LMFDB - Embedded newform 8004.2.a.f.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$63.9122617778$
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			1.5
Root			$-0.509166$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +0.509166 q^{5} +1.30726 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.509166 q^{5} +1.30726 q^{7} +1.00000 q^{9} +3.43078 q^{11} -4.04268 q^{13} +0.509166 q^{15} -3.71370 q^{17} -1.69392 q^{19} +1.30726 q^{21} -1.00000 q^{23} -4.74075 q^{25} +1.00000 q^{27} -1.00000 q^{29} -5.08739 q^{31} +3.43078 q^{33} +0.665612 q^{35} +6.50604 q^{37} -4.04268 q^{39} -5.79306 q^{41} -2.51385 q^{43} +0.509166 q^{45} -10.8573 q^{47} -5.29108 q^{49} -3.71370 q^{51} -8.24311 q^{53} +1.74684 q^{55} -1.69392 q^{57} -2.44828 q^{59} +4.30891 q^{61} +1.30726 q^{63} -2.05840 q^{65} -6.06826 q^{67} -1.00000 q^{69} +0.193315 q^{71} +10.7151 q^{73} -4.74075 q^{75} +4.48492 q^{77} -7.99866 q^{79} +1.00000 q^{81} -5.51057 q^{83} -1.89089 q^{85} -1.00000 q^{87} +3.60084 q^{89} -5.28483 q^{91} -5.08739 q^{93} -0.862487 q^{95} -1.88484 q^{97} +3.43078 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0.509166	0.227706	0.113853	−	0.993498i	$-0.463681\pi$
$5$	0.509166	0.227706	0.113853	+	0.993498i	$0.463681\pi$
$6$	0	0
$7$	1.30726	0.494097	0.247049	−	0.969003i	$-0.420539\pi$
$7$	1.30726	0.494097	0.247049	+	0.969003i	$0.420539\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.43078	1.03442	0.517210	−	0.855858i	$-0.326971\pi$
$11$	3.43078	1.03442	0.517210	+	0.855858i	$0.326971\pi$
$12$	0	0
$13$	−4.04268	−1.12124	−0.560619	−	0.828074i	$-0.689437\pi$
$13$	−4.04268	−1.12124	−0.560619	+	0.828074i	$0.689437\pi$
$14$	0	0
$15$	0.509166	0.131466
$16$	0	0
$17$	−3.71370	−0.900704	−0.450352	−	0.892851i	$-0.648701\pi$
$17$	−3.71370	−0.900704	−0.450352	+	0.892851i	$0.648701\pi$
$18$	0	0
$19$	−1.69392	−0.388612	−0.194306	−	0.980941i	$-0.562245\pi$
$19$	−1.69392	−0.388612	−0.194306	+	0.980941i	$0.562245\pi$
$20$	0	0
$21$	1.30726	0.285267
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.74075	−0.948150
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−5.08739	−0.913722	−0.456861	−	0.889538i	$-0.651026\pi$
$31$	−5.08739	−0.913722	−0.456861	+	0.889538i	$0.651026\pi$
$32$	0	0
$33$	3.43078	0.597223
$34$	0	0
$35$	0.665612	0.112509
$36$	0	0
$37$	6.50604	1.06959	0.534793	−	0.844983i	$-0.320389\pi$
$37$	6.50604	1.06959	0.534793	+	0.844983i	$0.320389\pi$
$38$	0	0
$39$	−4.04268	−0.647347
$40$	0	0
$41$	−5.79306	−0.904723	−0.452362	−	0.891835i	$-0.649418\pi$
$41$	−5.79306	−0.904723	−0.452362	+	0.891835i	$0.649418\pi$
$42$	0	0
$43$	−2.51385	−0.383358	−0.191679	−	0.981458i	$-0.561393\pi$
$43$	−2.51385	−0.383358	−0.191679	+	0.981458i	$0.561393\pi$
$44$	0	0
$45$	0.509166	0.0759020
$46$	0	0
$47$	−10.8573	−1.58369	−0.791847	−	0.610720i	$-0.790880\pi$
$47$	−10.8573	−1.58369	−0.791847	+	0.610720i	$0.790880\pi$
$48$	0	0
$49$	−5.29108	−0.755868
$50$	0	0
$51$	−3.71370	−0.520021
$52$	0	0
$53$	−8.24311	−1.13228	−0.566139	−	0.824310i	$-0.691563\pi$
$53$	−8.24311	−1.13228	−0.566139	+	0.824310i	$0.691563\pi$
$54$	0	0
$55$	1.74684	0.235544
$56$	0	0
$57$	−1.69392	−0.224365
$58$	0	0
$59$	−2.44828	−0.318738	−0.159369	−	0.987219i	$-0.550946\pi$
$59$	−2.44828	−0.318738	−0.159369	+	0.987219i	$0.550946\pi$
$60$	0	0
$61$	4.30891	0.551700	0.275850	−	0.961201i	$-0.411041\pi$
$61$	4.30891	0.551700	0.275850	+	0.961201i	$0.411041\pi$
$62$	0	0
$63$	1.30726	0.164699
$64$	0	0
$65$	−2.05840	−0.255313
$66$	0	0
$67$	−6.06826	−0.741356	−0.370678	−	0.928761i	$-0.620875\pi$
$67$	−6.06826	−0.741356	−0.370678	+	0.928761i	$0.620875\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	0.193315	0.0229423	0.0114711	−	0.999934i	$-0.496349\pi$
$71$	0.193315	0.0229423	0.0114711	+	0.999934i	$0.496349\pi$
$72$	0	0
$73$	10.7151	1.25411	0.627053	−	0.778977i	$-0.284261\pi$
$73$	10.7151	1.25411	0.627053	+	0.778977i	$0.284261\pi$
$74$	0	0
$75$	−4.74075	−0.547415
$76$	0	0
$77$	4.48492	0.511104
$78$	0	0
$79$	−7.99866	−0.899920	−0.449960	−	0.893049i	$-0.648562\pi$
$79$	−7.99866	−0.899920	−0.449960	+	0.893049i	$0.648562\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.51057	−0.604864	−0.302432	−	0.953171i	$-0.597798\pi$
$83$	−5.51057	−0.604864	−0.302432	+	0.953171i	$0.597798\pi$
$84$	0	0
$85$	−1.89089	−0.205096
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	3.60084	0.381689	0.190844	−	0.981620i	$-0.438877\pi$
$89$	3.60084	0.381689	0.190844	+	0.981620i	$0.438877\pi$
$90$	0	0
$91$	−5.28483	−0.554000
$92$	0	0
