LMFDB - Embedded newform 6004.2.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 15x^{6} + 56x^{5} + 87x^{4} - 248x^{3} - 241x^{2} + 340x + 248 $ x^8 - 4x^7 - 15x^6 + 56x^5 + 87x^4 - 248x^3 - 241x^2 + 340*x + 248
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.76571$ of defining polynomial
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.76571 q^{5} +2.40269 q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.76571 q^{5} +2.40269 q^{7} -3.00000 q^{9} +4.64261 q^{11} +3.69939 q^{13} +1.00852 q^{17} +1.00000 q^{19} -2.66776 q^{23} -1.88227 q^{25} -0.522363 q^{29} +7.54466 q^{31} -4.24245 q^{35} -4.36796 q^{37} +9.51751 q^{41} -0.856909 q^{43} +5.29713 q^{45} +9.01404 q^{47} -1.22708 q^{49} -12.1682 q^{53} -8.19750 q^{55} +12.6765 q^{59} -2.58820 q^{61} -7.20807 q^{63} -6.53204 q^{65} +6.77423 q^{67} -2.48726 q^{71} -5.73865 q^{73} +11.1548 q^{77} +1.00000 q^{79} +9.00000 q^{81} +5.22392 q^{83} -1.78076 q^{85} -10.0219 q^{89} +8.88848 q^{91} -1.76571 q^{95} -4.55616 q^{97} -13.9278 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{5} + q^{7} - 24 q^{9}+O(q^{10}) $ 8 * q + 4 * q^5 + q^7 - 24 * q^9	$ 8 q + 4 q^{5} + q^{7} - 24 q^{9} - 7 q^{11} - 12 q^{13} - 7 q^{17} + 8 q^{19} + 10 q^{23} + 6 q^{25} + 5 q^{29} + 3 q^{31} - 11 q^{35} - 15 q^{37} - 5 q^{41} + 10 q^{43} - 12 q^{45} + 18 q^{47} + 31 q^{49} + 9 q^{53} - 17 q^{55} + 8 q^{59} + 13 q^{61} - 3 q^{63} + 4 q^{65} + 21 q^{67} + 44 q^{71} - 20 q^{73} + 15 q^{77} + 8 q^{79} + 72 q^{81} - 4 q^{83} - 9 q^{85} - 10 q^{89} + 48 q^{91} + 4 q^{95} - 22 q^{97} + 21 q^{99}+O(q^{100}) $ 8 * q + 4 * q^5 + q^7 - 24 * q^9 - 7 * q^11 - 12 * q^13 - 7 * q^17 + 8 * q^19 + 10 * q^23 + 6 * q^25 + 5 * q^29 + 3 * q^31 - 11 * q^35 - 15 * q^37 - 5 * q^41 + 10 * q^43 - 12 * q^45 + 18 * q^47 + 31 * q^49 + 9 * q^53 - 17 * q^55 + 8 * q^59 + 13 * q^61 - 3 * q^63 + 4 * q^65 + 21 * q^67 + 44 * q^71 - 20 * q^73 + 15 * q^77 + 8 * q^79 + 72 * q^81 - 4 * q^83 - 9 * q^85 - 10 * q^89 + 48 * q^91 + 4 * q^95 - 22 * q^97 + 21 * 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
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	−1.76571	−0.789649	−0.394824	−	0.918757i	$-0.629195\pi$
$5$	−1.76571	−0.789649	−0.394824	+	0.918757i	$0.629195\pi$
$6$	0	0
$7$	2.40269	0.908132	0.454066	−	0.890968i	$-0.349973\pi$
$7$	2.40269	0.908132	0.454066	+	0.890968i	$0.349973\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	4.64261	1.39980	0.699900	−	0.714241i	$-0.253228\pi$
$11$	4.64261	1.39980	0.699900	+	0.714241i	$0.253228\pi$
$12$	0	0
$13$	3.69939	1.02603	0.513013	−	0.858381i	$-0.328529\pi$
$13$	3.69939	1.02603	0.513013	+	0.858381i	$0.328529\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00852	0.244603	0.122302	−	0.992493i	$-0.460972\pi$
$17$	1.00852	0.244603	0.122302	+	0.992493i	$0.460972\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.66776	−0.556267	−0.278134	−	0.960542i	$-0.589716\pi$
$23$	−2.66776	−0.556267	−0.278134	+	0.960542i	$0.589716\pi$
$24$	0	0
$25$	−1.88227	−0.376455
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.522363	−0.0970004	−0.0485002	−	0.998823i	$-0.515444\pi$
$29$	−0.522363	−0.0970004	−0.0485002	+	0.998823i	$0.515444\pi$
$30$	0	0
$31$	7.54466	1.35506	0.677531	−	0.735494i	$-0.263050\pi$
$31$	7.54466	1.35506	0.677531	+	0.735494i	$0.263050\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.24245	−0.717105
$36$	0	0
$37$	−4.36796	−0.718088	−0.359044	−	0.933321i	$-0.616897\pi$
$37$	−4.36796	−0.718088	−0.359044	+	0.933321i	$0.616897\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.51751	1.48639	0.743193	−	0.669077i	$-0.233311\pi$
$41$	9.51751	1.48639	0.743193	+	0.669077i	$0.233311\pi$
$42$	0	0
$43$	−0.856909	−0.130677	−0.0653387	−	0.997863i	$-0.520813\pi$
$43$	−0.856909	−0.130677	−0.0653387	+	0.997863i	$0.520813\pi$
$44$	0	0
$45$	5.29713	0.789649
$46$	0	0
$47$	9.01404	1.31483	0.657416	−	0.753528i	$-0.271649\pi$
$47$	9.01404	1.31483	0.657416	+	0.753528i	$0.271649\pi$
$48$	0	0
$49$	−1.22708	−0.175297
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−12.1682	−1.67144	−0.835718	−	0.549159i	$-0.814948\pi$
$53$	−12.1682	−1.67144	−0.835718	+	0.549159i	$0.814948\pi$
$54$	0	0
$55$	−8.19750	−1.10535
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.6765	1.65035	0.825173	−	0.564881i	$-0.191078\pi$
$59$	12.6765	1.65035	0.825173	+	0.564881i	$0.191078\pi$
$60$	0	0
$61$	−2.58820	−0.331384	−0.165692	−	0.986178i	$-0.552986\pi$
$61$	−2.58820	−0.331384	−0.165692	+	0.986178i	$0.552986\pi$
$62$	0	0
$63$	−7.20807	−0.908132
$64$	0	0
$65$	−6.53204	−0.810200
$66$	0	0
