LMFDB - Embedded newform 8004.2.a.f.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8004,2,Mod(1,8004)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8004.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8004, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,0,9,0,-1,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.9122617778$
Analytic rank:	$1$
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} - x^{8} - 17x^{7} + 4x^{6} + 75x^{5} + x^{4} - 118x^{3} - 26x^{2} + 60x + 24 $ x^9 - x^8 - 17x^7 + 4x^6 + 75x^5 + x^4 - 118x^3 - 26x^2 + 60x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.91513$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.91513 q^{5} -2.97719 q^{7} +1.00000 q^{9} +2.04500 q^{11} +1.36883 q^{13} +1.91513 q^{15} -6.26690 q^{17} -2.09894 q^{19} -2.97719 q^{21} -1.00000 q^{23} -1.33227 q^{25} +1.00000 q^{27} -1.00000 q^{29} +1.34340 q^{31} +2.04500 q^{33} -5.70171 q^{35} -5.21257 q^{37} +1.36883 q^{39} +7.87339 q^{41} -1.58152 q^{43} +1.91513 q^{45} +3.90675 q^{47} +1.86366 q^{49} -6.26690 q^{51} -9.86566 q^{53} +3.91645 q^{55} -2.09894 q^{57} -2.00323 q^{59} -9.80143 q^{61} -2.97719 q^{63} +2.62148 q^{65} +2.45384 q^{67} -1.00000 q^{69} -8.65971 q^{71} +14.2737 q^{73} -1.33227 q^{75} -6.08837 q^{77} +0.878902 q^{79} +1.00000 q^{81} -13.3255 q^{83} -12.0019 q^{85} -1.00000 q^{87} -5.49647 q^{89} -4.07526 q^{91} +1.34340 q^{93} -4.01975 q^{95} -8.26837 q^{97} +2.04500 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9} - 8 q^{11} - q^{13} - q^{15} - 2 q^{17} - 11 q^{19} - 5 q^{21} - 9 q^{23} - 10 q^{25} + 9 q^{27} - 9 q^{29} - 8 q^{33} + q^{35} - 2 q^{37} - q^{39} - 3 q^{41}+ \cdots - 8 q^{99}+O(q^{100}) $ 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9 - 8 * q^11 - q^13 - q^15 - 2 * q^17 - 11 * q^19 - 5 * q^21 - 9 * q^23 - 10 * q^25 + 9 * q^27 - 9 * q^29 - 8 * q^33 + q^35 - 2 * q^37 - q^39 - 3 * q^41 - 19 * q^43 - q^45 - 3 * q^47 - 6 * q^49 - 2 * q^51 - 9 * q^53 - 7 * q^55 - 11 * q^57 - 2 * q^59 - 25 * q^61 - 5 * q^63 - 12 * q^65 - 20 * q^67 - 9 * q^69 + 9 * q^71 - 11 * q^73 - 10 * q^75 - 19 * q^77 + 4 * q^79 + 9 * q^81 - 9 * q^83 - 50 * q^85 - 9 * q^87 - 29 * q^89 - 38 * q^91 + 23 * q^95 - 43 * q^97 - 8 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.91513	0.856473	0.428236	−	0.903667i	$-0.359135\pi$
$5$	1.91513	0.856473	0.428236	+	0.903667i	$0.359135\pi$
$6$	0	0
$7$	−2.97719	−1.12527	−0.562636	−	0.826705i	$-0.690213\pi$
$7$	−2.97719	−1.12527	−0.562636	+	0.826705i	$0.690213\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.04500	0.616592	0.308296	−	0.951290i	$-0.400241\pi$
$11$	2.04500	0.616592	0.308296	+	0.951290i	$0.400241\pi$
$12$	0	0
$13$	1.36883	0.379645	0.189822	−	0.981818i	$-0.439209\pi$
$13$	1.36883	0.379645	0.189822	+	0.981818i	$0.439209\pi$
$14$	0	0
$15$	1.91513	0.494485
$16$	0	0
$17$	−6.26690	−1.51995	−0.759974	−	0.649954i	$-0.774788\pi$
$17$	−6.26690	−1.51995	−0.759974	+	0.649954i	$0.774788\pi$
$18$	0	0
$19$	−2.09894	−0.481530	−0.240765	−	0.970583i	$-0.577398\pi$
$19$	−2.09894	−0.481530	−0.240765	+	0.970583i	$0.577398\pi$
$20$	0	0
$21$	−2.97719	−0.649676
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−1.33227	−0.266455
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	1.34340	0.241282	0.120641	−	0.992696i	$-0.461505\pi$
$31$	1.34340	0.241282	0.120641	+	0.992696i	$0.461505\pi$
$32$	0	0
$33$	2.04500	0.355990
$34$	0	0
$35$	−5.70171	−0.963765
$36$	0	0
$37$	−5.21257	−0.856942	−0.428471	−	0.903556i	$-0.640948\pi$
$37$	−5.21257	−0.856942	−0.428471	+	0.903556i	$0.640948\pi$
$38$	0	0
$39$	1.36883	0.219188
$40$	0	0
$41$	7.87339	1.22962	0.614808	−	0.788677i	$-0.289233\pi$
$41$	7.87339	1.22962	0.614808	+	0.788677i	$0.289233\pi$
$42$	0	0
$43$	−1.58152	−0.241180	−0.120590	−	0.992702i	$-0.538479\pi$
$43$	−1.58152	−0.241180	−0.120590	+	0.992702i	$0.538479\pi$
$44$	0	0
$45$	1.91513	0.285491
$46$	0	0
$47$	3.90675	0.569858	0.284929	−	0.958549i	$-0.408030\pi$
$47$	3.90675	0.569858	0.284929	+	0.958549i	$0.408030\pi$
$48$	0	0
$49$	1.86366	0.266237
$50$	0	0
$51$	−6.26690	−0.877542
$52$	0	0
$53$	−9.86566	−1.35515	−0.677576	−	0.735453i	$-0.736970\pi$
$53$	−9.86566	−1.35515	−0.677576	+	0.735453i	$0.736970\pi$
$54$	0	0
$55$	3.91645	0.528094
$56$	0	0
$57$	−2.09894	−0.278012
$58$	0	0
$59$	−2.00323	−0.260799	−0.130399	−	0.991462i	$-0.541626\pi$
$59$	−2.00323	−0.260799	−0.130399	+	0.991462i	$0.541626\pi$
$60$	0	0
$61$	−9.80143	−1.25494	−0.627472	−	0.778639i	$-0.715910\pi$
$61$	−9.80143	−1.25494	−0.627472	+	0.778639i	$0.715910\pi$
$62$	0	0
$63$	−2.97719	−0.375091
$64$	0	0
$65$	2.62148	0.325155
$66$	0	0
