LMFDB - Embedded newform 220.2.g.a.219.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [220,2,Mod(219,220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(220, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("220.219");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 220 = 2^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	220.g (of order $2$, degree $1$, 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:	$1.75670884447$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.2342560000.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{4} + 16 $ x^8 + 3*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			219.6
Root			$0.664066 + 1.24861i$ of defining polynomial
Character	$\chi$	$=$	220.219
Dual form			220.2.g.a.219.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.664066 + 1.24861i) q^{2} +(-1.11803 + 1.65831i) q^{4} -2.23607 q^{5} +3.08672i q^{7} +(-2.81303 - 0.294756i) q^{8} -3.00000 q^{9} +(-1.48490 - 2.79197i) q^{10} +3.31662i q^{11} +6.95418 q^{13} +(-3.85410 + 2.04979i) q^{14} +(-1.50000 - 3.70810i) q^{16} +1.64166 q^{17} +(-1.99220 - 3.74582i) q^{18} +(2.50000 - 3.70810i) q^{20} +(-4.14116 + 2.20246i) q^{22} +5.00000 q^{25} +(4.61803 + 8.68304i) q^{26} +(-5.11875 - 3.45106i) q^{28} -6.63325i q^{31} +(3.63386 - 4.33533i) q^{32} +(1.09017 + 2.04979i) q^{34} -6.90212i q^{35} +(3.35410 - 4.97494i) q^{36} +(6.29012 + 0.659094i) q^{40} +13.0756i q^{43} +(-5.50000 - 3.70810i) q^{44} +6.70820 q^{45} -2.52786 q^{49} +(3.32033 + 6.24303i) q^{50} +(-7.77501 + 11.5322i) q^{52} -7.41620i q^{55} +(0.909830 - 8.68304i) q^{56} +14.8324i q^{59} +(8.28232 - 4.40491i) q^{62} -9.26017i q^{63} +(7.82624 + 1.65831i) q^{64} -15.5500 q^{65} +(-1.83543 + 2.72239i) q^{68} +(8.61803 - 4.58346i) q^{70} -14.8324i q^{71} +(8.43908 + 0.884268i) q^{72} +12.2667 q^{73} -10.2375 q^{77} +(3.35410 + 8.29156i) q^{80} +9.00000 q^{81} +0.728677i q^{83} -3.67086 q^{85} +(-16.3262 + 8.68304i) q^{86} +(0.977595 - 9.32975i) q^{88} -13.4164 q^{89} +(4.45469 + 8.37590i) q^{90} +21.4656i q^{91} +(-1.67867 - 3.15631i) q^{98} -9.94987i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9} - 4 q^{14} - 12 q^{16} + 20 q^{20} + 40 q^{25} + 28 q^{26} - 36 q^{34} - 44 q^{44} - 56 q^{49} + 52 q^{56} + 60 q^{70} + 72 q^{81} - 68 q^{86}+O(q^{100}) $ 8 * q - 24 * q^9 - 4 * q^14 - 12 * q^16 + 20 * q^20 + 40 * q^25 + 28 * q^26 - 36 * q^34 - 44 * q^44 - 56 * q^49 + 52 * q^56 + 60 * q^70 + 72 * q^81 - 68 * q^86

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/220\mathbb{Z}\right)^\times$.

