LMFDB - Embedded newform 804.2.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [804,2,Mod(1,804)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(804, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("804.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 804 = 2^{2} \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	804.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:	$6.41997232251$
Analytic rank:	$0$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.789611$ of defining polynomial
Character	$\chi$	$=$	804.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.78961 q^{5} +4.79729 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.78961 q^{5} +4.79729 q^{7} +1.00000 q^{9} +5.70891 q^{11} -3.70891 q^{13} +1.78961 q^{15} -5.05120 q^{17} -2.12261 q^{19} +4.79729 q^{21} -8.04190 q^{23} -1.79729 q^{25} +1.00000 q^{27} -6.25229 q^{29} +5.99838 q^{31} +5.70891 q^{33} +8.58529 q^{35} +5.45500 q^{37} -3.70891 q^{39} +8.42711 q^{41} -10.0854 q^{43} +1.78961 q^{45} -6.36661 q^{47} +16.0140 q^{49} -5.05120 q^{51} -7.49852 q^{53} +10.2167 q^{55} -2.12261 q^{57} -1.24623 q^{59} +12.3749 q^{61} +4.79729 q^{63} -6.63750 q^{65} +1.00000 q^{67} -8.04190 q^{69} -0.291093 q^{71} +3.32531 q^{73} -1.79729 q^{75} +27.3873 q^{77} -6.01536 q^{79} +1.00000 q^{81} -6.34007 q^{83} -9.03968 q^{85} -6.25229 q^{87} -8.63042 q^{89} -17.7927 q^{91} +5.99838 q^{93} -3.79864 q^{95} -17.1421 q^{97} +5.70891 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + 3 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 + 3 * q^5 + 5 * q^7 + 5 * q^9	$ 5 q + 5 q^{3} + 3 q^{5} + 5 q^{7} + 5 q^{9} + 6 q^{11} + 4 q^{13} + 3 q^{15} + q^{17} + 13 q^{19} + 5 q^{21} + 10 q^{25} + 5 q^{27} + 3 q^{29} + 3 q^{31} + 6 q^{33} - 9 q^{35} + 12 q^{37} + 4 q^{39} + 11 q^{41} + 3 q^{43} + 3 q^{45} - 13 q^{47} + 24 q^{49} + q^{51} - 9 q^{53} + 14 q^{55} + 13 q^{57} - 12 q^{59} + 4 q^{61} + 5 q^{63} - 8 q^{65} + 5 q^{67} - 24 q^{71} + 12 q^{73} + 10 q^{75} - 4 q^{77} - 4 q^{79} + 5 q^{81} - 27 q^{83} - 4 q^{85} + 3 q^{87} - 5 q^{89} + 14 q^{91} + 3 q^{93} - 30 q^{95} + 8 q^{97} + 6 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + 3 * q^5 + 5 * q^7 + 5 * q^9 + 6 * q^11 + 4 * q^13 + 3 * q^15 + q^17 + 13 * q^19 + 5 * q^21 + 10 * q^25 + 5 * q^27 + 3 * q^29 + 3 * q^31 + 6 * q^33 - 9 * q^35 + 12 * q^37 + 4 * q^39 + 11 * q^41 + 3 * q^43 + 3 * q^45 - 13 * q^47 + 24 * q^49 + q^51 - 9 * q^53 + 14 * q^55 + 13 * q^57 - 12 * q^59 + 4 * q^61 + 5 * q^63 - 8 * q^65 + 5 * q^67 - 24 * q^71 + 12 * q^73 + 10 * q^75 - 4 * q^77 - 4 * q^79 + 5 * q^81 - 27 * q^83 - 4 * q^85 + 3 * q^87 - 5 * q^89 + 14 * q^91 + 3 * q^93 - 30 * q^95 + 8 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.78961	0.800338	0.400169	−	0.916441i	$-0.368951\pi$
$5$	1.78961	0.800338	0.400169	+	0.916441i	$0.368951\pi$
$6$	0	0
$7$	4.79729	1.81321	0.906603	−	0.421984i	$-0.138666\pi$
$7$	4.79729	1.81321	0.906603	+	0.421984i	$0.138666\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	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
$15$	1.78961	0.462076
$16$	0	0
$17$	−5.05120	−1.22510	−0.612548	−	0.790433i	$-0.709855\pi$
$17$	−5.05120	−1.22510	−0.612548	+	0.790433i	$0.709855\pi$
$18$	0	0
$19$	−2.12261	−0.486959	−0.243480	−	0.969906i	$-0.578289\pi$
$19$	−2.12261	−0.486959	−0.243480	+	0.969906i	$0.578289\pi$
$20$	0	0
$21$	4.79729	1.04686
$22$	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
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.25229	−1.16102	−0.580511	−	0.814253i	$-0.697147\pi$
$29$	−6.25229	−1.16102	−0.580511	+	0.814253i	$0.697147\pi$
$30$	0	0
$31$	5.99838	1.07734	0.538671	−	0.842516i	$-0.318927\pi$
