LMFDB - Embedded newform 6160.2.a.bh.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6160,2,Mod(1,6160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6160, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6160.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:	$49.1878476451$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1540)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	6160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.21432 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.90321 q^{9} +O(q^{10})$	$q+2.21432 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.90321 q^{9} -1.00000 q^{11} +2.21432 q^{13} -2.21432 q^{15} -2.96989 q^{17} -8.23506 q^{19} +2.21432 q^{21} +3.65878 q^{23} +1.00000 q^{25} -2.42864 q^{27} -0.193576 q^{29} -2.06668 q^{31} -2.21432 q^{33} -1.00000 q^{35} -9.19850 q^{37} +4.90321 q^{39} -8.36196 q^{41} +0.668149 q^{43} -1.90321 q^{45} -11.3985 q^{47} +1.00000 q^{49} -6.57628 q^{51} -8.90321 q^{53} +1.00000 q^{55} -18.2351 q^{57} +9.11753 q^{59} +4.55554 q^{61} +1.90321 q^{63} -2.21432 q^{65} +14.5763 q^{67} +8.10171 q^{69} -3.05086 q^{71} +7.82717 q^{73} +2.21432 q^{75} -1.00000 q^{77} -1.71900 q^{79} -11.0874 q^{81} +2.29529 q^{83} +2.96989 q^{85} -0.428639 q^{87} -16.8573 q^{89} +2.21432 q^{91} -4.57628 q^{93} +8.23506 q^{95} -10.3684 q^{97} -1.90321 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 3 q^{7} - q^{9}+O(q^{10}) $ 3 * q - 3 * q^5 + 3 * q^7 - q^9	$ 3 q - 3 q^{5} + 3 q^{7} - q^{9} - 3 q^{11} - 2 q^{17} + 2 q^{19} + 4 q^{23} + 3 q^{25} + 6 q^{27} - 14 q^{29} - 6 q^{31} - 3 q^{35} - 8 q^{37} + 8 q^{39} - 12 q^{41} + 22 q^{43} + q^{45} - 14 q^{47} + 3 q^{49} - 20 q^{53} + 3 q^{55} - 28 q^{57} + 14 q^{59} + 14 q^{61} - q^{63} + 24 q^{67} - 2 q^{69} + 4 q^{71} - 10 q^{73} - 3 q^{77} - 12 q^{79} - 13 q^{81} - 6 q^{83} + 2 q^{85} + 12 q^{87} - 24 q^{89} + 6 q^{93} - 2 q^{95} - 4 q^{97} + q^{99}+O(q^{100}) $ 3 * q - 3 * q^5 + 3 * q^7 - q^9 - 3 * q^11 - 2 * q^17 + 2 * q^19 + 4 * q^23 + 3 * q^25 + 6 * q^27 - 14 * q^29 - 6 * q^31 - 3 * q^35 - 8 * q^37 + 8 * q^39 - 12 * q^41 + 22 * q^43 + q^45 - 14 * q^47 + 3 * q^49 - 20 * q^53 + 3 * q^55 - 28 * q^57 + 14 * q^59 + 14 * q^61 - q^63 + 24 * q^67 - 2 * q^69 + 4 * q^71 - 10 * q^73 - 3 * q^77 - 12 * q^79 - 13 * q^81 - 6 * q^83 + 2 * q^85 + 12 * q^87 - 24 * q^89 + 6 * q^93 - 2 * q^95 - 4 * q^97 + 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
$3$	2.21432	1.27844	0.639219	−	0.769025i	$-0.279258\pi$
$3$	2.21432	1.27844	0.639219	+	0.769025i	$0.279258\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.90321	0.634404
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.21432	0.614142	0.307071	−	0.951687i	$-0.400651\pi$
$13$	2.21432	0.614142	0.307071	+	0.951687i	$0.400651\pi$
$14$	0	0
$15$	−2.21432	−0.571735
$16$	0	0
$17$	−2.96989	−0.720304	−0.360152	−	0.932894i	$-0.617275\pi$
$17$	−2.96989	−0.720304	−0.360152	+	0.932894i	$0.617275\pi$
$18$	0	0
$19$	−8.23506	−1.88925	−0.944627	−	0.328147i	$-0.893576\pi$
$19$	−8.23506	−1.88925	−0.944627	+	0.328147i	$0.893576\pi$
$20$	0	0
$21$	2.21432	0.483204
$22$	0	0
$23$	3.65878	0.762909	0.381454	−	0.924388i	$-0.375423\pi$
$23$	3.65878	0.762909	0.381454	+	0.924388i	$0.375423\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.42864	−0.467392
$28$	0	0
$29$	−0.193576	−0.0359462	−0.0179731	−	0.999838i	$-0.505721\pi$
$29$	−0.193576	−0.0359462	−0.0179731	+	0.999838i	$0.505721\pi$
$30$	0	0
$31$	−2.06668	−0.371186	−0.185593	−	0.982627i	$-0.559421\pi$
$31$	−2.06668	−0.371186	−0.185593	+	0.982627i	$0.559421\pi$
$32$	0	0
$33$	−2.21432	−0.385464
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−9.19850	−1.51222	−0.756112	−	0.654442i	$-0.772903\pi$
$37$	−9.19850	−1.51222	−0.756112	+	0.654442i	$0.772903\pi$
$38$	0	0
$39$	4.90321	0.785142
$40$	0	0
$41$	−8.36196	−1.30592	−0.652960	−	0.757393i	$-0.726473\pi$
$41$	−8.36196	−1.30592	−0.652960	+	0.757393i	$0.726473\pi$
$42$	0	0
$43$	0.668149	0.101892	0.0509459	−	0.998701i	$-0.483776\pi$
$43$	0.668149	0.101892	0.0509459	+	0.998701i	$0.483776\pi$
$44$	0	0
$45$	−1.90321	−0.283714
$46$	0	0
$47$	−11.3985	−1.66265	−0.831323	−	0.555789i	$-0.812416\pi$
$47$	−11.3985	−1.66265	−0.831323	+	0.555789i	$0.812416\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.57628	−0.920864
$52$	0	0
$53$	−8.90321	−1.22295	−0.611475	−	0.791264i	$-0.709424\pi$
$53$	−8.90321	−1.22295	−0.611475	+	0.791264i	$0.709424\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−18.2351	−2.41529
$58$	0	0
$59$	9.11753	1.18700	0.593501	−	0.804833i	$-0.297745\pi$
$59$	9.11753	1.18700	0.593501	+	0.804833i	$0.297745\pi$
$60$	0	0
$61$	4.55554	0.583277	0.291639	−	0.956529i	$-0.405800\pi$
$61$	4.55554	0.583277	0.291639	+	0.956529i	$0.405800\pi$
$62$	0	0