$n$	$101$	$111$	$177$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	0.664066	+	1.24861i	0.469565	+	0.882898i
$2$	0.664066	+	1.24861i	0.469565	+	0.882898i
$3$		0			0			−	1.00000i	$-0.5\pi$
$3$		0			0				1.00000i	$0.5\pi$
$4$	−1.11803	+	1.65831i	−0.559017	+	0.829156i
$5$	−2.23607			−1.00000
$5$	−2.23607			−1.00000
$6$		0			0
$7$			3.08672i			1.16667i	0.812231	+	0.583336i	$0.198253\pi$
$7$			3.08672i			1.16667i	−0.812231	+	0.583336i	$0.801747\pi$
$8$	−2.81303	−	0.294756i	−0.994555	−	0.104212i
$9$	−3.00000			−1.00000
$10$	−1.48490	−	2.79197i	−0.469565	−	0.882898i
$11$			3.31662i			1.00000i
$11$			3.31662i			1.00000i
$12$		0			0
$13$	6.95418			1.92874			0.964372	−	0.264550i	$-0.0852236\pi$
$13$	6.95418			1.92874			0.964372	+	0.264550i	$0.0852236\pi$
$14$	−3.85410	+	2.04979i	−1.03005	+	0.547829i
$15$		0			0
$16$	−1.50000	−	3.70810i	−0.375000	−	0.927025i
$17$	1.64166			0.398161			0.199081	−	0.979983i	$-0.436204\pi$
$17$	1.64166			0.398161			0.199081	+	0.979983i	$0.436204\pi$
$18$	−1.99220	−	3.74582i	−0.469565	−	0.882898i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	2.50000	−	3.70810i	0.559017	−	0.829156i
$21$		0			0
$22$	−4.14116	+	2.20246i	−0.882898	+	0.469565i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$	4.61803	+	8.68304i	0.905671	+	1.70288i
$27$		0			0
$28$	−5.11875	−	3.45106i	−0.967353	−	0.652189i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		−	6.63325i		−	1.19137i	−0.803219	−	0.595683i	$-0.796881\pi$
$31$		−	6.63325i		−	1.19137i	0.803219	−	0.595683i	$-0.203119\pi$
$32$	3.63386	−	4.33533i	0.642381	−	0.766385i
$33$		0			0
$34$	1.09017	+	2.04979i	0.186963	+	0.351536i
$35$		−	6.90212i		−	1.16667i
$36$	3.35410	−	4.97494i	0.559017	−	0.829156i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$	6.29012	+	0.659094i	0.994555	+	0.104212i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$			13.0756i			1.99401i	0.0773627	+	0.997003i	$0.475350\pi$
$43$			13.0756i			1.99401i	−0.0773627	+	0.997003i	$0.524650\pi$
$44$	−5.50000	−	3.70810i	−0.829156	−	0.559017i
$45$	6.70820			1.00000
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−2.52786			−0.361123
$50$	3.32033	+	6.24303i	0.469565	+	0.882898i
$51$		0			0
$52$	−7.77501	+	11.5322i	−1.07820	+	1.59923i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		−	7.41620i		−	1.00000i
$56$	0.909830	−	8.68304i	0.121581	−	1.16032i
$57$		0			0
$58$		0			0
$59$			14.8324i			1.93101i	0.260378	+	0.965507i	$0.416153\pi$
$59$			14.8324i			1.93101i	−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	8.28232	−	4.40491i	1.05186	−	0.559424i
$63$		−	9.26017i		−	1.16667i
$64$	7.82624	+	1.65831i	0.978280	+	0.207289i
$65$	−15.5500			−1.92874
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	−1.83543	+	2.72239i	−0.222579	+	0.330138i
$69$		0			0
$70$	8.61803	−	4.58346i	1.03005	−	0.547829i
$71$		−	14.8324i		−	1.76028i	−0.474713	−	0.880141i	$-0.657448\pi$
$71$		−	14.8324i		−	1.76028i	0.474713	−	0.880141i	$-0.342552\pi$
$72$	8.43908	+	0.884268i	0.994555	+	0.104212i
$73$	12.2667			1.43571			0.717855	−	0.696193i	$-0.245124\pi$
$73$	12.2667			1.43571			0.717855	+	0.696193i	$0.245124\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.2375			−1.16667
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	3.35410	+	8.29156i	0.375000	+	0.927025i
$81$	9.00000			1.00000
$82$		0			0
$83$			0.728677i			0.0799827i	0.999200	+	0.0399913i	$0.0127330\pi$
$83$			0.728677i			0.0799827i	−0.999200	+	0.0399913i	$0.987267\pi$
$84$		0			0
$85$	−3.67086			−0.398161
$86$	−16.3262	+	8.68304i	−1.76050	+	0.936316i
