LMFDB - Embedded newform 8004.2.a.e.1.8

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} - 3x^{8} - 13x^{7} + 32x^{6} + 40x^{5} - 79x^{4} - 39x^{3} + 58x^{2} + 9x - 11 $ x^9 - 3x^8 - 13x^7 + 32x^6 + 40x^5 - 79x^4 - 39x^3 + 58x^2 + 9x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.511914$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.34732 q^{5} +2.91839 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.34732 q^{5} +2.91839 q^{7} +1.00000 q^{9} +3.69746 q^{11} -6.03680 q^{13} -1.34732 q^{15} +5.75034 q^{17} -6.30556 q^{19} -2.91839 q^{21} -1.00000 q^{23} -3.18473 q^{25} -1.00000 q^{27} +1.00000 q^{29} -0.671792 q^{31} -3.69746 q^{33} +3.93200 q^{35} -4.42860 q^{37} +6.03680 q^{39} -2.20091 q^{41} -8.58949 q^{43} +1.34732 q^{45} +0.549485 q^{47} +1.51697 q^{49} -5.75034 q^{51} -2.93706 q^{53} +4.98166 q^{55} +6.30556 q^{57} -4.20516 q^{59} -10.6176 q^{61} +2.91839 q^{63} -8.13349 q^{65} -3.98884 q^{67} +1.00000 q^{69} -9.08197 q^{71} -2.11062 q^{73} +3.18473 q^{75} +10.7906 q^{77} -12.8630 q^{79} +1.00000 q^{81} -6.50563 q^{83} +7.74754 q^{85} -1.00000 q^{87} +4.64297 q^{89} -17.6177 q^{91} +0.671792 q^{93} -8.49560 q^{95} -1.50834 q^{97} +3.69746 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9	$ 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9} - 2 q^{11} - 7 q^{13} + 3 q^{15} - q^{19} - 7 q^{21} - 9 q^{23} + 2 q^{25} - 9 q^{27} + 9 q^{29} + 8 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} + 7 q^{39} - 19 q^{41} - 3 q^{43} - 3 q^{45} - 3 q^{47} - 18 q^{49} - 17 q^{53} + 9 q^{55} + q^{57} - 10 q^{59} + q^{61} + 7 q^{63} - 16 q^{65} + 12 q^{67} + 9 q^{69} - 7 q^{71} + 13 q^{73} - 2 q^{75} - 15 q^{77} - 10 q^{79} + 9 q^{81} + 9 q^{83} - 6 q^{85} - 9 q^{87} - 5 q^{89} - 18 q^{91} - 8 q^{93} + 31 q^{95} - 7 q^{97} - 2 q^{99}+O(q^{100}) $ 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9 - 2 * q^11 - 7 * q^13 + 3 * q^15 - q^19 - 7 * q^21 - 9 * q^23 + 2 * q^25 - 9 * q^27 + 9 * q^29 + 8 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 + 7 * q^39 - 19 * q^41 - 3 * q^43 - 3 * q^45 - 3 * q^47 - 18 * q^49 - 17 * q^53 + 9 * q^55 + q^57 - 10 * q^59 + q^61 + 7 * q^63 - 16 * q^65 + 12 * q^67 + 9 * q^69 - 7 * q^71 + 13 * q^73 - 2 * q^75 - 15 * q^77 - 10 * q^79 + 9 * q^81 + 9 * q^83 - 6 * q^85 - 9 * q^87 - 5 * q^89 - 18 * q^91 - 8 * q^93 + 31 * q^95 - 7 * q^97 - 2 * 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
$5$	1.34732	0.602539	0.301270	−	0.953539i	$-0.402590\pi$
$5$	1.34732	0.602539	0.301270	+	0.953539i	$0.402590\pi$
$6$	0	0
$7$	2.91839	1.10305	0.551523	−	0.834160i	$-0.314047\pi$
$7$	2.91839	1.10305	0.551523	+	0.834160i	$0.314047\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.69746	1.11483	0.557413	−	0.830235i	$-0.311794\pi$
$11$	3.69746	1.11483	0.557413	+	0.830235i	$0.311794\pi$
$12$	0	0
$13$	−6.03680	−1.67431	−0.837153	−	0.546969i	$-0.815782\pi$
$13$	−6.03680	−1.67431	−0.837153	+	0.546969i	$0.815782\pi$
$14$	0	0
$15$	−1.34732	−0.347876
$16$	0	0
$17$	5.75034	1.39466	0.697331	−	0.716750i	$-0.254371\pi$
$17$	5.75034	1.39466	0.697331	+	0.716750i	$0.254371\pi$
$18$	0	0
$19$	−6.30556	−1.44659	−0.723297	−	0.690537i	$-0.757374\pi$
$19$	−6.30556	−1.44659	−0.723297	+	0.690537i	$0.757374\pi$
$20$	0	0
$21$	−2.91839	−0.636844
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.18473	−0.636946
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−0.671792	−0.120657	−0.0603287	−	0.998179i	$-0.519215\pi$
$31$	−0.671792	−0.120657	−0.0603287	+	0.998179i	$0.519215\pi$
$32$	0	0
$33$	−3.69746	−0.643645
$34$	0	0
$35$	3.93200	0.664629
$36$	0	0
$37$	−4.42860	−0.728058	−0.364029	−	0.931388i	$-0.618599\pi$
$37$	−4.42860	−0.728058	−0.364029	+	0.931388i	$0.618599\pi$
$38$	0	0
$39$	6.03680	0.966661
$40$	0	0
$41$	−2.20091	−0.343725	−0.171862	−	0.985121i	$-0.554978\pi$
$41$	−2.20091	−0.343725	−0.171862	+	0.985121i	$0.554978\pi$
$42$	0	0
$43$	−8.58949	−1.30988	−0.654942	−	0.755679i	$-0.727307\pi$
$43$	−8.58949	−1.30988	−0.654942	+	0.755679i	$0.727307\pi$
$44$	0	0
$45$	1.34732	0.200846
$46$	0	0
$47$	0.549485	0.0801506	0.0400753	−	0.999197i	$-0.487240\pi$
$47$	0.549485	0.0801506	0.0400753	+	0.999197i	$0.487240\pi$
$48$	0	0
$49$	1.51697	0.216711
$50$	0	0
$51$	−5.75034	−0.805208
$52$	0	0
$53$	−2.93706	−0.403436	−0.201718	−	0.979444i	$-0.564652\pi$
$53$	−2.93706	−0.403436	−0.201718	+	0.979444i	$0.564652\pi$
$54$	0	0
$55$	4.98166	0.671726
$56$	0	0
$57$	6.30556	0.835191
$58$	0	0
$59$	−4.20516	−0.547465	−0.273733	−	0.961806i	$-0.588258\pi$
$59$	−4.20516	−0.547465	−0.273733	+	0.961806i	$0.588258\pi$
$60$	0	0
$61$	−10.6176	−1.35945	−0.679724	−	0.733468i	$-0.737900\pi$
$61$	−10.6176	−1.35945	−0.679724	+	0.733468i	$0.737900\pi$
$62$	0	0
