LMFDB - Embedded newform 9576.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9576.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:	$76.4647449756$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
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			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	9576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-2.00000 q^{5} -1.00000 q^{7} +2.00000 q^{11} +5.12311 q^{13} +5.12311 q^{17} -1.00000 q^{19} +1.12311 q^{23} -1.00000 q^{25} +0.876894 q^{29} +7.12311 q^{31} +2.00000 q^{35} -9.12311 q^{37} +8.24621 q^{41} +4.00000 q^{43} +1.00000 q^{49} -3.12311 q^{53} -4.00000 q^{55} -10.2462 q^{59} -2.00000 q^{61} -10.2462 q^{65} +2.24621 q^{67} +15.3693 q^{71} -10.0000 q^{73} -2.00000 q^{77} +4.87689 q^{79} -0.876894 q^{83} -10.2462 q^{85} +16.2462 q^{89} -5.12311 q^{91} +2.00000 q^{95} -8.24621 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q - 4 * q^5 - 2 * q^7	$ 2 q - 4 q^{5} - 2 q^{7} + 4 q^{11} + 2 q^{13} + 2 q^{17} - 2 q^{19} - 6 q^{23} - 2 q^{25} + 10 q^{29} + 6 q^{31} + 4 q^{35} - 10 q^{37} + 8 q^{43} + 2 q^{49} + 2 q^{53} - 8 q^{55} - 4 q^{59} - 4 q^{61} - 4 q^{65} - 12 q^{67} + 6 q^{71} - 20 q^{73} - 4 q^{77} + 18 q^{79} - 10 q^{83} - 4 q^{85} + 16 q^{89} - 2 q^{91} + 4 q^{95}+O(q^{100}) $ 2 * q - 4 * q^5 - 2 * q^7 + 4 * q^11 + 2 * q^13 + 2 * q^17 - 2 * q^19 - 6 * q^23 - 2 * q^25 + 10 * q^29 + 6 * q^31 + 4 * q^35 - 10 * q^37 + 8 * q^43 + 2 * q^49 + 2 * q^53 - 8 * q^55 - 4 * q^59 - 4 * q^61 - 4 * q^65 - 12 * q^67 + 6 * q^71 - 20 * q^73 - 4 * q^77 + 18 * q^79 - 10 * q^83 - 4 * q^85 + 16 * q^89 - 2 * q^91 + 4 * q^95

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
$3$	0	0
$4$	0	0
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	5.12311	1.42089	0.710447	−	0.703751i	$-0.248493\pi$
$13$	5.12311	1.42089	0.710447	+	0.703751i	$0.248493\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.12311	1.24254	0.621268	−	0.783598i	$-0.286618\pi$
$17$	5.12311	1.24254	0.621268	+	0.783598i	$0.286618\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.12311	0.234184	0.117092	−	0.993121i	$-0.462643\pi$
$23$	1.12311	0.234184	0.117092	+	0.993121i	$0.462643\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.876894	0.162835	0.0814176	−	0.996680i	$-0.474055\pi$
$29$	0.876894	0.162835	0.0814176	+	0.996680i	$0.474055\pi$
$30$	0	0
$31$	7.12311	1.27935	0.639674	−	0.768647i	$-0.279069\pi$
$31$	7.12311	1.27935	0.639674	+	0.768647i	$0.279069\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	−9.12311	−1.49983	−0.749915	−	0.661535i	$-0.769905\pi$
$37$	−9.12311	−1.49983	−0.749915	+	0.661535i	$0.769905\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.24621	1.28784	0.643921	−	0.765092i	$-0.277307\pi$
$41$	8.24621	1.28784	0.643921	+	0.765092i	$0.277307\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.12311	−0.428992	−0.214496	−	0.976725i	$-0.568811\pi$
$53$	−3.12311	−0.428992	−0.214496	+	0.976725i	$0.568811\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.2462	−1.33394	−0.666972	−	0.745083i	$-0.732410\pi$
$59$	−10.2462	−1.33394	−0.666972	+	0.745083i	$0.732410\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−10.2462	−1.27089
$66$	0	0
$67$	2.24621	0.274418	0.137209	−	0.990542i	$-0.456187\pi$
$67$	2.24621	0.274418	0.137209	+	0.990542i	$0.456187\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	15.3693	1.82400	0.912001	−	0.410188i	$-0.134537\pi$
$71$	15.3693	1.82400	0.912001	+	0.410188i	$0.134537\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	4.87689	0.548693	0.274347	−	0.961631i	$-0.411538\pi$
$79$	4.87689	0.548693	0.274347	+	0.961631i	$0.411538\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.876894	−0.0962517	−0.0481258	−	0.998841i	$-0.515325\pi$
$83$	−0.876894	−0.0962517	−0.0481258	+	0.998841i	$0.515325\pi$
$84$	0	0
$85$	−10.2462	−1.11136
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.2462	1.72209	0.861047	−	0.508525i	$-0.169809\pi$
$89$	16.2462	1.72209	0.861047	+	0.508525i	$0.169809\pi$