$87$		0			0
$88$	0.977595	−	9.32975i	0.104212	−	0.994555i
$89$	−13.4164			−1.42214			−0.711068	−	0.703123i	$-0.751788\pi$
$89$	−13.4164			−1.42214			−0.711068	+	0.703123i	$0.751788\pi$
$90$	4.45469	+	8.37590i	0.469565	+	0.882898i
$91$			21.4656i			2.25021i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−1.67867	−	3.15631i	−0.169571	−	0.318835i
$99$		−	9.94987i		−	1.00000i
$100$	−5.59017	+	8.29156i	−0.559017	+	0.829156i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$	−19.5623	−	2.04979i	−1.91824	−	0.200998i
$105$		0			0
$106$		0			0
$107$		−	19.2490i		−	1.86087i	−0.366453	−	0.930436i	$-0.619428\pi$
$107$		−	19.2490i		−	1.86087i	0.366453	−	0.930436i	$-0.380572\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	9.25991	−	4.92484i	0.882898	−	0.469565i
$111$		0			0
$112$	11.4459	−	4.63009i	1.08153	−	0.437502i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−20.8626			−1.92874
$118$	−18.5198	+	9.84968i	−1.70489	+	0.906737i
$119$			5.06735i			0.464523i
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		0			0
$124$	11.0000	+	7.41620i	0.987829	+	0.665994i
$125$	−11.1803			−1.00000
$126$	11.5623	−	6.14936i	1.03005	−	0.547829i
$127$		−	16.8910i		−	1.49883i	−0.662100	−	0.749416i	$-0.730334\pi$
$127$		−	16.8910i		−	1.49883i	0.662100	−	0.749416i	$-0.269666\pi$
$128$	3.12656	+	10.8731i	0.276351	+	0.961057i
$129$		0			0
$130$	−10.3262	−	19.4159i	−0.905671	−	1.70288i
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−4.61803	−	0.483889i	−0.395993	−	0.0414931i
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$	11.4459	+	7.71681i	0.967353	+	0.652189i
$141$		0			0
$142$	18.5198	−	9.84968i	1.55415	−	0.826567i
$143$			23.0644i			1.92874i
$144$	4.50000	+	11.1243i	0.375000	+	0.927025i
$145$		0			0
$146$	8.14590	+	15.3163i	0.674159	+	1.26758i
$147$		0			0
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	−4.92498			−0.398161
$154$	−6.79837	−	12.7826i	−0.547829	−	1.03005i
$155$			14.8324i			1.19137i
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$	−8.12555	+	9.69409i	−0.642381	+	0.766385i
$161$		0			0
$162$	5.97659	+	11.2375i	0.469565	+	0.882898i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$	−0.909830	+	0.483889i	−0.0706165	+	0.0375571i
$167$			10.7175i			0.829347i	0.909970	+	0.414673i	$0.136104\pi$
$167$			10.7175i			0.829347i	−0.909970	+	0.414673i	$0.863896\pi$
$168$		0			0
$169$	35.3607			2.72005
$170$	−2.43769	−	4.58346i	−0.186963	−	0.351536i
$171$		0			0
$172$	−21.6834	−	14.6189i	−1.65334	−	1.11468i
$173$	24.1459			1.83578			0.917888	−	0.396839i	$-0.129893\pi$
$173$	24.1459			1.83578			0.917888	+	0.396839i	$0.129893\pi$
$174$		0			0
$175$			15.4336i			1.16667i
$176$	12.2984	−	4.97494i	0.927025	−	0.375000i
$177$		0			0
$178$	−8.90937	−	16.7518i	−0.667786	−	1.25560i
$179$		−	19.8997i		−	1.48738i	−0.668526	−	0.743689i	$-0.733075\pi$
$179$		−	19.8997i		−	1.48738i	0.668526	−	0.743689i	$-0.266925\pi$
$180$	−7.50000	+	11.1243i	−0.559017	+	0.829156i
$181$	−4.47214			−0.332411			−0.166206	−	0.986091i	$-0.553152\pi$
$181$	−4.47214			−0.332411			−0.166206	+	0.986091i	$0.553152\pi$
$182$	−26.8021	+	14.2546i	−1.98671	+	1.05662i
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$			5.44477i			0.398161i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	6.63325i		−	0.479965i	−0.970777	−	0.239983i	$-0.922858\pi$
$191$		−	6.63325i		−	0.479965i	0.970777	−	0.239983i	$-0.0771417\pi$
$192$		0			0
$193$	18.8333			1.35565			0.677827	−	0.735221i	$-0.262922\pi$
$193$	18.8333			1.35565			0.677827	+	0.735221i	$0.262922\pi$