$63$	2.91839	0.367682
$64$	0	0
$65$	−8.13349	−1.00884
$66$	0	0
$67$	−3.98884	−0.487314	−0.243657	−	0.969861i	$-0.578347\pi$
$67$	−3.98884	−0.487314	−0.243657	+	0.969861i	$0.578347\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−9.08197	−1.07783	−0.538916	−	0.842359i	$-0.681166\pi$
$71$	−9.08197	−1.07783	−0.538916	+	0.842359i	$0.681166\pi$
$72$	0	0
$73$	−2.11062	−0.247029	−0.123515	−	0.992343i	$-0.539417\pi$
$73$	−2.11062	−0.247029	−0.123515	+	0.992343i	$0.539417\pi$
$74$	0	0
$75$	3.18473	0.367741
$76$	0	0
$77$	10.7906	1.22970
$78$	0	0
$79$	−12.8630	−1.44720	−0.723598	−	0.690222i	$-0.757513\pi$
$79$	−12.8630	−1.44720	−0.723598	+	0.690222i	$0.757513\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.50563	−0.714085	−0.357043	−	0.934088i	$-0.616215\pi$
$83$	−6.50563	−0.714085	−0.357043	+	0.934088i	$0.616215\pi$
$84$	0	0
$85$	7.74754	0.840338
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	4.64297	0.492154	0.246077	−	0.969250i	$-0.420858\pi$
$89$	4.64297	0.492154	0.246077	+	0.969250i	$0.420858\pi$
$90$	0	0
$91$	−17.6177	−1.84684
$92$	0	0
$93$	0.671792	0.0696616
$94$	0	0
$95$	−8.49560	−0.871630
$96$	0	0
$97$	−1.50834	−0.153148	−0.0765742	−	0.997064i	$-0.524398\pi$
$97$	−1.50834	−0.153148	−0.0765742	+	0.997064i	$0.524398\pi$
$98$	0	0
$99$	3.69746	0.371609
$100$	0	0
$101$	3.26567	0.324946	0.162473	−	0.986713i	$-0.448053\pi$
$101$	3.26567	0.324946	0.162473	+	0.986713i	$0.448053\pi$
$102$	0	0
$103$	−5.93651	−0.584942	−0.292471	−	0.956274i	$-0.594478\pi$
$103$	−5.93651	−0.584942	−0.292471	+	0.956274i	$0.594478\pi$
$104$	0	0
$105$	−3.93200	−0.383724
$106$	0	0
$107$	11.1575	1.07863	0.539317	−	0.842103i	$-0.318682\pi$
$107$	11.1575	1.07863	0.539317	+	0.842103i	$0.318682\pi$
$108$	0	0
$109$	−1.50930	−0.144564	−0.0722822	−	0.997384i	$-0.523028\pi$
$109$	−1.50930	−0.144564	−0.0722822	+	0.997384i	$0.523028\pi$
$110$	0	0
$111$	4.42860	0.420345
$112$	0	0
$113$	−4.54937	−0.427968	−0.213984	−	0.976837i	$-0.568644\pi$
$113$	−4.54937	−0.427968	−0.213984	+	0.976837i	$0.568644\pi$
$114$	0	0
$115$	−1.34732	−0.125638
$116$	0	0
$117$	−6.03680	−0.558102
$118$	0	0
$119$	16.7817	1.53838
$120$	0	0
$121$	2.67121	0.242837
$122$	0	0
$123$	2.20091	0.198450
$124$	0	0
$125$	−11.0274	−0.986325
$126$	0	0
$127$	−3.98492	−0.353605	−0.176802	−	0.984246i	$-0.556575\pi$
$127$	−3.98492	−0.353605	−0.176802	+	0.984246i	$0.556575\pi$
$128$	0	0
$129$	8.58949	0.756262
$130$	0	0
$131$	1.73658	0.151726	0.0758628	−	0.997118i	$-0.475829\pi$
$131$	1.73658	0.151726	0.0758628	+	0.997118i	$0.475829\pi$
$132$	0	0
$133$	−18.4020	−1.59566
$134$	0	0
$135$	−1.34732	−0.115959
$136$	0	0
$137$	−8.15181	−0.696456	−0.348228	−	0.937410i	$-0.613217\pi$
$137$	−8.15181	−0.696456	−0.348228	+	0.937410i	$0.613217\pi$
$138$	0	0
$139$	−0.954858	−0.0809900	−0.0404950	−	0.999180i	$-0.512893\pi$
$139$	−0.954858	−0.0809900	−0.0404950	+	0.999180i	$0.512893\pi$
$140$	0	0
$141$	−0.549485	−0.0462750
$142$	0	0
$143$	−22.3208	−1.86656
$144$	0	0
$145$	1.34732	0.111889
$146$	0	0
$147$	−1.51697	−0.125118
$148$	0	0
$149$	−0.579094	−0.0474412	−0.0237206	−	0.999719i	$-0.507551\pi$
$149$	−0.579094	−0.0474412	−0.0237206	+	0.999719i	$0.507551\pi$
$150$	0	0
$151$	16.8135	1.36827	0.684134	−	0.729357i	$-0.260181\pi$
$151$	16.8135	1.36827	0.684134	+	0.729357i	$0.260181\pi$
$152$	0	0
$153$	5.75034	0.464887
$154$	0	0
$155$	−0.905118	−0.0727008
$156$	0	0
$157$	11.7695	0.939307	0.469654	−	0.882851i	$-0.344379\pi$
$157$	11.7695	0.939307	0.469654	+	0.882851i	$0.344379\pi$
$158$	0	0
$159$	2.93706	0.232924
$160$	0	0
$161$	−2.91839	−0.230001
$162$	0	0
$163$	−1.97311	−0.154546	−0.0772729	−	0.997010i	$-0.524621\pi$
$163$	−1.97311	−0.154546	−0.0772729	+	0.997010i	$0.524621\pi$
$164$	0	0
$165$	−4.98166	−0.387821
$166$	0	0
$167$	15.3458	1.18750	0.593748	−	0.804651i	$-0.297648\pi$
$167$	15.3458	1.18750	0.593748	+	0.804651i	$0.297648\pi$
$168$	0	0
$169$	23.4429	1.80330
$170$	0	0
$171$	−6.30556	−0.482198
$172$	0	0
$173$	0.507138	0.0385570	0.0192785	−	0.999814i	$-0.493863\pi$
$173$	0.507138	0.0385570	0.0192785	+	0.999814i	$0.493863\pi$
$174$	0	0
$175$	−9.29428	−0.702581
$176$	0	0
$177$	4.20516	0.316079
$178$	0	0
$179$	17.0584	1.27501	0.637503	−	0.770448i	$-0.279967\pi$
$179$	17.0584	1.27501	0.637503	+	0.770448i	$0.279967\pi$
$180$	0	0
$181$	−8.69457	−0.646262	−0.323131	−	0.946354i	$-0.604735\pi$
$181$	−8.69457	−0.646262	−0.323131	+	0.946354i	$0.604735\pi$
$182$	0	0
$183$	10.6176	0.784877
$184$	0	0
$185$	−5.96674	−0.438684
$186$	0	0
$187$	21.2616	1.55480
$188$	0	0
$189$	−2.91839	−0.212281
$190$	0	0
$191$	7.97988	0.577404	0.288702	−	0.957419i	$-0.406776\pi$
$191$	7.97988	0.577404	0.288702	+	0.957419i	$0.406776\pi$
$192$	0	0
$193$	16.3801	1.17906	0.589531	−	0.807746i	$-0.299312\pi$
