LMFDB - Embedded newform 3380.2.f.i.3041.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(3041,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("3380.3041");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.22581504.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 $ x^8 - 4x^7 + 5x^6 + 2x^5 - 11x^4 + 4x^3 + 20x^2 - 32*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3041.5
Root			$-1.27597 + 0.609843i$ of defining polynomial
Character	$\chi$	$=$	3380.3041
Dual form			3380.2.f.i.3041.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.60020 q^{3} -1.00000i q^{5} +4.33225i q^{7} -0.439374 q^{9} +O(q^{10})$	$q+1.60020 q^{3} -1.00000i q^{5} +4.33225i q^{7} -0.439374 q^{9} -1.73205i q^{11} -1.60020i q^{15} -7.50367 q^{17} +5.37182i q^{19} +6.93244i q^{21} +1.16082 q^{23} -1.00000 q^{25} -5.50367 q^{27} -2.02473 q^{29} -7.86488i q^{31} -2.77162i q^{33} +4.33225 q^{35} +9.53264i q^{37} -7.73205i q^{41} -4.18555 q^{43} +0.439374i q^{45} -3.46410i q^{47} -11.7684 q^{49} -12.0073 q^{51} -12.6471 q^{53} -1.73205 q^{55} +8.59596i q^{57} +6.34709i q^{59} +3.70308 q^{61} -1.90348i q^{63} +5.26045i q^{67} +1.85754 q^{69} -12.5143i q^{71} +5.23898i q^{73} -1.60020 q^{75} +7.50367 q^{77} -8.16719 q^{79} -7.48883 q^{81} +0.456760i q^{83} +7.50367i q^{85} -3.23996 q^{87} +13.2753i q^{89} -12.5854i q^{93} +5.37182 q^{95} -2.81021i q^{97} +0.761018i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} + 8 q^{9}+O(q^{10}) $ 8 * q + 4 * q^3 + 8 * q^9	$ 8 q + 4 q^{3} + 8 q^{9} - 12 q^{17} + 12 q^{23} - 8 q^{25} + 4 q^{27} + 12 q^{35} - 20 q^{43} + 8 q^{49} + 24 q^{53} + 8 q^{61} + 48 q^{69} - 4 q^{75} + 12 q^{77} - 16 q^{79} - 16 q^{81} + 12 q^{87}+O(q^{100}) $ 8 * q + 4 * q^3 + 8 * q^9 - 12 * q^17 + 12 * q^23 - 8 * q^25 + 4 * q^27 + 12 * q^35 - 20 * q^43 + 8 * q^49 + 24 * q^53 + 8 * q^61 + 48 * q^69 - 4 * q^75 + 12 * q^77 - 16 * q^79 - 16 * q^81 + 12 * q^87

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$.

$n$	$677$	$1691$	$1861$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.60020	0.923873	0.461937	−	0.886913i	$-0.347155\pi$
$3$	1.60020	0.923873	0.461937	+	0.886913i	$0.347155\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	4.33225i	1.63744i	0.574196	+	0.818718i	$0.305315\pi$
$7$	4.33225i	1.63744i	−0.574196	+	0.818718i	$0.694685\pi$
$8$	0	0
$9$	−0.439374	−0.146458
$10$	0	0
$11$	− 1.73205i	− 0.522233i	−0.965307	−	0.261116i	$-0.915909\pi$
$11$	− 1.73205i	− 0.522233i	0.965307	−	0.261116i	$-0.0840907\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 1.60020i	− 0.413169i
$16$	0	0
$17$	−7.50367	−1.81991	−0.909954	−	0.414710i	$-0.863883\pi$
$17$	−7.50367	−1.81991	−0.909954	+	0.414710i	$0.863883\pi$
$18$	0	0
$19$	5.37182i	1.23238i	0.787598	+	0.616190i	$0.211324\pi$
$19$	5.37182i	1.23238i	−0.787598	+	0.616190i	$0.788676\pi$
$20$	0	0
$21$	6.93244i	1.51278i
$22$	0	0
$23$	1.16082	0.242048	0.121024	−	0.992650i	$-0.461382\pi$
$23$	1.16082	0.242048	0.121024	+	0.992650i	$0.461382\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.50367	−1.05918
$28$	0	0
$29$	−2.02473	−0.375983	−0.187991	−	0.982171i	$-0.560198\pi$
$29$	−2.02473	−0.375983	−0.187991	+	0.982171i	$0.560198\pi$
$30$	0	0
$31$	− 7.86488i	− 1.41257i	−0.707925	−	0.706287i	$-0.750369\pi$
$31$	− 7.86488i	− 1.41257i	0.707925	−	0.706287i	$-0.249631\pi$
$32$	0	0
$33$	− 2.77162i	− 0.482477i
$34$	0	0
$35$	4.33225	0.732283
$36$	0	0
$37$	9.53264i	1.56716i	0.621293	+	0.783578i	$0.286608\pi$
$37$	9.53264i	1.56716i	−0.621293	+	0.783578i	$0.713392\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 7.73205i	− 1.20754i	−0.797157	−	0.603772i	$-0.793664\pi$
$41$	− 7.73205i	− 1.20754i	0.797157	−	0.603772i	$-0.206336\pi$
$42$	0	0
$43$	−4.18555	−0.638290	−0.319145	−	0.947706i	$-0.603396\pi$
$43$	−4.18555	−0.638290	−0.319145	+	0.947706i	$0.603396\pi$
$44$	0	0
$45$	0.439374i	0.0654980i
$46$	0	0
$47$	− 3.46410i	− 0.505291i	−0.967559	−	0.252646i	$-0.918699\pi$
$47$	− 3.46410i	− 0.505291i	0.967559	−	0.252646i	$-0.0813007\pi$
$48$	0	0
$49$	−11.7684	−1.68119
$50$	0	0
$51$	−12.0073	−1.68136
$52$	0	0
$53$	−12.6471	−1.73721	−0.868607	−	0.495502i	$-0.834984\pi$
$53$	−12.6471	−1.73721	−0.868607	+	0.495502i	$0.834984\pi$
$54$	0	0
$55$	−1.73205	−0.233550
$56$	0	0
$57$	8.59596i	1.13856i
$58$	0	0
$59$	6.34709i	0.826320i	0.910658	+	0.413160i	$0.135575\pi$
$59$	6.34709i	0.826320i	−0.910658	+	0.413160i	$0.864425\pi$
$60$	0	0
$61$	3.70308	0.474131	0.237066	−	0.971494i	$-0.423814\pi$
$61$	3.70308	0.474131	0.237066	+	0.971494i	$0.423814\pi$
$62$	0	0
$63$	− 1.90348i	− 0.239815i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.26045i	0.642666i	0.946966	+	0.321333i	$0.104131\pi$
$67$	5.26045i	0.642666i	−0.946966	+	0.321333i	$0.895869\pi$
$68$	0	0
$69$	1.85754	0.223622
$70$	0	0
$71$	− 12.5143i	− 1.48517i	−0.669751	−	0.742586i	$-0.733599\pi$