$67$	6.77423	0.827604	0.413802	−	0.910367i	$-0.364201\pi$
$67$	6.77423	0.827604	0.413802	+	0.910367i	$0.364201\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.48726	−0.295183	−0.147592	−	0.989048i	$-0.547152\pi$
$71$	−2.48726	−0.295183	−0.147592	+	0.989048i	$0.547152\pi$
$72$	0	0
$73$	−5.73865	−0.671659	−0.335829	−	0.941923i	$-0.609017\pi$
$73$	−5.73865	−0.671659	−0.335829	+	0.941923i	$0.609017\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	11.1548	1.27120
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	5.22392	0.573400	0.286700	−	0.958020i	$-0.407442\pi$
$83$	5.22392	0.573400	0.286700	+	0.958020i	$0.407442\pi$
$84$	0	0
$85$	−1.78076	−0.193151
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.0219	−1.06232	−0.531159	−	0.847272i	$-0.678243\pi$
$89$	−10.0219	−1.06232	−0.531159	+	0.847272i	$0.678243\pi$
$90$	0	0
$91$	8.88848	0.931766
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.76571	−0.181158
$96$	0	0
$97$	−4.55616	−0.462608	−0.231304	−	0.972882i	$-0.574299\pi$
$97$	−4.55616	−0.462608	−0.231304	+	0.972882i	$0.574299\pi$
$98$	0	0
$99$	−13.9278	−1.39980
$100$	0	0
$101$	14.0015	1.39320	0.696598	−	0.717461i	$-0.254696\pi$
$101$	14.0015	1.39320	0.696598	+	0.717461i	$0.254696\pi$
$102$	0	0
$103$	4.82282	0.475206	0.237603	−	0.971362i	$-0.423638\pi$
$103$	4.82282	0.475206	0.237603	+	0.971362i	$0.423638\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.03720	−0.680312	−0.340156	−	0.940369i	$-0.610480\pi$
$107$	−7.03720	−0.680312	−0.340156	+	0.940369i	$0.610480\pi$
$108$	0	0
$109$	−3.29838	−0.315928	−0.157964	−	0.987445i	$-0.550493\pi$
$109$	−3.29838	−0.315928	−0.157964	+	0.987445i	$0.550493\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.4066	−1.26119	−0.630595	−	0.776112i	$-0.717189\pi$
$113$	−13.4066	−1.26119	−0.630595	+	0.776112i	$0.717189\pi$
$114$	0	0
$115$	4.71049	0.439256
$116$	0	0
$117$	−11.0982	−1.02603
$118$	0	0
$119$	2.42317	0.222132
$120$	0	0
$121$	10.5538	0.959439
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.1521	1.08692
$126$	0	0
$127$	−4.60770	−0.408867	−0.204434	−	0.978880i	$-0.565535\pi$
$127$	−4.60770	−0.408867	−0.204434	+	0.978880i	$0.565535\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.88355	0.251937	0.125968	−	0.992034i	$-0.459796\pi$
$131$	2.88355	0.251937	0.125968	+	0.992034i	$0.459796\pi$
$132$	0	0
$133$	2.40269	0.208340
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.1456	−1.29397	−0.646987	−	0.762501i	$-0.723971\pi$
$137$	−15.1456	−1.29397	−0.646987	+	0.762501i	$0.723971\pi$
$138$	0	0
$139$	3.53468	0.299807	0.149904	−	0.988701i	$-0.452104\pi$
$139$	3.53468	0.299807	0.149904	+	0.988701i	$0.452104\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	17.1748	1.43623
$144$	0	0
$145$	0.922341	0.0765962
$146$	0	0
$147$	0	0
$148$	0	0
$149$	19.9674	1.63579	0.817897	−	0.575365i	$-0.195140\pi$
$149$	19.9674	1.63579	0.817897	+	0.575365i	$0.195140\pi$
$150$	0	0
$151$	−2.87784	−0.234196	−0.117098	−	0.993120i	$-0.537359\pi$
$151$	−2.87784	−0.234196	−0.117098	+	0.993120i	$0.537359\pi$
$152$	0	0
$153$	−3.02557	−0.244603
$154$	0	0
$155$	−13.3217	−1.07002
$156$	0	0
$157$	−17.4176	−1.39007	−0.695037	−	0.718974i	$-0.744612\pi$
$157$	−17.4176	−1.39007	−0.695037	+	0.718974i	$0.744612\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−6.40981	−0.505164
$162$	0	0
$163$	−6.34773	−0.497193	−0.248596	−	0.968607i	$-0.579969\pi$
$163$	−6.34773	−0.497193	−0.248596	+	0.968607i	$0.579969\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.3947	1.73296	0.866478	−	0.499215i	$-0.166378\pi$
$167$	22.3947	1.73296	0.866478	+	0.499215i	$0.166378\pi$
$168$	0	0
$169$	0.685473	0.0527287
$170$	0	0
$171$	−3.00000	−0.229416
$172$	0	0
$173$	−13.4210	−1.02038	−0.510190	−	0.860062i	$-0.670425\pi$
$173$	−13.4210	−1.02038	−0.510190	+	0.860062i	$0.670425\pi$
$174$	0	0
$175$	−4.52252	−0.341870
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.42358	0.181147	0.0905736	−	0.995890i	$-0.471130\pi$
$179$	2.42358	0.181147	0.0905736	+	0.995890i	$0.471130\pi$
$180$	0	0
$181$	0.605206	0.0449846	0.0224923	−	0.999747i	$-0.492840\pi$
$181$	0.605206	0.0449846	0.0224923	+	0.999747i	$0.492840\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.71254	0.567037
$186$	0	0
$187$	4.68219	0.342395
$188$	0	0
$189$	0	0
$190$	0	0
$191$	26.6573	1.92886	0.964428	−	0.264345i	$-0.0851558\pi$
$191$	26.6573	1.92886	0.964428	+	0.264345i	$0.0851558\pi$
$192$	0	0
$193$	6.15357	0.442943	0.221472	−	0.975167i	$-0.428914\pi$
$193$	6.15357	0.442943	0.221472	+	0.975167i	$0.428914\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	25.0310	1.78339	0.891693	−	0.452641i	$-0.149518\pi$
