LMFDB - Embedded newform 3640.2.a.s.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:	$1$
Dimension:	$4$
Coefficient field:	4.4.2225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
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			$-0.820249$ 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)

$f(q)$	$=$	$q+0.820249 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.32719 q^{9} +O(q^{10})$	$q+0.820249 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.32719 q^{9} +1.23607 q^{11} +1.00000 q^{13} +0.820249 q^{15} -0.313307 q^{17} -1.90888 q^{19} -0.820249 q^{21} -7.89045 q^{23} +1.00000 q^{25} -4.36962 q^{27} +3.42970 q^{29} -6.27850 q^{31} +1.01388 q^{33} -1.00000 q^{35} -4.05632 q^{37} +0.820249 q^{39} -7.79933 q^{41} +2.87657 q^{43} -2.32719 q^{45} +11.5987 q^{47} +1.00000 q^{49} -0.256990 q^{51} -4.65438 q^{53} +1.23607 q^{55} -1.56575 q^{57} -9.50240 q^{59} -3.01388 q^{61} +2.32719 q^{63} +1.00000 q^{65} -11.4398 q^{67} -6.47214 q^{69} -3.77782 q^{71} -5.66827 q^{73} +0.820249 q^{75} -1.23607 q^{77} +11.5525 q^{79} +3.39739 q^{81} +10.1404 q^{83} -0.313307 q^{85} +2.81321 q^{87} +15.2106 q^{89} -1.00000 q^{91} -5.14994 q^{93} -1.90888 q^{95} +12.7670 q^{97} -2.87657 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) $ 4 * q - q^3 + 4 * q^5 - 4 * q^7 - q^9	$ 4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9} - 4 q^{11} + 4 q^{13} - q^{15} - q^{17} - 7 q^{19} + q^{21} - 6 q^{23} + 4 q^{25} - 4 q^{27} + q^{29} - 11 q^{31} - 4 q^{33} - 4 q^{35} - 3 q^{37} - q^{39} - 5 q^{41} - 6 q^{43} - q^{45} - 6 q^{47} + 4 q^{49} - 14 q^{51} - 2 q^{53} - 4 q^{55} + 14 q^{57} - q^{59} - 4 q^{61} + q^{63} + 4 q^{65} - 11 q^{67} - 8 q^{69} - 16 q^{71} + 2 q^{73} - q^{75} + 4 q^{77} - 15 q^{79} - 16 q^{81} - 2 q^{83} - q^{85} - 23 q^{87} - 3 q^{89} - 4 q^{91} - 5 q^{93} - 7 q^{95} + 8 q^{97} + 6 q^{99}+O(q^{100}) $ 4 * q - q^3 + 4 * q^5 - 4 * q^7 - q^9 - 4 * q^11 + 4 * q^13 - q^15 - q^17 - 7 * q^19 + q^21 - 6 * q^23 + 4 * q^25 - 4 * q^27 + q^29 - 11 * q^31 - 4 * q^33 - 4 * q^35 - 3 * q^37 - q^39 - 5 * q^41 - 6 * q^43 - q^45 - 6 * q^47 + 4 * q^49 - 14 * q^51 - 2 * q^53 - 4 * q^55 + 14 * q^57 - q^59 - 4 * q^61 + q^63 + 4 * q^65 - 11 * q^67 - 8 * q^69 - 16 * q^71 + 2 * q^73 - q^75 + 4 * q^77 - 15 * q^79 - 16 * q^81 - 2 * q^83 - q^85 - 23 * q^87 - 3 * q^89 - 4 * q^91 - 5 * q^93 - 7 * q^95 + 8 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.820249	0.473571	0.236786	−	0.971562i	$-0.423906\pi$
$3$	0.820249	0.473571	0.236786	+	0.971562i	$0.423906\pi$
$4$	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
$9$	−2.32719	−0.775730
$10$	0	0
$11$	1.23607	0.372689	0.186344	−	0.982485i	$-0.440336\pi$
$11$	1.23607	0.372689	0.186344	+	0.982485i	$0.440336\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0.820249	0.211787
$16$	0	0
$17$	−0.313307	−0.0759882	−0.0379941	−	0.999278i	$-0.512097\pi$
$17$	−0.313307	−0.0759882	−0.0379941	+	0.999278i	$0.512097\pi$
$18$	0	0
$19$	−1.90888	−0.437926	−0.218963	−	0.975733i	$-0.570267\pi$
$19$	−1.90888	−0.437926	−0.218963	+	0.975733i	$0.570267\pi$
$20$	0	0
$21$	−0.820249	−0.178993
$22$	0	0
$23$	−7.89045	−1.64527	−0.822636	−	0.568568i	$-0.807498\pi$
$23$	−7.89045	−1.64527	−0.822636	+	0.568568i	$0.807498\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.36962	−0.840935
$28$	0	0
$29$	3.42970	0.636880	0.318440	−	0.947943i	$-0.396841\pi$
$29$	3.42970	0.636880	0.318440	+	0.947943i	$0.396841\pi$
$30$	0	0
$31$	−6.27850	−1.12765	−0.563826	−	0.825894i	$-0.690671\pi$
$31$	−6.27850	−1.12765	−0.563826	+	0.825894i	$0.690671\pi$
$32$	0	0
$33$	1.01388	0.176495
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−4.05632	−0.666854	−0.333427	−	0.942776i	$-0.608205\pi$
$37$	−4.05632	−0.666854	−0.333427	+	0.942776i	$0.608205\pi$
$38$	0	0
$39$	0.820249	0.131345
$40$	0	0
$41$	−7.79933	−1.21805	−0.609025	−	0.793151i	$-0.708439\pi$
$41$	−7.79933	−1.21805	−0.609025	+	0.793151i	$0.708439\pi$
$42$	0	0
$43$	2.87657	0.438672	0.219336	−	0.975649i	$-0.429611\pi$
$43$	2.87657	0.438672	0.219336	+	0.975649i	$0.429611\pi$
$44$	0	0
$45$	−2.32719	−0.346917
$46$	0	0
$47$	11.5987	1.69184	0.845919	−	0.533312i	$-0.179053\pi$
$47$	11.5987	1.69184	0.845919	+	0.533312i	$0.179053\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.256990	−0.0359858
$52$	0	0
$53$	−4.65438	−0.639329	−0.319664	−	0.947531i	$-0.603570\pi$
$53$	−4.65438	−0.639329	−0.319664	+	0.947531i	$0.603570\pi$
$54$	0	0
$55$	1.23607	0.166671
$56$	0	0
$57$	−1.56575	−0.207389
$58$	0	0
$59$	−9.50240	−1.23711	−0.618554	−	0.785743i	$-0.712281\pi$
$59$	−9.50240	−1.23711	−0.618554	+	0.785743i	$0.712281\pi$
$60$	0	0
$61$	−3.01388	−0.385888	−0.192944	−	0.981210i	$-0.561804\pi$
