LMFDB - Embedded newform 6080.2.a.cj.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, 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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6080 = 2^{6} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6080.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:	$48.5490444289$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.2363492.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 11x^{3} - 6x^{2} + 14x + 10 $ x^5 - x^4 - 11x^3 - 6x^2 + 14*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3040)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.82337$ of defining polynomial
Character	$\chi$	$=$	6080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.52807 q^{3} -1.00000 q^{5} -4.03789 q^{7} +3.39116 q^{9} +O(q^{10})$	$q+2.52807 q^{3} -1.00000 q^{5} -4.03789 q^{7} +3.39116 q^{9} +1.67610 q^{11} +4.79476 q^{13} -2.52807 q^{15} -3.28494 q^{17} -1.00000 q^{19} -10.2081 q^{21} -4.03789 q^{23} +1.00000 q^{25} +0.988885 q^{27} -1.14411 q^{29} -9.35221 q^{31} +4.23731 q^{33} +4.03789 q^{35} +4.43297 q^{37} +12.1215 q^{39} +3.21768 q^{41} +4.89378 q^{43} -3.39116 q^{45} -6.65647 q^{47} +9.30458 q^{49} -8.30458 q^{51} +0.224876 q^{53} -1.67610 q^{55} -2.52807 q^{57} +5.52226 q^{59} -12.9696 q^{61} -13.6931 q^{63} -4.79476 q^{65} -11.5555 q^{67} -10.2081 q^{69} -1.21768 q^{71} -4.98888 q^{73} +2.52807 q^{75} -6.76793 q^{77} -6.00783 q^{79} -7.67351 q^{81} -10.6813 q^{83} +3.28494 q^{85} -2.89240 q^{87} -4.37184 q^{89} -19.3607 q^{91} -23.6431 q^{93} +1.00000 q^{95} +1.93887 q^{97} +5.68394 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 19 q^{9}+O(q^{10}) $ 5 * q - 2 * q^3 - 5 * q^5 - 2 * q^7 + 19 * q^9	$ 5 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 19 q^{9} - 10 q^{11} + 2 q^{15} + 4 q^{17} - 5 q^{19} - 10 q^{21} - 2 q^{23} + 5 q^{25} - 8 q^{27} - 10 q^{29} - 10 q^{31} + 10 q^{33} + 2 q^{35} - 2 q^{37} - 10 q^{39} + 12 q^{41} + 2 q^{43} - 19 q^{45} - 22 q^{47} + 19 q^{49} - 14 q^{51} + 18 q^{53} + 10 q^{55} + 2 q^{57} - 4 q^{59} - 6 q^{61} - 10 q^{63} - 20 q^{67} - 10 q^{69} - 2 q^{71} - 12 q^{73} - 2 q^{75} + 4 q^{77} - 14 q^{79} + 13 q^{81} + 14 q^{83} - 4 q^{85} + 12 q^{87} + 22 q^{89} - 40 q^{91} - 8 q^{93} + 5 q^{95} - 10 q^{97} - 6 q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 - 5 * q^5 - 2 * q^7 + 19 * q^9 - 10 * q^11 + 2 * q^15 + 4 * q^17 - 5 * q^19 - 10 * q^21 - 2 * q^23 + 5 * q^25 - 8 * q^27 - 10 * q^29 - 10 * q^31 + 10 * q^33 + 2 * q^35 - 2 * q^37 - 10 * q^39 + 12 * q^41 + 2 * q^43 - 19 * q^45 - 22 * q^47 + 19 * q^49 - 14 * q^51 + 18 * q^53 + 10 * q^55 + 2 * q^57 - 4 * q^59 - 6 * q^61 - 10 * q^63 - 20 * q^67 - 10 * q^69 - 2 * q^71 - 12 * q^73 - 2 * q^75 + 4 * q^77 - 14 * q^79 + 13 * q^81 + 14 * q^83 - 4 * q^85 + 12 * q^87 + 22 * q^89 - 40 * q^91 - 8 * q^93 + 5 * q^95 - 10 * 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
$3$	2.52807	1.45958	0.729792	−	0.683669i	$-0.239617\pi$
$3$	2.52807	1.45958	0.729792	+	0.683669i	$0.239617\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.03789	−1.52618	−0.763090	−	0.646292i	$-0.776319\pi$
$7$	−4.03789	−1.52618	−0.763090	+	0.646292i	$0.776319\pi$
$8$	0	0
$9$	3.39116	1.13039
$10$	0	0
$11$	1.67610	0.505364	0.252682	−	0.967549i	$-0.418687\pi$
$11$	1.67610	0.505364	0.252682	+	0.967549i	$0.418687\pi$
$12$	0	0
$13$	4.79476	1.32983	0.664914	−	0.746920i	$-0.268468\pi$
$13$	4.79476	1.32983	0.664914	+	0.746920i	$0.268468\pi$
$14$	0	0
$15$	−2.52807	−0.652746
$16$	0	0
$17$	−3.28494	−0.796715	−0.398358	−	0.917230i	$-0.630420\pi$
$17$	−3.28494	−0.796715	−0.398358	+	0.917230i	$0.630420\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−10.2081	−2.22759
$22$	0	0
$23$	−4.03789	−0.841959	−0.420979	−	0.907070i	$-0.638314\pi$
$23$	−4.03789	−0.841959	−0.420979	+	0.907070i	$0.638314\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0.988885	0.190311
$28$	0	0
$29$	−1.14411	−0.212456	−0.106228	−	0.994342i	$-0.533877\pi$
$29$	−1.14411	−0.212456	−0.106228	+	0.994342i	$0.533877\pi$
$30$	0	0
$31$	−9.35221	−1.67971	−0.839853	−	0.542814i	$-0.817359\pi$
$31$	−9.35221	−1.67971	−0.839853	+	0.542814i	$0.817359\pi$
$32$	0	0
$33$	4.23731	0.737622
$34$	0	0
$35$	4.03789	0.682528
$36$	0	0
$37$	4.43297	0.728776	0.364388	−	0.931247i	$-0.381278\pi$
$37$	4.43297	0.728776	0.364388	+	0.931247i	$0.381278\pi$
$38$	0	0
$39$	12.1215	1.94100
$40$	0	0
$41$	3.21768	0.502517	0.251258	−	0.967920i	$-0.419156\pi$
$41$	3.21768	0.502517	0.251258	+	0.967920i	$0.419156\pi$
$42$	0	0
$43$	4.89378	0.746295	0.373147	−	0.927772i	$-0.378279\pi$
$43$	4.89378	0.746295	0.373147	+	0.927772i	$0.378279\pi$
$44$	0	0
$45$	−3.39116	−0.505524
$46$	0	0
$47$	−6.65647	−0.970945	−0.485473	−	0.874252i	$-0.661352\pi$
$47$	−6.65647	−0.970945	−0.485473	+	0.874252i	$0.661352\pi$
$48$	0	0
