LMFDB - Embedded newform 3680.2.a.bc.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3680,2,Mod(1,3680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3680, 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("3680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3680 = 2^{5} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3680.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:	$29.3849479438$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.255601784.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 5x^{3} + 25x^{2} - 4 $ x^6 - x^5 - 10x^4 + 5x^3 + 25*x^2 - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.397065$ of defining polynomial
Character	$\chi$	$=$	3680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.602935 q^{3} +1.00000 q^{5} -4.26364 q^{7} -2.63647 q^{9} +O(q^{10})$	$q-0.602935 q^{3} +1.00000 q^{5} -4.26364 q^{7} -2.63647 q^{9} +3.63990 q^{11} +4.55951 q^{13} -0.602935 q^{15} -1.85371 q^{17} -1.23973 q^{19} +2.57070 q^{21} -1.00000 q^{23} +1.00000 q^{25} +3.39843 q^{27} +5.38880 q^{29} -4.49050 q^{31} -2.19462 q^{33} -4.26364 q^{35} -1.78619 q^{37} -2.74909 q^{39} -2.90043 q^{41} +0.649335 q^{43} -2.63647 q^{45} -5.27132 q^{47} +11.1786 q^{49} +1.11767 q^{51} +4.45826 q^{53} +3.63990 q^{55} +0.747474 q^{57} -4.22978 q^{59} +13.6015 q^{61} +11.2409 q^{63} +4.55951 q^{65} -8.69262 q^{67} +0.602935 q^{69} -0.430737 q^{71} -12.8339 q^{73} -0.602935 q^{75} -15.5192 q^{77} -11.7426 q^{79} +5.86038 q^{81} +1.77643 q^{83} -1.85371 q^{85} -3.24910 q^{87} -9.83271 q^{89} -19.4401 q^{91} +2.70748 q^{93} -1.23973 q^{95} +1.29912 q^{97} -9.59647 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 5 q^{3} + 6 q^{5} - 4 q^{7} + 7 q^{9}+O(q^{10}) $ 6 * q - 5 * q^3 + 6 * q^5 - 4 * q^7 + 7 * q^9	$ 6 q - 5 q^{3} + 6 q^{5} - 4 q^{7} + 7 q^{9} - 7 q^{11} + q^{13} - 5 q^{15} - 5 q^{19} - 6 q^{23} + 6 q^{25} - 20 q^{27} - 3 q^{29} - 12 q^{31} - 3 q^{33} - 4 q^{35} + 7 q^{37} - 8 q^{39} + 8 q^{41} - 28 q^{43} + 7 q^{45} - 8 q^{47} + 6 q^{49} - 7 q^{51} - 5 q^{53} - 7 q^{55} + 18 q^{57} - 9 q^{59} - 3 q^{61} + 5 q^{63} + q^{65} - 27 q^{67} + 5 q^{69} + 2 q^{71} - 2 q^{73} - 5 q^{75} - 14 q^{77} - 4 q^{79} + 14 q^{81} - 23 q^{83} - 12 q^{87} - 4 q^{89} + q^{91} - 4 q^{93} - 5 q^{95} - q^{97} - q^{99}+O(q^{100}) $ 6 * q - 5 * q^3 + 6 * q^5 - 4 * q^7 + 7 * q^9 - 7 * q^11 + q^13 - 5 * q^15 - 5 * q^19 - 6 * q^23 + 6 * q^25 - 20 * q^27 - 3 * q^29 - 12 * q^31 - 3 * q^33 - 4 * q^35 + 7 * q^37 - 8 * q^39 + 8 * q^41 - 28 * q^43 + 7 * q^45 - 8 * q^47 + 6 * q^49 - 7 * q^51 - 5 * q^53 - 7 * q^55 + 18 * q^57 - 9 * q^59 - 3 * q^61 + 5 * q^63 + q^65 - 27 * q^67 + 5 * q^69 + 2 * q^71 - 2 * q^73 - 5 * q^75 - 14 * q^77 - 4 * q^79 + 14 * q^81 - 23 * q^83 - 12 * q^87 - 4 * q^89 + q^91 - 4 * q^93 - 5 * q^95 - 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
$3$	−0.602935	−0.348105	−0.174052	−	0.984736i	$-0.555686\pi$
$3$	−0.602935	−0.348105	−0.174052	+	0.984736i	$0.555686\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.26364	−1.61150	−0.805752	−	0.592253i	$-0.798238\pi$
$7$	−4.26364	−1.61150	−0.805752	+	0.592253i	$0.798238\pi$
$8$	0	0
$9$	−2.63647	−0.878823
$10$	0	0
$11$	3.63990	1.09747	0.548735	−	0.835996i	$-0.315110\pi$
$11$	3.63990	1.09747	0.548735	+	0.835996i	$0.315110\pi$
$12$	0	0
$13$	4.55951	1.26458	0.632291	−	0.774731i	$-0.282115\pi$
$13$	4.55951	1.26458	0.632291	+	0.774731i	$0.282115\pi$
$14$	0	0
$15$	−0.602935	−0.155677
$16$	0	0
$17$	−1.85371	−0.449590	−0.224795	−	0.974406i	$-0.572171\pi$
$17$	−1.85371	−0.449590	−0.224795	+	0.974406i	$0.572171\pi$
$18$	0	0
$19$	−1.23973	−0.284412	−0.142206	−	0.989837i	$-0.545420\pi$
$19$	−1.23973	−0.284412	−0.142206	+	0.989837i	$0.545420\pi$
$20$	0	0
$21$	2.57070	0.560972
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.39843	0.654027
$28$	0	0
$29$	5.38880	1.00068	0.500338	−	0.865830i	$-0.333209\pi$
$29$	5.38880	1.00068	0.500338	+	0.865830i	$0.333209\pi$
$30$	0	0
$31$	−4.49050	−0.806517	−0.403259	−	0.915086i	$-0.632122\pi$
$31$	−4.49050	−0.806517	−0.403259	+	0.915086i	$0.632122\pi$
$32$	0	0
$33$	−2.19462	−0.382035
$34$	0	0
$35$	−4.26364	−0.720686
$36$	0	0
$37$	−1.78619	−0.293648	−0.146824	−	0.989163i	$-0.546905\pi$
$37$	−1.78619	−0.293648	−0.146824	+	0.989163i	$0.546905\pi$
$38$	0	0
$39$	−2.74909	−0.440207
$40$	0	0
$41$	−2.90043	−0.452971	−0.226485	−	0.974015i	$-0.572724\pi$
$41$	−2.90043	−0.452971	−0.226485	+	0.974015i	$0.572724\pi$
$42$	0	0
$43$	0.649335	0.0990226	0.0495113	−	0.998774i	$-0.484234\pi$
$43$	0.649335	0.0990226	0.0495113	+	0.998774i	$0.484234\pi$
$44$	0	0
$45$	−2.63647	−0.393022
$46$	0	0
$47$	−5.27132	−0.768901	−0.384451	−	0.923146i	$-0.625609\pi$
$47$	−5.27132	−0.768901	−0.384451	+	0.923146i	$0.625609\pi$
$48$	0	0
$49$	11.1786	1.59694
$50$	0	0
$51$	1.11767	0.156504
$52$	0	0