$71$	− 12.5143i	− 1.48517i	0.669751	−	0.742586i	$-0.266401\pi$
$72$	0	0
$73$	5.23898i	0.613177i	0.951842	+	0.306588i	$0.0991875\pi$
$73$	5.23898i	0.613177i	−0.951842	+	0.306588i	$0.900813\pi$
$74$	0	0
$75$	−1.60020	−0.184775
$76$	0	0
$77$	7.50367	0.855123
$78$	0	0
$79$	−8.16719	−0.918880	−0.459440	−	0.888209i	$-0.651950\pi$
$79$	−8.16719	−0.918880	−0.459440	+	0.888209i	$0.651950\pi$
$80$	0	0
$81$	−7.48883	−0.832092
$82$	0	0
$83$	0.456760i	0.0501359i	0.999686	+	0.0250679i	$0.00798021\pi$
$83$	0.456760i	0.0501359i	−0.999686	+	0.0250679i	$0.992020\pi$
$84$	0	0
$85$	7.50367i	0.813887i
$86$	0	0
$87$	−3.23996	−0.347360
$88$	0	0
$89$	13.2753i	1.40718i	0.710607	+	0.703589i	$0.248420\pi$
$89$	13.2753i	1.40718i	−0.710607	+	0.703589i	$0.751580\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 12.5854i	− 1.30504i
$94$	0	0
$95$	5.37182	0.551137
$96$	0	0
$97$	− 2.81021i	− 0.285334i	−0.989771	−	0.142667i	$-0.954432\pi$
$97$	− 2.81021i	− 0.285334i	0.989771	−	0.142667i	$-0.0455678\pi$
$98$	0	0
$99$	0.761018i	0.0764852i
$100$	0	0
$101$	6.63977	0.660681	0.330341	−	0.943862i	$-0.392836\pi$
$101$	6.63977	0.660681	0.330341	+	0.943862i	$0.392836\pi$
$102$	0	0
$103$	7.37605	0.726784	0.363392	−	0.931636i	$-0.381619\pi$
$103$	7.37605	0.726784	0.363392	+	0.931636i	$0.381619\pi$
$104$	0	0
$105$	6.93244	0.676537
$106$	0	0
$107$	−7.91336	−0.765014	−0.382507	−	0.923953i	$-0.624939\pi$
$107$	−7.91336	−0.765014	−0.382507	+	0.923953i	$0.624939\pi$
$108$	0	0
$109$	9.18301i	0.879572i	0.898102	+	0.439786i	$0.144946\pi$
$109$	9.18301i	0.879572i	−0.898102	+	0.439786i	$0.855054\pi$
$110$	0	0
$111$	15.2541i	1.44785i
$112$	0	0
$113$	−5.77586	−0.543347	−0.271674	−	0.962389i	$-0.587577\pi$
$113$	−5.77586	−0.543347	−0.271674	+	0.962389i	$0.587577\pi$
$114$	0	0
$115$	− 1.16082i	− 0.108247i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 32.5078i	− 2.97998i
$120$	0	0
$121$	8.00000	0.727273
$122$	0	0
$123$	− 12.3728i	− 1.11562i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−18.9678	−1.68312	−0.841559	−	0.540165i	$-0.818362\pi$
$127$	−18.9678	−1.68312	−0.841559	+	0.540165i	$0.818362\pi$
$128$	0	0
$129$	−6.69770	−0.589699
$130$	0	0
$131$	−1.34637	−0.117633	−0.0588165	−	0.998269i	$-0.518733\pi$
$131$	−1.34637	−0.117633	−0.0588165	+	0.998269i	$0.518733\pi$
$132$	0	0
$133$	−23.2720	−2.01794
$134$	0	0
$135$	5.50367i	0.473681i
$136$	0	0
$137$	− 12.3728i	− 1.05708i	−0.848909	−	0.528540i	$-0.822740\pi$
$137$	− 12.3728i	− 1.05708i	0.848909	−	0.528540i	$-0.177260\pi$
$138$	0	0
$139$	8.76836	0.743723	0.371861	−	0.928288i	$-0.378720\pi$
$139$	8.76836	0.743723	0.371861	+	0.928288i	$0.378720\pi$
$140$	0	0
$141$	− 5.54324i	− 0.466825i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.02473i	0.168144i
$146$	0	0
$147$	−18.8317	−1.55321
$148$	0	0
$149$	4.73939i	0.388266i	0.980975	+	0.194133i	$0.0621894\pi$
$149$	4.73939i	0.388266i	−0.980975	+	0.194133i	$0.937811\pi$
$150$	0	0
$151$	1.56063i	0.127002i	0.997982	+	0.0635010i	$0.0202266\pi$
$151$	1.56063i	0.127002i	−0.997982	+	0.0635010i	$0.979773\pi$
$152$	0	0
$153$	3.29692	0.266540
$154$	0	0
$155$	−7.86488	−0.631723
$156$	0	0
$157$	−9.15332	−0.730515	−0.365257	−	0.930907i	$-0.619019\pi$
$157$	−9.15332	−0.730515	−0.365257	+	0.930907i	$0.619019\pi$
$158$	0	0
$159$	−20.2378	−1.60497
$160$	0	0
$161$	5.02897i	0.396338i
$162$	0	0
$163$	2.64303i	0.207018i	0.994629	+	0.103509i	$0.0330070\pi$
$163$	2.64303i	0.207018i	−0.994629	+	0.103509i	$0.966993\pi$
$164$	0	0
$165$	−2.77162	−0.215770
$166$	0	0
$167$	− 8.90869i	− 0.689375i	−0.938717	−	0.344688i	$-0.887985\pi$
$167$	− 8.90869i	− 0.689375i	0.938717	−	0.344688i	$-0.112015\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 2.36023i	− 0.180492i
$172$	0	0
$173$	−13.8079	−1.04980	−0.524899	−	0.851165i	$-0.675897\pi$
$173$	−13.8079	−1.04980	−0.524899	+	0.851165i	$0.675897\pi$
$174$	0	0
$175$	− 4.33225i	− 0.327487i
$176$	0	0
$177$	10.1566i	0.763416i
$178$	0	0
$179$	24.3502	1.82002	0.910009	−	0.414588i	$-0.136074\pi$
$179$	24.3502	1.82002	0.910009	+	0.414588i	$0.136074\pi$
$180$	0	0
$181$	24.8336	1.84587	0.922935	−	0.384956i	$-0.125783\pi$
$181$	24.8336	1.84587	0.922935	+	0.384956i	$0.125783\pi$
$182$	0	0
$183$	5.92566	0.438037
$184$	0	0
$185$	9.53264	0.700853
$186$	0	0
$187$	12.9967i	0.950416i
$188$	0	0
$189$	− 23.8433i	− 1.73434i
$190$	0	0
$191$	5.96875	0.431884	0.215942	−	0.976406i	$-0.430718\pi$
$191$	5.96875	0.431884	0.215942	+	0.976406i	$0.430718\pi$
$192$	0	0
$193$	9.05272i	0.651629i	0.945434	+	0.325814i	$0.105638\pi$
$193$	9.05272i	0.651629i	−0.945434	+	0.325814i	$0.894362\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.3890i	0.811436i	0.913998	+	0.405718i	$0.132978\pi$
$197$	11.3890i	0.811436i	−0.913998	+	0.405718i	$0.867022\pi$
$198$	0	0
$199$	−10.7773	−0.763980	−0.381990	−	0.924166i	$-0.624761\pi$
$199$	−10.7773	−0.763980	−0.381990	+	0.924166i	$0.624761\pi$
