LMFDB - Embedded newform 9648.2.a.bj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9648,2,Mod(1,9648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9648, 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("9648.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9648 = 2^{4} \cdot 3^{2} \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9648.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:	$77.0396678701$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 402)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.81361$ of defining polynomial
Character	$\chi$	$=$	9648.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.710831 q^{5} +1.81361 q^{7} +O(q^{10})$	$q-0.710831 q^{5} +1.81361 q^{7} +0.578337 q^{11} +4.72999 q^{13} -4.20555 q^{17} +2.00000 q^{19} -5.44082 q^{23} -4.49472 q^{25} +4.72999 q^{29} +2.18639 q^{31} -1.28917 q^{35} -8.07306 q^{37} -9.96526 q^{41} +3.75971 q^{43} -2.05390 q^{47} -3.71083 q^{49} -4.71083 q^{53} -0.411100 q^{55} +0.132494 q^{59} -6.15165 q^{61} -3.36222 q^{65} -1.00000 q^{67} +8.15165 q^{71} -5.49472 q^{73} +1.04888 q^{77} -2.35720 q^{79} -13.5925 q^{83} +2.98944 q^{85} +3.83276 q^{89} +8.57834 q^{91} -1.42166 q^{95} +6.78389 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} - q^{7}+O(q^{10}) $ 3 * q - 3 * q^5 - q^7	$ 3 q - 3 q^{5} - q^{7} - 6 q^{13} + 2 q^{17} + 6 q^{19} + 3 q^{23} + 2 q^{25} - 6 q^{29} + 13 q^{31} - 3 q^{35} - 7 q^{37} - 5 q^{41} + q^{43} - 10 q^{47} - 12 q^{49} - 15 q^{53} + 28 q^{55} + 3 q^{59} + 8 q^{65} - 3 q^{67} + 6 q^{71} - q^{73} - 8 q^{77} + 26 q^{79} - 3 q^{83} - 22 q^{85} - 16 q^{89} + 24 q^{91} - 6 q^{95} + 4 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 - q^7 - 6 * q^13 + 2 * q^17 + 6 * q^19 + 3 * q^23 + 2 * q^25 - 6 * q^29 + 13 * q^31 - 3 * q^35 - 7 * q^37 - 5 * q^41 + q^43 - 10 * q^47 - 12 * q^49 - 15 * q^53 + 28 * q^55 + 3 * q^59 + 8 * q^65 - 3 * q^67 + 6 * q^71 - q^73 - 8 * q^77 + 26 * q^79 - 3 * q^83 - 22 * q^85 - 16 * q^89 + 24 * q^91 - 6 * q^95 + 4 * q^97

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
$5$	−0.710831	−0.317893	−0.158947	−	0.987287i	$-0.550810\pi$
$5$	−0.710831	−0.317893	−0.158947	+	0.987287i	$0.550810\pi$
$6$	0	0
$7$	1.81361	0.685479	0.342739	−	0.939431i	$-0.388645\pi$
$7$	1.81361	0.685479	0.342739	+	0.939431i	$0.388645\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.578337	0.174375	0.0871876	−	0.996192i	$-0.472212\pi$
$11$	0.578337	0.174375	0.0871876	+	0.996192i	$0.472212\pi$
$12$	0	0
$13$	4.72999	1.31186	0.655931	−	0.754821i	$-0.272276\pi$
$13$	4.72999	1.31186	0.655931	+	0.754821i	$0.272276\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.20555	−1.02000	−0.509998	−	0.860176i	$-0.670354\pi$
$17$	−4.20555	−1.02000	−0.509998	+	0.860176i	$0.670354\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.44082	−1.13449	−0.567245	−	0.823549i	$-0.691991\pi$
$23$	−5.44082	−1.13449	−0.567245	+	0.823549i	$0.691991\pi$
$24$	0	0
$25$	−4.49472	−0.898944
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.72999	0.878337	0.439168	−	0.898405i	$-0.355273\pi$
$29$	4.72999	0.878337	0.439168	+	0.898405i	$0.355273\pi$
$30$	0	0
$31$	2.18639	0.392688	0.196344	−	0.980535i	$-0.437093\pi$
$31$	2.18639	0.392688	0.196344	+	0.980535i	$0.437093\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.28917	−0.217909
$36$	0	0
$37$	−8.07306	−1.32720	−0.663601	−	0.748087i	$-0.730973\pi$
$37$	−8.07306	−1.32720	−0.663601	+	0.748087i	$0.730973\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.96526	−1.55631	−0.778156	−	0.628071i	$-0.783845\pi$
$41$	−9.96526	−1.55631	−0.778156	+	0.628071i	$0.783845\pi$
$42$	0	0
$43$	3.75971	0.573350	0.286675	−	0.958028i	$-0.407450\pi$
$43$	3.75971	0.573350	0.286675	+	0.958028i	$0.407450\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.05390	−0.299592	−0.149796	−	0.988717i	$-0.547862\pi$
$47$	−2.05390	−0.299592	−0.149796	+	0.988717i	$0.547862\pi$
$48$	0	0
$49$	−3.71083	−0.530119
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.71083	−0.647082	−0.323541	−	0.946214i	$-0.604873\pi$
$53$	−4.71083	−0.647082	−0.323541	+	0.946214i	$0.604873\pi$
$54$	0	0
$55$	−0.411100	−0.0554327
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.132494	0.0172493	0.00862465	−	0.999963i	$-0.497255\pi$
$59$	0.132494	0.0172493	0.00862465	+	0.999963i	$0.497255\pi$
$60$	0	0
$61$	−6.15165	−0.787638	−0.393819	−	0.919188i	$-0.628846\pi$
$61$	−6.15165	−0.787638	−0.393819	+	0.919188i	$0.628846\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.36222	−0.417033
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.15165	0.967423	0.483711	−	0.875228i	$-0.339288\pi$