$193$	16.3801	1.17906	0.589531	+	0.807746i	$0.299312\pi$
$194$	0	0
$195$	8.13349	0.582451
$196$	0	0
$197$	−10.4305	−0.743140	−0.371570	−	0.928405i	$-0.621180\pi$
$197$	−10.4305	−0.743140	−0.371570	+	0.928405i	$0.621180\pi$
$198$	0	0
$199$	21.3102	1.51064	0.755321	−	0.655355i	$-0.227481\pi$
$199$	21.3102	1.51064	0.755321	+	0.655355i	$0.227481\pi$
$200$	0	0
$201$	3.98884	0.281351
$202$	0	0
$203$	2.91839	0.204831
$204$	0	0
$205$	−2.96533	−0.207108
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−23.3145	−1.61270
$210$	0	0
$211$	−22.2304	−1.53040	−0.765201	−	0.643791i	$-0.777360\pi$
$211$	−22.2304	−1.53040	−0.765201	+	0.643791i	$0.777360\pi$
$212$	0	0
$213$	9.08197	0.622287
$214$	0	0
$215$	−11.5728	−0.789257
$216$	0	0
$217$	−1.96055	−0.133091
$218$	0	0
$219$	2.11062	0.142622
$220$	0	0
$221$	−34.7136	−2.33509
$222$	0	0
$223$	3.34435	0.223955	0.111977	−	0.993711i	$-0.464282\pi$
$223$	3.34435	0.223955	0.111977	+	0.993711i	$0.464282\pi$
$224$	0	0
$225$	−3.18473	−0.212315
$226$	0	0
$227$	−15.7776	−1.04720	−0.523598	−	0.851965i	$-0.675411\pi$
$227$	−15.7776	−1.04720	−0.523598	+	0.851965i	$0.675411\pi$
$228$	0	0
$229$	4.93012	0.325791	0.162896	−	0.986643i	$-0.447917\pi$
$229$	4.93012	0.325791	0.162896	+	0.986643i	$0.447917\pi$
$230$	0	0
$231$	−10.7906	−0.709970
$232$	0	0
$233$	−2.88297	−0.188869	−0.0944347	−	0.995531i	$-0.530104\pi$
$233$	−2.88297	−0.188869	−0.0944347	+	0.995531i	$0.530104\pi$
$234$	0	0
$235$	0.740332	0.0482939
$236$	0	0
$237$	12.8630	0.835539
$238$	0	0
$239$	−13.1677	−0.851749	−0.425874	−	0.904782i	$-0.640033\pi$
$239$	−13.1677	−0.851749	−0.425874	+	0.904782i	$0.640033\pi$
$240$	0	0
$241$	−1.71060	−0.110189	−0.0550947	−	0.998481i	$-0.517546\pi$
$241$	−1.71060	−0.110189	−0.0550947	+	0.998481i	$0.517546\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.04385	0.130577
$246$	0	0
$247$	38.0654	2.42204
$248$	0	0
$249$	6.50563	0.412277
$250$	0	0
$251$	8.52631	0.538176	0.269088	−	0.963116i	$-0.413278\pi$
$251$	8.52631	0.538176	0.269088	+	0.963116i	$0.413278\pi$
$252$	0	0
$253$	−3.69746	−0.232457
$254$	0	0
$255$	−7.74754	−0.485170
$256$	0	0
$257$	9.31579	0.581103	0.290551	−	0.956859i	$-0.406161\pi$
$257$	9.31579	0.581103	0.290551	+	0.956859i	$0.406161\pi$
$258$	0	0
$259$	−12.9244	−0.803082
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	31.3704	1.93438	0.967190	−	0.254054i	$-0.0817642\pi$
$263$	31.3704	1.93438	0.967190	+	0.254054i	$0.0817642\pi$
$264$	0	0
$265$	−3.95715	−0.243086
$266$	0	0
$267$	−4.64297	−0.284145
$268$	0	0
$269$	−4.98936	−0.304206	−0.152103	−	0.988365i	$-0.548605\pi$
$269$	−4.98936	−0.304206	−0.152103	+	0.988365i	$0.548605\pi$
$270$	0	0
$271$	28.3242	1.72057	0.860285	−	0.509813i	$-0.170286\pi$
$271$	28.3242	1.72057	0.860285	+	0.509813i	$0.170286\pi$
$272$	0	0
$273$	17.6177	1.06627
$274$	0	0
$275$	−11.7754	−0.710084
$276$	0	0
$277$	−4.33021	−0.260177	−0.130088	−	0.991502i	$-0.541526\pi$
$277$	−4.33021	−0.260177	−0.130088	+	0.991502i	$0.541526\pi$
$278$	0	0
$279$	−0.671792	−0.0402191
$280$	0	0
$281$	−9.22411	−0.550264	−0.275132	−	0.961406i	$-0.588722\pi$
$281$	−9.22411	−0.550264	−0.275132	+	0.961406i	$0.588722\pi$
$282$	0	0
$283$	20.4601	1.21623	0.608114	−	0.793850i	$-0.291927\pi$
$283$	20.4601	1.21623	0.608114	+	0.793850i	$0.291927\pi$
$284$	0	0
$285$	8.49560	0.503236
$286$	0	0
$287$	−6.42311	−0.379144
$288$	0	0
$289$	16.0664	0.945080
$290$	0	0
$291$	1.50834	0.0884203
$292$	0	0
$293$	4.53864	0.265150	0.132575	−	0.991173i	$-0.457675\pi$
$293$	4.53864	0.265150	0.132575	+	0.991173i	$0.457675\pi$
$294$	0	0
$295$	−5.66569	−0.329869
$296$	0	0
$297$	−3.69746	−0.214548
$298$	0	0
$299$	6.03680	0.349117
$300$	0	0
$301$	−25.0674	−1.44486
$302$	0	0
$303$	−3.26567	−0.187608
$304$	0	0
$305$	−14.3053	−0.819120
$306$	0	0
$307$	−17.5891	−1.00387	−0.501933	−	0.864907i	$-0.667378\pi$
$307$	−17.5891	−1.00387	−0.501933	+	0.864907i	$0.667378\pi$
$308$	0	0
$309$	5.93651	0.337717
$310$	0	0
$311$	−25.8397	−1.46524	−0.732618	−	0.680640i	$-0.761702\pi$
$311$	−25.8397	−1.46524	−0.732618	+	0.680640i	$0.761702\pi$
$312$	0	0
$313$	−7.28452	−0.411745	−0.205873	−	0.978579i	$-0.566003\pi$
$313$	−7.28452	−0.411745	−0.205873	+	0.978579i	$0.566003\pi$
$314$	0	0
$315$	3.93200	0.221543
$316$	0	0
$317$	−8.21892	−0.461620	−0.230810	−	0.972999i	$-0.574138\pi$
$317$	−8.21892	−0.461620	−0.230810	+	0.972999i	$0.574138\pi$
$318$	0	0
$319$	3.69746	0.207018
$320$	0	0
$321$	−11.1575	−0.622750
$322$	0	0
$323$	−36.2591	−2.01751
$324$	0	0
$325$	19.2256	1.06644
$326$	0	0
$327$	1.50930	0.0834643
$328$	0	0
$329$	1.60361	0.0884098
$330$	0	0
$331$	3.55431	0.195363	0.0976813	−	0.995218i	$-0.468857\pi$
$331$	3.55431	0.195363	0.0976813	+	0.995218i	$0.468857\pi$
$332$	0	0