$197$	25.0310	1.78339	0.891693	+	0.452641i	$0.149518\pi$
$198$	0	0
$199$	5.76232	0.408480	0.204240	−	0.978921i	$-0.434528\pi$
$199$	5.76232	0.408480	0.204240	+	0.978921i	$0.434528\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.25508	−0.0880891
$204$	0	0
$205$	−16.8051	−1.17372
$206$	0	0
$207$	8.00329	0.556267
$208$	0	0
$209$	4.64261	0.321136
$210$	0	0
$211$	20.2654	1.39513	0.697563	−	0.716523i	$-0.254268\pi$
$211$	20.2654	1.39513	0.697563	+	0.716523i	$0.254268\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.51305	0.103189
$216$	0	0
$217$	18.1275	1.23057
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.73093	0.250969
$222$	0	0
$223$	19.8018	1.32603	0.663015	−	0.748606i	$-0.269277\pi$
$223$	19.8018	1.32603	0.663015	+	0.748606i	$0.269277\pi$
$224$	0	0
$225$	5.64682	0.376455
$226$	0	0
$227$	18.6869	1.24029	0.620147	−	0.784486i	$-0.287073\pi$
$227$	18.6869	1.24029	0.620147	+	0.784486i	$0.287073\pi$
$228$	0	0
$229$	7.50642	0.496038	0.248019	−	0.968755i	$-0.420220\pi$
$229$	7.50642	0.496038	0.248019	+	0.968755i	$0.420220\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.93127	0.257546	0.128773	−	0.991674i	$-0.458896\pi$
$233$	3.93127	0.257546	0.128773	+	0.991674i	$0.458896\pi$
$234$	0	0
$235$	−15.9162	−1.03826
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.20977	0.401676	0.200838	−	0.979624i	$-0.435633\pi$
$239$	6.20977	0.401676	0.200838	+	0.979624i	$0.435633\pi$
$240$	0	0
$241$	3.97170	0.255840	0.127920	−	0.991785i	$-0.459170\pi$
$241$	3.97170	0.255840	0.127920	+	0.991785i	$0.459170\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.16667	0.138423
$246$	0	0
$247$	3.69939	0.235386
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.0936	1.52078	0.760389	−	0.649468i	$-0.225008\pi$
$251$	24.0936	1.52078	0.760389	+	0.649468i	$0.225008\pi$
$252$	0	0
$253$	−12.3854	−0.778662
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.9229	1.18038	0.590190	−	0.807265i	$-0.299053\pi$
$257$	18.9229	1.18038	0.590190	+	0.807265i	$0.299053\pi$
$258$	0	0
$259$	−10.4948	−0.652118
$260$	0	0
$261$	1.56709	0.0970004
$262$	0	0
$263$	7.10061	0.437842	0.218921	−	0.975743i	$-0.429746\pi$
$263$	7.10061	0.437842	0.218921	+	0.975743i	$0.429746\pi$
$264$	0	0
$265$	21.4856	1.31985
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−19.1418	−1.16710	−0.583550	−	0.812078i	$-0.698337\pi$
$269$	−19.1418	−1.16710	−0.583550	+	0.812078i	$0.698337\pi$
$270$	0	0
$271$	−13.4138	−0.814829	−0.407415	−	0.913243i	$-0.633570\pi$
$271$	−13.4138	−0.814829	−0.407415	+	0.913243i	$0.633570\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.73866	−0.526961
$276$	0	0
$277$	13.6792	0.821901	0.410950	−	0.911658i	$-0.365197\pi$
$277$	13.6792	0.821901	0.410950	+	0.911658i	$0.365197\pi$
$278$	0	0
$279$	−22.6340	−1.35506
$280$	0	0
$281$	−6.24356	−0.372459	−0.186230	−	0.982506i	$-0.559627\pi$
$281$	−6.24356	−0.372459	−0.186230	+	0.982506i	$0.559627\pi$
$282$	0	0
$283$	6.82226	0.405542	0.202771	−	0.979226i	$-0.435005\pi$
$283$	6.82226	0.405542	0.202771	+	0.979226i	$0.435005\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	22.8676	1.34983
$288$	0	0
$289$	−15.9829	−0.940169
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.96757	−0.114947	−0.0574735	−	0.998347i	$-0.518304\pi$
$293$	−1.96757	−0.114947	−0.0574735	+	0.998347i	$0.518304\pi$
$294$	0	0
$295$	−22.3831	−1.30319
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−9.86909	−0.570744
$300$	0	0
$301$	−2.05889	−0.118672
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.57000	0.261677
$306$	0	0
$307$	16.9276	0.966107	0.483053	−	0.875591i	$-0.339528\pi$
$307$	16.9276	0.966107	0.483053	+	0.875591i	$0.339528\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	23.9384	1.35742	0.678712	−	0.734404i	$-0.262538\pi$
$311$	23.9384	1.35742	0.678712	+	0.734404i	$0.262538\pi$
$312$	0	0
$313$	28.5184	1.61195	0.805977	−	0.591947i	$-0.201640\pi$
$313$	28.5184	1.61195	0.805977	+	0.591947i	$0.201640\pi$
$314$	0	0
$315$	12.7274	0.717105
$316$	0	0
$317$	−35.4326	−1.99009	−0.995045	−	0.0994238i	$-0.968300\pi$
$317$	−35.4326	−1.99009	−0.995045	+	0.0994238i	$0.968300\pi$
$318$	0	0
$319$	−2.42513	−0.135781
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.00852	0.0561158
$324$	0	0
$325$	−6.96326	−0.386252
$326$	0	0
$327$	0	0
$328$	0	0
$329$	21.6579	1.19404
$330$	0	0
$331$	−19.7650	−1.08638	−0.543192	−	0.839609i	$-0.682784\pi$
$331$	−19.7650	−1.08638	−0.543192	+	0.839609i	$0.682784\pi$
$332$	0	0
$333$	13.1039	0.718088
$334$	0	0
$335$	−11.9613	−0.653517
$336$	0	0
$337$	−0.499954	−0.0272342	−0.0136171	−	0.999907i	$-0.504335\pi$