$71$	8.15165	0.967423	0.483711	+	0.875228i	$0.339288\pi$
$72$	0	0
$73$	−5.49472	−0.643108	−0.321554	−	0.946891i	$-0.604205\pi$
$73$	−5.49472	−0.643108	−0.321554	+	0.946891i	$0.604205\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.04888	0.119531
$78$	0	0
$79$	−2.35720	−0.265206	−0.132603	−	0.991169i	$-0.542334\pi$
$79$	−2.35720	−0.265206	−0.132603	+	0.991169i	$0.542334\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13.5925	−1.49197	−0.745984	−	0.665964i	$-0.768020\pi$
$83$	−13.5925	−1.49197	−0.745984	+	0.665964i	$0.768020\pi$
$84$	0	0
$85$	2.98944	0.324250
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.83276	0.406272	0.203136	−	0.979151i	$-0.434887\pi$
$89$	3.83276	0.406272	0.203136	+	0.979151i	$0.434887\pi$
$90$	0	0
$91$	8.57834	0.899254
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.42166	−0.145860
$96$	0	0
$97$	6.78389	0.688799	0.344400	−	0.938823i	$-0.388082\pi$
$97$	6.78389	0.688799	0.344400	+	0.938823i	$0.388082\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.25443	−0.920850	−0.460425	−	0.887699i	$-0.652303\pi$
$101$	−9.25443	−0.920850	−0.460425	+	0.887699i	$0.652303\pi$
$102$	0	0
$103$	9.45998	0.932119	0.466060	−	0.884753i	$-0.345673\pi$
$103$	9.45998	0.932119	0.466060	+	0.884753i	$0.345673\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.20555	0.599913	0.299957	−	0.953953i	$-0.403028\pi$
$107$	6.20555	0.599913	0.299957	+	0.953953i	$0.403028\pi$
$108$	0	0
$109$	2.15165	0.206091	0.103045	−	0.994677i	$-0.467141\pi$
$109$	2.15165	0.206091	0.103045	+	0.994677i	$0.467141\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−18.8816	−1.77624	−0.888118	−	0.459616i	$-0.847987\pi$
$113$	−18.8816	−1.77624	−0.888118	+	0.459616i	$0.847987\pi$
$114$	0	0
$115$	3.86751	0.360647
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.62721	−0.699185
$120$	0	0
$121$	−10.6655	−0.969593
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.74914	0.603662
$126$	0	0
$127$	−5.04888	−0.448015	−0.224008	−	0.974587i	$-0.571914\pi$
$127$	−5.04888	−0.448015	−0.224008	+	0.974587i	$0.571914\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.3380	1.25272	0.626360	−	0.779534i	$-0.284544\pi$
$131$	14.3380	1.25272	0.626360	+	0.779534i	$0.284544\pi$
$132$	0	0
$133$	3.62721	0.314519
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.9058	−1.18805	−0.594027	−	0.804445i	$-0.702463\pi$
$137$	−13.9058	−1.18805	−0.594027	+	0.804445i	$0.702463\pi$
$138$	0	0
$139$	15.3869	1.30510	0.652551	−	0.757745i	$-0.273699\pi$
$139$	15.3869	1.30510	0.652551	+	0.757745i	$0.273699\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.73553	0.228756
$144$	0	0
$145$	−3.36222	−0.279218
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.93554	0.240489	0.120244	−	0.992744i	$-0.461632\pi$
$149$	2.93554	0.240489	0.120244	+	0.992744i	$0.461632\pi$
$150$	0	0
$151$	8.67609	0.706050	0.353025	−	0.935614i	$-0.385153\pi$
$151$	8.67609	0.706050	0.353025	+	0.935614i	$0.385153\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.55416	−0.124833
$156$	0	0
$157$	−2.13249	−0.170192	−0.0850958	−	0.996373i	$-0.527120\pi$
$157$	−2.13249	−0.170192	−0.0850958	+	0.996373i	$0.527120\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−9.86751	−0.777668
$162$	0	0
$163$	−20.8222	−1.63092	−0.815460	−	0.578813i	$-0.803516\pi$
$163$	−20.8222	−1.63092	−0.815460	+	0.578813i	$0.803516\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.40251	−0.108529	−0.0542646	−	0.998527i	$-0.517281\pi$
$167$	−1.40251	−0.108529	−0.0542646	+	0.998527i	$0.517281\pi$
$168$	0	0
$169$	9.37279	0.720984
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−13.1028	−0.996186	−0.498093	−	0.867124i	$-0.665966\pi$
$173$	−13.1028	−0.996186	−0.498093	+	0.867124i	$0.665966\pi$
$174$	0	0
$175$	−8.15165	−0.616207
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.15667	0.235941	0.117970	−	0.993017i	$-0.462361\pi$
$179$	3.15667	0.235941	0.117970	+	0.993017i	$0.462361\pi$
$180$	0	0
$181$	−0.975820	−0.0725321	−0.0362661	−	0.999342i	$-0.511546\pi$
$181$	−0.975820	−0.0725321	−0.0362661	+	0.999342i	$0.511546\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.73858	0.421909
$186$	0	0
$187$	−2.43223	−0.177862
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.2056	−1.02788	−0.513939	−	0.857827i	$-0.671814\pi$
$191$	−14.2056	−1.02788	−0.513939	+	0.857827i	$0.671814\pi$
$192$	0	0
$193$	25.2197	1.81535	0.907676	−	0.419671i	$-0.137855\pi$
$193$	25.2197	1.81535	0.907676	+	0.419671i	$0.137855\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.5819	1.03892	0.519459	−	0.854495i	$-0.326134\pi$