$61$	−3.01388	−0.385888	−0.192944	+	0.981210i	$0.561804\pi$
$62$	0	0
$63$	2.32719	0.293199
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−11.4398	−1.39760	−0.698799	−	0.715319i	$-0.746282\pi$
$67$	−11.4398	−1.39760	−0.698799	+	0.715319i	$0.746282\pi$
$68$	0	0
$69$	−6.47214	−0.779154
$70$	0	0
$71$	−3.77782	−0.448344	−0.224172	−	0.974550i	$-0.571968\pi$
$71$	−3.77782	−0.448344	−0.224172	+	0.974550i	$0.571968\pi$
$72$	0	0
$73$	−5.66827	−0.663420	−0.331710	−	0.943381i	$-0.607626\pi$
$73$	−5.66827	−0.663420	−0.331710	+	0.943381i	$0.607626\pi$
$74$	0	0
$75$	0.820249	0.0947142
$76$	0	0
$77$	−1.23607	−0.140863
$78$	0	0
$79$	11.5525	1.29975	0.649877	−	0.760040i	$-0.274821\pi$
$79$	11.5525	1.29975	0.649877	+	0.760040i	$0.274821\pi$
$80$	0	0
$81$	3.39739	0.377488
$82$	0	0
$83$	10.1404	1.11305	0.556527	−	0.830830i	$-0.312134\pi$
$83$	10.1404	1.11305	0.556527	+	0.830830i	$0.312134\pi$
$84$	0	0
$85$	−0.313307	−0.0339830
$86$	0	0
$87$	2.81321	0.301608
$88$	0	0
$89$	15.2106	1.61232	0.806160	−	0.591697i	$-0.201542\pi$
$89$	15.2106	1.61232	0.806160	+	0.591697i	$0.201542\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−5.14994	−0.534023
$94$	0	0
$95$	−1.90888	−0.195847
$96$	0	0
$97$	12.7670	1.29629	0.648147	−	0.761515i	$-0.275544\pi$
$97$	12.7670	1.29629	0.648147	+	0.761515i	$0.275544\pi$
$98$	0	0
$99$	−2.87657	−0.289106
$100$	0	0
$101$	−18.2253	−1.81348	−0.906741	−	0.421688i	$-0.861438\pi$
$101$	−18.2253	−1.81348	−0.906741	+	0.421688i	$0.861438\pi$
$102$	0	0
$103$	−9.83926	−0.969492	−0.484746	−	0.874655i	$-0.661088\pi$
$103$	−9.83926	−0.969492	−0.484746	+	0.874655i	$0.661088\pi$
$104$	0	0
$105$	−0.820249	−0.0800481
$106$	0	0
$107$	−13.7132	−1.32570	−0.662852	−	0.748750i	$-0.730654\pi$
$107$	−13.7132	−1.32570	−0.662852	+	0.748750i	$0.730654\pi$
$108$	0	0
$109$	−5.48602	−0.525465	−0.262733	−	0.964869i	$-0.584624\pi$
$109$	−5.48602	−0.525465	−0.262733	+	0.964869i	$0.584624\pi$
$110$	0	0
$111$	−3.32719	−0.315803
$112$	0	0
$113$	−1.37339	−0.129197	−0.0645986	−	0.997911i	$-0.520577\pi$
$113$	−1.37339	−0.129197	−0.0645986	+	0.997911i	$0.520577\pi$
$114$	0	0
$115$	−7.89045	−0.735788
$116$	0	0
$117$	−2.32719	−0.215149
$118$	0	0
$119$	0.313307	0.0287208
$120$	0	0
$121$	−9.47214	−0.861103
$122$	0	0
$123$	−6.39739	−0.576833
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.23607	0.109683	0.0548416	−	0.998495i	$-0.482535\pi$
$127$	1.23607	0.109683	0.0548416	+	0.998495i	$0.482535\pi$
$128$	0	0
$129$	2.35950	0.207743
$130$	0	0
$131$	−18.9720	−1.65759	−0.828797	−	0.559549i	$-0.810974\pi$
$131$	−18.9720	−1.65759	−0.828797	+	0.559549i	$0.810974\pi$
$132$	0	0
$133$	1.90888	0.165521
$134$	0	0
$135$	−4.36962	−0.376077
$136$	0	0
$137$	−11.0056	−0.940270	−0.470135	−	0.882594i	$-0.655795\pi$
$137$	−11.0056	−0.940270	−0.470135	+	0.882594i	$0.655795\pi$
$138$	0	0
$139$	−11.9304	−1.01192	−0.505961	−	0.862556i	$-0.668862\pi$
$139$	−11.9304	−1.01192	−0.505961	+	0.862556i	$0.668862\pi$
$140$	0	0
$141$	9.51379	0.801205
$142$	0	0
$143$	1.23607	0.103365
$144$	0	0
$145$	3.42970	0.284821
$146$	0	0
$147$	0.820249	0.0676530
$148$	0	0
$149$	20.9923	1.71976	0.859878	−	0.510500i	$-0.170540\pi$
$149$	20.9923	1.71976	0.859878	+	0.510500i	$0.170540\pi$
$150$	0	0
$151$	−18.2131	−1.48216	−0.741080	−	0.671416i	$-0.765686\pi$
$151$	−18.2131	−1.48216	−0.741080	+	0.671416i	$0.765686\pi$
$152$	0	0
$153$	0.729126	0.0589463
$154$	0	0
$155$	−6.27850	−0.504301
$156$	0	0
$157$	17.3879	1.38770	0.693851	−	0.720119i	$-0.255913\pi$
$157$	17.3879	1.38770	0.693851	+	0.720119i	$0.255913\pi$
$158$	0	0
$159$	−3.81775	−0.302768
$160$	0	0
$161$	7.89045	0.621855
$162$	0	0
$163$	1.54234	0.120805	0.0604026	−	0.998174i	$-0.480762\pi$
$163$	1.54234	0.120805	0.0604026	+	0.998174i	$0.480762\pi$
$164$	0	0
$165$	1.01388	0.0789307
$166$	0	0
$167$	−9.51379	−0.736199	−0.368099	−	0.929786i	$-0.619991\pi$
$167$	−9.51379	−0.736199	−0.368099	+	0.929786i	$0.619991\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.44232	0.339713
$172$	0	0
$173$	9.80945	0.745799	0.372899	−	0.927872i	$-0.378364\pi$
$173$	9.80945	0.745799	0.372899	+	0.927872i	$0.378364\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−7.79434	−0.585858
$178$	0	0
$179$	21.7625	1.62660	0.813302	−	0.581842i	$-0.197668\pi$
$179$	21.7625	1.62660	0.813302	+	0.581842i	$0.197668\pi$
$180$	0	0
$181$	21.1190	1.56976	0.784881	−	0.619646i	$-0.212724\pi$
$181$	21.1190	1.56976	0.784881	+	0.619646i	$0.212724\pi$
$182$	0	0
$183$	−2.47214	−0.182746
$184$	0	0
$185$	−4.05632	−0.298226
$186$	0	0
$187$	−0.387269	−0.0283199
$188$	0	0
$189$	4.36962	0.317843
$190$	0	0
$191$	−11.5842	−0.838202	−0.419101	−	0.907940i	$-0.637655\pi$
$191$	−11.5842	−0.838202	−0.419101	+	0.907940i	$0.637655\pi$
$192$	0	0
$193$	−17.5752	−1.26509	−0.632547	−	0.774522i	$-0.717990\pi$