$194$		0			0
$195$		0			0
$196$	2.82624	−	4.19199i	0.201874	−	0.299428i
$197$	17.5792			1.25247			0.626234	−	0.779635i	$-0.284595\pi$
$197$	17.5792			1.25247			0.626234	+	0.779635i	$0.284595\pi$
$198$	12.4235	−	6.60737i	0.882898	−	0.469565i
$199$		−	14.8324i		−	1.05144i	−0.850657	−	0.525720i	$-0.823796\pi$
$199$		−	14.8324i		−	1.05144i	0.850657	−	0.525720i	$-0.176204\pi$
$200$	−14.0651	−	1.47378i	−0.994555	−	0.104212i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−10.4313	−	25.7868i	−0.723279	−	1.78799i
$209$		0			0
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$		0			0
$214$	24.0344	−	12.7826i	1.64296	−	0.873801i
$215$		−	29.2379i		−	1.99401i
$216$		0			0
$217$	20.4750			1.38993
$218$		0			0
$219$		0			0
$220$	12.2984	+	8.29156i	0.829156	+	0.559017i
$221$	11.4164			0.767951
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	13.3820	+	11.2167i	0.894120	+	0.749448i
$225$	−15.0000			−1.00000
$226$		0			0
$227$			25.4225i			1.68735i	0.536855	+	0.843674i	$0.319612\pi$
$227$			25.4225i			1.68735i	−0.536855	+	0.843674i	$0.680388\pi$
$228$		0			0
$229$	6.00000			0.396491			0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000			0.396491			0.198246	+	0.980152i	$0.436476\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−26.1751			−1.71479			−0.857393	−	0.514662i	$-0.827917\pi$
$233$	−26.1751			−1.71479			−0.857393	+	0.514662i	$0.827917\pi$
$234$	−13.8541	−	26.0491i	−0.905671	−	1.70288i
$235$		0			0
$236$	−24.5967	−	16.5831i	−1.60111	−	1.07947i
$237$		0			0
$238$	−6.32713	+	3.36505i	−0.410127	+	0.218124i
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−7.30472	−	13.7347i	−0.469565	−	0.882898i
$243$		0			0
$244$		0			0
$245$	5.65248			0.361123
$246$		0			0
$247$		0			0
$248$	−1.95519	+	18.6595i	−0.124155	+	1.18488i
$249$		0			0
$250$	−7.42448	−	13.9598i	−0.469565	−	0.882898i
$251$			14.8324i			0.936213i	0.883672	+	0.468106i	$0.155064\pi$
$251$			14.8324i			0.936213i	−0.883672	+	0.468106i	$0.844936\pi$
$252$	15.3563	+	10.3532i	0.967353	+	0.652189i
$253$		0			0
$254$	21.0902	−	11.2167i	1.32331	−	0.703799i
$255$		0			0
$256$	−11.5000	+	11.1243i	−0.718750	+	0.695269i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$	17.3855	−	25.7868i	1.07820	−	1.59923i
$261$		0			0
$262$		0			0
$263$		−	4.54408i		−	0.280200i	−0.990137	−	0.140100i	$-0.955258\pi$
$263$		−	4.54408i		−	0.280200i	0.990137	−	0.140100i	$-0.0447424\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	14.0000			0.853595			0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000			0.853595			0.426798	+	0.904347i	$0.359642\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	−2.46249	−	6.08744i	−0.149310	−	0.369105i
$273$		0			0
$274$		0			0
$275$			16.5831i			1.00000i
$276$		0			0
$277$	13.5208			0.812388			0.406194	−	0.913787i	$-0.366856\pi$
$277$	13.5208			0.812388			0.406194	+	0.913787i	$0.366856\pi$
$278$		0			0
$279$			19.8997i			1.19137i
$280$	−2.03444	+	19.4159i	−0.121581	+	1.16032i
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		−	26.8798i		−	1.59784i	−0.601438	−	0.798920i	$-0.705405\pi$
$283$		−	26.8798i		−	1.59784i	0.601438	−	0.798920i	$-0.294595\pi$
$284$	24.5967	+	16.5831i	1.45955	+	0.984027i
$285$		0			0
$286$	−28.7984	+	15.3163i	−1.70288	+	0.905671i
$287$		0			0
$288$	−10.9016	+	13.0060i	−0.642381	+	0.766385i
$289$	−14.3050			−0.841468
$290$		0			0
$291$		0			0
$292$	−13.7146	+	20.3420i	−0.802586	+	1.19043i
$293$	0.387543			0.0226405			0.0113203	−	0.999936i	$-0.496397\pi$
$293$	0.387543			0.0226405			0.0113203	+	0.999936i	$0.496397\pi$