$197$	14.5819	1.03892	0.519459	+	0.854495i	$0.326134\pi$
$198$	0	0
$199$	15.2544	1.08136	0.540679	−	0.841229i	$-0.318167\pi$
$199$	15.2544	1.08136	0.540679	+	0.841229i	$0.318167\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.57834	0.602081
$204$	0	0
$205$	7.08362	0.494741
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.15667	0.0800088
$210$	0	0
$211$	−27.3522	−1.88300	−0.941501	−	0.337011i	$-0.890584\pi$
$211$	−27.3522	−1.88300	−0.941501	+	0.337011i	$0.890584\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.67252	−0.182264
$216$	0	0
$217$	3.96526	0.269179
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−19.8922	−1.33809
$222$	0	0
$223$	−14.5783	−0.976238	−0.488119	−	0.872777i	$-0.662317\pi$
$223$	−14.5783	−0.976238	−0.488119	+	0.872777i	$0.662317\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−24.2786	−1.61143	−0.805714	−	0.592305i	$-0.798218\pi$
$227$	−24.2786	−1.61143	−0.805714	+	0.592305i	$0.798218\pi$
$228$	0	0
$229$	15.6116	1.03165	0.515823	−	0.856695i	$-0.327486\pi$
$229$	15.6116	1.03165	0.515823	+	0.856695i	$0.327486\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.49472	−0.359971	−0.179985	−	0.983669i	$-0.557605\pi$
$233$	−5.49472	−0.359971	−0.179985	+	0.983669i	$0.557605\pi$
$234$	0	0
$235$	1.45998	0.0952383
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.98944	0.193371	0.0966853	−	0.995315i	$-0.469176\pi$
$239$	2.98944	0.193371	0.0966853	+	0.995315i	$0.469176\pi$
$240$	0	0
$241$	21.7003	1.39784	0.698919	−	0.715201i	$-0.253665\pi$
$241$	21.7003	1.39784	0.698919	+	0.715201i	$0.253665\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.63778	0.168521
$246$	0	0
$247$	9.45998	0.601924
$248$	0	0
$249$	0	0
$250$	0	0
$251$	9.15667	0.577964	0.288982	−	0.957335i	$-0.406683\pi$
$251$	9.15667	0.577964	0.288982	+	0.957335i	$0.406683\pi$
$252$	0	0
$253$	−3.14663	−0.197827
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.57834	−0.285589	−0.142794	−	0.989752i	$-0.545609\pi$
$257$	−4.57834	−0.285589	−0.142794	+	0.989752i	$0.545609\pi$
$258$	0	0
$259$	−14.6413	−0.909769
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0.921921	0.0568481	0.0284240	−	0.999596i	$-0.490951\pi$
$263$	0.921921	0.0568481	0.0284240	+	0.999596i	$0.490951\pi$
$264$	0	0
$265$	3.34861	0.205703
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−23.7194	−1.44620	−0.723099	−	0.690744i	$-0.757283\pi$
$269$	−23.7194	−1.44620	−0.723099	+	0.690744i	$0.757283\pi$
$270$	0	0
$271$	27.8172	1.68977	0.844887	−	0.534945i	$-0.179668\pi$
$271$	27.8172	1.68977	0.844887	+	0.534945i	$0.179668\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.59946	−0.156753
$276$	0	0
$277$	−6.17081	−0.370768	−0.185384	−	0.982666i	$-0.559353\pi$
$277$	−6.17081	−0.370768	−0.185384	+	0.982666i	$0.559353\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.5089	0.984836	0.492418	−	0.870359i	$-0.336113\pi$
$281$	16.5089	0.984836	0.492418	+	0.870359i	$0.336113\pi$
$282$	0	0
$283$	−9.42166	−0.560060	−0.280030	−	0.959991i	$-0.590344\pi$
$283$	−9.42166	−0.560060	−0.280030	+	0.959991i	$0.590344\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−18.0731	−1.06682
$288$	0	0
$289$	0.686652	0.0403913
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0.583877	0.0341104	0.0170552	−	0.999855i	$-0.494571\pi$
$293$	0.583877	0.0341104	0.0170552	+	0.999855i	$0.494571\pi$
$294$	0	0
$295$	−0.0941812	−0.00548344
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−25.7350	−1.48829
$300$	0	0
$301$	6.81863	0.393019
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.37279	0.250385
$306$	0	0
$307$	6.67609	0.381025	0.190512	−	0.981685i	$-0.438985\pi$
$307$	6.67609	0.381025	0.190512	+	0.981685i	$0.438985\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3.51941	−0.199568	−0.0997839	−	0.995009i	$-0.531815\pi$
$311$	−3.51941	−0.199568	−0.0997839	+	0.995009i	$0.531815\pi$
$312$	0	0
$313$	31.4600	1.77822	0.889111	−	0.457691i	$-0.151323\pi$
$313$	31.4600	1.77822	0.889111	+	0.457691i	$0.151323\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.799473	0.0449029	0.0224514	−	0.999748i	$-0.492853\pi$
$317$	0.799473	0.0449029	0.0224514	+	0.999748i	$0.492853\pi$
$318$	0	0
$319$	2.73553	0.153160
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−8.41110	−0.468006
$324$	0	0
$325$	−21.2600	−1.17929
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.72496	−0.205364
$330$	0	0
$331$	−18.2686	−1.00413	−0.502065	−	0.864830i	$-0.667426\pi$
$331$	−18.2686	−1.00413	−0.502065	+	0.864830i	$0.667426\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.710831	0.0388369