$93$	−5.08739	−0.527537
$94$	0	0
$95$	−0.862487	−0.0884893
$96$	0	0
$97$	−1.88484	−0.191376	−0.0956882	−	0.995411i	$-0.530505\pi$
$97$	−1.88484	−0.191376	−0.0956882	+	0.995411i	$0.530505\pi$
$98$	0	0
$99$	3.43078	0.344807
$100$	0	0
$101$	−0.243655	−0.0242446	−0.0121223	−	0.999927i	$-0.503859\pi$
$101$	−0.243655	−0.0242446	−0.0121223	+	0.999927i	$0.503859\pi$
$102$	0	0
$103$	9.52669	0.938693	0.469346	−	0.883014i	$-0.344490\pi$
$103$	9.52669	0.938693	0.469346	+	0.883014i	$0.344490\pi$
$104$	0	0
$105$	0.665612	0.0649571
$106$	0	0
$107$	3.32544	0.321482	0.160741	−	0.986997i	$-0.448612\pi$
$107$	3.32544	0.321482	0.160741	+	0.986997i	$0.448612\pi$
$108$	0	0
$109$	−5.03360	−0.482132	−0.241066	−	0.970509i	$-0.577497\pi$
$109$	−5.03360	−0.482132	−0.241066	+	0.970509i	$0.577497\pi$
$110$	0	0
$111$	6.50604	0.617526
$112$	0	0
$113$	−5.19281	−0.488499	−0.244249	−	0.969712i	$-0.578542\pi$
$113$	−5.19281	−0.488499	−0.244249	+	0.969712i	$0.578542\pi$
$114$	0	0
$115$	−0.509166	−0.0474800
$116$	0	0
$117$	−4.04268	−0.373746
$118$	0	0
$119$	−4.85476	−0.445035
$120$	0	0
$121$	0.770277	0.0700252
$122$	0	0
$123$	−5.79306	−0.522342
$124$	0	0
$125$	−4.95966	−0.443606
$126$	0	0
$127$	−11.9605	−1.06132	−0.530661	−	0.847584i	$-0.678056\pi$
$127$	−11.9605	−1.06132	−0.530661	+	0.847584i	$0.678056\pi$
$128$	0	0
$129$	−2.51385	−0.221332
$130$	0	0
$131$	−19.0584	−1.66514	−0.832570	−	0.553919i	$-0.813132\pi$
$131$	−19.0584	−1.66514	−0.832570	+	0.553919i	$0.813132\pi$
$132$	0	0
$133$	−2.21439	−0.192012
$134$	0	0
$135$	0.509166	0.0438221
$136$	0	0
$137$	3.28344	0.280523	0.140262	−	0.990114i	$-0.455206\pi$
$137$	3.28344	0.280523	0.140262	+	0.990114i	$0.455206\pi$
$138$	0	0
$139$	8.61444	0.730668	0.365334	−	0.930877i	$-0.380955\pi$
$139$	8.61444	0.730668	0.365334	+	0.930877i	$0.380955\pi$
$140$	0	0
$141$	−10.8573	−0.914346
$142$	0	0
$143$	−13.8696	−1.15983
$144$	0	0
$145$	−0.509166	−0.0422840
$146$	0	0
$147$	−5.29108	−0.436401
$148$	0	0
$149$	7.75834	0.635588	0.317794	−	0.948160i	$-0.397058\pi$
$149$	7.75834	0.635588	0.317794	+	0.948160i	$0.397058\pi$
$150$	0	0
$151$	−5.87766	−0.478317	−0.239159	−	0.970980i	$-0.576872\pi$
$151$	−5.87766	−0.478317	−0.239159	+	0.970980i	$0.576872\pi$
$152$	0	0
$153$	−3.71370	−0.300235
$154$	0	0
$155$	−2.59033	−0.208060
$156$	0	0
$157$	15.9870	1.27590	0.637952	−	0.770076i	$-0.279782\pi$
$157$	15.9870	1.27590	0.637952	+	0.770076i	$0.279782\pi$
$158$	0	0
$159$	−8.24311	−0.653721
$160$	0	0
$161$	−1.30726	−0.103026
$162$	0	0
$163$	−0.982301	−0.0769397	−0.0384699	−	0.999260i	$-0.512248\pi$
$163$	−0.982301	−0.0769397	−0.0384699	+	0.999260i	$0.512248\pi$
$164$	0	0
$165$	1.74684	0.135991
$166$	0	0
$167$	2.37639	0.183890	0.0919452	−	0.995764i	$-0.470692\pi$
$167$	2.37639	0.183890	0.0919452	+	0.995764i	$0.470692\pi$
$168$	0	0
$169$	3.34325	0.257173
$170$	0	0
$171$	−1.69392	−0.129537
$172$	0	0
$173$	13.1477	0.999603	0.499801	−	0.866140i	$-0.333406\pi$
$173$	13.1477	0.999603	0.499801	+	0.866140i	$0.333406\pi$
$174$	0	0
$175$	−6.19738	−0.468478
$176$	0	0
$177$	−2.44828	−0.184024
$178$	0	0
$179$	7.42512	0.554980	0.277490	−	0.960729i	$-0.410498\pi$
$179$	7.42512	0.554980	0.277490	+	0.960729i	$0.410498\pi$
$180$	0	0
$181$	11.9396	0.887464	0.443732	−	0.896160i	$-0.353654\pi$
$181$	11.9396	0.887464	0.443732	+	0.896160i	$0.353654\pi$
$182$	0	0
$183$	4.30891	0.318524
$184$	0	0
$185$	3.31266	0.243551
$186$	0	0
$187$	−12.7409	−0.931706
$188$	0	0
$189$	1.30726	0.0950891
$190$	0	0
$191$	5.62297	0.406864	0.203432	−	0.979089i	$-0.434790\pi$
$191$	5.62297	0.406864	0.203432	+	0.979089i	$0.434790\pi$
$192$	0	0
$193$	13.9968	1.00751	0.503757	−	0.863845i	$-0.331951\pi$
$193$	13.9968	1.00751	0.503757	+	0.863845i	$0.331951\pi$
$194$	0	0
$195$	−2.05840	−0.147405
$196$	0	0
$197$	−2.68306	−0.191160	−0.0955802	−	0.995422i	$-0.530471\pi$
$197$	−2.68306	−0.191160	−0.0955802	+	0.995422i	$0.530471\pi$
$198$	0	0
$199$	−6.11307	−0.433344	−0.216672	−	0.976244i	$-0.569520\pi$
$199$	−6.11307	−0.433344	−0.216672	+	0.976244i	$0.569520\pi$
$200$	0	0
$201$	−6.06826	−0.428022
$202$	0	0
$203$	−1.30726	−0.0917515
$204$	0	0
$205$	−2.94963	−0.206011
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−5.81148	−0.401988
$210$	0	0
$211$	−7.39854	−0.509336	−0.254668	−	0.967028i	$-0.581966\pi$
$211$	−7.39854	−0.509336	−0.254668	+	0.967028i	$0.581966\pi$
$212$	0	0
$213$	0.193315	0.0132457
$214$	0	0
$215$	−1.27997	−0.0872929
$216$	0	0
$217$	−6.65053	−0.451467
$218$	0	0
$219$	10.7151	0.724058
$220$	0	0
$221$	15.0133	1.00990