$333$	−4.42860	−0.242686
$334$	0	0
$335$	−5.37424	−0.293626
$336$	0	0
$337$	−20.7733	−1.13160	−0.565798	−	0.824544i	$-0.691432\pi$
$337$	−20.7733	−1.13160	−0.565798	+	0.824544i	$0.691432\pi$
$338$	0	0
$339$	4.54937	0.247088
$340$	0	0
$341$	−2.48392	−0.134512
$342$	0	0
$343$	−16.0016	−0.864004
$344$	0	0
$345$	1.34732	0.0725372
$346$	0	0
$347$	−7.56583	−0.406155	−0.203078	−	0.979163i	$-0.565094\pi$
$347$	−7.56583	−0.406155	−0.203078	+	0.979163i	$0.565094\pi$
$348$	0	0
$349$	9.14262	0.489393	0.244697	−	0.969600i	$-0.421312\pi$
$349$	9.14262	0.489393	0.244697	+	0.969600i	$0.421312\pi$
$350$	0	0
$351$	6.03680	0.322220
$352$	0	0
$353$	21.0243	1.11901	0.559506	−	0.828826i	$-0.310991\pi$
$353$	21.0243	1.11901	0.559506	+	0.828826i	$0.310991\pi$
$354$	0	0
$355$	−12.2363	−0.649436
$356$	0	0
$357$	−16.7817	−0.888182
$358$	0	0
$359$	18.8797	0.996433	0.498216	−	0.867053i	$-0.333989\pi$
$359$	18.8797	0.996433	0.498216	+	0.867053i	$0.333989\pi$
$360$	0	0
$361$	20.7600	1.09263
$362$	0	0
$363$	−2.67121	−0.140202
$364$	0	0
$365$	−2.84368	−0.148845
$366$	0	0
$367$	20.2455	1.05681	0.528404	−	0.848993i	$-0.322791\pi$
$367$	20.2455	1.05681	0.528404	+	0.848993i	$0.322791\pi$
$368$	0	0
$369$	−2.20091	−0.114575
$370$	0	0
$371$	−8.57147	−0.445008
$372$	0	0
$373$	−8.30473	−0.430003	−0.215001	−	0.976614i	$-0.568976\pi$
$373$	−8.30473	−0.430003	−0.215001	+	0.976614i	$0.568976\pi$
$374$	0	0
$375$	11.0274	0.569455
$376$	0	0
$377$	−6.03680	−0.310911
$378$	0	0
$379$	−23.1184	−1.18751	−0.593756	−	0.804645i	$-0.702356\pi$
$379$	−23.1184	−1.18751	−0.593756	+	0.804645i	$0.702356\pi$
$380$	0	0
$381$	3.98492	0.204154
$382$	0	0
$383$	−17.7677	−0.907889	−0.453945	−	0.891030i	$-0.649984\pi$
$383$	−17.7677	−0.907889	−0.453945	+	0.891030i	$0.649984\pi$
$384$	0	0
$385$	14.5384	0.740945
$386$	0	0
$387$	−8.58949	−0.436628
$388$	0	0
$389$	−4.16020	−0.210930	−0.105465	−	0.994423i	$-0.533633\pi$
$389$	−4.16020	−0.210930	−0.105465	+	0.994423i	$0.533633\pi$
$390$	0	0
$391$	−5.75034	−0.290807
$392$	0	0
$393$	−1.73658	−0.0875988
$394$	0	0
$395$	−17.3305	−0.871992
$396$	0	0
$397$	−1.81500	−0.0910922	−0.0455461	−	0.998962i	$-0.514503\pi$
$397$	−1.81500	−0.0910922	−0.0455461	+	0.998962i	$0.514503\pi$
$398$	0	0
$399$	18.4020	0.921255
$400$	0	0
$401$	−4.76103	−0.237754	−0.118877	−	0.992909i	$-0.537929\pi$
$401$	−4.76103	−0.237754	−0.118877	+	0.992909i	$0.537929\pi$
$402$	0	0
$403$	4.05547	0.202017
$404$	0	0
$405$	1.34732	0.0669488
$406$	0	0
$407$	−16.3746	−0.811658
$408$	0	0
$409$	3.00888	0.148780	0.0743898	−	0.997229i	$-0.476299\pi$
$409$	3.00888	0.148780	0.0743898	+	0.997229i	$0.476299\pi$
$410$	0	0
$411$	8.15181	0.402099
$412$	0	0
$413$	−12.2723	−0.603879
$414$	0	0
$415$	−8.76515	−0.430265
$416$	0	0
$417$	0.954858	0.0467596
$418$	0	0
$419$	6.26848	0.306236	0.153118	−	0.988208i	$-0.451069\pi$
$419$	6.26848	0.306236	0.153118	+	0.988208i	$0.451069\pi$
$420$	0	0
$421$	3.78433	0.184437	0.0922184	−	0.995739i	$-0.470604\pi$
$421$	3.78433	0.184437	0.0922184	+	0.995739i	$0.470604\pi$
$422$	0	0
$423$	0.549485	0.0267169
$424$	0	0
$425$	−18.3133	−0.888324
$426$	0	0
$427$	−30.9863	−1.49953
$428$	0	0
$429$	22.3208	1.07766
$430$	0	0
$431$	29.6641	1.42887	0.714435	−	0.699702i	$-0.246684\pi$
$431$	29.6641	1.42887	0.714435	+	0.699702i	$0.246684\pi$
$432$	0	0
$433$	−27.8508	−1.33843	−0.669213	−	0.743071i	$-0.733369\pi$
$433$	−27.8508	−1.33843	−0.669213	+	0.743071i	$0.733369\pi$
$434$	0	0
$435$	−1.34732	−0.0645990
$436$	0	0
$437$	6.30556	0.301636
$438$	0	0
$439$	−12.0457	−0.574910	−0.287455	−	0.957794i	$-0.592809\pi$
$439$	−12.0457	−0.574910	−0.287455	+	0.957794i	$0.592809\pi$
$440$	0	0
$441$	1.51697	0.0722369
$442$	0	0
$443$	1.13889	0.0541102	0.0270551	−	0.999634i	$-0.491387\pi$
$443$	1.13889	0.0541102	0.0270551	+	0.999634i	$0.491387\pi$
$444$	0	0
$445$	6.25556	0.296542
$446$	0	0
$447$	0.579094	0.0273902
$448$	0	0
$449$	−16.5017	−0.778764	−0.389382	−	0.921076i	$-0.627311\pi$
$449$	−16.5017	−0.778764	−0.389382	+	0.921076i	$0.627311\pi$
$450$	0	0
$451$	−8.13778	−0.383193
$452$	0	0
$453$	−16.8135	−0.789969
$454$	0	0
$455$	−23.7367	−1.11279
$456$	0	0
$457$	0.941855	0.0440581	0.0220291	−	0.999757i	$-0.492987\pi$
$457$	0.941855	0.0440581	0.0220291	+	0.999757i	$0.492987\pi$
$458$	0	0
$459$	−5.75034	−0.268403
$460$	0	0
$461$	−7.61417	−0.354627	−0.177314	−	0.984154i	$-0.556741\pi$
$461$	−7.61417	−0.354627	−0.177314	+	0.984154i	$0.556741\pi$
$462$	0	0
$463$	3.38347	0.157243	0.0786217	−	0.996905i	$-0.474948\pi$
$463$	3.38347	0.157243	0.0786217	+	0.996905i	$0.474948\pi$
$464$	0	0
$465$	0.905118	0.0419739
$466$	0	0
$467$	22.0548	1.02057	0.510286	−	0.860005i	$-0.329540\pi$
$467$	22.0548	1.02057	0.510286	+	0.860005i	$0.329540\pi$
$468$	0	0
$469$	−11.6410	−0.537530