$67$	2.45384	0.299784	0.149892	−	0.988702i	$-0.452107\pi$
$67$	2.45384	0.299784	0.149892	+	0.988702i	$0.452107\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−8.65971	−1.02772	−0.513859	−	0.857875i	$-0.671785\pi$
$71$	−8.65971	−1.02772	−0.513859	+	0.857875i	$0.671785\pi$
$72$	0	0
$73$	14.2737	1.67061	0.835303	−	0.549790i	$-0.185292\pi$
$73$	14.2737	1.67061	0.835303	+	0.549790i	$0.185292\pi$
$74$	0	0
$75$	−1.33227	−0.153838
$76$	0	0
$77$	−6.08837	−0.693834
$78$	0	0
$79$	0.878902	0.0988842	0.0494421	−	0.998777i	$-0.484256\pi$
$79$	0.878902	0.0988842	0.0494421	+	0.998777i	$0.484256\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.3255	−1.46266	−0.731330	−	0.682024i	$-0.761100\pi$
$83$	−13.3255	−1.46266	−0.731330	+	0.682024i	$0.761100\pi$
$84$	0	0
$85$	−12.0019	−1.30179
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−5.49647	−0.582624	−0.291312	−	0.956628i	$-0.594092\pi$
$89$	−5.49647	−0.582624	−0.291312	+	0.956628i	$0.594092\pi$
$90$	0	0
$91$	−4.07526	−0.427203
$92$	0	0
$93$	1.34340	0.139304
$94$	0	0
$95$	−4.01975	−0.412418
$96$	0	0
$97$	−8.26837	−0.839525	−0.419763	−	0.907634i	$-0.637887\pi$
$97$	−8.26837	−0.839525	−0.419763	+	0.907634i	$0.637887\pi$
$98$	0	0
$99$	2.04500	0.205531
$100$	0	0
$101$	10.3712	1.03197	0.515985	−	0.856597i	$-0.327426\pi$
$101$	10.3712	1.03197	0.515985	+	0.856597i	$0.327426\pi$
$102$	0	0
$103$	0.930722	0.0917067	0.0458534	−	0.998948i	$-0.485399\pi$
$103$	0.930722	0.0917067	0.0458534	+	0.998948i	$0.485399\pi$
$104$	0	0
$105$	−5.70171	−0.556430
$106$	0	0
$107$	−15.1650	−1.46606	−0.733028	−	0.680199i	$-0.761893\pi$
$107$	−15.1650	−1.46606	−0.733028	+	0.680199i	$0.761893\pi$
$108$	0	0
$109$	−1.05050	−0.100619	−0.0503096	−	0.998734i	$-0.516021\pi$
$109$	−1.05050	−0.100619	−0.0503096	+	0.998734i	$0.516021\pi$
$110$	0	0
$111$	−5.21257	−0.494756
$112$	0	0
$113$	−10.6446	−1.00136	−0.500678	−	0.865633i	$-0.666916\pi$
$113$	−10.6446	−1.00136	−0.500678	+	0.865633i	$0.666916\pi$
$114$	0	0
$115$	−1.91513	−0.178587
$116$	0	0
$117$	1.36883	0.126548
$118$	0	0
$119$	18.6578	1.71035
$120$	0	0
$121$	−6.81796	−0.619814
$122$	0	0
$123$	7.87339	0.709920
$124$	0	0
$125$	−12.1271	−1.08468
$126$	0	0
$127$	9.98416	0.885951	0.442976	−	0.896534i	$-0.353923\pi$
$127$	9.98416	0.885951	0.442976	+	0.896534i	$0.353923\pi$
$128$	0	0
$129$	−1.58152	−0.139245
$130$	0	0
$131$	6.00561	0.524713	0.262356	−	0.964971i	$-0.415500\pi$
$131$	6.00561	0.524713	0.262356	+	0.964971i	$0.415500\pi$
$132$	0	0
$133$	6.24895	0.541853
$134$	0	0
$135$	1.91513	0.164828
$136$	0	0
$137$	12.4401	1.06283	0.531413	−	0.847113i	$-0.321661\pi$
$137$	12.4401	1.06283	0.531413	+	0.847113i	$0.321661\pi$
$138$	0	0
$139$	−22.9607	−1.94750	−0.973750	−	0.227622i	$-0.926905\pi$
$139$	−22.9607	−1.94750	−0.973750	+	0.227622i	$0.926905\pi$
$140$	0	0
$141$	3.90675	0.329008
$142$	0	0
$143$	2.79926	0.234086
$144$	0	0
$145$	−1.91513	−0.159043
$146$	0	0
$147$	1.86366	0.153712
$148$	0	0
$149$	−14.9121	−1.22165	−0.610823	−	0.791767i	$-0.709161\pi$
$149$	−14.9121	−1.22165	−0.610823	+	0.791767i	$0.709161\pi$
$150$	0	0
$151$	−1.53402	−0.124837	−0.0624185	−	0.998050i	$-0.519881\pi$
$151$	−1.53402	−0.124837	−0.0624185	+	0.998050i	$0.519881\pi$
$152$	0	0
$153$	−6.26690	−0.506649
$154$	0	0
$155$	2.57279	0.206651
$156$	0	0
$157$	9.66353	0.771234	0.385617	−	0.922659i	$-0.373989\pi$
$157$	9.66353	0.771234	0.385617	+	0.922659i	$0.373989\pi$
$158$	0	0
$159$	−9.86566	−0.782398
$160$	0	0
$161$	2.97719	0.234635
$162$	0	0
$163$	12.9447	1.01391	0.506953	−	0.861974i	$-0.330772\pi$
$163$	12.9447	1.01391	0.506953	+	0.861974i	$0.330772\pi$
$164$	0	0
$165$	3.91645	0.304895
$166$	0	0
$167$	5.50077	0.425663	0.212831	−	0.977089i	$-0.431732\pi$
$167$	5.50077	0.425663	0.212831	+	0.977089i	$0.431732\pi$
$168$	0	0
$169$	−11.1263	−0.855870
$170$	0	0
$171$	−2.09894	−0.160510
$172$	0	0
$173$	−9.96116	−0.757333	−0.378666	−	0.925533i	$-0.623617\pi$
$173$	−9.96116	−0.757333	−0.378666	+	0.925533i	$0.623617\pi$
$174$	0	0
$175$	3.96643	0.299834
$176$	0	0
$177$	−2.00323	−0.150572
$178$	0	0
$179$	−13.0081	−0.972274	−0.486137	−	0.873883i	$-0.661594\pi$
$179$	−13.0081	−0.972274	−0.486137	+	0.873883i	$0.661594\pi$
$180$	0	0
$181$	9.81020	0.729186	0.364593	−	0.931167i	$-0.381208\pi$
$181$	9.81020	0.729186	0.364593	+	0.931167i	$0.381208\pi$
$182$	0	0
$183$	−9.80143	−0.724543
$184$	0	0
$185$	−9.98276	−0.733947
$186$	0	0
$187$	−12.8158	−0.937188
$188$	0	0
$189$	−2.97719	−0.216559
$190$	0	0
$191$	5.75152	0.416166	0.208083	−	0.978111i	$-0.433278\pi$
$191$	5.75152	0.416166	0.208083	+	0.978111i	$0.433278\pi$
$192$	0	0
$193$	11.3988	0.820502	0.410251	−	0.911973i	$-0.365441\pi$
$193$	11.3988	0.820502	0.410251	+	0.911973i	$0.365441\pi$
