LMFDB - Embedded newform 9680.2.a.cx.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9680,2,Mod(1,9680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, 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("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9680 = 2^{4} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9680.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:	$77.2951891566$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.45753625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 13x^{4} + 11x^{3} + 41x^{2} - 30x - 20 $ x^6 - x^5 - 13x^4 + 11x^3 + 41x^2 - 30x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.80540$ of defining polynomial
Character	$\chi$	$=$	9680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73383 q^{3} -1.00000 q^{5} +1.25225 q^{7} +0.00617996 q^{9} -2.50777 q^{13} -1.73383 q^{15} -3.60412 q^{17} +7.05765 q^{19} +2.17119 q^{21} +2.20737 q^{23} +1.00000 q^{25} -5.19079 q^{27} +2.69960 q^{29} +2.83345 q^{31} -1.25225 q^{35} -8.07294 q^{37} -4.34805 q^{39} -10.3266 q^{41} -6.88581 q^{43} -0.00617996 q^{45} -1.59656 q^{47} -5.43188 q^{49} -6.24894 q^{51} +1.77393 q^{53} +12.2368 q^{57} +4.36887 q^{59} -8.33324 q^{61} +0.00773884 q^{63} +2.50777 q^{65} -5.43293 q^{67} +3.82721 q^{69} +4.63664 q^{71} -2.03474 q^{73} +1.73383 q^{75} -16.7625 q^{79} -9.01850 q^{81} +5.62561 q^{83} +3.60412 q^{85} +4.68066 q^{87} -4.72832 q^{89} -3.14034 q^{91} +4.91274 q^{93} -7.05765 q^{95} +19.0943 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 6 q^{5} - 6 q^{7} + 10 q^{9} - 6 q^{13} + 2 q^{15} + 11 q^{17} + 11 q^{19} + 2 q^{21} - 18 q^{23} + 6 q^{25} + q^{27} - 6 q^{29} - q^{31} + 6 q^{35} + 4 q^{37} - 27 q^{39} - 4 q^{41} - 3 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) $ 6 * q - 2 * q^3 - 6 * q^5 - 6 * q^7 + 10 * q^9 - 6 * q^13 + 2 * q^15 + 11 * q^17 + 11 * q^19 + 2 * q^21 - 18 * q^23 + 6 * q^25 + q^27 - 6 * q^29 - q^31 + 6 * q^35 + 4 * q^37 - 27 * q^39 - 4 * q^41 - 3 * q^43 - 10 * q^45 - 14 * q^47 + 8 * q^49 - 31 * q^51 + 14 * q^53 - 5 * q^57 - 2 * q^59 - 4 * q^61 + 16 * q^63 + 6 * q^65 - 11 * q^67 + 8 * q^69 - 7 * q^71 - 9 * q^73 - 2 * q^75 - 36 * q^79 + 30 * q^81 + 45 * q^83 - 11 * q^85 - 25 * q^87 + q^89 + 8 * q^91 + 55 * q^93 - 11 * q^95 + 20 * q^97

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
$3$	1.73383	1.00103	0.500515	−	0.865728i	$-0.333144\pi$
$3$	1.73383	1.00103	0.500515	+	0.865728i	$0.333144\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.25225	0.473305	0.236652	−	0.971594i	$-0.423950\pi$
$7$	1.25225	0.473305	0.236652	+	0.971594i	$0.423950\pi$
$8$	0	0
$9$	0.00617996	0.00205999
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.50777	−0.695529	−0.347764	−	0.937582i	$-0.613059\pi$
$13$	−2.50777	−0.695529	−0.347764	+	0.937582i	$0.613059\pi$
$14$	0	0
$15$	−1.73383	−0.447674
$16$	0	0
$17$	−3.60412	−0.874126	−0.437063	−	0.899431i	$-0.643981\pi$
$17$	−3.60412	−0.874126	−0.437063	+	0.899431i	$0.643981\pi$
$18$	0	0
$19$	7.05765	1.61914	0.809568	−	0.587026i	$-0.199701\pi$
$19$	7.05765	1.61914	0.809568	+	0.587026i	$0.199701\pi$
$20$	0	0
$21$	2.17119	0.473792
$22$	0	0
$23$	2.20737	0.460268	0.230134	−	0.973159i	$-0.426084\pi$
$23$	2.20737	0.460268	0.230134	+	0.973159i	$0.426084\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.19079	−0.998967
$28$	0	0
$29$	2.69960	0.501303	0.250652	−	0.968077i	$-0.419355\pi$
$29$	2.69960	0.501303	0.250652	+	0.968077i	$0.419355\pi$
$30$	0	0
$31$	2.83345	0.508903	0.254452	−	0.967085i	$-0.418105\pi$
$31$	2.83345	0.508903	0.254452	+	0.967085i	$0.418105\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.25225	−0.211668
$36$	0	0
$37$	−8.07294	−1.32718	−0.663592	−	0.748095i	$-0.730969\pi$
$37$	−8.07294	−1.32718	−0.663592	+	0.748095i	$0.730969\pi$
$38$	0	0
$39$	−4.34805	−0.696245
$40$	0	0
$41$	−10.3266	−1.61274	−0.806371	−	0.591410i	$-0.798571\pi$
$41$	−10.3266	−1.61274	−0.806371	+	0.591410i	$0.798571\pi$
$42$	0	0
$43$	−6.88581	−1.05008	−0.525038	−	0.851079i	$-0.675949\pi$
$43$	−6.88581	−1.05008	−0.525038	+	0.851079i	$0.675949\pi$
$44$	0	0
$45$	−0.00617996	−0.000921254 0
$46$	0	0
$47$	−1.59656	−0.232883	−0.116441	−	0.993198i	$-0.537149\pi$
$47$	−1.59656	−0.232883	−0.116441	+	0.993198i	$0.537149\pi$
$48$	0	0
$49$	−5.43188	−0.775982
$50$	0	0
$51$	−6.24894	−0.875026
$52$	0	0
$53$	1.77393	0.243668	0.121834	−	0.992550i	$-0.461122\pi$
$53$	1.77393	0.243668	0.121834	+	0.992550i	$0.461122\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	12.2368	1.62080
$58$	0	0
$59$	4.36887	0.568779	0.284389	−	0.958709i	$-0.408209\pi$
$59$	4.36887	0.568779	0.284389	+	0.958709i	$0.408209\pi$
$60$	0	0
$61$	−8.33324	−1.06696	−0.533481	−	0.845812i	$-0.679116\pi$
$61$	−8.33324	−1.06696	−0.533481	+	0.845812i	$0.679116\pi$