$336$	0	0
$337$	18.8222	1.02531	0.512655	−	0.858595i	$-0.328662\pi$
$337$	18.8222	1.02531	0.512655	+	0.858595i	$0.328662\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.26447	0.0684750
$342$	0	0
$343$	−19.4252	−1.04886
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8.98944	−0.482578	−0.241289	−	0.970453i	$-0.577570\pi$
$347$	−8.98944	−0.482578	−0.241289	+	0.970453i	$0.577570\pi$
$348$	0	0
$349$	−23.7003	−1.26865	−0.634323	−	0.773068i	$-0.718721\pi$
$349$	−23.7003	−1.26865	−0.634323	+	0.773068i	$0.718721\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.96526	0.530397	0.265199	−	0.964194i	$-0.414562\pi$
$353$	9.96526	0.530397	0.265199	+	0.964194i	$0.414562\pi$
$354$	0	0
$355$	−5.79445	−0.307537
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−26.1864	−1.38206	−0.691032	−	0.722824i	$-0.742844\pi$
$359$	−26.1864	−1.38206	−0.691032	+	0.722824i	$0.742844\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.90582	0.204440
$366$	0	0
$367$	−3.51388	−0.183423	−0.0917114	−	0.995786i	$-0.529234\pi$
$367$	−3.51388	−0.183423	−0.0917114	+	0.995786i	$0.529234\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8.54359	−0.443561
$372$	0	0
$373$	−12.5728	−0.650995	−0.325497	−	0.945543i	$-0.605532\pi$
$373$	−12.5728	−0.650995	−0.325497	+	0.945543i	$0.605532\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	22.3728	1.15226
$378$	0	0
$379$	0.132494	0.00680578	0.00340289	−	0.999994i	$-0.498917\pi$
$379$	0.132494	0.00680578	0.00340289	+	0.999994i	$0.498917\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11.5577	−0.590572	−0.295286	−	0.955409i	$-0.595415\pi$
$383$	−11.5577	−0.590572	−0.295286	+	0.955409i	$0.595415\pi$
$384$	0	0
$385$	−0.745574	−0.0379980
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−17.5522	−0.889931	−0.444966	−	0.895548i	$-0.646784\pi$
$389$	−17.5522	−0.889931	−0.444966	+	0.895548i	$0.646784\pi$
$390$	0	0
$391$	22.8816	1.15717
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.67557	0.0843072
$396$	0	0
$397$	4.61665	0.231703	0.115852	−	0.993267i	$-0.463040\pi$
$397$	4.61665	0.231703	0.115852	+	0.993267i	$0.463040\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.65139	0.332155	0.166077	−	0.986113i	$-0.446890\pi$
$401$	6.65139	0.332155	0.166077	+	0.986113i	$0.446890\pi$
$402$	0	0
$403$	10.3416	0.515153
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.66895	−0.231431
$408$	0	0
$409$	−38.4494	−1.90120	−0.950601	−	0.310417i	$-0.899531\pi$
$409$	−38.4494	−1.90120	−0.950601	+	0.310417i	$0.899531\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0.240293	0.0118240
$414$	0	0
$415$	9.66196	0.474287
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−11.4947	−0.561554	−0.280777	−	0.959773i	$-0.590592\pi$
$419$	−11.4947	−0.561554	−0.280777	+	0.959773i	$0.590592\pi$
$420$	0	0
$421$	−5.19142	−0.253014	−0.126507	−	0.991966i	$-0.540377\pi$
$421$	−5.19142	−0.253014	−0.126507	+	0.991966i	$0.540377\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	18.9028	0.916919
$426$	0	0
$427$	−11.1567	−0.539909
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.34307	0.161030	0.0805150	−	0.996753i	$-0.474344\pi$
$431$	3.34307	0.161030	0.0805150	+	0.996753i	$0.474344\pi$
$432$	0	0
$433$	−37.2927	−1.79217	−0.896087	−	0.443878i	$-0.853602\pi$
$433$	−37.2927	−1.79217	−0.896087	+	0.443878i	$0.853602\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−10.8816	−0.520539
$438$	0	0
$439$	−2.61665	−0.124886	−0.0624430	−	0.998049i	$-0.519889\pi$
$439$	−2.61665	−0.124886	−0.0624430	+	0.998049i	$0.519889\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14.5783	0.692638	0.346319	−	0.938117i	$-0.387432\pi$
$443$	14.5783	0.692638	0.346319	+	0.938117i	$0.387432\pi$
$444$	0	0
$445$	−2.72445	−0.129151
$446$	0	0
$447$	0	0
$448$	0	0
$449$	21.2927	1.00487	0.502433	−	0.864616i	$-0.332438\pi$
$449$	21.2927	1.00487	0.502433	+	0.864616i	$0.332438\pi$
$450$	0	0
$451$	−5.76328	−0.271382
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.09775	−0.285867
$456$	0	0
$457$	−2.47054	−0.115567	−0.0577835	−	0.998329i	$-0.518403\pi$
$457$	−2.47054	−0.115567	−0.0577835	+	0.998329i	$0.518403\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15.3466	0.714764	0.357382	−	0.933958i	$-0.383669\pi$
$461$	15.3466	0.714764	0.357382	+	0.933958i	$0.383669\pi$
$462$	0	0
$463$	−29.0680	−1.35091	−0.675453	−	0.737403i	$-0.736052\pi$
$463$	−29.0680	−1.35091	−0.675453	+	0.737403i	$0.736052\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.9200	−1.06061	−0.530304	−	0.847807i	$-0.677922\pi$
$467$	−22.9200	−1.06061	−0.530304	+	0.847807i	$0.677922\pi$