$337$	−0.499954	−0.0272342	−0.0136171	+	0.999907i	$0.504335\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	35.0269	1.89681
$342$	0	0
$343$	−19.7671	−1.06732
$344$	0	0
$345$	0	0
$346$	0	0
$347$	34.7187	1.86380	0.931899	−	0.362717i	$-0.118151\pi$
$347$	34.7187	1.86380	0.931899	+	0.362717i	$0.118151\pi$
$348$	0	0
$349$	8.02757	0.429706	0.214853	−	0.976646i	$-0.431073\pi$
$349$	8.02757	0.429706	0.214853	+	0.976646i	$0.431073\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	35.2459	1.87595	0.937975	−	0.346703i	$-0.112699\pi$
$353$	35.2459	1.87595	0.937975	+	0.346703i	$0.112699\pi$
$354$	0	0
$355$	4.39177	0.233091
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−29.1604	−1.53902	−0.769512	−	0.638632i	$-0.779501\pi$
$359$	−29.1604	−1.53902	−0.769512	+	0.638632i	$0.779501\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.1328	0.530374
$366$	0	0
$367$	−16.8177	−0.877878	−0.438939	−	0.898517i	$-0.644646\pi$
$367$	−16.8177	−0.877878	−0.438939	+	0.898517i	$0.644646\pi$
$368$	0	0
$369$	−28.5525	−1.48639
$370$	0	0
$371$	−29.2365	−1.51788
$372$	0	0
$373$	−28.2832	−1.46445	−0.732223	−	0.681065i	$-0.761517\pi$
$373$	−28.2832	−1.46445	−0.732223	+	0.681065i	$0.761517\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.93242	−0.0995249
$378$	0	0
$379$	31.9641	1.64188	0.820942	−	0.571011i	$-0.193449\pi$
$379$	31.9641	1.64188	0.820942	+	0.571011i	$0.193449\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	20.3392	1.03928	0.519642	−	0.854384i	$-0.326065\pi$
$383$	20.3392	1.03928	0.519642	+	0.854384i	$0.326065\pi$
$384$	0	0
$385$	−19.6960	−1.00380
$386$	0	0
$387$	2.57073	0.130677
$388$	0	0
$389$	20.4904	1.03890	0.519451	−	0.854500i	$-0.326136\pi$
$389$	20.4904	1.03890	0.519451	+	0.854500i	$0.326136\pi$
$390$	0	0
$391$	−2.69051	−0.136065
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.76571	−0.0888424
$396$	0	0
$397$	8.73469	0.438381	0.219191	−	0.975682i	$-0.429658\pi$
$397$	8.73469	0.438381	0.219191	+	0.975682i	$0.429658\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.66736	0.332952	0.166476	−	0.986045i	$-0.446761\pi$
$401$	6.66736	0.332952	0.166476	+	0.986045i	$0.446761\pi$
$402$	0	0
$403$	27.9106	1.39033
$404$	0	0
$405$	−15.8914	−0.789649
$406$	0	0
$407$	−20.2787	−1.00518
$408$	0	0
$409$	29.9584	1.48135	0.740674	−	0.671864i	$-0.234506\pi$
$409$	29.9584	1.48135	0.740674	+	0.671864i	$0.234506\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	30.4578	1.49873
$414$	0	0
$415$	−9.22392	−0.452784
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.94327	0.143788	0.0718941	−	0.997412i	$-0.477096\pi$
$419$	2.94327	0.143788	0.0718941	+	0.997412i	$0.477096\pi$
$420$	0	0
$421$	10.6518	0.519138	0.259569	−	0.965725i	$-0.416420\pi$
$421$	10.6518	0.519138	0.259569	+	0.965725i	$0.416420\pi$
$422$	0	0
$423$	−27.0421	−1.31483
$424$	0	0
$425$	−1.89832	−0.0920820
$426$	0	0
$427$	−6.21863	−0.300941
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−0.628500	−0.0302738	−0.0151369	−	0.999885i	$-0.504818\pi$
$431$	−0.628500	−0.0302738	−0.0151369	+	0.999885i	$0.504818\pi$
$432$	0	0
$433$	−39.9076	−1.91784	−0.958919	−	0.283680i	$-0.908445\pi$
$433$	−39.9076	−1.91784	−0.958919	+	0.283680i	$0.908445\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.66776	−0.127616
$438$	0	0
$439$	26.7748	1.27789	0.638947	−	0.769251i	$-0.279370\pi$
$439$	26.7748	1.27789	0.638947	+	0.769251i	$0.279370\pi$
$440$	0	0
$441$	3.68124	0.175297
$442$	0	0
$443$	−36.5112	−1.73470	−0.867350	−	0.497698i	$-0.834179\pi$
$443$	−36.5112	−1.73470	−0.867350	+	0.497698i	$0.834179\pi$
$444$	0	0
$445$	17.6957	0.838858
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5.42675	−0.256104	−0.128052	−	0.991767i	$-0.540873\pi$
$449$	−5.42675	−0.256104	−0.128052	+	0.991767i	$0.540873\pi$
$450$	0	0
$451$	44.1861	2.08064
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−15.6945	−0.735768
$456$	0	0
$457$	11.1565	0.521879	0.260940	−	0.965355i	$-0.415968\pi$
$457$	11.1565	0.521879	0.260940	+	0.965355i	$0.415968\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−21.1546	−0.985269	−0.492635	−	0.870236i	$-0.663966\pi$
$461$	−21.1546	−0.985269	−0.492635	+	0.870236i	$0.663966\pi$
$462$	0	0
$463$	−26.5609	−1.23439	−0.617196	−	0.786810i	$-0.711731\pi$
$463$	−26.5609	−1.23439	−0.617196	+	0.786810i	$0.711731\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−31.9021	−1.47625	−0.738126	−	0.674663i	$-0.764289\pi$
$467$	−31.9021	−1.47625	−0.738126	+	0.674663i	$0.764289\pi$
$468$	0	0
$469$	16.2764	0.751574
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.97829	−0.182922