$90$	0	0
$91$	−5.12311	−0.537047
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	−8.24621	−0.837276	−0.418638	−	0.908153i	$-0.637492\pi$
$97$	−8.24621	−0.837276	−0.418638	+	0.908153i	$0.637492\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	−8.87689	−0.874666	−0.437333	−	0.899300i	$-0.644077\pi$
$103$	−8.87689	−0.874666	−0.437333	+	0.899300i	$0.644077\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.12311	−0.495269	−0.247635	−	0.968853i	$-0.579653\pi$
$107$	−5.12311	−0.495269	−0.247635	+	0.968853i	$0.579653\pi$
$108$	0	0
$109$	−9.12311	−0.873835	−0.436918	−	0.899502i	$-0.643930\pi$
$109$	−9.12311	−0.873835	−0.436918	+	0.899502i	$0.643930\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.876894	0.0824913	0.0412456	−	0.999149i	$-0.486867\pi$
$113$	0.876894	0.0824913	0.0412456	+	0.999149i	$0.486867\pi$
$114$	0	0
$115$	−2.24621	−0.209460
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.12311	−0.469634
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	12.8769	1.14264	0.571320	−	0.820728i	$-0.306432\pi$
$127$	12.8769	1.14264	0.571320	+	0.820728i	$0.306432\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−19.6155	−1.71382	−0.856908	−	0.515469i	$-0.827618\pi$
$131$	−19.6155	−1.71382	−0.856908	+	0.515469i	$0.827618\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.2462	−1.55888	−0.779440	−	0.626477i	$-0.784496\pi$
$137$	−18.2462	−1.55888	−0.779440	+	0.626477i	$0.784496\pi$
$138$	0	0
$139$	14.2462	1.20835	0.604174	−	0.796852i	$-0.293503\pi$
$139$	14.2462	1.20835	0.604174	+	0.796852i	$0.293503\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	10.2462	0.856831
$144$	0	0
$145$	−1.75379	−0.145644
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.87689	0.727224	0.363612	−	0.931551i	$-0.381543\pi$
$149$	8.87689	0.727224	0.363612	+	0.931551i	$0.381543\pi$
$150$	0	0
$151$	4.87689	0.396876	0.198438	−	0.980113i	$-0.436413\pi$
$151$	4.87689	0.396876	0.198438	+	0.980113i	$0.436413\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−14.2462	−1.14428
$156$	0	0
$157$	−24.2462	−1.93506	−0.967529	−	0.252759i	$-0.918662\pi$
$157$	−24.2462	−1.93506	−0.967529	+	0.252759i	$0.918662\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.12311	−0.0885131
$162$	0	0
$163$	8.49242	0.665178	0.332589	−	0.943072i	$-0.392078\pi$
$163$	8.49242	0.665178	0.332589	+	0.943072i	$0.392078\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.2462	1.41193	0.705967	−	0.708245i	$-0.250513\pi$
$167$	18.2462	1.41193	0.705967	+	0.708245i	$0.250513\pi$
$168$	0	0
$169$	13.2462	1.01894
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.2462	0.931062	0.465531	−	0.885032i	$-0.345863\pi$
$173$	12.2462	0.931062	0.465531	+	0.885032i	$0.345863\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.3693	0.849783	0.424891	−	0.905244i	$-0.360312\pi$
$179$	11.3693	0.849783	0.424891	+	0.905244i	$0.360312\pi$
$180$	0	0
$181$	19.3693	1.43971	0.719855	−	0.694124i	$-0.244208\pi$
$181$	19.3693	1.43971	0.719855	+	0.694124i	$0.244208\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	18.2462	1.34149
$186$	0	0
$187$	10.2462	0.749277
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−23.3693	−1.69094	−0.845472	−	0.534019i	$-0.820681\pi$
$191$	−23.3693	−1.69094	−0.845472	+	0.534019i	$0.820681\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.63068	0.187428	0.0937142	−	0.995599i	$-0.470126\pi$
$197$	2.63068	0.187428	0.0937142	+	0.995599i	$0.470126\pi$
$198$	0	0
$199$	−22.2462	−1.57699	−0.788496	−	0.615040i	$-0.789140\pi$
$199$	−22.2462	−1.57699	−0.788496	+	0.615040i	$0.789140\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.876894	−0.0615459
$204$	0	0
$205$	−16.4924	−1.15188
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−24.4924	−1.68613	−0.843064	−	0.537813i	$-0.819251\pi$
$211$	−24.4924	−1.68613	−0.843064	+	0.537813i	$0.819251\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	−7.12311	−0.483548
$218$	0	0
$219$	0	0
$220$	0	0
$221$	26.2462	1.76551
$222$	0	0