$194$	0	0
$195$	2.62148	0.187728
$196$	0	0
$197$	17.2757	1.23084	0.615422	−	0.788198i	$-0.288986\pi$
$197$	17.2757	1.23084	0.615422	+	0.788198i	$0.288986\pi$
$198$	0	0
$199$	−10.9978	−0.779613	−0.389807	−	0.920897i	$-0.627458\pi$
$199$	−10.9978	−0.779613	−0.389807	+	0.920897i	$0.627458\pi$
$200$	0	0
$201$	2.45384	0.173081
$202$	0	0
$203$	2.97719	0.208958
$204$	0	0
$205$	15.0786	1.05313
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−4.29235	−0.296908
$210$	0	0
$211$	−5.49631	−0.378382	−0.189191	−	0.981940i	$-0.560586\pi$
$211$	−5.49631	−0.378382	−0.189191	+	0.981940i	$0.560586\pi$
$212$	0	0
$213$	−8.65971	−0.593353
$214$	0	0
$215$	−3.02882	−0.206564
$216$	0	0
$217$	−3.99955	−0.271507
$218$	0	0
$219$	14.2737	0.964525
$220$	0	0
$221$	−8.57831	−0.577040
$222$	0	0
$223$	20.8557	1.39660	0.698300	−	0.715806i	$-0.253940\pi$
$223$	20.8557	1.39660	0.698300	+	0.715806i	$0.253940\pi$
$224$	0	0
$225$	−1.33227	−0.0888183
$226$	0	0
$227$	12.4917	0.829101	0.414550	−	0.910026i	$-0.363939\pi$
$227$	12.4917	0.829101	0.414550	+	0.910026i	$0.363939\pi$
$228$	0	0
$229$	−28.0229	−1.85180	−0.925902	−	0.377764i	$-0.876693\pi$
$229$	−28.0229	−1.85180	−0.925902	+	0.377764i	$0.876693\pi$
$230$	0	0
$231$	−6.08837	−0.400585
$232$	0	0
$233$	−12.2804	−0.804518	−0.402259	−	0.915526i	$-0.631775\pi$
$233$	−12.2804	−0.804518	−0.402259	+	0.915526i	$0.631775\pi$
$234$	0	0
$235$	7.48194	0.488068
$236$	0	0
$237$	0.878902	0.0570908
$238$	0	0
$239$	8.94247	0.578440	0.289220	−	0.957263i	$-0.406604\pi$
$239$	8.94247	0.578440	0.289220	+	0.957263i	$0.406604\pi$
$240$	0	0
$241$	−0.940096	−0.0605569	−0.0302785	−	0.999542i	$-0.509639\pi$
$241$	−0.940096	−0.0605569	−0.0302785	+	0.999542i	$0.509639\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.56915	0.228025
$246$	0	0
$247$	−2.87309	−0.182810
$248$	0	0
$249$	−13.3255	−0.844467
$250$	0	0
$251$	18.5996	1.17400	0.586998	−	0.809588i	$-0.300310\pi$
$251$	18.5996	1.17400	0.586998	+	0.809588i	$0.300310\pi$
$252$	0	0
$253$	−2.04500	−0.128568
$254$	0	0
$255$	−12.0019	−0.751591
$256$	0	0
$257$	2.55236	0.159212	0.0796058	−	0.996826i	$-0.474634\pi$
$257$	2.55236	0.159212	0.0796058	+	0.996826i	$0.474634\pi$
$258$	0	0
$259$	15.5188	0.964293
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	21.8897	1.34978	0.674889	−	0.737920i	$-0.264192\pi$
$263$	21.8897	1.34978	0.674889	+	0.737920i	$0.264192\pi$
$264$	0	0
$265$	−18.8940	−1.16065
$266$	0	0
$267$	−5.49647	−0.336378
$268$	0	0
$269$	−24.3149	−1.48251	−0.741254	−	0.671225i	$-0.765769\pi$
$269$	−24.3149	−1.48251	−0.741254	+	0.671225i	$0.765769\pi$
$270$	0	0
$271$	3.46398	0.210422	0.105211	−	0.994450i	$-0.466448\pi$
$271$	3.46398	0.210422	0.105211	+	0.994450i	$0.466448\pi$
$272$	0	0
$273$	−4.07526	−0.246646
$274$	0	0
$275$	−2.72451	−0.164294
$276$	0	0
$277$	20.3852	1.22483	0.612413	−	0.790538i	$-0.290199\pi$
$277$	20.3852	1.22483	0.612413	+	0.790538i	$0.290199\pi$
$278$	0	0
$279$	1.34340	0.0804272
$280$	0	0
$281$	−16.4489	−0.981260	−0.490630	−	0.871368i	$-0.663233\pi$
$281$	−16.4489	−0.981260	−0.490630	+	0.871368i	$0.663233\pi$
$282$	0	0
$283$	−5.85058	−0.347781	−0.173891	−	0.984765i	$-0.555634\pi$
$283$	−5.85058	−0.347781	−0.173891	+	0.984765i	$0.555634\pi$
$284$	0	0
$285$	−4.01975	−0.238109
$286$	0	0
$287$	−23.4406	−1.38365
$288$	0	0
$289$	22.2741	1.31024
$290$	0	0
$291$	−8.26837	−0.484700
$292$	0	0
$293$	−32.9715	−1.92621	−0.963107	−	0.269120i	$-0.913267\pi$
$293$	−32.9715	−1.92621	−0.963107	+	0.269120i	$0.913267\pi$
$294$	0	0
$295$	−3.83645	−0.223367
$296$	0	0
$297$	2.04500	0.118663
$298$	0	0
$299$	−1.36883	−0.0791614
$300$	0	0
$301$	4.70849	0.271393
$302$	0	0
$303$	10.3712	0.595809
$304$	0	0
$305$	−18.7710	−1.07483
$306$	0	0
$307$	−4.75728	−0.271512	−0.135756	−	0.990742i	$-0.543346\pi$
$307$	−4.75728	−0.271512	−0.135756	+	0.990742i	$0.543346\pi$
$308$	0	0
$309$	0.930722	0.0529469
$310$	0	0
$311$	−10.7276	−0.608305	−0.304153	−	0.952623i	$-0.598373\pi$
$311$	−10.7276	−0.608305	−0.304153	+	0.952623i	$0.598373\pi$
$312$	0	0
$313$	23.2780	1.31575	0.657876	−	0.753127i	$-0.271455\pi$
$313$	23.2780	1.31575	0.657876	+	0.753127i	$0.271455\pi$
$314$	0	0
$315$	−5.70171	−0.321255
$316$	0	0
$317$	−26.6155	−1.49488	−0.747438	−	0.664332i	$-0.768716\pi$
$317$	−26.6155	−1.49488	−0.747438	+	0.664332i	$0.768716\pi$
$318$	0	0
$319$	−2.04500	−0.114498
$320$	0	0
$321$	−15.1650	−0.846427
$322$	0	0
$323$	13.1539	0.731901
$324$	0	0
$325$	−1.82365	−0.101158
$326$	0	0
$327$	−1.05050	−0.0580925
$328$	0	0
$329$	−11.6311	−0.641245
$330$	0	0
$331$	−5.55686	−0.305432	−0.152716	−	0.988270i	$-0.548802\pi$
$331$	−5.55686	−0.305432	−0.152716	+	0.988270i	$0.548802\pi$