$53$	4.45826	0.612389	0.306194	−	0.951969i	$-0.400944\pi$
$53$	4.45826	0.612389	0.306194	+	0.951969i	$0.400944\pi$
$54$	0	0
$55$	3.63990	0.490803
$56$	0	0
$57$	0.747474	0.0990053
$58$	0	0
$59$	−4.22978	−0.550671	−0.275335	−	0.961348i	$-0.588789\pi$
$59$	−4.22978	−0.550671	−0.275335	+	0.961348i	$0.588789\pi$
$60$	0	0
$61$	13.6015	1.74150	0.870748	−	0.491730i	$-0.163635\pi$
$61$	13.6015	1.74150	0.870748	+	0.491730i	$0.163635\pi$
$62$	0	0
$63$	11.2409	1.41623
$64$	0	0
$65$	4.55951	0.565538
$66$	0	0
$67$	−8.69262	−1.06197	−0.530986	−	0.847380i	$-0.678178\pi$
$67$	−8.69262	−1.06197	−0.530986	+	0.847380i	$0.678178\pi$
$68$	0	0
$69$	0.602935	0.0725849
$70$	0	0
$71$	−0.430737	−0.0511190	−0.0255595	−	0.999673i	$-0.508137\pi$
$71$	−0.430737	−0.0511190	−0.0255595	+	0.999673i	$0.508137\pi$
$72$	0	0
$73$	−12.8339	−1.50210	−0.751050	−	0.660246i	$-0.770452\pi$
$73$	−12.8339	−1.50210	−0.751050	+	0.660246i	$0.770452\pi$
$74$	0	0
$75$	−0.602935	−0.0696210
$76$	0	0
$77$	−15.5192	−1.76858
$78$	0	0
$79$	−11.7426	−1.32115	−0.660575	−	0.750760i	$-0.729687\pi$
$79$	−11.7426	−1.32115	−0.660575	+	0.750760i	$0.729687\pi$
$80$	0	0
$81$	5.86038	0.651153
$82$	0	0
$83$	1.77643	0.194988	0.0974942	−	0.995236i	$-0.468917\pi$
$83$	1.77643	0.194988	0.0974942	+	0.995236i	$0.468917\pi$
$84$	0	0
$85$	−1.85371	−0.201063
$86$	0	0
$87$	−3.24910	−0.348340
$88$	0	0
$89$	−9.83271	−1.04226	−0.521132	−	0.853476i	$-0.674490\pi$
$89$	−9.83271	−1.04226	−0.521132	+	0.853476i	$0.674490\pi$
$90$	0	0
$91$	−19.4401	−2.03788
$92$	0	0
$93$	2.70748	0.280752
$94$	0	0
$95$	−1.23973	−0.127193
$96$	0	0
$97$	1.29912	0.131906	0.0659528	−	0.997823i	$-0.478991\pi$
$97$	1.29912	0.131906	0.0659528	+	0.997823i	$0.478991\pi$
$98$	0	0
$99$	−9.59647	−0.964482
$100$	0	0
$101$	−4.07587	−0.405564	−0.202782	−	0.979224i	$-0.564998\pi$
$101$	−4.07587	−0.405564	−0.202782	+	0.979224i	$0.564998\pi$
$102$	0	0
$103$	0.492251	0.0485030	0.0242515	−	0.999706i	$-0.492280\pi$
$103$	0.492251	0.0485030	0.0242515	+	0.999706i	$0.492280\pi$
$104$	0	0
$105$	2.57070	0.250874
$106$	0	0
$107$	−15.4936	−1.49782	−0.748912	−	0.662670i	$-0.769423\pi$
$107$	−15.4936	−1.49782	−0.748912	+	0.662670i	$0.769423\pi$
$108$	0	0
$109$	0.397130	0.0380381	0.0190191	−	0.999819i	$-0.493946\pi$
$109$	0.397130	0.0380381	0.0190191	+	0.999819i	$0.493946\pi$
$110$	0	0
$111$	1.07696	0.102220
$112$	0	0
$113$	−0.288008	−0.0270935	−0.0135468	−	0.999908i	$-0.504312\pi$
$113$	−0.288008	−0.0270935	−0.0135468	+	0.999908i	$0.504312\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−12.0210	−1.11134
$118$	0	0
$119$	7.90353	0.724516
$120$	0	0
$121$	2.24884	0.204440
$122$	0	0
$123$	1.74877	0.157681
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.36489	−0.387321	−0.193661	−	0.981069i	$-0.562036\pi$
$127$	−4.36489	−0.387321	−0.193661	+	0.981069i	$0.562036\pi$
$128$	0	0
$129$	−0.391507	−0.0344703
$130$	0	0
$131$	−12.7148	−1.11090	−0.555448	−	0.831551i	$-0.687453\pi$
$131$	−12.7148	−1.11090	−0.555448	+	0.831551i	$0.687453\pi$
$132$	0	0
$133$	5.28574	0.458332
$134$	0	0
$135$	3.39843	0.292490
$136$	0	0
$137$	−14.8502	−1.26874	−0.634368	−	0.773031i	$-0.718739\pi$
$137$	−14.8502	−1.26874	−0.634368	+	0.773031i	$0.718739\pi$
$138$	0	0
$139$	20.0447	1.70017	0.850085	−	0.526645i	$-0.176550\pi$
$139$	20.0447	1.70017	0.850085	+	0.526645i	$0.176550\pi$
$140$	0	0
$141$	3.17827	0.267658
$142$	0	0
$143$	16.5962	1.38784
$144$	0	0
$145$	5.38880	0.447516
$146$	0	0
$147$	−6.73997	−0.555904
$148$	0	0
$149$	−5.95975	−0.488242	−0.244121	−	0.969745i	$-0.578499\pi$
$149$	−5.95975	−0.488242	−0.244121	+	0.969745i	$0.578499\pi$
$150$	0	0
$151$	0.548585	0.0446432	0.0223216	−	0.999751i	$-0.492894\pi$
$151$	0.548585	0.0446432	0.0223216	+	0.999751i	$0.492894\pi$
$152$	0	0
$153$	4.88724	0.395110
$154$	0	0
$155$	−4.49050	−0.360685
$156$	0	0
$157$	−20.0659	−1.60143	−0.800715	−	0.599046i	$-0.795547\pi$
$157$	−20.0659	−1.60143	−0.800715	+	0.599046i	$0.795547\pi$
$158$	0	0
$159$	−2.68804	−0.213176
$160$	0	0
$161$	4.26364	0.336022
$162$	0	0
$163$	−12.2282	−0.957788	−0.478894	−	0.877873i	$-0.658962\pi$
$163$	−12.2282	−0.957788	−0.478894	+	0.877873i	$0.658962\pi$
$164$	0	0
$165$	−2.19462	−0.170851
$166$	0	0
$167$	−9.75239	−0.754663	−0.377331	−	0.926078i	$-0.623158\pi$
$167$	−9.75239	−0.754663	−0.377331	+	0.926078i	$0.623158\pi$
$168$	0	0
$169$	7.78916	0.599166
$170$	0	0
$171$	3.26850	0.249948
$172$	0	0
$173$	−21.5206	−1.63618	−0.818089	−	0.575092i	$-0.804966\pi$
$173$	−21.5206	−1.63618	−0.818089	+	0.575092i	$0.804966\pi$
$174$	0	0
$175$	−4.26364	−0.322301
$176$	0	0
$177$	2.55028	0.191691
$178$	0	0
$179$	−5.19087	−0.387984	−0.193992	−	0.981003i	$-0.562144\pi$
$179$	−5.19087	−0.387984	−0.193992	+	0.981003i	$0.562144\pi$
$180$	0	0
$181$	13.5415	1.00653	0.503265	−	0.864132i	$-0.332132\pi$
$181$	13.5415	1.00653	0.503265	+	0.864132i	$0.332132\pi$
$182$	0	0
$183$	−8.20083	−0.606223