$193$	−17.5752	−1.26509	−0.632547	+	0.774522i	$0.717990\pi$
$194$	0	0
$195$	0.820249	0.0587393
$196$	0	0
$197$	−17.9221	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$197$	−17.9221	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$198$	0	0
$199$	14.7873	1.04824	0.524121	−	0.851644i	$-0.324394\pi$
$199$	14.7873	1.04824	0.524121	+	0.851644i	$0.324394\pi$
$200$	0	0
$201$	−9.38351	−0.661862
$202$	0	0
$203$	−3.42970	−0.240718
$204$	0	0
$205$	−7.79933	−0.544729
$206$	0	0
$207$	18.3626	1.27629
$208$	0	0
$209$	−2.35950	−0.163210
$210$	0	0
$211$	−17.7069	−1.21900	−0.609498	−	0.792788i	$-0.708629\pi$
$211$	−17.7069	−1.21900	−0.609498	+	0.792788i	$0.708629\pi$
$212$	0	0
$213$	−3.09875	−0.212323
$214$	0	0
$215$	2.87657	0.196180
$216$	0	0
$217$	6.27850	0.426212
$218$	0	0
$219$	−4.64939	−0.314177
$220$	0	0
$221$	−0.313307	−0.0210753
$222$	0	0
$223$	−17.8885	−1.19791	−0.598953	−	0.800784i	$-0.704416\pi$
$223$	−17.8885	−1.19791	−0.598953	+	0.800784i	$0.704416\pi$
$224$	0	0
$225$	−2.32719	−0.155146
$226$	0	0
$227$	−21.8658	−1.45128	−0.725641	−	0.688074i	$-0.758457\pi$
$227$	−21.8658	−1.45128	−0.725641	+	0.688074i	$0.758457\pi$
$228$	0	0
$229$	−9.82709	−0.649393	−0.324696	−	0.945818i	$-0.605262\pi$
$229$	−9.82709	−0.649393	−0.324696	+	0.945818i	$0.605262\pi$
$230$	0	0
$231$	−1.01388	−0.0667087
$232$	0	0
$233$	24.7302	1.62013	0.810063	−	0.586342i	$-0.199433\pi$
$233$	24.7302	1.62013	0.810063	+	0.586342i	$0.199433\pi$
$234$	0	0
$235$	11.5987	0.756613
$236$	0	0
$237$	9.47590	0.615526
$238$	0	0
$239$	−2.51207	−0.162493	−0.0812463	−	0.996694i	$-0.525890\pi$
$239$	−2.51207	−0.162493	−0.0812463	+	0.996694i	$0.525890\pi$
$240$	0	0
$241$	22.0524	1.42052	0.710259	−	0.703941i	$-0.248578\pi$
$241$	22.0524	1.42052	0.710259	+	0.703941i	$0.248578\pi$
$242$	0	0
$243$	15.8956	1.01970
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−1.90888	−0.121459
$248$	0	0
$249$	8.31766	0.527110
$250$	0	0
$251$	21.2594	1.34188	0.670941	−	0.741511i	$-0.265891\pi$
$251$	21.2594	1.34188	0.670941	+	0.741511i	$0.265891\pi$
$252$	0	0
$253$	−9.75313	−0.613174
$254$	0	0
$255$	−0.256990	−0.0160933
$256$	0	0
$257$	−12.2530	−0.764323	−0.382162	−	0.924095i	$-0.624820\pi$
$257$	−12.2530	−0.764323	−0.382162	+	0.924095i	$0.624820\pi$
$258$	0	0
$259$	4.05632	0.252047
$260$	0	0
$261$	−7.98157	−0.494047
$262$	0	0
$263$	26.1435	1.61208	0.806038	−	0.591863i	$-0.201608\pi$
$263$	26.1435	1.61208	0.806038	+	0.591863i	$0.201608\pi$
$264$	0	0
$265$	−4.65438	−0.285916
$266$	0	0
$267$	12.4765	0.763549
$268$	0	0
$269$	−29.9734	−1.82751	−0.913756	−	0.406264i	$-0.866831\pi$
$269$	−29.9734	−1.82751	−0.913756	+	0.406264i	$0.866831\pi$
$270$	0	0
$271$	15.3967	0.935284	0.467642	−	0.883918i	$-0.345104\pi$
$271$	15.3967	0.935284	0.467642	+	0.883918i	$0.345104\pi$
$272$	0	0
$273$	−0.820249	−0.0496437
$274$	0	0
$275$	1.23607	0.0745377
$276$	0	0
$277$	6.56952	0.394724	0.197362	−	0.980331i	$-0.436763\pi$
$277$	6.56952	0.394724	0.197362	+	0.980331i	$0.436763\pi$
$278$	0	0
$279$	14.6113	0.874754
$280$	0	0
$281$	20.9923	1.25229	0.626147	−	0.779705i	$-0.284631\pi$
$281$	20.9923	1.25229	0.626147	+	0.779705i	$0.284631\pi$
$282$	0	0
$283$	11.0582	0.657340	0.328670	−	0.944445i	$-0.393400\pi$
$283$	11.0582	0.657340	0.328670	+	0.944445i	$0.393400\pi$
$284$	0	0
$285$	−1.56575	−0.0927473
$286$	0	0
$287$	7.79933	0.460380
$288$	0	0
$289$	−16.9018	−0.994226
$290$	0	0
$291$	10.4721	0.613887
$292$	0	0
$293$	0.267113	0.0156049	0.00780246	−	0.999970i	$-0.497516\pi$
$293$	0.267113	0.0156049	0.00780246	+	0.999970i	$0.497516\pi$
$294$	0	0
$295$	−9.50240	−0.553251
$296$	0	0
$297$	−5.40115	−0.313407
$298$	0	0
$299$	−7.89045	−0.456317
$300$	0	0
$301$	−2.87657	−0.165803
$302$	0	0
$303$	−14.9493	−0.858813
$304$	0	0
$305$	−3.01388	−0.172574
$306$	0	0
$307$	−5.57089	−0.317947	−0.158974	−	0.987283i	$-0.550818\pi$
$307$	−5.57089	−0.317947	−0.158974	+	0.987283i	$0.550818\pi$
$308$	0	0
$309$	−8.07065	−0.459123
$310$	0	0
$311$	−30.4353	−1.72583	−0.862913	−	0.505352i	$-0.831363\pi$
$311$	−30.4353	−1.72583	−0.862913	+	0.505352i	$0.831363\pi$
$312$	0	0
$313$	13.0333	0.736688	0.368344	−	0.929689i	$-0.379925\pi$
$313$	13.0333	0.736688	0.368344	+	0.929689i	$0.379925\pi$
$314$	0	0
$315$	2.32719	0.131122
$316$	0	0
$317$	30.3151	1.70267	0.851334	−	0.524625i	$-0.175794\pi$
$317$	30.3151	1.70267	0.851334	+	0.524625i	$0.175794\pi$
$318$	0	0
$319$	4.23935	0.237358
$320$	0	0
$321$	−11.2482	−0.627815
$322$	0	0
$323$	0.598065	0.0332772
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−4.49990	−0.248845
$328$	0	0
$329$	−11.5987	−0.639455
$330$	0	0
$331$	−35.5462	−1.95380	−0.976898	−	0.213706i	$-0.931447\pi$
$331$	−35.5462	−1.95380	−0.976898	+	0.213706i	$0.931447\pi$
