LMFDB - Embedded newform 9648.2.a.ca.1.3

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:	$5$
Coefficient field:	5.5.24571284.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 15x^{3} + 10x^{2} + 64x + 38 $ x^5 - 2x^4 - 15x^3 + 10x^2 + 64x + 38
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 804)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.789611$ 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)

$f(q)$	$=$	$q-1.78961 q^{5} -4.79729 q^{7} +O(q^{10})$	$q-1.78961 q^{5} -4.79729 q^{7} +5.70891 q^{11} -3.70891 q^{13} +5.05120 q^{17} +2.12261 q^{19} -8.04190 q^{23} -1.79729 q^{25} +6.25229 q^{29} -5.99838 q^{31} +8.58529 q^{35} +5.45500 q^{37} -8.42711 q^{41} +10.0854 q^{43} -6.36661 q^{47} +16.0140 q^{49} +7.49852 q^{53} -10.2167 q^{55} -1.24623 q^{59} +12.3749 q^{61} +6.63750 q^{65} -1.00000 q^{67} -0.291093 q^{71} +3.32531 q^{73} -27.3873 q^{77} +6.01536 q^{79} -6.34007 q^{83} -9.03968 q^{85} +8.63042 q^{89} +17.7927 q^{91} -3.79864 q^{95} -17.1421 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{5} - 5 q^{7}+O(q^{10}) $ 5 * q - 3 * q^5 - 5 * q^7	$ 5 q - 3 q^{5} - 5 q^{7} + 6 q^{11} + 4 q^{13} - q^{17} - 13 q^{19} + 10 q^{25} - 3 q^{29} - 3 q^{31} - 9 q^{35} + 12 q^{37} - 11 q^{41} - 3 q^{43} - 13 q^{47} + 24 q^{49} + 9 q^{53} - 14 q^{55} - 12 q^{59} + 4 q^{61} + 8 q^{65} - 5 q^{67} - 24 q^{71} + 12 q^{73} + 4 q^{77} + 4 q^{79} - 27 q^{83} - 4 q^{85} + 5 q^{89} - 14 q^{91} - 30 q^{95} + 8 q^{97}+O(q^{100}) $ 5 * q - 3 * q^5 - 5 * q^7 + 6 * q^11 + 4 * q^13 - q^17 - 13 * q^19 + 10 * q^25 - 3 * q^29 - 3 * q^31 - 9 * q^35 + 12 * q^37 - 11 * q^41 - 3 * q^43 - 13 * q^47 + 24 * q^49 + 9 * q^53 - 14 * q^55 - 12 * q^59 + 4 * q^61 + 8 * q^65 - 5 * q^67 - 24 * q^71 + 12 * q^73 + 4 * q^77 + 4 * q^79 - 27 * q^83 - 4 * q^85 + 5 * q^89 - 14 * q^91 - 30 * q^95 + 8 * 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
$5$	−1.78961	−0.800338	−0.400169	−	0.916441i	$-0.631049\pi$
$5$	−1.78961	−0.800338	−0.400169	+	0.916441i	$0.631049\pi$
$6$	0	0
$7$	−4.79729	−1.81321	−0.906603	−	0.421984i	$-0.861334\pi$
$7$	−4.79729	−1.81321	−0.906603	+	0.421984i	$0.861334\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.70891	1.72130	0.860650	−	0.509197i	$-0.170057\pi$
$11$	5.70891	1.72130	0.860650	+	0.509197i	$0.170057\pi$
$12$	0	0
$13$	−3.70891	−1.02867	−0.514333	−	0.857591i	$-0.671960\pi$
$13$	−3.70891	−1.02867	−0.514333	+	0.857591i	$0.671960\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.05120	1.22510	0.612548	−	0.790433i	$-0.290145\pi$
$17$	5.05120	1.22510	0.612548	+	0.790433i	$0.290145\pi$
$18$	0	0
$19$	2.12261	0.486959	0.243480	−	0.969906i	$-0.421711\pi$
$19$	2.12261	0.486959	0.243480	+	0.969906i	$0.421711\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.04190	−1.67685	−0.838426	−	0.545015i	$-0.816524\pi$
$23$	−8.04190	−1.67685	−0.838426	+	0.545015i	$0.816524\pi$
$24$	0	0
$25$	−1.79729	−0.359459
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.25229	1.16102	0.580511	−	0.814253i	$-0.302853\pi$
$29$	6.25229	1.16102	0.580511	+	0.814253i	$0.302853\pi$
$30$	0	0
$31$	−5.99838	−1.07734	−0.538671	−	0.842516i	$-0.681073\pi$
$31$	−5.99838	−1.07734	−0.538671	+	0.842516i	$0.681073\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.58529	1.45118
$36$	0	0
$37$	5.45500	0.896796	0.448398	−	0.893834i	$-0.351995\pi$
$37$	5.45500	0.896796	0.448398	+	0.893834i	$0.351995\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.42711	−1.31609	−0.658047	−	0.752977i	$-0.728617\pi$
$41$	−8.42711	−1.31609	−0.658047	+	0.752977i	$0.728617\pi$
$42$	0	0
$43$	10.0854	1.53801	0.769006	−	0.639241i	$-0.220752\pi$
$43$	10.0854	1.53801	0.769006	+	0.639241i	$0.220752\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.36661	−0.928666	−0.464333	−	0.885661i	$-0.653706\pi$
$47$	−6.36661	−0.928666	−0.464333	+	0.885661i	$0.653706\pi$
$48$	0	0
$49$	16.0140	2.28772
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.49852	1.03000	0.515000	−	0.857190i	$-0.327792\pi$
$53$	7.49852	1.03000	0.515000	+	0.857190i	$0.327792\pi$
$54$	0	0
$55$	−10.2167	−1.37762
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.24623	−0.162245	−0.0811224	−	0.996704i	$-0.525850\pi$
$59$	−1.24623	−0.162245	−0.0811224	+	0.996704i	$0.525850\pi$
$60$	0	0
$61$	12.3749	1.58444	0.792222	−	0.610233i	$-0.208924\pi$
$61$	12.3749	1.58444	0.792222	+	0.610233i	$0.208924\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.63750	0.823281
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.291093	−0.0345464	−0.0172732	−	0.999851i	$-0.505498\pi$