$49$	9.30458	1.32923
$50$	0	0
$51$	−8.30458	−1.16287
$52$	0	0
$53$	0.224876	0.0308891	0.0154446	−	0.999881i	$-0.495084\pi$
$53$	0.224876	0.0308891	0.0154446	+	0.999881i	$0.495084\pi$
$54$	0	0
$55$	−1.67610	−0.226006
$56$	0	0
$57$	−2.52807	−0.334852
$58$	0	0
$59$	5.52226	0.718936	0.359468	−	0.933157i	$-0.382958\pi$
$59$	5.52226	0.718936	0.359468	+	0.933157i	$0.382958\pi$
$60$	0	0
$61$	−12.9696	−1.66058	−0.830291	−	0.557330i	$-0.811826\pi$
$61$	−12.9696	−1.66058	−0.830291	+	0.557330i	$0.811826\pi$
$62$	0	0
$63$	−13.6931	−1.72517
$64$	0	0
$65$	−4.79476	−0.594717
$66$	0	0
$67$	−11.5555	−1.41173	−0.705867	−	0.708344i	$-0.749442\pi$
$67$	−11.5555	−1.41173	−0.705867	+	0.708344i	$0.749442\pi$
$68$	0	0
$69$	−10.2081	−1.22891
$70$	0	0
$71$	−1.21768	−0.144512	−0.0722559	−	0.997386i	$-0.523020\pi$
$71$	−1.21768	−0.144512	−0.0722559	+	0.997386i	$0.523020\pi$
$72$	0	0
$73$	−4.98888	−0.583905	−0.291952	−	0.956433i	$-0.594305\pi$
$73$	−4.98888	−0.583905	−0.291952	+	0.956433i	$0.594305\pi$
$74$	0	0
$75$	2.52807	0.291917
$76$	0	0
$77$	−6.76793	−0.771277
$78$	0	0
$79$	−6.00783	−0.675934	−0.337967	−	0.941158i	$-0.609739\pi$
$79$	−6.00783	−0.675934	−0.337967	+	0.941158i	$0.609739\pi$
$80$	0	0
$81$	−7.67351	−0.852612
$82$	0	0
$83$	−10.6813	−1.17243	−0.586215	−	0.810156i	$-0.699383\pi$
$83$	−10.6813	−1.17243	−0.586215	+	0.810156i	$0.699383\pi$
$84$	0	0
$85$	3.28494	0.356302
$86$	0	0
$87$	−2.89240	−0.310098
$88$	0	0
$89$	−4.37184	−0.463414	−0.231707	−	0.972786i	$-0.574431\pi$
$89$	−4.37184	−0.463414	−0.231707	+	0.972786i	$0.574431\pi$
$90$	0	0
$91$	−19.3607	−2.02956
$92$	0	0
$93$	−23.6431	−2.45167
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	1.93887	0.196863	0.0984313	−	0.995144i	$-0.468618\pi$
$97$	1.93887	0.196863	0.0984313	+	0.995144i	$0.468618\pi$
$98$	0	0
$99$	5.68394	0.571257
$100$	0	0
$101$	12.9774	1.29130	0.645650	−	0.763634i	$-0.276587\pi$
$101$	12.9774	1.29130	0.645650	+	0.763634i	$0.276587\pi$
$102$	0	0
$103$	17.8888	1.76264	0.881318	−	0.472524i	$-0.156657\pi$
$103$	17.8888	1.76264	0.881318	+	0.472524i	$0.156657\pi$
$104$	0	0
$105$	10.2081	0.996208
$106$	0	0
$107$	8.52807	0.824440	0.412220	−	0.911084i	$-0.364753\pi$
$107$	8.52807	0.824440	0.412220	+	0.911084i	$0.364753\pi$
$108$	0	0
$109$	1.71400	0.164171	0.0820855	−	0.996625i	$-0.473842\pi$
$109$	1.71400	0.164171	0.0820855	+	0.996625i	$0.473842\pi$
$110$	0	0
$111$	11.2069	1.06371
$112$	0	0
$113$	−14.0512	−1.32182	−0.660911	−	0.750464i	$-0.729830\pi$
$113$	−14.0512	−1.32182	−0.660911	+	0.750464i	$0.729830\pi$
$114$	0	0
$115$	4.03789	0.376535
$116$	0	0
$117$	16.2598	1.50322
$118$	0	0
$119$	13.2642	1.21593
$120$	0	0
$121$	−8.19068	−0.744607
$122$	0	0
$123$	8.13453	0.733466
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.1012	−0.896340	−0.448170	−	0.893948i	$-0.647924\pi$
$127$	−10.1012	−0.896340	−0.448170	+	0.893948i	$0.647924\pi$
$128$	0	0
$129$	12.3718	1.08928
$130$	0	0
$131$	1.45499	0.127123	0.0635616	−	0.997978i	$-0.479754\pi$
$131$	1.45499	0.127123	0.0635616	+	0.997978i	$0.479754\pi$
$132$	0	0
$133$	4.03789	0.350130
$134$	0	0
$135$	−0.988885	−0.0851097
$136$	0	0
$137$	0.0672644	0.00574679	0.00287339	−	0.999996i	$-0.499085\pi$
$137$	0.0672644	0.00574679	0.00287339	+	0.999996i	$0.499085\pi$
$138$	0	0
$139$	−18.2238	−1.54572	−0.772859	−	0.634577i	$-0.781174\pi$
$139$	−18.2238	−1.54572	−0.772859	+	0.634577i	$0.781174\pi$
$140$	0	0
$141$	−16.8280	−1.41718
$142$	0	0
$143$	8.03651	0.672047
$144$	0	0
$145$	1.14411	0.0950133
$146$	0	0
$147$	23.5227	1.94012
$148$	0	0
$149$	−2.85727	−0.234077	−0.117038	−	0.993127i	$-0.537340\pi$
$149$	−2.85727	−0.234077	−0.117038	+	0.993127i	$0.537340\pi$
$150$	0	0
$151$	−0.731412	−0.0595215	−0.0297607	−	0.999557i	$-0.509475\pi$
$151$	−0.731412	−0.0595215	−0.0297607	+	0.999557i	$0.509475\pi$
$152$	0	0
$153$	−11.1398	−0.900597
$154$	0	0
$155$	9.35221	0.751187
$156$	0	0
$157$	5.29346	0.422464	0.211232	−	0.977436i	$-0.432252\pi$
$157$	5.29346	0.422464	0.211232	+	0.977436i	$0.432252\pi$
$158$	0	0
$159$	0.568504	0.0450853
$160$	0	0
$161$	16.3046	1.28498
$162$	0	0
$163$	2.13369	0.167123	0.0835617	−	0.996503i	$-0.473370\pi$
$163$	2.13369	0.167123	0.0835617	+	0.996503i	$0.473370\pi$
$164$	0	0
$165$	−4.23731	−0.329874
$166$	0	0
$167$	17.1287	1.32546	0.662730	−	0.748859i	$-0.269398\pi$
$167$	17.1287	1.32546	0.662730	+	0.748859i	$0.269398\pi$
$168$	0	0
$169$	9.98972	0.768440
$170$	0	0
$171$	−3.39116	−0.259329
$172$	0	0
$173$	−23.1661	−1.76128	−0.880642	−	0.473783i	$-0.842888\pi$
$173$	−23.1661	−1.76128	−0.880642	+	0.473783i	$0.842888\pi$
$174$	0	0
$175$	−4.03789	−0.305236
$176$	0	0
$177$	13.9607	1.04935
$178$	0	0
$179$	−20.8437	−1.55793	−0.778966	−	0.627066i	$-0.784256\pi$
$179$	−20.8437	−1.55793	−0.778966	+	0.627066i	$0.784256\pi$
$180$	0	0
$181$	−20.3209	−1.51044	−0.755222	−	0.655470i	$-0.772471\pi$