$474$	0	0
$475$	−1.88227	−0.0863646
$476$	0	0
$477$	36.5047	1.67144
$478$	0	0
$479$	−8.39959	−0.383787	−0.191893	−	0.981416i	$-0.561463\pi$
$479$	−8.39959	−0.383787	−0.191893	+	0.981416i	$0.561463\pi$
$480$	0	0
$481$	−16.1588	−0.736777
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.04484	0.365298
$486$	0	0
$487$	15.4283	0.699122	0.349561	−	0.936914i	$-0.386331\pi$
$487$	15.4283	0.699122	0.349561	+	0.936914i	$0.386331\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.02221	−0.181520	−0.0907600	−	0.995873i	$-0.528930\pi$
$491$	−4.02221	−0.181520	−0.0907600	+	0.995873i	$0.528930\pi$
$492$	0	0
$493$	−0.526816	−0.0237266
$494$	0	0
$495$	24.5925	1.10535
$496$	0	0
$497$	−5.97611	−0.268065
$498$	0	0
$499$	−8.28394	−0.370840	−0.185420	−	0.982659i	$-0.559365\pi$
$499$	−8.28394	−0.370840	−0.185420	+	0.982659i	$0.559365\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−17.4311	−0.777214	−0.388607	−	0.921404i	$-0.627044\pi$
$503$	−17.4311	−0.777214	−0.388607	+	0.921404i	$0.627044\pi$
$504$	0	0
$505$	−24.7225	−1.10014
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−33.3089	−1.47639	−0.738196	−	0.674586i	$-0.764322\pi$
$509$	−33.3089	−1.47639	−0.738196	+	0.674586i	$0.764322\pi$
$510$	0	0
$511$	−13.7882	−0.609954
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.51569	−0.375246
$516$	0	0
$517$	41.8487	1.84050
$518$	0	0
$519$	0	0
$520$	0	0
$521$	26.7064	1.17003	0.585015	−	0.811022i	$-0.301089\pi$
$521$	26.7064	1.17003	0.585015	+	0.811022i	$0.301089\pi$
$522$	0	0
$523$	−28.0788	−1.22780	−0.613900	−	0.789384i	$-0.710400\pi$
$523$	−28.0788	−1.22780	−0.613900	+	0.789384i	$0.710400\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.60898	0.331452
$528$	0	0
$529$	−15.8830	−0.690567
$530$	0	0
$531$	−38.0296	−1.65035
$532$	0	0
$533$	35.2090	1.52507
$534$	0	0
$535$	12.4256	0.537207
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.69685	−0.245381
$540$	0	0
$541$	−30.1900	−1.29797	−0.648985	−	0.760801i	$-0.724806\pi$
$541$	−30.1900	−1.29797	−0.648985	+	0.760801i	$0.724806\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.82398	0.249472
$546$	0	0
$547$	5.29157	0.226251	0.113126	−	0.993581i	$-0.463914\pi$
$547$	5.29157	0.226251	0.113126	+	0.993581i	$0.463914\pi$
$548$	0	0
$549$	7.76459	0.331384
$550$	0	0
$551$	−0.522363	−0.0222534
$552$	0	0
$553$	2.40269	0.102173
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−44.8096	−1.89864	−0.949322	−	0.314305i	$-0.898228\pi$
$557$	−44.8096	−1.89864	−0.949322	+	0.314305i	$0.898228\pi$
$558$	0	0
$559$	−3.17004	−0.134078
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.3002	1.27700	0.638502	−	0.769620i	$-0.279555\pi$
$563$	30.3002	1.27700	0.638502	+	0.769620i	$0.279555\pi$
$564$	0	0
$565$	23.6722	0.995897
$566$	0	0
$567$	21.6242	0.908132
$568$	0	0
$569$	34.9579	1.46551	0.732756	−	0.680492i	$-0.238234\pi$
$569$	34.9579	1.46551	0.732756	+	0.680492i	$0.238234\pi$
$570$	0	0
$571$	3.85564	0.161353	0.0806767	−	0.996740i	$-0.474292\pi$
$571$	3.85564	0.161353	0.0806767	+	0.996740i	$0.474292\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.02146	0.209409
$576$	0	0
$577$	−10.4508	−0.435073	−0.217536	−	0.976052i	$-0.569802\pi$
$577$	−10.4508	−0.435073	−0.217536	+	0.976052i	$0.569802\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12.5515	0.520722
$582$	0	0
$583$	−56.4924	−2.33968
$584$	0	0
$585$	19.5961	0.810200
$586$	0	0
$587$	−14.6951	−0.606533	−0.303266	−	0.952906i	$-0.598077\pi$
$587$	−14.6951	−0.606533	−0.303266	+	0.952906i	$0.598077\pi$
$588$	0	0
$589$	7.54466	0.310872
$590$	0	0
$591$	0	0
$592$	0	0
$593$	42.3595	1.73950	0.869749	−	0.493495i	$-0.164281\pi$
$593$	42.3595	1.73950	0.869749	+	0.493495i	$0.164281\pi$
$594$	0	0
$595$	−4.27862	−0.175406
$596$	0	0
$597$	0	0
$598$	0	0
$599$	16.4475	0.672025	0.336012	−	0.941858i	$-0.390922\pi$
$599$	16.4475	0.672025	0.336012	+	0.941858i	$0.390922\pi$
$600$	0	0
$601$	−37.1503	−1.51539	−0.757697	−	0.652607i	$-0.773675\pi$
$601$	−37.1503	−1.51539	−0.757697	+	0.652607i	$0.773675\pi$
$602$	0	0
$603$	−20.3227	−0.827604
$604$	0	0
$605$	−18.6350	−0.757620
$606$	0	0
$607$	36.4957	1.48132	0.740658	−	0.671882i	$-0.234514\pi$
$607$	36.4957	1.48132	0.740658	+	0.671882i	$0.234514\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	33.3464	1.34905
$612$	0	0
$613$	−11.2547	−0.454572	−0.227286	−	0.973828i	$-0.572985\pi$
$613$	−11.2547	−0.454572	−0.227286	+	0.973828i	$0.572985\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.33341	0.174457	0.0872284	−	0.996188i	$-0.472199\pi$
$617$	4.33341	0.174457	0.0872284	+	0.996188i	$0.472199\pi$
$618$	0	0