$31$	5.99838	1.07734	0.538671	+	0.842516i	$0.318927\pi$
$32$	0	0
$33$	5.70891	0.993793
$34$	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
$39$	−3.70891	−0.593900
$40$	0	0
$41$	8.42711	1.31609	0.658047	−	0.752977i	$-0.271383\pi$
$41$	8.42711	1.31609	0.658047	+	0.752977i	$0.271383\pi$
$42$	0	0
$43$	−10.0854	−1.53801	−0.769006	−	0.639241i	$-0.779248\pi$
$43$	−10.0854	−1.53801	−0.769006	+	0.639241i	$0.779248\pi$
$44$	0	0
$45$	1.78961	0.266779
$46$	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
$51$	−5.05120	−0.707310
$52$	0	0
$53$	−7.49852	−1.03000	−0.515000	−	0.857190i	$-0.672208\pi$
$53$	−7.49852	−1.03000	−0.515000	+	0.857190i	$0.672208\pi$
$54$	0	0
$55$	10.2167	1.37762
$56$	0	0
$57$	−2.12261	−0.281146
$58$	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
$63$	4.79729	0.604402
$64$	0	0
$65$	−6.63750	−0.823281
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	−8.04190	−0.968131
$70$	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
$75$	−1.79729	−0.207533
$76$	0	0
$77$	27.3873	3.12107
$78$	0	0
$79$	−6.01536	−0.676781	−0.338391	−	0.941006i	$-0.609883\pi$
$79$	−6.01536	−0.676781	−0.338391	+	0.941006i	$0.609883\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	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
$87$	−6.25229	−0.670316
$88$	0	0
$89$	−8.63042	−0.914823	−0.457411	−	0.889255i	$-0.651223\pi$
$89$	−8.63042	−0.914823	−0.457411	+	0.889255i	$0.651223\pi$
$90$	0	0
$91$	−17.7927	−1.86518
$92$	0	0
$93$	5.99838	0.622003
$94$	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
$99$	5.70891	0.573767
$100$	0	0
$101$	6.63750	0.660456	0.330228	−	0.943901i	$-0.392874\pi$
$101$	6.63750	0.660456	0.330228	+	0.943901i	$0.392874\pi$
$102$	0	0
$103$	6.37490	0.628137	0.314069	−	0.949400i	$-0.398308\pi$
$103$	6.37490	0.628137	0.314069	+	0.949400i	$0.398308\pi$
$104$	0	0
$105$	8.58529	0.837838
$106$	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
$111$	5.45500	0.517766
$112$	0	0
$113$	−1.44954	−0.136361	−0.0681805	−	0.997673i	$-0.521719\pi$
$113$	−1.44954	−0.136361	−0.0681805	+	0.997673i	$0.521719\pi$
$114$	0	0
$115$	−14.3919	−1.34205
$116$	0	0
$117$	−3.70891	−0.342889
$118$	0	0
$119$	−24.2321	−2.22135
$120$	0	0
$121$	21.5916	1.96287
$122$	0	0
$123$	8.42711	0.759847
$124$	0	0
$125$	−12.1645	−1.08803
$126$	0	0
$127$	1.61479	0.143290	0.0716448	−	0.997430i	$-0.477175\pi$
$127$	1.61479	0.143290	0.0716448	+	0.997430i	$0.477175\pi$
$128$	0	0
$129$	−10.0854	−0.887972
$130$	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
$135$	1.78961	0.154025
$136$	0	0
$137$	7.49852	0.640642	0.320321	−	0.947309i	$-0.396209\pi$
$137$	7.49852	0.640642	0.320321	+	0.947309i	$0.396209\pi$
$138$	0	0
$139$	20.8811	1.77111	0.885556	−	0.464533i	$-0.153778\pi$
$139$	20.8811	1.77111	0.885556	+	0.464533i	$0.153778\pi$
$140$	0	0
$141$	−6.36661	−0.536166
$142$	0	0
$143$	−21.1738	−1.77064
$144$	0	0
$145$	−11.1892	−0.929210
$146$	0	0
$147$	16.0140	1.32081
$148$	0	0
$149$	−16.3547	−1.33983	−0.669914	−	0.742438i	$-0.733669\pi$
$149$	−16.3547	−1.33983	−0.669914	+	0.742438i	$0.733669\pi$
$150$	0	0
$151$	−4.72016	−0.384121	−0.192060	−	0.981383i	$-0.561517\pi$
$151$	−4.72016	−0.384121	−0.192060	+	0.981383i	$0.561517\pi$
$152$	0	0
$153$	−5.05120	−0.408365
$154$	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
$159$	−7.49852	−0.594671
$160$	0	0
$161$	−38.5794	−3.04048
$162$	0	0
$163$	23.0207	1.80312	0.901560	−	0.432655i	$-0.142423\pi$
$163$	23.0207	1.80312	0.901560	+	0.432655i	$0.142423\pi$
$164$	0	0
$165$	10.2167	0.795371
$166$	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
$171$	−2.12261	−0.162320
$172$	0	0
$173$	13.2964	1.01091	0.505454	−	0.862854i	$-0.331325\pi$
