	−29.1921	−1.01511	−0.507554	+	0.861620i	$0.669450\pi$
$828$	0	0
$829$	−33.0570	−1.14812	−0.574058	−	0.818815i	$-0.694631\pi$
$829$	−33.0570	−1.14812	−0.574058	+	0.818815i	$0.694631\pi$
$830$	0	0
$831$	−54.0058	−1.87344
$832$	0	0
$833$	−2.01508	−0.0698184
$834$	0	0
$835$	5.12373	0.177314
$836$	0	0
$837$	−13.4073	−0.463425
$838$	0	0
$839$	−28.1057	−0.970317	−0.485159	−	0.874426i	$-0.661238\pi$
$839$	−28.1057	−0.970317	−0.485159	+	0.874426i	$0.661238\pi$
$840$	0	0
$841$	−21.6967	−0.748164
$842$	0	0
$843$	17.2326	0.593521
$844$	0	0
$845$	1.08525	0.0373337
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	26.2350	0.900382
$850$	0	0
$851$	52.9030	1.81349
$852$	0	0
$853$	−32.7750	−1.12219	−0.561097	−	0.827750i	$-0.689621\pi$
$853$	−32.7750	−1.12219	−0.561097	+	0.827750i	$0.689621\pi$
$854$	0	0
$855$	−0.299649	−0.0102478
$856$	0	0
$857$	−1.48372	−0.0506828	−0.0253414	−	0.999679i	$-0.508067\pi$
$857$	−1.48372	−0.0506828	−0.0253414	+	0.999679i	$0.508067\pi$
$858$	0	0
$859$	15.6240	0.533084	0.266542	−	0.963823i	$-0.414119\pi$
$859$	15.6240	0.533084	0.266542	+	0.963823i	$0.414119\pi$
$860$	0	0
$861$	14.2681	0.486254
$862$	0	0
$863$	43.2220	1.47129	0.735646	−	0.677366i	$-0.236879\pi$
$863$	43.2220	1.47129	0.735646	+	0.677366i	$0.236879\pi$
$864$	0	0
$865$	−4.64689	−0.157999
$866$	0	0
$867$	−29.3739	−0.997592
$868$	0	0
$869$	4.99948	0.169596
$870$	0	0
$871$	11.2747	0.382029
$872$	0	0
$873$	−18.5528	−0.627919
$874$	0	0
$875$	−9.57432	−0.323671
$876$	0	0
$877$	37.5407	1.26766	0.633829	−	0.773473i	$-0.281482\pi$
$877$	37.5407	1.26766	0.633829	+	0.773473i	$0.281482\pi$
$878$	0	0
$879$	−30.8558	−1.04074
$880$	0	0
$881$	8.37195	0.282058	0.141029	−	0.990005i	$-0.454959\pi$
$881$	8.37195	0.282058	0.141029	+	0.990005i	$0.454959\pi$
$882$	0	0
$883$	−31.2211	−1.05067	−0.525337	−	0.850894i	$-0.676061\pi$
$883$	−31.2211	−1.05067	−0.525337	+	0.850894i	$0.676061\pi$
$884$	0	0
$885$	−24.3780	−0.819456
$886$	0	0
$887$	−42.5003	−1.42702	−0.713509	−	0.700646i	$-0.752895\pi$
$887$	−42.5003	−1.42702	−0.713509	+	0.700646i	$0.752895\pi$
$888$	0	0
$889$	−7.12479	−0.238958
$890$	0	0
$891$	−10.8231	−0.362587
$892$	0	0
$893$	0.863672	0.0289017
$894$	0	0
$895$	26.1698	0.874761
$896$	0	0
$897$	−15.2910	−0.510551
$898$	0	0
$899$	18.8527	0.628771
$900$	0	0
$901$	−13.5364	−0.450962
$902$	0	0
$903$	5.70169	0.189740
$904$	0	0
$905$	15.4647	0.514065
$906$	0	0
$907$	−41.0016	−1.36144	−0.680718	−	0.732546i	$-0.738332\pi$
$907$	−41.0016	−1.36144	−0.680718	+	0.732546i	$0.738332\pi$
$908$	0	0
$909$	−11.5530	−0.383187