$223$	−9.36932	−0.627416	−0.313708	−	0.949520i	$-0.601571\pi$
$223$	−9.36932	−0.627416	−0.313708	+	0.949520i	$0.601571\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.24621	−0.149086	−0.0745431	−	0.997218i	$-0.523750\pi$
$227$	−2.24621	−0.149086	−0.0745431	+	0.997218i	$0.523750\pi$
$228$	0	0
$229$	28.7386	1.89910	0.949551	−	0.313612i	$-0.101539\pi$
$229$	28.7386	1.89910	0.949551	+	0.313612i	$0.101539\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.24621	−0.147154	−0.0735771	−	0.997290i	$-0.523441\pi$
$233$	−2.24621	−0.147154	−0.0735771	+	0.997290i	$0.523441\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	17.1231	1.10760	0.553801	−	0.832649i	$-0.313177\pi$
$239$	17.1231	1.10760	0.553801	+	0.832649i	$0.313177\pi$
$240$	0	0
$241$	−28.7386	−1.85122	−0.925609	−	0.378481i	$-0.876447\pi$
$241$	−28.7386	−1.85122	−0.925609	+	0.378481i	$0.876447\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.00000	−0.127775
$246$	0	0
$247$	−5.12311	−0.325975
$248$	0	0
$249$	0	0
$250$	0	0
$251$	25.3693	1.60130	0.800649	−	0.599134i	$-0.204488\pi$
$251$	25.3693	1.60130	0.800649	+	0.599134i	$0.204488\pi$
$252$	0	0
$253$	2.24621	0.141218
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	9.12311	0.566882
$260$	0	0
$261$	0	0
$262$	0	0
$263$	29.6155	1.82617	0.913086	−	0.407767i	$-0.133693\pi$
$263$	29.6155	1.82617	0.913086	+	0.407767i	$0.133693\pi$
$264$	0	0
$265$	6.24621	0.383702
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.8769	0.768171	0.384086	−	0.923298i	$-0.374517\pi$
$281$	12.8769	0.768171	0.384086	+	0.923298i	$0.374517\pi$
$282$	0	0
$283$	24.4924	1.45592	0.727962	−	0.685618i	$-0.240468\pi$
$283$	24.4924	1.45592	0.727962	+	0.685618i	$0.240468\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.24621	−0.486758
$288$	0	0
$289$	9.24621	0.543895
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−0.246211	−0.0143838	−0.00719191	−	0.999974i	$-0.502289\pi$
$293$	−0.246211	−0.0143838	−0.00719191	+	0.999974i	$0.502289\pi$
$294$	0	0
$295$	20.4924	1.19311
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.75379	0.332750
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.00000	0.229039
$306$	0	0
$307$	−8.49242	−0.484688	−0.242344	−	0.970190i	$-0.577916\pi$
$307$	−8.49242	−0.484688	−0.242344	+	0.970190i	$0.577916\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	34.9848	1.97746	0.988730	−	0.149709i	$-0.0478335\pi$
$313$	34.9848	1.97746	0.988730	+	0.149709i	$0.0478335\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.36932	0.526233	0.263117	−	0.964764i	$-0.415250\pi$
$317$	9.36932	0.526233	0.263117	+	0.964764i	$0.415250\pi$
$318$	0	0
$319$	1.75379	0.0981933
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.12311	−0.285057
$324$	0	0
$325$	−5.12311	−0.284179
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.49242	−0.245447
$336$	0	0
$337$	26.4924	1.44313	0.721567	−	0.692345i	$-0.243422\pi$
$337$	26.4924	1.44313	0.721567	+	0.692345i	$0.243422\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	14.2462	0.771476
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.75379	0.416245	0.208123	−	0.978103i	$-0.433265\pi$
$347$	7.75379	0.416245	0.208123	+	0.978103i	$0.433265\pi$
$348$	0	0
$349$	20.2462	1.08375	0.541877	−	0.840458i	$-0.317714\pi$
$349$	20.2462	1.08375	0.541877	+	0.840458i	$0.317714\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	27.3693	1.45672	0.728361	−	0.685194i	$-0.240282\pi$
$353$	27.3693	1.45672	0.728361	+	0.685194i	$0.240282\pi$
$354$	0	0
$355$	−30.7386	−1.63144
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.36932	−0.388938	−0.194469	−	0.980909i	$-0.562298\pi$
$359$	−7.36932	−0.388938	−0.194469	+	0.980909i	$0.562298\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	20.0000	1.04685
$366$	0	0
$367$	−9.75379	−0.509144	−0.254572	−	0.967054i	$-0.581935\pi$
$367$	−9.75379	−0.509144	−0.254572	+	0.967054i	$0.581935\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.12311	0.162144
$372$	0	0
$373$	29.6155	1.53343	0.766717	−	0.641985i	$-0.221889\pi$