$181$	−20.3209	−1.51044	−0.755222	+	0.655470i	$0.772471\pi$
$182$	0	0
$183$	−32.7880	−2.42376
$184$	0	0
$185$	−4.43297	−0.325918
$186$	0	0
$187$	−5.50590	−0.402631
$188$	0	0
$189$	−3.99301	−0.290449
$190$	0	0
$191$	5.98782	0.433264	0.216632	−	0.976253i	$-0.430493\pi$
$191$	5.98782	0.433264	0.216632	+	0.976253i	$0.430493\pi$
$192$	0	0
$193$	−0.958509	−0.0689950	−0.0344975	−	0.999405i	$-0.510983\pi$
$193$	−0.958509	−0.0689950	−0.0344975	+	0.999405i	$0.510983\pi$
$194$	0	0
$195$	−12.1215	−0.868039
$196$	0	0
$197$	−16.0471	−1.14331	−0.571654	−	0.820494i	$-0.693698\pi$
$197$	−16.0471	−1.14331	−0.571654	+	0.820494i	$0.693698\pi$
$198$	0	0
$199$	5.34003	0.378545	0.189272	−	0.981925i	$-0.439387\pi$
$199$	5.34003	0.378545	0.189272	+	0.981925i	$0.439387\pi$
$200$	0	0
$201$	−29.2133	−2.06055
$202$	0	0
$203$	4.61980	0.324247
$204$	0	0
$205$	−3.21768	−0.224732
$206$	0	0
$207$	−13.6931	−0.951739
$208$	0	0
$209$	−1.67610	−0.115938
$210$	0	0
$211$	−25.6346	−1.76476	−0.882378	−	0.470542i	$-0.844059\pi$
$211$	−25.6346	−1.76476	−0.882378	+	0.470542i	$0.844059\pi$
$212$	0	0
$213$	−3.07838	−0.210927
$214$	0	0
$215$	−4.89378	−0.333753
$216$	0	0
$217$	37.7632	2.56353
$218$	0	0
$219$	−12.6123	−0.852258
$220$	0	0
$221$	−15.7505	−1.05949
$222$	0	0
$223$	18.7076	1.25276	0.626378	−	0.779519i	$-0.284537\pi$
$223$	18.7076	1.25276	0.626378	+	0.779519i	$0.284537\pi$
$224$	0	0
$225$	3.39116	0.226077
$226$	0	0
$227$	1.31040	0.0869741	0.0434871	−	0.999054i	$-0.486153\pi$
$227$	1.31040	0.0869741	0.0434871	+	0.999054i	$0.486153\pi$
$228$	0	0
$229$	17.1676	1.13447	0.567234	−	0.823557i	$-0.308014\pi$
$229$	17.1676	1.13447	0.567234	+	0.823557i	$0.308014\pi$
$230$	0	0
$231$	−17.1098	−1.12574
$232$	0	0
$233$	29.4529	1.92952	0.964761	−	0.263129i	$-0.0847544\pi$
$233$	29.4529	1.92952	0.964761	+	0.263129i	$0.0847544\pi$
$234$	0	0
$235$	6.65647	0.434220
$236$	0	0
$237$	−15.1883	−0.986583
$238$	0	0
$239$	3.67472	0.237698	0.118849	−	0.992912i	$-0.462080\pi$
$239$	3.67472	0.237698	0.118849	+	0.992912i	$0.462080\pi$
$240$	0	0
$241$	7.02747	0.452679	0.226340	−	0.974048i	$-0.427324\pi$
$241$	7.02747	0.452679	0.226340	+	0.974048i	$0.427324\pi$
$242$	0	0
$243$	−22.3659	−1.43477
$244$	0	0
$245$	−9.30458	−0.594448
$246$	0	0
$247$	−4.79476	−0.305083
$248$	0	0
$249$	−27.0032	−1.71126
$250$	0	0
$251$	−5.31293	−0.335349	−0.167675	−	0.985842i	$-0.553626\pi$
$251$	−5.31293	−0.335349	−0.167675	+	0.985842i	$0.553626\pi$
$252$	0	0
$253$	−6.76793	−0.425496
$254$	0	0
$255$	8.30458	0.520053
$256$	0	0
$257$	−13.5727	−0.846644	−0.423322	−	0.905979i	$-0.639136\pi$
$257$	−13.5727	−0.846644	−0.423322	+	0.905979i	$0.639136\pi$
$258$	0	0
$259$	−17.8999	−1.11224
$260$	0	0
$261$	−3.87987	−0.240158
$262$	0	0
$263$	−20.9187	−1.28990	−0.644950	−	0.764225i	$-0.723122\pi$
$263$	−20.9187	−1.28990	−0.644950	+	0.764225i	$0.723122\pi$
$264$	0	0
$265$	−0.224876	−0.0138140
$266$	0	0
$267$	−11.0523	−0.676393
$268$	0	0
$269$	15.1542	0.923966	0.461983	−	0.886889i	$-0.347138\pi$
$269$	15.1542	0.923966	0.461983	+	0.886889i	$0.347138\pi$
$270$	0	0
$271$	−20.7693	−1.26165	−0.630823	−	0.775927i	$-0.717282\pi$
$271$	−20.7693	−1.26165	−0.630823	+	0.775927i	$0.717282\pi$
$272$	0	0
$273$	−48.9454	−2.96231
$274$	0	0
$275$	1.67610	0.101073
$276$	0	0
$277$	22.4005	1.34592	0.672958	−	0.739680i	$-0.265023\pi$
$277$	22.4005	1.34592	0.672958	+	0.739680i	$0.265023\pi$
$278$	0	0
$279$	−31.7148	−1.89872
$280$	0	0
$281$	−1.32998	−0.0793397	−0.0396699	−	0.999213i	$-0.512631\pi$
$281$	−1.32998	−0.0793397	−0.0396699	+	0.999213i	$0.512631\pi$
$282$	0	0
$283$	−33.0454	−1.96434	−0.982171	−	0.187989i	$-0.939803\pi$
$283$	−33.0454	−1.96434	−0.982171	+	0.187989i	$0.939803\pi$
$284$	0	0
$285$	2.52807	0.149750
$286$	0	0
$287$	−12.9926	−0.766931
$288$	0	0
$289$	−6.20916	−0.365244
$290$	0	0
$291$	4.90161	0.287338
$292$	0	0
$293$	−14.4038	−0.841476	−0.420738	−	0.907182i	$-0.638229\pi$
$293$	−14.4038	−0.841476	−0.420738	+	0.907182i	$0.638229\pi$
$294$	0	0
$295$	−5.52226	−0.321518
$296$	0	0
$297$	1.65747	0.0961763
$298$	0	0
$299$	−19.3607	−1.11966
$300$	0	0
$301$	−19.7606	−1.13898
$302$	0	0
$303$	32.8078	1.88476
$304$	0	0
$305$	12.9696	0.742635
$306$	0	0
$307$	14.1705	0.808751	0.404375	−	0.914593i	$-0.367489\pi$
$307$	14.1705	0.808751	0.404375	+	0.914593i	$0.367489\pi$
$308$	0	0
$309$	45.2242	2.57272
$310$	0	0
$311$	−22.3339	−1.26644	−0.633221	−	0.773971i	$-0.718268\pi$
$311$	−22.3339	−1.26644	−0.633221	+	0.773971i	$0.718268\pi$
$312$	0	0
$313$	24.4756	1.38344	0.691722	−	0.722164i	$-0.256852\pi$
$313$	24.4756	1.38344	0.691722	+	0.722164i	$0.256852\pi$
$314$	0	0
$315$	13.6931	0.771521
$316$	0	0
$317$	31.9710	1.79567	0.897836	−	0.440329i	$-0.145138\pi$
$317$	31.9710	1.79567	0.897836	+	0.440329i	$0.145138\pi$
$318$	0	0
$319$	−1.91765	−0.107368
$320$	0	0
$321$	21.5596	1.20334
$322$	0	0