$222$	0	0
$223$	−25.2077	−1.68803	−0.844016	−	0.536317i	$-0.819815\pi$
$223$	−25.2077	−1.68803	−0.844016	+	0.536317i	$0.819815\pi$
$224$	0	0
$225$	−4.74075	−0.316050
$226$	0	0
$227$	9.41611	0.624969	0.312484	−	0.949923i	$-0.398839\pi$
$227$	9.41611	0.624969	0.312484	+	0.949923i	$0.398839\pi$
$228$	0	0
$229$	17.0488	1.12661	0.563307	−	0.826248i	$-0.309529\pi$
$229$	17.0488	1.12661	0.563307	+	0.826248i	$0.309529\pi$
$230$	0	0
$231$	4.48492	0.295086
$232$	0	0
$233$	9.65548	0.632551	0.316276	−	0.948667i	$-0.397568\pi$
$233$	9.65548	0.632551	0.316276	+	0.948667i	$0.397568\pi$
$234$	0	0
$235$	−5.52815	−0.360617
$236$	0	0
$237$	−7.99866	−0.519569
$238$	0	0
$239$	30.3586	1.96374	0.981869	−	0.189563i	$-0.0607070\pi$
$239$	30.3586	1.96374	0.981869	+	0.189563i	$0.0607070\pi$
$240$	0	0
$241$	−22.2497	−1.43323	−0.716615	−	0.697469i	$-0.754310\pi$
$241$	−22.2497	−1.43323	−0.716615	+	0.697469i	$0.754310\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.69404	−0.172116
$246$	0	0
$247$	6.84798	0.435726
$248$	0	0
$249$	−5.51057	−0.349218
$250$	0	0
$251$	29.4308	1.85766	0.928829	−	0.370509i	$-0.120817\pi$
$251$	29.4308	1.85766	0.928829	+	0.370509i	$0.120817\pi$
$252$	0	0
$253$	−3.43078	−0.215692
$254$	0	0
$255$	−1.89089	−0.118412
$256$	0	0
$257$	−23.1801	−1.44594	−0.722968	−	0.690882i	$-0.757222\pi$
$257$	−23.1801	−1.44594	−0.722968	+	0.690882i	$0.757222\pi$
$258$	0	0
$259$	8.50508	0.528480
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	15.3928	0.949163	0.474582	−	0.880212i	$-0.342599\pi$
$263$	15.3928	0.949163	0.474582	+	0.880212i	$0.342599\pi$
$264$	0	0
$265$	−4.19711	−0.257827
$266$	0	0
$267$	3.60084	0.220368
$268$	0	0
$269$	3.34502	0.203949	0.101975	−	0.994787i	$-0.467484\pi$
$269$	3.34502	0.203949	0.101975	+	0.994787i	$0.467484\pi$
$270$	0	0
$271$	−15.6481	−0.950554	−0.475277	−	0.879836i	$-0.657652\pi$
$271$	−15.6481	−0.950554	−0.475277	+	0.879836i	$0.657652\pi$
$272$	0	0
$273$	−5.28483	−0.319852
$274$	0	0
$275$	−16.2645	−0.980785
$276$	0	0
$277$	−23.7842	−1.42906	−0.714528	−	0.699607i	$-0.753358\pi$
$277$	−23.7842	−1.42906	−0.714528	+	0.699607i	$0.753358\pi$
$278$	0	0
$279$	−5.08739	−0.304574
$280$	0	0
$281$	19.6612	1.17289	0.586443	−	0.809990i	$-0.300528\pi$
$281$	19.6612	1.17289	0.586443	+	0.809990i	$0.300528\pi$
$282$	0	0
$283$	29.5806	1.75839	0.879193	−	0.476465i	$-0.158082\pi$
$283$	29.5806	1.75839	0.879193	+	0.476465i	$0.158082\pi$
$284$	0	0
$285$	−0.862487	−0.0510893
$286$	0	0
$287$	−7.57302	−0.447021
$288$	0	0
$289$	−3.20846	−0.188733
$290$	0	0
$291$	−1.88484	−0.110491
$292$	0	0
$293$	16.0867	0.939795	0.469898	−	0.882721i	$-0.344291\pi$
$293$	16.0867	0.939795	0.469898	+	0.882721i	$0.344291\pi$
$294$	0	0
$295$	−1.24658	−0.0725787
$296$	0	0
$297$	3.43078	0.199074
$298$	0	0
$299$	4.04268	0.233794
$300$	0	0
$301$	−3.28625	−0.189416
$302$	0	0
$303$	−0.243655	−0.0139976
$304$	0	0
$305$	2.19395	0.125625
$306$	0	0
$307$	32.3537	1.84652	0.923261	−	0.384172i	$-0.125513\pi$
$307$	32.3537	1.84652	0.923261	+	0.384172i	$0.125513\pi$
$308$	0	0
$309$	9.52669	0.541955
$310$	0	0
$311$	2.78216	0.157762	0.0788809	−	0.996884i	$-0.474865\pi$
$311$	2.78216	0.157762	0.0788809	+	0.996884i	$0.474865\pi$
$312$	0	0
$313$	−30.4052	−1.71860	−0.859301	−	0.511470i	$-0.829101\pi$
$313$	−30.4052	−1.71860	−0.859301	+	0.511470i	$0.829101\pi$
$314$	0	0
$315$	0.665612	0.0375030
$316$	0	0
$317$	−9.55833	−0.536849	−0.268425	−	0.963301i	$-0.586503\pi$
$317$	−9.55833	−0.536849	−0.268425	+	0.963301i	$0.586503\pi$
$318$	0	0
$319$	−3.43078	−0.192087
$320$	0	0
$321$	3.32544	0.185608
$322$	0	0
$323$	6.29071	0.350024
$324$	0	0
$325$	19.1653	1.06310
$326$	0	0
$327$	−5.03360	−0.278359
$328$	0	0
$329$	−14.1932	−0.782499
$330$	0	0
$331$	−16.6393	−0.914578	−0.457289	−	0.889318i	$-0.651179\pi$
$331$	−16.6393	−0.914578	−0.457289	+	0.889318i	$0.651179\pi$
$332$	0	0
$333$	6.50604	0.356529
$334$	0	0
$335$	−3.08975	−0.168811
$336$	0	0
$337$	−3.75618	−0.204612	−0.102306	−	0.994753i	$-0.532622\pi$
$337$	−3.75618	−0.204612	−0.102306	+	0.994753i	$0.532622\pi$
$338$	0	0
$339$	−5.19281	−0.282035
$340$	0	0
$341$	−17.4537	−0.945172
$342$	0	0
$343$	−16.0676	−0.867569
$344$	0	0
$345$	−0.509166	−0.0274126
$346$	0	0
$347$	−32.9016	−1.76625	−0.883126	−	0.469136i	$-0.844565\pi$
$347$	−32.9016	−1.76625	−0.883126	+	0.469136i	$0.844565\pi$
$348$	0	0
$349$	−13.3638	−0.715349	−0.357674	−	0.933846i	$-0.616430\pi$
$349$	−13.3638	−0.715349	−0.357674	+	0.933846i	$0.616430\pi$
$350$	0	0
$351$	−4.04268	−0.215782
$352$	0	0