$373$	29.6155	1.53343	0.766717	+	0.641985i	$0.221889\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.49242	0.231372
$378$	0	0
$379$	24.4924	1.25809	0.629046	−	0.777368i	$-0.283446\pi$
$379$	24.4924	1.25809	0.629046	+	0.777368i	$0.283446\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	32.0000	1.63512	0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000	1.63512	0.817562	+	0.575841i	$0.195325\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	0	0
$387$	0	0
$388$	0	0
$389$	11.1231	0.563964	0.281982	−	0.959420i	$-0.409008\pi$
$389$	11.1231	0.563964	0.281982	+	0.959420i	$0.409008\pi$
$390$	0	0
$391$	5.75379	0.290982
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−9.75379	−0.490766
$396$	0	0
$397$	−28.7386	−1.44235	−0.721175	−	0.692753i	$-0.756398\pi$
$397$	−28.7386	−1.44235	−0.721175	+	0.692753i	$0.756398\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	39.6155	1.97831	0.989153	−	0.146892i	$-0.0469270\pi$
$401$	39.6155	1.97831	0.989153	+	0.146892i	$0.0469270\pi$
$402$	0	0
$403$	36.4924	1.81782
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−18.2462	−0.904431
$408$	0	0
$409$	−3.75379	−0.185613	−0.0928065	−	0.995684i	$-0.529584\pi$
$409$	−3.75379	−0.185613	−0.0928065	+	0.995684i	$0.529584\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	10.2462	0.504183
$414$	0	0
$415$	1.75379	0.0860901
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.36932	0.0668955	0.0334478	−	0.999440i	$-0.489351\pi$
$419$	1.36932	0.0668955	0.0334478	+	0.999440i	$0.489351\pi$
$420$	0	0
$421$	26.8769	1.30990	0.654950	−	0.755672i	$-0.272690\pi$
$421$	26.8769	1.30990	0.654950	+	0.755672i	$0.272690\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.12311	−0.248507
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	0	0
$430$	0	0
$431$	26.8769	1.29461	0.647307	−	0.762229i	$-0.275895\pi$
$431$	26.8769	1.29461	0.647307	+	0.762229i	$0.275895\pi$
$432$	0	0
$433$	−24.7386	−1.18886	−0.594431	−	0.804146i	$-0.702623\pi$
$433$	−24.7386	−1.18886	−0.594431	+	0.804146i	$0.702623\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.12311	−0.0537254
$438$	0	0
$439$	14.6307	0.698284	0.349142	−	0.937070i	$-0.386473\pi$
$439$	14.6307	0.698284	0.349142	+	0.937070i	$0.386473\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.2462	0.581835	0.290918	−	0.956748i	$-0.406039\pi$
$443$	12.2462	0.581835	0.290918	+	0.956748i	$0.406039\pi$
$444$	0	0
$445$	−32.4924	−1.54029
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−24.8769	−1.17401	−0.587007	−	0.809582i	$-0.699694\pi$
$449$	−24.8769	−1.17401	−0.587007	+	0.809582i	$0.699694\pi$
$450$	0	0
$451$	16.4924	0.776598
$452$	0	0
$453$	0	0
$454$	0	0
$455$	10.2462	0.480350
$456$	0	0
$457$	15.7538	0.736931	0.368466	−	0.929641i	$-0.379883\pi$
$457$	15.7538	0.736931	0.368466	+	0.929641i	$0.379883\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−15.7538	−0.733727	−0.366864	−	0.930275i	$-0.619568\pi$
$461$	−15.7538	−0.733727	−0.366864	+	0.930275i	$0.619568\pi$
$462$	0	0
$463$	−18.7386	−0.870858	−0.435429	−	0.900223i	$-0.643403\pi$
$463$	−18.7386	−0.870858	−0.435429	+	0.900223i	$0.643403\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.36932	0.0633644	0.0316822	−	0.999498i	$-0.489914\pi$
$467$	1.36932	0.0633644	0.0316822	+	0.999498i	$0.489914\pi$
$468$	0	0
$469$	−2.24621	−0.103720
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.50758	0.160265	0.0801327	−	0.996784i	$-0.474466\pi$
$479$	3.50758	0.160265	0.0801327	+	0.996784i	$0.474466\pi$
$480$	0	0
$481$	−46.7386	−2.13110
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.4924	0.748882
$486$	0	0
$487$	−21.8617	−0.990650	−0.495325	−	0.868708i	$-0.664951\pi$
$487$	−21.8617	−0.990650	−0.495325	+	0.868708i	$0.664951\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	0	0
$493$	4.49242	0.202329
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−15.3693	−0.689408
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	−20.0000	−0.889988
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−40.7386	−1.80571	−0.902854	−	0.429947i	$-0.858532\pi$