$470$	0	0
$471$	−11.7695	−0.542309
$472$	0	0
$473$	−31.7593	−1.46029
$474$	0	0
$475$	20.0815	0.921403
$476$	0	0
$477$	−2.93706	−0.134479
$478$	0	0
$479$	22.8852	1.04565	0.522825	−	0.852440i	$-0.324878\pi$
$479$	22.8852	1.04565	0.522825	+	0.852440i	$0.324878\pi$
$480$	0	0
$481$	26.7346	1.21899
$482$	0	0
$483$	2.91839	0.132791
$484$	0	0
$485$	−2.03221	−0.0922780
$486$	0	0
$487$	32.7383	1.48351	0.741756	−	0.670670i	$-0.233993\pi$
$487$	32.7383	1.48351	0.741756	+	0.670670i	$0.233993\pi$
$488$	0	0
$489$	1.97311	0.0892271
$490$	0	0
$491$	−11.7397	−0.529805	−0.264902	−	0.964275i	$-0.585340\pi$
$491$	−11.7397	−0.529805	−0.264902	+	0.964275i	$0.585340\pi$
$492$	0	0
$493$	5.75034	0.258982
$494$	0	0
$495$	4.98166	0.223909
$496$	0	0
$497$	−26.5047	−1.18890
$498$	0	0
$499$	−33.6706	−1.50730	−0.753651	−	0.657275i	$-0.771709\pi$
$499$	−33.6706	−1.50730	−0.753651	+	0.657275i	$0.771709\pi$
$500$	0	0
$501$	−15.3458	−0.685601
$502$	0	0
$503$	−29.0415	−1.29490	−0.647448	−	0.762110i	$-0.724164\pi$
$503$	−29.0415	−1.29490	−0.647448	+	0.762110i	$0.724164\pi$
$504$	0	0
$505$	4.39990	0.195793
$506$	0	0
$507$	−23.4429	−1.04114
$508$	0	0
$509$	−34.1695	−1.51454	−0.757269	−	0.653103i	$-0.773467\pi$
$509$	−34.1695	−1.51454	−0.757269	+	0.653103i	$0.773467\pi$
$510$	0	0
$511$	−6.15960	−0.272485
$512$	0	0
$513$	6.30556	0.278397
$514$	0	0
$515$	−7.99838	−0.352451
$516$	0	0
$517$	2.03170	0.0893540
$518$	0	0
$519$	−0.507138	−0.0222609
$520$	0	0
$521$	−17.6254	−0.772185	−0.386092	−	0.922460i	$-0.626175\pi$
$521$	−17.6254	−0.772185	−0.386092	+	0.922460i	$0.626175\pi$
$522$	0	0
$523$	2.67968	0.117174	0.0585870	−	0.998282i	$-0.481340\pi$
$523$	2.67968	0.117174	0.0585870	+	0.998282i	$0.481340\pi$
$524$	0	0
$525$	9.29428	0.405635
$526$	0	0
$527$	−3.86303	−0.168276
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−4.20516	−0.182488
$532$	0	0
$533$	13.2865	0.575500
$534$	0	0
$535$	15.0327	0.649920
$536$	0	0
$537$	−17.0584	−0.736125
$538$	0	0
$539$	5.60895	0.241595
$540$	0	0
$541$	−0.691956	−0.0297495	−0.0148748	−	0.999889i	$-0.504735\pi$
$541$	−0.691956	−0.0297495	−0.0148748	+	0.999889i	$0.504735\pi$
$542$	0	0
$543$	8.69457	0.373120
$544$	0	0
$545$	−2.03350	−0.0871058
$546$	0	0
$547$	14.2353	0.608659	0.304330	−	0.952567i	$-0.401568\pi$
$547$	14.2353	0.608659	0.304330	+	0.952567i	$0.401568\pi$
$548$	0	0
$549$	−10.6176	−0.453149
$550$	0	0
$551$	−6.30556	−0.268626
$552$	0	0
$553$	−37.5391	−1.59632
$554$	0	0
$555$	5.96674	0.253274
$556$	0	0
$557$	−5.54928	−0.235130	−0.117565	−	0.993065i	$-0.537509\pi$
$557$	−5.54928	−0.235130	−0.117565	+	0.993065i	$0.537509\pi$
$558$	0	0
$559$	51.8530	2.19315
$560$	0	0
$561$	−21.2616	−0.897667
$562$	0	0
$563$	26.6802	1.12444	0.562219	−	0.826989i	$-0.309948\pi$
$563$	26.6802	1.12444	0.562219	+	0.826989i	$0.309948\pi$
$564$	0	0
$565$	−6.12945	−0.257868
$566$	0	0
$567$	2.91839	0.122561
$568$	0	0
$569$	−8.81271	−0.369448	−0.184724	−	0.982790i	$-0.559139\pi$
$569$	−8.81271	−0.369448	−0.184724	+	0.982790i	$0.559139\pi$
$570$	0	0
$571$	−41.6835	−1.74440	−0.872200	−	0.489150i	$-0.837307\pi$
$571$	−41.6835	−1.74440	−0.872200	+	0.489150i	$0.837307\pi$
$572$	0	0
$573$	−7.97988	−0.333364
$574$	0	0
$575$	3.18473	0.132812
$576$	0	0
$577$	−10.8443	−0.451452	−0.225726	−	0.974191i	$-0.572475\pi$
$577$	−10.8443	−0.451452	−0.225726	+	0.974191i	$0.572475\pi$
$578$	0	0
$579$	−16.3801	−0.680732
$580$	0	0
$581$	−18.9859	−0.787669
$582$	0	0
$583$	−10.8597	−0.449761
$584$	0	0
$585$	−8.13349	−0.336278
$586$	0	0
$587$	0.0198519	0.000819377 0	0.000409689	−	1.00000i	$-0.499870\pi$
$587$	0.0198519	0.000819377 0	0.000409689		1.00000i	$0.499870\pi$
$588$	0	0
$589$	4.23602	0.174542
$590$	0	0
$591$	10.4305	0.429052
$592$	0	0
$593$	6.59076	0.270650	0.135325	−	0.990801i	$-0.456792\pi$
$593$	6.59076	0.270650	0.135325	+	0.990801i	$0.456792\pi$
$594$	0	0
$595$	22.6103	0.926932
$596$	0	0
$597$	−21.3102	−0.872169
$598$	0	0
$599$	31.4616	1.28549	0.642743	−	0.766082i	$-0.277796\pi$
$599$	31.4616	1.28549	0.642743	+	0.766082i	$0.277796\pi$
$600$	0	0
$601$	−29.9067	−1.21992	−0.609961	−	0.792432i	$-0.708815\pi$
$601$	−29.9067	−1.21992	−0.609961	+	0.792432i	$0.708815\pi$
$602$	0	0
$603$	−3.98884	−0.162438
$604$	0	0
$605$	3.59897	0.146319
$606$	0	0
$607$	−5.08762	−0.206500	−0.103250	−	0.994655i	$-0.532924\pi$
$607$	−5.08762	−0.206500	−0.103250	+	0.994655i	$0.532924\pi$
$608$	0	0
$609$	−2.91839	−0.118259
$610$	0	0
$611$	−3.31713	−0.134197
$612$	0	0
$613$	35.8235	1.44690	0.723449	−	0.690378i	$-0.242556\pi$
$613$	35.8235	1.44690	0.723449	+	0.690378i	$0.242556\pi$
$614$	0	0
$615$	2.96533	0.119574
$616$	0	0
$617$	15.2128	0.612442	0.306221	−	0.951960i	$-0.400935\pi$
$617$	15.2128	0.612442	0.306221	+	0.951960i	$0.400935\pi$