$294$		0			0
$295$		−	33.1662i		−	1.93101i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−40.3607			−2.32635
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$	−3.27051	−	6.14936i	−0.186963	−	0.351536i
$307$			33.0533i			1.88645i	0.332155	+	0.943225i	$0.392224\pi$
$307$			33.0533i			1.88645i	−0.332155	+	0.943225i	$0.607776\pi$
$308$	11.4459	−	16.9770i	0.652189	−	0.967353i
$309$		0			0
$310$	−18.5198	+	9.84968i	−1.05186	+	0.559424i
$311$		−	14.8324i		−	0.841068i	−0.907277	−	0.420534i	$-0.861843\pi$
$311$		−	14.8324i		−	0.841068i	0.907277	−	0.420534i	$-0.138157\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$			20.7064i			1.16667i
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$	−17.5000	−	3.70810i	−0.978280	−	0.207289i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−10.0623	+	14.9248i	−0.559017	+	0.829156i
$325$	34.7709			1.92874
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$			6.63325i			0.364596i	0.983243	+	0.182298i	$0.0583536\pi$
$331$			6.63325i			0.364596i	−0.983243	+	0.182298i	$0.941646\pi$
$332$	−1.20837	−	0.814685i	−0.0663181	−	0.0447117i
$333$		0			0
$334$	−13.3820	+	7.11714i	−0.732229	+	0.389432i
$335$		0			0
$336$		0			0
$337$	−32.7417			−1.78356			−0.891778	−	0.452474i	$-0.850541\pi$
$337$	−32.7417			−1.78356			−0.891778	+	0.452474i	$0.850541\pi$
$338$	23.4818	+	44.1516i	1.27724	+	2.40153i
$339$		0			0
$340$	4.10415	−	6.08744i	0.222579	−	0.330138i
$341$	22.0000			1.19137
$342$		0			0
$343$			13.8042i			0.745359i
$344$	3.85410	−	36.7819i	0.207799	−	1.98315i
$345$		0			0
$346$	16.0344	+	30.1487i	0.862017	+	1.62080i
$347$		−	11.6182i		−	0.623699i	−0.950132	−	0.311849i	$-0.899052\pi$
$347$		−	11.6182i		−	0.623699i	0.950132	−	0.311849i	$-0.100948\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	−19.2705	+	10.2489i	−1.03005	+	0.547829i
$351$		0			0
$352$	14.3787	+	12.0521i	0.766385	+	0.642381i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$			33.1662i			1.76028i
$356$	15.0000	−	22.2486i	0.794998	−	1.17917i
$357$		0			0
$358$	24.8469	−	13.2147i	1.31320	−	0.698421i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−18.8704	−	1.97728i	−0.994555	−	0.104212i
$361$	−19.0000			−1.00000
$362$	−2.96979	−	5.58394i	−0.156089	−	0.293485i
$363$		0			0
$364$	−35.5967	−	23.9993i	−1.86578	−	1.25791i
$365$	−27.4292			−1.43571
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−38.0542			−1.97037			−0.985187	−	0.171484i	$-0.945144\pi$
$373$	−38.0542			−1.97037			−0.985187	+	0.171484i	$0.945144\pi$
$374$	−6.79837	+	3.61568i	−0.351536	+	0.186963i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			14.8324i			0.761889i	0.924598	+	0.380945i	$0.124401\pi$
$379$			14.8324i			0.761889i	−0.924598	+	0.380945i	$0.875599\pi$
$380$		0			0
$381$		0			0
$382$	8.28232	−	4.40491i	0.423760	−	0.225375i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$	22.8918			1.16667
$386$	12.5066	+	23.5154i	0.636568	+	1.19690i
$387$		−	39.2267i		−	1.99401i
$388$		0			0
$389$	−26.0000			−1.31825			−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000			−1.31825			−0.659126	+	0.752032i	$0.729074\pi$
$390$		0			0
$391$		0			0
$392$	7.11095	+	0.745103i	0.359157	+	0.0376334i
$393$		0			0
$394$	11.6738	+	21.9495i	0.588116	+	1.10580i
$395$		0			0
$396$	16.5000	+	11.1243i	0.829156	+	0.559017i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	18.5198	−	9.84968i	0.928315	−	0.493720i
$399$		0			0
$400$	−7.50000	−	18.5405i	−0.375000	−	0.927025i
$401$	4.47214			0.223328			0.111664	−	0.993746i	$-0.464382\pi$