$323$	3.28494	0.182779
$324$	0	0
$325$	4.79476	0.265965
$326$	0	0
$327$	4.33311	0.239622
$328$	0	0
$329$	26.8781	1.48184
$330$	0	0
$331$	−2.85742	−0.157058	−0.0785290	−	0.996912i	$-0.525022\pi$
$331$	−2.85742	−0.157058	−0.0785290	+	0.996912i	$0.525022\pi$
$332$	0	0
$333$	15.0329	0.823799
$334$	0	0
$335$	11.5555	0.631347
$336$	0	0
$337$	−27.1152	−1.47706	−0.738528	−	0.674223i	$-0.764479\pi$
$337$	−27.1152	−1.47706	−0.738528	+	0.674223i	$0.764479\pi$
$338$	0	0
$339$	−35.5224	−1.92931
$340$	0	0
$341$	−15.6753	−0.848863
$342$	0	0
$343$	−9.30564	−0.502457
$344$	0	0
$345$	10.2081	0.549585
$346$	0	0
$347$	14.6735	0.787715	0.393858	−	0.919171i	$-0.371140\pi$
$347$	14.6735	0.787715	0.393858	+	0.919171i	$0.371140\pi$
$348$	0	0
$349$	32.5964	1.74485	0.872424	−	0.488750i	$-0.162547\pi$
$349$	32.5964	1.74485	0.872424	+	0.488750i	$0.162547\pi$
$350$	0	0
$351$	4.74147	0.253081
$352$	0	0
$353$	0.580529	0.0308985	0.0154492	−	0.999881i	$-0.495082\pi$
$353$	0.580529	0.0308985	0.0154492	+	0.999881i	$0.495082\pi$
$354$	0	0
$355$	1.21768	0.0646276
$356$	0	0
$357$	33.5330	1.77475
$358$	0	0
$359$	−25.4372	−1.34252	−0.671262	−	0.741220i	$-0.734248\pi$
$359$	−25.4372	−1.34252	−0.671262	+	0.741220i	$0.734248\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−20.7066	−1.08682
$364$	0	0
$365$	4.98888	0.261130
$366$	0	0
$367$	−28.6465	−1.49534	−0.747668	−	0.664073i	$-0.768826\pi$
$367$	−28.6465	−1.49534	−0.747668	+	0.664073i	$0.768826\pi$
$368$	0	0
$369$	10.9117	0.568039
$370$	0	0
$371$	−0.908026	−0.0471424
$372$	0	0
$373$	30.0325	1.55503	0.777513	−	0.628867i	$-0.216481\pi$
$373$	30.0325	1.55503	0.777513	+	0.628867i	$0.216481\pi$
$374$	0	0
$375$	−2.52807	−0.130549
$376$	0	0
$377$	−5.48574	−0.282530
$378$	0	0
$379$	−17.3607	−0.891761	−0.445880	−	0.895093i	$-0.647109\pi$
$379$	−17.3607	−0.891761	−0.445880	+	0.895093i	$0.647109\pi$
$380$	0	0
$381$	−25.5367	−1.30828
$382$	0	0
$383$	28.0677	1.43419	0.717096	−	0.696975i	$-0.245471\pi$
$383$	28.0677	1.43419	0.717096	+	0.696975i	$0.245471\pi$
$384$	0	0
$385$	6.76793	0.344925
$386$	0	0
$387$	16.5956	0.843602
$388$	0	0
$389$	15.4396	0.782820	0.391410	−	0.920216i	$-0.371987\pi$
$389$	15.4396	0.782820	0.391410	+	0.920216i	$0.371987\pi$
$390$	0	0
$391$	13.2642	0.670802
$392$	0	0
$393$	3.67833	0.185547
$394$	0	0
$395$	6.00783	0.302287
$396$	0	0
$397$	−13.8193	−0.693571	−0.346786	−	0.937944i	$-0.612727\pi$
$397$	−13.8193	−0.693571	−0.346786	+	0.937944i	$0.612727\pi$
$398$	0	0
$399$	10.2081	0.511044
$400$	0	0
$401$	−3.39148	−0.169362	−0.0846812	−	0.996408i	$-0.526987\pi$
$401$	−3.39148	−0.169362	−0.0846812	+	0.996408i	$0.526987\pi$
$402$	0	0
$403$	−44.8416	−2.23372
$404$	0	0
$405$	7.67351	0.381300
$406$	0	0
$407$	7.43012	0.368297
$408$	0	0
$409$	27.8534	1.37726	0.688631	−	0.725112i	$-0.258212\pi$
$409$	27.8534	1.37726	0.688631	+	0.725112i	$0.258212\pi$
$410$	0	0
$411$	0.170049	0.00838792
$412$	0	0
$413$	−22.2983	−1.09723
$414$	0	0
$415$	10.6813	0.524326
$416$	0	0
$417$	−46.0710	−2.25611
$418$	0	0
$419$	17.5338	0.856584	0.428292	−	0.903640i	$-0.359115\pi$
$419$	17.5338	0.856584	0.428292	+	0.903640i	$0.359115\pi$
$420$	0	0
$421$	−28.3675	−1.38255	−0.691274	−	0.722593i	$-0.742950\pi$
$421$	−28.3675	−1.38255	−0.691274	+	0.722593i	$0.742950\pi$
$422$	0	0
$423$	−22.5732	−1.09754
$424$	0	0
$425$	−3.28494	−0.159343
$426$	0	0
$427$	52.3697	2.53435
$428$	0	0
$429$	20.3169	0.980909
$430$	0	0
$431$	15.5530	0.749162	0.374581	−	0.927194i	$-0.377787\pi$
$431$	15.5530	0.749162	0.374581	+	0.927194i	$0.377787\pi$
$432$	0	0
$433$	−15.6284	−0.751054	−0.375527	−	0.926811i	$-0.622538\pi$
$433$	−15.6284	−0.751054	−0.375527	+	0.926811i	$0.622538\pi$
$434$	0	0
$435$	2.89240	0.138680
$436$	0	0
$437$	4.03789	0.193159
$438$	0	0
$439$	−9.34437	−0.445983	−0.222991	−	0.974820i	$-0.571582\pi$
$439$	−9.34437	−0.445983	−0.222991	+	0.974820i	$0.571582\pi$
$440$	0	0
$441$	31.5533	1.50254
$442$	0	0
$443$	7.85330	0.373121	0.186561	−	0.982443i	$-0.440266\pi$
$443$	7.85330	0.373121	0.186561	+	0.982443i	$0.440266\pi$
$444$	0	0
$445$	4.37184	0.207245
$446$	0	0
$447$	−7.22339	−0.341654
$448$	0	0
$449$	11.8750	0.560415	0.280208	−	0.959939i	$-0.409597\pi$
$449$	11.8750	0.560415	0.280208	+	0.959939i	$0.409597\pi$
$450$	0	0
$451$	5.39316	0.253954
$452$	0	0
$453$	−1.84906	−0.0868766
$454$	0	0
$455$	19.3607	0.907645
$456$	0	0
$457$	16.7338	0.782773	0.391387	−	0.920226i	$-0.371996\pi$
$457$	16.7338	0.782773	0.391387	+	0.920226i	$0.371996\pi$
$458$	0	0
$459$	−3.24843	−0.151624
$460$	0	0
$461$	−35.0253	−1.63129	−0.815646	−	0.578551i	$-0.803619\pi$
$461$	−35.0253	−1.63129	−0.815646	+	0.578551i	$0.803619\pi$
$462$	0	0
$463$	30.6349	1.42372	0.711862	−	0.702320i	$-0.247852\pi$
$463$	30.6349	1.42372	0.711862	+	0.702320i	$0.247852\pi$
$464$	0	0
$465$	23.6431	1.09642
$466$	0	0
$467$	−16.0479	−0.742610	−0.371305	−	0.928511i	$-0.621090\pi$