$200$	0	0
$201$	8.41775i	0.593742i
$202$	0	0
$203$	− 8.77162i	− 0.615647i
$204$	0	0
$205$	−7.73205	−0.540030
$206$	0	0
$207$	−0.510035	−0.0354499
$208$	0	0
$209$	9.30426	0.643589
$210$	0	0
$211$	−3.22512	−0.222026	−0.111013	−	0.993819i	$-0.535410\pi$
$211$	−3.22512	−0.222026	−0.111013	+	0.993819i	$0.535410\pi$
$212$	0	0
$213$	− 20.0253i	− 1.37211i
$214$	0	0
$215$	4.18555i	0.285452i
$216$	0	0
$217$	34.0726	2.31300
$218$	0	0
$219$	8.38340i	0.566497i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	16.9794i	1.13702i	0.822676	+	0.568511i	$0.192480\pi$
$223$	16.9794i	1.13702i	−0.822676	+	0.568511i	$0.807520\pi$
$224$	0	0
$225$	0.439374	0.0292916
$226$	0	0
$227$	2.40052i	0.159328i	0.996822	+	0.0796641i	$0.0253848\pi$
$227$	2.40052i	0.159328i	−0.996822	+	0.0796641i	$0.974615\pi$
$228$	0	0
$229$	7.35671i	0.486145i	0.970008	+	0.243073i	$0.0781553\pi$
$229$	7.35671i	0.486145i	−0.970008	+	0.243073i	$0.921845\pi$
$230$	0	0
$231$	12.0073	0.790025
$232$	0	0
$233$	−3.37410	−0.221045	−0.110522	−	0.993874i	$-0.535252\pi$
$233$	−3.37410	−0.221045	−0.110522	+	0.993874i	$0.535252\pi$
$234$	0	0
$235$	−3.46410	−0.225973
$236$	0	0
$237$	−13.0691	−0.848929
$238$	0	0
$239$	23.7807i	1.53824i	0.639103	+	0.769121i	$0.279306\pi$
$239$	23.7807i	1.53824i	−0.639103	+	0.769121i	$0.720694\pi$
$240$	0	0
$241$	22.3212i	1.43784i	0.695095	+	0.718918i	$0.255362\pi$
$241$	22.3212i	1.43784i	−0.695095	+	0.718918i	$0.744638\pi$
$242$	0	0
$243$	4.52742	0.290434
$244$	0	0
$245$	11.7684i	0.751853i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0.730905i	0.0463192i
$250$	0	0
$251$	4.62238	0.291762	0.145881	−	0.989302i	$-0.453398\pi$
$251$	4.62238	0.291762	0.145881	+	0.989302i	$0.453398\pi$
$252$	0	0
$253$	− 2.01060i	− 0.126406i
$254$	0	0
$255$	12.0073i	0.751929i
$256$	0	0
$257$	−21.8639	−1.36383	−0.681916	−	0.731430i	$-0.738853\pi$
$257$	−21.8639	−1.36383	−0.681916	+	0.731430i	$0.738853\pi$
$258$	0	0
$259$	−41.2977	−2.56612
$260$	0	0
$261$	0.889612	0.0550656
$262$	0	0
$263$	−20.1790	−1.24429	−0.622146	−	0.782901i	$-0.713739\pi$
$263$	−20.1790	−1.24429	−0.622146	+	0.782901i	$0.713739\pi$
$264$	0	0
$265$	12.6471i	0.776906i
$266$	0	0
$267$	21.2431i	1.30005i
$268$	0	0
$269$	7.28687	0.444288	0.222144	−	0.975014i	$-0.428694\pi$
$269$	7.28687	0.444288	0.222144	+	0.975014i	$0.428694\pi$
$270$	0	0
$271$	4.89189i	0.297161i	0.988900	+	0.148581i	$0.0474705\pi$
$271$	4.89189i	0.297161i	−0.988900	+	0.148581i	$0.952530\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.73205i	0.104447i
$276$	0	0
$277$	−8.80059	−0.528776	−0.264388	−	0.964416i	$-0.585170\pi$
$277$	−8.80059	−0.528776	−0.264388	+	0.964416i	$0.585170\pi$
$278$	0	0
$279$	3.45562i	0.206883i
$280$	0	0
$281$	− 12.6085i	− 0.752161i	−0.926587	−	0.376081i	$-0.877272\pi$
$281$	− 12.6085i	− 0.752161i	0.926587	−	0.376081i	$-0.122728\pi$
$282$	0	0
$283$	20.9009	1.24243	0.621216	−	0.783640i	$-0.286639\pi$
$283$	20.9009	1.24243	0.621216	+	0.783640i	$0.286639\pi$
$284$	0	0
$285$	8.59596	0.509181
$286$	0	0
$287$	33.4972	1.97727
$288$	0	0
$289$	39.3051	2.31206
$290$	0	0
$291$	− 4.49689i	− 0.263612i
$292$	0	0
$293$	− 0.692481i	− 0.0404552i	−0.999795	−	0.0202276i	$-0.993561\pi$
$293$	− 0.692481i	− 0.0404552i	0.999795	−	0.0202276i	$-0.00643908\pi$
$294$	0	0
$295$	6.34709	0.369542
$296$	0	0
$297$	9.53264i	0.553140i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 18.1328i	− 1.04516i
$302$	0	0
$303$	10.6249	0.610386
$304$	0	0
$305$	− 3.70308i	− 0.212038i
$306$	0	0
$307$	8.83168i	0.504051i	0.967721	+	0.252025i	$0.0810966\pi$
$307$	8.83168i	0.504051i	−0.967721	+	0.252025i	$0.918903\pi$
$308$	0	0
$309$	11.8031	0.671457
$310$	0	0
$311$	−11.3958	−0.646198	−0.323099	−	0.946365i	$-0.604725\pi$
$311$	−11.3958	−0.646198	−0.323099	+	0.946365i	$0.604725\pi$
$312$	0	0
$313$	−3.38496	−0.191329	−0.0956647	−	0.995414i	$-0.530498\pi$
$313$	−3.38496	−0.191329	−0.0956647	+	0.995414i	$0.530498\pi$
$314$	0	0
$315$	−1.90348	−0.107249
$316$	0	0
$317$	− 0.557104i	− 0.0312901i	−0.999878	−	0.0156450i	$-0.995020\pi$
$317$	− 0.557104i	− 0.0312901i	0.999878	−	0.0156450i	$-0.00498017\pi$
$318$	0	0
$319$	3.50693i	0.196350i
$320$	0	0
$321$	−12.6629	−0.706776
$322$	0	0
$323$	− 40.3083i	− 2.24282i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	14.6946i	0.812614i
$328$	0	0
$329$	15.0073	0.827382
$330$	0	0
$331$	3.96369i	0.217864i	0.994049	+	0.108932i	$0.0347431\pi$
$331$	3.96369i	0.217864i	−0.994049	+	0.108932i	$0.965257\pi$
$332$	0	0
$333$	− 4.18839i	− 0.229522i
$334$	0	0
$335$	5.26045	0.287409
$336$	0	0
$337$	16.7243	0.911030	0.455515	−	0.890228i	$-0.349455\pi$
$337$	16.7243	0.911030	0.455515	+	0.890228i	$0.349455\pi$
$338$	0	0
$339$	−9.24251	−0.501984
$340$	0	0
$341$	−13.6224	−0.737693
$342$	0	0
$343$	− 20.6577i	− 1.11541i
$344$	0	0
$345$	− 1.85754i	− 0.100007i
$346$	0	0
$347$	7.58085	0.406962	0.203481	−	0.979079i	$-0.434775\pi$