$618$	0	0
$619$	30.5644	1.22849	0.614244	−	0.789116i	$-0.289461\pi$
$619$	30.5644	1.22849	0.614244	+	0.789116i	$0.289461\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	13.5500	0.542869
$624$	0	0
$625$	1.06617	0.0426470
$626$	0	0
$627$	23.3145	0.931093
$628$	0	0
$629$	−25.4660	−1.01539
$630$	0	0
$631$	−9.73458	−0.387527	−0.193764	−	0.981048i	$-0.562070\pi$
$631$	−9.73458	−0.387527	−0.193764	+	0.981048i	$0.562070\pi$
$632$	0	0
$633$	22.2304	0.883578
$634$	0	0
$635$	−5.36896	−0.213061
$636$	0	0
$637$	−9.15767	−0.362840
$638$	0	0
$639$	−9.08197	−0.359277
$640$	0	0
$641$	−38.9947	−1.54020	−0.770099	−	0.637924i	$-0.779793\pi$
$641$	−38.9947	−1.54020	−0.770099	+	0.637924i	$0.779793\pi$
$642$	0	0
$643$	4.49902	0.177424	0.0887120	−	0.996057i	$-0.471725\pi$
$643$	4.49902	0.177424	0.0887120	+	0.996057i	$0.471725\pi$
$644$	0	0
$645$	11.5728	0.455678
$646$	0	0
$647$	−5.20597	−0.204668	−0.102334	−	0.994750i	$-0.532631\pi$
$647$	−5.20597	−0.204668	−0.102334	+	0.994750i	$0.532631\pi$
$648$	0	0
$649$	−15.5484	−0.610328
$650$	0	0
$651$	1.96055	0.0768400
$652$	0	0
$653$	−1.19073	−0.0465968	−0.0232984	−	0.999729i	$-0.507417\pi$
$653$	−1.19073	−0.0465968	−0.0232984	+	0.999729i	$0.507417\pi$
$654$	0	0
$655$	2.33972	0.0914206
$656$	0	0
$657$	−2.11062	−0.0823431
$658$	0	0
$659$	42.9964	1.67490	0.837450	−	0.546514i	$-0.184045\pi$
$659$	42.9964	1.67490	0.837450	+	0.546514i	$0.184045\pi$
$660$	0	0
$661$	−42.7967	−1.66460	−0.832300	−	0.554325i	$-0.812976\pi$
$661$	−42.7967	−1.66460	−0.832300	+	0.554325i	$0.812976\pi$
$662$	0	0
$663$	34.7136	1.34816
$664$	0	0
$665$	−24.7934	−0.961448
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−3.34435	−0.129300
$670$	0	0
$671$	−39.2582	−1.51555
$672$	0	0
$673$	−24.7493	−0.954015	−0.477007	−	0.878899i	$-0.658279\pi$
$673$	−24.7493	−0.954015	−0.477007	+	0.878899i	$0.658279\pi$
$674$	0	0
$675$	3.18473	0.122580
$676$	0	0
$677$	−24.9081	−0.957296	−0.478648	−	0.878007i	$-0.658873\pi$
$677$	−24.9081	−0.957296	−0.478648	+	0.878007i	$0.658873\pi$
$678$	0	0
$679$	−4.40191	−0.168930
$680$	0	0
$681$	15.7776	0.604599
$682$	0	0
$683$	16.7745	0.641860	0.320930	−	0.947103i	$-0.396005\pi$
$683$	16.7745	0.641860	0.320930	+	0.947103i	$0.396005\pi$
$684$	0	0
$685$	−10.9831	−0.419642
$686$	0	0
$687$	−4.93012	−0.188096
$688$	0	0
$689$	17.7304	0.675475
$690$	0	0
$691$	−2.32052	−0.0882768	−0.0441384	−	0.999025i	$-0.514054\pi$
$691$	−2.32052	−0.0882768	−0.0441384	+	0.999025i	$0.514054\pi$
$692$	0	0
$693$	10.7906	0.409901
$694$	0	0
$695$	−1.28650	−0.0487997
$696$	0	0
$697$	−12.6560	−0.479380
$698$	0	0
$699$	2.88297	0.109044
$700$	0	0
$701$	5.11535	0.193204	0.0966020	−	0.995323i	$-0.469203\pi$
$701$	5.11535	0.193204	0.0966020	+	0.995323i	$0.469203\pi$
$702$	0	0
$703$	27.9248	1.05320
$704$	0	0
$705$	−0.740332	−0.0278825
$706$	0	0
$707$	9.53049	0.358431
$708$	0	0
$709$	−30.4887	−1.14503	−0.572514	−	0.819895i	$-0.694032\pi$
$709$	−30.4887	−1.14503	−0.572514	+	0.819895i	$0.694032\pi$
$710$	0	0
$711$	−12.8630	−0.482399
$712$	0	0
$713$	0.671792	0.0251588
$714$	0	0
$715$	−30.0733	−1.12468
$716$	0	0
$717$	13.1677	0.491757
$718$	0	0
$719$	−7.50338	−0.279829	−0.139914	−	0.990164i	$-0.544683\pi$
$719$	−7.50338	−0.279829	−0.139914	+	0.990164i	$0.544683\pi$
$720$	0	0
$721$	−17.3250	−0.645218
$722$	0	0
$723$	1.71060	0.0636179
$724$	0	0
$725$	−3.18473	−0.118278
$726$	0	0
$727$	−25.1529	−0.932870	−0.466435	−	0.884555i	$-0.654462\pi$
$727$	−25.1529	−0.932870	−0.466435	+	0.884555i	$0.654462\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−49.3925	−1.82685
$732$	0	0
$733$	−18.5387	−0.684742	−0.342371	−	0.939565i	$-0.611230\pi$
$733$	−18.5387	−0.684742	−0.342371	+	0.939565i	$0.611230\pi$
$734$	0	0
$735$	−2.04385	−0.0753885
$736$	0	0
$737$	−14.7486	−0.543271
$738$	0	0
$739$	17.1856	0.632184	0.316092	−	0.948729i	$-0.397629\pi$
$739$	17.1856	0.632184	0.316092	+	0.948729i	$0.397629\pi$
$740$	0	0
$741$	−38.0654	−1.39837
$742$	0	0
$743$	−27.9275	−1.02456	−0.512280	−	0.858818i	$-0.671199\pi$
$743$	−27.9275	−1.02456	−0.512280	+	0.858818i	$0.671199\pi$
$744$	0	0
$745$	−0.780224	−0.0285852
$746$	0	0
$747$	−6.50563	−0.238028
$748$	0	0
$749$	32.5618	1.18978
$750$	0	0
$751$	−44.7691	−1.63365	−0.816824	−	0.576888i	$-0.804267\pi$
$751$	−44.7691	−1.63365	−0.816824	+	0.576888i	$0.804267\pi$
$752$	0	0
$753$	−8.52631	−0.310716
$754$	0	0
$755$	22.6532	0.824435
$756$	0	0
$757$	38.2317	1.38955	0.694777	−	0.719225i	$-0.255503\pi$
$757$	38.2317	1.38955	0.694777	+	0.719225i	$0.255503\pi$
$758$	0	0
$759$	3.69746	0.134209
$760$	0	0
$761$	−40.0194	−1.45070	−0.725350	−	0.688380i	$-0.758322\pi$
$761$	−40.0194	−1.45070	−0.725350	+	0.688380i	$0.758322\pi$
$762$	0	0
$763$	−4.40471	−0.159461
$764$	0	0