$467$	−16.0479	−0.742610	−0.371305	+	0.928511i	$0.621090\pi$
$468$	0	0
$469$	46.6600	2.15456
$470$	0	0
$471$	13.3823	0.616622
$472$	0	0
$473$	8.20248	0.377151
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	0.762592	0.0349167
$478$	0	0
$479$	−41.1385	−1.87967	−0.939833	−	0.341635i	$-0.889019\pi$
$479$	−41.1385	−1.87967	−0.939833	+	0.341635i	$0.889019\pi$
$480$	0	0
$481$	21.2550	0.969146
$482$	0	0
$483$	41.2192	1.87554
$484$	0	0
$485$	−1.93887	−0.0880397
$486$	0	0
$487$	8.84049	0.400601	0.200300	−	0.979735i	$-0.435808\pi$
$487$	8.84049	0.400601	0.200300	+	0.979735i	$0.435808\pi$
$488$	0	0
$489$	5.39412	0.243931
$490$	0	0
$491$	30.4985	1.37638	0.688190	−	0.725531i	$-0.258406\pi$
$491$	30.4985	1.37638	0.688190	+	0.725531i	$0.258406\pi$
$492$	0	0
$493$	3.75834	0.169267
$494$	0	0
$495$	−5.68394	−0.255474
$496$	0	0
$497$	4.91685	0.220551
$498$	0	0
$499$	−25.1628	−1.12644	−0.563222	−	0.826306i	$-0.690438\pi$
$499$	−25.1628	−1.12644	−0.563222	+	0.826306i	$0.690438\pi$
$500$	0	0
$501$	43.3027	1.93462
$502$	0	0
$503$	−13.9256	−0.620912	−0.310456	−	0.950588i	$-0.600482\pi$
$503$	−13.9256	−0.620912	−0.310456	+	0.950588i	$0.600482\pi$
$504$	0	0
$505$	−12.9774	−0.577487
$506$	0	0
$507$	25.2548	1.12160
$508$	0	0
$509$	25.6403	1.13649	0.568244	−	0.822860i	$-0.307623\pi$
$509$	25.6403	1.13649	0.568244	+	0.822860i	$0.307623\pi$
$510$	0	0
$511$	20.1446	0.891144
$512$	0	0
$513$	−0.988885	−0.0436603
$514$	0	0
$515$	−17.8888	−0.788275
$516$	0	0
$517$	−11.1569	−0.490681
$518$	0	0
$519$	−58.5655	−2.57074
$520$	0	0
$521$	25.5598	1.11979	0.559897	−	0.828562i	$-0.310841\pi$
$521$	25.5598	1.11979	0.559897	+	0.828562i	$0.310841\pi$
$522$	0	0
$523$	27.7515	1.21349	0.606743	−	0.794898i	$-0.292476\pi$
$523$	27.7515	1.21349	0.606743	+	0.794898i	$0.292476\pi$
$524$	0	0
$525$	−10.2081	−0.445518
$526$	0	0
$527$	30.7215	1.33825
$528$	0	0
$529$	−6.69542	−0.291105
$530$	0	0
$531$	18.7269	0.812677
$532$	0	0
$533$	15.4280	0.668261
$534$	0	0
$535$	−8.52807	−0.368701
$536$	0	0
$537$	−52.6945	−2.27393
$538$	0	0
$539$	15.5954	0.671743
$540$	0	0
$541$	32.8042	1.41036	0.705182	−	0.709026i	$-0.250865\pi$
$541$	32.8042	1.41036	0.705182	+	0.709026i	$0.250865\pi$
$542$	0	0
$543$	−51.3728	−2.20462
$544$	0	0
$545$	−1.71400	−0.0734195
$546$	0	0
$547$	14.6517	0.626459	0.313230	−	0.949677i	$-0.398589\pi$
$547$	14.6517	0.626459	0.313230	+	0.949677i	$0.398589\pi$
$548$	0	0
$549$	−43.9819	−1.87710
$550$	0	0
$551$	1.14411	0.0487408
$552$	0	0
$553$	24.2590	1.03160
$554$	0	0
$555$	−11.2069	−0.475706
$556$	0	0
$557$	28.8515	1.22248	0.611240	−	0.791446i	$-0.290671\pi$
$557$	28.8515	1.22248	0.611240	+	0.791446i	$0.290671\pi$
$558$	0	0
$559$	23.4645	0.992443
$560$	0	0
$561$	−13.9193	−0.587675
$562$	0	0
$563$	−25.2045	−1.06224	−0.531121	−	0.847296i	$-0.678229\pi$
$563$	−25.2045	−1.06224	−0.531121	+	0.847296i	$0.678229\pi$
$564$	0	0
$565$	14.0512	0.591137
$566$	0	0
$567$	30.9848	1.30124
$568$	0	0
$569$	−6.04404	−0.253379	−0.126690	−	0.991942i	$-0.540435\pi$
$569$	−6.04404	−0.253379	−0.126690	+	0.991942i	$0.540435\pi$
$570$	0	0
$571$	35.6125	1.49034	0.745168	−	0.666877i	$-0.232369\pi$
$571$	35.6125	1.49034	0.745168	+	0.666877i	$0.232369\pi$
$572$	0	0
$573$	15.1377	0.632385
$574$	0	0
$575$	−4.03789	−0.168392
$576$	0	0
$577$	−21.1791	−0.881697	−0.440849	−	0.897581i	$-0.645322\pi$
$577$	−21.1791	−0.881697	−0.440849	+	0.897581i	$0.645322\pi$
$578$	0	0
$579$	−2.42318	−0.100704
$580$	0	0
$581$	43.1301	1.78934
$582$	0	0
$583$	0.376916	0.0156103
$584$	0	0
$585$	−16.2598	−0.672260
$586$	0	0
$587$	7.81276	0.322467	0.161233	−	0.986916i	$-0.448453\pi$
$587$	7.81276	0.322467	0.161233	+	0.986916i	$0.448453\pi$
$588$	0	0
$589$	9.35221	0.385351
$590$	0	0
$591$	−40.5683	−1.66876
$592$	0	0
$593$	−3.78924	−0.155606	−0.0778028	−	0.996969i	$-0.524790\pi$
$593$	−3.78924	−0.155606	−0.0778028	+	0.996969i	$0.524790\pi$
$594$	0	0
$595$	−13.2642	−0.543781
$596$	0	0
$597$	13.5000	0.552518
$598$	0	0
$599$	47.1260	1.92552	0.962758	−	0.270364i	$-0.0871440\pi$
$599$	47.1260	1.92552	0.962758	+	0.270364i	$0.0871440\pi$
$600$	0	0
$601$	18.3380	0.748022	0.374011	−	0.927424i	$-0.377982\pi$
$601$	18.3380	0.748022	0.374011	+	0.927424i	$0.377982\pi$
$602$	0	0
$603$	−39.1867	−1.59581
$604$	0	0
$605$	8.19068	0.332998
$606$	0	0
$607$	−22.7934	−0.925155	−0.462577	−	0.886579i	$-0.653075\pi$
$607$	−22.7934	−0.925155	−0.462577	+	0.886579i	$0.653075\pi$
$608$	0	0
$609$	11.6792	0.473265
$610$	0	0
$611$	−31.9162	−1.29119
$612$	0	0
$613$	29.3771	1.18653	0.593265	−	0.805008i	$-0.297839\pi$
$613$	29.3771	1.18653	0.593265	+	0.805008i	$0.297839\pi$
$614$	0	0
$615$	−8.13453	−0.328016
$616$	0	0
$617$	10.5986	0.426682	0.213341	−	0.976978i	$-0.431565\pi$
$617$	10.5986	0.426682	0.213341	+	0.976978i	$0.431565\pi$
$618$	0	0
$619$	−35.2936	−1.41857	−0.709284	−	0.704923i	$-0.750982\pi$