$71$	−0.291093	−0.0345464	−0.0172732	+	0.999851i	$0.505498\pi$
$72$	0	0
$73$	3.32531	0.389199	0.194599	−	0.980883i	$-0.437659\pi$
$73$	3.32531	0.389199	0.194599	+	0.980883i	$0.437659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−27.3873	−3.12107
$78$	0	0
$79$	6.01536	0.676781	0.338391	−	0.941006i	$-0.390117\pi$
$79$	6.01536	0.676781	0.338391	+	0.941006i	$0.390117\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.34007	−0.695914	−0.347957	−	0.937511i	$-0.613124\pi$
$83$	−6.34007	−0.695914	−0.347957	+	0.937511i	$0.613124\pi$
$84$	0	0
$85$	−9.03968	−0.980491
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.63042	0.914823	0.457411	−	0.889255i	$-0.348777\pi$
$89$	8.63042	0.914823	0.457411	+	0.889255i	$0.348777\pi$
$90$	0	0
$91$	17.7927	1.86518
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.79864	−0.389732
$96$	0	0
$97$	−17.1421	−1.74051	−0.870257	−	0.492597i	$-0.836048\pi$
$97$	−17.1421	−1.74051	−0.870257	+	0.492597i	$0.836048\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.63750	−0.660456	−0.330228	−	0.943901i	$-0.607126\pi$
$101$	−6.63750	−0.660456	−0.330228	+	0.943901i	$0.607126\pi$
$102$	0	0
$103$	−6.37490	−0.628137	−0.314069	−	0.949400i	$-0.601692\pi$
$103$	−6.37490	−0.628137	−0.314069	+	0.949400i	$0.601692\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.97979	0.578089	0.289044	−	0.957316i	$-0.406663\pi$
$107$	5.97979	0.578089	0.289044	+	0.957316i	$0.406663\pi$
$108$	0	0
$109$	15.5475	1.48918	0.744590	−	0.667522i	$-0.232645\pi$
$109$	15.5475	1.48918	0.744590	+	0.667522i	$0.232645\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.44954	0.136361	0.0681805	−	0.997673i	$-0.478281\pi$
$113$	1.44954	0.136361	0.0681805	+	0.997673i	$0.478281\pi$
$114$	0	0
$115$	14.3919	1.34205
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−24.2321	−2.22135
$120$	0	0
$121$	21.5916	1.96287
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.1645	1.08803
$126$	0	0
$127$	−1.61479	−0.143290	−0.0716448	−	0.997430i	$-0.522825\pi$
$127$	−1.61479	−0.143290	−0.0716448	+	0.997430i	$0.522825\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.45883	−0.389570	−0.194785	−	0.980846i	$-0.562401\pi$
$131$	−4.45883	−0.389570	−0.194785	+	0.980846i	$0.562401\pi$
$132$	0	0
$133$	−10.1828	−0.882958
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.49852	−0.640642	−0.320321	−	0.947309i	$-0.603791\pi$
$137$	−7.49852	−0.640642	−0.320321	+	0.947309i	$0.603791\pi$
$138$	0	0
$139$	−20.8811	−1.77111	−0.885556	−	0.464533i	$-0.846222\pi$
$139$	−20.8811	−1.77111	−0.885556	+	0.464533i	$0.846222\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−21.1738	−1.77064
$144$	0	0
$145$	−11.1892	−0.929210
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.3547	1.33983	0.669914	−	0.742438i	$-0.266331\pi$
$149$	16.3547	1.33983	0.669914	+	0.742438i	$0.266331\pi$
$150$	0	0
$151$	4.72016	0.384121	0.192060	−	0.981383i	$-0.438483\pi$
$151$	4.72016	0.384121	0.192060	+	0.981383i	$0.438483\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	10.7348	0.862238
$156$	0	0
$157$	−13.0069	−1.03807	−0.519033	−	0.854754i	$-0.673708\pi$
$157$	−13.0069	−1.03807	−0.519033	+	0.854754i	$0.673708\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	38.5794	3.04048
$162$	0	0
$163$	−23.0207	−1.80312	−0.901560	−	0.432655i	$-0.857577\pi$
$163$	−23.0207	−1.80312	−0.901560	+	0.432655i	$0.857577\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.455602	0.0352556	0.0176278	−	0.999845i	$-0.494389\pi$
$167$	0.455602	0.0352556	0.0176278	+	0.999845i	$0.494389\pi$
$168$	0	0
$169$	0.755993	0.0581533
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−13.2964	−1.01091	−0.505454	−	0.862854i	$-0.668675\pi$
$173$	−13.2964	−1.01091	−0.505454	+	0.862854i	$0.668675\pi$
$174$	0	0
$175$	8.62214	0.651772
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.70594	0.650713	0.325356	−	0.945591i	$-0.394516\pi$
$179$	8.70594	0.650713	0.325356	+	0.945591i	$0.394516\pi$
$180$	0	0
$181$	8.80558	0.654513	0.327257	−	0.944935i	$-0.393876\pi$
$181$	8.80558	0.654513	0.327257	+	0.944935i	$0.393876\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−9.76233	−0.717741
$186$	0	0
$187$	28.8368	2.10876
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.7714	−0.851745	−0.425873	−	0.904783i	$-0.640033\pi$
$191$	−11.7714	−0.851745	−0.425873	+	0.904783i	$0.640033\pi$
$192$	0	0
$193$	24.2103	1.74269	0.871346	−	0.490668i	$-0.163247\pi$
$193$	24.2103	1.74269	0.871346	+	0.490668i	$0.163247\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.83373	0.629377	0.314689	−	0.949195i	$-0.398100\pi$