$173$	13.2964	1.01091	0.505454	+	0.862854i	$0.331325\pi$
$174$	0	0
$175$	−8.62214	−0.651772
$176$	0	0
$177$	−1.24623	−0.0936721
$178$	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
$183$	12.3749	0.914779
$184$	0	0
$185$	9.76233	0.717741
$186$	0	0
$187$	−28.8368	−2.10876
$188$	0	0
$189$	4.79729	0.348952
$190$	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
$195$	−6.63750	−0.475321
$196$	0	0
$197$	−8.83373	−0.629377	−0.314689	−	0.949195i	$-0.601900\pi$
$197$	−8.83373	−0.629377	−0.314689	+	0.949195i	$0.601900\pi$
$198$	0	0
$199$	2.27089	0.160979	0.0804895	−	0.996755i	$-0.474352\pi$
$199$	2.27089	0.160979	0.0804895	+	0.996755i	$0.474352\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	−29.9941	−2.10517
$204$	0	0
$205$	15.0813	1.05332
$206$	0	0
$207$	−8.04190	−0.558951
$208$	0	0
$209$	−12.1178	−0.838203
$210$	0	0
$211$	−9.98140	−0.687148	−0.343574	−	0.939126i	$-0.611638\pi$
$211$	−9.98140	−0.687148	−0.343574	+	0.939126i	$0.611638\pi$
$212$	0	0
$213$	−0.291093	−0.0199454
$214$	0	0
$215$	−18.0490	−1.23093
$216$	0	0
$217$	28.7760	1.95344
$218$	0	0
$219$	3.32531	0.224704
$220$	0	0
$221$	18.7344	1.26021
$222$	0	0
$223$	−5.92125	−0.396516	−0.198258	−	0.980150i	$-0.563528\pi$
$223$	−5.92125	−0.396516	−0.198258	+	0.980150i	$0.563528\pi$
$224$	0	0
$225$	−1.79729	−0.119820
$226$	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
$231$	27.3873	1.80195
$232$	0	0
$233$	16.5580	1.08475	0.542375	−	0.840136i	$-0.317525\pi$
$233$	16.5580	1.08475	0.542375	+	0.840136i	$0.317525\pi$
$234$	0	0
$235$	−11.3938	−0.743247
$236$	0	0
$237$	−6.01536	−0.390740
$238$	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
$243$	1.00000	0.0641500
$244$	0	0
$245$	28.6589	1.83095
$246$	0	0
$247$	7.87255	0.500918
$248$	0	0
$249$	−6.34007	−0.401786
$250$	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
$255$	−9.03968	−0.566087
$256$	0	0
$257$	−6.25229	−0.390007	−0.195004	−	0.980803i	$-0.562472\pi$
$257$	−6.25229	−0.390007	−0.195004	+	0.980803i	$0.562472\pi$
$258$	0	0
$259$	26.1692	1.62608
$260$	0	0
$261$	−6.25229	−0.387007
$262$	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
$267$	−8.63042	−0.528173
$268$	0	0
$269$	8.37813	0.510824	0.255412	−	0.966832i	$-0.417789\pi$
$269$	8.37813	0.510824	0.255412	+	0.966832i	$0.417789\pi$
$270$	0	0
$271$	−18.1664	−1.10353	−0.551765	−	0.833999i	$-0.686046\pi$
$271$	−18.1664	−1.10353	−0.551765	+	0.833999i	$0.686046\pi$
$272$	0	0
$273$	−17.7927	−1.07686
$274$	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
$279$	5.99838	0.359114
$280$	0	0
$281$	26.8827	1.60369	0.801844	−	0.597533i	$-0.203852\pi$
$281$	26.8827	1.60369	0.801844	+	0.597533i	$0.203852\pi$
$282$	0	0
$283$	1.86324	0.110758	0.0553789	−	0.998465i	$-0.482363\pi$
$283$	1.86324	0.110758	0.0553789	+	0.998465i	$0.482363\pi$
$284$	0	0
$285$	−3.79864	−0.225012
$286$	0	0
$287$	40.4273	2.38635
$288$	0	0
$289$	8.51463	0.500860
$290$	0	0
$291$	−17.1421	−1.00489
$292$	0	0
$293$	3.75599	0.219427	0.109714	−	0.993963i	$-0.465007\pi$
$293$	3.75599	0.219427	0.109714	+	0.993963i	$0.465007\pi$
$294$	0	0
$295$	−2.23026	−0.129851
$296$	0	0
$297$	5.70891	0.331264
$298$	0	0
$299$	29.8267	1.72492
$300$	0	0
$301$	−48.3827	−2.78873
$302$	0	0
$303$	6.63750	0.381314
$304$	0	0
$305$	22.1463	1.26809
$306$	0	0
$307$	10.3983	0.593464	0.296732	−	0.954961i	$-0.404103\pi$
$307$	10.3983	0.593464	0.296732	+	0.954961i	$0.404103\pi$
$308$	0	0
$309$	6.37490	0.362655
$310$	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
$315$	8.58529	0.483726
$316$	0	0
$317$	22.3935	1.25774	0.628872	−	0.777509i	$-0.283517\pi$
$317$	22.3935	1.25774	0.628872	+	0.777509i	$0.283517\pi$
$318$	0	0
$319$	−35.6938	−1.99847
$320$	0	0
$321$	5.97979	0.333760
$322$	0	0