$509$	−40.7386	−1.80571	−0.902854	+	0.429947i	$0.858532\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	0	0
$514$	0	0
$515$	17.7538	0.782325
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.50758	0.0660482	0.0330241	−	0.999455i	$-0.489486\pi$
$521$	1.50758	0.0660482	0.0330241	+	0.999455i	$0.489486\pi$
$522$	0	0
$523$	−24.9848	−1.09251	−0.546255	−	0.837619i	$-0.683947\pi$
$523$	−24.9848	−1.09251	−0.546255	+	0.837619i	$0.683947\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	36.4924	1.58963
$528$	0	0
$529$	−21.7386	−0.945158
$530$	0	0
$531$	0	0
$532$	0	0
$533$	42.2462	1.82989
$534$	0	0
$535$	10.2462	0.442982
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	−32.2462	−1.38637	−0.693186	−	0.720758i	$-0.743794\pi$
$541$	−32.2462	−1.38637	−0.693186	+	0.720758i	$0.743794\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	18.2462	0.781582
$546$	0	0
$547$	44.9848	1.92341	0.961707	−	0.274081i	$-0.0883737\pi$
$547$	44.9848	1.92341	0.961707	+	0.274081i	$0.0883737\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.876894	−0.0373570
$552$	0	0
$553$	−4.87689	−0.207387
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.63068	−0.111466	−0.0557328	−	0.998446i	$-0.517750\pi$
$557$	−2.63068	−0.111466	−0.0557328	+	0.998446i	$0.517750\pi$
$558$	0	0
$559$	20.4924	0.866737
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.4924	1.03223	0.516116	−	0.856519i	$-0.327377\pi$
$563$	24.4924	1.03223	0.516116	+	0.856519i	$0.327377\pi$
$564$	0	0
$565$	−1.75379	−0.0737824
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0.384472	0.0161179	0.00805895	−	0.999968i	$-0.497435\pi$
$569$	0.384472	0.0161179	0.00805895	+	0.999968i	$0.497435\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.12311	−0.0468367
$576$	0	0
$577$	0.246211	0.0102499	0.00512495	−	0.999987i	$-0.498369\pi$
$577$	0.246211	0.0102499	0.00512495	+	0.999987i	$0.498369\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0.876894	0.0363797
$582$	0	0
$583$	−6.24621	−0.258692
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−17.3693	−0.716908	−0.358454	−	0.933547i	$-0.616696\pi$
$587$	−17.3693	−0.716908	−0.358454	+	0.933547i	$0.616696\pi$
$588$	0	0
$589$	−7.12311	−0.293502
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−27.8617	−1.14414	−0.572072	−	0.820203i	$-0.693860\pi$
$593$	−27.8617	−1.14414	−0.572072	+	0.820203i	$0.693860\pi$
$594$	0	0
$595$	10.2462	0.420054
$596$	0	0
$597$	0	0
$598$	0	0
$599$	39.3693	1.60859	0.804293	−	0.594232i	$-0.202544\pi$
$599$	39.3693	1.60859	0.804293	+	0.594232i	$0.202544\pi$
$600$	0	0
$601$	−18.4924	−0.754322	−0.377161	−	0.926148i	$-0.623100\pi$
$601$	−18.4924	−0.754322	−0.377161	+	0.926148i	$0.623100\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	14.0000	0.569181
$606$	0	0
$607$	47.6155	1.93265	0.966327	−	0.257316i	$-0.0828381\pi$
$607$	47.6155	1.93265	0.966327	+	0.257316i	$0.0828381\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	24.7386	0.999184	0.499592	−	0.866261i	$-0.333483\pi$
$613$	24.7386	0.999184	0.499592	+	0.866261i	$0.333483\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−0.492423	−0.0198242	−0.00991209	−	0.999951i	$-0.503155\pi$
$617$	−0.492423	−0.0198242	−0.00991209	+	0.999951i	$0.503155\pi$
$618$	0	0
$619$	−22.2462	−0.894151	−0.447075	−	0.894496i	$-0.647534\pi$
$619$	−22.2462	−0.894151	−0.447075	+	0.894496i	$0.647534\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−16.2462	−0.650891
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−46.7386	−1.86359
$630$	0	0
$631$	−22.2462	−0.885608	−0.442804	−	0.896619i	$-0.646016\pi$
$631$	−22.2462	−0.885608	−0.442804	+	0.896619i	$0.646016\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−25.7538	−1.02201
$636$	0	0
$637$	5.12311	0.202985
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−20.8769	−0.824588	−0.412294	−	0.911051i	$-0.635272\pi$
$641$	−20.8769	−0.824588	−0.412294	+	0.911051i	$0.635272\pi$
$642$	0	0