$62$	0	0
$63$	0.00773884	0.000975002 0
$64$	0	0
$65$	2.50777	0.311050
$66$	0	0
$67$	−5.43293	−0.663738	−0.331869	−	0.943326i	$-0.607679\pi$
$67$	−5.43293	−0.663738	−0.331869	+	0.943326i	$0.607679\pi$
$68$	0	0
$69$	3.82721	0.460742
$70$	0	0
$71$	4.63664	0.550267	0.275134	−	0.961406i	$-0.411278\pi$
$71$	4.63664	0.550267	0.275134	+	0.961406i	$0.411278\pi$
$72$	0	0
$73$	−2.03474	−0.238148	−0.119074	−	0.992885i	$-0.537993\pi$
$73$	−2.03474	−0.238148	−0.119074	+	0.992885i	$0.537993\pi$
$74$	0	0
$75$	1.73383	0.200206
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−16.7625	−1.88593	−0.942967	−	0.332887i	$-0.891977\pi$
$79$	−16.7625	−1.88593	−0.942967	+	0.332887i	$0.891977\pi$
$80$	0	0
$81$	−9.01850	−1.00206
$82$	0	0
$83$	5.62561	0.617491	0.308746	−	0.951145i	$-0.400091\pi$
$83$	5.62561	0.617491	0.308746	+	0.951145i	$0.400091\pi$
$84$	0	0
$85$	3.60412	0.390921
$86$	0	0
$87$	4.68066	0.501820
$88$	0	0
$89$	−4.72832	−0.501200	−0.250600	−	0.968091i	$-0.580628\pi$
$89$	−4.72832	−0.501200	−0.250600	+	0.968091i	$0.580628\pi$
$90$	0	0
$91$	−3.14034	−0.329197
$92$	0	0
$93$	4.91274	0.509427
$94$	0	0
$95$	−7.05765	−0.724100
$96$	0	0
$97$	19.0943	1.93873	0.969365	−	0.245625i	$-0.0789933\pi$
$97$	19.0943	1.93873	0.969365	+	0.245625i	$0.0789933\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.2252	−1.41546	−0.707732	−	0.706481i	$-0.750281\pi$
$101$	−14.2252	−1.41546	−0.707732	+	0.706481i	$0.750281\pi$
$102$	0	0
$103$	15.8815	1.56485	0.782426	−	0.622744i	$-0.213982\pi$
$103$	15.8815	1.56485	0.782426	+	0.622744i	$0.213982\pi$
$104$	0	0
$105$	−2.17119	−0.211886
$106$	0	0
$107$	4.04935	0.391465	0.195733	−	0.980657i	$-0.437292\pi$
$107$	4.04935	0.391465	0.195733	+	0.980657i	$0.437292\pi$
$108$	0	0
$109$	5.89844	0.564968	0.282484	−	0.959272i	$-0.408842\pi$
$109$	5.89844	0.564968	0.282484	+	0.959272i	$0.408842\pi$
$110$	0	0
$111$	−13.9971	−1.32855
$112$	0	0
$113$	3.41401	0.321163	0.160582	−	0.987023i	$-0.448663\pi$
$113$	3.41401	0.321163	0.160582	+	0.987023i	$0.448663\pi$
$114$	0	0
$115$	−2.20737	−0.205838
$116$	0	0
$117$	−0.0154979	−0.00143278
$118$	0	0
$119$	−4.51324	−0.413728
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−17.9046	−1.61440
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−6.19208	−0.549458	−0.274729	−	0.961522i	$-0.588588\pi$
$127$	−6.19208	−0.549458	−0.274729	+	0.961522i	$0.588588\pi$
$128$	0	0
$129$	−11.9389	−1.05116
$130$	0	0
$131$	7.39924	0.646475	0.323237	−	0.946318i	$-0.395229\pi$
$131$	7.39924	0.646475	0.323237	+	0.946318i	$0.395229\pi$
$132$	0	0
$133$	8.83792	0.766345
$134$	0	0
$135$	5.19079	0.446752
$136$	0	0
$137$	18.4830	1.57911	0.789553	−	0.613683i	$-0.210313\pi$
$137$	18.4830	1.57911	0.789553	+	0.613683i	$0.210313\pi$
$138$	0	0
$139$	17.8424	1.51337	0.756687	−	0.653778i	$-0.226817\pi$
$139$	17.8424	1.51337	0.756687	+	0.653778i	$0.226817\pi$
$140$	0	0
$141$	−2.76817	−0.233122
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−2.69960	−0.224190
$146$	0	0
$147$	−9.41797	−0.776781
$148$	0	0
$149$	−7.39910	−0.606157	−0.303079	−	0.952966i	$-0.598015\pi$
$149$	−7.39910	−0.606157	−0.303079	+	0.952966i	$0.598015\pi$
$150$	0	0
$151$	−4.41515	−0.359300	−0.179650	−	0.983731i	$-0.557496\pi$
$151$	−4.41515	−0.359300	−0.179650	+	0.983731i	$0.557496\pi$
$152$	0	0
$153$	−0.0222733	−0.00180069
$154$	0	0
$155$	−2.83345	−0.227589
$156$	0	0
$157$	−2.33057	−0.186000	−0.0929998	−	0.995666i	$-0.529646\pi$
$157$	−2.33057	−0.186000	−0.0929998	+	0.995666i	$0.529646\pi$
$158$	0	0
$159$	3.07570	0.243919
$160$	0	0
$161$	2.76417	0.217847
$162$	0	0
$163$	−1.50615	−0.117971	−0.0589855	−	0.998259i	$-0.518787\pi$
$163$	−1.50615	−0.117971	−0.0589855	+	0.998259i	$0.518787\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.79732	−0.758139	−0.379070	−	0.925368i	$-0.623756\pi$
$167$	−9.79732	−0.758139	−0.379070	+	0.925368i	$0.623756\pi$
$168$	0	0
$169$	−6.71111	−0.516239
$170$	0	0
$171$	0.0436160	0.00333540
$172$	0	0
$173$	−20.0936	−1.52769	−0.763845	−	0.645400i	$-0.776691\pi$
$173$	−20.0936	−1.52769	−0.763845	+	0.645400i	$0.776691\pi$
$174$	0	0
$175$	1.25225	0.0946610
$176$	0	0
$177$	7.57490	0.569364
$178$	0	0
$179$	−16.2049	−1.21121	−0.605606	−	0.795764i	$-0.707069\pi$
$179$	−16.2049	−1.21121	−0.605606	+	0.795764i	$0.707069\pi$
$180$	0	0
$181$	−9.42873	−0.700832	−0.350416	−	0.936594i	$-0.613960\pi$
$181$	−9.42873	−0.700832	−0.350416	+	0.936594i	$0.613960\pi$
$182$	0	0
$183$	−14.4485	−1.06806
$184$	0	0
$185$	8.07294	0.593534
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−6.50015	−0.472816
$190$	0	0
$191$	−8.17556	−0.591563	−0.295781	−	0.955256i	$-0.595580\pi$
$191$	−8.17556	−0.591563	−0.295781	+	0.955256i	$0.595580\pi$
$192$	0	0