$619$	47.8805	1.92448	0.962240	−	0.272201i	$-0.0877516\pi$
$619$	47.8805	1.92448	0.962240	+	0.272201i	$0.0877516\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−24.0795	−0.964724
$624$	0	0
$625$	−12.0457	−0.481827
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.40519	−0.175647
$630$	0	0
$631$	−46.8786	−1.86621	−0.933103	−	0.359608i	$-0.882910\pi$
$631$	−46.8786	−1.86621	−0.933103	+	0.359608i	$0.882910\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	8.13586	0.322862
$636$	0	0
$637$	−4.53944	−0.179859
$638$	0	0
$639$	7.46177	0.295183
$640$	0	0
$641$	−14.6688	−0.579382	−0.289691	−	0.957120i	$-0.593552\pi$
$641$	−14.6688	−0.579382	−0.289691	+	0.957120i	$0.593552\pi$
$642$	0	0
$643$	−22.3936	−0.883118	−0.441559	−	0.897232i	$-0.645574\pi$
$643$	−22.3936	−0.883118	−0.441559	+	0.897232i	$0.645574\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−23.4630	−0.922425	−0.461213	−	0.887290i	$-0.652585\pi$
$647$	−23.4630	−0.922425	−0.461213	+	0.887290i	$0.652585\pi$
$648$	0	0
$649$	58.8522	2.31015
$650$	0	0
$651$	0	0
$652$	0	0
$653$	38.9099	1.52266	0.761331	−	0.648364i	$-0.224546\pi$
$653$	38.9099	1.52266	0.761331	+	0.648364i	$0.224546\pi$
$654$	0	0
$655$	−5.09150	−0.198942
$656$	0	0
$657$	17.2160	0.671659
$658$	0	0
$659$	9.66770	0.376600	0.188300	−	0.982112i	$-0.439702\pi$
$659$	9.66770	0.376600	0.188300	+	0.982112i	$0.439702\pi$
$660$	0	0
$661$	27.8753	1.08422	0.542112	−	0.840306i	$-0.317625\pi$
$661$	27.8753	1.08422	0.542112	+	0.840306i	$0.317625\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.24245	−0.164515
$666$	0	0
$667$	1.39354	0.0539581
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−12.0160	−0.463872
$672$	0	0
$673$	−20.9874	−0.809004	−0.404502	−	0.914537i	$-0.632555\pi$
$673$	−20.9874	−0.809004	−0.404502	+	0.914537i	$0.632555\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	34.2154	1.31501	0.657503	−	0.753452i	$-0.271613\pi$
$677$	34.2154	1.31501	0.657503	+	0.753452i	$0.271613\pi$
$678$	0	0
$679$	−10.9470	−0.420109
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0.276336	0.0105737	0.00528686	−	0.999986i	$-0.498317\pi$
$683$	0.276336	0.0105737	0.00528686	+	0.999986i	$0.498317\pi$
$684$	0	0
$685$	26.7427	1.02179
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−45.0150	−1.71494
$690$	0	0
$691$	−36.5978	−1.39225	−0.696124	−	0.717922i	$-0.745094\pi$
$691$	−36.5978	−1.39225	−0.696124	+	0.717922i	$0.745094\pi$
$692$	0	0
$693$	−33.4643	−1.27120
$694$	0	0
$695$	−6.24121	−0.236743
$696$	0	0
$697$	9.59865	0.363575
$698$	0	0
$699$	0	0
$700$	0	0
$701$	30.0319	1.13429	0.567144	−	0.823619i	$-0.308048\pi$
$701$	30.0319	1.13429	0.567144	+	0.823619i	$0.308048\pi$
$702$	0	0
$703$	−4.36796	−0.164741
$704$	0	0
$705$	0	0
$706$	0	0
$707$	33.6412	1.26521
$708$	0	0
$709$	41.5061	1.55879	0.779397	−	0.626530i	$-0.215525\pi$
$709$	41.5061	1.55879	0.779397	+	0.626530i	$0.215525\pi$
$710$	0	0
$711$	−3.00000	−0.112509
$712$	0	0
$713$	−20.1274	−0.753776
$714$	0	0
$715$	−30.3257	−1.13412
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−0.745956	−0.0278195	−0.0139097	−	0.999903i	$-0.504428\pi$
$719$	−0.745956	−0.0278195	−0.0139097	+	0.999903i	$0.504428\pi$
$720$	0	0
$721$	11.5877	0.431550
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0.983230	0.0365162
$726$	0	0
$727$	35.9346	1.33274	0.666370	−	0.745622i	$-0.267847\pi$
$727$	35.9346	1.33274	0.666370	+	0.745622i	$0.267847\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−0.864214	−0.0319641
$732$	0	0
$733$	34.6131	1.27846	0.639232	−	0.769014i	$-0.279252\pi$
$733$	34.6131	1.27846	0.639232	+	0.769014i	$0.279252\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	31.4501	1.15848
$738$	0	0
$739$	−36.9541	−1.35938	−0.679689	−	0.733500i	$-0.737885\pi$
$739$	−36.9541	−1.35938	−0.679689	+	0.733500i	$0.737885\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−49.0959	−1.80115	−0.900576	−	0.434698i	$-0.856855\pi$
$743$	−49.0959	−1.80115	−0.900576	+	0.434698i	$0.856855\pi$
$744$	0	0
$745$	−35.2566	−1.29170
$746$	0	0
$747$	−15.6718	−0.573400
$748$	0	0
$749$	−16.9082	−0.617812
$750$	0	0
$751$	2.21555	0.0808465	0.0404233	−	0.999183i	$-0.487129\pi$
$751$	2.21555	0.0808465	0.0404233	+	0.999183i	$0.487129\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.08143	0.184932
$756$	0	0
$757$	27.4263	0.996827	0.498413	−	0.866939i	$-0.333916\pi$
$757$	27.4263	0.996827	0.498413	+	0.866939i	$0.333916\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−21.1771	−0.767669	−0.383834	−	0.923402i	$-0.625397\pi$
$761$	−21.1771	−0.767669	−0.383834	+	0.923402i	$0.625397\pi$
$762$	0	0