$619$	−35.2936	−1.41857	−0.709284	+	0.704923i	$0.750982\pi$
$620$	0	0
$621$	−3.99301	−0.160234
$622$	0	0
$623$	17.6530	0.707254
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.23731	−0.169222
$628$	0	0
$629$	−14.5621	−0.580627
$630$	0	0
$631$	−38.2609	−1.52314	−0.761571	−	0.648081i	$-0.775572\pi$
$631$	−38.2609	−1.52314	−0.761571	+	0.648081i	$0.775572\pi$
$632$	0	0
$633$	−64.8061	−2.57581
$634$	0	0
$635$	10.1012	0.400856
$636$	0	0
$637$	44.6132	1.76764
$638$	0	0
$639$	−4.12934	−0.163354
$640$	0	0
$641$	33.5730	1.32605	0.663027	−	0.748596i	$-0.269272\pi$
$641$	33.5730	1.32605	0.663027	+	0.748596i	$0.269272\pi$
$642$	0	0
$643$	−4.61857	−0.182139	−0.0910693	−	0.995845i	$-0.529028\pi$
$643$	−4.61857	−0.182139	−0.0910693	+	0.995845i	$0.529028\pi$
$644$	0	0
$645$	−12.3718	−0.487141
$646$	0	0
$647$	−27.8312	−1.09416	−0.547078	−	0.837082i	$-0.684260\pi$
$647$	−27.8312	−1.09416	−0.547078	+	0.837082i	$0.684260\pi$
$648$	0	0
$649$	9.25587	0.363325
$650$	0	0
$651$	95.4682	3.74169
$652$	0	0
$653$	−29.3927	−1.15023	−0.575114	−	0.818074i	$-0.695042\pi$
$653$	−29.3927	−1.15023	−0.575114	+	0.818074i	$0.695042\pi$
$654$	0	0
$655$	−1.45499	−0.0568512
$656$	0	0
$657$	−16.9181	−0.660038
$658$	0	0
$659$	39.9207	1.55509	0.777544	−	0.628829i	$-0.216465\pi$
$659$	39.9207	1.55509	0.777544	+	0.628829i	$0.216465\pi$
$660$	0	0
$661$	34.9777	1.36047	0.680237	−	0.732992i	$-0.261877\pi$
$661$	34.9777	1.36047	0.680237	+	0.732992i	$0.261877\pi$
$662$	0	0
$663$	−39.8185	−1.54642
$664$	0	0
$665$	−4.03789	−0.156583
$666$	0	0
$667$	4.61980	0.178879
$668$	0	0
$669$	47.2943	1.82850
$670$	0	0
$671$	−21.7383	−0.839199
$672$	0	0
$673$	−24.4015	−0.940610	−0.470305	−	0.882504i	$-0.655856\pi$
$673$	−24.4015	−0.940610	−0.470305	+	0.882504i	$0.655856\pi$
$674$	0	0
$675$	0.988885	0.0380622
$676$	0	0
$677$	−28.0025	−1.07622	−0.538111	−	0.842874i	$-0.680862\pi$
$677$	−28.0025	−1.07622	−0.538111	+	0.842874i	$0.680862\pi$
$678$	0	0
$679$	−7.82896	−0.300448
$680$	0	0
$681$	3.31278	0.126946
$682$	0	0
$683$	7.14311	0.273323	0.136662	−	0.990618i	$-0.456363\pi$
$683$	7.14311	0.273323	0.136662	+	0.990618i	$0.456363\pi$
$684$	0	0
$685$	−0.0672644	−0.00257004
$686$	0	0
$687$	43.4010	1.65585
$688$	0	0
$689$	1.07823	0.0410772
$690$	0	0
$691$	21.6962	0.825363	0.412681	−	0.910875i	$-0.364592\pi$
$691$	21.6962	0.825363	0.412681	+	0.910875i	$0.364592\pi$
$692$	0	0
$693$	−22.9511	−0.871841
$694$	0	0
$695$	18.2238	0.691266
$696$	0	0
$697$	−10.5699	−0.400363
$698$	0	0
$699$	74.4590	2.81630
$700$	0	0
$701$	−43.9543	−1.66013	−0.830065	−	0.557667i	$-0.811697\pi$
$701$	−43.9543	−1.66013	−0.830065	+	0.557667i	$0.811697\pi$
$702$	0	0
$703$	−4.43297	−0.167193
$704$	0	0
$705$	16.8280	0.633781
$706$	0	0
$707$	−52.4013	−1.97076
$708$	0	0
$709$	−11.3051	−0.424572	−0.212286	−	0.977208i	$-0.568091\pi$
$709$	−11.3051	−0.424572	−0.212286	+	0.977208i	$0.568091\pi$
$710$	0	0
$711$	−20.3735	−0.764067
$712$	0	0
$713$	37.7632	1.41424
$714$	0	0
$715$	−8.03651	−0.300549
$716$	0	0
$717$	9.28997	0.346940
$718$	0	0
$719$	7.23069	0.269659	0.134830	−	0.990869i	$-0.456951\pi$
$719$	7.23069	0.269659	0.134830	+	0.990869i	$0.456951\pi$
$720$	0	0
$721$	−72.2331	−2.69010
$722$	0	0
$723$	17.7660	0.660723
$724$	0	0
$725$	−1.14411	−0.0424913
$726$	0	0
$727$	−11.8138	−0.438150	−0.219075	−	0.975708i	$-0.570304\pi$
$727$	−11.8138	−0.438150	−0.219075	+	0.975708i	$0.570304\pi$
$728$	0	0
$729$	−33.5220	−1.24156
$730$	0	0
$731$	−16.0758	−0.594584
$732$	0	0
$733$	13.2098	0.487917	0.243958	−	0.969786i	$-0.421554\pi$
$733$	13.2098	0.487917	0.243958	+	0.969786i	$0.421554\pi$
$734$	0	0
$735$	−23.5227	−0.867647
$736$	0	0
$737$	−19.3683	−0.713440
$738$	0	0
$739$	39.7827	1.46343	0.731715	−	0.681611i	$-0.238720\pi$
$739$	39.7827	1.46343	0.731715	+	0.681611i	$0.238720\pi$
$740$	0	0
$741$	−12.1215	−0.445295
$742$	0	0
$743$	40.2442	1.47642	0.738209	−	0.674572i	$-0.235672\pi$
$743$	40.2442	1.47642	0.738209	+	0.674572i	$0.235672\pi$
$744$	0	0
$745$	2.85727	0.104682
$746$	0	0
$747$	−36.2222	−1.32530
$748$	0	0
$749$	−34.4355	−1.25824
$750$	0	0
$751$	14.7850	0.539511	0.269756	−	0.962929i	$-0.413057\pi$
$751$	14.7850	0.539511	0.269756	+	0.962929i	$0.413057\pi$
$752$	0	0
$753$	−13.4315	−0.489471
$754$	0	0
$755$	0.731412	0.0266188
$756$	0	0
$757$	−10.7071	−0.389155	−0.194577	−	0.980887i	$-0.562334\pi$
$757$	−10.7071	−0.389155	−0.194577	+	0.980887i	$0.562334\pi$
$758$	0	0
$759$	−17.1098	−0.621047
$760$	0	0
$761$	23.6768	0.858283	0.429142	−	0.903237i	$-0.358816\pi$
$761$	23.6768	0.858283	0.429142	+	0.903237i	$0.358816\pi$
$762$	0	0
$763$	−6.92093	−0.250555
$764$	0	0
$765$	11.1398	0.402759
$766$	0	0
$767$	26.4779	0.956061
$768$	0	0
$769$	−15.5193	−0.559640	−0.279820	−	0.960052i	$-0.590275\pi$
$769$	−15.5193	−0.559640	−0.279820	+	0.960052i	$0.590275\pi$
$770$	0	0