$197$	8.83373	0.629377	0.314689	+	0.949195i	$0.398100\pi$
$198$	0	0
$199$	−2.27089	−0.160979	−0.0804895	−	0.996755i	$-0.525648\pi$
$199$	−2.27089	−0.160979	−0.0804895	+	0.996755i	$0.525648\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−29.9941	−2.10517
$204$	0	0
$205$	15.0813	1.05332
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.1178	0.838203
$210$	0	0
$211$	9.98140	0.687148	0.343574	−	0.939126i	$-0.388362\pi$
$211$	9.98140	0.687148	0.343574	+	0.939126i	$0.388362\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−18.0490	−1.23093
$216$	0	0
$217$	28.7760	1.95344
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−18.7344	−1.26021
$222$	0	0
$223$	5.92125	0.396516	0.198258	−	0.980150i	$-0.436472\pi$
$223$	5.92125	0.396516	0.198258	+	0.980150i	$0.436472\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.7662	−1.04644	−0.523219	−	0.852198i	$-0.675269\pi$
$227$	−15.7662	−1.04644	−0.523219	+	0.852198i	$0.675269\pi$
$228$	0	0
$229$	−28.1894	−1.86281	−0.931405	−	0.363984i	$-0.881416\pi$
$229$	−28.1894	−1.86281	−0.931405	+	0.363984i	$0.881416\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.5580	−1.08475	−0.542375	−	0.840136i	$-0.682475\pi$
$233$	−16.5580	−1.08475	−0.542375	+	0.840136i	$0.682475\pi$
$234$	0	0
$235$	11.3938	0.743247
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.3506	−0.992946	−0.496473	−	0.868052i	$-0.665372\pi$
$239$	−15.3506	−0.992946	−0.496473	+	0.868052i	$0.665372\pi$
$240$	0	0
$241$	17.2794	1.11307	0.556533	−	0.830825i	$-0.312131\pi$
$241$	17.2794	1.11307	0.556533	+	0.830825i	$0.312131\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−28.6589	−1.83095
$246$	0	0
$247$	−7.87255	−0.500918
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−12.5046	−0.789282	−0.394641	−	0.918835i	$-0.629131\pi$
$251$	−12.5046	−0.789282	−0.394641	+	0.918835i	$0.629131\pi$
$252$	0	0
$253$	−45.9105	−2.88637
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.25229	0.390007	0.195004	−	0.980803i	$-0.437528\pi$
$257$	6.25229	0.390007	0.195004	+	0.980803i	$0.437528\pi$
$258$	0	0
$259$	−26.1692	−1.62608
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.32767	−0.0818679	−0.0409340	−	0.999162i	$-0.513033\pi$
$263$	−1.32767	−0.0818679	−0.0409340	+	0.999162i	$0.513033\pi$
$264$	0	0
$265$	−13.4194	−0.824349
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−8.37813	−0.510824	−0.255412	−	0.966832i	$-0.582211\pi$
$269$	−8.37813	−0.510824	−0.255412	+	0.966832i	$0.582211\pi$
$270$	0	0
$271$	18.1664	1.10353	0.551765	−	0.833999i	$-0.313954\pi$
$271$	18.1664	1.10353	0.551765	+	0.833999i	$0.313954\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−10.2606	−0.618736
$276$	0	0
$277$	−1.33683	−0.0803224	−0.0401612	−	0.999193i	$-0.512787\pi$
$277$	−1.33683	−0.0803224	−0.0401612	+	0.999193i	$0.512787\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−26.8827	−1.60369	−0.801844	−	0.597533i	$-0.796148\pi$
$281$	−26.8827	−1.60369	−0.801844	+	0.597533i	$0.796148\pi$
$282$	0	0
$283$	−1.86324	−0.110758	−0.0553789	−	0.998465i	$-0.517637\pi$
$283$	−1.86324	−0.110758	−0.0553789	+	0.998465i	$0.517637\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	40.4273	2.38635
$288$	0	0
$289$	8.51463	0.500860
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3.75599	−0.219427	−0.109714	−	0.993963i	$-0.534993\pi$
$293$	−3.75599	−0.219427	−0.109714	+	0.993963i	$0.534993\pi$
$294$	0	0
$295$	2.23026	0.129851
$296$	0	0
$297$	0	0
$298$	0	0
$299$	29.8267	1.72492
$300$	0	0
$301$	−48.3827	−2.78873
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−22.1463	−1.26809
$306$	0	0
$307$	−10.3983	−0.593464	−0.296732	−	0.954961i	$-0.595897\pi$
$307$	−10.3983	−0.593464	−0.296732	+	0.954961i	$0.595897\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−25.6204	−1.45280	−0.726399	−	0.687273i	$-0.758808\pi$
$311$	−25.6204	−1.45280	−0.726399	+	0.687273i	$0.758808\pi$
$312$	0	0
$313$	−20.3749	−1.15166	−0.575829	−	0.817570i	$-0.695320\pi$
$313$	−20.3749	−1.15166	−0.575829	+	0.817570i	$0.695320\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−22.3935	−1.25774	−0.628872	−	0.777509i	$-0.716483\pi$
$317$	−22.3935	−1.25774	−0.628872	+	0.777509i	$0.716483\pi$
$318$	0	0
$319$	35.6938	1.99847
$320$	0	0
$321$	0	0
$322$	0	0
$323$	10.7217	0.596572
$324$	0	0
$325$	6.66599	0.369763
$326$	0	0
$327$	0	0
$328$	0	0
$329$	30.5425	1.68386
$330$	0	0
$331$	0.577874	0.0317629	0.0158814	−	0.999874i	$-0.494945\pi$
$331$	0.577874	0.0317629	0.0158814	+	0.999874i	$0.494945\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.78961	0.0977769