$184$	0	0
$185$	−1.78619	−0.131323
$186$	0	0
$187$	−6.74730	−0.493411
$188$	0	0
$189$	−14.4897	−1.05397
$190$	0	0
$191$	13.6811	0.989927	0.494963	−	0.868914i	$-0.335181\pi$
$191$	13.6811	0.989927	0.494963	+	0.868914i	$0.335181\pi$
$192$	0	0
$193$	4.77913	0.344010	0.172005	−	0.985096i	$-0.444976\pi$
$193$	4.77913	0.344010	0.172005	+	0.985096i	$0.444976\pi$
$194$	0	0
$195$	−2.74909	−0.196866
$196$	0	0
$197$	−12.1724	−0.867248	−0.433624	−	0.901094i	$-0.642765\pi$
$197$	−12.1724	−0.867248	−0.433624	+	0.901094i	$0.642765\pi$
$198$	0	0
$199$	−15.0712	−1.06837	−0.534183	−	0.845369i	$-0.679381\pi$
$199$	−15.0712	−1.06837	−0.534183	+	0.845369i	$0.679381\pi$
$200$	0	0
$201$	5.24109	0.369678
$202$	0	0
$203$	−22.9759	−1.61259
$204$	0	0
$205$	−2.90043	−0.202575
$206$	0	0
$207$	2.63647	0.183247
$208$	0	0
$209$	−4.51247	−0.312134
$210$	0	0
$211$	−26.1273	−1.79868	−0.899339	−	0.437253i	$-0.855951\pi$
$211$	−26.1273	−1.79868	−0.899339	+	0.437253i	$0.855951\pi$
$212$	0	0
$213$	0.259706	0.0177948
$214$	0	0
$215$	0.649335	0.0442843
$216$	0	0
$217$	19.1459	1.29970
$218$	0	0
$219$	7.73803	0.522888
$220$	0	0
$221$	−8.45200	−0.568543
$222$	0	0
$223$	24.1628	1.61806	0.809032	−	0.587765i	$-0.199992\pi$
$223$	24.1628	1.61806	0.809032	+	0.587765i	$0.199992\pi$
$224$	0	0
$225$	−2.63647	−0.175765
$226$	0	0
$227$	17.3473	1.15138	0.575691	−	0.817667i	$-0.304733\pi$
$227$	17.3473	1.15138	0.575691	+	0.817667i	$0.304733\pi$
$228$	0	0
$229$	7.10305	0.469383	0.234692	−	0.972070i	$-0.424592\pi$
$229$	7.10305	0.469383	0.234692	+	0.972070i	$0.424592\pi$
$230$	0	0
$231$	9.35707	0.615650
$232$	0	0
$233$	−24.0724	−1.57704	−0.788518	−	0.615012i	$-0.789151\pi$
$233$	−24.0724	−1.57704	−0.788518	+	0.615012i	$0.789151\pi$
$234$	0	0
$235$	−5.27132	−0.343863
$236$	0	0
$237$	7.08004	0.459898
$238$	0	0
$239$	17.6977	1.14477	0.572385	−	0.819985i	$-0.306018\pi$
$239$	17.6977	1.14477	0.572385	+	0.819985i	$0.306018\pi$
$240$	0	0
$241$	26.8297	1.72825	0.864126	−	0.503276i	$-0.167872\pi$
$241$	26.8297	1.72825	0.864126	+	0.503276i	$0.167872\pi$
$242$	0	0
$243$	−13.7287	−0.880697
$244$	0	0
$245$	11.1786	0.714175
$246$	0	0
$247$	−5.65254	−0.359663
$248$	0	0
$249$	−1.07107	−0.0678764
$250$	0	0
$251$	3.95975	0.249937	0.124969	−	0.992161i	$-0.460117\pi$
$251$	3.95975	0.249937	0.124969	+	0.992161i	$0.460117\pi$
$252$	0	0
$253$	−3.63990	−0.228838
$254$	0	0
$255$	1.11767	0.0699909
$256$	0	0
$257$	11.7963	0.735835	0.367917	−	0.929858i	$-0.380071\pi$
$257$	11.7963	0.735835	0.367917	+	0.929858i	$0.380071\pi$
$258$	0	0
$259$	7.61566	0.473214
$260$	0	0
$261$	−14.2074	−0.879417
$262$	0	0
$263$	−6.34551	−0.391281	−0.195640	−	0.980676i	$-0.562679\pi$
$263$	−6.34551	−0.391281	−0.195640	+	0.980676i	$0.562679\pi$
$264$	0	0
$265$	4.45826	0.273869
$266$	0	0
$267$	5.92849	0.362817
$268$	0	0
$269$	24.9417	1.52072	0.760360	−	0.649502i	$-0.225023\pi$
$269$	24.9417	1.52072	0.760360	+	0.649502i	$0.225023\pi$
$270$	0	0
$271$	−1.25103	−0.0759945	−0.0379973	−	0.999278i	$-0.512098\pi$
$271$	−1.25103	−0.0759945	−0.0379973	+	0.999278i	$0.512098\pi$
$272$	0	0
$273$	11.7211	0.709395
$274$	0	0
$275$	3.63990	0.219494
$276$	0	0
$277$	9.68726	0.582051	0.291026	−	0.956715i	$-0.406003\pi$
$277$	9.68726	0.582051	0.291026	+	0.956715i	$0.406003\pi$
$278$	0	0
$279$	11.8391	0.708786
$280$	0	0
$281$	−25.3407	−1.51170	−0.755850	−	0.654744i	$-0.772776\pi$
$281$	−25.3407	−1.51170	−0.755850	+	0.654744i	$0.772776\pi$
$282$	0	0
$283$	−10.2484	−0.609203	−0.304602	−	0.952480i	$-0.598523\pi$
$283$	−10.2484	−0.609203	−0.304602	+	0.952480i	$0.598523\pi$
$284$	0	0
$285$	0.747474	0.0442765
$286$	0	0
$287$	12.3664	0.729964
$288$	0	0
$289$	−13.5638	−0.797869
$290$	0	0
$291$	−0.783285	−0.0459170
$292$	0	0
$293$	21.4420	1.25265	0.626327	−	0.779561i	$-0.284558\pi$
$293$	21.4420	1.25265	0.626327	+	0.779561i	$0.284558\pi$
$294$	0	0
$295$	−4.22978	−0.246267
$296$	0	0
$297$	12.3699	0.717775
$298$	0	0
$299$	−4.55951	−0.263683
$300$	0	0
$301$	−2.76853	−0.159575
$302$	0	0
$303$	2.45749	0.141179
$304$	0	0
$305$	13.6015	0.778821
$306$	0	0
$307$	−33.8194	−1.93017	−0.965086	−	0.261932i	$-0.915640\pi$
$307$	−33.8194	−1.93017	−0.965086	+	0.261932i	$0.915640\pi$
$308$	0	0
$309$	−0.296796	−0.0168841
$310$	0	0
$311$	−28.5617	−1.61958	−0.809792	−	0.586718i	$-0.800420\pi$
$311$	−28.5617	−1.61958	−0.809792	+	0.586718i	$0.800420\pi$
$312$	0	0
$313$	−5.61379	−0.317310	−0.158655	−	0.987334i	$-0.550716\pi$
$313$	−5.61379	−0.317310	−0.158655	+	0.987334i	$0.550716\pi$
$314$	0	0
$315$	11.2409	0.633356
$316$	0	0
$317$	19.4931	1.09484	0.547421	−	0.836857i	$-0.315610\pi$
$317$	19.4931	1.09484	0.547421	+	0.836857i	$0.315610\pi$
$318$	0	0
$319$	19.6147	1.09821
$320$	0	0
$321$	9.34164	0.521399
$322$	0	0
$323$	2.29809	0.127869
$324$	0	0
$325$	4.55951	0.252916
$326$	0	0
$327$	−0.239444	−0.0132413
$328$	0	0
$329$	22.4750	1.23909