$193$	−4.99518	−0.359561	−0.179781	−	0.983707i	$-0.557539\pi$
$193$	−4.99518	−0.359561	−0.179781	+	0.983707i	$0.557539\pi$
$194$	0	0
$195$	4.34805	0.311370
$196$	0	0
$197$	−7.00790	−0.499292	−0.249646	−	0.968337i	$-0.580314\pi$
$197$	−7.00790	−0.499292	−0.249646	+	0.968337i	$0.580314\pi$
$198$	0	0
$199$	−18.8762	−1.33810	−0.669050	−	0.743217i	$-0.733299\pi$
$199$	−18.8762	−1.33810	−0.669050	+	0.743217i	$0.733299\pi$
$200$	0	0
$201$	−9.41979	−0.664421
$202$	0	0
$203$	3.38057	0.237269
$204$	0	0
$205$	10.3266	0.721240
$206$	0	0
$207$	0.0136414	0.000948145 0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−13.7688	−0.947881	−0.473940	−	0.880557i	$-0.657169\pi$
$211$	−13.7688	−0.947881	−0.473940	+	0.880557i	$0.657169\pi$
$212$	0	0
$213$	8.03916	0.550834
$214$	0	0
$215$	6.88581	0.469608
$216$	0	0
$217$	3.54819	0.240867
$218$	0	0
$219$	−3.52790	−0.238394
$220$	0	0
$221$	9.03828	0.607980
$222$	0	0
$223$	−28.7976	−1.92843	−0.964214	−	0.265125i	$-0.914587\pi$
$223$	−28.7976	−1.92843	−0.964214	+	0.265125i	$0.914587\pi$
$224$	0	0
$225$	0.00617996	0.000411997 0
$226$	0	0
$227$	−24.2769	−1.61131	−0.805656	−	0.592383i	$-0.798187\pi$
$227$	−24.2769	−1.61131	−0.805656	+	0.592383i	$0.798187\pi$
$228$	0	0
$229$	−16.3933	−1.08330	−0.541650	−	0.840604i	$-0.682200\pi$
$229$	−16.3933	−1.08330	−0.541650	+	0.840604i	$0.682200\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.51844	−0.164988	−0.0824941	−	0.996592i	$-0.526289\pi$
$233$	−2.51844	−0.164988	−0.0824941	+	0.996592i	$0.526289\pi$
$234$	0	0
$235$	1.59656	0.104148
$236$	0	0
$237$	−29.0635	−1.88787
$238$	0	0
$239$	−6.67824	−0.431979	−0.215990	−	0.976396i	$-0.569298\pi$
$239$	−6.67824	−0.431979	−0.215990	+	0.976396i	$0.569298\pi$
$240$	0	0
$241$	−2.14310	−0.138049	−0.0690247	−	0.997615i	$-0.521989\pi$
$241$	−2.14310	−0.138049	−0.0690247	+	0.997615i	$0.521989\pi$
$242$	0	0
$243$	−0.0642239	−0.00411997
$244$	0	0
$245$	5.43188	0.347030
$246$	0	0
$247$	−17.6989	−1.12616
$248$	0	0
$249$	9.75388	0.618127
$250$	0	0
$251$	−14.3906	−0.908325	−0.454162	−	0.890919i	$-0.650061\pi$
$251$	−14.3906	−0.908325	−0.454162	+	0.890919i	$0.650061\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	6.24894	0.391324
$256$	0	0
$257$	24.4034	1.52224	0.761120	−	0.648612i	$-0.224650\pi$
$257$	24.4034	1.52224	0.761120	+	0.648612i	$0.224650\pi$
$258$	0	0
$259$	−10.1093	−0.628162
$260$	0	0
$261$	0.0166834	0.00103268
$262$	0	0
$263$	−0.992957	−0.0612283	−0.0306142	−	0.999531i	$-0.509746\pi$
$263$	−0.992957	−0.0612283	−0.0306142	+	0.999531i	$0.509746\pi$
$264$	0	0
$265$	−1.77393	−0.108972
$266$	0	0
$267$	−8.19811	−0.501716
$268$	0	0
$269$	30.4230	1.85493	0.927463	−	0.373915i	$-0.121985\pi$
$269$	30.4230	1.85493	0.927463	+	0.373915i	$0.121985\pi$
$270$	0	0
$271$	−25.9425	−1.57589	−0.787946	−	0.615744i	$-0.788856\pi$
$271$	−25.9425	−1.57589	−0.787946	+	0.615744i	$0.788856\pi$
$272$	0	0
$273$	−5.44483	−0.329536
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−5.03851	−0.302735	−0.151367	−	0.988478i	$-0.548368\pi$
$277$	−5.03851	−0.302735	−0.151367	+	0.988478i	$0.548368\pi$
$278$	0	0
$279$	0.0175106	0.00104833
$280$	0	0
$281$	20.7739	1.23927	0.619634	−	0.784891i	$-0.287281\pi$
$281$	20.7739	1.23927	0.619634	+	0.784891i	$0.287281\pi$
$282$	0	0
$283$	4.55629	0.270843	0.135422	−	0.990788i	$-0.456761\pi$
$283$	4.55629	0.270843	0.135422	+	0.990788i	$0.456761\pi$
$284$	0	0
$285$	−12.2368	−0.724845
$286$	0	0
$287$	−12.9314	−0.763319
$288$	0	0
$289$	−4.01035	−0.235903
$290$	0	0
$291$	33.1063	1.94073
$292$	0	0
$293$	−2.39092	−0.139679	−0.0698394	−	0.997558i	$-0.522249\pi$
$293$	−2.39092	−0.139679	−0.0698394	+	0.997558i	$0.522249\pi$
$294$	0	0
$295$	−4.36887	−0.254366
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.53556	−0.320130
$300$	0	0
$301$	−8.62274	−0.497006
$302$	0	0
$303$	−24.6642	−1.41692
$304$	0	0
$305$	8.33324	0.477160
$306$	0	0
$307$	−3.26648	−0.186428	−0.0932138	−	0.995646i	$-0.529714\pi$
$307$	−3.26648	−0.186428	−0.0932138	+	0.995646i	$0.529714\pi$
$308$	0	0
$309$	27.5359	1.56646
$310$	0	0
$311$	−2.73567	−0.155125	−0.0775627	−	0.996987i	$-0.524714\pi$
$311$	−2.73567	−0.155125	−0.0775627	+	0.996987i	$0.524714\pi$
$312$	0	0
$313$	−10.3586	−0.585501	−0.292750	−	0.956189i	$-0.594570\pi$
$313$	−10.3586	−0.585501	−0.292750	+	0.956189i	$0.594570\pi$
$314$	0	0
$315$	−0.00773884	−0.000436034 0
$316$	0	0
$317$	31.2560	1.75551	0.877755	−	0.479110i	$-0.159041\pi$
$317$	31.2560	1.75551	0.877755	+	0.479110i	$0.159041\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	7.02090	0.391868
$322$	0	0
$323$	−25.4366	−1.41533
$324$	0	0
$325$	−2.50777	−0.139106
$326$	0	0
$327$	10.2269	0.565550
$328$	0	0
$329$	−1.99929	−0.110224
$330$	0	0
$331$	29.7878	1.63729	0.818644	−	0.574302i	$-0.194726\pi$