$336$	0	0
$337$	−17.1619	−0.934867	−0.467434	−	0.884028i	$-0.654821\pi$
$337$	−17.1619	−0.934867	−0.467434	+	0.884028i	$0.654821\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−34.2442	−1.85443
$342$	0	0
$343$	−43.2429	−2.33490
$344$	0	0
$345$	0	0
$346$	0	0
$347$	31.4344	1.68749	0.843743	−	0.536747i	$-0.180347\pi$
$347$	31.4344	1.68749	0.843743	+	0.536747i	$0.180347\pi$
$348$	0	0
$349$	20.9528	1.12158	0.560788	−	0.827959i	$-0.310498\pi$
$349$	20.9528	1.12158	0.560788	+	0.827959i	$0.310498\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−24.7834	−1.31909	−0.659544	−	0.751666i	$-0.729251\pi$
$353$	−24.7834	−1.31909	−0.659544	+	0.751666i	$0.729251\pi$
$354$	0	0
$355$	0.520943	0.0276488
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.97077	−0.209569	−0.104784	−	0.994495i	$-0.533415\pi$
$359$	−3.97077	−0.209569	−0.104784	+	0.994495i	$0.533415\pi$
$360$	0	0
$361$	−14.4945	−0.762871
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5.95102	−0.311491
$366$	0	0
$367$	−5.40245	−0.282006	−0.141003	−	0.990009i	$-0.545033\pi$
$367$	−5.40245	−0.282006	−0.141003	+	0.990009i	$0.545033\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−35.9726	−1.86760
$372$	0	0
$373$	1.14532	0.0593023	0.0296511	−	0.999560i	$-0.490560\pi$
$373$	1.14532	0.0593023	0.0296511	+	0.999560i	$0.490560\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−23.1892	−1.19430
$378$	0	0
$379$	−17.0457	−0.875581	−0.437790	−	0.899077i	$-0.644239\pi$
$379$	−17.0457	−0.875581	−0.437790	+	0.899077i	$0.644239\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.59755	−0.132729	−0.0663643	−	0.997795i	$-0.521140\pi$
$383$	−2.59755	−0.132729	−0.0663643	+	0.997795i	$0.521140\pi$
$384$	0	0
$385$	49.0126	2.49791
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10.1002	−0.512099	−0.256049	−	0.966664i	$-0.582421\pi$
$389$	−10.1002	−0.512099	−0.256049	+	0.966664i	$0.582421\pi$
$390$	0	0
$391$	−40.6213	−2.05431
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−10.7652	−0.541654
$396$	0	0
$397$	4.35792	0.218718	0.109359	−	0.994002i	$-0.465120\pi$
$397$	4.35792	0.218718	0.109359	+	0.994002i	$0.465120\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−27.8488	−1.39070	−0.695352	−	0.718669i	$-0.744752\pi$
$401$	−27.8488	−1.39070	−0.695352	+	0.718669i	$0.744752\pi$
$402$	0	0
$403$	22.2474	1.10822
$404$	0	0
$405$	0	0
$406$	0	0
$407$	31.1421	1.54366
$408$	0	0
$409$	−11.6087	−0.574015	−0.287008	−	0.957928i	$-0.592661\pi$
$409$	−11.6087	−0.574015	−0.287008	+	0.957928i	$0.592661\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.97851	0.294183
$414$	0	0
$415$	11.3463	0.556966
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−6.53112	−0.319066	−0.159533	−	0.987193i	$-0.550999\pi$
$419$	−6.53112	−0.319066	−0.159533	+	0.987193i	$0.550999\pi$
$420$	0	0
$421$	1.51301	0.0737396	0.0368698	−	0.999320i	$-0.488261\pi$
$421$	1.51301	0.0737396	0.0368698	+	0.999320i	$0.488261\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−9.07848	−0.440371
$426$	0	0
$427$	−59.3660	−2.87292
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−22.6936	−1.09311	−0.546555	−	0.837423i	$-0.684061\pi$
$431$	−22.6936	−1.09311	−0.546555	+	0.837423i	$0.684061\pi$
$432$	0	0
$433$	−20.3585	−0.978369	−0.489184	−	0.872180i	$-0.662706\pi$
$433$	−20.3585	−0.978369	−0.489184	+	0.872180i	$0.662706\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−17.0698	−0.816559
$438$	0	0
$439$	27.4945	1.31224	0.656121	−	0.754655i	$-0.272196\pi$
$439$	27.4945	1.31224	0.656121	+	0.754655i	$0.272196\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	17.3023	0.822056	0.411028	−	0.911623i	$-0.365170\pi$
$443$	17.3023	0.822056	0.411028	+	0.911623i	$0.365170\pi$
$444$	0	0
$445$	−15.4451	−0.732168
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−22.9193	−1.08163	−0.540815	−	0.841142i	$-0.681884\pi$
$449$	−22.9193	−1.08163	−0.540815	+	0.841142i	$0.681884\pi$
$450$	0	0
$451$	−48.1096	−2.26539
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−31.8420	−1.49278
$456$	0	0
$457$	−19.7527	−0.923993	−0.461996	−	0.886882i	$-0.652867\pi$
$457$	−19.7527	−0.923993	−0.461996	+	0.886882i	$0.652867\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15.6143	0.727232	0.363616	−	0.931549i	$-0.381542\pi$
$461$	15.6143	0.727232	0.363616	+	0.931549i	$0.381542\pi$
$462$	0	0
$463$	0.955468	0.0444044	0.0222022	−	0.999754i	$-0.492932\pi$
$463$	0.955468	0.0444044	0.0222022	+	0.999754i	$0.492932\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11.7089	0.541824	0.270912	−	0.962604i	$-0.412675\pi$