$332$	0	0
$333$	−5.21257	−0.285647
$334$	0	0
$335$	4.69942	0.256757
$336$	0	0
$337$	−18.0060	−0.980851	−0.490426	−	0.871483i	$-0.663159\pi$
$337$	−18.0060	−0.980851	−0.490426	+	0.871483i	$0.663159\pi$
$338$	0	0
$339$	−10.6446	−0.578134
$340$	0	0
$341$	2.74726	0.148772
$342$	0	0
$343$	15.2919	0.825683
$344$	0	0
$345$	−1.91513	−0.103107
$346$	0	0
$347$	17.1964	0.923150	0.461575	−	0.887101i	$-0.347284\pi$
$347$	17.1964	0.923150	0.461575	+	0.887101i	$0.347284\pi$
$348$	0	0
$349$	−20.4351	−1.09387	−0.546934	−	0.837176i	$-0.684205\pi$
$349$	−20.4351	−1.09387	−0.546934	+	0.837176i	$0.684205\pi$
$350$	0	0
$351$	1.36883	0.0730626
$352$	0	0
$353$	−6.80347	−0.362112	−0.181056	−	0.983473i	$-0.557952\pi$
$353$	−6.80347	−0.362112	−0.181056	+	0.983473i	$0.557952\pi$
$354$	0	0
$355$	−16.5845	−0.880212
$356$	0	0
$357$	18.6578	0.987474
$358$	0	0
$359$	−16.6516	−0.878836	−0.439418	−	0.898283i	$-0.644815\pi$
$359$	−16.6516	−0.878836	−0.439418	+	0.898283i	$0.644815\pi$
$360$	0	0
$361$	−14.5944	−0.768128
$362$	0	0
$363$	−6.81796	−0.357850
$364$	0	0
$365$	27.3359	1.43083
$366$	0	0
$367$	−7.95943	−0.415479	−0.207740	−	0.978184i	$-0.566611\pi$
$367$	−7.95943	−0.415479	−0.207740	+	0.978184i	$0.566611\pi$
$368$	0	0
$369$	7.87339	0.409872
$370$	0	0
$371$	29.3719	1.52492
$372$	0	0
$373$	25.8613	1.33905	0.669523	−	0.742791i	$-0.266499\pi$
$373$	25.8613	1.33905	0.669523	+	0.742791i	$0.266499\pi$
$374$	0	0
$375$	−12.1271	−0.626242
$376$	0	0
$377$	−1.36883	−0.0704982
$378$	0	0
$379$	−28.2449	−1.45084	−0.725422	−	0.688304i	$-0.758355\pi$
$379$	−28.2449	−1.45084	−0.725422	+	0.688304i	$0.758355\pi$
$380$	0	0
$381$	9.98416	0.511504
$382$	0	0
$383$	−36.8460	−1.88274	−0.941372	−	0.337370i	$-0.890463\pi$
$383$	−36.8460	−1.88274	−0.941372	+	0.337370i	$0.890463\pi$
$384$	0	0
$385$	−11.6600	−0.594250
$386$	0	0
$387$	−1.58152	−0.0803933
$388$	0	0
$389$	28.9688	1.46878	0.734389	−	0.678729i	$-0.237469\pi$
$389$	28.9688	1.46878	0.734389	+	0.678729i	$0.237469\pi$
$390$	0	0
$391$	6.26690	0.316931
$392$	0	0
$393$	6.00561	0.302943
$394$	0	0
$395$	1.68321	0.0846916
$396$	0	0
$397$	9.14013	0.458730	0.229365	−	0.973340i	$-0.426335\pi$
$397$	9.14013	0.458730	0.229365	+	0.973340i	$0.426335\pi$
$398$	0	0
$399$	6.24895	0.312839
$400$	0	0
$401$	−15.4623	−0.772150	−0.386075	−	0.922467i	$-0.626169\pi$
$401$	−15.4623	−0.772150	−0.386075	+	0.922467i	$0.626169\pi$
$402$	0	0
$403$	1.83888	0.0916013
$404$	0	0
$405$	1.91513	0.0951636
$406$	0	0
$407$	−10.6597	−0.528384
$408$	0	0
$409$	−22.2649	−1.10093	−0.550463	−	0.834859i	$-0.685549\pi$
$409$	−22.2649	−1.10093	−0.550463	+	0.834859i	$0.685549\pi$
$410$	0	0
$411$	12.4401	0.613623
$412$	0	0
$413$	5.96400	0.293469
$414$	0	0
$415$	−25.5200	−1.25273
$416$	0	0
$417$	−22.9607	−1.12439
$418$	0	0
$419$	3.87683	0.189395	0.0946977	−	0.995506i	$-0.469812\pi$
$419$	3.87683	0.189395	0.0946977	+	0.995506i	$0.469812\pi$
$420$	0	0
$421$	−3.50411	−0.170780	−0.0853900	−	0.996348i	$-0.527214\pi$
$421$	−3.50411	−0.170780	−0.0853900	+	0.996348i	$0.527214\pi$
$422$	0	0
$423$	3.90675	0.189953
$424$	0	0
$425$	8.34923	0.404997
$426$	0	0
$427$	29.1807	1.41215
$428$	0	0
$429$	2.79926	0.135150
$430$	0	0
$431$	35.8991	1.72920	0.864599	−	0.502462i	$-0.167572\pi$
$431$	35.8991	1.72920	0.864599	+	0.502462i	$0.167572\pi$
$432$	0	0
$433$	−33.9905	−1.63348	−0.816740	−	0.577006i	$-0.804221\pi$
$433$	−33.9905	−1.63348	−0.816740	+	0.577006i	$0.804221\pi$
$434$	0	0
$435$	−1.91513	−0.0918235
$436$	0	0
$437$	2.09894	0.100406
$438$	0	0
$439$	6.28540	0.299986	0.149993	−	0.988687i	$-0.452075\pi$
$439$	6.28540	0.299986	0.149993	+	0.988687i	$0.452075\pi$
$440$	0	0
$441$	1.86366	0.0887457
$442$	0	0
$443$	36.9811	1.75702	0.878512	−	0.477721i	$-0.158537\pi$
$443$	36.9811	1.75702	0.878512	+	0.477721i	$0.158537\pi$
$444$	0	0
$445$	−10.5265	−0.499002
$446$	0	0
$447$	−14.9121	−0.705318
$448$	0	0
$449$	−1.78344	−0.0841658	−0.0420829	−	0.999114i	$-0.513399\pi$
$449$	−1.78344	−0.0841658	−0.0420829	+	0.999114i	$0.513399\pi$
$450$	0	0
$451$	16.1011	0.758172
$452$	0	0
$453$	−1.53402	−0.0720747
$454$	0	0
$455$	−7.80466	−0.365888
$456$	0	0
$457$	7.05415	0.329979	0.164990	−	0.986295i	$-0.447241\pi$
$457$	7.05415	0.329979	0.164990	+	0.986295i	$0.447241\pi$
$458$	0	0
$459$	−6.26690	−0.292514
$460$	0	0
$461$	−32.5834	−1.51756	−0.758781	−	0.651346i	$-0.774205\pi$
$461$	−32.5834	−1.51756	−0.758781	+	0.651346i	$0.774205\pi$
$462$	0	0
$463$	−4.39669	−0.204331	−0.102166	−	0.994767i	$-0.532577\pi$
$463$	−4.39669	−0.204331	−0.102166	+	0.994767i	$0.532577\pi$
$464$	0	0
$465$	2.57279	0.119310
$466$	0	0
$467$	17.9704	0.831571	0.415786	−	0.909463i	$-0.363507\pi$