$331$	29.7878	1.63729	0.818644	+	0.574302i	$0.194726\pi$
$332$	0	0
$333$	−0.0498904	−0.00273398
$334$	0	0
$335$	5.43293	0.296832
$336$	0	0
$337$	−29.8351	−1.62522	−0.812610	−	0.582808i	$-0.801954\pi$
$337$	−29.8351	−1.62522	−0.812610	+	0.582808i	$0.801954\pi$
$338$	0	0
$339$	5.91933	0.321494
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−15.5678	−0.840581
$344$	0	0
$345$	−3.82721	−0.206050
$346$	0	0
$347$	22.2170	1.19267	0.596336	−	0.802735i	$-0.296623\pi$
$347$	22.2170	1.19267	0.596336	+	0.802735i	$0.296623\pi$
$348$	0	0
$349$	−19.2134	−1.02847	−0.514234	−	0.857650i	$-0.671924\pi$
$349$	−19.2134	−1.02847	−0.514234	+	0.857650i	$0.671924\pi$
$350$	0	0
$351$	13.0173	0.694811
$352$	0	0
$353$	31.0691	1.65364	0.826821	−	0.562466i	$-0.190147\pi$
$353$	31.0691	1.65364	0.826821	+	0.562466i	$0.190147\pi$
$354$	0	0
$355$	−4.63664	−0.246087
$356$	0	0
$357$	−7.82521	−0.414154
$358$	0	0
$359$	27.5365	1.45332	0.726660	−	0.686998i	$-0.241072\pi$
$359$	27.5365	1.45332	0.726660	+	0.686998i	$0.241072\pi$
$360$	0	0
$361$	30.8104	1.62160
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.03474	0.106503
$366$	0	0
$367$	−23.7891	−1.24178	−0.620890	−	0.783898i	$-0.713229\pi$
$367$	−23.7891	−1.24178	−0.620890	+	0.783898i	$0.713229\pi$
$368$	0	0
$369$	−0.0638179	−0.00332223
$370$	0	0
$371$	2.22140	0.115329
$372$	0	0
$373$	−15.5085	−0.803000	−0.401500	−	0.915859i	$-0.631511\pi$
$373$	−15.5085	−0.803000	−0.401500	+	0.915859i	$0.631511\pi$
$374$	0	0
$375$	−1.73383	−0.0895348
$376$	0	0
$377$	−6.76997	−0.348671
$378$	0	0
$379$	19.0461	0.978335	0.489167	−	0.872190i	$-0.337301\pi$
$379$	19.0461	0.978335	0.489167	+	0.872190i	$0.337301\pi$
$380$	0	0
$381$	−10.7360	−0.550023
$382$	0	0
$383$	15.8224	0.808488	0.404244	−	0.914651i	$-0.367535\pi$
$383$	15.8224	0.808488	0.404244	+	0.914651i	$0.367535\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.0425540	−0.00216314
$388$	0	0
$389$	−16.2272	−0.822752	−0.411376	−	0.911466i	$-0.634952\pi$
$389$	−16.2272	−0.822752	−0.411376	+	0.911466i	$0.634952\pi$
$390$	0	0
$391$	−7.95560	−0.402332
$392$	0	0
$393$	12.8291	0.647140
$394$	0	0
$395$	16.7625	0.843415
$396$	0	0
$397$	31.3002	1.57091	0.785455	−	0.618919i	$-0.212429\pi$
$397$	31.3002	1.57091	0.785455	+	0.618919i	$0.212429\pi$
$398$	0	0
$399$	15.3235	0.767134
$400$	0	0
$401$	−6.86096	−0.342620	−0.171310	−	0.985217i	$-0.554800\pi$
$401$	−6.86096	−0.342620	−0.171310	+	0.985217i	$0.554800\pi$
$402$	0	0
$403$	−7.10564	−0.353957
$404$	0	0
$405$	9.01850	0.448133
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−22.8291	−1.12882	−0.564412	−	0.825493i	$-0.690897\pi$
$409$	−22.8291	−1.12882	−0.564412	+	0.825493i	$0.690897\pi$
$410$	0	0
$411$	32.0464	1.58073
$412$	0	0
$413$	5.47091	0.269206
$414$	0	0
$415$	−5.62561	−0.276150
$416$	0	0
$417$	30.9358	1.51493
$418$	0	0
$419$	−5.98106	−0.292194	−0.146097	−	0.989270i	$-0.546671\pi$
$419$	−5.98106	−0.292194	−0.146097	+	0.989270i	$0.546671\pi$
$420$	0	0
$421$	37.8255	1.84350	0.921751	−	0.387781i	$-0.126758\pi$
$421$	37.8255	1.84350	0.921751	+	0.387781i	$0.126758\pi$
$422$	0	0
$423$	−0.00986669	−0.000479735 0
$424$	0	0
$425$	−3.60412	−0.174825
$426$	0	0
$427$	−10.4353	−0.504999
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6.51913	−0.314015	−0.157008	−	0.987597i	$-0.550185\pi$
$431$	−6.51913	−0.314015	−0.157008	+	0.987597i	$0.550185\pi$
$432$	0	0
$433$	−29.0198	−1.39460	−0.697301	−	0.716778i	$-0.745616\pi$
$433$	−29.0198	−1.39460	−0.697301	+	0.716778i	$0.745616\pi$
$434$	0	0
$435$	−4.68066	−0.224421
$436$	0	0
$437$	15.5788	0.745236
$438$	0	0
$439$	−12.2777	−0.585984	−0.292992	−	0.956115i	$-0.594651\pi$
$439$	−12.2777	−0.585984	−0.292992	+	0.956115i	$0.594651\pi$
$440$	0	0
$441$	−0.0335688	−0.00159851
$442$	0	0
$443$	−8.75445	−0.415936	−0.207968	−	0.978136i	$-0.566685\pi$
$443$	−8.75445	−0.415936	−0.207968	+	0.978136i	$0.566685\pi$
$444$	0	0
$445$	4.72832	0.224144
$446$	0	0
$447$	−12.8288	−0.606782
$448$	0	0
$449$	−21.0104	−0.991543	−0.495771	−	0.868453i	$-0.665115\pi$
$449$	−21.0104	−0.991543	−0.495771	+	0.868453i	$0.665115\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−7.65513	−0.359670
$454$	0	0
$455$	3.14034	0.147222
$456$	0	0
$457$	−13.8185	−0.646402	−0.323201	−	0.946330i	$-0.604759\pi$
$457$	−13.8185	−0.646402	−0.323201	+	0.946330i	$0.604759\pi$
$458$	0	0
$459$	18.7082	0.873224
$460$	0	0
$461$	−38.6522	−1.80021	−0.900106	−	0.435672i	$-0.856511\pi$
$461$	−38.6522	−1.80021	−0.900106	+	0.435672i	$0.856511\pi$
$462$	0	0
$463$	15.0206	0.698066	0.349033	−	0.937110i	$-0.386510\pi$
$463$	15.0206	0.698066	0.349033	+	0.937110i	$0.386510\pi$
$464$	0	0
$465$	−4.91274	−0.227823
$466$	0	0
$467$	−11.7903	−0.545591	−0.272796	−	0.962072i	$-0.587948\pi$
$467$	−11.7903	−0.545591	−0.272796	+	0.962072i	$0.587948\pi$