$332$	0	0
$333$	9.43983	0.517299
$334$	0	0
$335$	−11.4398	−0.625024
$336$	0	0
$337$	23.3088	1.26971	0.634855	−	0.772632i	$-0.281060\pi$
$337$	23.3088	1.26971	0.634855	+	0.772632i	$0.281060\pi$
$338$	0	0
$339$	−1.12652	−0.0611841
$340$	0	0
$341$	−7.76065	−0.420263
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−6.47214	−0.348448
$346$	0	0
$347$	−20.7423	−1.11351	−0.556753	−	0.830678i	$-0.687953\pi$
$347$	−20.7423	−1.11351	−0.556753	+	0.830678i	$0.687953\pi$
$348$	0	0
$349$	−4.18146	−0.223829	−0.111914	−	0.993718i	$-0.535698\pi$
$349$	−4.18146	−0.223829	−0.111914	+	0.993718i	$0.535698\pi$
$350$	0	0
$351$	−4.36962	−0.233233
$352$	0	0
$353$	21.6417	1.15187	0.575935	−	0.817495i	$-0.304638\pi$
$353$	21.6417	1.15187	0.575935	+	0.817495i	$0.304638\pi$
$354$	0	0
$355$	−3.77782	−0.200506
$356$	0	0
$357$	0.256990	0.0136014
$358$	0	0
$359$	−7.83335	−0.413428	−0.206714	−	0.978401i	$-0.566277\pi$
$359$	−7.83335	−0.413428	−0.206714	+	0.978401i	$0.566277\pi$
$360$	0	0
$361$	−15.3562	−0.808220
$362$	0	0
$363$	−7.76951	−0.407794
$364$	0	0
$365$	−5.66827	−0.296691
$366$	0	0
$367$	−28.4802	−1.48665	−0.743327	−	0.668928i	$-0.766754\pi$
$367$	−28.4802	−1.48665	−0.743327	+	0.668928i	$0.766754\pi$
$368$	0	0
$369$	18.1505	0.944879
$370$	0	0
$371$	4.65438	0.241643
$372$	0	0
$373$	20.0683	1.03910	0.519548	−	0.854442i	$-0.326101\pi$
$373$	20.0683	1.03910	0.519548	+	0.854442i	$0.326101\pi$
$374$	0	0
$375$	0.820249	0.0423575
$376$	0	0
$377$	3.42970	0.176639
$378$	0	0
$379$	8.81448	0.452769	0.226385	−	0.974038i	$-0.427309\pi$
$379$	8.81448	0.452769	0.226385	+	0.974038i	$0.427309\pi$
$380$	0	0
$381$	1.01388	0.0519428
$382$	0	0
$383$	5.21638	0.266544	0.133272	−	0.991079i	$-0.457452\pi$
$383$	5.21638	0.266544	0.133272	+	0.991079i	$0.457452\pi$
$384$	0	0
$385$	−1.23607	−0.0629959
$386$	0	0
$387$	−6.69432	−0.340291
$388$	0	0
$389$	6.85506	0.347565	0.173783	−	0.984784i	$-0.444401\pi$
$389$	6.85506	0.347565	0.173783	+	0.984784i	$0.444401\pi$
$390$	0	0
$391$	2.47214	0.125021
$392$	0	0
$393$	−15.5618	−0.784989
$394$	0	0
$395$	11.5525	0.581267
$396$	0	0
$397$	20.4150	1.02460	0.512301	−	0.858806i	$-0.328793\pi$
$397$	20.4150	1.02460	0.512301	+	0.858806i	$0.328793\pi$
$398$	0	0
$399$	1.56575	0.0783858
$400$	0	0
$401$	−30.4858	−1.52239	−0.761195	−	0.648523i	$-0.775387\pi$
$401$	−30.4858	−1.52239	−0.761195	+	0.648523i	$0.775387\pi$
$402$	0	0
$403$	−6.27850	−0.312754
$404$	0	0
$405$	3.39739	0.168818
$406$	0	0
$407$	−5.01388	−0.248529
$408$	0	0
$409$	−29.5063	−1.45899	−0.729495	−	0.683986i	$-0.760245\pi$
$409$	−29.5063	−1.45899	−0.729495	+	0.683986i	$0.760245\pi$
$410$	0	0
$411$	−9.02732	−0.445285
$412$	0	0
$413$	9.50240	0.467582
$414$	0	0
$415$	10.1404	0.497773
$416$	0	0
$417$	−9.78589	−0.479217
$418$	0	0
$419$	−10.1176	−0.494278	−0.247139	−	0.968980i	$-0.579490\pi$
$419$	−10.1176	−0.494278	−0.247139	+	0.968980i	$0.579490\pi$
$420$	0	0
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	0	0
$423$	−26.9923	−1.31241
$424$	0	0
$425$	−0.313307	−0.0151976
$426$	0	0
$427$	3.01388	0.145852
$428$	0	0
$429$	1.01388	0.0489508
$430$	0	0
$431$	5.97785	0.287943	0.143971	−	0.989582i	$-0.454013\pi$
$431$	5.97785	0.287943	0.143971	+	0.989582i	$0.454013\pi$
$432$	0	0
$433$	−2.52585	−0.121385	−0.0606923	−	0.998157i	$-0.519331\pi$
$433$	−2.52585	−0.121385	−0.0606923	+	0.998157i	$0.519331\pi$
$434$	0	0
$435$	2.81321	0.134883
$436$	0	0
$437$	15.0619	0.720508
$438$	0	0
$439$	−13.6835	−0.653079	−0.326539	−	0.945184i	$-0.605883\pi$
$439$	−13.6835	−0.653079	−0.326539	+	0.945184i	$0.605883\pi$
$440$	0	0
$441$	−2.32719	−0.110819
$442$	0	0
$443$	−37.0372	−1.75969	−0.879846	−	0.475260i	$-0.842354\pi$
$443$	−37.0372	−1.75969	−0.879846	+	0.475260i	$0.842354\pi$
$444$	0	0
$445$	15.2106	0.721052
$446$	0	0
$447$	17.2189	0.814426
$448$	0	0
$449$	16.6544	0.785969	0.392984	−	0.919545i	$-0.371443\pi$
$449$	16.6544	0.785969	0.392984	+	0.919545i	$0.371443\pi$
$450$	0	0
$451$	−9.64050	−0.453953
$452$	0	0
$453$	−14.9393	−0.701909
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	2.30318	0.107738	0.0538692	−	0.998548i	$-0.482845\pi$
$457$	2.30318	0.107738	0.0538692	+	0.998548i	$0.482845\pi$
$458$	0	0
$459$	1.36904	0.0639011
$460$	0	0
$461$	−24.8183	−1.15591	−0.577953	−	0.816070i	$-0.696148\pi$
$461$	−24.8183	−1.15591	−0.577953	+	0.816070i	$0.696148\pi$
$462$	0	0
$463$	20.8410	0.968562	0.484281	−	0.874912i	$-0.339081\pi$
$463$	20.8410	0.968562	0.484281	+	0.874912i	$0.339081\pi$
$464$	0	0
$465$	−5.14994	−0.238823
$466$	0	0
$467$	28.8668	1.33580	0.667899	−	0.744252i	$-0.267194\pi$
$467$	28.8668	1.33580	0.667899	+	0.744252i	$0.267194\pi$
$468$	0	0
$469$	11.4398	0.528242
$470$	0	0
$471$	14.2624	0.657176
$472$	0	0
$473$	3.55563	0.163488