$468$	0	0
$469$	−1.81361	−0.0837446
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.17438	0.0999780
$474$	0	0
$475$	−8.98944	−0.412464
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17.0297	−0.778108	−0.389054	−	0.921215i	$-0.627198\pi$
$479$	−17.0297	−0.778108	−0.389054	+	0.921215i	$0.627198\pi$
$480$	0	0
$481$	−38.1855	−1.74111
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.82220	−0.218965
$486$	0	0
$487$	16.3919	0.742790	0.371395	−	0.928475i	$-0.378880\pi$
$487$	16.3919	0.742790	0.371395	+	0.928475i	$0.378880\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	41.5230	1.87391	0.936953	−	0.349455i	$-0.113633\pi$
$491$	41.5230	1.87391	0.936953	+	0.349455i	$0.113633\pi$
$492$	0	0
$493$	−19.8922	−0.895900
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14.7839	0.663148
$498$	0	0
$499$	−8.63778	−0.386680	−0.193340	−	0.981132i	$-0.561932\pi$
$499$	−8.63778	−0.386680	−0.193340	+	0.981132i	$0.561932\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−24.9300	−1.11157	−0.555787	−	0.831325i	$-0.687583\pi$
$503$	−24.9300	−1.11157	−0.555787	+	0.831325i	$0.687583\pi$
$504$	0	0
$505$	6.57834	0.292732
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−27.3083	−1.21042	−0.605210	−	0.796066i	$-0.706911\pi$
$509$	−27.3083	−1.21042	−0.605210	+	0.796066i	$0.706911\pi$
$510$	0	0
$511$	−9.96526	−0.440837
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6.72445	−0.296315
$516$	0	0
$517$	−1.18785	−0.0522414
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−26.1361	−1.14504	−0.572521	−	0.819890i	$-0.694034\pi$
$521$	−26.1361	−1.14504	−0.572521	+	0.819890i	$0.694034\pi$
$522$	0	0
$523$	−28.8222	−1.26031	−0.630153	−	0.776471i	$-0.717008\pi$
$523$	−28.8222	−1.26031	−0.630153	+	0.776471i	$0.717008\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.19499	−0.400540
$528$	0	0
$529$	6.60252	0.287066
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−47.1355	−2.04167
$534$	0	0
$535$	−4.41110	−0.190708
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.14611	−0.0924396
$540$	0	0
$541$	−38.6705	−1.66258	−0.831288	−	0.555841i	$-0.812396\pi$
$541$	−38.6705	−1.66258	−0.831288	+	0.555841i	$0.812396\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.52946	−0.0655149
$546$	0	0
$547$	1.85746	0.0794192	0.0397096	−	0.999211i	$-0.487357\pi$
$547$	1.85746	0.0794192	0.0397096	+	0.999211i	$0.487357\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	9.45998	0.403009
$552$	0	0
$553$	−4.27504	−0.181793
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−17.5139	−0.742087	−0.371043	−	0.928616i	$-0.621000\pi$
$557$	−17.5139	−0.742087	−0.371043	+	0.928616i	$0.621000\pi$
$558$	0	0
$559$	17.7834	0.752156
$560$	0	0
$561$	0	0
$562$	0	0
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	0	0
$565$	13.4217	0.564654
$566$	0	0
$567$	0	0
$568$	0	0
$569$	31.6061	1.32500	0.662498	−	0.749064i	$-0.269496\pi$
$569$	31.6061	1.32500	0.662498	+	0.749064i	$0.269496\pi$
$570$	0	0
$571$	−33.4217	−1.39865	−0.699327	−	0.714802i	$-0.746517\pi$
$571$	−33.4217	−1.39865	−0.699327	+	0.714802i	$0.746517\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	24.4550	1.01984
$576$	0	0
$577$	−5.66553	−0.235859	−0.117929	−	0.993022i	$-0.537626\pi$
$577$	−5.66553	−0.235859	−0.117929	+	0.993022i	$0.537626\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−24.6514	−1.02271
$582$	0	0
$583$	−2.72445	−0.112835
$584$	0	0
$585$	0	0
$586$	0	0
$587$	47.6938	1.96853	0.984267	−	0.176689i	$-0.0565386\pi$
$587$	47.6938	1.96853	0.984267	+	0.176689i	$0.0565386\pi$
$588$	0	0
$589$	4.37279	0.180178
$590$	0	0
$591$	0	0
$592$	0	0
$593$	11.8081	0.484899	0.242450	−	0.970164i	$-0.422049\pi$
$593$	11.8081	0.484899	0.242450	+	0.970164i	$0.422049\pi$
$594$	0	0
$595$	5.42166	0.222267
$596$	0	0
$597$	0	0
$598$	0	0
$599$	33.2333	1.35788	0.678938	−	0.734196i	$-0.262441\pi$
$599$	33.2333	1.35788	0.678938	+	0.734196i	$0.262441\pi$
$600$	0	0
$601$	29.8016	1.21563	0.607816	−	0.794078i	$-0.292046\pi$
$601$	29.8016	1.21563	0.607816	+	0.794078i	$0.292046\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.58139	0.308227
$606$	0	0
$607$	−29.7250	−1.20650	−0.603250	−	0.797552i	$-0.706128\pi$
$607$	−29.7250	−1.20650	−0.603250	+	0.797552i	$0.706128\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9.71492	−0.393024
$612$	0	0
$613$	5.41809	0.218835	0.109417	−	0.993996i	$-0.465102\pi$
$613$	5.41809	0.218835	0.109417	+	0.993996i	$0.465102\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−0.313348	−0.0126149	−0.00630747	−	0.999980i	$-0.502008\pi$
$617$	−0.313348	−0.0126149	−0.00630747	+	0.999980i	$0.502008\pi$