$771$	−34.3129	−1.23575
$772$	0	0
$773$	16.4689	0.592347	0.296173	−	0.955134i	$-0.404289\pi$
$773$	16.4689	0.592347	0.296173	+	0.955134i	$0.404289\pi$
$774$	0	0
$775$	−9.35221	−0.335941
$776$	0	0
$777$	−45.2522	−1.62341
$778$	0	0
$779$	−3.21768	−0.115285
$780$	0	0
$781$	−2.04095	−0.0730310
$782$	0	0
$783$	−1.13140	−0.0404328
$784$	0	0
$785$	−5.29346	−0.188932
$786$	0	0
$787$	2.41059	0.0859282	0.0429641	−	0.999077i	$-0.486320\pi$
$787$	2.41059	0.0859282	0.0429641	+	0.999077i	$0.486320\pi$
$788$	0	0
$789$	−52.8839	−1.88272
$790$	0	0
$791$	56.7371	2.01734
$792$	0	0
$793$	−62.1860	−2.20829
$794$	0	0
$795$	−0.568504	−0.0201628
$796$	0	0
$797$	7.95157	0.281659	0.140830	−	0.990034i	$-0.455023\pi$
$797$	7.95157	0.281659	0.140830	+	0.990034i	$0.455023\pi$
$798$	0	0
$799$	21.8661	0.773567
$800$	0	0
$801$	−14.8256	−0.523838
$802$	0	0
$803$	−8.36189	−0.295085
$804$	0	0
$805$	−16.3046	−0.574661
$806$	0	0
$807$	38.3109	1.34861
$808$	0	0
$809$	−31.3269	−1.10139	−0.550697	−	0.834705i	$-0.685638\pi$
$809$	−31.3269	−1.10139	−0.550697	+	0.834705i	$0.685638\pi$
$810$	0	0
$811$	−21.7978	−0.765423	−0.382711	−	0.923868i	$-0.625010\pi$
$811$	−21.7978	−0.765423	−0.382711	+	0.923868i	$0.625010\pi$
$812$	0	0
$813$	−52.5064	−1.84148
$814$	0	0
$815$	−2.13369	−0.0747399
$816$	0	0
$817$	−4.89378	−0.171212
$818$	0	0
$819$	−65.6553	−2.29418
$820$	0	0
$821$	39.7195	1.38622	0.693110	−	0.720832i	$-0.256240\pi$
$821$	39.7195	1.38622	0.693110	+	0.720832i	$0.256240\pi$
$822$	0	0
$823$	9.92612	0.346003	0.173001	−	0.984922i	$-0.444654\pi$
$823$	9.92612	0.346003	0.173001	+	0.984922i	$0.444654\pi$
$824$	0	0
$825$	4.23731	0.147524
$826$	0	0
$827$	−3.13723	−0.109092	−0.0545461	−	0.998511i	$-0.517371\pi$
$827$	−3.13723	−0.109092	−0.0545461	+	0.998511i	$0.517371\pi$
$828$	0	0
$829$	16.4184	0.570235	0.285117	−	0.958493i	$-0.407967\pi$
$829$	16.4184	0.570235	0.285117	+	0.958493i	$0.407967\pi$
$830$	0	0
$831$	56.6302	1.96448
$832$	0	0
$833$	−30.5650	−1.05901
$834$	0	0
$835$	−17.1287	−0.592763
$836$	0	0
$837$	−9.24825	−0.319666
$838$	0	0
$839$	−34.5168	−1.19165	−0.595827	−	0.803113i	$-0.703176\pi$
$839$	−34.5168	−1.19165	−0.595827	+	0.803113i	$0.703176\pi$
$840$	0	0
$841$	−27.6910	−0.954862
$842$	0	0
$843$	−3.36228	−0.115803
$844$	0	0
$845$	−9.98972	−0.343657
$846$	0	0
$847$	33.0731	1.13640
$848$	0	0
$849$	−83.5411	−2.86712
$850$	0	0
$851$	−17.8999	−0.613599
$852$	0	0
$853$	13.0831	0.447958	0.223979	−	0.974594i	$-0.428095\pi$
$853$	13.0831	0.447958	0.223979	+	0.974594i	$0.428095\pi$
$854$	0	0
$855$	3.39116	0.115975
$856$	0	0
$857$	44.7999	1.53034	0.765168	−	0.643830i	$-0.222656\pi$
$857$	44.7999	1.53034	0.765168	+	0.643830i	$0.222656\pi$
$858$	0	0
$859$	−33.0161	−1.12650	−0.563248	−	0.826288i	$-0.690448\pi$
$859$	−33.0161	−1.12650	−0.563248	+	0.826288i	$0.690448\pi$
$860$	0	0
$861$	−32.8464	−1.11940
$862$	0	0
$863$	27.4407	0.934093	0.467047	−	0.884233i	$-0.345318\pi$
$863$	27.4407	0.934093	0.467047	+	0.884233i	$0.345318\pi$
$864$	0	0
$865$	23.1661	0.787670
$866$	0	0
$867$	−15.6972	−0.533105
$868$	0	0
$869$	−10.0697	−0.341593
$870$	0	0
$871$	−55.4061	−1.87736
$872$	0	0
$873$	6.57503	0.222531
$874$	0	0
$875$	4.03789	0.136506
$876$	0	0
$877$	−12.9907	−0.438664	−0.219332	−	0.975650i	$-0.570388\pi$
$877$	−12.9907	−0.438664	−0.219332	+	0.975650i	$0.570388\pi$
$878$	0	0
$879$	−36.4138	−1.22821
$880$	0	0
$881$	0.554761	0.0186904	0.00934519	−	0.999956i	$-0.497025\pi$
$881$	0.554761	0.0186904	0.00934519	+	0.999956i	$0.497025\pi$
$882$	0	0
$883$	36.1537	1.21667	0.608334	−	0.793681i	$-0.291838\pi$
$883$	36.1537	1.21667	0.608334	+	0.793681i	$0.291838\pi$
$884$	0	0
$885$	−13.9607	−0.469283
$886$	0	0
$887$	40.8100	1.37026	0.685132	−	0.728418i	$-0.259744\pi$
$887$	40.8100	1.37026	0.685132	+	0.728418i	$0.259744\pi$
$888$	0	0
$889$	40.7877	1.36798
$890$	0	0
$891$	−12.8616	−0.430880
$892$	0	0
$893$	6.65647	0.222750
$894$	0	0
$895$	20.8437	0.696728
$896$	0	0
$897$	−48.9454	−1.63424
$898$	0	0
$899$	10.7000	0.356864
$900$	0	0
$901$	−0.738705	−0.0246098
$902$	0	0
$903$	−49.9562	−1.66244
$904$	0	0
$905$	20.3209	0.675491
$906$	0	0
$907$	46.4115	1.54107	0.770535	−	0.637398i	$-0.219989\pi$
$907$	46.4115	1.54107	0.770535	+	0.637398i	$0.219989\pi$
$908$	0	0
$909$	44.0085	1.45967
$910$	0	0
$911$	18.8697	0.625183	0.312591	−	0.949888i	$-0.398803\pi$
$911$	18.8697	0.625183	0.312591	+	0.949888i	$0.398803\pi$
$912$	0	0
$913$	−17.9030	−0.592504
$914$	0	0
$915$	32.7880	1.08394
$916$	0	0
$917$	−5.87510	−0.194013
$918$	0	0
$919$	5.96295	0.196699	0.0983497	−	0.995152i	$-0.468644\pi$
$919$	5.96295	0.196699	0.0983497	+	0.995152i	$0.468644\pi$
$920$	0	0
$921$	35.8240	1.18044
$922$	0	0
$923$	−5.83847	−0.192176
$924$	0	0
$925$	4.43297	0.145755
$926$	0	0
$927$	60.6638	1.99246
$928$	0	0
$929$	49.0325	1.60870	0.804352	−	0.594153i	$-0.202513\pi$