$467$	11.7089	0.541824	0.270912	+	0.962604i	$0.412675\pi$
$468$	0	0
$469$	4.79729	0.221518
$470$	0	0
$471$	0	0
$472$	0	0
$473$	57.5767	2.64738
$474$	0	0
$475$	−3.81494	−0.175042
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−26.7325	−1.22144	−0.610719	−	0.791847i	$-0.709120\pi$
$479$	−26.7325	−1.22144	−0.610719	+	0.791847i	$0.709120\pi$
$480$	0	0
$481$	−20.2321	−0.922504
$482$	0	0
$483$	0	0
$484$	0	0
$485$	30.6777	1.39300
$486$	0	0
$487$	−11.4950	−0.520888	−0.260444	−	0.965489i	$-0.583869\pi$
$487$	−11.4950	−0.520888	−0.260444	+	0.965489i	$0.583869\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	31.9807	1.44327	0.721633	−	0.692275i	$-0.243392\pi$
$491$	31.9807	1.44327	0.721633	+	0.692275i	$0.243392\pi$
$492$	0	0
$493$	31.5816	1.42236
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.39646	0.0626397
$498$	0	0
$499$	−4.08704	−0.182961	−0.0914805	−	0.995807i	$-0.529160\pi$
$499$	−4.08704	−0.182961	−0.0914805	+	0.995807i	$0.529160\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−7.37786	−0.328963	−0.164481	−	0.986380i	$-0.552595\pi$
$503$	−7.37786	−0.328963	−0.164481	+	0.986380i	$0.552595\pi$
$504$	0	0
$505$	11.8785	0.528588
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−24.3678	−1.08008	−0.540042	−	0.841638i	$-0.681592\pi$
$509$	−24.3678	−1.08008	−0.540042	+	0.841638i	$0.681592\pi$
$510$	0	0
$511$	−15.9525	−0.705697
$512$	0	0
$513$	0	0
$514$	0	0
$515$	11.4086	0.502722
$516$	0	0
$517$	−36.3464	−1.59851
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−11.5448	−0.505787	−0.252894	−	0.967494i	$-0.581382\pi$
$521$	−11.5448	−0.505787	−0.252894	+	0.967494i	$0.581382\pi$
$522$	0	0
$523$	−4.78739	−0.209338	−0.104669	−	0.994507i	$-0.533378\pi$
$523$	−4.78739	−0.209338	−0.104669	+	0.994507i	$0.533378\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−30.2990	−1.31985
$528$	0	0
$529$	41.6722	1.81183
$530$	0	0
$531$	0	0
$532$	0	0
$533$	31.2554	1.35382
$534$	0	0
$535$	−10.7015	−0.462666
$536$	0	0
$537$	0	0
$538$	0	0
$539$	91.4225	3.93785
$540$	0	0
$541$	31.1880	1.34088	0.670438	−	0.741966i	$-0.266106\pi$
$541$	31.1880	1.34088	0.670438	+	0.741966i	$0.266106\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−27.8240	−1.19185
$546$	0	0
$547$	−20.6250	−0.881860	−0.440930	−	0.897542i	$-0.645351\pi$
$547$	−20.6250	−0.881860	−0.440930	+	0.897542i	$0.645351\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	13.2712	0.565370
$552$	0	0
$553$	−28.8575	−1.22714
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.96948	0.422421	0.211210	−	0.977441i	$-0.432260\pi$
$557$	9.96948	0.422421	0.211210	+	0.977441i	$0.432260\pi$
$558$	0	0
$559$	−37.4059	−1.58210
$560$	0	0
$561$	0	0
$562$	0	0
$563$	37.9943	1.60127	0.800634	−	0.599154i	$-0.204496\pi$
$563$	37.9943	1.60127	0.800634	+	0.599154i	$0.204496\pi$
$564$	0	0
$565$	−2.59411	−0.109135
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−45.4814	−1.90668	−0.953340	−	0.301899i	$-0.902380\pi$
$569$	−45.4814	−1.90668	−0.953340	+	0.301899i	$0.902380\pi$
$570$	0	0
$571$	30.3044	1.26820	0.634099	−	0.773252i	$-0.281371\pi$
$571$	30.3044	1.26820	0.634099	+	0.773252i	$0.281371\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	14.4537	0.602759
$576$	0	0
$577$	−19.9070	−0.828741	−0.414370	−	0.910108i	$-0.635998\pi$
$577$	−19.9070	−0.828741	−0.414370	+	0.910108i	$0.635998\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	30.4152	1.26183
$582$	0	0
$583$	42.8083	1.77294
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−33.7084	−1.39130	−0.695648	−	0.718383i	$-0.744883\pi$
$587$	−33.7084	−1.39130	−0.695648	+	0.718383i	$0.744883\pi$
$588$	0	0
$589$	−12.7322	−0.524622
$590$	0	0
$591$	0	0
$592$	0	0
$593$	42.1055	1.72907	0.864533	−	0.502576i	$-0.167614\pi$
$593$	42.1055	1.72907	0.864533	+	0.502576i	$0.167614\pi$
$594$	0	0
$595$	43.3660	1.77783
$596$	0	0
$597$	0	0
$598$	0	0
$599$	16.1577	0.660186	0.330093	−	0.943948i	$-0.392920\pi$
$599$	16.1577	0.660186	0.330093	+	0.943948i	$0.392920\pi$
$600$	0	0
$601$	−12.1626	−0.496121	−0.248061	−	0.968745i	$-0.579793\pi$
$601$	−12.1626	−0.496121	−0.248061	+	0.968745i	$0.579793\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−38.6406	−1.57096
$606$	0	0
$607$	35.3303	1.43401	0.717007	−	0.697066i	$-0.245512\pi$
$607$	35.3303	1.43401	0.717007	+	0.697066i	$0.245512\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	23.6132	0.955287
$612$	0	0
$613$	−25.8376	−1.04357	−0.521785	−	0.853077i	$-0.674734\pi$
$613$	−25.8376	−1.04357	−0.521785	+	0.853077i	$0.674734\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14.9582	−0.602196	−0.301098	−	0.953593i	$-0.597353\pi$