$468$	0	0
$469$	−6.80337	−0.314150
$470$	0	0
$471$	−4.04082	−0.186191
$472$	0	0
$473$	0	0
$474$	0	0
$475$	7.05765	0.323827
$476$	0	0
$477$	0.0109628	0.000501953 0
$478$	0	0
$479$	−18.3651	−0.839125	−0.419562	−	0.907726i	$-0.637816\pi$
$479$	−18.3651	−0.839125	−0.419562	+	0.907726i	$0.637816\pi$
$480$	0	0
$481$	20.2450	0.923094
$482$	0	0
$483$	4.79261	0.218071
$484$	0	0
$485$	−19.0943	−0.867026
$486$	0	0
$487$	−22.3361	−1.01214	−0.506072	−	0.862492i	$-0.668903\pi$
$487$	−22.3361	−1.01214	−0.506072	+	0.862492i	$0.668903\pi$
$488$	0	0
$489$	−2.61142	−0.118092
$490$	0	0
$491$	−8.71330	−0.393226	−0.196613	−	0.980481i	$-0.562994\pi$
$491$	−8.71330	−0.393226	−0.196613	+	0.980481i	$0.562994\pi$
$492$	0	0
$493$	−9.72968	−0.438203
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.80621	0.260444
$498$	0	0
$499$	−38.9860	−1.74525	−0.872626	−	0.488390i	$-0.837585\pi$
$499$	−38.9860	−1.74525	−0.872626	+	0.488390i	$0.837585\pi$
$500$	0	0
$501$	−16.9869	−0.758920
$502$	0	0
$503$	−12.7798	−0.569821	−0.284910	−	0.958554i	$-0.591964\pi$
$503$	−12.7798	−0.569821	−0.284910	+	0.958554i	$0.591964\pi$
$504$	0	0
$505$	14.2252	0.633015
$506$	0	0
$507$	−11.6360	−0.516771
$508$	0	0
$509$	36.9761	1.63894	0.819469	−	0.573124i	$-0.194269\pi$
$509$	36.9761	1.63894	0.819469	+	0.573124i	$0.194269\pi$
$510$	0	0
$511$	−2.54800	−0.112717
$512$	0	0
$513$	−36.6348	−1.61746
$514$	0	0
$515$	−15.8815	−0.699823
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−34.8390	−1.52926
$520$	0	0
$521$	−8.97119	−0.393035	−0.196518	−	0.980500i	$-0.562963\pi$
$521$	−8.97119	−0.393035	−0.196518	+	0.980500i	$0.562963\pi$
$522$	0	0
$523$	−3.09464	−0.135319	−0.0676596	−	0.997708i	$-0.521553\pi$
$523$	−3.09464	−0.135319	−0.0676596	+	0.997708i	$0.521553\pi$
$524$	0	0
$525$	2.17119	0.0947584
$526$	0	0
$527$	−10.2121	−0.444846
$528$	0	0
$529$	−18.1275	−0.788154
$530$	0	0
$531$	0.0269994	0.00117168
$532$	0	0
$533$	25.8967	1.12171
$534$	0	0
$535$	−4.04935	−0.175069
$536$	0	0
$537$	−28.0966	−1.21246
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−18.8882	−0.812065	−0.406033	−	0.913859i	$-0.633088\pi$
$541$	−18.8882	−0.812065	−0.406033	+	0.913859i	$0.633088\pi$
$542$	0	0
$543$	−16.3478	−0.701553
$544$	0	0
$545$	−5.89844	−0.252661
$546$	0	0
$547$	34.0440	1.45562	0.727808	−	0.685781i	$-0.240539\pi$
$547$	34.0440	1.45562	0.727808	+	0.685781i	$0.240539\pi$
$548$	0	0
$549$	−0.0514991	−0.00219793
$550$	0	0
$551$	19.0528	0.811678
$552$	0	0
$553$	−20.9908	−0.892622
$554$	0	0
$555$	13.9971	0.594145
$556$	0	0
$557$	19.8622	0.841589	0.420795	−	0.907156i	$-0.361751\pi$
$557$	19.8622	0.841589	0.420795	+	0.907156i	$0.361751\pi$
$558$	0	0
$559$	17.2680	0.730359
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.7516	1.00101	0.500505	−	0.865734i	$-0.333148\pi$
$563$	23.7516	1.00101	0.500505	+	0.865734i	$0.333148\pi$
$564$	0	0
$565$	−3.41401	−0.143628
$566$	0	0
$567$	−11.2934	−0.474278
$568$	0	0
$569$	36.4083	1.52632	0.763158	−	0.646211i	$-0.223648\pi$
$569$	36.4083	1.52632	0.763158	+	0.646211i	$0.223648\pi$
$570$	0	0
$571$	1.49673	0.0626362	0.0313181	−	0.999509i	$-0.490030\pi$
$571$	1.49673	0.0626362	0.0313181	+	0.999509i	$0.490030\pi$
$572$	0	0
$573$	−14.1751	−0.592172
$574$	0	0
$575$	2.20737	0.0920536
$576$	0	0
$577$	41.2565	1.71753	0.858766	−	0.512368i	$-0.171232\pi$
$577$	41.2565	1.71753	0.858766	+	0.512368i	$0.171232\pi$
$578$	0	0
$579$	−8.66082	−0.359931
$580$	0	0
$581$	7.04466	0.292262
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0.0154979	0.000640759 0
$586$	0	0
$587$	−41.6959	−1.72097	−0.860486	−	0.509474i	$-0.829840\pi$
$587$	−41.6959	−1.72097	−0.860486	+	0.509474i	$0.829840\pi$
$588$	0	0
$589$	19.9975	0.823984
$590$	0	0
$591$	−12.1505	−0.499806
$592$	0	0
$593$	16.0656	0.659736	0.329868	−	0.944027i	$-0.392996\pi$
$593$	16.0656	0.659736	0.329868	+	0.944027i	$0.392996\pi$
$594$	0	0
$595$	4.51324	0.185025
$596$	0	0
$597$	−32.7282	−1.33948
$598$	0	0
$599$	25.4316	1.03911	0.519553	−	0.854438i	$-0.326098\pi$
$599$	25.4316	1.03911	0.519553	+	0.854438i	$0.326098\pi$
$600$	0	0
$601$	13.4615	0.549104	0.274552	−	0.961572i	$-0.411470\pi$
$601$	13.4615	0.549104	0.274552	+	0.961572i	$0.411470\pi$
$602$	0	0
$603$	−0.0335753	−0.00136729
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2.05778	0.0835226	0.0417613	−	0.999128i	$-0.486703\pi$
$607$	2.05778	0.0835226	0.0417613	+	0.999128i	$0.486703\pi$
$608$	0	0
$609$	5.86134	0.237514
$610$	0	0
$611$	4.00380	0.161977
$612$	0	0
$613$	−38.3437	−1.54869	−0.774344	−	0.632765i	$-0.781920\pi$
$613$	−38.3437	−1.54869	−0.774344	+	0.632765i	$0.781920\pi$
$614$	0	0
$615$	17.9046	0.721982
$616$	0	0
$617$	7.25724	0.292166	0.146083	−	0.989272i	$-0.453333\pi$
$617$	7.25724	0.292166	0.146083	+	0.989272i	$0.453333\pi$