$63$	1.90321	0.239782
$64$	0	0
$65$	−2.21432	−0.274653
$66$	0	0
$67$	14.5763	1.78078	0.890388	−	0.455202i	$-0.150433\pi$
$67$	14.5763	1.78078	0.890388	+	0.455202i	$0.150433\pi$
$68$	0	0
$69$	8.10171	0.975331
$70$	0	0
$71$	−3.05086	−0.362070	−0.181035	−	0.983477i	$-0.557945\pi$
$71$	−3.05086	−0.362070	−0.181035	+	0.983477i	$0.557945\pi$
$72$	0	0
$73$	7.82717	0.916101	0.458050	−	0.888926i	$-0.348548\pi$
$73$	7.82717	0.916101	0.458050	+	0.888926i	$0.348548\pi$
$74$	0	0
$75$	2.21432	0.255688
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−1.71900	−0.193403	−0.0967015	−	0.995313i	$-0.530829\pi$
$79$	−1.71900	−0.193403	−0.0967015	+	0.995313i	$0.530829\pi$
$80$	0	0
$81$	−11.0874	−1.23194
$82$	0	0
$83$	2.29529	0.251940	0.125970	−	0.992034i	$-0.459796\pi$
$83$	2.29529	0.251940	0.125970	+	0.992034i	$0.459796\pi$
$84$	0	0
$85$	2.96989	0.322130
$86$	0	0
$87$	−0.428639	−0.0459550
$88$	0	0
$89$	−16.8573	−1.78687	−0.893434	−	0.449194i	$-0.851711\pi$
$89$	−16.8573	−1.78687	−0.893434	+	0.449194i	$0.851711\pi$
$90$	0	0
$91$	2.21432	0.232124
$92$	0	0
$93$	−4.57628	−0.474538
$94$	0	0
$95$	8.23506	0.844900
$96$	0	0
$97$	−10.3684	−1.05275	−0.526377	−	0.850252i	$-0.676450\pi$
$97$	−10.3684	−1.05275	−0.526377	+	0.850252i	$0.676450\pi$
$98$	0	0
$99$	−1.90321	−0.191280
$100$	0	0
$101$	−8.79060	−0.874698	−0.437349	−	0.899292i	$-0.644082\pi$
$101$	−8.79060	−0.874698	−0.437349	+	0.899292i	$0.644082\pi$
$102$	0	0
$103$	−3.88739	−0.383036	−0.191518	−	0.981489i	$-0.561341\pi$
$103$	−3.88739	−0.383036	−0.191518	+	0.981489i	$0.561341\pi$
$104$	0	0
$105$	−2.21432	−0.216095
$106$	0	0
$107$	18.6637	1.80429	0.902144	−	0.431435i	$-0.141992\pi$
$107$	18.6637	1.80429	0.902144	+	0.431435i	$0.141992\pi$
$108$	0	0
$109$	−14.0415	−1.34493	−0.672465	−	0.740129i	$-0.734765\pi$
$109$	−14.0415	−1.34493	−0.672465	+	0.740129i	$0.734765\pi$
$110$	0	0
$111$	−20.3684	−1.93328
$112$	0	0
$113$	11.2859	1.06169	0.530845	−	0.847469i	$-0.321875\pi$
$113$	11.2859	1.06169	0.530845	+	0.847469i	$0.321875\pi$
$114$	0	0
$115$	−3.65878	−0.341183
$116$	0	0
$117$	4.21432	0.389614
$118$	0	0
$119$	−2.96989	−0.272249
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−18.5161	−1.66954
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	10.2351	0.908215	0.454108	−	0.890947i	$-0.349958\pi$
$127$	10.2351	0.908215	0.454108	+	0.890947i	$0.349958\pi$
$128$	0	0
$129$	1.47949	0.130262
$130$	0	0
$131$	−14.1017	−1.23207	−0.616036	−	0.787718i	$-0.711262\pi$
$131$	−14.1017	−1.23207	−0.616036	+	0.787718i	$0.711262\pi$
$132$	0	0
$133$	−8.23506	−0.714071
$134$	0	0
$135$	2.42864	0.209024
$136$	0	0
$137$	0.903212	0.0771666	0.0385833	−	0.999255i	$-0.487716\pi$
$137$	0.903212	0.0771666	0.0385833	+	0.999255i	$0.487716\pi$
$138$	0	0
$139$	−2.91750	−0.247459	−0.123730	−	0.992316i	$-0.539486\pi$
$139$	−2.91750	−0.247459	−0.123730	+	0.992316i	$0.539486\pi$
$140$	0	0
$141$	−25.2400	−2.12559
$142$	0	0
$143$	−2.21432	−0.185171
$144$	0	0
$145$	0.193576	0.0160756
$146$	0	0
$147$	2.21432	0.182634
$148$	0	0
$149$	15.5526	1.27412	0.637060	−	0.770814i	$-0.280150\pi$
$149$	15.5526	1.27412	0.637060	+	0.770814i	$0.280150\pi$
$150$	0	0
$151$	8.51606	0.693027	0.346514	−	0.938045i	$-0.387365\pi$
$151$	8.51606	0.693027	0.346514	+	0.938045i	$0.387365\pi$
$152$	0	0
$153$	−5.65233	−0.456964
$154$	0	0
$155$	2.06668	0.165999
$156$	0	0
$157$	−23.3876	−1.86654	−0.933268	−	0.359181i	$-0.883056\pi$
$157$	−23.3876	−1.86654	−0.933268	+	0.359181i	$0.883056\pi$
$158$	0	0
$159$	−19.7146	−1.56347
$160$	0	0
$161$	3.65878	0.288352
$162$	0	0
$163$	11.9956	0.939564	0.469782	−	0.882782i	$-0.344332\pi$
$163$	11.9956	0.939564	0.469782	+	0.882782i	$0.344332\pi$
$164$	0	0
$165$	2.21432	0.172385
$166$	0	0
$167$	11.2859	0.873331	0.436665	−	0.899624i	$-0.356159\pi$
$167$	11.2859	0.873331	0.436665	+	0.899624i	$0.356159\pi$
$168$	0	0
$169$	−8.09679	−0.622830
$170$	0	0
$171$	−15.6731	−1.19855
$172$	0	0
$173$	−0.734825	−0.0558677	−0.0279339	−	0.999610i	$-0.508893\pi$
$173$	−0.734825	−0.0558677	−0.0279339	+	0.999610i	$0.508893\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	20.1891	1.51751
$178$	0	0
$179$	7.05086	0.527006	0.263503	−	0.964659i	$-0.415122\pi$
$179$	7.05086	0.527006	0.263503	+	0.964659i	$0.415122\pi$
$180$	0	0
$181$	19.4795	1.44790	0.723950	−	0.689853i	$-0.242325\pi$
$181$	19.4795	1.44790	0.723950	+	0.689853i	$0.242325\pi$
$182$	0	0
$183$	10.0874	0.745684
$184$	0	0
$185$	9.19850	0.676287
$186$	0	0
$187$	2.96989	0.217180
$188$	0	0
$189$	−2.42864	−0.176658
$190$	0	0
$191$	−15.7462	−1.13936	−0.569678	−	0.821868i	$-0.692932\pi$
$191$	−15.7462	−1.13936	−0.569678	+	0.821868i	$0.692932\pi$
$192$	0	0
$193$	23.8020	1.71330	0.856652	−	0.515895i	$-0.172540\pi$
$193$	23.8020	1.71330	0.856652	+	0.515895i	$0.172540\pi$
$194$	0	0