$618$	0	0
$619$	35.4983	1.42680	0.713398	−	0.700759i	$-0.247155\pi$
$619$	35.4983	1.42680	0.713398	+	0.700759i	$0.247155\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.95112	0.278491
$624$	0	0
$625$	17.6761	0.707044
$626$	0	0
$627$	0	0
$628$	0	0
$629$	33.9516	1.35374
$630$	0	0
$631$	31.4827	1.25331	0.626653	−	0.779298i	$-0.284424\pi$
$631$	31.4827	1.25331	0.626653	+	0.779298i	$0.284424\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.58890	0.142421
$636$	0	0
$637$	−17.5522	−0.695443
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.2686	0.484579	0.242289	−	0.970204i	$-0.422102\pi$
$641$	12.2686	0.484579	0.242289	+	0.970204i	$0.422102\pi$
$642$	0	0
$643$	−21.3239	−0.840933	−0.420466	−	0.907308i	$-0.638134\pi$
$643$	−21.3239	−0.840933	−0.420466	+	0.907308i	$0.638134\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.3728	0.486424	0.243212	−	0.969973i	$-0.421799\pi$
$647$	12.3728	0.486424	0.243212	+	0.969973i	$0.421799\pi$
$648$	0	0
$649$	0.0766264	0.00300785
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−47.5925	−1.86244	−0.931219	−	0.364461i	$-0.881253\pi$
$653$	−47.5925	−1.86244	−0.931219	+	0.364461i	$0.881253\pi$
$654$	0	0
$655$	−10.1919	−0.398232
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−6.37636	−0.248388	−0.124194	−	0.992258i	$-0.539634\pi$
$659$	−6.37636	−0.248388	−0.124194	+	0.992258i	$0.539634\pi$
$660$	0	0
$661$	−31.2005	−1.21356	−0.606780	−	0.794870i	$-0.707539\pi$
$661$	−31.2005	−1.21356	−0.606780	+	0.794870i	$0.707539\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.57834	−0.0999836
$666$	0	0
$667$	−25.7350	−0.996464
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3.55773	−0.137345
$672$	0	0
$673$	−7.31386	−0.281929	−0.140964	−	0.990015i	$-0.545020\pi$
$673$	−7.31386	−0.281929	−0.140964	+	0.990015i	$0.545020\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	44.4842	1.70966	0.854832	−	0.518904i	$-0.173660\pi$
$677$	44.4842	1.70966	0.854832	+	0.518904i	$0.173660\pi$
$678$	0	0
$679$	12.3033	0.472157
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8.67609	0.331981	0.165991	−	0.986127i	$-0.446918\pi$
$683$	8.67609	0.331981	0.165991	+	0.986127i	$0.446918\pi$
$684$	0	0
$685$	9.88469	0.377675
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−22.2822	−0.848883
$690$	0	0
$691$	−34.7738	−1.32286	−0.661430	−	0.750007i	$-0.730050\pi$
$691$	−34.7738	−1.32286	−0.661430	+	0.750007i	$0.730050\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10.9375	−0.414883
$696$	0	0
$697$	41.9094	1.58743
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−24.0036	−0.906602	−0.453301	−	0.891357i	$-0.649754\pi$
$701$	−24.0036	−0.906602	−0.453301	+	0.891357i	$0.649754\pi$
$702$	0	0
$703$	−16.1461	−0.608962
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−16.7839	−0.631223
$708$	0	0
$709$	−37.0141	−1.39009	−0.695047	−	0.718964i	$-0.744617\pi$
$709$	−37.0141	−1.39009	−0.695047	+	0.718964i	$0.744617\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−11.8958	−0.445500
$714$	0	0
$715$	−1.94450	−0.0727201
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−16.2147	−0.604705	−0.302352	−	0.953196i	$-0.597772\pi$
$719$	−16.2147	−0.604705	−0.302352	+	0.953196i	$0.597772\pi$
$720$	0	0
$721$	17.1567	0.638948
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−21.2600	−0.789575
$726$	0	0
$727$	8.49974	0.315238	0.157619	−	0.987500i	$-0.449618\pi$
$727$	8.49974	0.315238	0.157619	+	0.987500i	$0.449618\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−15.8116	−0.584815
$732$	0	0
$733$	1.10278	0.0407319	0.0203660	−	0.999793i	$-0.493517\pi$
$733$	1.10278	0.0407319	0.0203660	+	0.999793i	$0.493517\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.578337	−0.0213033
$738$	0	0
$739$	44.4247	1.63419	0.817095	−	0.576503i	$-0.195583\pi$
$739$	44.4247	1.63419	0.817095	+	0.576503i	$0.195583\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	44.6258	1.63716	0.818580	−	0.574392i	$-0.194762\pi$
$743$	44.6258	1.63716	0.818580	+	0.574392i	$0.194762\pi$
$744$	0	0
$745$	−2.08667	−0.0764498
$746$	0	0
$747$	0	0
$748$	0	0
$749$	11.2544	0.411228
$750$	0	0
$751$	25.1184	0.916582	0.458291	−	0.888802i	$-0.348462\pi$
$751$	25.1184	0.916582	0.458291	+	0.888802i	$0.348462\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−6.16724	−0.224449
$756$	0	0
$757$	27.7194	1.00748	0.503740	−	0.863855i	$-0.331957\pi$
$757$	27.7194	1.00748	0.503740	+	0.863855i	$0.331957\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	52.4011	1.89954	0.949768	−	0.312954i	$-0.101318\pi$
$761$	52.4011	1.89954	0.949768	+	0.312954i	$0.101318\pi$
