LMFDB - Embedded newform 3640.2.a.w.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3640,2,Mod(1,3640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, 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("3640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3640.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:	$29.0655463357$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.23724.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + 4x + 10 $ x^4 - x^3 - 7x^2 + 4x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.70103$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.10649 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.77568 q^{9} +O(q^{10})$	$q+1.10649 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.77568 q^{9} +3.40207 q^{11} +1.00000 q^{13} -1.10649 q^{15} -2.77568 q^{17} +6.39073 q^{19} +1.10649 q^{21} +7.61504 q^{23} +1.00000 q^{25} -5.28424 q^{27} -0.508553 q^{29} -0.893512 q^{31} +3.76434 q^{33} -1.00000 q^{35} -3.25579 q^{37} +1.10649 q^{39} -7.79279 q^{41} -1.61504 q^{43} +1.77568 q^{45} +5.55137 q^{47} +1.00000 q^{49} -3.07126 q^{51} +2.21298 q^{53} -3.40207 q^{55} +7.07126 q^{57} +6.65786 q^{59} -12.5685 q^{61} -1.77568 q^{63} -1.00000 q^{65} +4.98866 q^{67} +8.42595 q^{69} +5.82802 q^{71} +7.01711 q^{73} +1.10649 q^{75} +3.40207 q^{77} +7.20164 q^{79} -0.519893 q^{81} +2.21298 q^{83} +2.77568 q^{85} -0.562708 q^{87} +10.2729 q^{89} +1.00000 q^{91} -0.988660 q^{93} -6.39073 q^{95} +5.33839 q^{97} -6.04099 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} - 4 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) $ 4 * q + q^3 - 4 * q^5 + 4 * q^7 + 3 * q^9	$ 4 q + q^{3} - 4 q^{5} + 4 q^{7} + 3 q^{9} + 2 q^{11} + 4 q^{13} - q^{15} - q^{17} - 3 q^{19} + q^{21} + 12 q^{23} + 4 q^{25} + 4 q^{27} + 13 q^{29} - 7 q^{31} - 12 q^{33} - 4 q^{35} - q^{37} + q^{39} + 9 q^{41} + 12 q^{43} - 3 q^{45} + 2 q^{47} + 4 q^{49} + 6 q^{51} + 2 q^{53} - 2 q^{55} + 10 q^{57} + 3 q^{59} + 3 q^{63} - 4 q^{65} + 3 q^{67} + 20 q^{69} - 2 q^{71} - 2 q^{73} + q^{75} + 2 q^{77} + 5 q^{79} - 4 q^{81} + 2 q^{83} + q^{85} + q^{87} - q^{89} + 4 q^{91} + 13 q^{93} + 3 q^{95} + 8 q^{97} + 8 q^{99}+O(q^{100}) $ 4 * q + q^3 - 4 * q^5 + 4 * q^7 + 3 * q^9 + 2 * q^11 + 4 * q^13 - q^15 - q^17 - 3 * q^19 + q^21 + 12 * q^23 + 4 * q^25 + 4 * q^27 + 13 * q^29 - 7 * q^31 - 12 * q^33 - 4 * q^35 - q^37 + q^39 + 9 * q^41 + 12 * q^43 - 3 * q^45 + 2 * q^47 + 4 * q^49 + 6 * q^51 + 2 * q^53 - 2 * q^55 + 10 * q^57 + 3 * q^59 + 3 * q^63 - 4 * q^65 + 3 * q^67 + 20 * q^69 - 2 * q^71 - 2 * q^73 + q^75 + 2 * q^77 + 5 * q^79 - 4 * q^81 + 2 * q^83 + q^85 + q^87 - q^89 + 4 * q^91 + 13 * q^93 + 3 * q^95 + 8 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.10649	0.638831	0.319416	−	0.947615i	$-0.396513\pi$
$3$	1.10649	0.638831	0.319416	+	0.947615i	$0.396513\pi$
$4$	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
$9$	−1.77568	−0.591895
$10$	0	0
$11$	3.40207	1.02576	0.512881	−	0.858460i	$-0.328578\pi$
$11$	3.40207	1.02576	0.512881	+	0.858460i	$0.328578\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.10649	−0.285694
$16$	0	0
$17$	−2.77568	−0.673202	−0.336601	−	0.941647i	$-0.609277\pi$
$17$	−2.77568	−0.673202	−0.336601	+	0.941647i	$0.609277\pi$
$18$	0	0
$19$	6.39073	1.46613	0.733066	−	0.680157i	$-0.238088\pi$
$19$	6.39073	1.46613	0.733066	+	0.680157i	$0.238088\pi$
$20$	0	0
$21$	1.10649	0.241455
$22$	0	0
$23$	7.61504	1.58785	0.793923	−	0.608018i	$-0.208035\pi$
$23$	7.61504	1.58785	0.793923	+	0.608018i	$0.208035\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.28424	−1.01695
$28$	0	0
$29$	−0.508553	−0.0944360	−0.0472180	−	0.998885i	$-0.515036\pi$
$29$	−0.508553	−0.0944360	−0.0472180	+	0.998885i	$0.515036\pi$
$30$	0	0
$31$	−0.893512	−0.160479	−0.0802397	−	0.996776i	$-0.525569\pi$
$31$	−0.893512	−0.160479	−0.0802397	+	0.996776i	$0.525569\pi$
$32$	0	0
$33$	3.76434	0.655288
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−3.25579	−0.535249	−0.267624	−	0.963523i	$-0.586239\pi$
$37$	−3.25579	−0.535249	−0.267624	+	0.963523i	$0.586239\pi$
$38$	0	0
$39$	1.10649	0.177180
$40$	0	0
$41$	−7.79279	−1.21703	−0.608515	−	0.793543i	$-0.708234\pi$
$41$	−7.79279	−1.21703	−0.608515	+	0.793543i	$0.708234\pi$
$42$	0	0
$43$	−1.61504	−0.246291	−0.123146	−	0.992389i	$-0.539298\pi$
$43$	−1.61504	−0.246291	−0.123146	+	0.992389i	$0.539298\pi$
$44$	0	0
$45$	1.77568	0.264703
$46$	0	0
$47$	5.55137	0.809750	0.404875	−	0.914372i	$-0.367315\pi$
$47$	5.55137	0.809750	0.404875	+	0.914372i	$0.367315\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.07126	−0.430063
$52$	0	0
$53$	2.21298	0.303976	0.151988	−	0.988382i	$-0.451433\pi$
$53$	2.21298	0.303976	0.151988	+	0.988382i	$0.451433\pi$
$54$	0	0
$55$	−3.40207	−0.458734
$56$	0	0
$57$	7.07126	0.936611