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.75379	0.0689486	0.0344743	−	0.999406i	$-0.489024\pi$
$647$	1.75379	0.0689486	0.0344743	+	0.999406i	$0.489024\pi$
$648$	0	0
$649$	−20.4924	−0.804398
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−45.8617	−1.79471	−0.897354	−	0.441311i	$-0.854514\pi$
$653$	−45.8617	−1.79471	−0.897354	+	0.441311i	$0.854514\pi$
$654$	0	0
$655$	39.2311	1.53288
$656$	0	0
$657$	0	0
$658$	0	0
$659$	43.8617	1.70861	0.854305	−	0.519771i	$-0.173983\pi$
$659$	43.8617	1.70861	0.854305	+	0.519771i	$0.173983\pi$
$660$	0	0
$661$	5.61553	0.218419	0.109209	−	0.994019i	$-0.465168\pi$
$661$	5.61553	0.218419	0.109209	+	0.994019i	$0.465168\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.00000	−0.0775567
$666$	0	0
$667$	0.984845	0.0381334
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	−13.5076	−0.520679	−0.260339	−	0.965517i	$-0.583834\pi$
$673$	−13.5076	−0.520679	−0.260339	+	0.965517i	$0.583834\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.75379	−0.144270	−0.0721349	−	0.997395i	$-0.522981\pi$
$677$	−3.75379	−0.144270	−0.0721349	+	0.997395i	$0.522981\pi$
$678$	0	0
$679$	8.24621	0.316461
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−9.12311	−0.349086	−0.174543	−	0.984650i	$-0.555845\pi$
$683$	−9.12311	−0.349086	−0.174543	+	0.984650i	$0.555845\pi$
$684$	0	0
$685$	36.4924	1.39430
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−16.0000	−0.609551
$690$	0	0
$691$	−38.2462	−1.45495	−0.727477	−	0.686132i	$-0.759307\pi$
$691$	−38.2462	−1.45495	−0.727477	+	0.686132i	$0.759307\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−28.4924	−1.08078
$696$	0	0
$697$	42.2462	1.60019
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−7.61553	−0.287635	−0.143817	−	0.989604i	$-0.545938\pi$
$701$	−7.61553	−0.287635	−0.143817	+	0.989604i	$0.545938\pi$
$702$	0	0
$703$	9.12311	0.344084
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10.0000	−0.376089
$708$	0	0
$709$	−37.2311	−1.39824	−0.699121	−	0.715004i	$-0.746425\pi$
$709$	−37.2311	−1.39824	−0.699121	+	0.715004i	$0.746425\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	−20.4924	−0.766373
$716$	0	0
$717$	0	0
$718$	0	0
$719$	10.2462	0.382119	0.191060	−	0.981578i	$-0.438808\pi$
$719$	10.2462	0.382119	0.191060	+	0.981578i	$0.438808\pi$
$720$	0	0
$721$	8.87689	0.330593
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.876894	−0.0325670
$726$	0	0
$727$	38.7386	1.43674	0.718368	−	0.695663i	$-0.244889\pi$
$727$	38.7386	1.43674	0.718368	+	0.695663i	$0.244889\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	20.4924	0.757940
$732$	0	0
$733$	−26.4924	−0.978520	−0.489260	−	0.872138i	$-0.662733\pi$
$733$	−26.4924	−0.978520	−0.489260	+	0.872138i	$0.662733\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.49242	0.165481
$738$	0	0
$739$	−0.492423	−0.0181141	−0.00905703	−	0.999959i	$-0.502883\pi$
$739$	−0.492423	−0.0181141	−0.00905703	+	0.999959i	$0.502883\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	47.3693	1.73781	0.868906	−	0.494977i	$-0.164824\pi$
$743$	47.3693	1.73781	0.868906	+	0.494977i	$0.164824\pi$
$744$	0	0
$745$	−17.7538	−0.650448
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5.12311	0.187194
$750$	0	0
$751$	17.3693	0.633815	0.316908	−	0.948456i	$-0.397355\pi$
$751$	17.3693	0.633815	0.316908	+	0.948456i	$0.397355\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−9.75379	−0.354977
$756$	0	0
$757$	6.49242	0.235971	0.117986	−	0.993015i	$-0.462356\pi$
$757$	6.49242	0.235971	0.117986	+	0.993015i	$0.462356\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.8769	0.684287	0.342143	−	0.939648i	$-0.388847\pi$
$761$	18.8769	0.684287	0.342143	+	0.939648i	$0.388847\pi$
$762$	0	0
$763$	9.12311	0.330279
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−52.4924	−1.89539
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	9.50758	0.341964	0.170982	−	0.985274i	$-0.445306\pi$
$773$	9.50758	0.341964	0.170982	+	0.985274i	$0.445306\pi$
$774$	0	0
$775$	−7.12311	−0.255870
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−8.24621	−0.295451