$617$	−14.9582	−0.602196	−0.301098	+	0.953593i	$0.597353\pi$
$618$	0	0
$619$	−33.6653	−1.35312	−0.676561	−	0.736387i	$-0.736530\pi$
$619$	−33.6653	−1.35312	−0.676561	+	0.736387i	$0.736530\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−41.4027	−1.65876
$624$	0	0
$625$	−12.7833	−0.511331
$626$	0	0
$627$	0	0
$628$	0	0
$629$	27.5543	1.09866
$630$	0	0
$631$	26.0652	1.03764	0.518820	−	0.854884i	$-0.326372\pi$
$631$	26.0652	1.03764	0.518820	+	0.854884i	$0.326372\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.88985	0.114680
$636$	0	0
$637$	−59.3945	−2.35330
$638$	0	0
$639$	0	0
$640$	0	0
$641$	36.5548	1.44383	0.721913	−	0.691983i	$-0.243263\pi$
$641$	36.5548	1.44383	0.721913	+	0.691983i	$0.243263\pi$
$642$	0	0
$643$	−7.25861	−0.286252	−0.143126	−	0.989705i	$-0.545715\pi$
$643$	−7.25861	−0.286252	−0.143126	+	0.989705i	$0.545715\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8.41365	−0.330775	−0.165387	−	0.986229i	$-0.552887\pi$
$647$	−8.41365	−0.330775	−0.165387	+	0.986229i	$0.552887\pi$
$648$	0	0
$649$	−7.11459	−0.279272
$650$	0	0
$651$	0	0
$652$	0	0
$653$	16.3800	0.641000	0.320500	−	0.947249i	$-0.396149\pi$
$653$	16.3800	0.641000	0.320500	+	0.947249i	$0.396149\pi$
$654$	0	0
$655$	7.97958	0.311788
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−30.8245	−1.20075	−0.600375	−	0.799719i	$-0.704982\pi$
$659$	−30.8245	−1.20075	−0.600375	+	0.799719i	$0.704982\pi$
$660$	0	0
$661$	−18.4034	−0.715809	−0.357904	−	0.933758i	$-0.616509\pi$
$661$	−18.4034	−0.715809	−0.357904	+	0.933758i	$0.616509\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	18.2232	0.706665
$666$	0	0
$667$	−50.2803	−1.94686
$668$	0	0
$669$	0	0
$670$	0	0
$671$	70.6471	2.72730
$672$	0	0
$673$	10.1111	0.389754	0.194877	−	0.980828i	$-0.437569\pi$
$673$	10.1111	0.389754	0.194877	+	0.980828i	$0.437569\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	23.8215	0.915536	0.457768	−	0.889072i	$-0.348649\pi$
$677$	23.8215	0.915536	0.457768	+	0.889072i	$0.348649\pi$
$678$	0	0
$679$	82.2356	3.15591
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−35.7677	−1.36861	−0.684306	−	0.729195i	$-0.739895\pi$
$683$	−35.7677	−1.36861	−0.684306	+	0.729195i	$0.739895\pi$
$684$	0	0
$685$	13.4194	0.512730
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−27.8113	−1.05953
$690$	0	0
$691$	21.8667	0.831848	0.415924	−	0.909399i	$-0.363458\pi$
$691$	21.8667	0.831848	0.415924	+	0.909399i	$0.363458\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	37.3690	1.41749
$696$	0	0
$697$	−42.5670	−1.61234
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−36.9614	−1.39601	−0.698006	−	0.716092i	$-0.745929\pi$
$701$	−36.9614	−1.39601	−0.698006	+	0.716092i	$0.745929\pi$
$702$	0	0
$703$	11.5788	0.436703
$704$	0	0
$705$	0	0
$706$	0	0
$707$	31.8420	1.19754
$708$	0	0
$709$	1.75761	0.0660084	0.0330042	−	0.999455i	$-0.489493\pi$
$709$	1.75761	0.0660084	0.0330042	+	0.999455i	$0.489493\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	48.2384	1.80654
$714$	0	0
$715$	37.8929	1.41711
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−21.5363	−0.803169	−0.401585	−	0.915822i	$-0.631540\pi$
$719$	−21.5363	−0.803169	−0.401585	+	0.915822i	$0.631540\pi$
$720$	0	0
$721$	30.5823	1.13894
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−11.2372	−0.417339
$726$	0	0
$727$	29.1985	1.08291	0.541456	−	0.840729i	$-0.317873\pi$
$727$	29.1985	1.08291	0.541456	+	0.840729i	$0.317873\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	50.9435	1.88421
$732$	0	0
$733$	30.3724	1.12183	0.560915	−	0.827873i	$-0.310449\pi$
$733$	30.3724	1.12183	0.560915	+	0.827873i	$0.310449\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.70891	−0.210290
$738$	0	0
$739$	−31.5020	−1.15882	−0.579410	−	0.815036i	$-0.696717\pi$
$739$	−31.5020	−1.15882	−0.579410	+	0.815036i	$0.696717\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−31.4187	−1.15264	−0.576320	−	0.817224i	$-0.695512\pi$
$743$	−31.4187	−1.15264	−0.576320	+	0.817224i	$0.695512\pi$
$744$	0	0
$745$	−29.2685	−1.07232
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−28.6868	−1.04819
$750$	0	0
$751$	17.5344	0.639840	0.319920	−	0.947445i	$-0.396344\pi$
$751$	17.5344	0.639840	0.319920	+	0.947445i	$0.396344\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−8.44724	−0.307427
$756$	0	0
$757$	3.62854	0.131882	0.0659408	−	0.997824i	$-0.478995\pi$
$757$	3.62854	0.131882	0.0659408	+	0.997824i	$0.478995\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−23.7544	−0.861098	−0.430549	−	0.902567i	$-0.641680\pi$