$58$	0	0
$59$	6.65786	0.866779	0.433390	−	0.901207i	$-0.357317\pi$
$59$	6.65786	0.866779	0.433390	+	0.901207i	$0.357317\pi$
$60$	0	0
$61$	−12.5685	−1.60923	−0.804614	−	0.593798i	$-0.797628\pi$
$61$	−12.5685	−1.60923	−0.804614	+	0.593798i	$0.797628\pi$
$62$	0	0
$63$	−1.77568	−0.223715
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	4.98866	0.609462	0.304731	−	0.952438i	$-0.401433\pi$
$67$	4.98866	0.609462	0.304731	+	0.952438i	$0.401433\pi$
$68$	0	0
$69$	8.42595	1.01437
$70$	0	0
$71$	5.82802	0.691658	0.345829	−	0.938297i	$-0.387598\pi$
$71$	5.82802	0.691658	0.345829	+	0.938297i	$0.387598\pi$
$72$	0	0
$73$	7.01711	0.821290	0.410645	−	0.911795i	$-0.365304\pi$
$73$	7.01711	0.821290	0.410645	+	0.911795i	$0.365304\pi$
$74$	0	0
$75$	1.10649	0.127766
$76$	0	0
$77$	3.40207	0.387701
$78$	0	0
$79$	7.20164	0.810247	0.405124	−	0.914262i	$-0.367228\pi$
$79$	7.20164	0.810247	0.405124	+	0.914262i	$0.367228\pi$
$80$	0	0
$81$	−0.519893	−0.0577659
$82$	0	0
$83$	2.21298	0.242906	0.121453	−	0.992597i	$-0.461245\pi$
$83$	2.21298	0.242906	0.121453	+	0.992597i	$0.461245\pi$
$84$	0	0
$85$	2.77568	0.301065
$86$	0	0
$87$	−0.562708	−0.0603286
$88$	0	0
$89$	10.2729	1.08892	0.544462	−	0.838785i	$-0.316734\pi$
$89$	10.2729	1.08892	0.544462	+	0.838785i	$0.316734\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−0.988660	−0.102519
$94$	0	0
$95$	−6.39073	−0.655675
$96$	0	0
$97$	5.33839	0.542032	0.271016	−	0.962575i	$-0.412640\pi$
$97$	5.33839	0.542032	0.271016	+	0.962575i	$0.412640\pi$
$98$	0	0
$99$	−6.04099	−0.607143
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	2.83936	0.279770	0.139885	−	0.990168i	$-0.455327\pi$
$103$	2.83936	0.279770	0.139885	+	0.990168i	$0.455327\pi$
$104$	0	0
$105$	−1.10649	−0.107982
$106$	0	0
$107$	6.38496	0.617257	0.308629	−	0.951183i	$-0.400130\pi$
$107$	6.38496	0.617257	0.308629	+	0.951183i	$0.400130\pi$
$108$	0	0
$109$	14.8041	1.41798	0.708989	−	0.705219i	$-0.249151\pi$
$109$	14.8041	1.41798	0.708989	+	0.705219i	$0.249151\pi$
$110$	0	0
$111$	−3.60249	−0.341934
$112$	0	0
$113$	3.12542	0.294014	0.147007	−	0.989135i	$-0.453036\pi$
$113$	3.12542	0.294014	0.147007	+	0.989135i	$0.453036\pi$
$114$	0	0
$115$	−7.61504	−0.710106
$116$	0	0
$117$	−1.77568	−0.164162
$118$	0	0
$119$	−2.77568	−0.254447
$120$	0	0
$121$	0.574048	0.0521862
$122$	0	0
$123$	−8.62263	−0.777476
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	15.9705	1.41716	0.708578	−	0.705632i	$-0.249337\pi$
$127$	15.9705	1.41716	0.708578	+	0.705632i	$0.249337\pi$
$128$	0	0
$129$	−1.78702	−0.157339
$130$	0	0
$131$	7.23008	0.631695	0.315848	−	0.948810i	$-0.397711\pi$
$131$	7.23008	0.631695	0.315848	+	0.948810i	$0.397711\pi$
$132$	0	0
$133$	6.39073	0.554146
$134$	0	0
$135$	5.28424	0.454795
$136$	0	0
$137$	2.82984	0.241769	0.120885	−	0.992667i	$-0.461427\pi$
$137$	2.82984	0.241769	0.120885	+	0.992667i	$0.461427\pi$
$138$	0	0
$139$	−7.76434	−0.658563	−0.329282	−	0.944232i	$-0.606807\pi$
$139$	−7.76434	−0.658563	−0.329282	+	0.944232i	$0.606807\pi$
$140$	0	0
$141$	6.14252	0.517294
$142$	0	0
$143$	3.40207	0.284495
$144$	0	0
$145$	0.508553	0.0422331
$146$	0	0
$147$	1.10649	0.0912616
$148$	0	0
$149$	1.19587	0.0979695	0.0489847	−	0.998800i	$-0.484401\pi$
$149$	1.19587	0.0979695	0.0489847	+	0.998800i	$0.484401\pi$
$150$	0	0
$151$	−15.3794	−1.25156	−0.625778	−	0.780001i	$-0.715219\pi$
$151$	−15.3794	−1.25156	−0.625778	+	0.780001i	$0.715219\pi$
$152$	0	0
$153$	4.92874	0.398465
$154$	0	0
$155$	0.893512	0.0717686
$156$	0	0
$157$	−8.69885	−0.694244	−0.347122	−	0.937820i	$-0.612841\pi$
$157$	−8.69885	−0.694244	−0.347122	+	0.937820i	$0.612841\pi$
$158$	0	0
$159$	2.44863	0.194189
$160$	0	0
$161$	7.61504	0.600149
$162$	0	0
$163$	−17.1853	−1.34606	−0.673030	−	0.739615i	$-0.735007\pi$
$163$	−17.1853	−1.34606	−0.673030	+	0.739615i	$0.735007\pi$
$164$	0	0
$165$	−3.76434	−0.293054
$166$	0	0
$167$	25.3726	1.96339	0.981696	−	0.190457i	$-0.0609969\pi$
$167$	25.3726	1.96339	0.981696	+	0.190457i	$0.0609969\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−11.3479	−0.867796
$172$	0	0
$173$	−13.7160	−1.04280	−0.521402	−	0.853311i	$-0.674591\pi$
$173$	−13.7160	−1.04280	−0.521402	+	0.853311i	$0.674591\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	7.36684	0.553725
$178$	0	0
$179$	12.3271	0.921367	0.460684	−	0.887564i	$-0.347604\pi$
$179$	12.3271	0.921367	0.460684	+	0.887564i	$0.347604\pi$
$180$	0	0
$181$	15.0171	1.11621	0.558106	−	0.829769i	$-0.311528\pi$
$181$	15.0171	1.11621	0.558106	+	0.829769i	$0.311528\pi$
$182$	0	0
$183$	−13.9069	−1.02803
$184$	0	0
$185$	3.25579	0.239371
$186$	0	0
$187$	−9.44306	−0.690545
$188$	0	0
$189$	−5.28424	−0.384372
$190$	0	0
$191$	23.5886	1.70681	0.853406	−	0.521247i	$-0.174533\pi$
$191$	23.5886	1.70681	0.853406	+	0.521247i	$0.174533\pi$