$323$	10.7217	0.596572
$324$	0	0
$325$	6.66599	0.369763
$326$	0	0
$327$	15.5475	0.859779
$328$	0	0
$329$	−30.5425	−1.68386
$330$	0	0
$331$	−0.577874	−0.0317629	−0.0158814	−	0.999874i	$-0.505055\pi$
$331$	−0.577874	−0.0317629	−0.0158814	+	0.999874i	$0.505055\pi$
$332$	0	0
$333$	5.45500	0.298932
$334$	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
$339$	−1.44954	−0.0787280
$340$	0	0
$341$	34.2442	1.85443
$342$	0	0
$343$	43.2429	2.33490
$344$	0	0
$345$	−14.3919	−0.774833
$346$	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
$351$	−3.70891	−0.197967
$352$	0	0
$353$	24.7834	1.31909	0.659544	−	0.751666i	$-0.270749\pi$
$353$	24.7834	1.31909	0.659544	+	0.751666i	$0.270749\pi$
$354$	0	0
$355$	−0.520943	−0.0276488
$356$	0	0
$357$	−24.2321	−1.28250
$358$	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
$363$	21.5916	1.13327
$364$	0	0
$365$	5.95102	0.311491
$366$	0	0
$367$	5.40245	0.282006	0.141003	−	0.990009i	$-0.454967\pi$
$367$	5.40245	0.282006	0.141003	+	0.990009i	$0.454967\pi$
$368$	0	0
$369$	8.42711	0.438698
$370$	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
$375$	−12.1645	−0.628173
$376$	0	0
$377$	23.1892	1.19430
$378$	0	0
$379$	17.0457	0.875581	0.437790	−	0.899077i	$-0.355761\pi$
$379$	17.0457	0.875581	0.437790	+	0.899077i	$0.355761\pi$
$380$	0	0
$381$	1.61479	0.0827282
$382$	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
$387$	−10.0854	−0.512671
$388$	0	0
$389$	10.1002	0.512099	0.256049	−	0.966664i	$-0.417579\pi$
$389$	10.1002	0.512099	0.256049	+	0.966664i	$0.417579\pi$
$390$	0	0
$391$	40.6213	2.05431
$392$	0	0
$393$	−4.45883	−0.224918
$394$	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
$399$	−10.1828	−0.509776
$400$	0	0
$401$	27.8488	1.39070	0.695352	−	0.718669i	$-0.255248\pi$
$401$	27.8488	1.39070	0.695352	+	0.718669i	$0.255248\pi$
$402$	0	0
$403$	−22.2474	−1.10822
$404$	0	0
$405$	1.78961	0.0889265
$406$	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
$411$	7.49852	0.369875
$412$	0	0
$413$	−5.97851	−0.294183
$414$	0	0
$415$	−11.3463	−0.556966
$416$	0	0
$417$	20.8811	1.02255
$418$	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
$423$	−6.36661	−0.309555
$424$	0	0
$425$	9.07848	0.440371
$426$	0	0
$427$	59.3660	2.87292
$428$	0	0
$429$	−21.1738	−1.02228
$430$	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
$435$	−11.1892	−0.536480
$436$	0	0
$437$	17.0698	0.816559
$438$	0	0
$439$	−27.4945	−1.31224	−0.656121	−	0.754655i	$-0.727804\pi$
$439$	−27.4945	−1.31224	−0.656121	+	0.754655i	$0.727804\pi$
$440$	0	0
$441$	16.0140	0.762572
$442$	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
$447$	−16.3547	−0.773550
$448$	0	0
$449$	22.9193	1.08163	0.540815	−	0.841142i	$-0.318116\pi$
$449$	22.9193	1.08163	0.540815	+	0.841142i	$0.318116\pi$
$450$	0	0
$451$	48.1096	2.26539
$452$	0	0
$453$	−4.72016	−0.221772
$454$	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
$459$	−5.05120	−0.235770
$460$	0	0
$461$	−15.6143	−0.727232	−0.363616	−	0.931549i	$-0.618458\pi$
$461$	−15.6143	−0.727232	−0.363616	+	0.931549i	$0.618458\pi$
$462$	0	0
$463$	−0.955468	−0.0444044	−0.0222022	−	0.999754i	$-0.507068\pi$
$463$	−0.955468	−0.0444044	−0.0222022	+	0.999754i	$0.507068\pi$
$464$	0	0
$465$	10.7348	0.497813
$466$	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
$471$	−13.0069	−0.599328
$472$	0	0
$473$	−57.5767	−2.64738
$474$	0	0
$475$	3.81494	0.175042
$476$	0	0
$477$	−7.49852	−0.343334
$478$	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
$483$	−38.5794	−1.75542
$484$	0	0
$485$	−30.6777	−1.39300
$486$	0	0
$487$	11.4950	0.520888	0.260444	−	0.965489i	$-0.416131\pi$
$487$	11.4950	0.520888	0.260444	+	0.965489i	$0.416131\pi$
$488$	0	0