$765$	7.74754	0.280113
$766$	0	0
$767$	25.3857	0.916624
$768$	0	0
$769$	−33.6601	−1.21381	−0.606906	−	0.794773i	$-0.707590\pi$
$769$	−33.6601	−1.21381	−0.606906	+	0.794773i	$0.707590\pi$
$770$	0	0
$771$	−9.31579	−0.335500
$772$	0	0
$773$	2.00330	0.0720536	0.0360268	−	0.999351i	$-0.488530\pi$
$773$	2.00330	0.0720536	0.0360268	+	0.999351i	$0.488530\pi$
$774$	0	0
$775$	2.13948	0.0768523
$776$	0	0
$777$	12.9244	0.463659
$778$	0	0
$779$	13.8780	0.497230
$780$	0	0
$781$	−33.5802	−1.20160
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	15.8572	0.565969
$786$	0	0
$787$	−0.0531304	−0.00189389	−0.000946946	−	1.00000i	$-0.500301\pi$
$787$	−0.0531304	−0.00189389	−0.000946946		1.00000i	$0.500301\pi$
$788$	0	0
$789$	−31.3704	−1.11681
$790$	0	0
$791$	−13.2768	−0.472069
$792$	0	0
$793$	64.0964	2.27613
$794$	0	0
$795$	3.95715	0.140346
$796$	0	0
$797$	31.2146	1.10568	0.552839	−	0.833288i	$-0.313545\pi$
$797$	31.2146	1.10568	0.552839	+	0.833288i	$0.313545\pi$
$798$	0	0
$799$	3.15972	0.111783
$800$	0	0
$801$	4.64297	0.164051
$802$	0	0
$803$	−7.80393	−0.275395
$804$	0	0
$805$	−3.93200	−0.138585
$806$	0	0
$807$	4.98936	0.175634
$808$	0	0
$809$	−0.153360	−0.00539185	−0.00269592	−	0.999996i	$-0.500858\pi$
$809$	−0.153360	−0.00539185	−0.00269592	+	0.999996i	$0.500858\pi$
$810$	0	0
$811$	11.3984	0.400253	0.200126	−	0.979770i	$-0.435865\pi$
$811$	11.3984	0.400253	0.200126	+	0.979770i	$0.435865\pi$
$812$	0	0
$813$	−28.3242	−0.993371
$814$	0	0
$815$	−2.65841	−0.0931199
$816$	0	0
$817$	54.1615	1.89487
$818$	0	0
$819$	−17.6177	−0.615612
$820$	0	0
$821$	−18.9338	−0.660795	−0.330397	−	0.943842i	$-0.607183\pi$
$821$	−18.9338	−0.660795	−0.330397	+	0.943842i	$0.607183\pi$
$822$	0	0
$823$	42.4815	1.48081	0.740405	−	0.672161i	$-0.234634\pi$
$823$	42.4815	1.48081	0.740405	+	0.672161i	$0.234634\pi$
$824$	0	0
$825$	11.7754	0.409967
$826$	0	0
$827$	45.4445	1.58026	0.790129	−	0.612940i	$-0.210013\pi$
$827$	45.4445	1.58026	0.790129	+	0.612940i	$0.210013\pi$
$828$	0	0
$829$	49.6222	1.72345	0.861725	−	0.507375i	$-0.169384\pi$
$829$	49.6222	1.72345	0.861725	+	0.507375i	$0.169384\pi$
$830$	0	0
$831$	4.33021	0.150213
$832$	0	0
$833$	8.72311	0.302238
$834$	0	0
$835$	20.6757	0.715513
$836$	0	0
$837$	0.671792	0.0232205
$838$	0	0
$839$	53.2719	1.83915	0.919576	−	0.392913i	$-0.128533\pi$
$839$	53.2719	1.83915	0.919576	+	0.392913i	$0.128533\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	9.22411	0.317695
$844$	0	0
$845$	31.5851	1.08656
$846$	0	0
$847$	7.79561	0.267860
$848$	0	0
$849$	−20.4601	−0.702189
$850$	0	0
$851$	4.42860	0.151811
$852$	0	0
$853$	24.3344	0.833194	0.416597	−	0.909091i	$-0.363223\pi$
$853$	24.3344	0.833194	0.416597	+	0.909091i	$0.363223\pi$
$854$	0	0
$855$	−8.49560	−0.290543
$856$	0	0
$857$	4.88014	0.166702	0.0833512	−	0.996520i	$-0.473438\pi$
$857$	4.88014	0.166702	0.0833512	+	0.996520i	$0.473438\pi$
$858$	0	0
$859$	−42.3449	−1.44479	−0.722394	−	0.691481i	$-0.756958\pi$
$859$	−42.3449	−1.44479	−0.722394	+	0.691481i	$0.756958\pi$
$860$	0	0
$861$	6.42311	0.218899
$862$	0	0
$863$	−55.7214	−1.89678	−0.948389	−	0.317110i	$-0.897287\pi$
$863$	−55.7214	−1.89678	−0.948389	+	0.317110i	$0.897287\pi$
$864$	0	0
$865$	0.683277	0.0232321
$866$	0	0
$867$	−16.0664	−0.545642
$868$	0	0
$869$	−47.5603	−1.61337
$870$	0	0
$871$	24.0798	0.815913
$872$	0	0
$873$	−1.50834	−0.0510495
$874$	0	0
$875$	−32.1823	−1.08796
$876$	0	0
$877$	−41.5958	−1.40459	−0.702295	−	0.711886i	$-0.747841\pi$
$877$	−41.5958	−1.40459	−0.702295	+	0.711886i	$0.747841\pi$
$878$	0	0
$879$	−4.53864	−0.153084
$880$	0	0
$881$	51.9100	1.74889	0.874446	−	0.485122i	$-0.161225\pi$
$881$	51.9100	1.74889	0.874446	+	0.485122i	$0.161225\pi$
$882$	0	0
$883$	−8.84338	−0.297604	−0.148802	−	0.988867i	$-0.547542\pi$
$883$	−8.84338	−0.297604	−0.148802	+	0.988867i	$0.547542\pi$
$884$	0	0
$885$	5.66569	0.190450
$886$	0	0
$887$	14.7401	0.494925	0.247462	−	0.968897i	$-0.420403\pi$
$887$	14.7401	0.494925	0.247462	+	0.968897i	$0.420403\pi$
$888$	0	0
$889$	−11.6295	−0.390042
$890$	0	0
$891$	3.69746	0.123870
$892$	0	0
$893$	−3.46481	−0.115945
$894$	0	0
$895$	22.9831	0.768241
$896$	0	0
$897$	−6.03680	−0.201563
$898$	0	0
$899$	−0.671792	−0.0224055
$900$	0	0
$901$	−16.8891	−0.562657
$902$	0	0
$903$	25.0674	0.834192
$904$	0	0
$905$	−11.7144	−0.389398
$906$	0	0
$907$	−20.0429	−0.665515	−0.332757	−	0.943012i	$-0.607979\pi$
$907$	−20.0429	−0.665515	−0.332757	+	0.943012i	$0.607979\pi$
$908$	0	0
$909$	3.26567	0.108315
$910$	0	0
$911$	4.47857	0.148382	0.0741908	−	0.997244i	$-0.476363\pi$
$911$	4.47857	0.148382	0.0741908	+	0.997244i	$0.476363\pi$
$912$	0	0
$913$	−24.0543	−0.796081
$914$	0	0
$915$	14.3053	0.472919
$916$	0	0
$917$	5.06800	0.167360
$918$	0	0