$347$	7.58085	0.406962	0.203481	+	0.979079i	$0.434775\pi$
$348$	0	0
$349$	− 27.3356i	− 1.46324i	−0.681712	−	0.731621i	$-0.738764\pi$
$349$	− 27.3356i	− 1.46324i	0.681712	−	0.731621i	$-0.261236\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 26.0921i	− 1.38874i	−0.719616	−	0.694372i	$-0.755682\pi$
$353$	− 26.0921i	− 1.38874i	0.719616	−	0.694372i	$-0.244318\pi$
$354$	0	0
$355$	−12.5143	−0.664189
$356$	0	0
$357$	− 52.0188i	− 2.75313i
$358$	0	0
$359$	0.694176i	0.0366372i	0.999832	+	0.0183186i	$0.00583132\pi$
$359$	0.694176i	0.0366372i	−0.999832	+	0.0183186i	$0.994169\pi$
$360$	0	0
$361$	−9.85641	−0.518758
$362$	0	0
$363$	12.8016	0.671908
$364$	0	0
$365$	5.23898	0.274221
$366$	0	0
$367$	22.8218	1.19129	0.595644	−	0.803249i	$-0.296897\pi$
$367$	22.8218	1.19129	0.595644	+	0.803249i	$0.296897\pi$
$368$	0	0
$369$	3.39726i	0.176854i
$370$	0	0
$371$	− 54.7904i	− 2.84458i
$372$	0	0
$373$	−7.88129	−0.408078	−0.204039	−	0.978963i	$-0.565407\pi$
$373$	−7.88129	−0.408078	−0.204039	+	0.978963i	$0.565407\pi$
$374$	0	0
$375$	1.60020i	0.0826338i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 31.1426i	− 1.59969i	−0.600209	−	0.799843i	$-0.704916\pi$
$379$	− 31.1426i	− 1.59969i	0.600209	−	0.799843i	$-0.295084\pi$
$380$	0	0
$381$	−30.3521	−1.55499
$382$	0	0
$383$	34.5123i	1.76349i	0.471723	+	0.881747i	$0.343632\pi$
$383$	34.5123i	1.76349i	−0.471723	+	0.881747i	$0.656368\pi$
$384$	0	0
$385$	− 7.50367i	− 0.382422i
$386$	0	0
$387$	1.83902	0.0934827
$388$	0	0
$389$	−6.18414	−0.313548	−0.156774	−	0.987634i	$-0.550109\pi$
$389$	−6.18414	−0.313548	−0.156774	+	0.987634i	$0.550109\pi$
$390$	0	0
$391$	−8.71043	−0.440505
$392$	0	0
$393$	−2.15446	−0.108678
$394$	0	0
$395$	8.16719i	0.410936i
$396$	0	0
$397$	22.6563i	1.13709i	0.822654	+	0.568543i	$0.192493\pi$
$397$	22.6563i	1.13709i	−0.822654	+	0.568543i	$0.807507\pi$
$398$	0	0
$399$	−37.2398	−1.86432
$400$	0	0
$401$	− 3.87803i	− 0.193660i	−0.995301	−	0.0968298i	$-0.969130\pi$
$401$	− 3.87803i	− 0.193660i	0.995301	−	0.0968298i	$-0.0308703\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	7.48883i	0.372123i
$406$	0	0
$407$	16.5110	0.818421
$408$	0	0
$409$	13.8606i	0.685365i	0.939451	+	0.342682i	$0.111335\pi$
$409$	13.8606i	0.685365i	−0.939451	+	0.342682i	$0.888665\pi$
$410$	0	0
$411$	− 19.7989i	− 0.976607i
$412$	0	0
$413$	−27.4972	−1.35305
$414$	0	0
$415$	0.456760	0.0224214
$416$	0	0
$417$	14.0311	0.687106
$418$	0	0
$419$	−30.3502	−1.48270	−0.741352	−	0.671116i	$-0.765815\pi$
$419$	−30.3502	−1.48270	−0.741352	+	0.671116i	$0.765815\pi$
$420$	0	0
$421$	0.608516i	0.0296573i	0.999890	+	0.0148286i	$0.00472027\pi$
$421$	0.608516i	0.0296573i	−0.999890	+	0.0148286i	$0.995280\pi$
$422$	0	0
$423$	1.52204i	0.0740039i
$424$	0	0
$425$	7.50367	0.363982
$426$	0	0
$427$	16.0427i	0.776359i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	39.3544i	1.89564i	0.318812	+	0.947818i	$0.396716\pi$
$431$	39.3544i	1.89564i	−0.318812	+	0.947818i	$0.603284\pi$
$432$	0	0
$433$	−6.42949	−0.308981	−0.154491	−	0.987994i	$-0.549374\pi$
$433$	−6.42949	−0.308981	−0.154491	+	0.987994i	$0.549374\pi$
$434$	0	0
$435$	3.23996i	0.155344i
$436$	0	0
$437$	6.23572i	0.298295i
$438$	0	0
$439$	−1.47144	−0.0702282	−0.0351141	−	0.999383i	$-0.511179\pi$
$439$	−1.47144	−0.0702282	−0.0351141	+	0.999383i	$0.511179\pi$
$440$	0	0
$441$	5.17071	0.246224
$442$	0	0
$443$	−16.9193	−0.803860	−0.401930	−	0.915670i	$-0.631660\pi$
$443$	−16.9193	−0.803860	−0.401930	+	0.915670i	$0.631660\pi$
$444$	0	0
$445$	13.2753	0.629309
$446$	0	0
$447$	7.58396i	0.358709i
$448$	0	0
$449$	− 9.11701i	− 0.430258i	−0.976586	−	0.215129i	$-0.930983\pi$
$449$	− 9.11701i	− 0.430258i	0.976586	−	0.215129i	$-0.0690173\pi$
$450$	0	0
$451$	−13.3923	−0.630619
$452$	0	0
$453$	2.49731i	0.117334i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.5562i	0.821246i	0.911805	+	0.410623i	$0.134689\pi$
$457$	17.5562i	0.821246i	−0.911805	+	0.410623i	$0.865311\pi$
$458$	0	0
$459$	41.2977	1.92761
$460$	0	0
$461$	− 15.3544i	− 0.715127i	−0.933889	−	0.357564i	$-0.883608\pi$
$461$	− 15.3544i	− 0.715127i	0.933889	−	0.357564i	$-0.116392\pi$
$462$	0	0
$463$	− 18.8614i	− 0.876562i	−0.898838	−	0.438281i	$-0.855588\pi$
$463$	− 18.8614i	− 0.876562i	0.898838	−	0.438281i	$-0.144412\pi$
$464$	0	0
$465$	−12.5854	−0.583632
$466$	0	0
$467$	32.9302	1.52383	0.761913	−	0.647679i	$-0.224260\pi$
$467$	32.9302	1.52383	0.761913	+	0.647679i	$0.224260\pi$
$468$	0	0
$469$	−22.7896	−1.05232
$470$	0	0
$471$	−14.6471	−0.674903
$472$	0	0
$473$	7.24958i	0.333336i
$474$	0	0
$475$	− 5.37182i	− 0.246476i
$476$	0	0
$477$	5.55681	0.254429
$478$	0	0
$479$	17.2476i	0.788061i	0.919097	+	0.394031i	$0.128920\pi$
$479$	17.2476i	0.788061i	−0.919097	+	0.394031i	$0.871080\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	8.04733i	0.366166i
$484$	0	0
$485$	−2.81021	−0.127605
$486$	0	0