$489$	23.0207	1.04103
$490$	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
$495$	10.2167	0.459208
$496$	0	0
$497$	−1.39646	−0.0626397
$498$	0	0
$499$	4.08704	0.182961	0.0914805	−	0.995807i	$-0.470840\pi$
$499$	4.08704	0.182961	0.0914805	+	0.995807i	$0.470840\pi$
$500$	0	0
$501$	0.455602	0.0203548
$502$	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
$507$	0.755993	0.0335748
$508$	0	0
$509$	24.3678	1.08008	0.540042	−	0.841638i	$-0.318408\pi$
$509$	24.3678	1.08008	0.540042	+	0.841638i	$0.318408\pi$
$510$	0	0
$511$	15.9525	0.705697
$512$	0	0
$513$	−2.12261	−0.0937154
$514$	0	0
$515$	11.4086	0.502722
$516$	0	0
$517$	−36.3464	−1.59851
$518$	0	0
$519$	13.2964	0.583648
$520$	0	0
$521$	11.5448	0.505787	0.252894	−	0.967494i	$-0.418618\pi$
$521$	11.5448	0.505787	0.252894	+	0.967494i	$0.418618\pi$
$522$	0	0
$523$	4.78739	0.209338	0.104669	−	0.994507i	$-0.466622\pi$
$523$	4.78739	0.209338	0.104669	+	0.994507i	$0.466622\pi$
$524$	0	0
$525$	−8.62214	−0.376301
$526$	0	0
$527$	−30.2990	−1.31985
$528$	0	0
$529$	41.6722	1.81183
$530$	0	0
$531$	−1.24623	−0.0540816
$532$	0	0
$533$	−31.2554	−1.35382
$534$	0	0
$535$	10.7015	0.462666
$536$	0	0
$537$	8.70594	0.375689
$538$	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
$543$	8.80558	0.377883
$544$	0	0
$545$	27.8240	1.19185
$546$	0	0
$547$	20.6250	0.881860	0.440930	−	0.897542i	$-0.354649\pi$
$547$	20.6250	0.881860	0.440930	+	0.897542i	$0.354649\pi$
$548$	0	0
$549$	12.3749	0.528148
$550$	0	0
$551$	13.2712	0.565370
$552$	0	0
$553$	−28.8575	−1.22714
$554$	0	0
$555$	9.76233	0.414388
$556$	0	0
$557$	−9.96948	−0.422421	−0.211210	−	0.977441i	$-0.567740\pi$
$557$	−9.96948	−0.422421	−0.211210	+	0.977441i	$0.567740\pi$
$558$	0	0
$559$	37.4059	1.58210
$560$	0	0
$561$	−28.8368	−1.21749
$562$	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
$567$	4.79729	0.201467
$568$	0	0
$569$	45.4814	1.90668	0.953340	−	0.301899i	$-0.0976204\pi$
$569$	45.4814	1.90668	0.953340	+	0.301899i	$0.0976204\pi$
$570$	0	0
$571$	−30.3044	−1.26820	−0.634099	−	0.773252i	$-0.718629\pi$
$571$	−30.3044	−1.26820	−0.634099	+	0.773252i	$0.718629\pi$
$572$	0	0
$573$	−11.7714	−0.491755
$574$	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
$579$	24.2103	1.00614
$580$	0	0
$581$	−30.4152	−1.26183
$582$	0	0
$583$	−42.8083	−1.77294
$584$	0	0
$585$	−6.63750	−0.274427
$586$	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
$591$	−8.83373	−0.363371
$592$	0	0
$593$	−42.1055	−1.72907	−0.864533	−	0.502576i	$-0.832386\pi$
$593$	−42.1055	−1.72907	−0.864533	+	0.502576i	$0.832386\pi$
$594$	0	0
$595$	−43.3660	−1.77783
$596$	0	0
$597$	2.27089	0.0929412
$598$	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
$603$	1.00000	0.0407231
$604$	0	0
$605$	38.6406	1.57096
$606$	0	0
$607$	−35.3303	−1.43401	−0.717007	−	0.697066i	$-0.754488\pi$
$607$	−35.3303	−1.43401	−0.717007	+	0.697066i	$0.754488\pi$
$608$	0	0
$609$	−29.9941	−1.21542
$610$	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
$615$	15.0813	0.608135
$616$	0	0
$617$	14.9582	0.602196	0.301098	−	0.953593i	$-0.402647\pi$
$617$	14.9582	0.602196	0.301098	+	0.953593i	$0.402647\pi$
$618$	0	0
$619$	33.6653	1.35312	0.676561	−	0.736387i	$-0.263470\pi$
$619$	33.6653	1.35312	0.676561	+	0.736387i	$0.263470\pi$
$620$	0	0
$621$	−8.04190	−0.322710
$622$	0	0
$623$	−41.4027	−1.65876
$624$	0	0
$625$	−12.7833	−0.511331
$626$	0	0
$627$	−12.1178	−0.483937
$628$	0	0
$629$	−27.5543	−1.09866
$630$	0	0
$631$	−26.0652	−1.03764	−0.518820	−	0.854884i	$-0.673628\pi$
$631$	−26.0652	−1.03764	−0.518820	+	0.854884i	$0.673628\pi$
$632$	0	0
$633$	−9.98140	−0.396725
$634$	0	0
$635$	2.88985	0.114680
$636$	0	0
$637$	−59.3945	−2.35330
$638$	0	0
$639$	−0.291093	−0.0115155