$618$	0	0
$619$	17.6113	0.707860	0.353930	−	0.935272i	$-0.384845\pi$
$619$	17.6113	0.707860	0.353930	+	0.935272i	$0.384845\pi$
$620$	0	0
$621$	−11.4580	−0.459793
$622$	0	0
$623$	−5.92102	−0.237221
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	29.0958	1.16013
$630$	0	0
$631$	28.4353	1.13199	0.565996	−	0.824408i	$-0.308492\pi$
$631$	28.4353	1.13199	0.565996	+	0.824408i	$0.308492\pi$
$632$	0	0
$633$	−23.8727	−0.948857
$634$	0	0
$635$	6.19208	0.245725
$636$	0	0
$637$	13.6219	0.539718
$638$	0	0
$639$	0.0286542	0.00113354
$640$	0	0
$641$	−42.5129	−1.67916	−0.839579	−	0.543237i	$-0.817199\pi$
$641$	−42.5129	−1.67916	−0.839579	+	0.543237i	$0.817199\pi$
$642$	0	0
$643$	22.9742	0.906015	0.453007	−	0.891507i	$-0.350351\pi$
$643$	22.9742	0.906015	0.453007	+	0.891507i	$0.350351\pi$
$644$	0	0
$645$	11.9389	0.470092
$646$	0	0
$647$	14.2632	0.560746	0.280373	−	0.959891i	$-0.409542\pi$
$647$	14.2632	0.560746	0.280373	+	0.959891i	$0.409542\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	6.15196	0.241115
$652$	0	0
$653$	−26.8378	−1.05024	−0.525122	−	0.851027i	$-0.675980\pi$
$653$	−26.8378	−1.05024	−0.525122	+	0.851027i	$0.675980\pi$
$654$	0	0
$655$	−7.39924	−0.289112
$656$	0	0
$657$	−0.0125746	−0.000490582 0
$658$	0	0
$659$	−28.8747	−1.12480	−0.562400	−	0.826865i	$-0.690122\pi$
$659$	−28.8747	−1.12480	−0.562400	+	0.826865i	$0.690122\pi$
$660$	0	0
$661$	−23.6938	−0.921582	−0.460791	−	0.887509i	$-0.652434\pi$
$661$	−23.6938	−0.921582	−0.460791	+	0.887509i	$0.652434\pi$
$662$	0	0
$663$	15.6709	0.608606
$664$	0	0
$665$	−8.83792	−0.342720
$666$	0	0
$667$	5.95901	0.230734
$668$	0	0
$669$	−49.9302	−1.93041
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−5.90541	−0.227637	−0.113818	−	0.993502i	$-0.536308\pi$
$673$	−5.90541	−0.227637	−0.113818	+	0.993502i	$0.536308\pi$
$674$	0	0
$675$	−5.19079	−0.199793
$676$	0	0
$677$	30.4595	1.17065	0.585327	−	0.810797i	$-0.300966\pi$
$677$	30.4595	1.17065	0.585327	+	0.810797i	$0.300966\pi$
$678$	0	0
$679$	23.9108	0.917610
$680$	0	0
$681$	−42.0921	−1.61297
$682$	0	0
$683$	2.95068	0.112905	0.0564524	−	0.998405i	$-0.482021\pi$
$683$	2.95068	0.112905	0.0564524	+	0.998405i	$0.482021\pi$
$684$	0	0
$685$	−18.4830	−0.706198
$686$	0	0
$687$	−28.4233	−1.08442
$688$	0	0
$689$	−4.44860	−0.169478
$690$	0	0
$691$	38.6147	1.46897	0.734487	−	0.678623i	$-0.237423\pi$
$691$	38.6147	1.46897	0.734487	+	0.678623i	$0.237423\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−17.8424	−0.676801
$696$	0	0
$697$	37.2182	1.40974
$698$	0	0
$699$	−4.36655	−0.165158
$700$	0	0
$701$	33.1129	1.25066	0.625328	−	0.780362i	$-0.284965\pi$
$701$	33.1129	1.25066	0.625328	+	0.780362i	$0.284965\pi$
$702$	0	0
$703$	−56.9760	−2.14889
$704$	0	0
$705$	2.76817	0.104255
$706$	0	0
$707$	−17.8135	−0.669946
$708$	0	0
$709$	21.9947	0.826030	0.413015	−	0.910724i	$-0.364476\pi$
$709$	21.9947	0.826030	0.413015	+	0.910724i	$0.364476\pi$
$710$	0	0
$711$	−0.103592	−0.00388500
$712$	0	0
$713$	6.25447	0.234232
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−11.5790	−0.432424
$718$	0	0
$719$	5.99121	0.223434	0.111717	−	0.993740i	$-0.464365\pi$
$719$	5.99121	0.223434	0.111717	+	0.993740i	$0.464365\pi$
$720$	0	0
$721$	19.8876	0.740652
$722$	0	0
$723$	−3.71579	−0.138192
$724$	0	0
$725$	2.69960	0.100261
$726$	0	0
$727$	4.06512	0.150767	0.0753834	−	0.997155i	$-0.475982\pi$
$727$	4.06512	0.150767	0.0753834	+	0.997155i	$0.475982\pi$
$728$	0	0
$729$	26.9442	0.997932
$730$	0	0
$731$	24.8173	0.917899
$732$	0	0
$733$	−29.9274	−1.10539	−0.552697	−	0.833382i	$-0.686401\pi$
$733$	−29.9274	−1.10539	−0.552697	+	0.833382i	$0.686401\pi$
$734$	0	0
$735$	9.41797	0.347387
$736$	0	0
$737$	0	0
$738$	0	0
$739$	19.6749	0.723755	0.361877	−	0.932226i	$-0.382136\pi$
$739$	19.6749	0.723755	0.361877	+	0.932226i	$0.382136\pi$
$740$	0	0
$741$	−30.6870	−1.12732
$742$	0	0
$743$	26.6733	0.978548	0.489274	−	0.872130i	$-0.337262\pi$
$743$	26.6733	0.978548	0.489274	+	0.872130i	$0.337262\pi$
$744$	0	0
$745$	7.39910	0.271082
$746$	0	0
$747$	0.0347660	0.00127202
$748$	0	0
$749$	5.07079	0.185282
$750$	0	0
$751$	−7.86537	−0.287011	−0.143506	−	0.989649i	$-0.545838\pi$
$751$	−7.86537	−0.287011	−0.143506	+	0.989649i	$0.545838\pi$
$752$	0	0
$753$	−24.9509	−0.909260
$754$	0	0
$755$	4.41515	0.160684
$756$	0	0
$757$	−24.4559	−0.888866	−0.444433	−	0.895812i	$-0.646595\pi$
$757$	−24.4559	−0.888866	−0.444433	+	0.895812i	$0.646595\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−52.2028	−1.89235	−0.946174	−	0.323658i	$-0.895087\pi$
$761$	−52.2028	−1.89235	−0.946174	+	0.323658i	$0.895087\pi$
$762$	0	0
$763$	7.38631	0.267402
$764$	0	0
$765$	0.0222733	0.000805292 0
$766$	0	0
$767$	−10.9561	−0.395602
$768$	0	0