$487$	− 17.9744i	− 0.814498i	−0.913317	−	0.407249i	$-0.866488\pi$
$487$	− 17.9744i	− 0.814498i	0.913317	−	0.407249i	$-0.133512\pi$
$488$	0	0
$489$	4.22936i	0.191258i
$490$	0	0
$491$	−26.1953	−1.18218	−0.591089	−	0.806607i	$-0.701302\pi$
$491$	−26.1953	−1.18218	−0.591089	+	0.806607i	$0.701302\pi$
$492$	0	0
$493$	15.1929	0.684253
$494$	0	0
$495$	0.761018	0.0342052
$496$	0	0
$497$	54.2149	2.43187
$498$	0	0
$499$	− 18.3428i	− 0.821139i	−0.911829	−	0.410569i	$-0.865330\pi$
$499$	− 18.3428i	− 0.821139i	0.911829	−	0.410569i	$-0.134670\pi$
$500$	0	0
$501$	− 14.2557i	− 0.636896i
$502$	0	0
$503$	−14.2562	−0.635653	−0.317827	−	0.948149i	$-0.602953\pi$
$503$	−14.2562	−0.635653	−0.317827	+	0.948149i	$0.602953\pi$
$504$	0	0
$505$	− 6.63977i	− 0.295466i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 31.7981i	− 1.40943i	−0.709491	−	0.704714i	$-0.751075\pi$
$509$	− 31.7981i	− 1.40943i	0.709491	−	0.704714i	$-0.248925\pi$
$510$	0	0
$511$	−22.6966	−1.00404
$512$	0	0
$513$	− 29.5647i	− 1.30531i
$514$	0	0
$515$	− 7.37605i	− 0.325028i
$516$	0	0
$517$	−6.00000	−0.263880
$518$	0	0
$519$	−22.0954	−0.969880
$520$	0	0
$521$	−45.1676	−1.97883	−0.989414	−	0.145122i	$-0.953642\pi$
$521$	−45.1676	−1.97883	−0.989414	+	0.145122i	$0.953642\pi$
$522$	0	0
$523$	26.1146	1.14191	0.570955	−	0.820981i	$-0.306573\pi$
$523$	26.1146	1.14191	0.570955	+	0.820981i	$0.306573\pi$
$524$	0	0
$525$	− 6.93244i	− 0.302557i
$526$	0	0
$527$	59.0155i	2.57076i
$528$	0	0
$529$	−21.6525	−0.941413
$530$	0	0
$531$	− 2.78874i	− 0.121021i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	7.91336i	0.342124i
$536$	0	0
$537$	38.9651	1.68147
$538$	0	0
$539$	20.3834i	0.877975i
$540$	0	0
$541$	31.4750i	1.35321i	0.736344	+	0.676607i	$0.236550\pi$
$541$	31.4750i	1.35321i	−0.736344	+	0.676607i	$0.763450\pi$
$542$	0	0
$543$	39.7387	1.70535
$544$	0	0
$545$	9.18301	0.393357
$546$	0	0
$547$	−5.70391	−0.243881	−0.121941	−	0.992537i	$-0.538912\pi$
$547$	−5.70391	−0.243881	−0.121941	+	0.992537i	$0.538912\pi$
$548$	0	0
$549$	−1.62704	−0.0694403
$550$	0	0
$551$	− 10.8765i	− 0.463353i
$552$	0	0
$553$	− 35.3823i	− 1.50461i
$554$	0	0
$555$	15.2541	0.647500
$556$	0	0
$557$	− 20.8720i	− 0.884374i	−0.896923	−	0.442187i	$-0.854203\pi$
$557$	− 20.8720i	− 0.884374i	0.896923	−	0.442187i	$-0.145797\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	20.7973i	0.878064i
$562$	0	0
$563$	33.2315	1.40054	0.700270	−	0.713878i	$-0.253063\pi$
$563$	33.2315	1.40054	0.700270	+	0.713878i	$0.253063\pi$
$564$	0	0
$565$	5.77586i	0.242992i
$566$	0	0
$567$	− 32.4435i	− 1.36250i
$568$	0	0
$569$	40.0671	1.67970	0.839851	−	0.542817i	$-0.182642\pi$
$569$	40.0671	1.67970	0.839851	+	0.542817i	$0.182642\pi$
$570$	0	0
$571$	−4.53590	−0.189821	−0.0949107	−	0.995486i	$-0.530257\pi$
$571$	−4.53590	−0.189821	−0.0949107	+	0.995486i	$0.530257\pi$
$572$	0	0
$573$	9.55117	0.399006
$574$	0	0
$575$	−1.16082	−0.0484096
$576$	0	0
$577$	− 35.2706i	− 1.46834i	−0.678968	−	0.734168i	$-0.737573\pi$
$577$	− 35.2706i	− 1.46834i	0.678968	−	0.734168i	$-0.262427\pi$
$578$	0	0
$579$	14.4861i	0.602023i
$580$	0	0
$581$	−1.97879	−0.0820942
$582$	0	0
$583$	21.9054i	0.907230i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	41.1663i	1.69912i	0.527496	+	0.849558i	$0.323131\pi$
$587$	41.1663i	1.69912i	−0.527496	+	0.849558i	$0.676869\pi$
$588$	0	0
$589$	42.2487	1.74083
$590$	0	0
$591$	18.2247i	0.749664i
$592$	0	0
$593$	9.39726i	0.385899i	0.981209	+	0.192950i	$0.0618054\pi$
$593$	9.39726i	0.385899i	−0.981209	+	0.192950i	$0.938195\pi$
$594$	0	0
$595$	−32.5078	−1.33269
$596$	0	0
$597$	−17.2457	−0.705821
$598$	0	0
$599$	21.7881	0.890239	0.445119	−	0.895471i	$-0.353161\pi$
$599$	21.7881	0.890239	0.445119	+	0.895471i	$0.353161\pi$
$600$	0	0
$601$	25.0962	1.02370	0.511848	−	0.859076i	$-0.328961\pi$
$601$	25.0962	1.02370	0.511848	+	0.859076i	$0.328961\pi$
$602$	0	0
$603$	− 2.31130i	− 0.0941236i
$604$	0	0
$605$	− 8.00000i	− 0.325246i
$606$	0	0
$607$	−33.9775	−1.37910	−0.689552	−	0.724236i	$-0.742193\pi$
$607$	−33.9775	−1.37910	−0.689552	+	0.724236i	$0.742193\pi$
$608$	0	0
$609$	− 14.0363i	− 0.568780i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 21.4470i	− 0.866235i	−0.901337	−	0.433118i	$-0.857413\pi$
$613$	− 21.4470i	− 0.866235i	0.901337	−	0.433118i	$-0.142587\pi$
$614$	0	0
$615$	−12.3728	−0.498919
$616$	0	0
$617$	34.0743i	1.37178i	0.727705	+	0.685890i	$0.240587\pi$
$617$	34.0743i	1.37178i	−0.727705	+	0.685890i	$0.759413\pi$
$618$	0	0
$619$	− 26.9791i	− 1.08438i	−0.840256	−	0.542191i	$-0.817595\pi$
$619$	− 26.9791i	− 1.08438i	0.840256	−	0.542191i	$-0.182405\pi$
$620$	0	0
$621$	−6.38878	−0.256373
$622$	0	0
$623$	−57.5118	−2.30416
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	14.8886	0.594595
$628$	0	0
$629$	− 71.5298i	− 2.85208i
$630$	0	0
$631$	− 37.9738i	− 1.51171i	−0.654737	−	0.755857i	$-0.727221\pi$