$330$	0	0
$331$	21.7710	1.19664	0.598321	−	0.801256i	$-0.295835\pi$
$331$	21.7710	1.19664	0.598321	+	0.801256i	$0.295835\pi$
$332$	0	0
$333$	4.70923	0.258064
$334$	0	0
$335$	−8.69262	−0.474929
$336$	0	0
$337$	14.2835	0.778074	0.389037	−	0.921222i	$-0.372808\pi$
$337$	14.2835	0.778074	0.389037	+	0.921222i	$0.372808\pi$
$338$	0	0
$339$	0.173650	0.00943138
$340$	0	0
$341$	−16.3449	−0.885128
$342$	0	0
$343$	−17.8160	−0.961976
$344$	0	0
$345$	0.602935	0.0324609
$346$	0	0
$347$	−24.2248	−1.30045	−0.650226	−	0.759740i	$-0.725326\pi$
$347$	−24.2248	−1.30045	−0.650226	+	0.759740i	$0.725326\pi$
$348$	0	0
$349$	−12.7265	−0.681232	−0.340616	−	0.940203i	$-0.610636\pi$
$349$	−12.7265	−0.681232	−0.340616	+	0.940203i	$0.610636\pi$
$350$	0	0
$351$	15.4952	0.827071
$352$	0	0
$353$	−26.6419	−1.41801	−0.709003	−	0.705205i	$-0.750855\pi$
$353$	−26.6419	−1.41801	−0.709003	+	0.705205i	$0.750855\pi$
$354$	0	0
$355$	−0.430737	−0.0228611
$356$	0	0
$357$	−4.76532	−0.252207
$358$	0	0
$359$	17.4725	0.922165	0.461083	−	0.887357i	$-0.347461\pi$
$359$	17.4725	0.922165	0.461083	+	0.887357i	$0.347461\pi$
$360$	0	0
$361$	−17.4631	−0.919110
$362$	0	0
$363$	−1.35591	−0.0711666
$364$	0	0
$365$	−12.8339	−0.671759
$366$	0	0
$367$	−32.1078	−1.67601	−0.838007	−	0.545660i	$-0.816279\pi$
$367$	−32.1078	−1.67601	−0.838007	+	0.545660i	$0.816279\pi$
$368$	0	0
$369$	7.64689	0.398081
$370$	0	0
$371$	−19.0084	−0.986867
$372$	0	0
$373$	−0.273085	−0.0141398	−0.00706991	−	0.999975i	$-0.502250\pi$
$373$	−0.273085	−0.0141398	−0.00706991	+	0.999975i	$0.502250\pi$
$374$	0	0
$375$	−0.602935	−0.0311354
$376$	0	0
$377$	24.5703	1.26544
$378$	0	0
$379$	25.1573	1.29224	0.646121	−	0.763235i	$-0.276390\pi$
$379$	25.1573	1.29224	0.646121	+	0.763235i	$0.276390\pi$
$380$	0	0
$381$	2.63175	0.134828
$382$	0	0
$383$	−23.5544	−1.20357	−0.601786	−	0.798657i	$-0.705544\pi$
$383$	−23.5544	−1.20357	−0.601786	+	0.798657i	$0.705544\pi$
$384$	0	0
$385$	−15.5192	−0.790931
$386$	0	0
$387$	−1.71195	−0.0870234
$388$	0	0
$389$	4.09913	0.207834	0.103917	−	0.994586i	$-0.466862\pi$
$389$	4.09913	0.207834	0.103917	+	0.994586i	$0.466862\pi$
$390$	0	0
$391$	1.85371	0.0937460
$392$	0	0
$393$	7.66619	0.386708
$394$	0	0
$395$	−11.7426	−0.590836
$396$	0	0
$397$	23.3298	1.17089	0.585444	−	0.810713i	$-0.300920\pi$
$397$	23.3298	1.17089	0.585444	+	0.810713i	$0.300920\pi$
$398$	0	0
$399$	−3.18696	−0.159547
$400$	0	0
$401$	−0.820503	−0.0409740	−0.0204870	−	0.999790i	$-0.506522\pi$
$401$	−0.820503	−0.0409740	−0.0204870	+	0.999790i	$0.506522\pi$
$402$	0	0
$403$	−20.4745	−1.01991
$404$	0	0
$405$	5.86038	0.291204
$406$	0	0
$407$	−6.50154	−0.322270
$408$	0	0
$409$	−33.5791	−1.66038	−0.830190	−	0.557481i	$-0.811768\pi$
$409$	−33.5791	−1.66038	−0.830190	+	0.557481i	$0.811768\pi$
$410$	0	0
$411$	8.95369	0.441653
$412$	0	0
$413$	18.0343	0.887408
$414$	0	0
$415$	1.77643	0.0872015
$416$	0	0
$417$	−12.0857	−0.591838
$418$	0	0
$419$	23.5624	1.15110	0.575548	−	0.817768i	$-0.304789\pi$
$419$	23.5624	1.15110	0.575548	+	0.817768i	$0.304789\pi$
$420$	0	0
$421$	0.354746	0.0172893	0.00864463	−	0.999963i	$-0.497248\pi$
$421$	0.354746	0.0172893	0.00864463	+	0.999963i	$0.497248\pi$
$422$	0	0
$423$	13.8977	0.675728
$424$	0	0
$425$	−1.85371	−0.0899180
$426$	0	0
$427$	−57.9919	−2.80643
$428$	0	0
$429$	−10.0064	−0.483114
$430$	0	0
$431$	8.67271	0.417750	0.208875	−	0.977942i	$-0.433020\pi$
$431$	8.67271	0.417750	0.208875	+	0.977942i	$0.433020\pi$
$432$	0	0
$433$	−40.3628	−1.93971	−0.969857	−	0.243675i	$-0.921647\pi$
$433$	−40.3628	−1.93971	−0.969857	+	0.243675i	$0.921647\pi$
$434$	0	0
$435$	−3.24910	−0.155782
$436$	0	0
$437$	1.23973	0.0593041
$438$	0	0
$439$	−8.63987	−0.412359	−0.206179	−	0.978514i	$-0.566103\pi$
$439$	−8.63987	−0.412359	−0.206179	+	0.978514i	$0.566103\pi$
$440$	0	0
$441$	−29.4720	−1.40343
$442$	0	0
$443$	−1.56964	−0.0745757	−0.0372879	−	0.999305i	$-0.511872\pi$
$443$	−1.56964	−0.0745757	−0.0372879	+	0.999305i	$0.511872\pi$
$444$	0	0
$445$	−9.83271	−0.466115
$446$	0	0
$447$	3.59334	0.169959
$448$	0	0
$449$	−33.4502	−1.57861	−0.789306	−	0.614000i	$-0.789559\pi$
$449$	−33.4502	−1.57861	−0.789306	+	0.614000i	$0.789559\pi$
$450$	0	0
$451$	−10.5573	−0.497122
$452$	0	0
$453$	−0.330761	−0.0155405
$454$	0	0
$455$	−19.4401	−0.911366
$456$	0	0
$457$	−13.4322	−0.628330	−0.314165	−	0.949368i	$-0.601725\pi$
$457$	−13.4322	−0.628330	−0.314165	+	0.949368i	$0.601725\pi$
$458$	0	0
$459$	−6.29968	−0.294044
$460$	0	0
$461$	−22.2689	−1.03717	−0.518583	−	0.855027i	$-0.673540\pi$
$461$	−22.2689	−1.03717	−0.518583	+	0.855027i	$0.673540\pi$
$462$	0	0
$463$	38.7201	1.79948	0.899738	−	0.436431i	$-0.143758\pi$
$463$	38.7201	1.79948	0.899738	+	0.436431i	$0.143758\pi$
$464$	0	0
$465$	2.70748	0.125556
$466$	0	0
$467$	1.55132	0.0717864	0.0358932	−	0.999356i	$-0.488572\pi$
$467$	1.55132	0.0717864	0.0358932	+	0.999356i	$0.488572\pi$