$353$	−17.1289	−0.911679	−0.455839	−	0.890062i	$-0.650661\pi$
$353$	−17.1289	−0.911679	−0.455839	+	0.890062i	$0.650661\pi$
$354$	0	0
$355$	0.0984294	0.00522409
$356$	0	0
$357$	−4.85476	−0.256941
$358$	0	0
$359$	−15.6112	−0.823927	−0.411963	−	0.911200i	$-0.635157\pi$
$359$	−15.6112	−0.823927	−0.411963	+	0.911200i	$0.635157\pi$
$360$	0	0
$361$	−16.1306	−0.848981
$362$	0	0
$363$	0.770277	0.0404291
$364$	0	0
$365$	5.45576	0.285568
$366$	0	0
$367$	−25.1722	−1.31398	−0.656990	−	0.753899i	$-0.728171\pi$
$367$	−25.1722	−1.31398	−0.656990	+	0.753899i	$0.728171\pi$
$368$	0	0
$369$	−5.79306	−0.301574
$370$	0	0
$371$	−10.7759	−0.559455
$372$	0	0
$373$	−12.9847	−0.672321	−0.336161	−	0.941805i	$-0.609128\pi$
$373$	−12.9847	−0.672321	−0.336161	+	0.941805i	$0.609128\pi$
$374$	0	0
$375$	−4.95966	−0.256116
$376$	0	0
$377$	4.04268	0.208209
$378$	0	0
$379$	−16.3268	−0.838649	−0.419325	−	0.907836i	$-0.637733\pi$
$379$	−16.3268	−0.838649	−0.419325	+	0.907836i	$0.637733\pi$
$380$	0	0
$381$	−11.9605	−0.612754
$382$	0	0
$383$	37.6998	1.92637	0.963186	−	0.268836i	$-0.0866388\pi$
$383$	37.6998	1.92637	0.963186	+	0.268836i	$0.0866388\pi$
$384$	0	0
$385$	2.28357	0.116382
$386$	0	0
$387$	−2.51385	−0.127786
$388$	0	0
$389$	1.73187	0.0878095	0.0439048	−	0.999036i	$-0.486020\pi$
$389$	1.73187	0.0878095	0.0439048	+	0.999036i	$0.486020\pi$
$390$	0	0
$391$	3.71370	0.187810
$392$	0	0
$393$	−19.0584	−0.961369
$394$	0	0
$395$	−4.07265	−0.204917
$396$	0	0
$397$	11.2598	0.565113	0.282556	−	0.959251i	$-0.408818\pi$
$397$	11.2598	0.565113	0.282556	+	0.959251i	$0.408818\pi$
$398$	0	0
$399$	−2.21439	−0.110858
$400$	0	0
$401$	37.6093	1.87812	0.939060	−	0.343752i	$-0.111698\pi$
$401$	37.6093	1.87812	0.939060	+	0.343752i	$0.111698\pi$
$402$	0	0
$403$	20.5667	1.02450
$404$	0	0
$405$	0.509166	0.0253007
$406$	0	0
$407$	22.3208	1.10640
$408$	0	0
$409$	−24.0726	−1.19032	−0.595158	−	0.803609i	$-0.702910\pi$
$409$	−24.0726	−1.19032	−0.595158	+	0.803609i	$0.702910\pi$
$410$	0	0
$411$	3.28344	0.161960
$412$	0	0
$413$	−3.20053	−0.157488
$414$	0	0
$415$	−2.80580	−0.137731
$416$	0	0
$417$	8.61444	0.421851
$418$	0	0
$419$	−12.2253	−0.597246	−0.298623	−	0.954371i	$-0.596527\pi$
$419$	−12.2253	−0.597246	−0.298623	+	0.954371i	$0.596527\pi$
$420$	0	0
$421$	−5.49466	−0.267793	−0.133897	−	0.990995i	$-0.542749\pi$
$421$	−5.49466	−0.267793	−0.133897	+	0.990995i	$0.542749\pi$
$422$	0	0
$423$	−10.8573	−0.527898
$424$	0	0
$425$	17.6057	0.854002
$426$	0	0
$427$	5.63286	0.272593
$428$	0	0
$429$	−13.8696	−0.669629
$430$	0	0
$431$	−31.8503	−1.53417	−0.767087	−	0.641543i	$-0.778294\pi$
$431$	−31.8503	−1.53417	−0.767087	+	0.641543i	$0.778294\pi$
$432$	0	0
$433$	32.8919	1.58068	0.790341	−	0.612667i	$-0.209903\pi$
$433$	32.8919	1.58068	0.790341	+	0.612667i	$0.209903\pi$
$434$	0	0
$435$	−0.509166	−0.0244127
$436$	0	0
$437$	1.69392	0.0810312
$438$	0	0
$439$	18.0168	0.859894	0.429947	−	0.902854i	$-0.358532\pi$
$439$	18.0168	0.859894	0.429947	+	0.902854i	$0.358532\pi$
$440$	0	0
$441$	−5.29108	−0.251956
$442$	0	0
$443$	−14.8304	−0.704616	−0.352308	−	0.935884i	$-0.614603\pi$
$443$	−14.8304	−0.704616	−0.352308	+	0.935884i	$0.614603\pi$
$444$	0	0
$445$	1.83343	0.0869128
$446$	0	0
$447$	7.75834	0.366957
$448$	0	0
$449$	−19.4148	−0.916240	−0.458120	−	0.888890i	$-0.651477\pi$
$449$	−19.4148	−0.916240	−0.458120	+	0.888890i	$0.651477\pi$
$450$	0	0
$451$	−19.8747	−0.935864
$452$	0	0
$453$	−5.87766	−0.276157
$454$	0	0
$455$	−2.69085	−0.126149
$456$	0	0
$457$	10.8838	0.509121	0.254560	−	0.967057i	$-0.418069\pi$
$457$	10.8838	0.509121	0.254560	+	0.967057i	$0.418069\pi$
$458$	0	0
$459$	−3.71370	−0.173340
$460$	0	0
$461$	−8.09627	−0.377081	−0.188540	−	0.982065i	$-0.560376\pi$
$461$	−8.09627	−0.377081	−0.188540	+	0.982065i	$0.560376\pi$
$462$	0	0
$463$	12.6008	0.585611	0.292805	−	0.956172i	$-0.405411\pi$
$463$	12.6008	0.585611	0.292805	+	0.956172i	$0.405411\pi$
$464$	0	0
$465$	−2.59033	−0.120123
$466$	0	0
$467$	27.0875	1.25346	0.626730	−	0.779236i	$-0.284393\pi$
$467$	27.0875	1.25346	0.626730	+	0.779236i	$0.284393\pi$
$468$	0	0
$469$	−7.93279	−0.366302
$470$	0	0
$471$	15.9870	0.736643
$472$	0	0
$473$	−8.62446	−0.396553
$474$	0	0
$475$	8.03046	0.368463
$476$	0	0
$477$	−8.24311	−0.377426
$478$	0	0
$479$	−2.01566	−0.0920979	−0.0460489	−	0.998939i	$-0.514663\pi$
$479$	−2.01566	−0.0920979	−0.0460489	+	0.998939i	$0.514663\pi$
$480$	0	0
$481$	−26.3018	−1.19926
$482$	0	0
$483$	−1.30726	−0.0594823
$484$	0	0
$485$	−0.959696	−0.0435776
$486$	0	0