$919$	20.4388	0.674214	0.337107	−	0.941466i	$-0.390552\pi$
$919$	20.4388	0.674214	0.337107	+	0.941466i	$0.390552\pi$
$920$	0	0
$921$	17.5891	0.579582
$922$	0	0
$923$	54.8260	1.80462
$924$	0	0
$925$	14.1039	0.463734
$926$	0	0
$927$	−5.93651	−0.194981
$928$	0	0
$929$	−31.2202	−1.02430	−0.512151	−	0.858895i	$-0.671151\pi$
$929$	−31.2202	−1.02430	−0.512151	+	0.858895i	$0.671151\pi$
$930$	0	0
$931$	−9.56537	−0.313492
$932$	0	0
$933$	25.8397	0.845955
$934$	0	0
$935$	28.6462	0.936831
$936$	0	0
$937$	−35.5696	−1.16201	−0.581005	−	0.813900i	$-0.697340\pi$
$937$	−35.5696	−1.16201	−0.581005	+	0.813900i	$0.697340\pi$
$938$	0	0
$939$	7.28452	0.237721
$940$	0	0
$941$	0.409490	0.0133490	0.00667449	−	0.999978i	$-0.497875\pi$
$941$	0.409490	0.0133490	0.00667449	+	0.999978i	$0.497875\pi$
$942$	0	0
$943$	2.20091	0.0716716
$944$	0	0
$945$	−3.93200	−0.127908
$946$	0	0
$947$	43.0270	1.39819	0.699095	−	0.715029i	$-0.253586\pi$
$947$	43.0270	1.39819	0.699095	+	0.715029i	$0.253586\pi$
$948$	0	0
$949$	12.7414	0.413603
$950$	0	0
$951$	8.21892	0.266517
$952$	0	0
$953$	17.1891	0.556808	0.278404	−	0.960464i	$-0.410195\pi$
$953$	17.1891	0.556808	0.278404	+	0.960464i	$0.410195\pi$
$954$	0	0
$955$	10.7514	0.347908
$956$	0	0
$957$	−3.69746	−0.119522
$958$	0	0
$959$	−23.7901	−0.768223
$960$	0	0
$961$	−30.5487	−0.985442
$962$	0	0
$963$	11.1575	0.359545
$964$	0	0
$965$	22.0692	0.710432
$966$	0	0
$967$	8.03839	0.258497	0.129249	−	0.991612i	$-0.458743\pi$
$967$	8.03839	0.258497	0.129249	+	0.991612i	$0.458743\pi$
$968$	0	0
$969$	36.2591	1.16481
$970$	0	0
$971$	45.4945	1.45999	0.729995	−	0.683453i	$-0.239522\pi$
$971$	45.4945	1.45999	0.729995	+	0.683453i	$0.239522\pi$
$972$	0	0
$973$	−2.78664	−0.0893357
$974$	0	0
$975$	−19.2256	−0.615711
$976$	0	0
$977$	−15.6237	−0.499846	−0.249923	−	0.968266i	$-0.580405\pi$
$977$	−15.6237	−0.499846	−0.249923	+	0.968266i	$0.580405\pi$
$978$	0	0
$979$	17.1672	0.548666
$980$	0	0
$981$	−1.50930	−0.0481882
$982$	0	0
$983$	13.8606	0.442084	0.221042	−	0.975264i	$-0.429054\pi$
$983$	13.8606	0.442084	0.221042	+	0.975264i	$0.429054\pi$
$984$	0	0
$985$	−14.0532	−0.447771
$986$	0	0
$987$	−1.60361	−0.0510434
$988$	0	0
$989$	8.58949	0.273130
$990$	0	0
$991$	−32.4027	−1.02930	−0.514652	−	0.857399i	$-0.672079\pi$
$991$	−32.4027	−1.02930	−0.514652	+	0.857399i	$0.672079\pi$
$992$	0	0
$993$	−3.55431	−0.112793
$994$	0	0
$995$	28.7117	0.910221
$996$	0	0
$997$	−32.7250	−1.03641	−0.518206	−	0.855256i	$-0.673400\pi$
$997$	−32.7250	−1.03641	−0.518206	+	0.855256i	$0.673400\pi$
$998$	0	0
$999$	4.42860	0.140115

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.e.1.8	✓	9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.34732 q^{5} +2.91839 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.34732 q^{5} +2.91839 q^{7} +1.00000 q^{9} +3.69746 q^{11} -6.03680 q^{13} -1.34732 q^{15} +5.75034 q^{17} -6.30556 q^{19} -2.91839 q^{21} -1.00000 q^{23} -3.18473 q^{25} -1.00000 q^{27} +1.00000 q^{29} -0.671792 q^{31} -3.69746 q^{33} +3.93200 q^{35} -4.42860 q^{37} +6.03680 q^{39} -2.20091 q^{41} -8.58949 q^{43} +1.34732 q^{45} +0.549485 q^{47} +1.51697 q^{49} -5.75034 q^{51} -2.93706 q^{53} +4.98166 q^{55} +6.30556 q^{57} -4.20516 q^{59} -10.6176 q^{61} +2.91839 q^{63} -8.13349 q^{65} -3.98884 q^{67} +1.00000 q^{69} -9.08197 q^{71} -2.11062 q^{73} +3.18473 q^{75} +10.7906 q^{77} -12.8630 q^{79} +1.00000 q^{81} -6.50563 q^{83} +7.74754 q^{85} -1.00000 q^{87} +4.64297 q^{89} -17.6177 q^{91} +0.671792 q^{93} -8.49560 q^{95} -1.50834 q^{97} +3.69746 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9	\( 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9} - 2 q^{11} - 7 q^{13} + 3 q^{15} - q^{19} - 7 q^{21} - 9 q^{23} + 2 q^{25} - 9 q^{27} + 9 q^{29} + 8 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} + 7 q^{39} - 19 q^{41} - 3 q^{43} - 3 q^{45} - 3 q^{47} - 18 q^{49} - 17 q^{53} + 9 q^{55} + q^{57} - 10 q^{59} + q^{61} + 7 q^{63} - 16 q^{65} + 12 q^{67} + 9 q^{69} - 7 q^{71} + 13 q^{73} - 2 q^{75} - 15 q^{77} - 10 q^{79} + 9 q^{81} + 9 q^{83} - 6 q^{85} - 9 q^{87} - 5 q^{89} - 18 q^{91} - 8 q^{93} + 31 q^{95} - 7 q^{97} - 2 q^{99}+O(q^{100}) \) 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9 - 2 * q^11 - 7 * q^13 + 3 * q^15 - q^19 - 7 * q^21 - 9 * q^23 + 2 * q^25 - 9 * q^27 + 9 * q^29 + 8 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 + 7 * q^39 - 19 * q^41 - 3 * q^43 - 3 * q^45 - 3 * q^47 - 18 * q^49 - 17 * q^53 + 9 * q^55 + q^57 - 10 * q^59 + q^61 + 7 * q^63 - 16 * q^65 + 12 * q^67 + 9 * q^69 - 7 * q^71 + 13 * q^73 - 2 * q^75 - 15 * q^77 - 10 * q^79 + 9 * q^81 + 9 * q^83 - 6 * q^85 - 9 * q^87 - 5 * q^89 - 18 * q^91 - 8 * q^93 + 31 * q^95 - 7 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more