$468$	0	0
$469$	37.0622	1.71137
$470$	0	0
$471$	12.0984	0.557465
$472$	0	0
$473$	2.36351	0.108674
$474$	0	0
$475$	−1.23973	−0.0568825
$476$	0	0
$477$	−11.7541	−0.538182
$478$	0	0
$479$	12.5953	0.575492	0.287746	−	0.957707i	$-0.407094\pi$
$479$	12.5953	0.575492	0.287746	+	0.957707i	$0.407094\pi$
$480$	0	0
$481$	−8.14415	−0.371341
$482$	0	0
$483$	−2.57070	−0.116971
$484$	0	0
$485$	1.29912	0.0589900
$486$	0	0
$487$	17.2243	0.780507	0.390253	−	0.920708i	$-0.372387\pi$
$487$	17.2243	0.780507	0.390253	+	0.920708i	$0.372387\pi$
$488$	0	0
$489$	7.37282	0.333411
$490$	0	0
$491$	−19.0641	−0.860351	−0.430176	−	0.902745i	$-0.641548\pi$
$491$	−19.0641	−0.860351	−0.430176	+	0.902745i	$0.641548\pi$
$492$	0	0
$493$	−9.98926	−0.449894
$494$	0	0
$495$	−9.59647	−0.431329
$496$	0	0
$497$	1.83650	0.0823785
$498$	0	0
$499$	−3.60752	−0.161495	−0.0807474	−	0.996735i	$-0.525731\pi$
$499$	−3.60752	−0.161495	−0.0807474	+	0.996735i	$0.525731\pi$
$500$	0	0
$501$	5.88006	0.262702
$502$	0	0
$503$	16.2314	0.723721	0.361861	−	0.932232i	$-0.382142\pi$
$503$	16.2314	0.723721	0.361861	+	0.932232i	$0.382142\pi$
$504$	0	0
$505$	−4.07587	−0.181374
$506$	0	0
$507$	−4.69636	−0.208573
$508$	0	0
$509$	42.6827	1.89188	0.945939	−	0.324345i	$-0.105144\pi$
$509$	42.6827	1.89188	0.945939	+	0.324345i	$0.105144\pi$
$510$	0	0
$511$	54.7193	2.42064
$512$	0	0
$513$	−4.21311	−0.186014
$514$	0	0
$515$	0.492251	0.0216912
$516$	0	0
$517$	−19.1871	−0.843846
$518$	0	0
$519$	12.9755	0.569561
$520$	0	0
$521$	−20.5335	−0.899588	−0.449794	−	0.893132i	$-0.648503\pi$
$521$	−20.5335	−0.899588	−0.449794	+	0.893132i	$0.648503\pi$
$522$	0	0
$523$	−5.54661	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$523$	−5.54661	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$524$	0	0
$525$	2.57070	0.112194
$526$	0	0
$527$	8.32406	0.362602
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	11.1517	0.483942
$532$	0	0
$533$	−13.2245	−0.572818
$534$	0	0
$535$	−15.4936	−0.669847
$536$	0	0
$537$	3.12976	0.135059
$538$	0	0
$539$	40.6890	1.75260
$540$	0	0
$541$	−6.16238	−0.264942	−0.132471	−	0.991187i	$-0.542291\pi$
$541$	−6.16238	−0.264942	−0.132471	+	0.991187i	$0.542291\pi$
$542$	0	0
$543$	−8.16464	−0.350378
$544$	0	0
$545$	0.397130	0.0170112
$546$	0	0
$547$	23.5084	1.00515	0.502574	−	0.864534i	$-0.332386\pi$
$547$	23.5084	1.00515	0.502574	+	0.864534i	$0.332386\pi$
$548$	0	0
$549$	−35.8600	−1.53047
$550$	0	0
$551$	−6.68064	−0.284605
$552$	0	0
$553$	50.0663	2.12904
$554$	0	0
$555$	1.07696	0.0457143
$556$	0	0
$557$	1.79332	0.0759853	0.0379927	−	0.999278i	$-0.487904\pi$
$557$	1.79332	0.0759853	0.0379927	+	0.999278i	$0.487904\pi$
$558$	0	0
$559$	2.96065	0.125222
$560$	0	0
$561$	4.06818	0.171759
$562$	0	0
$563$	16.1239	0.679543	0.339771	−	0.940508i	$-0.389650\pi$
$563$	16.1239	0.679543	0.339771	+	0.940508i	$0.389650\pi$
$564$	0	0
$565$	−0.288008	−0.0121166
$566$	0	0
$567$	−24.9865	−1.04934
$568$	0	0
$569$	12.0589	0.505536	0.252768	−	0.967527i	$-0.418659\pi$
$569$	12.0589	0.505536	0.252768	+	0.967527i	$0.418659\pi$
$570$	0	0
$571$	23.9032	1.00032	0.500159	−	0.865934i	$-0.333275\pi$
$571$	23.9032	1.00032	0.500159	+	0.865934i	$0.333275\pi$
$572$	0	0
$573$	−8.24879	−0.344598
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	6.96576	0.289988	0.144994	−	0.989433i	$-0.453684\pi$
$577$	6.96576	0.289988	0.144994	+	0.989433i	$0.453684\pi$
$578$	0	0
$579$	−2.88151	−0.119751
$580$	0	0
$581$	−7.57405	−0.314225
$582$	0	0
$583$	16.2276	0.672078
$584$	0	0
$585$	−12.0210	−0.497008
$586$	0	0
$587$	3.60990	0.148996	0.0744982	−	0.997221i	$-0.476264\pi$
$587$	3.60990	0.148996	0.0744982	+	0.997221i	$0.476264\pi$
$588$	0	0
$589$	5.56698	0.229384
$590$	0	0
$591$	7.33917	0.301893
$592$	0	0
$593$	−7.74967	−0.318241	−0.159120	−	0.987259i	$-0.550866\pi$
$593$	−7.74967	−0.318241	−0.159120	+	0.987259i	$0.550866\pi$
$594$	0	0
$595$	7.90353	0.324013
$596$	0	0
$597$	9.08693	0.371903
$598$	0	0
$599$	−19.1300	−0.781631	−0.390816	−	0.920469i	$-0.627807\pi$
$599$	−19.1300	−0.781631	−0.390816	+	0.920469i	$0.627807\pi$
$600$	0	0
$601$	5.55815	0.226722	0.113361	−	0.993554i	$-0.463838\pi$
$601$	5.55815	0.226722	0.113361	+	0.993554i	$0.463838\pi$
$602$	0	0
$603$	22.9178	0.933286
$604$	0	0
$605$	2.24884	0.0914285
$606$	0	0
$607$	2.45607	0.0996890	0.0498445	−	0.998757i	$-0.484127\pi$
$607$	2.45607	0.0996890	0.0498445	+	0.998757i	$0.484127\pi$
$608$	0	0
$609$	13.8530	0.561351
$610$	0	0
$611$	−24.0347	−0.972338
$612$	0	0
$613$	−8.75548	−0.353630	−0.176815	−	0.984244i	$-0.556579\pi$
$613$	−8.75548	−0.353630	−0.176815	+	0.984244i	$0.556579\pi$
$614$	0	0
$615$	1.74877	0.0705172
$616$	0	0
$617$	−23.4835	−0.945411	−0.472706	−	0.881220i	$-0.656723\pi$
$617$	−23.4835	−0.945411	−0.472706	+	0.881220i	$0.656723\pi$
$618$	0	0
$619$	20.3874	0.819438	0.409719	−	0.912212i	$-0.365627\pi$