$640$	0	0
$641$	−36.5548	−1.44383	−0.721913	−	0.691983i	$-0.756737\pi$
$641$	−36.5548	−1.44383	−0.721913	+	0.691983i	$0.756737\pi$
$642$	0	0
$643$	7.25861	0.286252	0.143126	−	0.989705i	$-0.454285\pi$
$643$	7.25861	0.286252	0.143126	+	0.989705i	$0.454285\pi$
$644$	0	0
$645$	−18.0490	−0.710678
$646$	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
$651$	28.7760	1.12782
$652$	0	0
$653$	−16.3800	−0.641000	−0.320500	−	0.947249i	$-0.603851\pi$
$653$	−16.3800	−0.641000	−0.320500	+	0.947249i	$0.603851\pi$
$654$	0	0
$655$	−7.97958	−0.311788
$656$	0	0
$657$	3.32531	0.129733
$658$	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
$663$	18.7344	0.727585
$664$	0	0
$665$	−18.2232	−0.706665
$666$	0	0
$667$	50.2803	1.94686
$668$	0	0
$669$	−5.92125	−0.228929
$670$	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
$675$	−1.79729	−0.0691778
$676$	0	0
$677$	−23.8215	−0.915536	−0.457768	−	0.889072i	$-0.651351\pi$
$677$	−23.8215	−0.915536	−0.457768	+	0.889072i	$0.651351\pi$
$678$	0	0
$679$	−82.2356	−3.15591
$680$	0	0
$681$	−15.7662	−0.604161
$682$	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
$687$	−28.1894	−1.07549
$688$	0	0
$689$	27.8113	1.05953
$690$	0	0
$691$	−21.8667	−0.831848	−0.415924	−	0.909399i	$-0.636542\pi$
$691$	−21.8667	−0.831848	−0.415924	+	0.909399i	$0.636542\pi$
$692$	0	0
$693$	27.3873	1.04036
$694$	0	0
$695$	37.3690	1.41749
$696$	0	0
$697$	−42.5670	−1.61234
$698$	0	0
$699$	16.5580	0.626281
$700$	0	0
$701$	36.9614	1.39601	0.698006	−	0.716092i	$-0.254071\pi$
$701$	36.9614	1.39601	0.698006	+	0.716092i	$0.254071\pi$
$702$	0	0
$703$	−11.5788	−0.436703
$704$	0	0
$705$	−11.3938	−0.429114
$706$	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
$711$	−6.01536	−0.225594
$712$	0	0
$713$	−48.2384	−1.80654
$714$	0	0
$715$	−37.8929	−1.41711
$716$	0	0
$717$	−15.3506	−0.573278
$718$	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
$723$	17.2794	0.642629
$724$	0	0
$725$	11.2372	0.417339
$726$	0	0
$727$	−29.1985	−1.08291	−0.541456	−	0.840729i	$-0.682127\pi$
$727$	−29.1985	−1.08291	−0.541456	+	0.840729i	$0.682127\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	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
$735$	28.6589	1.05710
$736$	0	0
$737$	5.70891	0.210290
$738$	0	0
$739$	31.5020	1.15882	0.579410	−	0.815036i	$-0.303283\pi$
$739$	31.5020	1.15882	0.579410	+	0.815036i	$0.303283\pi$
$740$	0	0
$741$	7.87255	0.289205
$742$	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
$747$	−6.34007	−0.231971
$748$	0	0
$749$	28.6868	1.04819
$750$	0	0
$751$	−17.5344	−0.639840	−0.319920	−	0.947445i	$-0.603656\pi$
$751$	−17.5344	−0.639840	−0.319920	+	0.947445i	$0.603656\pi$
$752$	0	0
$753$	−12.5046	−0.455692
$754$	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
$759$	−45.9105	−1.66644
$760$	0	0
$761$	23.7544	0.861098	0.430549	−	0.902567i	$-0.358320\pi$
$761$	23.7544	0.861098	0.430549	+	0.902567i	$0.358320\pi$
$762$	0	0
$763$	74.5859	2.70019
$764$	0	0
$765$	−9.03968	−0.326830
$766$	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
$771$	−6.25229	−0.225171
$772$	0	0
$773$	−31.0227	−1.11581	−0.557905	−	0.829905i	$-0.688395\pi$
$773$	−31.0227	−1.11581	−0.557905	+	0.829905i	$0.688395\pi$
$774$	0	0
$775$	−10.7809	−0.387260
$776$	0	0
$777$	26.1692	0.938816
$778$	0	0
$779$	−17.8874	−0.640884
$780$	0	0
$781$	−1.66182	−0.0594647
$782$	0	0
$783$	−6.25229	−0.223439
$784$	0	0
$785$	−23.2774	−0.830805
$786$	0	0
$787$	17.9382	0.639427	0.319713	−	0.947514i	$-0.396413\pi$
$787$	17.9382	0.639427	0.319713	+	0.947514i	$0.396413\pi$
$788$	0	0
$789$	−1.32767	−0.0472665
$790$	0	0
$791$	−6.95385	−0.247251
$792$	0	0
$793$	−45.8973	−1.62986
$794$	0	0
$795$	−13.4194	−0.475938