$195$	−4.90321	−0.351126
$196$	0	0
$197$	−20.2810	−1.44496	−0.722481	−	0.691391i	$-0.756998\pi$
$197$	−20.2810	−1.44496	−0.722481	+	0.691391i	$0.756998\pi$
$198$	0	0
$199$	−10.8222	−0.767169	−0.383584	−	0.923506i	$-0.625310\pi$
$199$	−10.8222	−0.767169	−0.383584	+	0.923506i	$0.625310\pi$
$200$	0	0
$201$	32.2766	2.27661
$202$	0	0
$203$	−0.193576	−0.0135864
$204$	0	0
$205$	8.36196	0.584025
$206$	0	0
$207$	6.96343	0.483992
$208$	0	0
$209$	8.23506	0.569631
$210$	0	0
$211$	−11.4795	−0.790281	−0.395141	−	0.918621i	$-0.629304\pi$
$211$	−11.4795	−0.790281	−0.395141	+	0.918621i	$0.629304\pi$
$212$	0	0
$213$	−6.75557	−0.462884
$214$	0	0
$215$	−0.668149	−0.0455674
$216$	0	0
$217$	−2.06668	−0.140295
$218$	0	0
$219$	17.3319	1.17118
$220$	0	0
$221$	−6.57628	−0.442369
$222$	0	0
$223$	−6.17283	−0.413363	−0.206682	−	0.978408i	$-0.566266\pi$
$223$	−6.17283	−0.413363	−0.206682	+	0.978408i	$0.566266\pi$
$224$	0	0
$225$	1.90321	0.126881
$226$	0	0
$227$	14.7556	0.979361	0.489681	−	0.871902i	$-0.337113\pi$
$227$	14.7556	0.979361	0.489681	+	0.871902i	$0.337113\pi$
$228$	0	0
$229$	−1.47949	−0.0977678	−0.0488839	−	0.998804i	$-0.515566\pi$
$229$	−1.47949	−0.0977678	−0.0488839	+	0.998804i	$0.515566\pi$
$230$	0	0
$231$	−2.21432	−0.145692
$232$	0	0
$233$	−11.4652	−0.751111	−0.375555	−	0.926800i	$-0.622548\pi$
$233$	−11.4652	−0.751111	−0.375555	+	0.926800i	$0.622548\pi$
$234$	0	0
$235$	11.3985	0.743558
$236$	0	0
$237$	−3.80642	−0.247254
$238$	0	0
$239$	−13.5812	−0.878495	−0.439248	−	0.898366i	$-0.644755\pi$
$239$	−13.5812	−0.878495	−0.439248	+	0.898366i	$0.644755\pi$
$240$	0	0
$241$	−22.1684	−1.42799	−0.713996	−	0.700150i	$-0.753116\pi$
$241$	−22.1684	−1.42799	−0.713996	+	0.700150i	$0.753116\pi$
$242$	0	0
$243$	−17.2652	−1.10756
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−18.2351	−1.16027
$248$	0	0
$249$	5.08250	0.322090
$250$	0	0
$251$	21.7210	1.37102	0.685509	−	0.728064i	$-0.259580\pi$
$251$	21.7210	1.37102	0.685509	+	0.728064i	$0.259580\pi$
$252$	0	0
$253$	−3.65878	−0.230026
$254$	0	0
$255$	6.57628	0.411823
$256$	0	0
$257$	−7.86665	−0.490708	−0.245354	−	0.969434i	$-0.578904\pi$
$257$	−7.86665	−0.490708	−0.245354	+	0.969434i	$0.578904\pi$
$258$	0	0
$259$	−9.19850	−0.571567
$260$	0	0
$261$	−0.368416	−0.0228044
$262$	0	0
$263$	−21.8479	−1.34720	−0.673600	−	0.739096i	$-0.735253\pi$
$263$	−21.8479	−1.34720	−0.673600	+	0.739096i	$0.735253\pi$
$264$	0	0
$265$	8.90321	0.546920
$266$	0	0
$267$	−37.3274	−2.28440
$268$	0	0
$269$	4.16193	0.253758	0.126879	−	0.991918i	$-0.459504\pi$
$269$	4.16193	0.253758	0.126879	+	0.991918i	$0.459504\pi$
$270$	0	0
$271$	−21.5526	−1.30923	−0.654614	−	0.755963i	$-0.727169\pi$
$271$	−21.5526	−1.30923	−0.654614	+	0.755963i	$0.727169\pi$
$272$	0	0
$273$	4.90321	0.296756
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−8.44293	−0.507286	−0.253643	−	0.967298i	$-0.581629\pi$
$277$	−8.44293	−0.507286	−0.253643	+	0.967298i	$0.581629\pi$
$278$	0	0
$279$	−3.93332	−0.235482
$280$	0	0
$281$	8.36842	0.499218	0.249609	−	0.968347i	$-0.419698\pi$
$281$	8.36842	0.499218	0.249609	+	0.968347i	$0.419698\pi$
$282$	0	0
$283$	16.7654	0.996600	0.498300	−	0.867005i	$-0.333958\pi$
$283$	16.7654	0.996600	0.498300	+	0.867005i	$0.333958\pi$
$284$	0	0
$285$	18.2351	1.08015
$286$	0	0
$287$	−8.36196	−0.493591
$288$	0	0
$289$	−8.17976	−0.481162
$290$	0	0
$291$	−22.9590	−1.34588
$292$	0	0
$293$	8.93825	0.522178	0.261089	−	0.965315i	$-0.415918\pi$
$293$	8.93825	0.522178	0.261089	+	0.965315i	$0.415918\pi$
$294$	0	0
$295$	−9.11753	−0.530843
$296$	0	0
$297$	2.42864	0.140924
$298$	0	0
$299$	8.10171	0.468534
$300$	0	0
$301$	0.668149	0.0385114
$302$	0	0
$303$	−19.4652	−1.11825
$304$	0	0
$305$	−4.55554	−0.260849
$306$	0	0
$307$	−31.4608	−1.79556	−0.897780	−	0.440444i	$-0.854821\pi$
$307$	−31.4608	−1.79556	−0.897780	+	0.440444i	$0.854821\pi$
$308$	0	0
$309$	−8.60793	−0.489688
$310$	0	0
$311$	9.91459	0.562205	0.281102	−	0.959678i	$-0.409300\pi$
$311$	9.91459	0.562205	0.281102	+	0.959678i	$0.409300\pi$
$312$	0	0
$313$	19.4193	1.09764	0.548821	−	0.835940i	$-0.315077\pi$
$313$	19.4193	1.09764	0.548821	+	0.835940i	$0.315077\pi$
$314$	0	0
$315$	−1.90321	−0.107234
$316$	0	0
$317$	−19.5111	−1.09585	−0.547927	−	0.836526i	$-0.684583\pi$
$317$	−19.5111	−1.09585	−0.547927	+	0.836526i	$0.684583\pi$
$318$	0	0
$319$	0.193576	0.0108382
$320$	0	0
$321$	41.3274	2.30667
$322$	0	0
$323$	24.4572	1.36084
$324$	0	0
$325$	2.21432	0.122828
$326$	0	0
$327$	−31.0923	−1.71941
$328$	0	0
$329$	−11.3985	−0.628421
$330$	0	0
$331$	6.53035	0.358940	0.179470	−	0.983763i	$-0.442562\pi$
$331$	6.53035	0.358940	0.179470	+	0.983763i	$0.442562\pi$
$332$	0	0
$333$	−17.5067	−0.959361
$334$	0	0
$335$	−14.5763	−0.796387
$336$	0	0
$337$	15.8020	0.860789	0.430394	−	0.902641i	$-0.358375\pi$