$763$	−7.92499	−0.286904
$764$	0	0
$765$	5.34228	0.193151
$766$	0	0
$767$	46.8955	1.69330
$768$	0	0
$769$	−11.7155	−0.422471	−0.211235	−	0.977435i	$-0.567749\pi$
$769$	−11.7155	−0.422471	−0.211235	+	0.977435i	$0.567749\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	13.7546	0.494720	0.247360	−	0.968924i	$-0.420437\pi$
$773$	13.7546	0.494720	0.247360	+	0.968924i	$0.420437\pi$
$774$	0	0
$775$	−14.2011	−0.510119
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.51751	0.341000
$780$	0	0
$781$	−11.5474	−0.413197
$782$	0	0
$783$	0	0
$784$	0	0
$785$	30.7544	1.09767
$786$	0	0
$787$	−46.5685	−1.65999	−0.829994	−	0.557772i	$-0.811656\pi$
$787$	−46.5685	−1.65999	−0.829994	+	0.557772i	$0.811656\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−32.2120	−1.14533
$792$	0	0
$793$	−9.57474	−0.340009
$794$	0	0
$795$	0	0
$796$	0	0
$797$	39.1260	1.38591	0.692957	−	0.720979i	$-0.256308\pi$
$797$	39.1260	1.38591	0.692957	+	0.720979i	$0.256308\pi$
$798$	0	0
$799$	9.09088	0.321612
$800$	0	0
$801$	30.0656	1.06232
$802$	0	0
$803$	−26.6423	−0.940187
$804$	0	0
$805$	11.3179	0.398902
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−13.2904	−0.467264	−0.233632	−	0.972325i	$-0.575061\pi$
$809$	−13.2904	−0.467264	−0.233632	+	0.972325i	$0.575061\pi$
$810$	0	0
$811$	−38.3097	−1.34523	−0.672617	−	0.739991i	$-0.734830\pi$
$811$	−38.3097	−1.34523	−0.672617	+	0.739991i	$0.734830\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	11.2082	0.392608
$816$	0	0
$817$	−0.856909	−0.0299794
$818$	0	0
$819$	−26.6655	−0.931766
$820$	0	0
$821$	−29.6151	−1.03357	−0.516787	−	0.856114i	$-0.672872\pi$
$821$	−29.6151	−1.03357	−0.516787	+	0.856114i	$0.672872\pi$
$822$	0	0
$823$	6.59329	0.229827	0.114914	−	0.993375i	$-0.463341\pi$
$823$	6.59329	0.229827	0.114914	+	0.993375i	$0.463341\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−48.8894	−1.70005	−0.850025	−	0.526742i	$-0.823413\pi$
$827$	−48.8894	−1.70005	−0.850025	+	0.526742i	$0.823413\pi$
$828$	0	0
$829$	−50.8481	−1.76603	−0.883013	−	0.469349i	$-0.844489\pi$
$829$	−50.8481	−1.76603	−0.883013	+	0.469349i	$0.844489\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.23754	−0.0428782
$834$	0	0
$835$	−39.5426	−1.36843
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−39.9160	−1.37805	−0.689027	−	0.724735i	$-0.741962\pi$
$839$	−39.9160	−1.37805	−0.689027	+	0.724735i	$0.741962\pi$
$840$	0	0
$841$	−28.7271	−0.990591
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.21034	−0.0416371
$846$	0	0
$847$	25.3576	0.871296
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11.6527	0.399449
$852$	0	0
$853$	−17.9827	−0.615717	−0.307859	−	0.951432i	$-0.599612\pi$
$853$	−17.9827	−0.615717	−0.307859	+	0.951432i	$0.599612\pi$
$854$	0	0
$855$	5.29713	0.181158
$856$	0	0
$857$	32.4455	1.10832	0.554158	−	0.832411i	$-0.313040\pi$
$857$	32.4455	1.10832	0.554158	+	0.832411i	$0.313040\pi$
$858$	0	0
$859$	−53.1071	−1.81199	−0.905996	−	0.423287i	$-0.860876\pi$
$859$	−53.1071	−1.81199	−0.905996	+	0.423287i	$0.860876\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26.9923	0.918829	0.459414	−	0.888222i	$-0.348059\pi$
$863$	26.9923	0.918829	0.459414	+	0.888222i	$0.348059\pi$
$864$	0	0
$865$	23.6976	0.805742
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.64261	0.157490
$870$	0	0
$871$	25.0605	0.849143
$872$	0	0
$873$	13.6685	0.462608
$874$	0	0
$875$	29.1977	0.987063
$876$	0	0
$877$	−17.5137	−0.591395	−0.295697	−	0.955282i	$-0.595552\pi$
$877$	−17.5137	−0.591395	−0.295697	+	0.955282i	$0.595552\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	8.96224	0.301946	0.150973	−	0.988538i	$-0.451759\pi$
$881$	8.96224	0.301946	0.150973	+	0.988538i	$0.451759\pi$
$882$	0	0
$883$	−53.0134	−1.78404	−0.892021	−	0.451995i	$-0.850713\pi$
$883$	−53.0134	−1.78404	−0.892021	+	0.451995i	$0.850713\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7.75659	0.260441	0.130220	−	0.991485i	$-0.458432\pi$
$887$	7.75659	0.260441	0.130220	+	0.991485i	$0.458432\pi$
$888$	0	0
$889$	−11.0709	−0.371305
$890$	0	0
$891$	41.7835	1.39980
$892$	0	0
$893$	9.01404	0.301643
$894$	0	0
$895$	−4.27934	−0.143043
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.94105	−0.131441
$900$	0	0
$901$	−12.2720	−0.408839
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.06862	−0.0355220
$906$	0	0
$907$	21.9437	0.728629	0.364314	−	0.931276i	$-0.381303\pi$
$907$	21.9437	0.728629	0.364314	+	0.931276i	$0.381303\pi$
$908$	0	0
$909$	−42.0044	−1.39320
$910$	0	0
$911$	−37.1133	−1.22962	−0.614809	−	0.788676i	$-0.710767\pi$
$911$	−37.1133	−1.22962	−0.614809	+	0.788676i	$0.710767\pi$
$912$	0	0