$467$	17.9704	0.831571	0.415786	+	0.909463i	$0.363507\pi$
$468$	0	0
$469$	−7.30555	−0.337339
$470$	0	0
$471$	9.66353	0.445272
$472$	0	0
$473$	−3.23422	−0.148710
$474$	0	0
$475$	2.79637	0.128306
$476$	0	0
$477$	−9.86566	−0.451717
$478$	0	0
$479$	12.4889	0.570633	0.285316	−	0.958433i	$-0.407901\pi$
$479$	12.4889	0.570633	0.285316	+	0.958433i	$0.407901\pi$
$480$	0	0
$481$	−7.13512	−0.325333
$482$	0	0
$483$	2.97719	0.135467
$484$	0	0
$485$	−15.8350	−0.719030
$486$	0	0
$487$	−41.0747	−1.86127	−0.930637	−	0.365944i	$-0.880746\pi$
$487$	−41.0747	−1.86127	−0.930637	+	0.365944i	$0.880746\pi$
$488$	0	0
$489$	12.9447	0.585379
$490$	0	0
$491$	17.4708	0.788448	0.394224	−	0.919014i	$-0.371013\pi$
$491$	17.4708	0.788448	0.394224	+	0.919014i	$0.371013\pi$
$492$	0	0
$493$	6.26690	0.282247
$494$	0	0
$495$	3.91645	0.176031
$496$	0	0
$497$	25.7816	1.15646
$498$	0	0
$499$	16.5429	0.740563	0.370281	−	0.928920i	$-0.379261\pi$
$499$	16.5429	0.740563	0.370281	+	0.928920i	$0.379261\pi$
$500$	0	0
$501$	5.50077	0.245756
$502$	0	0
$503$	28.5878	1.27467	0.637334	−	0.770587i	$-0.280037\pi$
$503$	28.5878	1.27467	0.637334	+	0.770587i	$0.280037\pi$
$504$	0	0
$505$	19.8622	0.883855
$506$	0	0
$507$	−11.1263	−0.494137
$508$	0	0
$509$	−43.6303	−1.93388	−0.966940	−	0.255004i	$-0.917923\pi$
$509$	−43.6303	−1.93388	−0.966940	+	0.255004i	$0.917923\pi$
$510$	0	0
$511$	−42.4954	−1.87989
$512$	0	0
$513$	−2.09894	−0.0926706
$514$	0	0
$515$	1.78245	0.0785443
$516$	0	0
$517$	7.98932	0.351370
$518$	0	0
$519$	−9.96116	−0.437246
$520$	0	0
$521$	3.35339	0.146915	0.0734573	−	0.997298i	$-0.476597\pi$
$521$	3.35339	0.146915	0.0734573	+	0.997298i	$0.476597\pi$
$522$	0	0
$523$	23.9658	1.04795	0.523976	−	0.851733i	$-0.324448\pi$
$523$	23.9658	1.04795	0.523976	+	0.851733i	$0.324448\pi$
$524$	0	0
$525$	3.96643	0.173109
$526$	0	0
$527$	−8.41895	−0.366735
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−2.00323	−0.0869328
$532$	0	0
$533$	10.7773	0.466817
$534$	0	0
$535$	−29.0429	−1.25564
$536$	0	0
$537$	−13.0081	−0.561342
$538$	0	0
$539$	3.81119	0.164160
$540$	0	0
$541$	3.93074	0.168996	0.0844978	−	0.996424i	$-0.473071\pi$
$541$	3.93074	0.168996	0.0844978	+	0.996424i	$0.473071\pi$
$542$	0	0
$543$	9.81020	0.420996
$544$	0	0
$545$	−2.01184	−0.0861776
$546$	0	0
$547$	36.3389	1.55374	0.776869	−	0.629662i	$-0.216807\pi$
$547$	36.3389	1.55374	0.776869	+	0.629662i	$0.216807\pi$
$548$	0	0
$549$	−9.80143	−0.418315
$550$	0	0
$551$	2.09894	0.0894180
$552$	0	0
$553$	−2.61666	−0.111272
$554$	0	0
$555$	−9.98276	−0.423745
$556$	0	0
$557$	−23.2713	−0.986035	−0.493018	−	0.870019i	$-0.664106\pi$
$557$	−23.2713	−0.986035	−0.493018	+	0.870019i	$0.664106\pi$
$558$	0	0
$559$	−2.16483	−0.0915626
$560$	0	0
$561$	−12.8158	−0.541086
$562$	0	0
$563$	26.3970	1.11250	0.556251	−	0.831015i	$-0.312239\pi$
$563$	26.3970	1.11250	0.556251	+	0.831015i	$0.312239\pi$
$564$	0	0
$565$	−20.3857	−0.857635
$566$	0	0
$567$	−2.97719	−0.125030
$568$	0	0
$569$	24.0833	1.00962	0.504811	−	0.863230i	$-0.331562\pi$
$569$	24.0833	1.00962	0.504811	+	0.863230i	$0.331562\pi$
$570$	0	0
$571$	12.0239	0.503185	0.251592	−	0.967833i	$-0.419046\pi$
$571$	12.0239	0.503185	0.251592	+	0.967833i	$0.419046\pi$
$572$	0	0
$573$	5.75152	0.240273
$574$	0	0
$575$	1.33227	0.0555597
$576$	0	0
$577$	−41.9631	−1.74695	−0.873473	−	0.486873i	$-0.838137\pi$
$577$	−41.9631	−1.74695	−0.873473	+	0.486873i	$0.838137\pi$
$578$	0	0
$579$	11.3988	0.473717
$580$	0	0
$581$	39.6724	1.64589
$582$	0	0
$583$	−20.1753	−0.835576
$584$	0	0
$585$	2.62148	0.108385
$586$	0	0
$587$	−6.05295	−0.249832	−0.124916	−	0.992167i	$-0.539866\pi$
$587$	−6.05295	−0.249832	−0.124916	+	0.992167i	$0.539866\pi$
$588$	0	0
$589$	−2.81972	−0.116184
$590$	0	0
$591$	17.2757	0.710628
$592$	0	0
$593$	13.5815	0.557726	0.278863	−	0.960331i	$-0.410042\pi$
$593$	13.5815	0.557726	0.278863	+	0.960331i	$0.410042\pi$
$594$	0	0
$595$	35.7321	1.46487
$596$	0	0
$597$	−10.9978	−0.450110
$598$	0	0
$599$	34.9387	1.42756	0.713778	−	0.700372i	$-0.246982\pi$
$599$	34.9387	1.42756	0.713778	+	0.700372i	$0.246982\pi$
$600$	0	0
$601$	−4.53774	−0.185098	−0.0925491	−	0.995708i	$-0.529502\pi$
$601$	−4.53774	−0.185098	−0.0925491	+	0.995708i	$0.529502\pi$
$602$	0	0
$603$	2.45384	0.0999281
$604$	0	0
$605$	−13.0573	−0.530854
$606$	0	0
$607$	−37.1037	−1.50599	−0.752996	−	0.658025i	$-0.771392\pi$
$607$	−37.1037	−1.50599	−0.752996	+	0.658025i	$0.771392\pi$
$608$	0	0
$609$	2.97719	0.120642
$610$	0	0
$611$	5.34767	0.216344
$612$	0	0
$613$	16.2866	0.657811	0.328906	−	0.944363i	$-0.393320\pi$
$613$	16.2866	0.657811	0.328906	+	0.944363i	$0.393320\pi$
$614$	0	0
$615$	15.0786	0.608027