$619$	20.3874	0.819438	0.409719	+	0.912212i	$0.365627\pi$
$620$	0	0
$621$	−3.39843	−0.136374
$622$	0	0
$623$	41.9231	1.67961
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.72073	0.108655
$628$	0	0
$629$	3.31107	0.132021
$630$	0	0
$631$	39.7165	1.58109	0.790545	−	0.612403i	$-0.209797\pi$
$631$	39.7165	1.58109	0.790545	+	0.612403i	$0.209797\pi$
$632$	0	0
$633$	15.7531	0.626128
$634$	0	0
$635$	−4.36489	−0.173215
$636$	0	0
$637$	50.9690	2.01946
$638$	0	0
$639$	1.13562	0.0449246
$640$	0	0
$641$	3.49864	0.138188	0.0690940	−	0.997610i	$-0.477989\pi$
$641$	3.49864	0.138188	0.0690940	+	0.997610i	$0.477989\pi$
$642$	0	0
$643$	33.9716	1.33971	0.669855	−	0.742492i	$-0.266356\pi$
$643$	33.9716	1.33971	0.669855	+	0.742492i	$0.266356\pi$
$644$	0	0
$645$	−0.391507	−0.0154156
$646$	0	0
$647$	21.0113	0.826039	0.413020	−	0.910722i	$-0.364474\pi$
$647$	21.0113	0.826039	0.413020	+	0.910722i	$0.364474\pi$
$648$	0	0
$649$	−15.3960	−0.604345
$650$	0	0
$651$	−11.5437	−0.452434
$652$	0	0
$653$	6.94516	0.271785	0.135892	−	0.990724i	$-0.456610\pi$
$653$	6.94516	0.271785	0.135892	+	0.990724i	$0.456610\pi$
$654$	0	0
$655$	−12.7148	−0.496808
$656$	0	0
$657$	33.8363	1.32008
$658$	0	0
$659$	−10.5141	−0.409572	−0.204786	−	0.978807i	$-0.565650\pi$
$659$	−10.5141	−0.409572	−0.204786	+	0.978807i	$0.565650\pi$
$660$	0	0
$661$	−33.7748	−1.31369	−0.656843	−	0.754027i	$-0.728109\pi$
$661$	−33.7748	−1.31369	−0.656843	+	0.754027i	$0.728109\pi$
$662$	0	0
$663$	5.09601	0.197913
$664$	0	0
$665$	5.28574	0.204972
$666$	0	0
$667$	−5.38880	−0.208655
$668$	0	0
$669$	−14.5686	−0.563256
$670$	0	0
$671$	49.5081	1.91124
$672$	0	0
$673$	−17.3866	−0.670203	−0.335102	−	0.942182i	$-0.608771\pi$
$673$	−17.3866	−0.670203	−0.335102	+	0.942182i	$0.608771\pi$
$674$	0	0
$675$	3.39843	0.130805
$676$	0	0
$677$	3.84504	0.147777	0.0738884	−	0.997267i	$-0.476459\pi$
$677$	3.84504	0.147777	0.0738884	+	0.997267i	$0.476459\pi$
$678$	0	0
$679$	−5.53897	−0.212566
$680$	0	0
$681$	−10.4593	−0.400802
$682$	0	0
$683$	24.0383	0.919799	0.459900	−	0.887971i	$-0.347885\pi$
$683$	24.0383	0.919799	0.459900	+	0.887971i	$0.347885\pi$
$684$	0	0
$685$	−14.8502	−0.567396
$686$	0	0
$687$	−4.28268	−0.163394
$688$	0	0
$689$	20.3275	0.774416
$690$	0	0
$691$	27.2386	1.03621	0.518103	−	0.855319i	$-0.326639\pi$
$691$	27.2386	1.03621	0.518103	+	0.855319i	$0.326639\pi$
$692$	0	0
$693$	40.9159	1.55427
$694$	0	0
$695$	20.0447	0.760339
$696$	0	0
$697$	5.37654	0.203651
$698$	0	0
$699$	14.5141	0.548973
$700$	0	0
$701$	21.1776	0.799867	0.399934	−	0.916544i	$-0.369033\pi$
$701$	21.1776	0.799867	0.399934	+	0.916544i	$0.369033\pi$
$702$	0	0
$703$	2.21438	0.0835171
$704$	0	0
$705$	3.17827	0.119700
$706$	0	0
$707$	17.3780	0.653568
$708$	0	0
$709$	33.7589	1.26784	0.633922	−	0.773397i	$-0.281444\pi$
$709$	33.7589	1.26784	0.633922	+	0.773397i	$0.281444\pi$
$710$	0	0
$711$	30.9591	1.16106
$712$	0	0
$713$	4.49050	0.168170
$714$	0	0
$715$	16.5962	0.620661
$716$	0	0
$717$	−10.6706	−0.398500
$718$	0	0
$719$	14.3506	0.535186	0.267593	−	0.963532i	$-0.413772\pi$
$719$	14.3506	0.535186	0.267593	+	0.963532i	$0.413772\pi$
$720$	0	0
$721$	−2.09878	−0.0781627
$722$	0	0
$723$	−16.1766	−0.601612
$724$	0	0
$725$	5.38880	0.200135
$726$	0	0
$727$	−23.1234	−0.857600	−0.428800	−	0.903399i	$-0.641063\pi$
$727$	−23.1234	−0.857600	−0.428800	+	0.903399i	$0.641063\pi$
$728$	0	0
$729$	−9.30361	−0.344578
$730$	0	0
$731$	−1.20368	−0.0445196
$732$	0	0
$733$	−9.30758	−0.343783	−0.171892	−	0.985116i	$-0.554988\pi$
$733$	−9.30758	−0.343783	−0.171892	+	0.985116i	$0.554988\pi$
$734$	0	0
$735$	−6.73997	−0.248608
$736$	0	0
$737$	−31.6402	−1.16548
$738$	0	0
$739$	29.8573	1.09832	0.549160	−	0.835717i	$-0.314948\pi$
$739$	29.8573	1.09832	0.549160	+	0.835717i	$0.314948\pi$
$740$	0	0
$741$	3.40812	0.125200
$742$	0	0
$743$	35.3524	1.29695	0.648476	−	0.761235i	$-0.275407\pi$
$743$	35.3524	1.29695	0.648476	+	0.761235i	$0.275407\pi$
$744$	0	0
$745$	−5.95975	−0.218348
$746$	0	0
$747$	−4.68350	−0.171360
$748$	0	0
$749$	66.0591	2.41375
$750$	0	0
$751$	−49.2370	−1.79668	−0.898342	−	0.439298i	$-0.855227\pi$
$751$	−49.2370	−1.79668	−0.898342	+	0.439298i	$0.855227\pi$
$752$	0	0
$753$	−2.38747	−0.0870043
$754$	0	0
$755$	0.548585	0.0199650
$756$	0	0
$757$	−12.1108	−0.440175	−0.220087	−	0.975480i	$-0.570634\pi$
$757$	−12.1108	−0.440175	−0.220087	+	0.975480i	$0.570634\pi$
$758$	0	0
$759$	2.19462	0.0796597
$760$	0	0
$761$	47.4310	1.71937	0.859686	−	0.510822i	$-0.170659\pi$
$761$	47.4310	1.71937	0.859686	+	0.510822i	$0.170659\pi$
$762$	0	0
$763$	−1.69322	−0.0612986
$764$	0	0
$765$	4.88724	0.176699
$766$	0	0
$767$	−19.2857	−0.696368
$768$	0	0
$769$	5.63608	0.203242	0.101621	−	0.994823i	$-0.467597\pi$
$769$	5.63608	0.203242	0.101621	+	0.994823i	$0.467597\pi$
$770$	0	0
$771$	−7.11242	−0.256148
$772$	0	0
$773$	20.4841	0.736760	0.368380	−	0.929675i	$-0.379913\pi$