$796$	0	0
$797$	37.3533	1.32312	0.661562	−	0.749891i	$-0.269894\pi$
$797$	37.3533	1.32312	0.661562	+	0.749891i	$0.269894\pi$
$798$	0	0
$799$	32.1590	1.13771
$800$	0	0
$801$	−8.63042	−0.304941
$802$	0	0
$803$	18.9839	0.669928
$804$	0	0
$805$	−69.0420	−2.43341
$806$	0	0
$807$	8.37813	0.294924
$808$	0	0
$809$	−8.07357	−0.283852	−0.141926	−	0.989877i	$-0.545329\pi$
$809$	−8.07357	−0.283852	−0.141926	+	0.989877i	$0.545329\pi$
$810$	0	0
$811$	34.4690	1.21037	0.605185	−	0.796085i	$-0.293099\pi$
$811$	34.4690	1.21037	0.605185	+	0.796085i	$0.293099\pi$
$812$	0	0
$813$	−18.1664	−0.637124
$814$	0	0
$815$	41.1981	1.44311
$816$	0	0
$817$	21.4074	0.748950
$818$	0	0
$819$	−17.7927	−0.621728
$820$	0	0
$821$	28.7082	1.00192	0.500962	−	0.865469i	$-0.332980\pi$
$821$	28.7082	1.00192	0.500962	+	0.865469i	$0.332980\pi$
$822$	0	0
$823$	−50.3061	−1.75356	−0.876781	−	0.480891i	$-0.840313\pi$
$823$	−50.3061	−1.75356	−0.876781	+	0.480891i	$0.840313\pi$
$824$	0	0
$825$	−10.2606	−0.357227
$826$	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
$831$	−1.33683	−0.0463741
$832$	0	0
$833$	−80.8900	−2.80267
$834$	0	0
$835$	0.815351	0.0282164
$836$	0	0
$837$	5.99838	0.207334
$838$	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
$843$	26.8827	0.925890
$844$	0	0
$845$	1.35293	0.0465423
$846$	0	0
$847$	103.581	3.55910
$848$	0	0
$849$	1.86324	0.0639461
$850$	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
$855$	−3.79864	−0.129911
$856$	0	0
$857$	3.57633	0.122165	0.0610825	−	0.998133i	$-0.480545\pi$
$857$	3.57633	0.122165	0.0610825	+	0.998133i	$0.480545\pi$
$858$	0	0
$859$	−20.0400	−0.683754	−0.341877	−	0.939745i	$-0.611063\pi$
$859$	−20.0400	−0.683754	−0.341877	+	0.939745i	$0.611063\pi$
$860$	0	0
$861$	40.4273	1.37776
$862$	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
$867$	8.51463	0.289172
$868$	0	0
$869$	−34.3411	−1.16494
$870$	0	0
$871$	−3.70891	−0.125672
$872$	0	0
$873$	−17.1421	−0.580172
$874$	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
$879$	3.75599	0.126686
$880$	0	0
$881$	−9.07848	−0.305862	−0.152931	−	0.988237i	$-0.548871\pi$
$881$	−9.07848	−0.305862	−0.152931	+	0.988237i	$0.548871\pi$
$882$	0	0
$883$	−18.3557	−0.617719	−0.308860	−	0.951108i	$-0.599947\pi$
$883$	−18.3557	−0.617719	−0.308860	+	0.951108i	$0.599947\pi$
$884$	0	0
$885$	−2.23026	−0.0749694
$886$	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
$891$	5.70891	0.191256
$892$	0	0
$893$	13.5138	0.452223
$894$	0	0
$895$	15.5803	0.520790
$896$	0	0
$897$	29.8267	0.995884
$898$	0	0
$899$	−37.5036	−1.25082
$900$	0	0
$901$	37.8765	1.26185
$902$	0	0
$903$	−48.3827	−1.61008
$904$	0	0
$905$	15.7586	0.523832
$906$	0	0
$907$	14.1933	0.471280	0.235640	−	0.971840i	$-0.424281\pi$
$907$	14.1933	0.471280	0.235640	+	0.971840i	$0.424281\pi$
$908$	0	0
$909$	6.63750	0.220152
$910$	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
$915$	22.1463	0.732133
$916$	0	0
$917$	−21.3903	−0.706371
$918$	0	0
$919$	−3.81069	−0.125703	−0.0628515	−	0.998023i	$-0.520019\pi$
$919$	−3.81069	−0.125703	−0.0628515	+	0.998023i	$0.520019\pi$
$920$	0	0
$921$	10.3983	0.342637
$922$	0	0
$923$	1.07964	0.0355367
$924$	0	0
$925$	−9.80423	−0.322361
$926$	0	0
$927$	6.37490	0.209379
$928$	0	0
$929$	−20.1436	−0.660889	−0.330444	−	0.943825i	$-0.607199\pi$
$929$	−20.1436	−0.660889	−0.330444	+	0.943825i	$0.607199\pi$
$930$	0	0
$931$	−33.9915	−1.11402
$932$	0	0
$933$	−25.6204	−0.838774
$934$	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
$939$	−20.3749	−0.664910
$940$	0	0
$941$	−43.8373	−1.42906	−0.714528	−	0.699607i	$-0.753358\pi$
$941$	−43.8373	−1.42906	−0.714528	+	0.699607i	$0.753358\pi$
$942$	0	0
$943$	−67.7700	−2.20690
$944$	0	0