$910$	0	0
$911$	7.64361	0.253244	0.126622	−	0.991951i	$-0.459586\pi$
$911$	7.64361	0.253244	0.126622	+	0.991951i	$0.459586\pi$
$912$	0	0
$913$	16.0328	0.530608
$914$	0	0
$915$	5.97032	0.197373
$916$	0	0
$917$	8.26930	0.273076
$918$	0	0
$919$	46.3353	1.52846	0.764231	−	0.644943i	$-0.223119\pi$
$919$	46.3353	1.52846	0.764231	+	0.644943i	$0.223119\pi$
$920$	0	0
$921$	47.7289	1.57272
$922$	0	0
$923$	3.36875	0.110884
$924$	0	0
$925$	−30.0199	−0.987048
$926$	0	0
$927$	12.2689	0.402965
$928$	0	0
$929$	−18.4188	−0.604300	−0.302150	−	0.953260i	$-0.597704\pi$
$929$	−18.4188	−0.604300	−0.302150	+	0.953260i	$0.597704\pi$
$930$	0	0
$931$	−0.128221	−0.00420228
$932$	0	0
$933$	−44.3483	−1.45190
$934$	0	0
$935$	−2.18687	−0.0715181
$936$	0	0
$937$	0.520065	0.0169898	0.00849489	−	0.999964i	$-0.497296\pi$
$937$	0.520065	0.0169898	0.00849489	+	0.999964i	$0.497296\pi$
$938$	0	0
$939$	23.3905	0.763320
$940$	0	0
$941$	−33.3047	−1.08570	−0.542851	−	0.839829i	$-0.682655\pi$
$941$	−33.3047	−1.08570	−0.542851	+	0.839829i	$0.682655\pi$
$942$	0	0
$943$	42.3357	1.37864
$944$	0	0
$945$	−2.08573	−0.0678487
$946$	0	0
$947$	−10.3402	−0.336011	−0.168006	−	0.985786i	$-0.553733\pi$
$947$	−10.3402	−0.336011	−0.168006	+	0.985786i	$0.553733\pi$
$948$	0	0
$949$	−9.69249	−0.314631
$950$	0	0
$951$	11.9331	0.386957
$952$	0	0
$953$	−6.72930	−0.217984	−0.108992	−	0.994043i	$-0.534762\pi$
$953$	−6.72930	−0.217984	−0.108992	+	0.994043i	$0.534762\pi$
$954$	0	0
$955$	−26.8056	−0.867410
$956$	0	0
$957$	6.13486	0.198312
$958$	0	0
$959$	−9.57317	−0.309134
$960$	0	0
$961$	17.6663	0.569881
$962$	0	0
$963$	13.1925	0.425121
$964$	0	0
$965$	−2.67273	−0.0860381
$966$	0	0
$967$	28.5310	0.917495	0.458747	−	0.888567i	$-0.348298\pi$
$967$	28.5310	0.917495	0.458747	+	0.888567i	$0.348298\pi$
$968$	0	0
$969$	0.586543	0.0188425
$970$	0	0
$971$	−7.56323	−0.242716	−0.121358	−	0.992609i	$-0.538725\pi$
$971$	−7.56323	−0.242716	−0.121358	+	0.992609i	$0.538725\pi$
$972$	0	0
$973$	12.7851	0.409870
$974$	0	0
$975$	8.67689	0.277883
$976$	0	0
$977$	24.9984	0.799770	0.399885	−	0.916565i	$-0.369050\pi$
$977$	24.9984	0.799770	0.399885	+	0.916565i	$0.369050\pi$
$978$	0	0
$979$	4.67643	0.149459
$980$	0	0
$981$	10.5502	0.336840
$982$	0	0
$983$	31.1356	0.993070	0.496535	−	0.868017i	$-0.334605\pi$
$983$	31.1356	0.993070	0.496535	+	0.868017i	$0.334605\pi$
$984$	0	0
$985$	−26.3702	−0.840225
$986$	0	0
$987$	−15.2910	−0.486717
$988$	0	0
$989$	16.9179	0.537957
$990$	0	0
$991$	57.0721	1.81295	0.906477	−	0.422255i	$-0.138761\pi$