$913$	24.2526	0.802645
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.92827	0.228792
$918$	0	0
$919$	−4.63759	−0.152980	−0.0764900	−	0.997070i	$-0.524371\pi$
$919$	−4.63759	−0.152980	−0.0764900	+	0.997070i	$0.524371\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−9.20133	−0.302866
$924$	0	0
$925$	8.22169	0.270328
$926$	0	0
$927$	−14.4685	−0.475206
$928$	0	0
$929$	−6.33317	−0.207785	−0.103892	−	0.994589i	$-0.533130\pi$
$929$	−6.33317	−0.207785	−0.103892	+	0.994589i	$0.533130\pi$
$930$	0	0
$931$	−1.22708	−0.0402159
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.26738	−0.270372
$936$	0	0
$937$	8.80575	0.287671	0.143836	−	0.989602i	$-0.454056\pi$
$937$	8.80575	0.287671	0.143836	+	0.989602i	$0.454056\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	41.7928	1.36240	0.681202	−	0.732095i	$-0.261457\pi$
$941$	41.7928	1.36240	0.681202	+	0.732095i	$0.261457\pi$
$942$	0	0
$943$	−25.3905	−0.826827
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7.25168	−0.235648	−0.117824	−	0.993034i	$-0.537592\pi$
$947$	−7.25168	−0.235648	−0.117824	+	0.993034i	$0.537592\pi$
$948$	0	0
$949$	−21.2295	−0.689139
$950$	0	0
$951$	0	0
$952$	0	0
$953$	39.2211	1.27050	0.635248	−	0.772308i	$-0.280898\pi$
$953$	39.2211	1.27050	0.635248	+	0.772308i	$0.280898\pi$
$954$	0	0
$955$	−47.0691	−1.52312
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−36.3902	−1.17510
$960$	0	0
$961$	25.9220	0.836192
$962$	0	0
$963$	21.1116	0.680312
$964$	0	0
$965$	−10.8654	−0.349770
$966$	0	0
$967$	15.2063	0.489003	0.244502	−	0.969649i	$-0.421376\pi$
$967$	15.2063	0.489003	0.244502	+	0.969649i	$0.421376\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−25.5272	−0.819206	−0.409603	−	0.912264i	$-0.634333\pi$
$971$	−25.5272	−0.819206	−0.409603	+	0.912264i	$0.634333\pi$
$972$	0	0
$973$	8.49274	0.272265
$974$	0	0
$975$	0	0
$976$	0	0
$977$	23.1833	0.741701	0.370850	−	0.928693i	$-0.379066\pi$
$977$	23.1833	0.741701	0.370850	+	0.928693i	$0.379066\pi$
$978$	0	0
$979$	−46.5277	−1.48703
$980$	0	0
$981$	9.89515	0.315928
$982$	0	0
$983$	−20.0431	−0.639277	−0.319638	−	0.947540i	$-0.603561\pi$
$983$	−20.0431	−0.639277	−0.319638	+	0.947540i	$0.603561\pi$
$984$	0	0
$985$	−44.1975	−1.40825
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.28603	0.0726915
$990$	0	0
$991$	5.32064	0.169016	0.0845079	−	0.996423i	$-0.473068\pi$
$991$	5.32064	0.169016	0.0845079	+	0.996423i	$0.473068\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−10.1746	−0.322556
$996$	0	0
$997$	−11.5793	−0.366721	−0.183360	−	0.983046i	$-0.558698\pi$
$997$	−11.5793	−0.366721	−0.183360	+	0.983046i	$0.558698\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.d.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.d.1.3	✓	8	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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.76571 q^{5} +2.40269 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.76571 q^{5} +2.40269 q^{7} -3.00000 q^{9} +4.64261 q^{11} +3.69939 q^{13} +1.00852 q^{17} +1.00000 q^{19} -2.66776 q^{23} -1.88227 q^{25} -0.522363 q^{29} +7.54466 q^{31} -4.24245 q^{35} -4.36796 q^{37} +9.51751 q^{41} -0.856909 q^{43} +5.29713 q^{45} +9.01404 q^{47} -1.22708 q^{49} -12.1682 q^{53} -8.19750 q^{55} +12.6765 q^{59} -2.58820 q^{61} -7.20807 q^{63} -6.53204 q^{65} +6.77423 q^{67} -2.48726 q^{71} -5.73865 q^{73} +11.1548 q^{77} +1.00000 q^{79} +9.00000 q^{81} +5.22392 q^{83} -1.78076 q^{85} -10.0219 q^{89} +8.88848 q^{91} -1.76571 q^{95} -4.55616 q^{97} -13.9278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{5} + q^{7} - 24 q^{9}+O(q^{10}) \) 8 * q + 4 * q^5 + q^7 - 24 * q^9	\( 8 q + 4 q^{5} + q^{7} - 24 q^{9} - 7 q^{11} - 12 q^{13} - 7 q^{17} + 8 q^{19} + 10 q^{23} + 6 q^{25} + 5 q^{29} + 3 q^{31} - 11 q^{35} - 15 q^{37} - 5 q^{41} + 10 q^{43} - 12 q^{45} + 18 q^{47} + 31 q^{49} + 9 q^{53} - 17 q^{55} + 8 q^{59} + 13 q^{61} - 3 q^{63} + 4 q^{65} + 21 q^{67} + 44 q^{71} - 20 q^{73} + 15 q^{77} + 8 q^{79} + 72 q^{81} - 4 q^{83} - 9 q^{85} - 10 q^{89} + 48 q^{91} + 4 q^{95} - 22 q^{97} + 21 q^{99}+O(q^{100}) \) 8 * q + 4 * q^5 + q^7 - 24 * q^9 - 7 * q^11 - 12 * q^13 - 7 * q^17 + 8 * q^19 + 10 * q^23 + 6 * q^25 + 5 * q^29 + 3 * q^31 - 11 * q^35 - 15 * q^37 - 5 * q^41 + 10 * q^43 - 12 * q^45 + 18 * q^47 + 31 * q^49 + 9 * q^53 - 17 * q^55 + 8 * q^59 + 13 * q^61 - 3 * q^63 + 4 * q^65 + 21 * q^67 + 44 * q^71 - 20 * q^73 + 15 * q^77 + 8 * q^79 + 72 * q^81 - 4 * q^83 - 9 * q^85 - 10 * q^89 + 48 * q^91 + 4 * q^95 - 22 * q^97 + 21 * q^99

Introduction

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