$773$	20.4841	0.736760	0.368380	+	0.929675i	$0.379913\pi$
$774$	0	0
$775$	−4.49050	−0.161303
$776$	0	0
$777$	−4.59175	−0.164728
$778$	0	0
$779$	3.59573	0.128831
$780$	0	0
$781$	−1.56784	−0.0561016
$782$	0	0
$783$	18.3134	0.654469
$784$	0	0
$785$	−20.0659	−0.716181
$786$	0	0
$787$	−43.2633	−1.54217	−0.771084	−	0.636733i	$-0.780285\pi$
$787$	−43.2633	−1.54217	−0.771084	+	0.636733i	$0.780285\pi$
$788$	0	0
$789$	3.82593	0.136207
$790$	0	0
$791$	1.22796	0.0436613
$792$	0	0
$793$	62.0163	2.20226
$794$	0	0
$795$	−2.68804	−0.0953350
$796$	0	0
$797$	27.9666	0.990628	0.495314	−	0.868714i	$-0.335053\pi$
$797$	27.9666	0.990628	0.495314	+	0.868714i	$0.335053\pi$
$798$	0	0
$799$	9.77148	0.345690
$800$	0	0
$801$	25.9236	0.915966
$802$	0	0
$803$	−46.7142	−1.64851
$804$	0	0
$805$	4.26364	0.150273
$806$	0	0
$807$	−15.0382	−0.529370
$808$	0	0
$809$	31.5540	1.10938	0.554691	−	0.832057i	$-0.312837\pi$
$809$	31.5540	1.10938	0.554691	+	0.832057i	$0.312837\pi$
$810$	0	0
$811$	8.46732	0.297328	0.148664	−	0.988888i	$-0.452503\pi$
$811$	8.46732	0.297328	0.148664	+	0.988888i	$0.452503\pi$
$812$	0	0
$813$	0.754289	0.0264541
$814$	0	0
$815$	−12.2282	−0.428336
$816$	0	0
$817$	−0.804997	−0.0281633
$818$	0	0
$819$	51.2532	1.79093
$820$	0	0
$821$	0.659828	0.0230281	0.0115141	−	0.999934i	$-0.496335\pi$
$821$	0.659828	0.0230281	0.0115141	+	0.999934i	$0.496335\pi$
$822$	0	0
$823$	−42.9987	−1.49884	−0.749419	−	0.662096i	$-0.769667\pi$
$823$	−42.9987	−1.49884	−0.749419	+	0.662096i	$0.769667\pi$
$824$	0	0
$825$	−2.19462	−0.0764069
$826$	0	0
$827$	−9.52132	−0.331089	−0.165544	−	0.986202i	$-0.552938\pi$
$827$	−9.52132	−0.331089	−0.165544	+	0.986202i	$0.552938\pi$
$828$	0	0
$829$	−46.3024	−1.60815	−0.804074	−	0.594529i	$-0.797339\pi$
$829$	−46.3024	−1.60815	−0.804074	+	0.594529i	$0.797339\pi$
$830$	0	0
$831$	−5.84079	−0.202615
$832$	0	0
$833$	−20.7219	−0.717970
$834$	0	0
$835$	−9.75239	−0.337495
$836$	0	0
$837$	−15.2606	−0.527484
$838$	0	0
$839$	−1.55100	−0.0535465	−0.0267732	−	0.999642i	$-0.508523\pi$
$839$	−1.55100	−0.0535465	−0.0267732	+	0.999642i	$0.508523\pi$
$840$	0	0
$841$	0.0392028	0.00135182
$842$	0	0
$843$	15.2788	0.526230
$844$	0	0
$845$	7.78916	0.267955
$846$	0	0
$847$	−9.58825	−0.329456
$848$	0	0
$849$	6.17911	0.212067
$850$	0	0
$851$	1.78619	0.0612298
$852$	0	0
$853$	7.61471	0.260723	0.130361	−	0.991467i	$-0.458386\pi$
$853$	7.61471	0.260723	0.130361	+	0.991467i	$0.458386\pi$
$854$	0	0
$855$	3.26850	0.111780
$856$	0	0
$857$	13.0077	0.444336	0.222168	−	0.975008i	$-0.428687\pi$
$857$	13.0077	0.444336	0.222168	+	0.975008i	$0.428687\pi$
$858$	0	0
$859$	14.1260	0.481973	0.240987	−	0.970528i	$-0.422529\pi$
$859$	14.1260	0.481973	0.240987	+	0.970528i	$0.422529\pi$
$860$	0	0
$861$	−7.45612	−0.254104
$862$	0	0
$863$	49.0516	1.66974	0.834868	−	0.550450i	$-0.185544\pi$
$863$	49.0516	1.66974	0.834868	+	0.550450i	$0.185544\pi$
$864$	0	0
$865$	−21.5206	−0.731721
$866$	0	0
$867$	8.17807	0.277742
$868$	0	0
$869$	−42.7419	−1.44992
$870$	0	0
$871$	−39.6341	−1.34295
$872$	0	0
$873$	−3.42509	−0.115922
$874$	0	0
$875$	−4.26364	−0.144137
$876$	0	0
$877$	−2.02340	−0.0683255	−0.0341628	−	0.999416i	$-0.510876\pi$
$877$	−2.02340	−0.0683255	−0.0341628	+	0.999416i	$0.510876\pi$
$878$	0	0
$879$	−12.9281	−0.436055
$880$	0	0
$881$	−2.62923	−0.0885810	−0.0442905	−	0.999019i	$-0.514103\pi$
$881$	−2.62923	−0.0885810	−0.0442905	+	0.999019i	$0.514103\pi$
$882$	0	0
$883$	54.1687	1.82292	0.911462	−	0.411385i	$-0.134955\pi$
$883$	54.1687	1.82292	0.911462	+	0.411385i	$0.134955\pi$
$884$	0	0
$885$	2.55028	0.0857269
$886$	0	0
$887$	49.2234	1.65276	0.826380	−	0.563112i	$-0.190396\pi$
$887$	49.2234	1.65276	0.826380	+	0.563112i	$0.190396\pi$
$888$	0	0
$889$	18.6103	0.624170
$890$	0	0
$891$	21.3312	0.714621
$892$	0	0
$893$	6.53499	0.218685
$894$	0	0
$895$	−5.19087	−0.173512
$896$	0	0
$897$	2.74909	0.0917895
$898$	0	0
$899$	−24.1984	−0.807062
$900$	0	0
$901$	−8.26430	−0.275324
$902$	0	0
$903$	1.66924	0.0555489
$904$	0	0
$905$	13.5415	0.450134
$906$	0	0
$907$	−5.30775	−0.176241	−0.0881204	−	0.996110i	$-0.528086\pi$
$907$	−5.30775	−0.176241	−0.0881204	+	0.996110i	$0.528086\pi$
$908$	0	0
$909$	10.7459	0.356419
$910$	0	0
$911$	−4.02578	−0.133380	−0.0666901	−	0.997774i	$-0.521244\pi$
$911$	−4.02578	−0.133380	−0.0666901	+	0.997774i	$0.521244\pi$
$912$	0	0
$913$	6.46602	0.213994
$914$	0	0
$915$	−8.20083	−0.271111
$916$	0	0
$917$	54.2112	1.79021
$918$	0	0
$919$	−44.4178	−1.46521	−0.732604	−	0.680655i	$-0.761695\pi$
$919$	−44.4178	−1.46521	−0.732604	+	0.680655i	$0.761695\pi$
$920$	0	0
$921$	20.3909	0.671902
$922$	0	0
$923$	−1.96395	−0.0646442
$924$	0	0
$925$	−1.78619	−0.0587295
$926$	0	0
$927$	−1.29781	−0.0426255
$928$	0	0
$929$	0.622945	0.0204382	0.0102191	−	0.999948i	$-0.496747\pi$
$929$	0.622945	0.0204382	0.0102191	+	0.999948i	$0.496747\pi$