$991$	57.0721	1.81295	0.906477	+	0.422255i	$0.138761\pi$
$992$	0	0
$993$	33.3153	1.05723
$994$	0	0
$995$	−18.5533	−0.588180
$996$	0	0
$997$	−18.7037	−0.592352	−0.296176	−	0.955133i	$-0.595711\pi$
$997$	−18.7037	−0.592352	−0.296176	+	0.955133i	$0.595711\pi$
$998$	0	0
$999$	−15.0945	−0.477570

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.q.1.7	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.q.1.7	✓	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)

\(f(q)\)	\(=\)	\(q+2.27011 q^{3} +1.08525 q^{5} +1.00000 q^{7} +2.15339 q^{9} +O(q^{10})\)	\(q+2.27011 q^{3} +1.08525 q^{5} +1.00000 q^{7} +2.15339 q^{9} +1.00000 q^{11} -1.00000 q^{13} +2.46363 q^{15} -2.01508 q^{17} -0.128221 q^{19} +2.27011 q^{21} +6.73579 q^{23} -3.82224 q^{25} -1.92189 q^{27} +2.70245 q^{29} +6.97612 q^{31} +2.27011 q^{33} +1.08525 q^{35} +7.85402 q^{37} -2.27011 q^{39} +6.28519 q^{41} +2.51164 q^{43} +2.33697 q^{45} -6.73579 q^{47} +1.00000 q^{49} -4.57445 q^{51} +6.71754 q^{53} +1.08525 q^{55} -0.291076 q^{57} -9.89513 q^{59} +2.42338 q^{61} +2.15339 q^{63} -1.08525 q^{65} -11.2747 q^{67} +15.2910 q^{69} -3.36875 q^{71} +9.69249 q^{73} -8.67689 q^{75} +1.00000 q^{77} +4.99948 q^{79} -10.8231 q^{81} +16.0328 q^{83} -2.18687 q^{85} +6.13486 q^{87} +4.67643 q^{89} -1.00000 q^{91} +15.8366 q^{93} -0.139152 q^{95} -8.61563 q^{97} +2.15339 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9	\( 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + q^{17} + 16 q^{19} + 3 q^{21} + 6 q^{23} + 11 q^{25} + 9 q^{27} - 2 q^{29} + 3 q^{31} + 3 q^{33} + 8 q^{35} - 4 q^{37} - 3 q^{39} + 20 q^{41} + 20 q^{43} + 8 q^{45} - 6 q^{47} + 9 q^{49} + 15 q^{51} + 15 q^{53} + 8 q^{55} + 16 q^{57} + 21 q^{59} + 12 q^{63} - 8 q^{65} + 18 q^{67} + 10 q^{69} + 12 q^{71} + 29 q^{73} + 12 q^{75} + 9 q^{77} - 6 q^{79} + 5 q^{81} + 25 q^{83} + 13 q^{85} + 17 q^{87} + 32 q^{89} - 9 q^{91} - 13 q^{93} + 38 q^{95} + 17 q^{97} + 12 q^{99}+O(q^{100}) \) 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 - 9 * q^15 + q^17 + 16 * q^19 + 3 * q^21 + 6 * q^23 + 11 * q^25 + 9 * q^27 - 2 * q^29 + 3 * q^31 + 3 * q^33 + 8 * q^35 - 4 * q^37 - 3 * q^39 + 20 * q^41 + 20 * q^43 + 8 * q^45 - 6 * q^47 + 9 * q^49 + 15 * q^51 + 15 * q^53 + 8 * q^55 + 16 * q^57 + 21 * q^59 + 12 * q^63 - 8 * q^65 + 18 * q^67 + 10 * q^69 + 12 * q^71 + 29 * q^73 + 12 * q^75 + 9 * q^77 - 6 * q^79 + 5 * q^81 + 25 * q^83 + 13 * q^85 + 17 * q^87 + 32 * q^89 - 9 * q^91 - 13 * q^93 + 38 * q^95 + 17 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more