$487$	38.4993	1.74457	0.872285	−	0.488998i	$-0.162638\pi$
$487$	38.4993	1.74457	0.872285	+	0.488998i	$0.162638\pi$
$488$	0	0
$489$	−0.982301	−0.0444212
$490$	0	0
$491$	−1.19556	−0.0539547	−0.0269773	−	0.999636i	$-0.508588\pi$
$491$	−1.19556	−0.0539547	−0.0269773	+	0.999636i	$0.508588\pi$
$492$	0	0
$493$	3.71370	0.167256
$494$	0	0
$495$	1.74684	0.0785146
$496$	0	0
$497$	0.252713	0.0113357
$498$	0	0
$499$	−14.2632	−0.638509	−0.319255	−	0.947669i	$-0.603433\pi$
$499$	−14.2632	−0.638509	−0.319255	+	0.947669i	$0.603433\pi$
$500$	0	0
$501$	2.37639	0.106169
$502$	0	0
$503$	10.2134	0.455393	0.227697	−	0.973732i	$-0.426881\pi$
$503$	10.2134	0.455393	0.227697	+	0.973732i	$0.426881\pi$
$504$	0	0
$505$	−0.124061	−0.00552065
$506$	0	0
$507$	3.34325	0.148479
$508$	0	0
$509$	4.76047	0.211004	0.105502	−	0.994419i	$-0.466355\pi$
$509$	4.76047	0.211004	0.105502	+	0.994419i	$0.466355\pi$
$510$	0	0
$511$	14.0074	0.619650
$512$	0	0
$513$	−1.69392	−0.0747884
$514$	0	0
$515$	4.85067	0.213746
$516$	0	0
$517$	−37.2489	−1.63821
$518$	0	0
$519$	13.1477	0.577121
$520$	0	0
$521$	12.8085	0.561151	0.280576	−	0.959832i	$-0.409475\pi$
$521$	12.8085	0.561151	0.280576	+	0.959832i	$0.409475\pi$
$522$	0	0
$523$	10.3067	0.450682	0.225341	−	0.974280i	$-0.427650\pi$
$523$	10.3067	0.450682	0.225341	+	0.974280i	$0.427650\pi$
$524$	0	0
$525$	−6.19738	−0.270476
$526$	0	0
$527$	18.8930	0.822992
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−2.44828	−0.106246
$532$	0	0
$533$	23.4195	1.01441
$534$	0	0
$535$	1.69320	0.0732035
$536$	0	0
$537$	7.42512	0.320418
$538$	0	0
$539$	−18.1525	−0.781885
$540$	0	0
$541$	−27.6637	−1.18936	−0.594678	−	0.803964i	$-0.702721\pi$
$541$	−27.6637	−1.18936	−0.594678	+	0.803964i	$0.702721\pi$
$542$	0	0
$543$	11.9396	0.512378
$544$	0	0
$545$	−2.56294	−0.109784
$546$	0	0
$547$	−43.5276	−1.86110	−0.930552	−	0.366159i	$-0.880673\pi$
$547$	−43.5276	−1.86110	−0.930552	+	0.366159i	$0.880673\pi$
$548$	0	0
$549$	4.30891	0.183900
$550$	0	0
$551$	1.69392	0.0721635
$552$	0	0
$553$	−10.4563	−0.444648
$554$	0	0
$555$	3.31266	0.140615
$556$	0	0
$557$	3.31648	0.140524	0.0702619	−	0.997529i	$-0.477616\pi$
$557$	3.31648	0.140524	0.0702619	+	0.997529i	$0.477616\pi$
$558$	0	0
$559$	10.1627	0.429835
$560$	0	0
$561$	−12.7409	−0.537921
$562$	0	0
$563$	−36.5082	−1.53864	−0.769319	−	0.638865i	$-0.779404\pi$
$563$	−36.5082	−1.53864	−0.769319	+	0.638865i	$0.779404\pi$
$564$	0	0
$565$	−2.64400	−0.111234
$566$	0	0
$567$	1.30726	0.0548997
$568$	0	0
$569$	0.0512148	0.00214703	0.00107352	−	0.999999i	$-0.499658\pi$
$569$	0.0512148	0.00214703	0.00107352	+	0.999999i	$0.499658\pi$
$570$	0	0
$571$	4.89820	0.204983	0.102492	−	0.994734i	$-0.467319\pi$
$571$	4.89820	0.204983	0.102492	+	0.994734i	$0.467319\pi$
$572$	0	0
$573$	5.62297	0.234903
$574$	0	0
$575$	4.74075	0.197703
$576$	0	0
$577$	−34.0432	−1.41724	−0.708618	−	0.705592i	$-0.750681\pi$
$577$	−34.0432	−1.41724	−0.708618	+	0.705592i	$0.750681\pi$
$578$	0	0
$579$	13.9968	0.581688
$580$	0	0
$581$	−7.20374	−0.298861
$582$	0	0
$583$	−28.2803	−1.17125
$584$	0	0
$585$	−2.05840	−0.0851042
$586$	0	0
$587$	23.4134	0.966373	0.483187	−	0.875517i	$-0.339479\pi$
$587$	23.4134	0.966373	0.483187	+	0.875517i	$0.339479\pi$
$588$	0	0
$589$	8.61763	0.355083
$590$	0	0
$591$	−2.68306	−0.110367
$592$	0	0
$593$	−47.3215	−1.94326	−0.971631	−	0.236503i	$-0.923999\pi$
$593$	−47.3215	−1.94326	−0.971631	+	0.236503i	$0.923999\pi$
$594$	0	0
$595$	−2.47188	−0.101337
$596$	0	0
$597$	−6.11307	−0.250191
$598$	0	0
$599$	24.4802	1.00023	0.500117	−	0.865958i	$-0.333290\pi$
$599$	24.4802	1.00023	0.500117	+	0.865958i	$0.333290\pi$
$600$	0	0
$601$	−17.5545	−0.716063	−0.358031	−	0.933710i	$-0.616552\pi$
$601$	−17.5545	−0.716063	−0.358031	+	0.933710i	$0.616552\pi$
$602$	0	0
$603$	−6.06826	−0.247119
$604$	0	0
$605$	0.392199	0.0159452
$606$	0	0
$607$	−8.40501	−0.341149	−0.170574	−	0.985345i	$-0.554562\pi$
$607$	−8.40501	−0.341149	−0.170574	+	0.985345i	$0.554562\pi$
$608$	0	0
$609$	−1.30726	−0.0529728
$610$	0	0
$611$	43.8924	1.77570
$612$	0	0
$613$	−17.1520	−0.692761	−0.346380	−	0.938094i	$-0.612589\pi$
$613$	−17.1520	−0.692761	−0.346380	+	0.938094i	$0.612589\pi$
$614$	0	0
$615$	−2.94963	−0.118941
$616$	0	0
$617$	−10.9296	−0.440007	−0.220004	−	0.975499i	$-0.570607\pi$
$617$	−10.9296	−0.440007	−0.220004	+	0.975499i	$0.570607\pi$
$618$	0	0
$619$	17.2540	0.693498	0.346749	−	0.937958i	$-0.387286\pi$
$619$	17.2540	0.693498	0.346749	+	0.937958i	$0.387286\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	4.70723	0.188591