$780$	0	0
$781$	30.7386	1.09991
$782$	0	0
$783$	0	0
$784$	0	0
$785$	48.4924	1.73077
$786$	0	0
$787$	−50.7386	−1.80864	−0.904318	−	0.426858i	$-0.859620\pi$
$787$	−50.7386	−1.80864	−0.904318	+	0.426858i	$0.859620\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−0.876894	−0.0311788
$792$	0	0
$793$	−10.2462	−0.363854
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−42.9848	−1.52260	−0.761301	−	0.648399i	$-0.775439\pi$
$797$	−42.9848	−1.52260	−0.761301	+	0.648399i	$0.775439\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−20.0000	−0.705785
$804$	0	0
$805$	2.24621	0.0791685
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−34.7386	−1.22135	−0.610673	−	0.791883i	$-0.709101\pi$
$809$	−34.7386	−1.22135	−0.610673	+	0.791883i	$0.709101\pi$
$810$	0	0
$811$	2.73863	0.0961664	0.0480832	−	0.998843i	$-0.484689\pi$
$811$	2.73863	0.0961664	0.0480832	+	0.998843i	$0.484689\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−16.9848	−0.594953
$816$	0	0
$817$	−4.00000	−0.139942
$818$	0	0
$819$	0	0
$820$	0	0
$821$	24.1080	0.841373	0.420687	−	0.907206i	$-0.361789\pi$
$821$	24.1080	0.841373	0.420687	+	0.907206i	$0.361789\pi$
$822$	0	0
$823$	42.7386	1.48978	0.744888	−	0.667190i	$-0.232503\pi$
$823$	42.7386	1.48978	0.744888	+	0.667190i	$0.232503\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−7.86174	−0.273379	−0.136690	−	0.990614i	$-0.543646\pi$
$827$	−7.86174	−0.273379	−0.136690	+	0.990614i	$0.543646\pi$
$828$	0	0
$829$	−37.6155	−1.30644	−0.653221	−	0.757168i	$-0.726583\pi$
$829$	−37.6155	−1.30644	−0.653221	+	0.757168i	$0.726583\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.12311	0.177505
$834$	0	0
$835$	−36.4924	−1.26287
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−22.7386	−0.785025	−0.392512	−	0.919747i	$-0.628394\pi$
$839$	−22.7386	−0.785025	−0.392512	+	0.919747i	$0.628394\pi$
$840$	0	0
$841$	−28.2311	−0.973485
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−26.4924	−0.911367
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−10.2462	−0.351236
$852$	0	0
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	36.7386	1.25497	0.627484	−	0.778630i	$-0.284085\pi$
$857$	36.7386	1.25497	0.627484	+	0.778630i	$0.284085\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.61553	0.327316	0.163658	−	0.986517i	$-0.447671\pi$
$863$	9.61553	0.327316	0.163658	+	0.986517i	$0.447671\pi$
$864$	0	0
$865$	−24.4924	−0.832767
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9.75379	0.330875
$870$	0	0
$871$	11.5076	0.389919
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−12.0000	−0.405674
$876$	0	0
$877$	21.6155	0.729905	0.364952	−	0.931026i	$-0.381085\pi$
$877$	21.6155	0.729905	0.364952	+	0.931026i	$0.381085\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−33.1231	−1.11595	−0.557973	−	0.829859i	$-0.688421\pi$
$881$	−33.1231	−1.11595	−0.557973	+	0.829859i	$0.688421\pi$
$882$	0	0
$883$	−12.9848	−0.436975	−0.218487	−	0.975840i	$-0.570112\pi$
$883$	−12.9848	−0.436975	−0.218487	+	0.975840i	$0.570112\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−26.2462	−0.881262	−0.440631	−	0.897688i	$-0.645245\pi$
$887$	−26.2462	−0.881262	−0.440631	+	0.897688i	$0.645245\pi$
$888$	0	0
$889$	−12.8769	−0.431877
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−22.7386	−0.760069
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.24621	0.208323
$900$	0	0
$901$	−16.0000	−0.533037
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−38.7386	−1.28772
$906$	0	0
$907$	10.2462	0.340220	0.170110	−	0.985425i	$-0.445588\pi$
$907$	10.2462	0.340220	0.170110	+	0.985425i	$0.445588\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	59.8617	1.98331	0.991654	−	0.128928i	$-0.0411536\pi$
$911$	59.8617	1.98331	0.991654	+	0.128928i	$0.0411536\pi$
$912$	0	0
$913$	−1.75379	−0.0580419
$914$	0	0
$915$	0	0
$916$	0	0
$917$	19.6155	0.647762
$918$	0	0
$919$	−22.2462	−0.733835	−0.366917	−	0.930254i	$-0.619587\pi$