$192$	0	0
$193$	−23.7701	−1.71101	−0.855505	−	0.517795i	$-0.826753\pi$
$193$	−23.7701	−1.71101	−0.855505	+	0.517795i	$0.826753\pi$
$194$	0	0
$195$	−1.10649	−0.0792373
$196$	0	0
$197$	6.53123	0.465331	0.232666	−	0.972557i	$-0.425255\pi$
$197$	6.53123	0.465331	0.232666	+	0.972557i	$0.425255\pi$
$198$	0	0
$199$	−5.62182	−0.398520	−0.199260	−	0.979947i	$-0.563854\pi$
$199$	−5.62182	−0.398520	−0.199260	+	0.979947i	$0.563854\pi$
$200$	0	0
$201$	5.51989	0.389343
$202$	0	0
$203$	−0.508553	−0.0356934
$204$	0	0
$205$	7.79279	0.544272
$206$	0	0
$207$	−13.5219	−0.939838
$208$	0	0
$209$	21.7417	1.50390
$210$	0	0
$211$	5.57981	0.384130	0.192065	−	0.981382i	$-0.438482\pi$
$211$	5.57981	0.384130	0.192065	+	0.981382i	$0.438482\pi$
$212$	0	0
$213$	6.44863	0.441853
$214$	0	0
$215$	1.61504	0.110145
$216$	0	0
$217$	−0.893512	−0.0606555
$218$	0	0
$219$	7.76434	0.524666
$220$	0	0
$221$	−2.77568	−0.186713
$222$	0	0
$223$	18.7588	1.25618	0.628090	−	0.778141i	$-0.283837\pi$
$223$	18.7588	1.25618	0.628090	+	0.778141i	$0.283837\pi$
$224$	0	0
$225$	−1.77568	−0.118379
$226$	0	0
$227$	−27.7759	−1.84355	−0.921775	−	0.387725i	$-0.873261\pi$
$227$	−27.7759	−1.84355	−0.921775	+	0.387725i	$0.873261\pi$
$228$	0	0
$229$	28.7303	1.89855	0.949277	−	0.314442i	$-0.101817\pi$
$229$	28.7303	1.89855	0.949277	+	0.314442i	$0.101817\pi$
$230$	0	0
$231$	3.76434	0.247676
$232$	0	0
$233$	−7.01711	−0.459706	−0.229853	−	0.973225i	$-0.573825\pi$
$233$	−7.01711	−0.459706	−0.229853	+	0.973225i	$0.573825\pi$
$234$	0	0
$235$	−5.55137	−0.362131
$236$	0	0
$237$	7.96852	0.517611
$238$	0	0
$239$	−2.95343	−0.191042	−0.0955209	−	0.995427i	$-0.530452\pi$
$239$	−2.95343	−0.191042	−0.0955209	+	0.995427i	$0.530452\pi$
$240$	0	0
$241$	−21.7701	−1.40234	−0.701168	−	0.712996i	$-0.747338\pi$
$241$	−21.7701	−1.40234	−0.701168	+	0.712996i	$0.747338\pi$
$242$	0	0
$243$	15.2775	0.980049
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	6.39073	0.406632
$248$	0	0
$249$	2.44863	0.155176
$250$	0	0
$251$	−7.52869	−0.475207	−0.237603	−	0.971362i	$-0.576362\pi$
$251$	−7.52869	−0.475207	−0.237603	+	0.971362i	$0.576362\pi$
$252$	0	0
$253$	25.9069	1.62875
$254$	0	0
$255$	3.07126	0.192330
$256$	0	0
$257$	4.96021	0.309410	0.154705	−	0.987961i	$-0.450557\pi$
$257$	4.96021	0.309410	0.154705	+	0.987961i	$0.450557\pi$
$258$	0	0
$259$	−3.25579	−0.202305
$260$	0	0
$261$	0.903030	0.0558962
$262$	0	0
$263$	−3.61504	−0.222913	−0.111456	−	0.993769i	$-0.535552\pi$
$263$	−3.61504	−0.222913	−0.111456	+	0.993769i	$0.535552\pi$
$264$	0	0
$265$	−2.21298	−0.135942
$266$	0	0
$267$	11.3668	0.695639
$268$	0	0
$269$	−17.6939	−1.07882	−0.539408	−	0.842045i	$-0.681352\pi$
$269$	−17.6939	−1.07882	−0.539408	+	0.842045i	$0.681352\pi$
$270$	0	0
$271$	−14.2766	−0.867245	−0.433622	−	0.901095i	$-0.642765\pi$
$271$	−14.2766	−0.867245	−0.433622	+	0.901095i	$0.642765\pi$
$272$	0	0
$273$	1.10649	0.0669677
$274$	0	0
$275$	3.40207	0.205152
$276$	0	0
$277$	25.9069	1.55659	0.778296	−	0.627897i	$-0.216084\pi$
$277$	25.9069	1.55659	0.778296	+	0.627897i	$0.216084\pi$
$278$	0	0
$279$	1.58660	0.0949870
$280$	0	0
$281$	−3.38617	−0.202002	−0.101001	−	0.994886i	$-0.532204\pi$
$281$	−3.38617	−0.202002	−0.101001	+	0.994886i	$0.532204\pi$
$282$	0	0
$283$	−13.0750	−0.777229	−0.388614	−	0.921400i	$-0.627046\pi$
$283$	−13.0750	−0.777229	−0.388614	+	0.921400i	$0.627046\pi$
$284$	0	0
$285$	−7.07126	−0.418865
$286$	0	0
$287$	−7.79279	−0.459994
$288$	0	0
$289$	−9.29558	−0.546799
$290$	0	0
$291$	5.90687	0.346267
$292$	0	0
$293$	−1.89169	−0.110514	−0.0552569	−	0.998472i	$-0.517598\pi$
$293$	−1.89169	−0.110514	−0.0552569	+	0.998472i	$0.517598\pi$
$294$	0	0
$295$	−6.65786	−0.387635
$296$	0	0
$297$	−17.9773	−1.04315
$298$	0	0
$299$	7.61504	0.440389
$300$	0	0
$301$	−1.61504	−0.0930894
$302$	0	0
$303$	6.63893	0.381396
$304$	0	0
$305$	12.5685	0.719669
$306$	0	0
$307$	−6.87458	−0.392353	−0.196177	−	0.980569i	$-0.562853\pi$
$307$	−6.87458	−0.392353	−0.196177	+	0.980569i	$0.562853\pi$
$308$	0	0
$309$	3.14171	0.178726
$310$	0	0
$311$	−29.0801	−1.64898	−0.824489	−	0.565877i	$-0.808538\pi$
$311$	−29.0801	−1.64898	−0.824489	+	0.565877i	$0.808538\pi$
$312$	0	0
$313$	1.14748	0.0648595	0.0324297	−	0.999474i	$-0.489675\pi$
$313$	1.14748	0.0648595	0.0324297	+	0.999474i	$0.489675\pi$
$314$	0	0
$315$	1.77568	0.100048
$316$	0	0
$317$	−1.52869	−0.0858597	−0.0429299	−	0.999078i	$-0.513669\pi$
$317$	−1.52869	−0.0858597	−0.0429299	+	0.999078i	$0.513669\pi$
$318$	0	0
$319$	−1.73013	−0.0968688
$320$	0	0
$321$	7.06488	0.394323
$322$	0	0
$323$	−17.7386	−0.987004
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	16.3806	0.905849
$328$	0	0
$329$	5.55137	0.306057
$330$	0	0
$331$	−20.8451	−1.14575	−0.572876	−	0.819642i	$-0.694172\pi$
$331$	−20.8451	−1.14575	−0.572876	+	0.819642i	$0.694172\pi$