$624$	0	0
$625$	21.1785	0.847138
$626$	0	0
$627$	−5.81148	−0.232088
$628$	0	0
$629$	−24.1615	−0.963381
$630$	0	0
$631$	−25.5266	−1.01620	−0.508099	−	0.861299i	$-0.669652\pi$
$631$	−25.5266	−1.01620	−0.508099	+	0.861299i	$0.669652\pi$
$632$	0	0
$633$	−7.39854	−0.294066
$634$	0	0
$635$	−6.08988	−0.241669
$636$	0	0
$637$	21.3901	0.847507
$638$	0	0
$639$	0.193315	0.00764742
$640$	0	0
$641$	−32.1532	−1.26998	−0.634988	−	0.772522i	$-0.718995\pi$
$641$	−32.1532	−1.26998	−0.634988	+	0.772522i	$0.718995\pi$
$642$	0	0
$643$	20.8761	0.823275	0.411637	−	0.911348i	$-0.364957\pi$
$643$	20.8761	0.823275	0.411637	+	0.911348i	$0.364957\pi$
$644$	0	0
$645$	−1.27997	−0.0503986
$646$	0	0
$647$	−9.52129	−0.374321	−0.187160	−	0.982329i	$-0.559928\pi$
$647$	−9.52129	−0.374321	−0.187160	+	0.982329i	$0.559928\pi$
$648$	0	0
$649$	−8.39951	−0.329709
$650$	0	0
$651$	−6.65053	−0.260655
$652$	0	0
$653$	46.8745	1.83434	0.917171	−	0.398493i	$-0.130467\pi$
$653$	46.8745	1.83434	0.917171	+	0.398493i	$0.130467\pi$
$654$	0	0
$655$	−9.70390	−0.379163
$656$	0	0
$657$	10.7151	0.418035
$658$	0	0
$659$	26.9143	1.04843	0.524216	−	0.851585i	$-0.324358\pi$
$659$	26.9143	1.04843	0.524216	+	0.851585i	$0.324358\pi$
$660$	0	0
$661$	9.10472	0.354132	0.177066	−	0.984199i	$-0.443339\pi$
$661$	9.10472	0.354132	0.177066	+	0.984199i	$0.443339\pi$
$662$	0	0
$663$	15.0133	0.583067
$664$	0	0
$665$	−1.12749	−0.0437223
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	−25.2077	−0.974586
$670$	0	0
$671$	14.7829	0.570689
$672$	0	0
$673$	−5.89361	−0.227182	−0.113591	−	0.993528i	$-0.536235\pi$
$673$	−5.89361	−0.227182	−0.113591	+	0.993528i	$0.536235\pi$
$674$	0	0
$675$	−4.74075	−0.182472
$676$	0	0
$677$	−19.4660	−0.748138	−0.374069	−	0.927401i	$-0.622038\pi$
$677$	−19.4660	−0.748138	−0.374069	+	0.927401i	$0.622038\pi$
$678$	0	0
$679$	−2.46397	−0.0945585
$680$	0	0
$681$	9.41611	0.360826
$682$	0	0
$683$	−19.2913	−0.738161	−0.369081	−	0.929397i	$-0.620327\pi$
$683$	−19.2913	−0.738161	−0.369081	+	0.929397i	$0.620327\pi$
$684$	0	0
$685$	1.67182	0.0638768
$686$	0	0
$687$	17.0488	0.650450
$688$	0	0
$689$	33.3242	1.26955
$690$	0	0
$691$	−14.1687	−0.539002	−0.269501	−	0.963000i	$-0.586859\pi$
$691$	−14.1687	−0.539002	−0.269501	+	0.963000i	$0.586859\pi$
$692$	0	0
$693$	4.48492	0.170368
$694$	0	0
$695$	4.38618	0.166377
$696$	0	0
$697$	21.5136	0.814887
$698$	0	0
$699$	9.65548	0.365204
$700$	0	0
$701$	−22.6534	−0.855606	−0.427803	−	0.903872i	$-0.640712\pi$
$701$	−22.6534	−0.855606	−0.427803	+	0.903872i	$0.640712\pi$
$702$	0	0
$703$	−11.0207	−0.415655
$704$	0	0
$705$	−5.52815	−0.208202
$706$	0	0
$707$	−0.318521	−0.0119792
$708$	0	0
$709$	−45.1718	−1.69646	−0.848232	−	0.529626i	$-0.822332\pi$
$709$	−45.1718	−1.69646	−0.848232	+	0.529626i	$0.822332\pi$
$710$	0	0
$711$	−7.99866	−0.299973
$712$	0	0
$713$	5.08739	0.190524
$714$	0	0
$715$	−7.06191	−0.264100
$716$	0	0
$717$	30.3586	1.13376
$718$	0	0
$719$	36.6236	1.36583	0.682916	−	0.730497i	$-0.260712\pi$
$719$	36.6236	1.36583	0.682916	+	0.730497i	$0.260712\pi$
$720$	0	0
$721$	12.4538	0.463805
$722$	0	0
$723$	−22.2497	−0.827476
$724$	0	0
$725$	4.74075	0.176067
$726$	0	0
$727$	38.6477	1.43337	0.716683	−	0.697399i	$-0.245660\pi$
$727$	38.6477	1.43337	0.716683	+	0.697399i	$0.245660\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	9.33566	0.345292
$732$	0	0
$733$	14.7267	0.543943	0.271972	−	0.962305i	$-0.412324\pi$
$733$	14.7267	0.543943	0.271972	+	0.962305i	$0.412324\pi$
$734$	0	0
$735$	−2.69404	−0.0993711
$736$	0	0
$737$	−20.8189	−0.766874
$738$	0	0
$739$	−43.6807	−1.60682	−0.803410	−	0.595426i	$-0.796983\pi$
$739$	−43.6807	−1.60682	−0.803410	+	0.595426i	$0.796983\pi$
$740$	0	0
$741$	6.84798	0.251567
$742$	0	0
$743$	40.5041	1.48595	0.742975	−	0.669319i	$-0.233414\pi$
$743$	40.5041	1.48595	0.742975	+	0.669319i	$0.233414\pi$
$744$	0	0
$745$	3.95029	0.144727
$746$	0	0
$747$	−5.51057	−0.201621
$748$	0	0
$749$	4.34721	0.158843
$750$	0	0
$751$	−2.36035	−0.0861305	−0.0430653	−	0.999072i	$-0.513712\pi$
$751$	−2.36035	−0.0861305	−0.0430653	+	0.999072i	$0.513712\pi$
$752$	0	0
$753$	29.4308	1.07252
$754$	0	0
$755$	−2.99271	−0.108916
$756$	0	0
$757$	−43.9668	−1.59800	−0.799001	−	0.601330i	$-0.794638\pi$
$757$	−43.9668	−1.59800	−0.799001	+	0.601330i	$0.794638\pi$
$758$	0	0
$759$	−3.43078	−0.124530
$760$	0	0
$761$	−22.0030	−0.797609	−0.398805	−	0.917036i	$-0.630575\pi$
$761$	−22.0030	−0.797609	−0.398805	+	0.917036i	$0.630575\pi$
$762$	0	0
$763$	−6.58022	−0.238220
$764$	0	0