$337$	15.8020	0.860789	0.430394	+	0.902641i	$0.358375\pi$
$338$	0	0
$339$	24.9906	1.35730
$340$	0	0
$341$	2.06668	0.111917
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−8.10171	−0.436181
$346$	0	0
$347$	15.0366	0.807205	0.403603	−	0.914934i	$-0.367758\pi$
$347$	15.0366	0.807205	0.403603	+	0.914934i	$0.367758\pi$
$348$	0	0
$349$	1.38424	0.0740966	0.0370483	−	0.999313i	$-0.488204\pi$
$349$	1.38424	0.0740966	0.0370483	+	0.999313i	$0.488204\pi$
$350$	0	0
$351$	−5.37778	−0.287045
$352$	0	0
$353$	−23.4193	−1.24648	−0.623241	−	0.782030i	$-0.714184\pi$
$353$	−23.4193	−1.24648	−0.623241	+	0.782030i	$0.714184\pi$
$354$	0	0
$355$	3.05086	0.161923
$356$	0	0
$357$	−6.57628	−0.348054
$358$	0	0
$359$	−19.9956	−1.05532	−0.527662	−	0.849454i	$-0.676931\pi$
$359$	−19.9956	−1.05532	−0.527662	+	0.849454i	$0.676931\pi$
$360$	0	0
$361$	48.8163	2.56928
$362$	0	0
$363$	2.21432	0.116222
$364$	0	0
$365$	−7.82717	−0.409693
$366$	0	0
$367$	7.53188	0.393161	0.196580	−	0.980488i	$-0.437016\pi$
$367$	7.53188	0.393161	0.196580	+	0.980488i	$0.437016\pi$
$368$	0	0
$369$	−15.9146	−0.828480
$370$	0	0
$371$	−8.90321	−0.462232
$372$	0	0
$373$	14.3827	0.744708	0.372354	−	0.928091i	$-0.378551\pi$
$373$	14.3827	0.744708	0.372354	+	0.928091i	$0.378551\pi$
$374$	0	0
$375$	−2.21432	−0.114347
$376$	0	0
$377$	−0.428639	−0.0220761
$378$	0	0
$379$	10.3970	0.534058	0.267029	−	0.963689i	$-0.413958\pi$
$379$	10.3970	0.534058	0.267029	+	0.963689i	$0.413958\pi$
$380$	0	0
$381$	22.6637	1.16110
$382$	0	0
$383$	−5.06175	−0.258644	−0.129322	−	0.991603i	$-0.541280\pi$
$383$	−5.06175	−0.258644	−0.129322	+	0.991603i	$0.541280\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	1.27163	0.0646405
$388$	0	0
$389$	−25.6543	−1.30073	−0.650363	−	0.759623i	$-0.725383\pi$
$389$	−25.6543	−1.30073	−0.650363	+	0.759623i	$0.725383\pi$
$390$	0	0
$391$	−10.8662	−0.549526
$392$	0	0
$393$	−31.2257	−1.57513
$394$	0	0
$395$	1.71900	0.0864925
$396$	0	0
$397$	11.8666	0.595570	0.297785	−	0.954633i	$-0.403752\pi$
$397$	11.8666	0.595570	0.297785	+	0.954633i	$0.403752\pi$
$398$	0	0
$399$	−18.2351	−0.912895
$400$	0	0
$401$	−14.0830	−0.703270	−0.351635	−	0.936137i	$-0.614374\pi$
$401$	−14.0830	−0.703270	−0.351635	+	0.936137i	$0.614374\pi$
$402$	0	0
$403$	−4.57628	−0.227961
$404$	0	0
$405$	11.0874	0.550938
$406$	0	0
$407$	9.19850	0.455953
$408$	0	0
$409$	22.0163	1.08864	0.544318	−	0.838879i	$-0.316789\pi$
$409$	22.0163	1.08864	0.544318	+	0.838879i	$0.316789\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	0	0
$413$	9.11753	0.448644
$414$	0	0
$415$	−2.29529	−0.112671
$416$	0	0
$417$	−6.46028	−0.316361
$418$	0	0
$419$	7.84146	0.383080	0.191540	−	0.981485i	$-0.438652\pi$
$419$	7.84146	0.383080	0.191540	+	0.981485i	$0.438652\pi$
$420$	0	0
$421$	−23.6271	−1.15152	−0.575758	−	0.817620i	$-0.695293\pi$
$421$	−23.6271	−1.15152	−0.575758	+	0.817620i	$0.695293\pi$
$422$	0	0
$423$	−21.6938	−1.05479
$424$	0	0
$425$	−2.96989	−0.144061
$426$	0	0
$427$	4.55554	0.220458
$428$	0	0
$429$	−4.90321	−0.236729
$430$	0	0
$431$	0.649413	0.0312811	0.0156406	−	0.999878i	$-0.495021\pi$
$431$	0.649413	0.0312811	0.0156406	+	0.999878i	$0.495021\pi$
$432$	0	0
$433$	20.1432	0.968020	0.484010	−	0.875062i	$-0.339180\pi$
$433$	20.1432	0.968020	0.484010	+	0.875062i	$0.339180\pi$
$434$	0	0
$435$	0.428639	0.0205517
$436$	0	0
$437$	−30.1303	−1.44133
$438$	0	0
$439$	28.3368	1.35244	0.676220	−	0.736700i	$-0.263617\pi$
$439$	28.3368	1.35244	0.676220	+	0.736700i	$0.263617\pi$
$440$	0	0
$441$	1.90321	0.0906291
$442$	0	0
$443$	−20.9862	−0.997084	−0.498542	−	0.866866i	$-0.666131\pi$
$443$	−20.9862	−0.997084	−0.498542	+	0.866866i	$0.666131\pi$
$444$	0	0
$445$	16.8573	0.799112
$446$	0	0
$447$	34.4385	1.62888
$448$	0	0
$449$	−26.7699	−1.26335	−0.631674	−	0.775234i	$-0.717632\pi$
$449$	−26.7699	−1.26335	−0.631674	+	0.775234i	$0.717632\pi$
$450$	0	0
$451$	8.36196	0.393749
$452$	0	0
$453$	18.8573	0.885992
$454$	0	0
$455$	−2.21432	−0.103809
$456$	0	0
$457$	13.0464	0.610285	0.305143	−	0.952307i	$-0.401296\pi$
$457$	13.0464	0.610285	0.305143	+	0.952307i	$0.401296\pi$
$458$	0	0
$459$	7.21279	0.336664
$460$	0	0
$461$	−40.2415	−1.87423	−0.937117	−	0.349015i	$-0.886516\pi$
$461$	−40.2415	−1.87423	−0.937117	+	0.349015i	$0.886516\pi$
$462$	0	0
$463$	9.80198	0.455537	0.227768	−	0.973715i	$-0.426857\pi$
$463$	9.80198	0.455537	0.227768	+	0.973715i	$0.426857\pi$
$464$	0	0
$465$	4.57628	0.212220
$466$	0	0
$467$	−37.5605	−1.73809	−0.869045	−	0.494732i	$-0.835266\pi$
$467$	−37.5605	−1.73809	−0.869045	+	0.494732i	$0.835266\pi$
$468$	0	0
$469$	14.5763	0.673070
$470$	0	0
$471$	−51.7877	−2.38625
$472$	0	0
$473$	−0.668149	−0.0307215
$474$	0	0
$475$	−8.23506	−0.377851
$476$	0	0
$477$	−16.9447	−0.775844
$478$	0	0