$332$	0	0
$333$	5.78126	0.316811
$334$	0	0
$335$	−4.98866	−0.272560
$336$	0	0
$337$	−24.4602	−1.33243	−0.666215	−	0.745760i	$-0.732087\pi$
$337$	−24.4602	−1.33243	−0.666215	+	0.745760i	$0.732087\pi$
$338$	0	0
$339$	3.45824	0.187826
$340$	0	0
$341$	−3.03979	−0.164614
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−8.42595	−0.453638
$346$	0	0
$347$	−11.4841	−0.616496	−0.308248	−	0.951306i	$-0.599743\pi$
$347$	−11.4841	−0.616496	−0.308248	+	0.951306i	$0.599743\pi$
$348$	0	0
$349$	12.3812	0.662751	0.331375	−	0.943499i	$-0.392487\pi$
$349$	12.3812	0.662751	0.331375	+	0.943499i	$0.392487\pi$
$350$	0	0
$351$	−5.28424	−0.282052
$352$	0	0
$353$	23.5629	1.25413	0.627063	−	0.778968i	$-0.284257\pi$
$353$	23.5629	1.25413	0.627063	+	0.778968i	$0.284257\pi$
$354$	0	0
$355$	−5.82802	−0.309319
$356$	0	0
$357$	−3.07126	−0.162548
$358$	0	0
$359$	−27.7349	−1.46379	−0.731896	−	0.681417i	$-0.761364\pi$
$359$	−27.7349	−1.46379	−0.731896	+	0.681417i	$0.761364\pi$
$360$	0	0
$361$	21.8414	1.14955
$362$	0	0
$363$	0.635177	0.0333381
$364$	0	0
$365$	−7.01711	−0.367292
$366$	0	0
$367$	−10.2766	−0.536437	−0.268218	−	0.963358i	$-0.586435\pi$
$367$	−10.2766	−0.536437	−0.268218	+	0.963358i	$0.586435\pi$
$368$	0	0
$369$	13.8375	0.720353
$370$	0	0
$371$	2.21298	0.114892
$372$	0	0
$373$	31.2930	1.62029	0.810146	−	0.586228i	$-0.199388\pi$
$373$	31.2930	1.62029	0.810146	+	0.586228i	$0.199388\pi$
$374$	0	0
$375$	−1.10649	−0.0571388
$376$	0	0
$377$	−0.508553	−0.0261918
$378$	0	0
$379$	4.41917	0.226998	0.113499	−	0.993538i	$-0.463794\pi$
$379$	4.41917	0.226998	0.113499	+	0.993538i	$0.463794\pi$
$380$	0	0
$381$	17.6712	0.905324
$382$	0	0
$383$	4.82681	0.246638	0.123319	−	0.992367i	$-0.460646\pi$
$383$	4.82681	0.246638	0.123319	+	0.992367i	$0.460646\pi$
$384$	0	0
$385$	−3.40207	−0.173385
$386$	0	0
$387$	2.86780	0.145779
$388$	0	0
$389$	5.67295	0.287630	0.143815	−	0.989605i	$-0.454063\pi$
$389$	5.67295	0.287630	0.143815	+	0.989605i	$0.454063\pi$
$390$	0	0
$391$	−21.1369	−1.06894
$392$	0	0
$393$	8.00000	0.403547
$394$	0	0
$395$	−7.20164	−0.362354
$396$	0	0
$397$	−32.6254	−1.63742	−0.818710	−	0.574207i	$-0.805310\pi$
$397$	−32.6254	−1.63742	−0.818710	+	0.574207i	$0.805310\pi$
$398$	0	0
$399$	7.07126	0.354006
$400$	0	0
$401$	−15.3862	−0.768348	−0.384174	−	0.923261i	$-0.625514\pi$
$401$	−15.3862	−0.768348	−0.384174	+	0.923261i	$0.625514\pi$
$402$	0	0
$403$	−0.893512	−0.0445090
$404$	0	0
$405$	0.519893	0.0258337
$406$	0	0
$407$	−11.0764	−0.549037
$408$	0	0
$409$	−22.5231	−1.11370	−0.556848	−	0.830614i	$-0.687989\pi$
$409$	−22.5231	−1.11370	−0.556848	+	0.830614i	$0.687989\pi$
$410$	0	0
$411$	3.13118	0.154450
$412$	0	0
$413$	6.65786	0.327612
$414$	0	0
$415$	−2.21298	−0.108631
$416$	0	0
$417$	−8.59115	−0.420711
$418$	0	0
$419$	8.55330	0.417856	0.208928	−	0.977931i	$-0.433003\pi$
$419$	8.55330	0.417856	0.208928	+	0.977931i	$0.433003\pi$
$420$	0	0
$421$	−35.4068	−1.72562	−0.862811	−	0.505526i	$-0.831298\pi$
$421$	−35.4068	−1.72562	−0.862811	+	0.505526i	$0.831298\pi$
$422$	0	0
$423$	−9.85748	−0.479287
$424$	0	0
$425$	−2.77568	−0.134640
$426$	0	0
$427$	−12.5685	−0.608231
$428$	0	0
$429$	3.76434	0.181744
$430$	0	0
$431$	−2.65483	−0.127879	−0.0639393	−	0.997954i	$-0.520366\pi$
$431$	−2.65483	−0.127879	−0.0639393	+	0.997954i	$0.520366\pi$
$432$	0	0
$433$	0.798364	0.0383669	0.0191835	−	0.999816i	$-0.493893\pi$
$433$	0.798364	0.0383669	0.0191835	+	0.999816i	$0.493893\pi$
$434$	0	0
$435$	0.562708	0.0269798
$436$	0	0
$437$	48.6656	2.32799
$438$	0	0
$439$	−30.9809	−1.47864	−0.739318	−	0.673356i	$-0.764852\pi$
$439$	−30.9809	−1.47864	−0.739318	+	0.673356i	$0.764852\pi$
$440$	0	0
$441$	−1.77568	−0.0845564
$442$	0	0
$443$	24.0183	1.14114	0.570572	−	0.821247i	$-0.306721\pi$
$443$	24.0183	1.14114	0.570572	+	0.821247i	$0.306721\pi$
$444$	0	0
$445$	−10.2729	−0.486982
$446$	0	0
$447$	1.32322	0.0625859
$448$	0	0
$449$	0.448632	0.0211722	0.0105861	−	0.999944i	$-0.496630\pi$
$449$	0.448632	0.0211722	0.0105861	+	0.999944i	$0.496630\pi$
$450$	0	0
$451$	−26.5116	−1.24838
$452$	0	0
$453$	−17.0171	−0.799533
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	8.86405	0.414643	0.207321	−	0.978273i	$-0.433525\pi$
$457$	8.86405	0.414643	0.207321	+	0.978273i	$0.433525\pi$
$458$	0	0
$459$	14.6674	0.684614
$460$	0	0
$461$	41.0408	1.91146	0.955730	−	0.294245i	$-0.0950682\pi$
$461$	41.0408	1.91146	0.955730	+	0.294245i	$0.0950682\pi$
$462$	0	0
$463$	22.4696	1.04425	0.522125	−	0.852869i	$-0.325139\pi$
$463$	22.4696	1.04425	0.522125	+	0.852869i	$0.325139\pi$
$464$	0	0
$465$	0.988660	0.0458480
$466$	0	0
$467$	5.30317	0.245401	0.122701	−	0.992444i	$-0.460845\pi$
$467$	5.30317	0.245401	0.122701	+	0.992444i	$0.460845\pi$
$468$	0	0
$469$	4.98866	0.230355
$470$	0	0
$471$	−9.62517	−0.443505
$472$	0	0
$473$	−5.49448	−0.252636