$761$	−23.7544	−0.861098	−0.430549	+	0.902567i	$0.641680\pi$
$762$	0	0
$763$	−74.5859	−2.70019
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.62214	0.166896
$768$	0	0
$769$	49.0804	1.76988	0.884942	−	0.465701i	$-0.154198\pi$
$769$	49.0804	1.76988	0.884942	+	0.465701i	$0.154198\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	31.0227	1.11581	0.557905	−	0.829905i	$-0.311605\pi$
$773$	31.0227	1.11581	0.557905	+	0.829905i	$0.311605\pi$
$774$	0	0
$775$	10.7809	0.387260
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−17.8874	−0.640884
$780$	0	0
$781$	−1.66182	−0.0594647
$782$	0	0
$783$	0	0
$784$	0	0
$785$	23.2774	0.830805
$786$	0	0
$787$	−17.9382	−0.639427	−0.319713	−	0.947514i	$-0.603587\pi$
$787$	−17.9382	−0.639427	−0.319713	+	0.947514i	$0.603587\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.95385	−0.247251
$792$	0	0
$793$	−45.8973	−1.62986
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−37.3533	−1.32312	−0.661562	−	0.749891i	$-0.730106\pi$
$797$	−37.3533	−1.32312	−0.661562	+	0.749891i	$0.730106\pi$
$798$	0	0
$799$	−32.1590	−1.13771
$800$	0	0
$801$	0	0
$802$	0	0
$803$	18.9839	0.669928
$804$	0	0
$805$	−69.0420	−2.43341
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8.07357	0.283852	0.141926	−	0.989877i	$-0.454671\pi$
$809$	8.07357	0.283852	0.141926	+	0.989877i	$0.454671\pi$
$810$	0	0
$811$	−34.4690	−1.21037	−0.605185	−	0.796085i	$-0.706901\pi$
$811$	−34.4690	−1.21037	−0.605185	+	0.796085i	$0.706901\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	41.1981	1.44311
$816$	0	0
$817$	21.4074	0.748950
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−28.7082	−1.00192	−0.500962	−	0.865469i	$-0.667020\pi$
$821$	−28.7082	−1.00192	−0.500962	+	0.865469i	$0.667020\pi$
$822$	0	0
$823$	50.3061	1.75356	0.876781	−	0.480891i	$-0.159687\pi$
$823$	50.3061	1.75356	0.876781	+	0.480891i	$0.159687\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	21.8479	0.759725	0.379862	−	0.925043i	$-0.375971\pi$
$827$	21.8479	0.759725	0.379862	+	0.925043i	$0.375971\pi$
$828$	0	0
$829$	−33.5394	−1.16487	−0.582436	−	0.812876i	$-0.697900\pi$
$829$	−33.5394	−1.16487	−0.582436	+	0.812876i	$0.697900\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	80.8900	2.80267
$834$	0	0
$835$	−0.815351	−0.0282164
$836$	0	0
$837$	0	0
$838$	0	0
$839$	35.6352	1.23026	0.615131	−	0.788425i	$-0.289103\pi$
$839$	35.6352	1.23026	0.615131	+	0.788425i	$0.289103\pi$
$840$	0	0
$841$	10.0912	0.347971
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.35293	−0.0465423
$846$	0	0
$847$	−103.581	−3.55910
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−43.8686	−1.50380
$852$	0	0
$853$	−13.0912	−0.448233	−0.224116	−	0.974562i	$-0.571950\pi$
$853$	−13.0912	−0.448233	−0.224116	+	0.974562i	$0.571950\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.57633	−0.122165	−0.0610825	−	0.998133i	$-0.519455\pi$
$857$	−3.57633	−0.122165	−0.0610825	+	0.998133i	$0.519455\pi$
$858$	0	0
$859$	20.0400	0.683754	0.341877	−	0.939745i	$-0.388937\pi$
$859$	20.0400	0.683754	0.341877	+	0.939745i	$0.388937\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	37.1813	1.26567	0.632833	−	0.774288i	$-0.281892\pi$
$863$	37.1813	1.26567	0.632833	+	0.774288i	$0.281892\pi$
$864$	0	0
$865$	23.7954	0.809068
$866$	0	0
$867$	0	0
$868$	0	0
$869$	34.3411	1.16494
$870$	0	0
$871$	3.70891	0.125672
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−58.3567	−1.97282
$876$	0	0
$877$	43.9126	1.48282	0.741412	−	0.671050i	$-0.234156\pi$
$877$	43.9126	1.48282	0.741412	+	0.671050i	$0.234156\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	9.07848	0.305862	0.152931	−	0.988237i	$-0.451129\pi$
$881$	9.07848	0.305862	0.152931	+	0.988237i	$0.451129\pi$
$882$	0	0
$883$	18.3557	0.617719	0.308860	−	0.951108i	$-0.400053\pi$
$883$	18.3557	0.617719	0.308860	+	0.951108i	$0.400053\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0.159457	0.00535405	0.00267702	−	0.999996i	$-0.499148\pi$
$887$	0.159457	0.00535405	0.00267702	+	0.999996i	$0.499148\pi$
$888$	0	0
$889$	7.74662	0.259813
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−13.5138	−0.452223
$894$	0	0
$895$	−15.5803	−0.520790
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−37.5036	−1.25082
$900$	0	0
$901$	37.8765	1.26185
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−15.7586	−0.523832
$906$	0	0
$907$	−14.1933	−0.471280	−0.235640	−	0.971840i	$-0.575719\pi$
$907$	−14.1933	−0.471280	−0.235640	+	0.971840i	$0.575719\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−21.2253	−0.703225	−0.351612	−	0.936146i	$-0.614366\pi$