$479$	18.6321	0.851321	0.425660	−	0.904883i	$-0.360042\pi$
$479$	18.6321	0.851321	0.425660	+	0.904883i	$0.360042\pi$
$480$	0	0
$481$	−20.3684	−0.928720
$482$	0	0
$483$	8.10171	0.368641
$484$	0	0
$485$	10.3684	0.470806
$486$	0	0
$487$	28.7828	1.30427	0.652136	−	0.758102i	$-0.273873\pi$
$487$	28.7828	1.30427	0.652136	+	0.758102i	$0.273873\pi$
$488$	0	0
$489$	26.5620	1.20117
$490$	0	0
$491$	0.474572	0.0214172	0.0107086	−	0.999943i	$-0.496591\pi$
$491$	0.474572	0.0214172	0.0107086	+	0.999943i	$0.496591\pi$
$492$	0	0
$493$	0.574900	0.0258922
$494$	0	0
$495$	1.90321	0.0855430
$496$	0	0
$497$	−3.05086	−0.136850
$498$	0	0
$499$	6.36842	0.285089	0.142545	−	0.989788i	$-0.454472\pi$
$499$	6.36842	0.285089	0.142545	+	0.989788i	$0.454472\pi$
$500$	0	0
$501$	24.9906	1.11650
$502$	0	0
$503$	27.2355	1.21437	0.607186	−	0.794559i	$-0.292298\pi$
$503$	27.2355	1.21437	0.607186	+	0.794559i	$0.292298\pi$
$504$	0	0
$505$	8.79060	0.391177
$506$	0	0
$507$	−17.9289	−0.796249
$508$	0	0
$509$	20.5303	0.909992	0.454996	−	0.890494i	$-0.349641\pi$
$509$	20.5303	0.909992	0.454996	+	0.890494i	$0.349641\pi$
$510$	0	0
$511$	7.82717	0.346254
$512$	0	0
$513$	20.0000	0.883022
$514$	0	0
$515$	3.88739	0.171299
$516$	0	0
$517$	11.3985	0.501307
$518$	0	0
$519$	−1.62714	−0.0714234
$520$	0	0
$521$	−22.2351	−0.974136	−0.487068	−	0.873364i	$-0.661934\pi$
$521$	−22.2351	−0.974136	−0.487068	+	0.873364i	$0.661934\pi$
$522$	0	0
$523$	25.7462	1.12580	0.562901	−	0.826524i	$-0.309685\pi$
$523$	25.7462	1.12580	0.562901	+	0.826524i	$0.309685\pi$
$524$	0	0
$525$	2.21432	0.0966408
$526$	0	0
$527$	6.13780	0.267367
$528$	0	0
$529$	−9.61332	−0.417971
$530$	0	0
$531$	17.3526	0.753038
$532$	0	0
$533$	−18.5161	−0.802020
$534$	0	0
$535$	−18.6637	−0.806902
$536$	0	0
$537$	15.6128	0.673744
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−19.0192	−0.817700	−0.408850	−	0.912602i	$-0.634070\pi$
$541$	−19.0192	−0.817700	−0.408850	+	0.912602i	$0.634070\pi$
$542$	0	0
$543$	43.1338	1.85105
$544$	0	0
$545$	14.0415	0.601471
$546$	0	0
$547$	24.6953	1.05590	0.527948	−	0.849276i	$-0.322961\pi$
$547$	24.6953	1.05590	0.527948	+	0.849276i	$0.322961\pi$
$548$	0	0
$549$	8.67016	0.370033
$550$	0	0
$551$	1.59411	0.0679114
$552$	0	0
$553$	−1.71900	−0.0730995
$554$	0	0
$555$	20.3684	0.864591
$556$	0	0
$557$	31.4019	1.33054	0.665271	−	0.746602i	$-0.268316\pi$
$557$	31.4019	1.33054	0.665271	+	0.746602i	$0.268316\pi$
$558$	0	0
$559$	1.47949	0.0625760
$560$	0	0
$561$	6.57628	0.277651
$562$	0	0
$563$	13.8479	0.583620	0.291810	−	0.956476i	$-0.405743\pi$
$563$	13.8479	0.583620	0.291810	+	0.956476i	$0.405743\pi$
$564$	0	0
$565$	−11.2859	−0.474802
$566$	0	0
$567$	−11.0874	−0.465628
$568$	0	0
$569$	0.0602231	0.00252468	0.00126234	−	0.999999i	$-0.499598\pi$
$569$	0.0602231	0.00252468	0.00126234	+	0.999999i	$0.499598\pi$
$570$	0	0
$571$	27.2128	1.13882	0.569410	−	0.822054i	$-0.307172\pi$
$571$	27.2128	1.13882	0.569410	+	0.822054i	$0.307172\pi$
$572$	0	0
$573$	−34.8671	−1.45659
$574$	0	0
$575$	3.65878	0.152582
$576$	0	0
$577$	18.3970	0.765877	0.382938	−	0.923774i	$-0.374912\pi$
$577$	18.3970	0.765877	0.382938	+	0.923774i	$0.374912\pi$
$578$	0	0
$579$	52.7052	2.19035
$580$	0	0
$581$	2.29529	0.0952245
$582$	0	0
$583$	8.90321	0.368733
$584$	0	0
$585$	−4.21432	−0.174241
$586$	0	0
$587$	−7.75404	−0.320043	−0.160022	−	0.987113i	$-0.551156\pi$
$587$	−7.75404	−0.320043	−0.160022	+	0.987113i	$0.551156\pi$
$588$	0	0
$589$	17.0192	0.701264
$590$	0	0
$591$	−44.9086	−1.84729
$592$	0	0
$593$	−26.2558	−1.07820	−0.539098	−	0.842243i	$-0.681235\pi$
$593$	−26.2558	−1.07820	−0.539098	+	0.842243i	$0.681235\pi$
$594$	0	0
$595$	2.96989	0.121754
$596$	0	0
$597$	−23.9639	−0.980778
$598$	0	0
$599$	37.0607	1.51426	0.757130	−	0.653265i	$-0.226601\pi$
$599$	37.0607	1.51426	0.757130	+	0.653265i	$0.226601\pi$
$600$	0	0
$601$	−22.2514	−0.907652	−0.453826	−	0.891090i	$-0.649941\pi$
$601$	−22.2514	−0.907652	−0.453826	+	0.891090i	$0.649941\pi$
$602$	0	0
$603$	27.7418	1.12973
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	29.8578	1.21189	0.605944	−	0.795507i	$-0.292795\pi$
$607$	29.8578	1.21189	0.605944	+	0.795507i	$0.292795\pi$
$608$	0	0
$609$	−0.428639	−0.0173693
$610$	0	0
$611$	−25.2400	−1.02110
$612$	0	0
$613$	33.0879	1.33641	0.668204	−	0.743978i	$-0.267063\pi$
$613$	33.0879	1.33641	0.668204	+	0.743978i	$0.267063\pi$
$614$	0	0
$615$	18.5161	0.746640
$616$	0	0
$617$	8.78277	0.353581	0.176790	−	0.984249i	$-0.443429\pi$
$617$	8.78277	0.353581	0.176790	+	0.984249i	$0.443429\pi$
$618$	0	0
$619$	−30.1180	−1.21054	−0.605272	−	0.796018i	$-0.706936\pi$
$619$	−30.1180	−1.21054	−0.605272	+	0.796018i	$0.706936\pi$
$620$	0	0
$621$	−8.88586	−0.356577
$622$	0	0
$623$	−16.8573	−0.675373
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	18.2351	0.728238
$628$	0	0