$474$	0	0
$475$	−1.90888	−0.0875853
$476$	0	0
$477$	10.8316	0.495947
$478$	0	0
$479$	14.4810	0.661653	0.330827	−	0.943692i	$-0.392672\pi$
$479$	14.4810	0.661653	0.330827	+	0.943692i	$0.392672\pi$
$480$	0	0
$481$	−4.05632	−0.184952
$482$	0	0
$483$	6.47214	0.294492
$484$	0	0
$485$	12.7670	0.579720
$486$	0	0
$487$	20.1043	0.911014	0.455507	−	0.890232i	$-0.349458\pi$
$487$	20.1043	0.911014	0.455507	+	0.890232i	$0.349458\pi$
$488$	0	0
$489$	1.26510	0.0572098
$490$	0	0
$491$	−40.9049	−1.84601	−0.923006	−	0.384786i	$-0.874275\pi$
$491$	−40.9049	−1.84601	−0.923006	+	0.384786i	$0.874275\pi$
$492$	0	0
$493$	−1.07455	−0.0483953
$494$	0	0
$495$	−2.87657	−0.129292
$496$	0	0
$497$	3.77782	0.169458
$498$	0	0
$499$	17.5184	0.784233	0.392116	−	0.919916i	$-0.371743\pi$
$499$	17.5184	0.784233	0.392116	+	0.919916i	$0.371743\pi$
$500$	0	0
$501$	−7.80368	−0.348643
$502$	0	0
$503$	37.1155	1.65490	0.827450	−	0.561540i	$-0.189791\pi$
$503$	37.1155	1.65490	0.827450	+	0.561540i	$0.189791\pi$
$504$	0	0
$505$	−18.2253	−0.811014
$506$	0	0
$507$	0.820249	0.0364285
$508$	0	0
$509$	−19.6701	−0.871861	−0.435931	−	0.899980i	$-0.643581\pi$
$509$	−19.6701	−0.871861	−0.435931	+	0.899980i	$0.643581\pi$
$510$	0	0
$511$	5.66827	0.250749
$512$	0	0
$513$	8.34107	0.368267
$514$	0	0
$515$	−9.83926	−0.433570
$516$	0	0
$517$	14.3367	0.630528
$518$	0	0
$519$	8.04619	0.353189
$520$	0	0
$521$	0.640306	0.0280523	0.0140262	−	0.999902i	$-0.495535\pi$
$521$	0.640306	0.0280523	0.0140262	+	0.999902i	$0.495535\pi$
$522$	0	0
$523$	−31.7269	−1.38732	−0.693660	−	0.720303i	$-0.744003\pi$
$523$	−31.7269	−1.38732	−0.693660	+	0.720303i	$0.744003\pi$
$524$	0	0
$525$	−0.820249	−0.0357986
$526$	0	0
$527$	1.96710	0.0856882
$528$	0	0
$529$	39.2592	1.70692
$530$	0	0
$531$	22.1139	0.959662
$532$	0	0
$533$	−7.79933	−0.337826
$534$	0	0
$535$	−13.7132	−0.592873
$536$	0	0
$537$	17.8507	0.770313
$538$	0	0
$539$	1.23607	0.0532412
$540$	0	0
$541$	32.2733	1.38754	0.693768	−	0.720198i	$-0.255949\pi$
$541$	32.2733	1.38754	0.693768	+	0.720198i	$0.255949\pi$
$542$	0	0
$543$	17.3228	0.743394
$544$	0	0
$545$	−5.48602	−0.234995
$546$	0	0
$547$	−26.1082	−1.11631	−0.558153	−	0.829738i	$-0.688490\pi$
$547$	−26.1082	−1.11631	−0.558153	+	0.829738i	$0.688490\pi$
$548$	0	0
$549$	7.01388	0.299345
$550$	0	0
$551$	−6.54688	−0.278906
$552$	0	0
$553$	−11.5525	−0.491261
$554$	0	0
$555$	−3.32719	−0.141231
$556$	0	0
$557$	40.8437	1.73060	0.865302	−	0.501251i	$-0.167127\pi$
$557$	40.8437	1.73060	0.865302	+	0.501251i	$0.167127\pi$
$558$	0	0
$559$	2.87657	0.121666
$560$	0	0
$561$	−0.317657	−0.0134115
$562$	0	0
$563$	−41.0366	−1.72949	−0.864743	−	0.502215i	$-0.832519\pi$
$563$	−41.0366	−1.72949	−0.864743	+	0.502215i	$0.832519\pi$
$564$	0	0
$565$	−1.37339	−0.0577788
$566$	0	0
$567$	−3.39739	−0.142677
$568$	0	0
$569$	−17.9917	−0.754251	−0.377126	−	0.926162i	$-0.623087\pi$
$569$	−17.9917	−0.754251	−0.377126	+	0.926162i	$0.623087\pi$
$570$	0	0
$571$	−8.56121	−0.358276	−0.179138	−	0.983824i	$-0.557331\pi$
$571$	−8.56121	−0.358276	−0.179138	+	0.983824i	$0.557331\pi$
$572$	0	0
$573$	−9.50192	−0.396948
$574$	0	0
$575$	−7.89045	−0.329055
$576$	0	0
$577$	−31.7821	−1.32311	−0.661553	−	0.749899i	$-0.730102\pi$
$577$	−31.7821	−1.32311	−0.661553	+	0.749899i	$0.730102\pi$
$578$	0	0
$579$	−14.4161	−0.599112
$580$	0	0
$581$	−10.1404	−0.420695
$582$	0	0
$583$	−5.75313	−0.238270
$584$	0	0
$585$	−2.32719	−0.0962175
$586$	0	0
$587$	20.2025	0.833846	0.416923	−	0.908942i	$-0.363108\pi$
$587$	20.2025	0.833846	0.416923	+	0.908942i	$0.363108\pi$
$588$	0	0
$589$	11.9849	0.493829
$590$	0	0
$591$	−14.7006	−0.604701
$592$	0	0
$593$	−4.17569	−0.171475	−0.0857375	−	0.996318i	$-0.527325\pi$
$593$	−4.17569	−0.171475	−0.0857375	+	0.996318i	$0.527325\pi$
$594$	0	0
$595$	0.313307	0.0128443
$596$	0	0
$597$	12.1292	0.496417
$598$	0	0
$599$	13.8924	0.567626	0.283813	−	0.958880i	$-0.408400\pi$
$599$	13.8924	0.567626	0.283813	+	0.958880i	$0.408400\pi$
$600$	0	0
$601$	29.2176	1.19181	0.595904	−	0.803055i	$-0.296794\pi$
$601$	29.2176	1.19181	0.595904	+	0.803055i	$0.296794\pi$
$602$	0	0
$603$	26.6227	1.08416
$604$	0	0
$605$	−9.47214	−0.385097
$606$	0	0
$607$	−3.67965	−0.149353	−0.0746763	−	0.997208i	$-0.523792\pi$
$607$	−3.67965	−0.149353	−0.0746763	+	0.997208i	$0.523792\pi$
$608$	0	0
$609$	−2.81321	−0.113997
$610$	0	0
$611$	11.5987	0.469231
$612$	0	0
$613$	3.49354	0.141103	0.0705514	−	0.997508i	$-0.477524\pi$
$613$	3.49354	0.141103	0.0705514	+	0.997508i	$0.477524\pi$
$614$	0	0
$615$	−6.39739	−0.257968
$616$	0	0
$617$	40.1928	1.61810	0.809050	−	0.587740i	$-0.199982\pi$
$617$	40.1928	1.61810	0.809050	+	0.587740i	$0.199982\pi$
$618$	0	0
$619$	29.0314	1.16687	0.583436	−	0.812159i	$-0.301708\pi$