$911$	−21.2253	−0.703225	−0.351612	+	0.936146i	$0.614366\pi$
$912$	0	0
$913$	−36.1949	−1.19788
$914$	0	0
$915$	0	0
$916$	0	0
$917$	21.3903	0.706371
$918$	0	0
$919$	3.81069	0.125703	0.0628515	−	0.998023i	$-0.479981\pi$
$919$	3.81069	0.125703	0.0628515	+	0.998023i	$0.479981\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.07964	0.0355367
$924$	0	0
$925$	−9.80423	−0.322361
$926$	0	0
$927$	0	0
$928$	0	0
$929$	20.1436	0.660889	0.330444	−	0.943825i	$-0.392801\pi$
$929$	20.1436	0.660889	0.330444	+	0.943825i	$0.392801\pi$
$930$	0	0
$931$	33.9915	1.11402
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−51.6067	−1.68772
$936$	0	0
$937$	−11.1235	−0.363389	−0.181694	−	0.983355i	$-0.558158\pi$
$937$	−11.1235	−0.363389	−0.181694	+	0.983355i	$0.558158\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	43.8373	1.42906	0.714528	−	0.699607i	$-0.246642\pi$
$941$	43.8373	1.42906	0.714528	+	0.699607i	$0.246642\pi$
$942$	0	0
$943$	67.7700	2.20690
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5.25290	−0.170696	−0.0853482	−	0.996351i	$-0.527200\pi$
$947$	−5.25290	−0.170696	−0.0853482	+	0.996351i	$0.527200\pi$
$948$	0	0
$949$	−12.3333	−0.400355
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−49.0974	−1.59042	−0.795210	−	0.606335i	$-0.792639\pi$
$953$	−49.0974	−1.59042	−0.795210	+	0.606335i	$0.792639\pi$
$954$	0	0
$955$	21.0661	0.681685
$956$	0	0
$957$	0	0
$958$	0	0
$959$	35.9726	1.16162
$960$	0	0
$961$	4.98061	0.160665
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−43.3270	−1.39474
$966$	0	0
$967$	7.27667	0.234002	0.117001	−	0.993132i	$-0.462672\pi$
$967$	7.27667	0.234002	0.117001	+	0.993132i	$0.462672\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	22.6656	0.727373	0.363687	−	0.931521i	$-0.381518\pi$
$971$	22.6656	0.727373	0.363687	+	0.931521i	$0.381518\pi$
$972$	0	0
$973$	100.173	3.21139
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8.16969	0.261372	0.130686	−	0.991424i	$-0.458282\pi$
$977$	8.16969	0.261372	0.130686	+	0.991424i	$0.458282\pi$
$978$	0	0
$979$	49.2703	1.57468
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−6.23457	−0.198852	−0.0994260	−	0.995045i	$-0.531701\pi$
$983$	−6.23457	−0.198852	−0.0994260	+	0.995045i	$0.531701\pi$
$984$	0	0
$985$	−15.8089	−0.503715
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−81.1060	−2.57902
$990$	0	0
$991$	−53.0257	−1.68442	−0.842208	−	0.539152i	$-0.818745\pi$
$991$	−53.0257	−1.68442	−0.842208	+	0.539152i	$0.818745\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.06400	0.128838
$996$	0	0
$997$	32.8823	1.04139	0.520697	−	0.853742i	$-0.325672\pi$
$997$	32.8823	1.04139	0.520697	+	0.853742i	$0.325672\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9648.2.a.ca.1.3		5
3.2	odd	2		3216.2.a.ba.1.3		5
4.3	odd	2		2412.2.a.j.1.3		5
12.11	even	2		804.2.a.f.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.a.f.1.3	✓	5	12.11	even	2
2412.2.a.j.1.3		5	4.3	odd	2
3216.2.a.ba.1.3		5	3.2	odd	2
9648.2.a.ca.1.3		5	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)

\(f(q)\)	\(=\)	\(q-1.78961 q^{5} -4.79729 q^{7} +O(q^{10})\)	\(q-1.78961 q^{5} -4.79729 q^{7} +5.70891 q^{11} -3.70891 q^{13} +5.05120 q^{17} +2.12261 q^{19} -8.04190 q^{23} -1.79729 q^{25} +6.25229 q^{29} -5.99838 q^{31} +8.58529 q^{35} +5.45500 q^{37} -8.42711 q^{41} +10.0854 q^{43} -6.36661 q^{47} +16.0140 q^{49} +7.49852 q^{53} -10.2167 q^{55} -1.24623 q^{59} +12.3749 q^{61} +6.63750 q^{65} -1.00000 q^{67} -0.291093 q^{71} +3.32531 q^{73} -27.3873 q^{77} +6.01536 q^{79} -6.34007 q^{83} -9.03968 q^{85} +8.63042 q^{89} +17.7927 q^{91} -3.79864 q^{95} -17.1421 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{5} - 5 q^{7}+O(q^{10}) \) 5 * q - 3 * q^5 - 5 * q^7	\( 5 q - 3 q^{5} - 5 q^{7} + 6 q^{11} + 4 q^{13} - q^{17} - 13 q^{19} + 10 q^{25} - 3 q^{29} - 3 q^{31} - 9 q^{35} + 12 q^{37} - 11 q^{41} - 3 q^{43} - 13 q^{47} + 24 q^{49} + 9 q^{53} - 14 q^{55} - 12 q^{59} + 4 q^{61} + 8 q^{65} - 5 q^{67} - 24 q^{71} + 12 q^{73} + 4 q^{77} + 4 q^{79} - 27 q^{83} - 4 q^{85} + 5 q^{89} - 14 q^{91} - 30 q^{95} + 8 q^{97}+O(q^{100}) \) 5 * q - 3 * q^5 - 5 * q^7 + 6 * q^11 + 4 * q^13 - q^17 - 13 * q^19 + 10 * q^25 - 3 * q^29 - 3 * q^31 - 9 * q^35 + 12 * q^37 - 11 * q^41 - 3 * q^43 - 13 * q^47 + 24 * q^49 + 9 * q^53 - 14 * q^55 - 12 * q^59 + 4 * q^61 + 8 * q^65 - 5 * q^67 - 24 * q^71 + 12 * q^73 + 4 * q^77 + 4 * q^79 - 27 * q^83 - 4 * q^85 + 5 * q^89 - 14 * q^91 - 30 * q^95 + 8 * q^97

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