$629$	27.3185	1.08926
$630$	0	0
$631$	−21.3274	−0.849031	−0.424515	−	0.905421i	$-0.639555\pi$
$631$	−21.3274	−0.849031	−0.424515	+	0.905421i	$0.639555\pi$
$632$	0	0
$633$	−25.4193	−1.01033
$634$	0	0
$635$	−10.2351	−0.406166
$636$	0	0
$637$	2.21432	0.0877345
$638$	0	0
$639$	−5.80642	−0.229699
$640$	0	0
$641$	39.3862	1.55566	0.777832	−	0.628473i	$-0.216320\pi$
$641$	39.3862	1.55566	0.777832	+	0.628473i	$0.216320\pi$
$642$	0	0
$643$	−19.2652	−0.759744	−0.379872	−	0.925039i	$-0.624032\pi$
$643$	−19.2652	−0.759744	−0.379872	+	0.925039i	$0.624032\pi$
$644$	0	0
$645$	−1.47949	−0.0582550
$646$	0	0
$647$	−35.8183	−1.40816	−0.704081	−	0.710120i	$-0.748641\pi$
$647$	−35.8183	−1.40816	−0.704081	+	0.710120i	$0.748641\pi$
$648$	0	0
$649$	−9.11753	−0.357894
$650$	0	0
$651$	−4.57628	−0.179359
$652$	0	0
$653$	−21.5714	−0.844153	−0.422076	−	0.906560i	$-0.638699\pi$
$653$	−21.5714	−0.844153	−0.422076	+	0.906560i	$0.638699\pi$
$654$	0	0
$655$	14.1017	0.550999
$656$	0	0
$657$	14.8968	0.581178
$658$	0	0
$659$	37.8524	1.47452	0.737259	−	0.675610i	$-0.236120\pi$
$659$	37.8524	1.47452	0.737259	+	0.675610i	$0.236120\pi$
$660$	0	0
$661$	13.2730	0.516260	0.258130	−	0.966110i	$-0.416894\pi$
$661$	13.2730	0.516260	0.258130	+	0.966110i	$0.416894\pi$
$662$	0	0
$663$	−14.5620	−0.565541
$664$	0	0
$665$	8.23506	0.319342
$666$	0	0
$667$	−0.708253	−0.0274237
$668$	0	0
$669$	−13.6686	−0.528460
$670$	0	0
$671$	−4.55554	−0.175865
$672$	0	0
$673$	−41.7288	−1.60853	−0.804264	−	0.594272i	$-0.797440\pi$
$673$	−41.7288	−1.60853	−0.804264	+	0.594272i	$0.797440\pi$
$674$	0	0
$675$	−2.42864	−0.0934784
$676$	0	0
$677$	7.43017	0.285565	0.142782	−	0.989754i	$-0.454395\pi$
$677$	7.43017	0.285565	0.142782	+	0.989754i	$0.454395\pi$
$678$	0	0
$679$	−10.3684	−0.397903
$680$	0	0
$681$	32.6735	1.25205
$682$	0	0
$683$	−22.6593	−0.867032	−0.433516	−	0.901146i	$-0.642727\pi$
$683$	−22.6593	−0.867032	−0.433516	+	0.901146i	$0.642727\pi$
$684$	0	0
$685$	−0.903212	−0.0345100
$686$	0	0
$687$	−3.27607	−0.124990
$688$	0	0
$689$	−19.7146	−0.751065
$690$	0	0
$691$	−0.0666765	−0.00253650	−0.00126825	−	0.999999i	$-0.500404\pi$
$691$	−0.0666765	−0.00253650	−0.00126825	+	0.999999i	$0.500404\pi$
$692$	0	0
$693$	−1.90321	−0.0722970
$694$	0	0
$695$	2.91750	0.110667
$696$	0	0
$697$	24.8341	0.940659
$698$	0	0
$699$	−25.3876	−0.960248
$700$	0	0
$701$	−19.7649	−0.746511	−0.373256	−	0.927729i	$-0.621759\pi$
$701$	−19.7649	−0.746511	−0.373256	+	0.927729i	$0.621759\pi$
$702$	0	0
$703$	75.7502	2.85697
$704$	0	0
$705$	25.2400	0.950593
$706$	0	0
$707$	−8.79060	−0.330605
$708$	0	0
$709$	9.52987	0.357902	0.178951	−	0.983858i	$-0.442730\pi$
$709$	9.52987	0.357902	0.178951	+	0.983858i	$0.442730\pi$
$710$	0	0
$711$	−3.27163	−0.122696
$712$	0	0
$713$	−7.56152	−0.283181
$714$	0	0
$715$	2.21432	0.0828109
$716$	0	0
$717$	−30.0731	−1.12310
$718$	0	0
$719$	32.3620	1.20690	0.603449	−	0.797402i	$-0.293793\pi$
$719$	32.3620	1.20690	0.603449	+	0.797402i	$0.293793\pi$
$720$	0	0
$721$	−3.88739	−0.144774
$722$	0	0
$723$	−49.0879	−1.82560
$724$	0	0
$725$	−0.193576	−0.00718924
$726$	0	0
$727$	−53.4499	−1.98235	−0.991173	−	0.132576i	$-0.957675\pi$
$727$	−53.4499	−1.98235	−0.991173	+	0.132576i	$0.957675\pi$
$728$	0	0
$729$	−4.96836	−0.184013
$730$	0	0
$731$	−1.98433	−0.0733930
$732$	0	0
$733$	5.88739	0.217456	0.108728	−	0.994072i	$-0.465322\pi$
$733$	5.88739	0.217456	0.108728	+	0.994072i	$0.465322\pi$
$734$	0	0
$735$	−2.21432	−0.0816764
$736$	0	0
$737$	−14.5763	−0.536924
$738$	0	0
$739$	−29.4479	−1.08326	−0.541628	−	0.840618i	$-0.682192\pi$
$739$	−29.4479	−1.08326	−0.541628	+	0.840618i	$0.682192\pi$
$740$	0	0
$741$	−40.3783	−1.48333
$742$	0	0
$743$	−14.6637	−0.537959	−0.268980	−	0.963146i	$-0.586686\pi$
$743$	−14.6637	−0.537959	−0.268980	+	0.963146i	$0.586686\pi$
$744$	0	0
$745$	−15.5526	−0.569804
$746$	0	0
$747$	4.36842	0.159832
$748$	0	0
$749$	18.6637	0.681957
$750$	0	0
$751$	−30.3466	−1.10736	−0.553682	−	0.832728i	$-0.686778\pi$
$751$	−30.3466	−1.10736	−0.553682	+	0.832728i	$0.686778\pi$
$752$	0	0
$753$	48.0973	1.75276
$754$	0	0
$755$	−8.51606	−0.309931
$756$	0	0
$757$	0.825636	0.0300083	0.0150041	−	0.999887i	$-0.495224\pi$
$757$	0.825636	0.0300083	0.0150041	+	0.999887i	$0.495224\pi$
$758$	0	0
$759$	−8.10171	−0.294073
$760$	0	0
$761$	−34.7116	−1.25830	−0.629148	−	0.777285i	$-0.716596\pi$
$761$	−34.7116	−1.25830	−0.629148	+	0.777285i	$0.716596\pi$
$762$	0	0
$763$	−14.0415	−0.508336
$764$	0	0
$765$	5.65233	0.204360
$766$	0	0
$767$	20.1891	0.728987
$768$	0	0
$769$	13.6983	0.493972	0.246986	−	0.969019i	$-0.420560\pi$
$769$	13.6983	0.493972	0.246986	+	0.969019i	$0.420560\pi$
$770$	0	0
$771$	−17.4193	−0.627340
$772$	0	0
$773$	11.6128	0.417685	0.208843	−	0.977949i	$-0.433030\pi$
$773$	11.6128	0.417685	0.208843	+	0.977949i	$0.433030\pi$