$619$	29.0314	1.16687	0.583436	+	0.812159i	$0.301708\pi$
$620$	0	0
$621$	34.4783	1.38357
$622$	0	0
$623$	−15.2106	−0.609400
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.93538	−0.0772916
$628$	0	0
$629$	1.27087	0.0506731
$630$	0	0
$631$	14.6943	0.584972	0.292486	−	0.956270i	$-0.405518\pi$
$631$	14.6943	0.584972	0.292486	+	0.956270i	$0.405518\pi$
$632$	0	0
$633$	−14.5241	−0.577281
$634$	0	0
$635$	1.23607	0.0490519
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	8.79170	0.347794
$640$	0	0
$641$	−28.8183	−1.13826	−0.569128	−	0.822249i	$-0.692719\pi$
$641$	−28.8183	−1.13826	−0.569128	+	0.822249i	$0.692719\pi$
$642$	0	0
$643$	7.06599	0.278656	0.139328	−	0.990246i	$-0.455506\pi$
$643$	7.06599	0.278656	0.139328	+	0.990246i	$0.455506\pi$
$644$	0	0
$645$	2.35950	0.0929053
$646$	0	0
$647$	−5.81545	−0.228629	−0.114314	−	0.993445i	$-0.536467\pi$
$647$	−5.81545	−0.228629	−0.114314	+	0.993445i	$0.536467\pi$
$648$	0	0
$649$	−11.7456	−0.461056
$650$	0	0
$651$	5.14994	0.201842
$652$	0	0
$653$	21.1315	0.826940	0.413470	−	0.910518i	$-0.364317\pi$
$653$	21.1315	0.826940	0.413470	+	0.910518i	$0.364317\pi$
$654$	0	0
$655$	−18.9720	−0.741299
$656$	0	0
$657$	13.1911	0.514635
$658$	0	0
$659$	21.7095	0.845681	0.422840	−	0.906204i	$-0.361033\pi$
$659$	21.7095	0.845681	0.422840	+	0.906204i	$0.361033\pi$
$660$	0	0
$661$	34.1079	1.32664	0.663322	−	0.748334i	$-0.269146\pi$
$661$	34.1079	1.32664	0.663322	+	0.748334i	$0.269146\pi$
$662$	0	0
$663$	−0.256990	−0.00998067
$664$	0	0
$665$	1.90888	0.0740231
$666$	0	0
$667$	−27.0619	−1.04784
$668$	0	0
$669$	−14.6731	−0.567293
$670$	0	0
$671$	−3.72537	−0.143816
$672$	0	0
$673$	22.7024	0.875113	0.437557	−	0.899191i	$-0.355844\pi$
$673$	22.7024	0.875113	0.437557	+	0.899191i	$0.355844\pi$
$674$	0	0
$675$	−4.36962	−0.168187
$676$	0	0
$677$	11.6733	0.448640	0.224320	−	0.974516i	$-0.427984\pi$
$677$	11.6733	0.448640	0.224320	+	0.974516i	$0.427984\pi$
$678$	0	0
$679$	−12.7670	−0.489953
$680$	0	0
$681$	−17.9354	−0.687285
$682$	0	0
$683$	13.5600	0.518858	0.259429	−	0.965762i	$-0.416466\pi$
$683$	13.5600	0.518858	0.259429	+	0.965762i	$0.416466\pi$
$684$	0	0
$685$	−11.0056	−0.420502
$686$	0	0
$687$	−8.06067	−0.307534
$688$	0	0
$689$	−4.65438	−0.177318
$690$	0	0
$691$	9.05803	0.344584	0.172292	−	0.985046i	$-0.444883\pi$
$691$	9.05803	0.344584	0.172292	+	0.985046i	$0.444883\pi$
$692$	0	0
$693$	2.87657	0.109272
$694$	0	0
$695$	−11.9304	−0.452545
$696$	0	0
$697$	2.44359	0.0925574
$698$	0	0
$699$	20.2849	0.767245
$700$	0	0
$701$	−2.05236	−0.0775167	−0.0387584	−	0.999249i	$-0.512340\pi$
$701$	−2.05236	−0.0775167	−0.0387584	+	0.999249i	$0.512340\pi$
$702$	0	0
$703$	7.74301	0.292033
$704$	0	0
$705$	9.51379	0.358310
$706$	0	0
$707$	18.2253	0.685432
$708$	0	0
$709$	15.1593	0.569319	0.284659	−	0.958629i	$-0.408120\pi$
$709$	15.1593	0.569319	0.284659	+	0.958629i	$0.408120\pi$
$710$	0	0
$711$	−26.8848	−1.00826
$712$	0	0
$713$	49.5402	1.85530
$714$	0	0
$715$	1.23607	0.0462263
$716$	0	0
$717$	−2.06053	−0.0769518
$718$	0	0
$719$	12.2808	0.457997	0.228998	−	0.973427i	$-0.426455\pi$
$719$	12.2808	0.457997	0.228998	+	0.973427i	$0.426455\pi$
$720$	0	0
$721$	9.83926	0.366433
$722$	0	0
$723$	18.0884	0.672716
$724$	0	0
$725$	3.42970	0.127376
$726$	0	0
$727$	35.4949	1.31643	0.658216	−	0.752829i	$-0.271311\pi$
$727$	35.4949	1.31643	0.658216	+	0.752829i	$0.271311\pi$
$728$	0	0
$729$	2.84616	0.105413
$730$	0	0
$731$	−0.901249	−0.0333339
$732$	0	0
$733$	26.0202	0.961080	0.480540	−	0.876973i	$-0.340441\pi$
$733$	26.0202	0.961080	0.480540	+	0.876973i	$0.340441\pi$
$734$	0	0
$735$	0.820249	0.0302553
$736$	0	0
$737$	−14.1404	−0.520868
$738$	0	0
$739$	50.2889	1.84991	0.924953	−	0.380081i	$-0.124104\pi$
$739$	50.2889	1.84991	0.924953	+	0.380081i	$0.124104\pi$
$740$	0	0
$741$	−1.56575	−0.0575194
$742$	0	0
$743$	15.6121	0.572752	0.286376	−	0.958117i	$-0.407549\pi$
$743$	15.6121	0.572752	0.286376	+	0.958117i	$0.407549\pi$
$744$	0	0
$745$	20.9923	0.769098
$746$	0	0
$747$	−23.5987	−0.863430
$748$	0	0
$749$	13.7132	0.501069
$750$	0	0
$751$	−23.2929	−0.849971	−0.424986	−	0.905200i	$-0.639721\pi$
$751$	−23.2929	−0.849971	−0.424986	+	0.905200i	$0.639721\pi$
$752$	0	0
$753$	17.4380	0.635476
$754$	0	0
$755$	−18.2131	−0.662843
$756$	0	0
$757$	19.1961	0.697695	0.348848	−	0.937179i	$-0.386573\pi$
$757$	19.1961	0.697695	0.348848	+	0.937179i	$0.386573\pi$
$758$	0	0
$759$	−8.00000	−0.290382
$760$	0	0
$761$	−38.6598	−1.40142	−0.700708	−	0.713448i	$-0.747132\pi$
$761$	−38.6598	−1.40142	−0.700708	+	0.713448i	$0.747132\pi$
$762$	0	0
$763$	5.48602	0.198607
$764$	0	0
$765$	0.729126	0.0263616
$766$	0	0
$767$	−9.50240	−0.343112
$768$	0	0
$769$	−6.94427	−0.250417	−0.125208	−	0.992130i	$-0.539960\pi$
$769$	−6.94427	−0.250417	−0.125208	+	0.992130i	$0.539960\pi$