$616$	0	0
$617$	16.9355	0.681799	0.340899	−	0.940100i	$-0.389268\pi$
$617$	16.9355	0.681799	0.340899	+	0.940100i	$0.389268\pi$
$618$	0	0
$619$	−41.7152	−1.67668	−0.838338	−	0.545151i	$-0.816472\pi$
$619$	−41.7152	−1.67668	−0.838338	+	0.545151i	$0.816472\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	16.3640	0.655611
$624$	0	0
$625$	−16.5637	−0.662547
$626$	0	0
$627$	−4.29235	−0.171420
$628$	0	0
$629$	32.6667	1.30251
$630$	0	0
$631$	12.9485	0.515472	0.257736	−	0.966215i	$-0.417024\pi$
$631$	12.9485	0.515472	0.257736	+	0.966215i	$0.417024\pi$
$632$	0	0
$633$	−5.49631	−0.218459
$634$	0	0
$635$	19.1210	0.758793
$636$	0	0
$637$	2.55103	0.101076
$638$	0	0
$639$	−8.65971	−0.342573
$640$	0	0
$641$	−22.2148	−0.877431	−0.438716	−	0.898626i	$-0.644566\pi$
$641$	−22.2148	−0.877431	−0.438716	+	0.898626i	$0.644566\pi$
$642$	0	0
$643$	−9.67998	−0.381741	−0.190871	−	0.981615i	$-0.561131\pi$
$643$	−9.67998	−0.381741	−0.190871	+	0.981615i	$0.561131\pi$
$644$	0	0
$645$	−3.02882	−0.119260
$646$	0	0
$647$	36.9496	1.45264	0.726320	−	0.687357i	$-0.241229\pi$
$647$	36.9496	1.45264	0.726320	+	0.687357i	$0.241229\pi$
$648$	0	0
$649$	−4.09662	−0.160806
$650$	0	0
$651$	−3.99955	−0.156755
$652$	0	0
$653$	46.7622	1.82995	0.914973	−	0.403515i	$-0.132212\pi$
$653$	46.7622	1.82995	0.914973	+	0.403515i	$0.132212\pi$
$654$	0	0
$655$	11.5015	0.449402
$656$	0	0
$657$	14.2737	0.556869
$658$	0	0
$659$	19.2858	0.751267	0.375633	−	0.926768i	$-0.377425\pi$
$659$	19.2858	0.751267	0.375633	+	0.926768i	$0.377425\pi$
$660$	0	0
$661$	−29.0834	−1.13121	−0.565607	−	0.824675i	$-0.691358\pi$
$661$	−29.0834	−1.13121	−0.565607	+	0.824675i	$0.691358\pi$
$662$	0	0
$663$	−8.57831	−0.333154
$664$	0	0
$665$	11.9676	0.464082
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	20.8557	0.806327
$670$	0	0
$671$	−20.0440	−0.773789
$672$	0	0
$673$	30.0182	1.15712	0.578558	−	0.815641i	$-0.303616\pi$
$673$	30.0182	1.15712	0.578558	+	0.815641i	$0.303616\pi$
$674$	0	0
$675$	−1.33227	−0.0512792
$676$	0	0
$677$	45.6725	1.75534	0.877669	−	0.479267i	$-0.159097\pi$
$677$	45.6725	1.75534	0.877669	+	0.479267i	$0.159097\pi$
$678$	0	0
$679$	24.6165	0.944694
$680$	0	0
$681$	12.4917	0.478682
$682$	0	0
$683$	−16.7995	−0.642816	−0.321408	−	0.946941i	$-0.604156\pi$
$683$	−16.7995	−0.642816	−0.321408	+	0.946941i	$0.604156\pi$
$684$	0	0
$685$	23.8243	0.910282
$686$	0	0
$687$	−28.0229	−1.06914
$688$	0	0
$689$	−13.5044	−0.514476
$690$	0	0
$691$	40.6370	1.54591	0.772953	−	0.634463i	$-0.218779\pi$
$691$	40.6370	1.54591	0.772953	+	0.634463i	$0.218779\pi$
$692$	0	0
$693$	−6.08837	−0.231278
$694$	0	0
$695$	−43.9727	−1.66798
$696$	0	0
$697$	−49.3418	−1.86895
$698$	0	0
$699$	−12.2804	−0.464488
$700$	0	0
$701$	−24.8484	−0.938513	−0.469256	−	0.883062i	$-0.655478\pi$
$701$	−24.8484	−0.938513	−0.469256	+	0.883062i	$0.655478\pi$
$702$	0	0
$703$	10.9409	0.412644
$704$	0	0
$705$	7.48194	0.281786
$706$	0	0
$707$	−30.8770	−1.16125
$708$	0	0
$709$	39.6290	1.48830	0.744150	−	0.668012i	$-0.232855\pi$
$709$	39.6290	1.48830	0.744150	+	0.668012i	$0.232855\pi$
$710$	0	0
$711$	0.878902	0.0329614
$712$	0	0
$713$	−1.34340	−0.0503107
$714$	0	0
$715$	5.36095	0.200488
$716$	0	0
$717$	8.94247	0.333963
$718$	0	0
$719$	17.5417	0.654197	0.327098	−	0.944990i	$-0.393929\pi$
$719$	17.5417	0.654197	0.327098	+	0.944990i	$0.393929\pi$
$720$	0	0
$721$	−2.77094	−0.103195
$722$	0	0
$723$	−0.940096	−0.0349626
$724$	0	0
$725$	1.33227	0.0494794
$726$	0	0
$727$	17.6238	0.653631	0.326815	−	0.945088i	$-0.394024\pi$
$727$	17.6238	0.653631	0.326815	+	0.945088i	$0.394024\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	9.91125	0.366581
$732$	0	0
$733$	29.7967	1.10057	0.550284	−	0.834978i	$-0.314520\pi$
$733$	29.7967	1.10057	0.550284	+	0.834978i	$0.314520\pi$
$734$	0	0
$735$	3.56915	0.131650
$736$	0	0
$737$	5.01811	0.184845
$738$	0	0
$739$	−44.6690	−1.64318	−0.821588	−	0.570082i	$-0.806911\pi$
$739$	−44.6690	−1.64318	−0.821588	+	0.570082i	$0.806911\pi$
$740$	0	0
$741$	−2.87309	−0.105546
$742$	0	0
$743$	30.2257	1.10887	0.554436	−	0.832226i	$-0.312934\pi$
$743$	30.2257	1.10887	0.554436	+	0.832226i	$0.312934\pi$
$744$	0	0
$745$	−28.5586	−1.04631
$746$	0	0
$747$	−13.3255	−0.487553
$748$	0	0
$749$	45.1491	1.64971
$750$	0	0
$751$	−0.702893	−0.0256489	−0.0128245	−	0.999918i	$-0.504082\pi$
$751$	−0.702893	−0.0256489	−0.0128245	+	0.999918i	$0.504082\pi$
$752$	0	0
$753$	18.5996	0.677807
$754$	0	0
$755$	−2.93785	−0.106919
$756$	0	0
$757$	21.2356	0.771820	0.385910	−	0.922537i	$-0.373888\pi$
$757$	21.2356	0.771820	0.385910	+	0.922537i	$0.373888\pi$
$758$	0	0
$759$	−2.04500	−0.0742290
$760$	0	0
$761$	21.2752	0.771227	0.385613	−	0.922660i	$-0.373990\pi$