$474$	0	0
$475$	6.39073	0.293227
$476$	0	0
$477$	−3.92955	−0.179922
$478$	0	0
$479$	−18.1714	−0.830271	−0.415136	−	0.909760i	$-0.636266\pi$
$479$	−18.1714	−0.830271	−0.415136	+	0.909760i	$0.636266\pi$
$480$	0	0
$481$	−3.25579	−0.148451
$482$	0	0
$483$	8.42595	0.383394
$484$	0	0
$485$	−5.33839	−0.242404
$486$	0	0
$487$	40.5353	1.83683	0.918414	−	0.395621i	$-0.129470\pi$
$487$	40.5353	1.83683	0.918414	+	0.395621i	$0.129470\pi$
$488$	0	0
$489$	−19.0154	−0.859905
$490$	0	0
$491$	23.2165	1.04775	0.523873	−	0.851796i	$-0.324486\pi$
$491$	23.2165	1.04775	0.523873	+	0.851796i	$0.324486\pi$
$492$	0	0
$493$	1.41158	0.0635745
$494$	0	0
$495$	6.04099	0.271522
$496$	0	0
$497$	5.82802	0.261422
$498$	0	0
$499$	−1.61504	−0.0722992	−0.0361496	−	0.999346i	$-0.511509\pi$
$499$	−1.61504	−0.0722992	−0.0361496	+	0.999346i	$0.511509\pi$
$500$	0	0
$501$	28.0745	1.25428
$502$	0	0
$503$	16.3109	0.727265	0.363633	−	0.931542i	$-0.381536\pi$
$503$	16.3109	0.727265	0.363633	+	0.931542i	$0.381536\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	1.10649	0.0491409
$508$	0	0
$509$	24.5591	1.08856	0.544281	−	0.838903i	$-0.316803\pi$
$509$	24.5591	1.08856	0.544281	+	0.838903i	$0.316803\pi$
$510$	0	0
$511$	7.01711	0.310419
$512$	0	0
$513$	−33.7701	−1.49099
$514$	0	0
$515$	−2.83936	−0.125117
$516$	0	0
$517$	18.8861	0.830610
$518$	0	0
$519$	−15.1765	−0.666176
$520$	0	0
$521$	−27.3157	−1.19672	−0.598362	−	0.801226i	$-0.704181\pi$
$521$	−27.3157	−1.19672	−0.598362	+	0.801226i	$0.704181\pi$
$522$	0	0
$523$	−14.7178	−0.643564	−0.321782	−	0.946814i	$-0.604282\pi$
$523$	−14.7178	−0.643564	−0.321782	+	0.946814i	$0.604282\pi$
$524$	0	0
$525$	1.10649	0.0482911
$526$	0	0
$527$	2.48011	0.108035
$528$	0	0
$529$	34.9889	1.52125
$530$	0	0
$531$	−11.8223	−0.513042
$532$	0	0
$533$	−7.79279	−0.337543
$534$	0	0
$535$	−6.38496	−0.276046
$536$	0	0
$537$	13.6397	0.588598
$538$	0	0
$539$	3.40207	0.146537
$540$	0	0
$541$	−3.70139	−0.159135	−0.0795677	−	0.996829i	$-0.525354\pi$
$541$	−3.70139	−0.159135	−0.0795677	+	0.996829i	$0.525354\pi$
$542$	0	0
$543$	16.6162	0.713071
$544$	0	0
$545$	−14.8041	−0.634139
$546$	0	0
$547$	22.8109	0.975324	0.487662	−	0.873032i	$-0.337850\pi$
$547$	22.8109	0.975324	0.487662	+	0.873032i	$0.337850\pi$
$548$	0	0
$549$	22.3176	0.952494
$550$	0	0
$551$	−3.25002	−0.138456
$552$	0	0
$553$	7.20164	0.306245
$554$	0	0
$555$	3.60249	0.152917
$556$	0	0
$557$	−38.4469	−1.62905	−0.814524	−	0.580130i	$-0.803002\pi$
$557$	−38.4469	−1.62905	−0.814524	+	0.580130i	$0.803002\pi$
$558$	0	0
$559$	−1.61504	−0.0683090
$560$	0	0
$561$	−10.4486	−0.441142
$562$	0	0
$563$	15.1531	0.638629	0.319314	−	0.947649i	$-0.396547\pi$
$563$	15.1531	0.638629	0.319314	+	0.947649i	$0.396547\pi$
$564$	0	0
$565$	−3.12542	−0.131487
$566$	0	0
$567$	−0.519893	−0.0218335
$568$	0	0
$569$	−15.0066	−0.629108	−0.314554	−	0.949240i	$-0.601855\pi$
$569$	−15.0066	−0.629108	−0.314554	+	0.949240i	$0.601855\pi$
$570$	0	0
$571$	26.5579	1.11142	0.555708	−	0.831378i	$-0.312447\pi$
$571$	26.5579	1.11142	0.555708	+	0.831378i	$0.312447\pi$
$572$	0	0
$573$	26.1005	1.09036
$574$	0	0
$575$	7.61504	0.317569
$576$	0	0
$577$	−32.6254	−1.35821	−0.679106	−	0.734040i	$-0.737632\pi$
$577$	−32.6254	−1.35821	−0.679106	+	0.734040i	$0.737632\pi$
$578$	0	0
$579$	−26.3013	−1.09305
$580$	0	0
$581$	2.21298	0.0918097
$582$	0	0
$583$	7.52869	0.311806
$584$	0	0
$585$	1.77568	0.0734155
$586$	0	0
$587$	25.5250	1.05353	0.526766	−	0.850011i	$-0.323405\pi$
$587$	25.5250	1.05353	0.526766	+	0.850011i	$0.323405\pi$
$588$	0	0
$589$	−5.71019	−0.235284
$590$	0	0
$591$	7.22673	0.297268
$592$	0	0
$593$	−21.0323	−0.863692	−0.431846	−	0.901947i	$-0.642138\pi$
$593$	−21.0323	−0.863692	−0.431846	+	0.901947i	$0.642138\pi$
$594$	0	0
$595$	2.77568	0.113792
$596$	0	0
$597$	−6.22048	−0.254587
$598$	0	0
$599$	−23.3121	−0.952505	−0.476253	−	0.879308i	$-0.658005\pi$
$599$	−23.3121	−0.952505	−0.476253	+	0.879308i	$0.658005\pi$
$600$	0	0
$601$	2.31764	0.0945386	0.0472693	−	0.998882i	$-0.484948\pi$
$601$	2.31764	0.0945386	0.0472693	+	0.998882i	$0.484948\pi$
$602$	0	0
$603$	−8.85829	−0.360737
$604$	0	0
$605$	−0.574048	−0.0233384
$606$	0	0
$607$	−37.7494	−1.53220	−0.766101	−	0.642720i	$-0.777806\pi$
$607$	−37.7494	−1.53220	−0.766101	+	0.642720i	$0.777806\pi$
$608$	0	0
$609$	−0.562708	−0.0228021
$610$	0	0
$611$	5.55137	0.224584
$612$	0	0
$613$	19.6218	0.792518	0.396259	−	0.918139i	$-0.370308\pi$
$613$	19.6218	0.792518	0.396259	+	0.918139i	$0.370308\pi$
$614$	0	0
$615$	8.62263	0.347698
$616$	0	0
$617$	7.38952	0.297491	0.148745	−	0.988876i	$-0.452477\pi$
$617$	7.38952	0.297491	0.148745	+	0.988876i	$0.452477\pi$
$618$	0	0
$619$	−45.2514	−1.81881	−0.909404	−	0.415913i	$-0.863462\pi$
$619$	−45.2514	−1.81881	−0.909404	+	0.415913i	$0.863462\pi$
$620$	0	0