$631$	− 37.9738i	− 1.51171i	0.654737	−	0.755857i	$-0.272779\pi$
$632$	0	0
$633$	−5.16082	−0.205124
$634$	0	0
$635$	18.9678i	0.752713i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	5.49844i	0.217515i
$640$	0	0
$641$	8.25180	0.325927	0.162963	−	0.986632i	$-0.447895\pi$
$641$	8.25180	0.325927	0.162963	+	0.986632i	$0.447895\pi$
$642$	0	0
$643$	48.1501i	1.89885i	0.313989	+	0.949427i	$0.398334\pi$
$643$	48.1501i	1.89885i	−0.313989	+	0.949427i	$0.601666\pi$
$644$	0	0
$645$	6.69770i	0.263722i
$646$	0	0
$647$	2.17903	0.0856665	0.0428332	−	0.999082i	$-0.486362\pi$
$647$	2.17903	0.0856665	0.0428332	+	0.999082i	$0.486362\pi$
$648$	0	0
$649$	10.9935	0.431532
$650$	0	0
$651$	54.5229	2.13692
$652$	0	0
$653$	33.8682	1.32537	0.662684	−	0.748900i	$-0.269417\pi$
$653$	33.8682	1.32537	0.662684	+	0.748900i	$0.269417\pi$
$654$	0	0
$655$	1.34637i	0.0526071i
$656$	0	0
$657$	− 2.30187i	− 0.0898046i
$658$	0	0
$659$	26.4491	1.03031	0.515155	−	0.857097i	$-0.327734\pi$
$659$	26.4491	1.03031	0.515155	+	0.857097i	$0.327734\pi$
$660$	0	0
$661$	30.3985i	1.18236i	0.806538	+	0.591182i	$0.201339\pi$
$661$	30.3985i	1.18236i	−0.806538	+	0.591182i	$0.798661\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	23.2720i	0.902451i
$666$	0	0
$667$	−2.35035	−0.0910059
$668$	0	0
$669$	27.1703i	1.05046i
$670$	0	0
$671$	− 6.41393i	− 0.247607i
$672$	0	0
$673$	36.3952	1.40293	0.701467	−	0.712702i	$-0.252529\pi$
$673$	36.3952	1.40293	0.701467	+	0.712702i	$0.252529\pi$
$674$	0	0
$675$	5.50367	0.211836
$676$	0	0
$677$	36.0043	1.38376	0.691880	−	0.722013i	$-0.256783\pi$
$677$	36.0043	1.38376	0.691880	+	0.722013i	$0.256783\pi$
$678$	0	0
$679$	12.1745	0.467215
$680$	0	0
$681$	3.84130i	0.147199i
$682$	0	0
$683$	− 12.8296i	− 0.490909i	−0.969408	−	0.245455i	$-0.921063\pi$
$683$	− 12.8296i	− 0.490909i	0.969408	−	0.245455i	$-0.0789372\pi$
$684$	0	0
$685$	−12.3728	−0.472740
$686$	0	0
$687$	11.7722i	0.449137i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 32.9792i	− 1.25459i	−0.778783	−	0.627294i	$-0.784162\pi$
$691$	− 32.9792i	− 1.25459i	0.778783	−	0.627294i	$-0.215838\pi$
$692$	0	0
$693$	−3.29692	−0.125239
$694$	0	0
$695$	− 8.76836i	− 0.332603i
$696$	0	0
$697$	58.0188i	2.19762i
$698$	0	0
$699$	−5.39922	−0.204217
$700$	0	0
$701$	40.7352	1.53855	0.769273	−	0.638921i	$-0.220619\pi$
$701$	40.7352	1.53855	0.769273	+	0.638921i	$0.220619\pi$
$702$	0	0
$703$	−51.2076	−1.93133
$704$	0	0
$705$	−5.54324	−0.208771
$706$	0	0
$707$	28.7651i	1.08182i
$708$	0	0
$709$	− 17.6591i	− 0.663202i	−0.943420	−	0.331601i	$-0.892411\pi$
$709$	− 17.6591i	− 0.663202i	0.943420	−	0.331601i	$-0.107589\pi$
$710$	0	0
$711$	3.58845	0.134577
$712$	0	0
$713$	− 9.12973i	− 0.341911i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	38.0537i	1.42114i
$718$	0	0
$719$	−49.5672	−1.84855	−0.924273	−	0.381733i	$-0.875327\pi$
$719$	−49.5672	−1.84855	−0.924273	+	0.381733i	$0.875327\pi$
$720$	0	0
$721$	31.9549i	1.19006i
$722$	0	0
$723$	35.7183i	1.32838i
$724$	0	0
$725$	2.02473	0.0751965
$726$	0	0
$727$	−36.4738	−1.35274	−0.676370	−	0.736562i	$-0.736448\pi$
$727$	−36.4738	−1.35274	−0.676370	+	0.736562i	$0.736448\pi$
$728$	0	0
$729$	29.7112	1.10042
$730$	0	0
$731$	31.4070	1.16163
$732$	0	0
$733$	− 28.8491i	− 1.06556i	−0.846252	−	0.532782i	$-0.821146\pi$
$733$	− 28.8491i	− 1.06556i	0.846252	−	0.532782i	$-0.178854\pi$
$734$	0	0
$735$	18.8317i	0.694617i
$736$	0	0
$737$	9.11137	0.335621
$738$	0	0
$739$	− 19.7835i	− 0.727746i	−0.931448	−	0.363873i	$-0.881454\pi$
$739$	− 19.7835i	− 0.727746i	0.931448	−	0.363873i	$-0.118546\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	26.4005i	0.968541i	0.874918	+	0.484271i	$0.160915\pi$
$743$	26.4005i	0.968541i	−0.874918	+	0.484271i	$0.839085\pi$
$744$	0	0
$745$	4.73939	0.173638
$746$	0	0
$747$	− 0.200688i	− 0.00734279i
$748$	0	0
$749$	− 34.2826i	− 1.25266i
$750$	0	0
$751$	13.8034	0.503694	0.251847	−	0.967767i	$-0.418962\pi$
$751$	13.8034	0.503694	0.251847	+	0.967767i	$0.418962\pi$
$752$	0	0
$753$	7.39671	0.269551
$754$	0	0
$755$	1.56063	0.0567970
$756$	0	0
$757$	−5.18160	−0.188328	−0.0941642	−	0.995557i	$-0.530018\pi$
$757$	−5.18160	−0.188328	−0.0941642	+	0.995557i	$0.530018\pi$
$758$	0	0
$759$	− 3.21736i	− 0.116783i
$760$	0	0
$761$	0.00858275i	0 0.000311124i	1.00000		0.000155562i	$4.95170e-5\pi$
$761$	0.00858275i	0 0.000311124i	−1.00000		0.000155562i	$0.999950\pi$
$762$	0	0
$763$	−39.7830	−1.44024
$764$	0	0
$765$	− 3.29692i	− 0.119200i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 11.8483i	− 0.427262i	−0.976914	−	0.213631i	$-0.931471\pi$
$769$	− 11.8483i	− 0.427262i	0.976914	−	0.213631i	$-0.0685291\pi$
$770$	0	0
$771$	−34.9865	−1.26001
$772$	0	0
$773$	− 21.0458i	− 0.756964i	−0.925609	−	0.378482i	$-0.876446\pi$
$773$	− 21.0458i	− 0.756964i	0.925609	−	0.378482i	$-0.123554\pi$
$774$	0	0
$775$	7.86488i	0.282515i
$776$	0	0
$777$	−66.0845	−2.37077
$778$	0	0