$761$	21.2752	0.771227	0.385613	+	0.922660i	$0.373990\pi$
$762$	0	0
$763$	3.12753	0.113224
$764$	0	0
$765$	−12.0019	−0.433931
$766$	0	0
$767$	−2.74208	−0.0990107
$768$	0	0
$769$	−0.507069	−0.0182854	−0.00914268	−	0.999958i	$-0.502910\pi$
$769$	−0.507069	−0.0182854	−0.00914268	+	0.999958i	$0.502910\pi$
$770$	0	0
$771$	2.55236	0.0919209
$772$	0	0
$773$	17.7540	0.638565	0.319283	−	0.947660i	$-0.396558\pi$
$773$	17.7540	0.638565	0.319283	+	0.947660i	$0.396558\pi$
$774$	0	0
$775$	−1.78978	−0.0642906
$776$	0	0
$777$	15.5188	0.556735
$778$	0	0
$779$	−16.5258	−0.592098
$780$	0	0
$781$	−17.7091	−0.633683
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	18.5069	0.660541
$786$	0	0
$787$	−27.1313	−0.967127	−0.483563	−	0.875309i	$-0.660658\pi$
$787$	−27.1313	−0.967127	−0.483563	+	0.875309i	$0.660658\pi$
$788$	0	0
$789$	21.8897	0.779294
$790$	0	0
$791$	31.6909	1.12680
$792$	0	0
$793$	−13.4165	−0.476433
$794$	0	0
$795$	−18.8940	−0.670102
$796$	0	0
$797$	0.564690	0.0200023	0.0100012	−	0.999950i	$-0.496816\pi$
$797$	0.564690	0.0200023	0.0100012	+	0.999950i	$0.496816\pi$
$798$	0	0
$799$	−24.4832	−0.866154
$800$	0	0
$801$	−5.49647	−0.194208
$802$	0	0
$803$	29.1897	1.03008
$804$	0	0
$805$	5.70171	0.200959
$806$	0	0
$807$	−24.3149	−0.855926
$808$	0	0
$809$	−13.6861	−0.481177	−0.240588	−	0.970627i	$-0.577340\pi$
$809$	−13.6861	−0.481177	−0.240588	+	0.970627i	$0.577340\pi$
$810$	0	0
$811$	−19.4358	−0.682483	−0.341241	−	0.939976i	$-0.610847\pi$
$811$	−19.4358	−0.682483	−0.341241	+	0.939976i	$0.610847\pi$
$812$	0	0
$813$	3.46398	0.121487
$814$	0	0
$815$	24.7908	0.868382
$816$	0	0
$817$	3.31952	0.116135
$818$	0	0
$819$	−4.07526	−0.142401
$820$	0	0
$821$	−18.3709	−0.641148	−0.320574	−	0.947223i	$-0.603876\pi$
$821$	−18.3709	−0.641148	−0.320574	+	0.947223i	$0.603876\pi$
$822$	0	0
$823$	−23.1554	−0.807147	−0.403574	−	0.914947i	$-0.632232\pi$
$823$	−23.1554	−0.807147	−0.403574	+	0.914947i	$0.632232\pi$
$824$	0	0
$825$	−2.72451	−0.0948551
$826$	0	0
$827$	−57.0284	−1.98307	−0.991536	−	0.129832i	$-0.958556\pi$
$827$	−57.0284	−1.98307	−0.991536	+	0.129832i	$0.958556\pi$
$828$	0	0
$829$	−56.2996	−1.95536	−0.977682	−	0.210089i	$-0.932625\pi$
$829$	−56.2996	−1.95536	−0.977682	+	0.210089i	$0.932625\pi$
$830$	0	0
$831$	20.3852	0.707154
$832$	0	0
$833$	−11.6794	−0.404667
$834$	0	0
$835$	10.5347	0.364568
$836$	0	0
$837$	1.34340	0.0464347
$838$	0	0
$839$	−55.4700	−1.91504	−0.957519	−	0.288371i	$-0.906886\pi$
$839$	−55.4700	−1.91504	−0.957519	+	0.288371i	$0.906886\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−16.4489	−0.566531
$844$	0	0
$845$	−21.3083	−0.733029
$846$	0	0
$847$	20.2984	0.697460
$848$	0	0
$849$	−5.85058	−0.200792
$850$	0	0
$851$	5.21257	0.178685
$852$	0	0
$853$	−22.7217	−0.777977	−0.388988	−	0.921243i	$-0.627175\pi$
$853$	−22.7217	−0.777977	−0.388988	+	0.921243i	$0.627175\pi$
$854$	0	0
$855$	−4.01975	−0.137473
$856$	0	0
$857$	−40.9883	−1.40014	−0.700068	−	0.714077i	$-0.746847\pi$
$857$	−40.9883	−1.40014	−0.700068	+	0.714077i	$0.746847\pi$
$858$	0	0
$859$	−50.0067	−1.70621	−0.853104	−	0.521741i	$-0.825283\pi$
$859$	−50.0067	−1.70621	−0.853104	+	0.521741i	$0.825283\pi$
$860$	0	0
$861$	−23.4406	−0.798853
$862$	0	0
$863$	20.8350	0.709232	0.354616	−	0.935012i	$-0.384612\pi$
$863$	20.8350	0.709232	0.354616	+	0.935012i	$0.384612\pi$
$864$	0	0
$865$	−19.0769	−0.648635
$866$	0	0
$867$	22.2741	0.756468
$868$	0	0
$869$	1.79736	0.0609712
$870$	0	0
$871$	3.35888	0.113811
$872$	0	0
$873$	−8.26837	−0.279842
$874$	0	0
$875$	36.1048	1.22056
$876$	0	0
$877$	−7.70605	−0.260215	−0.130107	−	0.991500i	$-0.541532\pi$
$877$	−7.70605	−0.260215	−0.130107	+	0.991500i	$0.541532\pi$
$878$	0	0
$879$	−32.9715	−1.11210
$880$	0	0
$881$	−48.1678	−1.62281	−0.811407	−	0.584482i	$-0.801298\pi$
$881$	−48.1678	−1.62281	−0.811407	+	0.584482i	$0.801298\pi$
$882$	0	0
$883$	11.7355	0.394932	0.197466	−	0.980310i	$-0.436729\pi$
$883$	11.7355	0.394932	0.197466	+	0.980310i	$0.436729\pi$
$884$	0	0
$885$	−3.83645	−0.128961
$886$	0	0
$887$	29.9496	1.00561	0.502804	−	0.864400i	$-0.332302\pi$
$887$	29.9496	1.00561	0.502804	+	0.864400i	$0.332302\pi$
$888$	0	0
$889$	−29.7248	−0.996936
$890$	0	0
$891$	2.04500	0.0685102
$892$	0	0
$893$	−8.20004	−0.274404
$894$	0	0
$895$	−24.9123	−0.832726
$896$	0	0
$897$	−1.36883	−0.0457038
$898$	0	0
$899$	−1.34340	−0.0448049
$900$	0	0
$901$	61.8271	2.05976
$902$	0	0
$903$	4.70849	0.156689
$904$	0	0
$905$	18.7878	0.624528
$906$	0	0
$907$	−54.0361	−1.79424	−0.897120	−	0.441786i	$-0.854345\pi$
$907$	−54.0361	−1.79424	−0.897120	+	0.441786i	$0.854345\pi$
$908$	0	0
$909$	10.3712	0.343990
$910$	0	0