$621$	−40.2397	−1.61476
$622$	0	0
$623$	10.2729	0.411575
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	24.0569	0.960740
$628$	0	0
$629$	9.03705	0.360331
$630$	0	0
$631$	−44.7898	−1.78306	−0.891528	−	0.452966i	$-0.850366\pi$
$631$	−44.7898	−1.78306	−0.891528	+	0.452966i	$0.850366\pi$
$632$	0	0
$633$	6.17400	0.245394
$634$	0	0
$635$	−15.9705	−0.633772
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−10.3487	−0.409389
$640$	0	0
$641$	−9.22705	−0.364447	−0.182223	−	0.983257i	$-0.558329\pi$
$641$	−9.22705	−0.364447	−0.182223	+	0.983257i	$0.558329\pi$
$642$	0	0
$643$	−39.4068	−1.55405	−0.777027	−	0.629468i	$-0.783273\pi$
$643$	−39.4068	−1.55405	−0.777027	+	0.629468i	$0.783273\pi$
$644$	0	0
$645$	1.78702	0.0703640
$646$	0	0
$647$	0.0667021	0.00262233	0.00131116	−	0.999999i	$-0.499583\pi$
$647$	0.0667021	0.00262233	0.00131116	+	0.999999i	$0.499583\pi$
$648$	0	0
$649$	22.6505	0.889108
$650$	0	0
$651$	−0.988660	−0.0387487
$652$	0	0
$653$	31.1847	1.22035	0.610176	−	0.792266i	$-0.291099\pi$
$653$	31.1847	1.22035	0.610176	+	0.792266i	$0.291099\pi$
$654$	0	0
$655$	−7.23008	−0.282503
$656$	0	0
$657$	−12.4602	−0.486117
$658$	0	0
$659$	11.0743	0.431393	0.215697	−	0.976460i	$-0.430798\pi$
$659$	11.0743	0.431393	0.215697	+	0.976460i	$0.430798\pi$
$660$	0	0
$661$	−47.8331	−1.86049	−0.930245	−	0.366938i	$-0.880406\pi$
$661$	−47.8331	−1.86049	−0.930245	+	0.366938i	$0.880406\pi$
$662$	0	0
$663$	−3.07126	−0.119278
$664$	0	0
$665$	−6.39073	−0.247822
$666$	0	0
$667$	−3.87265	−0.149950
$668$	0	0
$669$	20.7564	0.802487
$670$	0	0
$671$	−42.7588	−1.65068
$672$	0	0
$673$	−30.5004	−1.17571	−0.587853	−	0.808968i	$-0.700027\pi$
$673$	−30.5004	−1.17571	−0.587853	+	0.808968i	$0.700027\pi$
$674$	0	0
$675$	−5.28424	−0.203390
$676$	0	0
$677$	11.6083	0.446142	0.223071	−	0.974802i	$-0.428392\pi$
$677$	11.6083	0.446142	0.223071	+	0.974802i	$0.428392\pi$
$678$	0	0
$679$	5.33839	0.204869
$680$	0	0
$681$	−30.7337	−1.17772
$682$	0	0
$683$	−22.3673	−0.855862	−0.427931	−	0.903811i	$-0.640757\pi$
$683$	−22.3673	−0.855862	−0.427931	+	0.903811i	$0.640757\pi$
$684$	0	0
$685$	−2.82984	−0.108123
$686$	0	0
$687$	31.7898	1.21286
$688$	0	0
$689$	2.21298	0.0843077
$690$	0	0
$691$	−39.9222	−1.51871	−0.759355	−	0.650676i	$-0.774485\pi$
$691$	−39.9222	−1.51871	−0.759355	+	0.650676i	$0.774485\pi$
$692$	0	0
$693$	−6.04099	−0.229478
$694$	0	0
$695$	7.76434	0.294518
$696$	0	0
$697$	21.6303	0.819307
$698$	0	0
$699$	−7.76434	−0.293674
$700$	0	0
$701$	12.3613	0.466879	0.233439	−	0.972371i	$-0.425002\pi$
$701$	12.3613	0.466879	0.233439	+	0.972371i	$0.425002\pi$
$702$	0	0
$703$	−20.8069	−0.784746
$704$	0	0
$705$	−6.14252	−0.231341
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	5.48092	0.205840	0.102920	−	0.994690i	$-0.467181\pi$
$709$	5.48092	0.205840	0.102920	+	0.994690i	$0.467181\pi$
$710$	0	0
$711$	−12.7878	−0.479581
$712$	0	0
$713$	−6.80413	−0.254817
$714$	0	0
$715$	−3.40207	−0.127230
$716$	0	0
$717$	−3.26794	−0.122043
$718$	0	0
$719$	0.471311	0.0175769	0.00878847	−	0.999961i	$-0.497203\pi$
$719$	0.471311	0.0175769	0.00878847	+	0.999961i	$0.497203\pi$
$720$	0	0
$721$	2.83936	0.105743
$722$	0	0
$723$	−24.0884	−0.895856
$724$	0	0
$725$	−0.508553	−0.0188872
$726$	0	0
$727$	7.11802	0.263993	0.131996	−	0.991250i	$-0.457861\pi$
$727$	7.11802	0.263993	0.131996	+	0.991250i	$0.457861\pi$
$728$	0	0
$729$	18.4640	0.683852
$730$	0	0
$731$	4.48284	0.165804
$732$	0	0
$733$	29.2795	1.08146	0.540731	−	0.841196i	$-0.318148\pi$
$733$	29.2795	1.08146	0.540731	+	0.841196i	$0.318148\pi$
$734$	0	0
$735$	−1.10649	−0.0408134
$736$	0	0
$737$	16.9717	0.625162
$738$	0	0
$739$	−40.2655	−1.48119	−0.740595	−	0.671951i	$-0.765456\pi$
$739$	−40.2655	−1.48119	−0.740595	+	0.671951i	$0.765456\pi$
$740$	0	0
$741$	7.07126	0.259769
$742$	0	0
$743$	−10.5400	−0.386676	−0.193338	−	0.981132i	$-0.561931\pi$
$743$	−10.5400	−0.386676	−0.193338	+	0.981132i	$0.561931\pi$
$744$	0	0
$745$	−1.19587	−0.0438133
$746$	0	0
$747$	−3.92955	−0.143775
$748$	0	0
$749$	6.38496	0.233301
$750$	0	0
$751$	8.02207	0.292729	0.146365	−	0.989231i	$-0.453243\pi$
$751$	8.02207	0.292729	0.146365	+	0.989231i	$0.453243\pi$
$752$	0	0
$753$	−8.33040	−0.303577
$754$	0	0
$755$	15.3794	0.559713
$756$	0	0
$757$	16.0705	0.584091	0.292045	−	0.956404i	$-0.405664\pi$
$757$	16.0705	0.584091	0.292045	+	0.956404i	$0.405664\pi$
$758$	0	0
$759$	28.6656	1.04050
$760$	0	0
$761$	16.3071	0.591132	0.295566	−	0.955322i	$-0.404492\pi$
$761$	16.3071	0.591132	0.295566	+	0.955322i	$0.404492\pi$
$762$	0	0
$763$	14.8041	0.535946
$764$	0	0
$765$	−4.92874	−0.178199
$766$	0	0
$767$	6.65786	0.240401
$768$	0	0
$769$	18.2381	0.657682	0.328841	−	0.944385i	$-0.393342\pi$
$769$	18.2381	0.657682	0.328841	+	0.944385i	$0.393342\pi$
$770$	0	0
$771$	5.48842	0.197661
$772$	0	0