$762$	0	0
$763$	3.90225	0.141271
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.626697	0.0226287
$768$	0	0
$769$	−18.7839	−0.677364	−0.338682	−	0.940901i	$-0.609981\pi$
$769$	−18.7839	−0.677364	−0.338682	+	0.940901i	$0.609981\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−28.8378	−1.03722	−0.518612	−	0.855010i	$-0.673551\pi$
$773$	−28.8378	−1.03722	−0.518612	+	0.855010i	$0.673551\pi$
$774$	0	0
$775$	−9.82722	−0.353004
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−19.9305	−0.714085
$780$	0	0
$781$	4.71440	0.168695
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.51584	0.0541028
$786$	0	0
$787$	22.0419	0.785708	0.392854	−	0.919601i	$-0.371488\pi$
$787$	22.0419	0.785708	0.392854	+	0.919601i	$0.371488\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−34.2439	−1.21757
$792$	0	0
$793$	−29.0972	−1.03327
$794$	0	0
$795$	0	0
$796$	0	0
$797$	40.3955	1.43088	0.715441	−	0.698673i	$-0.246226\pi$
$797$	40.3955	1.43088	0.715441	+	0.698673i	$0.246226\pi$
$798$	0	0
$799$	8.63778	0.305583
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−3.17780	−0.112142
$804$	0	0
$805$	7.01413	0.247216
$806$	0	0
$807$	0	0
$808$	0	0
$809$	19.1814	0.674381	0.337191	−	0.941436i	$-0.390523\pi$
$809$	19.1814	0.674381	0.337191	+	0.941436i	$0.390523\pi$
$810$	0	0
$811$	3.98638	0.139981	0.0699904	−	0.997548i	$-0.477703\pi$
$811$	3.98638	0.139981	0.0699904	+	0.997548i	$0.477703\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	14.8011	0.518459
$816$	0	0
$817$	7.51941	0.263071
$818$	0	0
$819$	0	0
$820$	0	0
$821$	46.1588	1.61095	0.805476	−	0.592628i	$-0.201909\pi$
$821$	46.1588	1.61095	0.805476	+	0.592628i	$0.201909\pi$
$822$	0	0
$823$	2.35166	0.0819738	0.0409869	−	0.999160i	$-0.486950\pi$
$823$	2.35166	0.0819738	0.0409869	+	0.999160i	$0.486950\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.73553	0.0951236	0.0475618	−	0.998868i	$-0.484855\pi$
$827$	2.73553	0.0951236	0.0475618	+	0.998868i	$0.484855\pi$
$828$	0	0
$829$	19.0177	0.660512	0.330256	−	0.943891i	$-0.392865\pi$
$829$	19.0177	0.660512	0.330256	+	0.943891i	$0.392865\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	15.6061	0.540719
$834$	0	0
$835$	0.996946	0.0345007
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−36.5628	−1.26229	−0.631143	−	0.775666i	$-0.717414\pi$
$839$	−36.5628	−1.26229	−0.631143	+	0.775666i	$0.717414\pi$
$840$	0	0
$841$	−6.62721	−0.228525
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−6.66247	−0.229196
$846$	0	0
$847$	−19.3431	−0.664636
$848$	0	0
$849$	0	0
$850$	0	0
$851$	43.9240	1.50570
$852$	0	0
$853$	−53.6344	−1.83641	−0.918203	−	0.396111i	$-0.870360\pi$
$853$	−53.6344	−1.83641	−0.918203	+	0.396111i	$0.870360\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−28.1325	−0.960988	−0.480494	−	0.876998i	$-0.659543\pi$
$857$	−28.1325	−0.960988	−0.480494	+	0.876998i	$0.659543\pi$
$858$	0	0
$859$	1.37330	0.0468565	0.0234283	−	0.999726i	$-0.492542\pi$
$859$	1.37330	0.0468565	0.0234283	+	0.999726i	$0.492542\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−6.34358	−0.215938	−0.107969	−	0.994154i	$-0.534435\pi$
$863$	−6.34358	−0.215938	−0.107969	+	0.994154i	$0.534435\pi$
$864$	0	0
$865$	9.31386	0.316681
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.36326	−0.0462453
$870$	0	0
$871$	−4.72999	−0.160270
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12.2403	0.413797
$876$	0	0
$877$	−19.6555	−0.663718	−0.331859	−	0.943329i	$-0.607676\pi$
$877$	−19.6555	−0.663718	−0.331859	+	0.943329i	$0.607676\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−14.7527	−0.497032	−0.248516	−	0.968628i	$-0.579943\pi$
$881$	−14.7527	−0.497032	−0.248516	+	0.968628i	$0.579943\pi$
$882$	0	0
$883$	−13.7497	−0.462713	−0.231356	−	0.972869i	$-0.574316\pi$
$883$	−13.7497	−0.462713	−0.231356	+	0.972869i	$0.574316\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−45.2141	−1.51814	−0.759071	−	0.651008i	$-0.774347\pi$
$887$	−45.2141	−1.51814	−0.759071	+	0.651008i	$0.774347\pi$
$888$	0	0
$889$	−9.15667	−0.307105
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4.10780	−0.137462
$894$	0	0
$895$	−2.24386	−0.0750041
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.3416	0.344912
$900$	0	0
$901$	19.8116	0.660021
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0.693644	0.0230575
$906$	0	0
$907$	−58.0071	−1.92610	−0.963048	−	0.269331i	$-0.913198\pi$
$907$	−58.0071	−1.92610	−0.963048	+	0.269331i	$0.913198\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−20.2283	−0.670193	−0.335096	−	0.942184i	$-0.608769\pi$
$911$	−20.2283	−0.670193	−0.335096	+	0.942184i	$0.608769\pi$