$770$	0	0
$771$	−10.0505	−0.361962
$772$	0	0
$773$	−51.3618	−1.84735	−0.923677	−	0.383171i	$-0.874832\pi$
$773$	−51.3618	−1.84735	−0.923677	+	0.383171i	$0.874832\pi$
$774$	0	0
$775$	−6.27850	−0.225530
$776$	0	0
$777$	3.32719	0.119362
$778$	0	0
$779$	14.8880	0.533416
$780$	0	0
$781$	−4.66964	−0.167093
$782$	0	0
$783$	−14.9865	−0.535574
$784$	0	0
$785$	17.3879	0.620599
$786$	0	0
$787$	19.3365	0.689273	0.344636	−	0.938736i	$-0.388002\pi$
$787$	19.3365	0.689273	0.344636	+	0.938736i	$0.388002\pi$
$788$	0	0
$789$	21.4442	0.763433
$790$	0	0
$791$	1.37339	0.0488320
$792$	0	0
$793$	−3.01388	−0.107026
$794$	0	0
$795$	−3.81775	−0.135402
$796$	0	0
$797$	−4.19173	−0.148479	−0.0742393	−	0.997240i	$-0.523653\pi$
$797$	−4.19173	−0.148479	−0.0742393	+	0.997240i	$0.523653\pi$
$798$	0	0
$799$	−3.63394	−0.128560
$800$	0	0
$801$	−35.3980	−1.25073
$802$	0	0
$803$	−7.00636	−0.247249
$804$	0	0
$805$	7.89045	0.278102
$806$	0	0
$807$	−24.5857	−0.865456
$808$	0	0
$809$	−48.5800	−1.70798	−0.853992	−	0.520287i	$-0.825825\pi$
$809$	−48.5800	−1.70798	−0.853992	+	0.520287i	$0.825825\pi$
$810$	0	0
$811$	−23.6220	−0.829480	−0.414740	−	0.909940i	$-0.636127\pi$
$811$	−23.6220	−0.829480	−0.414740	+	0.909940i	$0.636127\pi$
$812$	0	0
$813$	12.6291	0.442923
$814$	0	0
$815$	1.54234	0.0540257
$816$	0	0
$817$	−5.49101	−0.192106
$818$	0	0
$819$	2.32719	0.0813186
$820$	0	0
$821$	24.8328	0.866671	0.433336	−	0.901233i	$-0.357336\pi$
$821$	24.8328	0.866671	0.433336	+	0.901233i	$0.357336\pi$
$822$	0	0
$823$	45.5230	1.58683	0.793417	−	0.608679i	$-0.208300\pi$
$823$	45.5230	1.58683	0.793417	+	0.608679i	$0.208300\pi$
$824$	0	0
$825$	1.01388	0.0352989
$826$	0	0
$827$	−49.2806	−1.71365	−0.856827	−	0.515603i	$-0.827568\pi$
$827$	−49.2806	−1.71365	−0.856827	+	0.515603i	$0.827568\pi$
$828$	0	0
$829$	29.3225	1.01841	0.509205	−	0.860645i	$-0.329939\pi$
$829$	29.3225	1.01841	0.509205	+	0.860645i	$0.329939\pi$
$830$	0	0
$831$	5.38864	0.186930
$832$	0	0
$833$	−0.313307	−0.0108555
$834$	0	0
$835$	−9.51379	−0.329238
$836$	0	0
$837$	27.4347	0.948282
$838$	0	0
$839$	41.1649	1.42117	0.710585	−	0.703611i	$-0.248430\pi$
$839$	41.1649	1.42117	0.710585	+	0.703611i	$0.248430\pi$
$840$	0	0
$841$	−17.2371	−0.594384
$842$	0	0
$843$	17.2189	0.593051
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	9.47214	0.325466
$848$	0	0
$849$	9.07046	0.311297
$850$	0	0
$851$	32.0062	1.09716
$852$	0	0
$853$	23.1365	0.792179	0.396089	−	0.918212i	$-0.370367\pi$
$853$	23.1365	0.792179	0.396089	+	0.918212i	$0.370367\pi$
$854$	0	0
$855$	4.44232	0.151924
$856$	0	0
$857$	4.07909	0.139339	0.0696696	−	0.997570i	$-0.477806\pi$
$857$	4.07909	0.139339	0.0696696	+	0.997570i	$0.477806\pi$
$858$	0	0
$859$	7.24667	0.247253	0.123627	−	0.992329i	$-0.460547\pi$
$859$	7.24667	0.247253	0.123627	+	0.992329i	$0.460547\pi$
$860$	0	0
$861$	6.39739	0.218023
$862$	0	0
$863$	−23.6776	−0.805995	−0.402998	−	0.915201i	$-0.632032\pi$
$863$	−23.6776	−0.805995	−0.402998	+	0.915201i	$0.632032\pi$
$864$	0	0
$865$	9.80945	0.333531
$866$	0	0
$867$	−13.8637	−0.470837
$868$	0	0
$869$	14.2796	0.484403
$870$	0	0
$871$	−11.4398	−0.387624
$872$	0	0
$873$	−29.7113	−1.00557
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	44.8107	1.51315	0.756575	−	0.653907i	$-0.226871\pi$
$877$	44.8107	1.51315	0.756575	+	0.653907i	$0.226871\pi$
$878$	0	0
$879$	0.219099	0.00739004
$880$	0	0
$881$	−53.8935	−1.81572	−0.907860	−	0.419274i	$-0.862285\pi$
$881$	−53.8935	−1.81572	−0.907860	+	0.419274i	$0.862285\pi$
$882$	0	0
$883$	−18.8256	−0.633533	−0.316767	−	0.948504i	$-0.602597\pi$
$883$	−18.8256	−0.633533	−0.316767	+	0.948504i	$0.602597\pi$
$884$	0	0
$885$	−7.79434	−0.262004
$886$	0	0
$887$	−11.9331	−0.400673	−0.200337	−	0.979727i	$-0.564204\pi$
$887$	−11.9331	−0.400673	−0.200337	+	0.979727i	$0.564204\pi$
$888$	0	0
$889$	−1.23607	−0.0414564
$890$	0	0
$891$	4.19941	0.140685
$892$	0	0
$893$	−22.1404	−0.740900
$894$	0	0
$895$	21.7625	0.727439
$896$	0	0
$897$	−6.47214	−0.216098
$898$	0	0
$899$	−21.5334	−0.718179
$900$	0	0
$901$	1.45825	0.0485814
$902$	0	0
$903$	−2.35950	−0.0785193
$904$	0	0
$905$	21.1190	0.702019
$906$	0	0
$907$	45.8916	1.52381	0.761903	−	0.647691i	$-0.224265\pi$
$907$	45.8916	1.52381	0.761903	+	0.647691i	$0.224265\pi$
$908$	0	0
$909$	42.4137	1.40677
$910$	0	0
$911$	17.7625	0.588497	0.294249	−	0.955729i	$-0.404931\pi$
$911$	17.7625	0.588497	0.294249	+	0.955729i	$0.404931\pi$
$912$	0	0
$913$	12.5342	0.414822
$914$	0	0
$915$	−2.47214	−0.0817263
$916$	0	0
$917$	18.9720	0.626512
$918$	0	0
$919$	−29.6181	−0.977012	−0.488506	−	0.872561i	$-0.662458\pi$
$919$	−29.6181	−0.977012	−0.488506	+	0.872561i	$0.662458\pi$
$920$	0	0
$921$	−4.56952	−0.150571
$922$	0	0
$923$	−3.77782	−0.124348