$773$	−6.69946	−0.240963	−0.120481	−	0.992716i	$-0.538444\pi$
$773$	−6.69946	−0.240963	−0.120481	+	0.992716i	$0.538444\pi$
$774$	0	0
$775$	−0.893512	−0.0320959
$776$	0	0
$777$	−3.60249	−0.129239
$778$	0	0
$779$	−49.8016	−1.78433
$780$	0	0
$781$	19.8273	0.709476
$782$	0	0
$783$	2.68732	0.0960368
$784$	0	0
$785$	8.69885	0.310475
$786$	0	0
$787$	31.6425	1.12793	0.563966	−	0.825798i	$-0.309275\pi$
$787$	31.6425	1.12793	0.563966	+	0.825798i	$0.309275\pi$
$788$	0	0
$789$	−4.00000	−0.142404
$790$	0	0
$791$	3.12542	0.111127
$792$	0	0
$793$	−12.5685	−0.446320
$794$	0	0
$795$	−2.44863	−0.0868440
$796$	0	0
$797$	9.55198	0.338349	0.169174	−	0.985586i	$-0.445890\pi$
$797$	9.55198	0.338349	0.169174	+	0.985586i	$0.445890\pi$
$798$	0	0
$799$	−15.4088	−0.545126
$800$	0	0
$801$	−18.2414	−0.644529
$802$	0	0
$803$	23.8727	0.842448
$804$	0	0
$805$	−7.61504	−0.268395
$806$	0	0
$807$	−19.5781	−0.689181
$808$	0	0
$809$	3.39558	0.119382	0.0596911	−	0.998217i	$-0.480988\pi$
$809$	3.39558	0.119382	0.0596911	+	0.998217i	$0.480988\pi$
$810$	0	0
$811$	−18.8587	−0.662218	−0.331109	−	0.943593i	$-0.607423\pi$
$811$	−18.8587	−0.662218	−0.331109	+	0.943593i	$0.607423\pi$
$812$	0	0
$813$	−15.7969	−0.554023
$814$	0	0
$815$	17.1853	0.601976
$816$	0	0
$817$	−10.3213	−0.361096
$818$	0	0
$819$	−1.77568	−0.0620474
$820$	0	0
$821$	−37.7986	−1.31918	−0.659589	−	0.751626i	$-0.729270\pi$
$821$	−37.7986	−1.31918	−0.659589	+	0.751626i	$0.729270\pi$
$822$	0	0
$823$	31.2006	1.08759	0.543793	−	0.839220i	$-0.316988\pi$
$823$	31.2006	1.08759	0.543793	+	0.839220i	$0.316988\pi$
$824$	0	0
$825$	3.76434	0.131058
$826$	0	0
$827$	22.6314	0.786972	0.393486	−	0.919331i	$-0.371269\pi$
$827$	22.6314	0.786972	0.393486	+	0.919331i	$0.371269\pi$
$828$	0	0
$829$	38.1314	1.32436	0.662179	−	0.749346i	$-0.269632\pi$
$829$	38.1314	1.32436	0.662179	+	0.749346i	$0.269632\pi$
$830$	0	0
$831$	28.6656	0.994400
$832$	0	0
$833$	−2.77568	−0.0961718
$834$	0	0
$835$	−25.3726	−0.878055
$836$	0	0
$837$	4.72153	0.163200
$838$	0	0
$839$	−3.51788	−0.121451	−0.0607253	−	0.998155i	$-0.519341\pi$
$839$	−3.51788	−0.121451	−0.0607253	+	0.998155i	$0.519341\pi$
$840$	0	0
$841$	−28.7414	−0.991082
$842$	0	0
$843$	−3.74675	−0.129045
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	0.574048	0.0197245
$848$	0	0
$849$	−14.4673	−0.496518
$850$	0	0
$851$	−24.7930	−0.849893
$852$	0	0
$853$	5.56251	0.190457	0.0952284	−	0.995455i	$-0.469642\pi$
$853$	5.56251	0.190457	0.0952284	+	0.995455i	$0.469642\pi$
$854$	0	0
$855$	11.3479	0.388090
$856$	0	0
$857$	10.9270	0.373259	0.186630	−	0.982430i	$-0.440244\pi$
$857$	10.9270	0.373259	0.186630	+	0.982430i	$0.440244\pi$
$858$	0	0
$859$	29.9889	1.02321	0.511603	−	0.859222i	$-0.329052\pi$
$859$	29.9889	1.02321	0.511603	+	0.859222i	$0.329052\pi$
$860$	0	0
$861$	−8.62263	−0.293858
$862$	0	0
$863$	−0.861314	−0.0293195	−0.0146597	−	0.999893i	$-0.504667\pi$
$863$	−0.861314	−0.0293195	−0.0146597	+	0.999893i	$0.504667\pi$
$864$	0	0
$865$	13.7160	0.466357
$866$	0	0
$867$	−10.2854	−0.349312
$868$	0	0
$869$	24.5004	0.831120
$870$	0	0
$871$	4.98866	0.169034
$872$	0	0
$873$	−9.47930	−0.320826
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−21.9201	−0.740190	−0.370095	−	0.928994i	$-0.620675\pi$
$877$	−21.9201	−0.740190	−0.370095	+	0.928994i	$0.620675\pi$
$878$	0	0
$879$	−2.09313	−0.0705996
$880$	0	0
$881$	−9.34993	−0.315007	−0.157504	−	0.987518i	$-0.550345\pi$
$881$	−9.34993	−0.315007	−0.157504	+	0.987518i	$0.550345\pi$
$882$	0	0
$883$	3.57333	0.120252	0.0601260	−	0.998191i	$-0.480850\pi$
$883$	3.57333	0.120252	0.0601260	+	0.998191i	$0.480850\pi$
$884$	0	0
$885$	−7.36684	−0.247634
$886$	0	0
$887$	12.0985	0.406228	0.203114	−	0.979155i	$-0.434894\pi$
$887$	12.0985	0.406228	0.203114	+	0.979155i	$0.434894\pi$
$888$	0	0
$889$	15.9705	0.535635
$890$	0	0
$891$	−1.76871	−0.0592540
$892$	0	0
$893$	35.4773	1.18720
$894$	0	0
$895$	−12.3271	−0.412048
$896$	0	0
$897$	8.42595	0.281334
$898$	0	0
$899$	0.454398	0.0151550
$900$	0	0
$901$	−6.14252	−0.204637
$902$	0	0
$903$	−1.78702	−0.0594684
$904$	0	0
$905$	−15.0171	−0.499186
$906$	0	0
$907$	24.0637	0.799021	0.399511	−	0.916729i	$-0.369180\pi$
$907$	24.0637	0.799021	0.399511	+	0.916729i	$0.369180\pi$
$908$	0	0
$909$	−10.6541	−0.353374
$910$	0	0
$911$	27.5707	0.913458	0.456729	−	0.889606i	$-0.349021\pi$
$911$	27.5707	0.913458	0.456729	+	0.889606i	$0.349021\pi$
$912$	0	0
$913$	7.52869	0.249163
$914$	0	0
$915$	13.9069	0.459747
$916$	0	0
$917$	7.23008	0.238758
$918$	0	0
$919$	25.8394	0.852365	0.426182	−	0.904637i	$-0.359858\pi$
$919$	25.8394	0.852365	0.426182	+	0.904637i	$0.359858\pi$
$920$	0	0
$921$	−7.60664	−0.250647
$922$	0	0
$923$	5.82802	0.191832
$924$	0	0
$925$	−3.25579	−0.107050
$926$	0	0
$927$	−5.04180	−0.165594
$928$	0	0
$929$	28.0221	0.919374	0.459687	−	0.888081i	$-0.347962\pi$