$769$	26.7480	0.964558	0.482279	−	0.876018i	$-0.339809\pi$
$769$	26.7480	0.964558	0.482279	+	0.876018i	$0.339809\pi$
$770$	0	0
$771$	42.3114	1.52381
$772$	0	0
$773$	−50.2424	−1.80709	−0.903546	−	0.428490i	$-0.859046\pi$
$773$	−50.2424	−1.80709	−0.903546	+	0.428490i	$0.859046\pi$
$774$	0	0
$775$	2.83345	0.101781
$776$	0	0
$777$	−17.5279	−0.628809
$778$	0	0
$779$	−72.8814	−2.61125
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−14.0131	−0.500786
$784$	0	0
$785$	2.33057	0.0831816
$786$	0	0
$787$	32.3414	1.15285	0.576423	−	0.817152i	$-0.304448\pi$
$787$	32.3414	1.15285	0.576423	+	0.817152i	$0.304448\pi$
$788$	0	0
$789$	−1.72162	−0.0612914
$790$	0	0
$791$	4.27518	0.152008
$792$	0	0
$793$	20.8978	0.742103
$794$	0	0
$795$	−3.07570	−0.109084
$796$	0	0
$797$	38.6252	1.36817	0.684087	−	0.729400i	$-0.260201\pi$
$797$	38.6252	1.36817	0.684087	+	0.729400i	$0.260201\pi$
$798$	0	0
$799$	5.75420	0.203569
$800$	0	0
$801$	−0.0292208	−0.00103247
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−2.76417	−0.0974242
$806$	0	0
$807$	52.7485	1.85684
$808$	0	0
$809$	−29.6195	−1.04137	−0.520684	−	0.853750i	$-0.674323\pi$
$809$	−29.6195	−1.04137	−0.520684	+	0.853750i	$0.674323\pi$
$810$	0	0
$811$	14.6182	0.513315	0.256657	−	0.966502i	$-0.417379\pi$
$811$	14.6182	0.513315	0.256657	+	0.966502i	$0.417379\pi$
$812$	0	0
$813$	−44.9799	−1.57751
$814$	0	0
$815$	1.50615	0.0527583
$816$	0	0
$817$	−48.5976	−1.70022
$818$	0	0
$819$	−0.0194072	−0.000678142 0
$820$	0	0
$821$	27.9890	0.976822	0.488411	−	0.872614i	$-0.337577\pi$
$821$	27.9890	0.976822	0.488411	+	0.872614i	$0.337577\pi$
$822$	0	0
$823$	−17.6590	−0.615556	−0.307778	−	0.951458i	$-0.599585\pi$
$823$	−17.6590	−0.615556	−0.307778	+	0.951458i	$0.599585\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.87712	0.134821	0.0674103	−	0.997725i	$-0.478526\pi$
$827$	3.87712	0.134821	0.0674103	+	0.997725i	$0.478526\pi$
$828$	0	0
$829$	−20.8274	−0.723366	−0.361683	−	0.932301i	$-0.617798\pi$
$829$	−20.8274	−0.723366	−0.361683	+	0.932301i	$0.617798\pi$
$830$	0	0
$831$	−8.73593	−0.303046
$832$	0	0
$833$	19.5771	0.678307
$834$	0	0
$835$	9.79732	0.339050
$836$	0	0
$837$	−14.7079	−0.508378
$838$	0	0
$839$	29.2063	1.00831	0.504157	−	0.863612i	$-0.331803\pi$
$839$	29.2063	1.00831	0.504157	+	0.863612i	$0.331803\pi$
$840$	0	0
$841$	−21.7122	−0.748695
$842$	0	0
$843$	36.0185	1.24054
$844$	0	0
$845$	6.71111	0.230869
$846$	0	0
$847$	0	0
$848$	0	0
$849$	7.89985	0.271122
$850$	0	0
$851$	−17.8199	−0.610860
$852$	0	0
$853$	57.3270	1.96284	0.981420	−	0.191870i	$-0.0614553\pi$
$853$	57.3270	1.96284	0.981420	+	0.191870i	$0.0614553\pi$
$854$	0	0
$855$	−0.0436160	−0.00149163
$856$	0	0
$857$	43.7938	1.49597	0.747984	−	0.663717i	$-0.231022\pi$
$857$	43.7938	1.49597	0.747984	+	0.663717i	$0.231022\pi$
$858$	0	0
$859$	32.0382	1.09313	0.546564	−	0.837417i	$-0.315935\pi$
$859$	32.0382	1.09313	0.546564	+	0.837417i	$0.315935\pi$
$860$	0	0
$861$	−22.4210	−0.764105
$862$	0	0
$863$	−50.5619	−1.72115	−0.860574	−	0.509325i	$-0.829895\pi$
$863$	−50.5619	−1.72115	−0.860574	+	0.509325i	$0.829895\pi$
$864$	0	0
$865$	20.0936	0.683204
$866$	0	0
$867$	−6.95329	−0.236146
$868$	0	0
$869$	0	0
$870$	0	0
$871$	13.6245	0.461649
$872$	0	0
$873$	0.118002	0.00399376
$874$	0	0
$875$	−1.25225	−0.0423337
$876$	0	0
$877$	19.9976	0.675270	0.337635	−	0.941277i	$-0.390373\pi$
$877$	19.9976	0.675270	0.337635	+	0.941277i	$0.390373\pi$
$878$	0	0
$879$	−4.14545	−0.139823
$880$	0	0
$881$	−19.7712	−0.666108	−0.333054	−	0.942908i	$-0.608079\pi$
$881$	−19.7712	−0.666108	−0.333054	+	0.942908i	$0.608079\pi$
$882$	0	0
$883$	8.72958	0.293774	0.146887	−	0.989153i	$-0.453075\pi$
$883$	8.72958	0.293774	0.146887	+	0.989153i	$0.453075\pi$
$884$	0	0
$885$	−7.57490	−0.254627
$886$	0	0
$887$	50.1444	1.68368	0.841842	−	0.539724i	$-0.181471\pi$
$887$	50.1444	1.68368	0.841842	+	0.539724i	$0.181471\pi$
$888$	0	0
$889$	−7.75401	−0.260061
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−11.2680	−0.377069
$894$	0	0
$895$	16.2049	0.541671
$896$	0	0
$897$	−9.59774	−0.320459
$898$	0	0
$899$	7.64920	0.255115
$900$	0	0
$901$	−6.39345	−0.212997
$902$	0	0
$903$	−14.9504	−0.497518
$904$	0	0
$905$	9.42873	0.313422
$906$	0	0
$907$	−21.5643	−0.716031	−0.358015	−	0.933716i	$-0.616547\pi$
$907$	−21.5643	−0.716031	−0.358015	+	0.933716i	$0.616547\pi$
$908$	0	0
$909$	−0.0879113	−0.00291584
$910$	0	0
$911$	−25.9189	−0.858731	−0.429365	−	0.903131i	$-0.641263\pi$
$911$	−25.9189	−0.858731	−0.429365	+	0.903131i	$0.641263\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	14.4485	0.477651
$916$	0	0
$917$	9.26568	0.305980
$918$	0	0
$919$	−13.1654	−0.434286	−0.217143	−	0.976140i	$-0.569674\pi$
$919$	−13.1654	−0.434286	−0.217143	+	0.976140i	$0.569674\pi$
$920$	0	0
$921$	−5.66353	−0.186620