$929$	49.0325	1.60870	0.804352	+	0.594153i	$0.202513\pi$
$930$	0	0
$931$	−9.30458	−0.304945
$932$	0	0
$933$	−56.4619	−1.84848
$934$	0	0
$935$	5.50590	0.180062
$936$	0	0
$937$	−22.6477	−0.739869	−0.369935	−	0.929058i	$-0.620620\pi$
$937$	−22.6477	−0.739869	−0.369935	+	0.929058i	$0.620620\pi$
$938$	0	0
$939$	61.8762	2.01925
$940$	0	0
$941$	20.9975	0.684499	0.342250	−	0.939609i	$-0.388811\pi$
$941$	20.9975	0.684499	0.342250	+	0.939609i	$0.388811\pi$
$942$	0	0
$943$	−12.9926	−0.423098
$944$	0	0
$945$	3.99301	0.129893
$946$	0	0
$947$	−36.0466	−1.17136	−0.585678	−	0.810544i	$-0.699172\pi$
$947$	−36.0466	−1.17136	−0.585678	+	0.810544i	$0.699172\pi$
$948$	0	0
$949$	−23.9205	−0.776492
$950$	0	0
$951$	80.8252	2.62094
$952$	0	0
$953$	37.7137	1.22167	0.610834	−	0.791759i	$-0.290834\pi$
$953$	37.7137	1.22167	0.610834	+	0.791759i	$0.290834\pi$
$954$	0	0
$955$	−5.98782	−0.193761
$956$	0	0
$957$	−4.84796	−0.156712
$958$	0	0
$959$	−0.271607	−0.00877063
$960$	0	0
$961$	56.4638	1.82141
$962$	0	0
$963$	28.9201	0.931936
$964$	0	0
$965$	0.958509	0.0308555
$966$	0	0
$967$	21.1768	0.681001	0.340500	−	0.940244i	$-0.389404\pi$
$967$	21.1768	0.681001	0.340500	+	0.940244i	$0.389404\pi$
$968$	0	0
$969$	8.30458	0.266781
$970$	0	0
$971$	−14.6324	−0.469577	−0.234789	−	0.972046i	$-0.575440\pi$
$971$	−14.6324	−0.469577	−0.234789	+	0.972046i	$0.575440\pi$
$972$	0	0
$973$	73.5856	2.35905
$974$	0	0
$975$	12.1215	0.388199
$976$	0	0
$977$	−43.3160	−1.38580	−0.692900	−	0.721034i	$-0.743667\pi$
$977$	−43.3160	−1.38580	−0.692900	+	0.721034i	$0.743667\pi$
$978$	0	0
$979$	−7.32766	−0.234193
$980$	0	0
$981$	5.81244	0.185577
$982$	0	0
$983$	23.1679	0.738940	0.369470	−	0.929243i	$-0.379539\pi$
$983$	23.1679	0.738940	0.369470	+	0.929243i	$0.379539\pi$
$984$	0	0
$985$	16.0471	0.511303
$986$	0	0
$987$	67.9498	2.16287
$988$	0	0
$989$	−19.7606	−0.628349
$990$	0	0
$991$	−23.8607	−0.757960	−0.378980	−	0.925405i	$-0.623725\pi$
$991$	−23.8607	−0.757960	−0.378980	+	0.925405i	$0.623725\pi$
$992$	0	0
$993$	−7.22377	−0.229239
$994$	0	0
$995$	−5.34003	−0.169290
$996$	0	0
$997$	−35.2357	−1.11592	−0.557962	−	0.829866i	$-0.688417\pi$
$997$	−35.2357	−1.11592	−0.557962	+	0.829866i	$0.688417\pi$
$998$	0	0
$999$	4.38370	0.138694

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.cj.1.4		5
4.3	odd	2		6080.2.a.ck.1.2		5
8.3	odd	2		3040.2.a.v.1.4	✓	5
8.5	even	2		3040.2.a.w.1.2	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.v.1.4	✓	5	8.3	odd	2
3040.2.a.w.1.2	yes	5	8.5	even	2
6080.2.a.cj.1.4		5	1.1	even	1	trivial
6080.2.a.ck.1.2		5	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6080.a (trivial)

\(f(q)\)	\(=\)	\(q+2.52807 q^{3} -1.00000 q^{5} -4.03789 q^{7} +3.39116 q^{9} +O(q^{10})\)	\(q+2.52807 q^{3} -1.00000 q^{5} -4.03789 q^{7} +3.39116 q^{9} +1.67610 q^{11} +4.79476 q^{13} -2.52807 q^{15} -3.28494 q^{17} -1.00000 q^{19} -10.2081 q^{21} -4.03789 q^{23} +1.00000 q^{25} +0.988885 q^{27} -1.14411 q^{29} -9.35221 q^{31} +4.23731 q^{33} +4.03789 q^{35} +4.43297 q^{37} +12.1215 q^{39} +3.21768 q^{41} +4.89378 q^{43} -3.39116 q^{45} -6.65647 q^{47} +9.30458 q^{49} -8.30458 q^{51} +0.224876 q^{53} -1.67610 q^{55} -2.52807 q^{57} +5.52226 q^{59} -12.9696 q^{61} -13.6931 q^{63} -4.79476 q^{65} -11.5555 q^{67} -10.2081 q^{69} -1.21768 q^{71} -4.98888 q^{73} +2.52807 q^{75} -6.76793 q^{77} -6.00783 q^{79} -7.67351 q^{81} -10.6813 q^{83} +3.28494 q^{85} -2.89240 q^{87} -4.37184 q^{89} -19.3607 q^{91} -23.6431 q^{93} +1.00000 q^{95} +1.93887 q^{97} +5.68394 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 19 q^{9}+O(q^{10}) \) 5 * q - 2 * q^3 - 5 * q^5 - 2 * q^7 + 19 * q^9	\( 5 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 19 q^{9} - 10 q^{11} + 2 q^{15} + 4 q^{17} - 5 q^{19} - 10 q^{21} - 2 q^{23} + 5 q^{25} - 8 q^{27} - 10 q^{29} - 10 q^{31} + 10 q^{33} + 2 q^{35} - 2 q^{37} - 10 q^{39} + 12 q^{41} + 2 q^{43} - 19 q^{45} - 22 q^{47} + 19 q^{49} - 14 q^{51} + 18 q^{53} + 10 q^{55} + 2 q^{57} - 4 q^{59} - 6 q^{61} - 10 q^{63} - 20 q^{67} - 10 q^{69} - 2 q^{71} - 12 q^{73} - 2 q^{75} + 4 q^{77} - 14 q^{79} + 13 q^{81} + 14 q^{83} - 4 q^{85} + 12 q^{87} + 22 q^{89} - 40 q^{91} - 8 q^{93} + 5 q^{95} - 10 q^{97} - 6 q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 - 5 * q^5 - 2 * q^7 + 19 * q^9 - 10 * q^11 + 2 * q^15 + 4 * q^17 - 5 * q^19 - 10 * q^21 - 2 * q^23 + 5 * q^25 - 8 * q^27 - 10 * q^29 - 10 * q^31 + 10 * q^33 + 2 * q^35 - 2 * q^37 - 10 * q^39 + 12 * q^41 + 2 * q^43 - 19 * q^45 - 22 * q^47 + 19 * q^49 - 14 * q^51 + 18 * q^53 + 10 * q^55 + 2 * q^57 - 4 * q^59 - 6 * q^61 - 10 * q^63 - 20 * q^67 - 10 * q^69 - 2 * q^71 - 12 * q^73 - 2 * q^75 + 4 * q^77 - 14 * q^79 + 13 * q^81 + 14 * q^83 - 4 * q^85 + 12 * q^87 + 22 * q^89 - 40 * q^91 - 8 * q^93 + 5 * q^95 - 10 * q^97 - 6 * q^99

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