$945$	8.58529	0.279279
$946$	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
$951$	22.3935	0.726159
$952$	0	0
$953$	49.0974	1.59042	0.795210	−	0.606335i	$-0.207361\pi$
$953$	49.0974	1.59042	0.795210	+	0.606335i	$0.207361\pi$
$954$	0	0
$955$	−21.0661	−0.681685
$956$	0	0
$957$	−35.6938	−1.15382
$958$	0	0
$959$	35.9726	1.16162
$960$	0	0
$961$	4.98061	0.160665
$962$	0	0
$963$	5.97979	0.192696
$964$	0	0
$965$	43.3270	1.39474
$966$	0	0
$967$	−7.27667	−0.234002	−0.117001	−	0.993132i	$-0.537328\pi$
$967$	−7.27667	−0.234002	−0.117001	+	0.993132i	$0.537328\pi$
$968$	0	0
$969$	10.7217	0.344431
$970$	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
$975$	6.66599	0.213483
$976$	0	0
$977$	−8.16969	−0.261372	−0.130686	−	0.991424i	$-0.541718\pi$
$977$	−8.16969	−0.261372	−0.130686	+	0.991424i	$0.541718\pi$
$978$	0	0
$979$	−49.2703	−1.57468
$980$	0	0
$981$	15.5475	0.496393
$982$	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
$987$	−30.5425	−0.972179
$988$	0	0
$989$	81.1060	2.57902
$990$	0	0
$991$	53.0257	1.68442	0.842208	−	0.539152i	$-0.181255\pi$
$991$	53.0257	1.68442	0.842208	+	0.539152i	$0.181255\pi$
$992$	0	0
$993$	−0.577874	−0.0183383
$994$	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
$999$	5.45500	0.172589

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.a.f.1.3	✓	5	1.1	even	1	trivial
2412.2.a.j.1.3		5	3.2	odd	2
3216.2.a.ba.1.3		5	4.3	odd	2
9648.2.a.ca.1.3		5	12.11	even	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:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.78961 q^{5} +4.79729 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.78961 q^{5} +4.79729 q^{7} +1.00000 q^{9} +5.70891 q^{11} -3.70891 q^{13} +1.78961 q^{15} -5.05120 q^{17} -2.12261 q^{19} +4.79729 q^{21} -8.04190 q^{23} -1.79729 q^{25} +1.00000 q^{27} -6.25229 q^{29} +5.99838 q^{31} +5.70891 q^{33} +8.58529 q^{35} +5.45500 q^{37} -3.70891 q^{39} +8.42711 q^{41} -10.0854 q^{43} +1.78961 q^{45} -6.36661 q^{47} +16.0140 q^{49} -5.05120 q^{51} -7.49852 q^{53} +10.2167 q^{55} -2.12261 q^{57} -1.24623 q^{59} +12.3749 q^{61} +4.79729 q^{63} -6.63750 q^{65} +1.00000 q^{67} -8.04190 q^{69} -0.291093 q^{71} +3.32531 q^{73} -1.79729 q^{75} +27.3873 q^{77} -6.01536 q^{79} +1.00000 q^{81} -6.34007 q^{83} -9.03968 q^{85} -6.25229 q^{87} -8.63042 q^{89} -17.7927 q^{91} +5.99838 q^{93} -3.79864 q^{95} -17.1421 q^{97} +5.70891 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 3 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 + 3 * q^5 + 5 * q^7 + 5 * q^9	\( 5 q + 5 q^{3} + 3 q^{5} + 5 q^{7} + 5 q^{9} + 6 q^{11} + 4 q^{13} + 3 q^{15} + q^{17} + 13 q^{19} + 5 q^{21} + 10 q^{25} + 5 q^{27} + 3 q^{29} + 3 q^{31} + 6 q^{33} - 9 q^{35} + 12 q^{37} + 4 q^{39} + 11 q^{41} + 3 q^{43} + 3 q^{45} - 13 q^{47} + 24 q^{49} + q^{51} - 9 q^{53} + 14 q^{55} + 13 q^{57} - 12 q^{59} + 4 q^{61} + 5 q^{63} - 8 q^{65} + 5 q^{67} - 24 q^{71} + 12 q^{73} + 10 q^{75} - 4 q^{77} - 4 q^{79} + 5 q^{81} - 27 q^{83} - 4 q^{85} + 3 q^{87} - 5 q^{89} + 14 q^{91} + 3 q^{93} - 30 q^{95} + 8 q^{97} + 6 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + 3 * q^5 + 5 * q^7 + 5 * q^9 + 6 * q^11 + 4 * q^13 + 3 * q^15 + q^17 + 13 * q^19 + 5 * q^21 + 10 * q^25 + 5 * q^27 + 3 * q^29 + 3 * q^31 + 6 * q^33 - 9 * q^35 + 12 * q^37 + 4 * q^39 + 11 * q^41 + 3 * q^43 + 3 * q^45 - 13 * q^47 + 24 * q^49 + q^51 - 9 * q^53 + 14 * q^55 + 13 * q^57 - 12 * q^59 + 4 * q^61 + 5 * q^63 - 8 * q^65 + 5 * q^67 - 24 * q^71 + 12 * q^73 + 10 * q^75 - 4 * q^77 - 4 * q^79 + 5 * q^81 - 27 * q^83 - 4 * q^85 + 3 * q^87 - 5 * q^89 + 14 * q^91 + 3 * q^93 - 30 * q^95 + 8 * q^97 + 6 * q^99

Introduction

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