$924$	0	0
$925$	−4.05632	−0.133371
$926$	0	0
$927$	22.8979	0.752064
$928$	0	0
$929$	−43.1940	−1.41715	−0.708575	−	0.705636i	$-0.750661\pi$
$929$	−43.1940	−1.41715	−0.708575	+	0.705636i	$0.750661\pi$
$930$	0	0
$931$	−1.90888	−0.0625609
$932$	0	0
$933$	−24.9645	−0.817302
$934$	0	0
$935$	−0.387269	−0.0126651
$936$	0	0
$937$	−2.04210	−0.0667125	−0.0333563	−	0.999444i	$-0.510620\pi$
$937$	−2.04210	−0.0667125	−0.0333563	+	0.999444i	$0.510620\pi$
$938$	0	0
$939$	10.6906	0.348874
$940$	0	0
$941$	−17.4170	−0.567780	−0.283890	−	0.958857i	$-0.591625\pi$
$941$	−17.4170	−0.567780	−0.283890	+	0.958857i	$0.591625\pi$
$942$	0	0
$943$	61.5402	2.00402
$944$	0	0
$945$	4.36962	0.142144
$946$	0	0
$947$	35.5852	1.15636	0.578182	−	0.815908i	$-0.303762\pi$
$947$	35.5852	1.15636	0.578182	+	0.815908i	$0.303762\pi$
$948$	0	0
$949$	−5.66827	−0.184000
$950$	0	0
$951$	24.8660	0.806334
$952$	0	0
$953$	−10.2621	−0.332423	−0.166211	−	0.986090i	$-0.553153\pi$
$953$	−10.2621	−0.332423	−0.166211	+	0.986090i	$0.553153\pi$
$954$	0	0
$955$	−11.5842	−0.374855
$956$	0	0
$957$	3.47732	0.112406
$958$	0	0
$959$	11.0056	0.355389
$960$	0	0
$961$	8.41958	0.271599
$962$	0	0
$963$	31.9132	1.02839
$964$	0	0
$965$	−17.5752	−0.565767
$966$	0	0
$967$	11.2727	0.362505	0.181253	−	0.983437i	$-0.441985\pi$
$967$	11.2727	0.362505	0.181253	+	0.983437i	$0.441985\pi$
$968$	0	0
$969$	0.490562	0.0157591
$970$	0	0
$971$	−12.1579	−0.390166	−0.195083	−	0.980787i	$-0.562498\pi$
$971$	−12.1579	−0.390166	−0.195083	+	0.980787i	$0.562498\pi$
$972$	0	0
$973$	11.9304	0.382471
$974$	0	0
$975$	0.820249	0.0262690
$976$	0	0
$977$	8.62843	0.276048	0.138024	−	0.990429i	$-0.455925\pi$
$977$	8.62843	0.276048	0.138024	+	0.990429i	$0.455925\pi$
$978$	0	0
$979$	18.8013	0.600893
$980$	0	0
$981$	12.7670	0.407619
$982$	0	0
$983$	−50.1566	−1.59975	−0.799873	−	0.600170i	$-0.795100\pi$
$983$	−50.1566	−1.59975	−0.799873	+	0.600170i	$0.795100\pi$
$984$	0	0
$985$	−17.9221	−0.571045
$986$	0	0
$987$	−9.51379	−0.302827
$988$	0	0
$989$	−22.6974	−0.721735
$990$	0	0
$991$	−36.9042	−1.17230	−0.586151	−	0.810202i	$-0.699357\pi$
$991$	−36.9042	−1.17230	−0.586151	+	0.810202i	$0.699357\pi$
$992$	0	0
$993$	−29.1567	−0.925261
$994$	0	0
$995$	14.7873	0.468788
$996$	0	0
$997$	−33.0141	−1.04557	−0.522784	−	0.852465i	$-0.675107\pi$
$997$	−33.0141	−1.04557	−0.522784	+	0.852465i	$0.675107\pi$
$998$	0	0
$999$	17.7246	0.560781

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.s.1.3	✓	4	1.1	even	1	trivial
7280.2.a.ca.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)

\(f(q)\)	\(=\)	\(q+0.820249 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.32719 q^{9} +O(q^{10})\)	\(q+0.820249 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.32719 q^{9} +1.23607 q^{11} +1.00000 q^{13} +0.820249 q^{15} -0.313307 q^{17} -1.90888 q^{19} -0.820249 q^{21} -7.89045 q^{23} +1.00000 q^{25} -4.36962 q^{27} +3.42970 q^{29} -6.27850 q^{31} +1.01388 q^{33} -1.00000 q^{35} -4.05632 q^{37} +0.820249 q^{39} -7.79933 q^{41} +2.87657 q^{43} -2.32719 q^{45} +11.5987 q^{47} +1.00000 q^{49} -0.256990 q^{51} -4.65438 q^{53} +1.23607 q^{55} -1.56575 q^{57} -9.50240 q^{59} -3.01388 q^{61} +2.32719 q^{63} +1.00000 q^{65} -11.4398 q^{67} -6.47214 q^{69} -3.77782 q^{71} -5.66827 q^{73} +0.820249 q^{75} -1.23607 q^{77} +11.5525 q^{79} +3.39739 q^{81} +10.1404 q^{83} -0.313307 q^{85} +2.81321 q^{87} +15.2106 q^{89} -1.00000 q^{91} -5.14994 q^{93} -1.90888 q^{95} +12.7670 q^{97} -2.87657 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) \) 4 * q - q^3 + 4 * q^5 - 4 * q^7 - q^9	\( 4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9} - 4 q^{11} + 4 q^{13} - q^{15} - q^{17} - 7 q^{19} + q^{21} - 6 q^{23} + 4 q^{25} - 4 q^{27} + q^{29} - 11 q^{31} - 4 q^{33} - 4 q^{35} - 3 q^{37} - q^{39} - 5 q^{41} - 6 q^{43} - q^{45} - 6 q^{47} + 4 q^{49} - 14 q^{51} - 2 q^{53} - 4 q^{55} + 14 q^{57} - q^{59} - 4 q^{61} + q^{63} + 4 q^{65} - 11 q^{67} - 8 q^{69} - 16 q^{71} + 2 q^{73} - q^{75} + 4 q^{77} - 15 q^{79} - 16 q^{81} - 2 q^{83} - q^{85} - 23 q^{87} - 3 q^{89} - 4 q^{91} - 5 q^{93} - 7 q^{95} + 8 q^{97} + 6 q^{99}+O(q^{100}) \) 4 * q - q^3 + 4 * q^5 - 4 * q^7 - q^9 - 4 * q^11 + 4 * q^13 - q^15 - q^17 - 7 * q^19 + q^21 - 6 * q^23 + 4 * q^25 - 4 * q^27 + q^29 - 11 * q^31 - 4 * q^33 - 4 * q^35 - 3 * q^37 - q^39 - 5 * q^41 - 6 * q^43 - q^45 - 6 * q^47 + 4 * q^49 - 14 * q^51 - 2 * q^53 - 4 * q^55 + 14 * q^57 - q^59 - 4 * q^61 + q^63 + 4 * q^65 - 11 * q^67 - 8 * q^69 - 16 * q^71 + 2 * q^73 - q^75 + 4 * q^77 - 15 * q^79 - 16 * q^81 - 2 * q^83 - q^85 - 23 * q^87 - 3 * q^89 - 4 * q^91 - 5 * q^93 - 7 * q^95 + 8 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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