$919$	−22.2462	−0.733835	−0.366917	+	0.930254i	$0.619587\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	78.7386	2.59171
$924$	0	0
$925$	9.12311	0.299966
$926$	0	0
$927$	0	0
$928$	0	0
$929$	45.6155	1.49660	0.748298	−	0.663362i	$-0.230871\pi$
$929$	45.6155	1.49660	0.748298	+	0.663362i	$0.230871\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−20.4924	−0.670174
$936$	0	0
$937$	−11.2614	−0.367893	−0.183946	−	0.982936i	$-0.558887\pi$
$937$	−11.2614	−0.367893	−0.183946	+	0.982936i	$0.558887\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	33.2311	1.08330	0.541651	−	0.840604i	$-0.317800\pi$
$941$	33.2311	1.08330	0.541651	+	0.840604i	$0.317800\pi$
$942$	0	0
$943$	9.26137	0.301592
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16.7386	−0.543933	−0.271966	−	0.962307i	$-0.587674\pi$
$947$	−16.7386	−0.543933	−0.271966	+	0.962307i	$0.587674\pi$
$948$	0	0
$949$	−51.2311	−1.66303
$950$	0	0
$951$	0	0
$952$	0	0
$953$	32.1080	1.04008	0.520039	−	0.854142i	$-0.325917\pi$
$953$	32.1080	1.04008	0.520039	+	0.854142i	$0.325917\pi$
$954$	0	0
$955$	46.7386	1.51243
$956$	0	0
$957$	0	0
$958$	0	0
$959$	18.2462	0.589201
$960$	0	0
$961$	19.7386	0.636730
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.00000	0.128765
$966$	0	0
$967$	24.0000	0.771788	0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000	0.771788	0.385894	+	0.922543i	$0.373893\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12.4924	0.400901	0.200450	−	0.979704i	$-0.435759\pi$
$971$	12.4924	0.400901	0.200450	+	0.979704i	$0.435759\pi$
$972$	0	0
$973$	−14.2462	−0.456713
$974$	0	0
$975$	0	0
$976$	0	0
$977$	23.6155	0.755528	0.377764	−	0.925902i	$-0.376693\pi$
$977$	23.6155	0.755528	0.377764	+	0.925902i	$0.376693\pi$
$978$	0	0
$979$	32.4924	1.03846
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−8.00000	−0.255160	−0.127580	−	0.991828i	$-0.540721\pi$
$983$	−8.00000	−0.255160	−0.127580	+	0.991828i	$0.540721\pi$
$984$	0	0
$985$	−5.26137	−0.167641
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.49242	0.142851
$990$	0	0
$991$	9.36932	0.297626	0.148813	−	0.988865i	$-0.452455\pi$
$991$	9.36932	0.297626	0.148813	+	0.988865i	$0.452455\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	44.4924	1.41050
$996$	0	0
$997$	−11.7538	−0.372246	−0.186123	−	0.982526i	$-0.559592\pi$
$997$	−11.7538	−0.372246	−0.186123	+	0.982526i	$0.559592\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.bd.1.2	✓	2
3.2	odd	2		9576.2.a.bx.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9576.2.a.bd.1.2	✓	2	1.1	even	1	trivial
9576.2.a.bx.1.2	yes	2	3.2	odd	2

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:	\( N \)	\(=\)	\( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9576.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-2.00000 q^{5} -1.00000 q^{7} +2.00000 q^{11} +5.12311 q^{13} +5.12311 q^{17} -1.00000 q^{19} +1.12311 q^{23} -1.00000 q^{25} +0.876894 q^{29} +7.12311 q^{31} +2.00000 q^{35} -9.12311 q^{37} +8.24621 q^{41} +4.00000 q^{43} +1.00000 q^{49} -3.12311 q^{53} -4.00000 q^{55} -10.2462 q^{59} -2.00000 q^{61} -10.2462 q^{65} +2.24621 q^{67} +15.3693 q^{71} -10.0000 q^{73} -2.00000 q^{77} +4.87689 q^{79} -0.876894 q^{83} -10.2462 q^{85} +16.2462 q^{89} -5.12311 q^{91} +2.00000 q^{95} -8.24621 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - 4 * q^5 - 2 * q^7	\( 2 q - 4 q^{5} - 2 q^{7} + 4 q^{11} + 2 q^{13} + 2 q^{17} - 2 q^{19} - 6 q^{23} - 2 q^{25} + 10 q^{29} + 6 q^{31} + 4 q^{35} - 10 q^{37} + 8 q^{43} + 2 q^{49} + 2 q^{53} - 8 q^{55} - 4 q^{59} - 4 q^{61} - 4 q^{65} - 12 q^{67} + 6 q^{71} - 20 q^{73} - 4 q^{77} + 18 q^{79} - 10 q^{83} - 4 q^{85} + 16 q^{89} - 2 q^{91} + 4 q^{95}+O(q^{100}) \) 2 * q - 4 * q^5 - 2 * q^7 + 4 * q^11 + 2 * q^13 + 2 * q^17 - 2 * q^19 - 6 * q^23 - 2 * q^25 + 10 * q^29 + 6 * q^31 + 4 * q^35 - 10 * q^37 + 8 * q^43 + 2 * q^49 + 2 * q^53 - 8 * q^55 - 4 * q^59 - 4 * q^61 - 4 * q^65 - 12 * q^67 + 6 * q^71 - 20 * q^73 - 4 * q^77 + 18 * q^79 - 10 * q^83 - 4 * q^85 + 16 * q^89 - 2 * q^91 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more