$765$	−1.89089	−0.0683652
$766$	0	0
$767$	9.89759	0.357381
$768$	0	0
$769$	10.1762	0.366964	0.183482	−	0.983023i	$-0.441263\pi$
$769$	10.1762	0.366964	0.183482	+	0.983023i	$0.441263\pi$
$770$	0	0
$771$	−23.1801	−0.834811
$772$	0	0
$773$	−27.0046	−0.971288	−0.485644	−	0.874157i	$-0.661415\pi$
$773$	−27.0046	−0.971288	−0.485644	+	0.874157i	$0.661415\pi$
$774$	0	0
$775$	24.1180	0.866345
$776$	0	0
$777$	8.50508	0.305118
$778$	0	0
$779$	9.81298	0.351586
$780$	0	0
$781$	0.663222	0.0237319
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	8.14005	0.290531
$786$	0	0
$787$	−9.24689	−0.329616	−0.164808	−	0.986326i	$-0.552700\pi$
$787$	−9.24689	−0.329616	−0.164808	+	0.986326i	$0.552700\pi$
$788$	0	0
$789$	15.3928	0.548000
$790$	0	0
$791$	−6.78834	−0.241366
$792$	0	0
$793$	−17.4195	−0.618586
$794$	0	0
$795$	−4.19711	−0.148856
$796$	0	0
$797$	−29.9893	−1.06228	−0.531138	−	0.847285i	$-0.678235\pi$
$797$	−29.9893	−1.06228	−0.531138	+	0.847285i	$0.678235\pi$
$798$	0	0
$799$	40.3206	1.42644
$800$	0	0
$801$	3.60084	0.127230
$802$	0	0
$803$	36.7611	1.29727
$804$	0	0
$805$	−0.665612	−0.0234597
$806$	0	0
$807$	3.34502	0.117750
$808$	0	0
$809$	38.0904	1.33919	0.669593	−	0.742728i	$-0.266468\pi$
$809$	38.0904	1.33919	0.669593	+	0.742728i	$0.266468\pi$
$810$	0	0
$811$	12.1044	0.425044	0.212522	−	0.977156i	$-0.431832\pi$
$811$	12.1044	0.425044	0.212522	+	0.977156i	$0.431832\pi$
$812$	0	0
$813$	−15.6481	−0.548803
$814$	0	0
$815$	−0.500154	−0.0175196
$816$	0	0
$817$	4.25826	0.148978
$818$	0	0
$819$	−5.28483	−0.184667
$820$	0	0
$821$	26.6470	0.929985	0.464993	−	0.885315i	$-0.346057\pi$
$821$	26.6470	0.929985	0.464993	+	0.885315i	$0.346057\pi$
$822$	0	0
$823$	−28.9945	−1.01069	−0.505343	−	0.862919i	$-0.668634\pi$
$823$	−28.9945	−1.01069	−0.505343	+	0.862919i	$0.668634\pi$
$824$	0	0
$825$	−16.2645	−0.566257
$826$	0	0
$827$	−18.9682	−0.659590	−0.329795	−	0.944053i	$-0.606980\pi$
$827$	−18.9682	−0.659590	−0.329795	+	0.944053i	$0.606980\pi$
$828$	0	0
$829$	31.8321	1.10557	0.552787	−	0.833322i	$-0.313564\pi$
$829$	31.8321	1.10557	0.552787	+	0.833322i	$0.313564\pi$
$830$	0	0
$831$	−23.7842	−0.825065
$832$	0	0
$833$	19.6494	0.680813
$834$	0	0
$835$	1.20998	0.0418730
$836$	0	0
$837$	−5.08739	−0.175846
$838$	0	0
$839$	15.2293	0.525774	0.262887	−	0.964827i	$-0.415325\pi$
$839$	15.2293	0.525774	0.262887	+	0.964827i	$0.415325\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	19.6612	0.677166
$844$	0	0
$845$	1.70227	0.0585599
$846$	0	0
$847$	1.00695	0.0345993
$848$	0	0
$849$	29.5806	1.01521
$850$	0	0
$851$	−6.50604	−0.223024
$852$	0	0
$853$	27.8289	0.952843	0.476421	−	0.879217i	$-0.341934\pi$
$853$	27.8289	0.952843	0.476421	+	0.879217i	$0.341934\pi$
$854$	0	0
$855$	−0.862487	−0.0294964
$856$	0	0
$857$	26.5110	0.905598	0.452799	−	0.891613i	$-0.350426\pi$
$857$	26.5110	0.905598	0.452799	+	0.891613i	$0.350426\pi$
$858$	0	0
$859$	−15.5338	−0.530005	−0.265003	−	0.964248i	$-0.585373\pi$
$859$	−15.5338	−0.530005	−0.265003	+	0.964248i	$0.585373\pi$
$860$	0	0
$861$	−7.57302	−0.258088
$862$	0	0
$863$	7.18241	0.244492	0.122246	−	0.992500i	$-0.460990\pi$
$863$	7.18241	0.244492	0.122246	+	0.992500i	$0.460990\pi$
$864$	0	0
$865$	6.69438	0.227616
$866$	0	0
$867$	−3.20846	−0.108965
$868$	0	0
$869$	−27.4417	−0.930895
$870$	0	0
$871$	24.5320	0.831236
$872$	0	0
$873$	−1.88484	−0.0637921
$874$	0	0
$875$	−6.48356	−0.219184
$876$	0	0
$877$	42.2360	1.42621	0.713105	−	0.701058i	$-0.247288\pi$
$877$	42.2360	1.42621	0.713105	+	0.701058i	$0.247288\pi$
$878$	0	0
$879$	16.0867	0.542591
$880$	0	0
$881$	13.5643	0.456991	0.228496	−	0.973545i	$-0.426619\pi$
$881$	13.5643	0.456991	0.228496	+	0.973545i	$0.426619\pi$
$882$	0	0
$883$	−7.09760	−0.238853	−0.119427	−	0.992843i	$-0.538106\pi$
$883$	−7.09760	−0.238853	−0.119427	+	0.992843i	$0.538106\pi$
$884$	0	0
$885$	−1.24658	−0.0419033
$886$	0	0
$887$	17.1885	0.577132	0.288566	−	0.957460i	$-0.406821\pi$
$887$	17.1885	0.577132	0.288566	+	0.957460i	$0.406821\pi$
$888$	0	0
$889$	−15.6354	−0.524396
$890$	0	0
$891$	3.43078	0.114936
$892$	0	0
$893$	18.3913	0.615443
$894$	0	0
$895$	3.78062	0.126372
$896$	0	0
$897$	4.04268	0.134981
$898$	0	0
$899$	5.08739	0.169674
$900$	0	0
$901$	30.6124	1.01985
$902$	0	0
$903$	−3.28625	−0.109359
$904$	0	0
$905$	6.07924	0.202081
$906$	0	0
$907$	−12.2060	−0.405292	−0.202646	−	0.979252i	$-0.564954\pi$
$907$	−12.2060	−0.405292	−0.202646	+	0.979252i	$0.564954\pi$
$908$	0	0
$909$	−0.243655	−0.00808154
$910$	0	0
$911$	−14.2461	−0.471996	−0.235998	−	0.971754i	$-0.575836\pi$