$911$	−5.74195	−0.190239	−0.0951197	−	0.995466i	$-0.530323\pi$
$911$	−5.74195	−0.190239	−0.0951197	+	0.995466i	$0.530323\pi$
$912$	0	0
$913$	−27.2506	−0.901864
$914$	0	0
$915$	−18.7710	−0.620551
$916$	0	0
$917$	−17.8799	−0.590445
$918$	0	0
$919$	−37.0026	−1.22060	−0.610302	−	0.792169i	$-0.708952\pi$
$919$	−37.0026	−1.22060	−0.610302	+	0.792169i	$0.708952\pi$
$920$	0	0
$921$	−4.75728	−0.156758
$922$	0	0
$923$	−11.8536	−0.390168
$924$	0	0
$925$	6.94458	0.228336
$926$	0	0
$927$	0.930722	0.0305689
$928$	0	0
$929$	5.62066	0.184408	0.0922040	−	0.995740i	$-0.470609\pi$
$929$	5.62066	0.184408	0.0922040	+	0.995740i	$0.470609\pi$
$930$	0	0
$931$	−3.91172	−0.128201
$932$	0	0
$933$	−10.7276	−0.351205
$934$	0	0
$935$	−24.5440	−0.802675
$936$	0	0
$937$	−18.5337	−0.605471	−0.302736	−	0.953075i	$-0.597900\pi$
$937$	−18.5337	−0.605471	−0.302736	+	0.953075i	$0.597900\pi$
$938$	0	0
$939$	23.2780	0.759649
$940$	0	0
$941$	39.7857	1.29698	0.648489	−	0.761224i	$-0.275401\pi$
$941$	39.7857	1.29698	0.648489	+	0.761224i	$0.275401\pi$
$942$	0	0
$943$	−7.87339	−0.256393
$944$	0	0
$945$	−5.70171	−0.185477
$946$	0	0
$947$	48.2941	1.56935	0.784673	−	0.619910i	$-0.212831\pi$
$947$	48.2941	1.56935	0.784673	+	0.619910i	$0.212831\pi$
$948$	0	0
$949$	19.5382	0.634237
$950$	0	0
$951$	−26.6155	−0.863067
$952$	0	0
$953$	−1.78374	−0.0577809	−0.0288905	−	0.999583i	$-0.509197\pi$
$953$	−1.78374	−0.0577809	−0.0288905	+	0.999583i	$0.509197\pi$
$954$	0	0
$955$	11.0149	0.356434
$956$	0	0
$957$	−2.04500	−0.0661056
$958$	0	0
$959$	−37.0364	−1.19597
$960$	0	0
$961$	−29.1953	−0.941783
$962$	0	0
$963$	−15.1650	−0.488685
$964$	0	0
$965$	21.8302	0.702737
$966$	0	0
$967$	12.8378	0.412836	0.206418	−	0.978464i	$-0.433819\pi$
$967$	12.8378	0.412836	0.206418	+	0.978464i	$0.433819\pi$
$968$	0	0
$969$	13.1539	0.422563
$970$	0	0
$971$	42.7856	1.37306	0.686528	−	0.727103i	$-0.259134\pi$
$971$	42.7856	1.37306	0.686528	+	0.727103i	$0.259134\pi$
$972$	0	0
$973$	68.3583	2.19147
$974$	0	0
$975$	−1.82365	−0.0584037
$976$	0	0
$977$	48.2828	1.54470	0.772351	−	0.635195i	$-0.219080\pi$
$977$	48.2828	1.54470	0.772351	+	0.635195i	$0.219080\pi$
$978$	0	0
$979$	−11.2403	−0.359242
$980$	0	0
$981$	−1.05050	−0.0335397
$982$	0	0
$983$	−18.0755	−0.576520	−0.288260	−	0.957552i	$-0.593077\pi$
$983$	−18.0755	−0.576520	−0.288260	+	0.957552i	$0.593077\pi$
$984$	0	0
$985$	33.0852	1.05418
$986$	0	0
$987$	−11.6311	−0.370223
$988$	0	0
$989$	1.58152	0.0502895
$990$	0	0
$991$	−59.8805	−1.90217	−0.951084	−	0.308932i	$-0.900028\pi$
$991$	−59.8805	−1.90217	−0.951084	+	0.308932i	$0.900028\pi$
$992$	0	0
$993$	−5.55686	−0.176342
$994$	0	0
$995$	−21.0622	−0.667717
$996$	0	0
$997$	1.18469	0.0375195	0.0187598	−	0.999824i	$-0.494028\pi$
$997$	1.18469	0.0375195	0.0187598	+	0.999824i	$0.494028\pi$
$998$	0	0
$999$	−5.21257	−0.164919

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.f.1.8	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.f.1.8	✓	9	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.91513 q^{5} -2.97719 q^{7} +1.00000 q^{9} +2.04500 q^{11} +1.36883 q^{13} +1.91513 q^{15} -6.26690 q^{17} -2.09894 q^{19} -2.97719 q^{21} -1.00000 q^{23} -1.33227 q^{25} +1.00000 q^{27} -1.00000 q^{29} +1.34340 q^{31} +2.04500 q^{33} -5.70171 q^{35} -5.21257 q^{37} +1.36883 q^{39} +7.87339 q^{41} -1.58152 q^{43} +1.91513 q^{45} +3.90675 q^{47} +1.86366 q^{49} -6.26690 q^{51} -9.86566 q^{53} +3.91645 q^{55} -2.09894 q^{57} -2.00323 q^{59} -9.80143 q^{61} -2.97719 q^{63} +2.62148 q^{65} +2.45384 q^{67} -1.00000 q^{69} -8.65971 q^{71} +14.2737 q^{73} -1.33227 q^{75} -6.08837 q^{77} +0.878902 q^{79} +1.00000 q^{81} -13.3255 q^{83} -12.0019 q^{85} -1.00000 q^{87} -5.49647 q^{89} -4.07526 q^{91} +1.34340 q^{93} -4.01975 q^{95} -8.26837 q^{97} +2.04500 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9} - 8 q^{11} - q^{13} - q^{15} - 2 q^{17} - 11 q^{19} - 5 q^{21} - 9 q^{23} - 10 q^{25} + 9 q^{27} - 9 q^{29} - 8 q^{33} + q^{35} - 2 q^{37} - q^{39} - 3 q^{41}+ \cdots - 8 q^{99}+O(q^{100}) \) 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9 - 8 * q^11 - q^13 - q^15 - 2 * q^17 - 11 * q^19 - 5 * q^21 - 9 * q^23 - 10 * q^25 + 9 * q^27 - 9 * q^29 - 8 * q^33 + q^35 - 2 * q^37 - q^39 - 3 * q^41 - 19 * q^43 - q^45 - 3 * q^47 - 6 * q^49 - 2 * q^51 - 9 * q^53 - 7 * q^55 - 11 * q^57 - 2 * q^59 - 25 * q^61 - 5 * q^63 - 12 * q^65 - 20 * q^67 - 9 * q^69 + 9 * q^71 - 11 * q^73 - 10 * q^75 - 19 * q^77 + 4 * q^79 + 9 * q^81 - 9 * q^83 - 50 * q^85 - 9 * q^87 - 29 * q^89 - 38 * q^91 + 23 * q^95 - 43 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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