$779$	41.5352	1.48815
$780$	0	0
$781$	−21.6754	−0.775605
$782$	0	0
$783$	11.1434	0.398234
$784$	0	0
$785$	9.15332i	0.326696i
$786$	0	0
$787$	− 5.01174i	− 0.178649i	−0.996003	−	0.0893246i	$-0.971529\pi$
$787$	− 5.01174i	− 0.178649i	0.996003	−	0.0893246i	$-0.0284708\pi$
$788$	0	0
$789$	−32.2904	−1.14957
$790$	0	0
$791$	− 25.0224i	− 0.889696i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	20.2378i	0.717762i
$796$	0	0
$797$	−40.6566	−1.44013	−0.720065	−	0.693907i	$-0.755888\pi$
$797$	−40.6566	−1.44013	−0.720065	+	0.693907i	$0.755888\pi$
$798$	0	0
$799$	25.9935i	0.919583i
$800$	0	0
$801$	− 5.83281i	− 0.206092i
$802$	0	0
$803$	9.07418	0.320221
$804$	0	0
$805$	5.02897	0.177248
$806$	0	0
$807$	11.6604	0.410466
$808$	0	0
$809$	5.41351	0.190329	0.0951644	−	0.995462i	$-0.469662\pi$
$809$	5.41351	0.190329	0.0951644	+	0.995462i	$0.469662\pi$
$810$	0	0
$811$	0.616994i	0.0216656i	0.999941	+	0.0108328i	$0.00344825\pi$
$811$	0.616994i	0.0216656i	−0.999941	+	0.0108328i	$0.996552\pi$
$812$	0	0
$813$	7.82799i	0.274540i
$814$	0	0
$815$	2.64303	0.0925811
$816$	0	0
$817$	− 22.4840i	− 0.786616i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	22.9066i	0.799445i	0.916636	+	0.399723i	$0.130894\pi$
$821$	22.9066i	0.799445i	−0.916636	+	0.399723i	$0.869106\pi$
$822$	0	0
$823$	−9.65856	−0.336676	−0.168338	−	0.985729i	$-0.553840\pi$
$823$	−9.65856	−0.336676	−0.168338	+	0.985729i	$0.553840\pi$
$824$	0	0
$825$	2.77162i	0.0964954i
$826$	0	0
$827$	− 21.2602i	− 0.739289i	−0.929173	−	0.369645i	$-0.879479\pi$
$827$	− 21.2602i	− 0.739289i	0.929173	−	0.369645i	$-0.120521\pi$
$828$	0	0
$829$	23.9895	0.833191	0.416595	−	0.909092i	$-0.363223\pi$
$829$	23.9895	0.833191	0.416595	+	0.909092i	$0.363223\pi$
$830$	0	0
$831$	−14.0827	−0.488522
$832$	0	0
$833$	88.3059	3.05962
$834$	0	0
$835$	−8.90869	−0.308298
$836$	0	0
$837$	43.2857i	1.49617i
$838$	0	0
$839$	− 21.3055i	− 0.735548i	−0.929915	−	0.367774i	$-0.880120\pi$
$839$	− 21.3055i	− 0.735548i	0.929915	−	0.367774i	$-0.119880\pi$
$840$	0	0
$841$	−24.9005	−0.858637
$842$	0	0
$843$	− 20.1761i	− 0.694902i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	34.6580i	1.19086i
$848$	0	0
$849$	33.4456	1.14785
$850$	0	0
$851$	11.0657i	0.379327i
$852$	0	0
$853$	− 12.1098i	− 0.414631i	−0.978274	−	0.207315i	$-0.933527\pi$
$853$	− 12.1098i	− 0.414631i	0.978274	−	0.207315i	$-0.0664726\pi$
$854$	0	0
$855$	−2.36023	−0.0807183
$856$	0	0
$857$	35.4491	1.21092	0.605459	−	0.795876i	$-0.292990\pi$
$857$	35.4491	1.21092	0.605459	+	0.795876i	$0.292990\pi$
$858$	0	0
$859$	−47.4600	−1.61931	−0.809657	−	0.586904i	$-0.800347\pi$
$859$	−47.4600	−1.61931	−0.809657	+	0.586904i	$0.800347\pi$
$860$	0	0
$861$	53.6020	1.82675
$862$	0	0
$863$	18.7268i	0.637467i	0.947844	+	0.318733i	$0.103257\pi$
$863$	18.7268i	0.637467i	−0.947844	+	0.318733i	$0.896743\pi$
$864$	0	0
$865$	13.8079i	0.469484i
$866$	0	0
$867$	62.8958	2.13605
$868$	0	0
$869$	14.1460i	0.479870i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	1.23473i	0.0417894i
$874$	0	0
$875$	−4.33225	−0.146457
$876$	0	0
$877$	1.81517i	0.0612938i	0.999530	+	0.0306469i	$0.00975674\pi$
$877$	1.81517i	0.0612938i	−0.999530	+	0.0306469i	$0.990243\pi$
$878$	0	0
$879$	− 1.10811i	− 0.0373755i
$880$	0	0
$881$	50.3437	1.69612	0.848061	−	0.529899i	$-0.177770\pi$
$881$	50.3437	1.69612	0.848061	+	0.529899i	$0.177770\pi$
$882$	0	0
$883$	4.42003	0.148746	0.0743729	−	0.997230i	$-0.476304\pi$
$883$	4.42003	0.148746	0.0743729	+	0.997230i	$0.476304\pi$
$884$	0	0
$885$	10.1566	0.341410
$886$	0	0
$887$	2.69134	0.0903662	0.0451831	−	0.998979i	$-0.485613\pi$
$887$	2.69134	0.0903662	0.0451831	+	0.998979i	$0.485613\pi$
$888$	0	0
$889$	− 82.1731i	− 2.75600i
$890$	0	0
$891$	12.9710i	0.434546i
$892$	0	0
$893$	18.6085	0.622710
$894$	0	0
$895$	− 24.3502i	− 0.813937i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	15.9243i	0.531103i
$900$	0	0
$901$	94.8997	3.16157
$902$	0	0
$903$	− 29.0161i	− 0.965595i
$904$	0	0
$905$	− 24.8336i	− 0.825498i
$906$	0	0
$907$	−14.4017	−0.478199	−0.239100	−	0.970995i	$-0.576852\pi$
$907$	−14.4017	−0.478199	−0.239100	+	0.970995i	$0.576852\pi$
$908$	0	0
$909$	−2.91734	−0.0967620
$910$	0	0
$911$	−8.88723	−0.294447	−0.147223	−	0.989103i	$-0.547034\pi$
$911$	−8.88723	−0.294447	−0.147223	+	0.989103i	$0.547034\pi$
$912$	0	0
$913$	0.791131	0.0261826
$914$	0	0
$915$	− 5.92566i	− 0.195896i
$916$	0	0
$917$	− 5.83281i	− 0.192617i
$918$	0	0
$919$	−42.9747	−1.41760	−0.708802	−	0.705408i	$-0.750764\pi$
$919$	−42.9747	−1.41760	−0.708802	+	0.705408i	$0.750764\pi$
$920$	0	0
$921$	14.1324i	0.465679i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 9.53264i	− 0.313431i
$926$	0	0
$927$	−3.24084	−0.106443
$928$	0	0
$929$	− 29.7378i	− 0.975666i	−0.872937	−	0.487833i	$-0.837787\pi$
$929$	− 29.7378i	− 0.975666i	0.872937	−	0.487833i	$-0.162213\pi$
$930$	0	0
$931$	− 63.2175i	− 2.07187i