$774$	0	0
$775$	−2.06668	−0.0742372
$776$	0	0
$777$	−20.3684	−0.730713
$778$	0	0
$779$	68.8613	2.46721
$780$	0	0
$781$	3.05086	0.109168
$782$	0	0
$783$	0.470127	0.0168010
$784$	0	0
$785$	23.3876	0.834740
$786$	0	0
$787$	25.3145	0.902364	0.451182	−	0.892432i	$-0.351002\pi$
$787$	25.3145	0.902364	0.451182	+	0.892432i	$0.351002\pi$
$788$	0	0
$789$	−48.3783	−1.72231
$790$	0	0
$791$	11.2859	0.401281
$792$	0	0
$793$	10.0874	0.358215
$794$	0	0
$795$	19.7146	0.699203
$796$	0	0
$797$	26.4099	0.935487	0.467743	−	0.883864i	$-0.345067\pi$
$797$	26.4099	0.935487	0.467743	+	0.883864i	$0.345067\pi$
$798$	0	0
$799$	33.8524	1.19761
$800$	0	0
$801$	−32.0830	−1.13360
$802$	0	0
$803$	−7.82717	−0.276215
$804$	0	0
$805$	−3.65878	−0.128955
$806$	0	0
$807$	9.21585	0.324413
$808$	0	0
$809$	−17.5843	−0.618230	−0.309115	−	0.951025i	$-0.600033\pi$
$809$	−17.5843	−0.618230	−0.309115	+	0.951025i	$0.600033\pi$
$810$	0	0
$811$	40.6923	1.42890	0.714450	−	0.699687i	$-0.246677\pi$
$811$	40.6923	1.42890	0.714450	+	0.699687i	$0.246677\pi$
$812$	0	0
$813$	−47.7244	−1.67377
$814$	0	0
$815$	−11.9956	−0.420186
$816$	0	0
$817$	−5.50225	−0.192499
$818$	0	0
$819$	4.21432	0.147260
$820$	0	0
$821$	−36.5847	−1.27682	−0.638408	−	0.769698i	$-0.720407\pi$
$821$	−36.5847	−1.27682	−0.638408	+	0.769698i	$0.720407\pi$
$822$	0	0
$823$	−4.50315	−0.156970	−0.0784850	−	0.996915i	$-0.525008\pi$
$823$	−4.50315	−0.156970	−0.0784850	+	0.996915i	$0.525008\pi$
$824$	0	0
$825$	−2.21432	−0.0770927
$826$	0	0
$827$	20.3096	0.706233	0.353117	−	0.935579i	$-0.385122\pi$
$827$	20.3096	0.706233	0.353117	+	0.935579i	$0.385122\pi$
$828$	0	0
$829$	41.2257	1.43183	0.715914	−	0.698189i	$-0.246010\pi$
$829$	41.2257	1.43183	0.715914	+	0.698189i	$0.246010\pi$
$830$	0	0
$831$	−18.6953	−0.648534
$832$	0	0
$833$	−2.96989	−0.102901
$834$	0	0
$835$	−11.2859	−0.390565
$836$	0	0
$837$	5.01921	0.173489
$838$	0	0
$839$	26.4637	0.913627	0.456814	−	0.889562i	$-0.348991\pi$
$839$	26.4637	0.913627	0.456814	+	0.889562i	$0.348991\pi$
$840$	0	0
$841$	−28.9625	−0.998708
$842$	0	0
$843$	18.5303	0.638219
$844$	0	0
$845$	8.09679	0.278538
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	37.1240	1.27409
$850$	0	0
$851$	−33.6553	−1.15369
$852$	0	0
$853$	−0.928401	−0.0317879	−0.0158939	−	0.999874i	$-0.505059\pi$
$853$	−0.928401	−0.0317879	−0.0158939	+	0.999874i	$0.505059\pi$
$854$	0	0
$855$	15.6731	0.536008
$856$	0	0
$857$	2.49087	0.0850865	0.0425433	−	0.999095i	$-0.486454\pi$
$857$	2.49087	0.0850865	0.0425433	+	0.999095i	$0.486454\pi$
$858$	0	0
$859$	22.5239	0.768505	0.384253	−	0.923228i	$-0.374459\pi$
$859$	22.5239	0.768505	0.384253	+	0.923228i	$0.374459\pi$
$860$	0	0
$861$	−18.5161	−0.631026
$862$	0	0
$863$	−8.04593	−0.273887	−0.136943	−	0.990579i	$-0.543728\pi$
$863$	−8.04593	−0.273887	−0.136943	+	0.990579i	$0.543728\pi$
$864$	0	0
$865$	0.734825	0.0249848
$866$	0	0
$867$	−18.1126	−0.615136
$868$	0	0
$869$	1.71900	0.0583132
$870$	0	0
$871$	32.2766	1.09365
$872$	0	0
$873$	−19.7333	−0.667871
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	4.73822	0.159998	0.0799991	−	0.996795i	$-0.474508\pi$
$877$	4.73822	0.159998	0.0799991	+	0.996795i	$0.474508\pi$
$878$	0	0
$879$	19.7921	0.667572
$880$	0	0
$881$	17.8479	0.601311	0.300656	−	0.953733i	$-0.402794\pi$
$881$	17.8479	0.601311	0.300656	+	0.953733i	$0.402794\pi$
$882$	0	0
$883$	−25.3230	−0.852185	−0.426093	−	0.904679i	$-0.640110\pi$
$883$	−25.3230	−0.852185	−0.426093	+	0.904679i	$0.640110\pi$
$884$	0	0
$885$	−20.1891	−0.678650
$886$	0	0
$887$	−7.30465	−0.245266	−0.122633	−	0.992452i	$-0.539134\pi$
$887$	−7.30465	−0.245266	−0.122633	+	0.992452i	$0.539134\pi$
$888$	0	0
$889$	10.2351	0.343273
$890$	0	0
$891$	11.0874	0.371443
$892$	0	0
$893$	93.8676	3.14116
$894$	0	0
$895$	−7.05086	−0.235684
$896$	0	0
$897$	17.9398	0.598992
$898$	0	0
$899$	0.400059	0.0133427
$900$	0	0
$901$	26.4415	0.880896
$902$	0	0
$903$	1.47949	0.0492345
$904$	0	0
$905$	−19.4795	−0.647520
$906$	0	0
$907$	−15.3002	−0.508035	−0.254018	−	0.967200i	$-0.581752\pi$
$907$	−15.3002	−0.508035	−0.254018	+	0.967200i	$0.581752\pi$
$908$	0	0
$909$	−16.7304	−0.554912
$910$	0	0
$911$	−15.0509	−0.498657	−0.249329	−	0.968419i	$-0.580210\pi$
$911$	−15.0509	−0.498657	−0.249329	+	0.968419i	$0.580210\pi$
$912$	0	0
$913$	−2.29529	−0.0759629
$914$	0	0
$915$	−10.0874	−0.333480
$916$	0	0
$917$	−14.1017	−0.465679
$918$	0	0
$919$	−1.37778	−0.0454489	−0.0227245	−	0.999742i	$-0.507234\pi$
$919$	−1.37778	−0.0454489	−0.0227245	+	0.999742i	$0.507234\pi$
$920$	0	0
$921$	−69.6642	−2.29551
$922$	0	0
$923$	−6.75557	−0.222362
$924$	0	0
$925$	−9.19850	−0.302445
$926$	0	0
$927$	−7.39853	−0.243000
$928$	0	0
$929$	−41.2770	−1.35426	−0.677128	−	0.735866i	$-0.736776\pi$