$912$	0	0
$913$	−7.86103	−0.260162
$914$	0	0
$915$	0	0
$916$	0	0
$917$	26.0036	0.858714
$918$	0	0
$919$	39.2736	1.29552	0.647758	−	0.761846i	$-0.275707\pi$
$919$	39.2736	1.29552	0.647758	+	0.761846i	$0.275707\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	38.5572	1.26913
$924$	0	0
$925$	36.2861	1.19308
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−26.9583	−0.884472	−0.442236	−	0.896899i	$-0.645815\pi$
$929$	−26.9583	−0.884472	−0.442236	+	0.896899i	$0.645815\pi$
$930$	0	0
$931$	−7.42166	−0.243235
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.72890	0.0565412
$936$	0	0
$937$	−39.9789	−1.30605	−0.653026	−	0.757335i	$-0.726501\pi$
$937$	−39.9789	−1.30605	−0.653026	+	0.757335i	$0.726501\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−43.0177	−1.40234	−0.701169	−	0.712996i	$-0.747338\pi$
$941$	−43.0177	−1.40234	−0.701169	+	0.712996i	$0.747338\pi$
$942$	0	0
$943$	54.2192	1.76562
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14.8953	0.484031	0.242015	−	0.970272i	$-0.422192\pi$
$947$	14.8953	0.484031	0.242015	+	0.970272i	$0.422192\pi$
$948$	0	0
$949$	−25.9900	−0.843670
$950$	0	0
$951$	0	0
$952$	0	0
$953$	33.6272	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$953$	33.6272	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$954$	0	0
$955$	10.0978	0.326756
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−25.2197	−0.814386
$960$	0	0
$961$	−26.2197	−0.845796
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−17.9269	−0.577089
$966$	0	0
$967$	−51.5194	−1.65675	−0.828376	−	0.560172i	$-0.810735\pi$
$967$	−51.5194	−1.65675	−0.828376	+	0.560172i	$0.810735\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	41.1255	1.31978	0.659890	−	0.751362i	$-0.270603\pi$
$971$	41.1255	1.31978	0.659890	+	0.751362i	$0.270603\pi$
$972$	0	0
$973$	27.9058	0.894619
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.4423	−0.334078	−0.167039	−	0.985950i	$-0.553421\pi$
$977$	−10.4423	−0.334078	−0.167039	+	0.985950i	$0.553421\pi$
$978$	0	0
$979$	2.21663	0.0708438
$980$	0	0
$981$	0	0
$982$	0	0
$983$	25.9789	0.828597	0.414299	−	0.910141i	$-0.364027\pi$
$983$	25.9789	0.828597	0.414299	+	0.910141i	$0.364027\pi$
$984$	0	0
$985$	−10.3653	−0.330265
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−20.4559	−0.650459
$990$	0	0
$991$	−31.8419	−1.01149	−0.505745	−	0.862683i	$-0.668782\pi$
$991$	−31.8419	−1.01149	−0.505745	+	0.862683i	$0.668782\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−10.8433	−0.343757
$996$	0	0
$997$	−57.6591	−1.82608	−0.913040	−	0.407870i	$-0.866272\pi$
$997$	−57.6591	−1.82608	−0.913040	+	0.407870i	$0.866272\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9648.2.a.bj.1.2		3
3.2	odd	2		3216.2.a.s.1.2		3
4.3	odd	2		1206.2.a.m.1.2		3
12.11	even	2		402.2.a.g.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
402.2.a.g.1.2	✓	3	12.11	even	2
1206.2.a.m.1.2		3	4.3	odd	2
3216.2.a.s.1.2		3	3.2	odd	2
9648.2.a.bj.1.2		3	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:	\( N \)	\(=\)	\( 9648 = 2^{4} \cdot 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9648.a (trivial)

\(f(q)\)	\(=\)	\(q-0.710831 q^{5} +1.81361 q^{7} +O(q^{10})\)	\(q-0.710831 q^{5} +1.81361 q^{7} +0.578337 q^{11} +4.72999 q^{13} -4.20555 q^{17} +2.00000 q^{19} -5.44082 q^{23} -4.49472 q^{25} +4.72999 q^{29} +2.18639 q^{31} -1.28917 q^{35} -8.07306 q^{37} -9.96526 q^{41} +3.75971 q^{43} -2.05390 q^{47} -3.71083 q^{49} -4.71083 q^{53} -0.411100 q^{55} +0.132494 q^{59} -6.15165 q^{61} -3.36222 q^{65} -1.00000 q^{67} +8.15165 q^{71} -5.49472 q^{73} +1.04888 q^{77} -2.35720 q^{79} -13.5925 q^{83} +2.98944 q^{85} +3.83276 q^{89} +8.57834 q^{91} -1.42166 q^{95} +6.78389 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} - q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 - q^7	\( 3 q - 3 q^{5} - q^{7} - 6 q^{13} + 2 q^{17} + 6 q^{19} + 3 q^{23} + 2 q^{25} - 6 q^{29} + 13 q^{31} - 3 q^{35} - 7 q^{37} - 5 q^{41} + q^{43} - 10 q^{47} - 12 q^{49} - 15 q^{53} + 28 q^{55} + 3 q^{59} + 8 q^{65} - 3 q^{67} + 6 q^{71} - q^{73} - 8 q^{77} + 26 q^{79} - 3 q^{83} - 22 q^{85} - 16 q^{89} + 24 q^{91} - 6 q^{95} + 4 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 - q^7 - 6 * q^13 + 2 * q^17 + 6 * q^19 + 3 * q^23 + 2 * q^25 - 6 * q^29 + 13 * q^31 - 3 * q^35 - 7 * q^37 - 5 * q^41 + q^43 - 10 * q^47 - 12 * q^49 - 15 * q^53 + 28 * q^55 + 3 * q^59 + 8 * q^65 - 3 * q^67 + 6 * q^71 - q^73 - 8 * q^77 + 26 * q^79 - 3 * q^83 - 22 * q^85 - 16 * q^89 + 24 * q^91 - 6 * q^95 + 4 * q^97

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