$922$	0	0
$923$	−11.6276	−0.382727
$924$	0	0
$925$	−8.07294	−0.265437
$926$	0	0
$927$	0.0981471	0.00322357
$928$	0	0
$929$	40.2771	1.32145	0.660725	−	0.750628i	$-0.270249\pi$
$929$	40.2771	1.32145	0.660725	+	0.750628i	$0.270249\pi$
$930$	0	0
$931$	−38.3363	−1.25642
$932$	0	0
$933$	−4.74319	−0.155285
$934$	0	0
$935$	0	0
$936$	0	0
$937$	37.3771	1.22106	0.610528	−	0.791995i	$-0.290957\pi$
$937$	37.3771	1.22106	0.610528	+	0.791995i	$0.290957\pi$
$938$	0	0
$939$	−17.9600	−0.586103
$940$	0	0
$941$	−38.2224	−1.24601	−0.623007	−	0.782216i	$-0.714089\pi$
$941$	−38.2224	−1.24601	−0.623007	+	0.782216i	$0.714089\pi$
$942$	0	0
$943$	−22.7946	−0.742293
$944$	0	0
$945$	6.50015	0.211450
$946$	0	0
$947$	58.9505	1.91564	0.957818	−	0.287377i	$-0.0927832\pi$
$947$	58.9505	1.91564	0.957818	+	0.287377i	$0.0927832\pi$
$948$	0	0
$949$	5.10265	0.165639
$950$	0	0
$951$	54.1926	1.75732
$952$	0	0
$953$	20.6255	0.668126	0.334063	−	0.942551i	$-0.391580\pi$
$953$	20.6255	0.668126	0.334063	+	0.942551i	$0.391580\pi$
$954$	0	0
$955$	8.17556	0.264555
$956$	0	0
$957$	0	0
$958$	0	0
$959$	23.1452	0.747399
$960$	0	0
$961$	−22.9715	−0.741017
$962$	0	0
$963$	0.0250248	0.000806413 0
$964$	0	0
$965$	4.99518	0.160801
$966$	0	0
$967$	35.8174	1.15181	0.575904	−	0.817517i	$-0.304650\pi$
$967$	35.8174	1.15181	0.575904	+	0.817517i	$0.304650\pi$
$968$	0	0
$969$	−44.1028	−1.41679
$970$	0	0
$971$	−11.3094	−0.362937	−0.181469	−	0.983397i	$-0.558085\pi$
$971$	−11.3094	−0.362937	−0.181469	+	0.983397i	$0.558085\pi$
$972$	0	0
$973$	22.3431	0.716287
$974$	0	0
$975$	−4.34805	−0.139249
$976$	0	0
$977$	23.2137	0.742671	0.371336	−	0.928499i	$-0.378900\pi$
$977$	23.2137	0.742671	0.371336	+	0.928499i	$0.378900\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0.0364521	0.00116383
$982$	0	0
$983$	−34.1621	−1.08960	−0.544801	−	0.838565i	$-0.683395\pi$
$983$	−34.1621	−1.08960	−0.544801	+	0.838565i	$0.683395\pi$
$984$	0	0
$985$	7.00790	0.223290
$986$	0	0
$987$	−3.46644	−0.110338
$988$	0	0
$989$	−15.1995	−0.483316
$990$	0	0
$991$	27.7516	0.881560	0.440780	−	0.897615i	$-0.354702\pi$
$991$	27.7516	0.881560	0.440780	+	0.897615i	$0.354702\pi$
$992$	0	0
$993$	51.6472	1.63897
$994$	0	0
$995$	18.8762	0.598416
$996$	0	0
$997$	−43.1978	−1.36809	−0.684044	−	0.729440i	$-0.739781\pi$
$997$	−43.1978	−1.36809	−0.684044	+	0.729440i	$0.739781\pi$
$998$	0	0
$999$	41.9049	1.32581

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.cx.1.5		6
4.3	odd	2		4840.2.a.bf.1.2		6
11.5	even	5		880.2.bo.j.641.3		12
11.9	even	5		880.2.bo.j.81.3		12
11.10	odd	2		9680.2.a.cy.1.5		6
44.27	odd	10		440.2.y.b.201.1	yes	12
44.31	odd	10		440.2.y.b.81.1	✓	12
44.43	even	2		4840.2.a.be.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.y.b.81.1	✓	12	44.31	odd	10
440.2.y.b.201.1	yes	12	44.27	odd	10
880.2.bo.j.81.3		12	11.9	even	5
880.2.bo.j.641.3		12	11.5	even	5
4840.2.a.be.1.2		6	44.43	even	2
4840.2.a.bf.1.2		6	4.3	odd	2
9680.2.a.cx.1.5		6	1.1	even	1	trivial
9680.2.a.cy.1.5		6	11.10	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:	\( N \)	\(=\)	\( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9680.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73383 q^{3} -1.00000 q^{5} +1.25225 q^{7} +0.00617996 q^{9} -2.50777 q^{13} -1.73383 q^{15} -3.60412 q^{17} +7.05765 q^{19} +2.17119 q^{21} +2.20737 q^{23} +1.00000 q^{25} -5.19079 q^{27} +2.69960 q^{29} +2.83345 q^{31} -1.25225 q^{35} -8.07294 q^{37} -4.34805 q^{39} -10.3266 q^{41} -6.88581 q^{43} -0.00617996 q^{45} -1.59656 q^{47} -5.43188 q^{49} -6.24894 q^{51} +1.77393 q^{53} +12.2368 q^{57} +4.36887 q^{59} -8.33324 q^{61} +0.00773884 q^{63} +2.50777 q^{65} -5.43293 q^{67} +3.82721 q^{69} +4.63664 q^{71} -2.03474 q^{73} +1.73383 q^{75} -16.7625 q^{79} -9.01850 q^{81} +5.62561 q^{83} +3.60412 q^{85} +4.68066 q^{87} -4.72832 q^{89} -3.14034 q^{91} +4.91274 q^{93} -7.05765 q^{95} +19.0943 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 6 q^{5} - 6 q^{7} + 10 q^{9} - 6 q^{13} + 2 q^{15} + 11 q^{17} + 11 q^{19} + 2 q^{21} - 18 q^{23} + 6 q^{25} + q^{27} - 6 q^{29} - q^{31} + 6 q^{35} + 4 q^{37} - 27 q^{39} - 4 q^{41} - 3 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) \) 6 * q - 2 * q^3 - 6 * q^5 - 6 * q^7 + 10 * q^9 - 6 * q^13 + 2 * q^15 + 11 * q^17 + 11 * q^19 + 2 * q^21 - 18 * q^23 + 6 * q^25 + q^27 - 6 * q^29 - q^31 + 6 * q^35 + 4 * q^37 - 27 * q^39 - 4 * q^41 - 3 * q^43 - 10 * q^45 - 14 * q^47 + 8 * q^49 - 31 * q^51 + 14 * q^53 - 5 * q^57 - 2 * q^59 - 4 * q^61 + 16 * q^63 + 6 * q^65 - 11 * q^67 + 8 * q^69 - 7 * q^71 - 9 * q^73 - 2 * q^75 - 36 * q^79 + 30 * q^81 + 45 * q^83 - 11 * q^85 - 25 * q^87 + q^89 + 8 * q^91 + 55 * q^93 - 11 * q^95 + 20 * q^97

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