$929$	−41.2770	−1.35426	−0.677128	+	0.735866i	$0.736776\pi$
$930$	0	0
$931$	−8.23506	−0.269893
$932$	0	0
$933$	21.9541	0.718744
$934$	0	0
$935$	−2.96989	−0.0971257
$936$	0	0
$937$	−7.68397	−0.251024	−0.125512	−	0.992092i	$-0.540057\pi$
$937$	−7.68397	−0.251024	−0.125512	+	0.992092i	$0.540057\pi$
$938$	0	0
$939$	43.0005	1.40327
$940$	0	0
$941$	49.0257	1.59819	0.799096	−	0.601204i	$-0.205312\pi$
$941$	49.0257	1.59819	0.799096	+	0.601204i	$0.205312\pi$
$942$	0	0
$943$	−30.5946	−0.996297
$944$	0	0
$945$	2.42864	0.0790036
$946$	0	0
$947$	9.37334	0.304593	0.152296	−	0.988335i	$-0.451333\pi$
$947$	9.37334	0.304593	0.152296	+	0.988335i	$0.451333\pi$
$948$	0	0
$949$	17.3319	0.562616
$950$	0	0
$951$	−43.2039	−1.40098
$952$	0	0
$953$	−10.0745	−0.326345	−0.163173	−	0.986598i	$-0.552173\pi$
$953$	−10.0745	−0.326345	−0.163173	+	0.986598i	$0.552173\pi$
$954$	0	0
$955$	15.7462	0.509535
$956$	0	0
$957$	0.428639	0.0138559
$958$	0	0
$959$	0.903212	0.0291662
$960$	0	0
$961$	−26.7288	−0.862221
$962$	0	0
$963$	35.5210	1.14465
$964$	0	0
$965$	−23.8020	−0.766213
$966$	0	0
$967$	−46.8243	−1.50577	−0.752883	−	0.658154i	$-0.771338\pi$
$967$	−46.8243	−1.50577	−0.752883	+	0.658154i	$0.771338\pi$
$968$	0	0
$969$	54.1561	1.73974
$970$	0	0
$971$	19.8544	0.637157	0.318578	−	0.947897i	$-0.396795\pi$
$971$	19.8544	0.637157	0.318578	+	0.947897i	$0.396795\pi$
$972$	0	0
$973$	−2.91750	−0.0935308
$974$	0	0
$975$	4.90321	0.157028
$976$	0	0
$977$	21.2672	0.680397	0.340199	−	0.940354i	$-0.389506\pi$
$977$	21.2672	0.680397	0.340199	+	0.940354i	$0.389506\pi$
$978$	0	0
$979$	16.8573	0.538761
$980$	0	0
$981$	−26.7239	−0.853229
$982$	0	0
$983$	−52.9926	−1.69020	−0.845101	−	0.534606i	$-0.820460\pi$
$983$	−52.9926	−1.69020	−0.845101	+	0.534606i	$0.820460\pi$
$984$	0	0
$985$	20.2810	0.646206
$986$	0	0
$987$	−25.2400	−0.803398
$988$	0	0
$989$	2.44461	0.0777340
$990$	0	0
$991$	21.4608	0.681723	0.340862	−	0.940113i	$-0.389281\pi$
$991$	21.4608	0.681723	0.340862	+	0.940113i	$0.389281\pi$
$992$	0	0
$993$	14.4603	0.458883
$994$	0	0
$995$	10.8222	0.343088
$996$	0	0
$997$	−47.7482	−1.51220	−0.756100	−	0.654456i	$-0.772898\pi$
$997$	−47.7482	−1.51220	−0.756100	+	0.654456i	$0.772898\pi$
$998$	0	0
$999$	22.3398	0.706801

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bh.1.3		3
4.3	odd	2		1540.2.a.g.1.1	✓	3
20.3	even	4		7700.2.e.r.1849.1		6
20.7	even	4		7700.2.e.r.1849.6		6
20.19	odd	2		7700.2.a.x.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1540.2.a.g.1.1	✓	3	4.3	odd	2
6160.2.a.bh.1.3		3	1.1	even	1	trivial
7700.2.a.x.1.3		3	20.19	odd	2
7700.2.e.r.1849.1		6	20.3	even	4
7700.2.e.r.1849.6		6	20.7	even	4

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:	\( N \)	\(=\)	\( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6160.a (trivial)

\(f(q)\)	\(=\)	\(q+2.21432 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.90321 q^{9} +O(q^{10})\)	\(q+2.21432 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.90321 q^{9} -1.00000 q^{11} +2.21432 q^{13} -2.21432 q^{15} -2.96989 q^{17} -8.23506 q^{19} +2.21432 q^{21} +3.65878 q^{23} +1.00000 q^{25} -2.42864 q^{27} -0.193576 q^{29} -2.06668 q^{31} -2.21432 q^{33} -1.00000 q^{35} -9.19850 q^{37} +4.90321 q^{39} -8.36196 q^{41} +0.668149 q^{43} -1.90321 q^{45} -11.3985 q^{47} +1.00000 q^{49} -6.57628 q^{51} -8.90321 q^{53} +1.00000 q^{55} -18.2351 q^{57} +9.11753 q^{59} +4.55554 q^{61} +1.90321 q^{63} -2.21432 q^{65} +14.5763 q^{67} +8.10171 q^{69} -3.05086 q^{71} +7.82717 q^{73} +2.21432 q^{75} -1.00000 q^{77} -1.71900 q^{79} -11.0874 q^{81} +2.29529 q^{83} +2.96989 q^{85} -0.428639 q^{87} -16.8573 q^{89} +2.21432 q^{91} -4.57628 q^{93} +8.23506 q^{95} -10.3684 q^{97} -1.90321 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 3 q^{7} - q^{9}+O(q^{10}) \) 3 * q - 3 * q^5 + 3 * q^7 - q^9	\( 3 q - 3 q^{5} + 3 q^{7} - q^{9} - 3 q^{11} - 2 q^{17} + 2 q^{19} + 4 q^{23} + 3 q^{25} + 6 q^{27} - 14 q^{29} - 6 q^{31} - 3 q^{35} - 8 q^{37} + 8 q^{39} - 12 q^{41} + 22 q^{43} + q^{45} - 14 q^{47} + 3 q^{49} - 20 q^{53} + 3 q^{55} - 28 q^{57} + 14 q^{59} + 14 q^{61} - q^{63} + 24 q^{67} - 2 q^{69} + 4 q^{71} - 10 q^{73} - 3 q^{77} - 12 q^{79} - 13 q^{81} - 6 q^{83} + 2 q^{85} + 12 q^{87} - 24 q^{89} + 6 q^{93} - 2 q^{95} - 4 q^{97} + q^{99}+O(q^{100}) \) 3 * q - 3 * q^5 + 3 * q^7 - q^9 - 3 * q^11 - 2 * q^17 + 2 * q^19 + 4 * q^23 + 3 * q^25 + 6 * q^27 - 14 * q^29 - 6 * q^31 - 3 * q^35 - 8 * q^37 + 8 * q^39 - 12 * q^41 + 22 * q^43 + q^45 - 14 * q^47 + 3 * q^49 - 20 * q^53 + 3 * q^55 - 28 * q^57 + 14 * q^59 + 14 * q^61 - q^63 + 24 * q^67 - 2 * q^69 + 4 * q^71 - 10 * q^73 - 3 * q^77 - 12 * q^79 - 13 * q^81 - 6 * q^83 + 2 * q^85 + 12 * q^87 - 24 * q^89 + 6 * q^93 - 2 * q^95 - 4 * q^97 + q^99

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