$932$	0	0
$933$	−18.2356	−0.597005
$934$	0	0
$935$	12.9967	0.425039
$936$	0	0
$937$	17.1355	0.559793	0.279896	−	0.960030i	$-0.409700\pi$
$937$	17.1355	0.559793	0.279896	+	0.960030i	$0.409700\pi$
$938$	0	0
$939$	−5.41660	−0.176764
$940$	0	0
$941$	23.5767i	0.768580i	0.923212	+	0.384290i	$0.125554\pi$
$941$	23.5767i	0.768580i	−0.923212	+	0.384290i	$0.874446\pi$
$942$	0	0
$943$	− 8.97553i	− 0.292284i
$944$	0	0
$945$	−23.8433	−0.775621
$946$	0	0
$947$	− 17.6272i	− 0.572807i	−0.958109	−	0.286404i	$-0.907540\pi$
$947$	− 17.6272i	− 0.572807i	0.958109	−	0.286404i	$-0.0924598\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 0.891475i	− 0.0289080i
$952$	0	0
$953$	58.7631	1.90352	0.951762	−	0.306837i	$-0.0992707\pi$
$953$	58.7631	1.90352	0.951762	+	0.306837i	$0.0992707\pi$
$954$	0	0
$955$	− 5.96875i	− 0.193144i
$956$	0	0
$957$	5.61178i	0.181403i
$958$	0	0
$959$	53.6020	1.73090
$960$	0	0
$961$	−30.8564	−0.995368
$962$	0	0
$963$	3.47692	0.112042
$964$	0	0
$965$	9.05272	0.291417
$966$	0	0
$967$	2.92168i	0.0939550i	0.998896	+	0.0469775i	$0.0149589\pi$
$967$	2.92168i	0.0939550i	−0.998896	+	0.0469775i	$0.985041\pi$
$968$	0	0
$969$	− 64.5012i	− 2.07208i
$970$	0	0
$971$	−10.0139	−0.321360	−0.160680	−	0.987007i	$-0.551369\pi$
$971$	−10.0139	−0.321360	−0.160680	+	0.987007i	$0.551369\pi$
$972$	0	0
$973$	37.9867i	1.21780i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.5301i	0.848772i	0.905481	+	0.424386i	$0.139510\pi$
$977$	26.5301i	0.848772i	−0.905481	+	0.424386i	$0.860490\pi$
$978$	0	0
$979$	22.9935	0.734875
$980$	0	0
$981$	− 4.03477i	− 0.128820i
$982$	0	0
$983$	4.69317i	0.149689i	0.997195	+	0.0748445i	$0.0238460\pi$
$983$	4.69317i	0.149689i	−0.997195	+	0.0748445i	$0.976154\pi$
$984$	0	0
$985$	11.3890	0.362885
$986$	0	0
$987$	24.0147	0.764396
$988$	0	0
$989$	−4.85868	−0.154497
$990$	0	0
$991$	−22.9201	−0.728081	−0.364041	−	0.931383i	$-0.618603\pi$
$991$	−22.9201	−0.728081	−0.364041	+	0.931383i	$0.618603\pi$
$992$	0	0
$993$	6.34268i	0.201279i
$994$	0	0
$995$	10.7773i	0.341662i
$996$	0	0
$997$	49.4426	1.56586	0.782932	−	0.622108i	$-0.213723\pi$
$997$	49.4426	1.56586	0.782932	+	0.622108i	$0.213723\pi$
$998$	0	0
$999$	− 52.4645i	− 1.65990i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.f.i.3041.5		8
13.3	even	3		260.2.x.a.121.2	yes	8
13.4	even	6		260.2.x.a.101.2	✓	8
13.5	odd	4		3380.2.a.p.1.3		4
13.8	odd	4		3380.2.a.q.1.3		4
13.12	even	2	inner	3380.2.f.i.3041.6		8
39.17	odd	6		2340.2.dj.d.361.2		8
39.29	odd	6		2340.2.dj.d.901.4		8
52.3	odd	6		1040.2.da.c.641.3		8
52.43	odd	6		1040.2.da.c.881.3		8
65.3	odd	12		1300.2.ba.c.849.1		8
65.4	even	6		1300.2.y.b.101.3		8
65.17	odd	12		1300.2.ba.c.49.1		8
65.29	even	6		1300.2.y.b.901.3		8
65.42	odd	12		1300.2.ba.b.849.4		8
65.43	odd	12		1300.2.ba.b.49.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.x.a.101.2	✓	8	13.4	even	6
260.2.x.a.121.2	yes	8	13.3	even	3
1040.2.da.c.641.3		8	52.3	odd	6
1040.2.da.c.881.3		8	52.43	odd	6
1300.2.y.b.101.3		8	65.4	even	6
1300.2.y.b.901.3		8	65.29	even	6
1300.2.ba.b.49.4		8	65.43	odd	12
1300.2.ba.b.849.4		8	65.42	odd	12
1300.2.ba.c.49.1		8	65.17	odd	12
1300.2.ba.c.849.1		8	65.3	odd	12
2340.2.dj.d.361.2		8	39.17	odd	6
2340.2.dj.d.901.4		8	39.29	odd	6
3380.2.a.p.1.3		4	13.5	odd	4
3380.2.a.q.1.3		4	13.8	odd	4
3380.2.f.i.3041.5		8	1.1	even	1	trivial
3380.2.f.i.3041.6		8	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.60020 q^{3} -1.00000i q^{5} +4.33225i q^{7} -0.439374 q^{9} +O(q^{10})\)	\(q+1.60020 q^{3} -1.00000i q^{5} +4.33225i q^{7} -0.439374 q^{9} -1.73205i q^{11} -1.60020i q^{15} -7.50367 q^{17} +5.37182i q^{19} +6.93244i q^{21} +1.16082 q^{23} -1.00000 q^{25} -5.50367 q^{27} -2.02473 q^{29} -7.86488i q^{31} -2.77162i q^{33} +4.33225 q^{35} +9.53264i q^{37} -7.73205i q^{41} -4.18555 q^{43} +0.439374i q^{45} -3.46410i q^{47} -11.7684 q^{49} -12.0073 q^{51} -12.6471 q^{53} -1.73205 q^{55} +8.59596i q^{57} +6.34709i q^{59} +3.70308 q^{61} -1.90348i q^{63} +5.26045i q^{67} +1.85754 q^{69} -12.5143i q^{71} +5.23898i q^{73} -1.60020 q^{75} +7.50367 q^{77} -8.16719 q^{79} -7.48883 q^{81} +0.456760i q^{83} +7.50367i q^{85} -3.23996 q^{87} +13.2753i q^{89} -12.5854i q^{93} +5.37182 q^{95} -2.81021i q^{97} +0.761018i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} + 8 q^{9}+O(q^{10}) \) 8 * q + 4 * q^3 + 8 * q^9	\( 8 q + 4 q^{3} + 8 q^{9} - 12 q^{17} + 12 q^{23} - 8 q^{25} + 4 q^{27} + 12 q^{35} - 20 q^{43} + 8 q^{49} + 24 q^{53} + 8 q^{61} + 48 q^{69} - 4 q^{75} + 12 q^{77} - 16 q^{79} - 16 q^{81} + 12 q^{87}+O(q^{100}) \) 8 * q + 4 * q^3 + 8 * q^9 - 12 * q^17 + 12 * q^23 - 8 * q^25 + 4 * q^27 + 12 * q^35 - 20 * q^43 + 8 * q^49 + 24 * q^53 + 8 * q^61 + 48 * q^69 - 4 * q^75 + 12 * q^77 - 16 * q^79 - 16 * q^81 + 12 * q^87

Introduction

L-functions

Modular forms

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