$930$	0	0
$931$	−13.8584	−0.454191
$932$	0	0
$933$	17.2208	0.563785
$934$	0	0
$935$	−6.74730	−0.220660
$936$	0	0
$937$	−42.5551	−1.39021	−0.695107	−	0.718906i	$-0.744643\pi$
$937$	−42.5551	−1.39021	−0.695107	+	0.718906i	$0.744643\pi$
$938$	0	0
$939$	3.38475	0.110457
$940$	0	0
$941$	4.54974	0.148317	0.0741587	−	0.997246i	$-0.476373\pi$
$941$	4.54974	0.148317	0.0741587	+	0.997246i	$0.476373\pi$
$942$	0	0
$943$	2.90043	0.0944509
$944$	0	0
$945$	−14.4897	−0.471348
$946$	0	0
$947$	−7.65595	−0.248785	−0.124392	−	0.992233i	$-0.539698\pi$
$947$	−7.65595	−0.248785	−0.124392	+	0.992233i	$0.539698\pi$
$948$	0	0
$949$	−58.5165	−1.89953
$950$	0	0
$951$	−11.7531	−0.381120
$952$	0	0
$953$	29.2923	0.948872	0.474436	−	0.880290i	$-0.342652\pi$
$953$	29.2923	0.948872	0.474436	+	0.880290i	$0.342652\pi$
$954$	0	0
$955$	13.6811	0.442709
$956$	0	0
$957$	−11.8264	−0.382293
$958$	0	0
$959$	63.3157	2.04457
$960$	0	0
$961$	−10.8354	−0.349530
$962$	0	0
$963$	40.8484	1.31632
$964$	0	0
$965$	4.77913	0.153846
$966$	0	0
$967$	−20.0342	−0.644256	−0.322128	−	0.946696i	$-0.604398\pi$
$967$	−20.0342	−0.644256	−0.322128	+	0.946696i	$0.604398\pi$
$968$	0	0
$969$	−1.38560	−0.0445118
$970$	0	0
$971$	29.6018	0.949968	0.474984	−	0.879994i	$-0.342454\pi$
$971$	29.6018	0.949968	0.474984	+	0.879994i	$0.342454\pi$
$972$	0	0
$973$	−85.4634	−2.73983
$974$	0	0
$975$	−2.74909	−0.0880414
$976$	0	0
$977$	2.22512	0.0711878	0.0355939	−	0.999366i	$-0.488668\pi$
$977$	2.22512	0.0711878	0.0355939	+	0.999366i	$0.488668\pi$
$978$	0	0
$979$	−35.7900	−1.14385
$980$	0	0
$981$	−1.04702	−0.0334288
$982$	0	0
$983$	−8.94663	−0.285353	−0.142677	−	0.989769i	$-0.545571\pi$
$983$	−8.94663	−0.285353	−0.142677	+	0.989769i	$0.545571\pi$
$984$	0	0
$985$	−12.1724	−0.387845
$986$	0	0
$987$	−13.5510	−0.431332
$988$	0	0
$989$	−0.649335	−0.0206476
$990$	0	0
$991$	−38.6304	−1.22714	−0.613568	−	0.789642i	$-0.710266\pi$
$991$	−38.6304	−1.22714	−0.613568	+	0.789642i	$0.710266\pi$
$992$	0	0
$993$	−13.1265	−0.416557
$994$	0	0
$995$	−15.0712	−0.477788
$996$	0	0
$997$	−40.9937	−1.29828	−0.649141	−	0.760668i	$-0.724872\pi$
$997$	−40.9937	−1.29828	−0.649141	+	0.760668i	$0.724872\pi$
$998$	0	0
$999$	−6.07023	−0.192054

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3680.2.a.bc.1.4	✓	6
4.3	odd	2		3680.2.a.bd.1.3	yes	6
8.3	odd	2		7360.2.a.cu.1.4		6
8.5	even	2		7360.2.a.cv.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.a.bc.1.4	✓	6	1.1	even	1	trivial
3680.2.a.bd.1.3	yes	6	4.3	odd	2
7360.2.a.cu.1.4		6	8.3	odd	2
7360.2.a.cv.1.3		6	8.5	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:	\( N \)	\(=\)	\( 3680 = 2^{5} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3680.a (trivial)

\(f(q)\)	\(=\)	\(q-0.602935 q^{3} +1.00000 q^{5} -4.26364 q^{7} -2.63647 q^{9} +O(q^{10})\)	\(q-0.602935 q^{3} +1.00000 q^{5} -4.26364 q^{7} -2.63647 q^{9} +3.63990 q^{11} +4.55951 q^{13} -0.602935 q^{15} -1.85371 q^{17} -1.23973 q^{19} +2.57070 q^{21} -1.00000 q^{23} +1.00000 q^{25} +3.39843 q^{27} +5.38880 q^{29} -4.49050 q^{31} -2.19462 q^{33} -4.26364 q^{35} -1.78619 q^{37} -2.74909 q^{39} -2.90043 q^{41} +0.649335 q^{43} -2.63647 q^{45} -5.27132 q^{47} +11.1786 q^{49} +1.11767 q^{51} +4.45826 q^{53} +3.63990 q^{55} +0.747474 q^{57} -4.22978 q^{59} +13.6015 q^{61} +11.2409 q^{63} +4.55951 q^{65} -8.69262 q^{67} +0.602935 q^{69} -0.430737 q^{71} -12.8339 q^{73} -0.602935 q^{75} -15.5192 q^{77} -11.7426 q^{79} +5.86038 q^{81} +1.77643 q^{83} -1.85371 q^{85} -3.24910 q^{87} -9.83271 q^{89} -19.4401 q^{91} +2.70748 q^{93} -1.23973 q^{95} +1.29912 q^{97} -9.59647 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 5 q^{3} + 6 q^{5} - 4 q^{7} + 7 q^{9}+O(q^{10}) \) 6 * q - 5 * q^3 + 6 * q^5 - 4 * q^7 + 7 * q^9	\( 6 q - 5 q^{3} + 6 q^{5} - 4 q^{7} + 7 q^{9} - 7 q^{11} + q^{13} - 5 q^{15} - 5 q^{19} - 6 q^{23} + 6 q^{25} - 20 q^{27} - 3 q^{29} - 12 q^{31} - 3 q^{33} - 4 q^{35} + 7 q^{37} - 8 q^{39} + 8 q^{41} - 28 q^{43} + 7 q^{45} - 8 q^{47} + 6 q^{49} - 7 q^{51} - 5 q^{53} - 7 q^{55} + 18 q^{57} - 9 q^{59} - 3 q^{61} + 5 q^{63} + q^{65} - 27 q^{67} + 5 q^{69} + 2 q^{71} - 2 q^{73} - 5 q^{75} - 14 q^{77} - 4 q^{79} + 14 q^{81} - 23 q^{83} - 12 q^{87} - 4 q^{89} + q^{91} - 4 q^{93} - 5 q^{95} - q^{97} - q^{99}+O(q^{100}) \) 6 * q - 5 * q^3 + 6 * q^5 - 4 * q^7 + 7 * q^9 - 7 * q^11 + q^13 - 5 * q^15 - 5 * q^19 - 6 * q^23 + 6 * q^25 - 20 * q^27 - 3 * q^29 - 12 * q^31 - 3 * q^33 - 4 * q^35 + 7 * q^37 - 8 * q^39 + 8 * q^41 - 28 * q^43 + 7 * q^45 - 8 * q^47 + 6 * q^49 - 7 * q^51 - 5 * q^53 - 7 * q^55 + 18 * q^57 - 9 * q^59 - 3 * q^61 + 5 * q^63 + q^65 - 27 * q^67 + 5 * q^69 + 2 * q^71 - 2 * q^73 - 5 * q^75 - 14 * q^77 - 4 * q^79 + 14 * q^81 - 23 * q^83 - 12 * q^87 - 4 * q^89 + q^91 - 4 * q^93 - 5 * q^95 - q^97 - q^99

Introduction

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