$929$	28.0221	0.919374	0.459687	+	0.888081i	$0.347962\pi$
$930$	0	0
$931$	6.39073	0.209448
$932$	0	0
$933$	−32.1767	−1.05342
$934$	0	0
$935$	9.44306	0.308821
$936$	0	0
$937$	−16.6961	−0.545438	−0.272719	−	0.962094i	$-0.587923\pi$
$937$	−16.6961	−0.545438	−0.272719	+	0.962094i	$0.587923\pi$
$938$	0	0
$939$	1.26967	0.0414343
$940$	0	0
$941$	−37.0419	−1.20753	−0.603766	−	0.797162i	$-0.706334\pi$
$941$	−37.0419	−1.20753	−0.603766	+	0.797162i	$0.706334\pi$
$942$	0	0
$943$	−59.3424	−1.93246
$944$	0	0
$945$	5.28424	0.171896
$946$	0	0
$947$	33.1275	1.07650	0.538250	−	0.842785i	$-0.319086\pi$
$947$	33.1275	1.07650	0.538250	+	0.842785i	$0.319086\pi$
$948$	0	0
$949$	7.01711	0.227785
$950$	0	0
$951$	−1.69148	−0.0548499
$952$	0	0
$953$	−30.7497	−0.996079	−0.498039	−	0.867154i	$-0.665947\pi$
$953$	−30.7497	−0.996079	−0.498039	+	0.867154i	$0.665947\pi$
$954$	0	0
$955$	−23.5886	−0.763310
$956$	0	0
$957$	−1.91437	−0.0618828
$958$	0	0
$959$	2.82984	0.0913803
$960$	0	0
$961$	−30.2016	−0.974246
$962$	0	0
$963$	−11.3377	−0.365351
$964$	0	0
$965$	23.7701	0.765187
$966$	0	0
$967$	23.2331	0.747127	0.373563	−	0.927605i	$-0.378136\pi$
$967$	23.2331	0.747127	0.373563	+	0.927605i	$0.378136\pi$
$968$	0	0
$969$	−19.6276	−0.630529
$970$	0	0
$971$	−42.9430	−1.37811	−0.689053	−	0.724711i	$-0.741973\pi$
$971$	−42.9430	−1.37811	−0.689053	+	0.724711i	$0.741973\pi$
$972$	0	0
$973$	−7.76434	−0.248913
$974$	0	0
$975$	1.10649	0.0354360
$976$	0	0
$977$	−9.84968	−0.315119	−0.157560	−	0.987509i	$-0.550363\pi$
$977$	−9.84968	−0.315119	−0.157560	+	0.987509i	$0.550363\pi$
$978$	0	0
$979$	34.9491	1.11698
$980$	0	0
$981$	−26.2875	−0.839294
$982$	0	0
$983$	−2.52514	−0.0805396	−0.0402698	−	0.999189i	$-0.512822\pi$
$983$	−2.52514	−0.0805396	−0.0402698	+	0.999189i	$0.512822\pi$
$984$	0	0
$985$	−6.53123	−0.208102
$986$	0	0
$987$	6.14252	0.195519
$988$	0	0
$989$	−12.2986	−0.391073
$990$	0	0
$991$	13.9922	0.444477	0.222239	−	0.974992i	$-0.428664\pi$
$991$	13.9922	0.444477	0.222239	+	0.974992i	$0.428664\pi$
$992$	0	0
$993$	−23.0649	−0.731942
$994$	0	0
$995$	5.62182	0.178224
$996$	0	0
$997$	−57.6634	−1.82622	−0.913109	−	0.407715i	$-0.866326\pi$
$997$	−57.6634	−1.82622	−0.913109	+	0.407715i	$0.866326\pi$
$998$	0	0
$999$	17.2044	0.544322

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.w.1.3	✓	4
4.3	odd	2		7280.2.a.bs.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.w.1.3	✓	4	1.1	even	1	trivial
7280.2.a.bs.1.2		4	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:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)

\(f(q)\)	\(=\)	\(q+1.10649 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.77568 q^{9} +O(q^{10})\)	\(q+1.10649 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.77568 q^{9} +3.40207 q^{11} +1.00000 q^{13} -1.10649 q^{15} -2.77568 q^{17} +6.39073 q^{19} +1.10649 q^{21} +7.61504 q^{23} +1.00000 q^{25} -5.28424 q^{27} -0.508553 q^{29} -0.893512 q^{31} +3.76434 q^{33} -1.00000 q^{35} -3.25579 q^{37} +1.10649 q^{39} -7.79279 q^{41} -1.61504 q^{43} +1.77568 q^{45} +5.55137 q^{47} +1.00000 q^{49} -3.07126 q^{51} +2.21298 q^{53} -3.40207 q^{55} +7.07126 q^{57} +6.65786 q^{59} -12.5685 q^{61} -1.77568 q^{63} -1.00000 q^{65} +4.98866 q^{67} +8.42595 q^{69} +5.82802 q^{71} +7.01711 q^{73} +1.10649 q^{75} +3.40207 q^{77} +7.20164 q^{79} -0.519893 q^{81} +2.21298 q^{83} +2.77568 q^{85} -0.562708 q^{87} +10.2729 q^{89} +1.00000 q^{91} -0.988660 q^{93} -6.39073 q^{95} +5.33839 q^{97} -6.04099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} - 4 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) \) 4 * q + q^3 - 4 * q^5 + 4 * q^7 + 3 * q^9	\( 4 q + q^{3} - 4 q^{5} + 4 q^{7} + 3 q^{9} + 2 q^{11} + 4 q^{13} - q^{15} - q^{17} - 3 q^{19} + q^{21} + 12 q^{23} + 4 q^{25} + 4 q^{27} + 13 q^{29} - 7 q^{31} - 12 q^{33} - 4 q^{35} - q^{37} + q^{39} + 9 q^{41} + 12 q^{43} - 3 q^{45} + 2 q^{47} + 4 q^{49} + 6 q^{51} + 2 q^{53} - 2 q^{55} + 10 q^{57} + 3 q^{59} + 3 q^{63} - 4 q^{65} + 3 q^{67} + 20 q^{69} - 2 q^{71} - 2 q^{73} + q^{75} + 2 q^{77} + 5 q^{79} - 4 q^{81} + 2 q^{83} + q^{85} + q^{87} - q^{89} + 4 q^{91} + 13 q^{93} + 3 q^{95} + 8 q^{97} + 8 q^{99}+O(q^{100}) \) 4 * q + q^3 - 4 * q^5 + 4 * q^7 + 3 * q^9 + 2 * q^11 + 4 * q^13 - q^15 - q^17 - 3 * q^19 + q^21 + 12 * q^23 + 4 * q^25 + 4 * q^27 + 13 * q^29 - 7 * q^31 - 12 * q^33 - 4 * q^35 - q^37 + q^39 + 9 * q^41 + 12 * q^43 - 3 * q^45 + 2 * q^47 + 4 * q^49 + 6 * q^51 + 2 * q^53 - 2 * q^55 + 10 * q^57 + 3 * q^59 + 3 * q^63 - 4 * q^65 + 3 * q^67 + 20 * q^69 - 2 * q^71 - 2 * q^73 + q^75 + 2 * q^77 + 5 * q^79 - 4 * q^81 + 2 * q^83 + q^85 + q^87 - q^89 + 4 * q^91 + 13 * q^93 + 3 * q^95 + 8 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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