$911$	−14.2461	−0.471996	−0.235998	+	0.971754i	$0.575836\pi$
$912$	0	0
$913$	−18.9056	−0.625683
$914$	0	0
$915$	2.19395	0.0725298
$916$	0	0
$917$	−24.9143	−0.822741
$918$	0	0
$919$	−58.7651	−1.93848	−0.969241	−	0.246115i	$-0.920846\pi$
$919$	−58.7651	−1.93848	−0.969241	+	0.246115i	$0.920846\pi$
$920$	0	0
$921$	32.3537	1.06609
$922$	0	0
$923$	−0.781510	−0.0257237
$924$	0	0
$925$	−30.8435	−1.01413
$926$	0	0
$927$	9.52669	0.312898
$928$	0	0
$929$	44.6357	1.46445	0.732225	−	0.681063i	$-0.238482\pi$
$929$	44.6357	1.46445	0.732225	+	0.681063i	$0.238482\pi$
$930$	0	0
$931$	8.96266	0.293739
$932$	0	0
$933$	2.78216	0.0910838
$934$	0	0
$935$	−6.48723	−0.212155
$936$	0	0
$937$	34.3892	1.12345	0.561723	−	0.827325i	$-0.310139\pi$
$937$	34.3892	1.12345	0.561723	+	0.827325i	$0.310139\pi$
$938$	0	0
$939$	−30.4052	−0.992236
$940$	0	0
$941$	50.2715	1.63880	0.819402	−	0.573219i	$-0.194306\pi$
$941$	50.2715	1.63880	0.819402	+	0.573219i	$0.194306\pi$
$942$	0	0
$943$	5.79306	0.188648
$944$	0	0
$945$	0.665612	0.0216524
$946$	0	0
$947$	−37.2293	−1.20979	−0.604896	−	0.796305i	$-0.706785\pi$
$947$	−37.2293	−1.20979	−0.604896	+	0.796305i	$0.706785\pi$
$948$	0	0
$949$	−43.3176	−1.40615
$950$	0	0
$951$	−9.55833	−0.309950
$952$	0	0
$953$	17.9919	0.582814	0.291407	−	0.956599i	$-0.405877\pi$
$953$	17.9919	0.582814	0.291407	+	0.956599i	$0.405877\pi$
$954$	0	0
$955$	2.86303	0.0926454
$956$	0	0
$957$	−3.43078	−0.110901
$958$	0	0
$959$	4.29230	0.138606
$960$	0	0
$961$	−5.11850	−0.165113
$962$	0	0
$963$	3.32544	0.107161
$964$	0	0
$965$	7.12671	0.229417
$966$	0	0
$967$	52.3076	1.68210	0.841049	−	0.540959i	$-0.181939\pi$
$967$	52.3076	1.68210	0.841049	+	0.540959i	$0.181939\pi$
$968$	0	0
$969$	6.29071	0.202087
$970$	0	0
$971$	−30.8754	−0.990840	−0.495420	−	0.868654i	$-0.664986\pi$
$971$	−30.8754	−0.990840	−0.495420	+	0.868654i	$0.664986\pi$
$972$	0	0
$973$	11.2613	0.361021
$974$	0	0
$975$	19.1653	0.613782
$976$	0	0
$977$	−12.7762	−0.408746	−0.204373	−	0.978893i	$-0.565516\pi$
$977$	−12.7762	−0.408746	−0.204373	+	0.978893i	$0.565516\pi$
$978$	0	0
$979$	12.3537	0.394826
$980$	0	0
$981$	−5.03360	−0.160711
$982$	0	0
$983$	−3.94722	−0.125897	−0.0629483	−	0.998017i	$-0.520050\pi$
$983$	−3.94722	−0.125897	−0.0629483	+	0.998017i	$0.520050\pi$
$984$	0	0
$985$	−1.36613	−0.0435284
$986$	0	0
$987$	−14.1932	−0.451776
$988$	0	0
$989$	2.51385	0.0799356
$990$	0	0
$991$	−36.9887	−1.17499	−0.587493	−	0.809230i	$-0.699885\pi$
$991$	−36.9887	−1.17499	−0.587493	+	0.809230i	$0.699885\pi$
$992$	0	0
$993$	−16.6393	−0.528032
$994$	0	0
$995$	−3.11257	−0.0986751
$996$	0	0
$997$	30.1581	0.955117	0.477559	−	0.878600i	$-0.341522\pi$
$997$	30.1581	0.955117	0.477559	+	0.878600i	$0.341522\pi$
$998$	0	0
$999$	6.50604	0.205842

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.f.1.5	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.f.1.5	✓	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.509166 q^{5} +1.30726 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.509166 q^{5} +1.30726 q^{7} +1.00000 q^{9} +3.43078 q^{11} -4.04268 q^{13} +0.509166 q^{15} -3.71370 q^{17} -1.69392 q^{19} +1.30726 q^{21} -1.00000 q^{23} -4.74075 q^{25} +1.00000 q^{27} -1.00000 q^{29} -5.08739 q^{31} +3.43078 q^{33} +0.665612 q^{35} +6.50604 q^{37} -4.04268 q^{39} -5.79306 q^{41} -2.51385 q^{43} +0.509166 q^{45} -10.8573 q^{47} -5.29108 q^{49} -3.71370 q^{51} -8.24311 q^{53} +1.74684 q^{55} -1.69392 q^{57} -2.44828 q^{59} +4.30891 q^{61} +1.30726 q^{63} -2.05840 q^{65} -6.06826 q^{67} -1.00000 q^{69} +0.193315 q^{71} +10.7151 q^{73} -4.74075 q^{75} +4.48492 q^{77} -7.99866 q^{79} +1.00000 q^{81} -5.51057 q^{83} -1.89089 q^{85} -1.00000 q^{87} +3.60084 q^{89} -5.28483 q^{91} -5.08739 q^{93} -0.862487 q^{95} -1.88484 q^{97} +3.43078 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9	\( 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9} - 8 q^{11} - q^{13} - q^{15} - 2 q^{17} - 11 q^{19} - 5 q^{21} - 9 q^{23} - 10 q^{25} + 9 q^{27} - 9 q^{29} - 8 q^{33} + q^{35} - 2 q^{37} - q^{39} - 3 q^{41} - 19 q^{43} - q^{45} - 3 q^{47} - 6 q^{49} - 2 q^{51} - 9 q^{53} - 7 q^{55} - 11 q^{57} - 2 q^{59} - 25 q^{61} - 5 q^{63} - 12 q^{65} - 20 q^{67} - 9 q^{69} + 9 q^{71} - 11 q^{73} - 10 q^{75} - 19 q^{77} + 4 q^{79} + 9 q^{81} - 9 q^{83} - 50 q^{85} - 9 q^{87} - 29 q^{89} - 38 q^{91} + 23 q^{95} - 43 q^{97} - 8 q^{99}+O(q^{100}) \) 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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