$401$	4.47214			0.223328			0.111664	+	0.993746i	$0.464382\pi$
$402$		0			0
$403$		−	46.1288i		−	2.29784i
$404$		0			0
$405$	−20.1246			−1.00000
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−45.7835			−2.25286
$414$		0			0
$415$		−	1.62937i		−	0.0799827i
$416$	25.2705	−	30.1487i	1.23899	−	1.47816i
$417$		0			0
$418$		0			0
$419$		−	19.8997i		−	0.972166i	−0.873913	−	0.486083i	$-0.838425\pi$
$419$		−	19.8997i		−	0.972166i	0.873913	−	0.486083i	$-0.161575\pi$
$420$		0			0
$421$	31.3050			1.52571			0.762855	−	0.646570i	$-0.223797\pi$
$421$	31.3050			1.52571			0.762855	+	0.646570i	$0.223797\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	8.20830			0.398161
$426$		0			0
$427$		0			0
$428$	31.9209	+	21.5211i	1.54295	+	1.04026i
$429$		0			0
$430$	36.5066	−	19.4159i	1.76050	−	0.936316i
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$	13.5967	+	25.5652i	0.652665	+	1.22717i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$	−2.18597	+	20.8620i	−0.104212	+	0.994555i
$441$	7.58359			0.361123
$442$	7.58124	+	14.2546i	0.360603	+	0.678022i
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$	30.0000			1.42214
$446$		0			0
$447$		0			0
$448$	−5.11875	+	24.1574i	−0.241838	+	1.14133i
$449$	40.2492			1.89948			0.949739	−	0.313042i	$-0.101348\pi$
$449$	40.2492			1.89948			0.949739	+	0.313042i	$0.101348\pi$
$450$	−9.96098	−	18.7291i	−0.469565	−	0.882898i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	−31.7426	+	16.8822i	−1.48976	+	0.792320i
$455$		−	47.9986i		−	2.25021i
$456$		0			0
$457$	−22.1167			−1.03457			−0.517287	−	0.855812i	$-0.673058\pi$
$457$	−22.1167			−1.03457			−0.517287	+	0.855812i	$0.673058\pi$
$458$	3.98439	+	7.49164i	0.186178	+	0.350061i
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$	−17.3820	−	32.6824i	−0.805204	−	1.51398i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	23.3250	−	34.5966i	1.07820	−	1.59923i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	4.37194	−	41.7239i	0.201235	−	1.92050i
$473$	−43.3668			−1.99401
$474$		0			0
$475$		0			0
$476$	−8.40325	−	5.66547i	−0.385162	−	0.259676i
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	12.2984	−	18.2414i	0.559017	−	0.829156i
$485$		0			0
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$	3.75361	+	7.05772i	0.169571	+	0.318835i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$			22.2486i			1.00000i
$496$	−24.5967	+	9.94987i	−1.10443	+	0.446763i
$497$	45.7835			2.05367
$498$		0			0
$499$		−	44.4972i		−	1.99197i	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$		−	44.4972i		−	1.99197i	0.0895323	−	0.995984i	$-0.471463\pi$
$500$	12.5000	−	18.5405i	0.559017	−	0.829156i
$501$		0			0
$502$	−18.5198	+	9.84968i	−0.826580	+	0.439613i
$503$		−	36.8687i		−	1.64389i	−0.569565	−	0.821946i	$-0.692888\pi$
$503$		−	36.8687i		−	1.64389i	0.569565	−	0.821946i	$-0.307112\pi$
$504$	−2.72949	+	26.0491i	−0.121581	+	1.16032i
$505$		0			0
$506$		0			0
$507$		0			0
$508$	28.0105	+	18.8847i	1.24277	+	0.837872i
$509$	−34.0000			−1.50702			−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000			−1.50702			−0.753512	+	0.657434i	$0.771642\pi$
$510$		0			0
$511$			37.8639i			1.67500i
$512$	−21.5266	−	6.97171i	−0.951351	−	0.308109i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	43.7426	+	4.58346i	1.91824	+	0.200998i
$521$	22.3607			0.979639			0.489820	−	0.871824i	$-0.337063\pi$
$521$	22.3607			0.979639			0.489820	+	0.871824i	$0.337063\pi$
$522$		0			0
$523$			45.4002i			1.98521i	0.121387	+	0.992605i	$0.461266\pi$
$523$			45.4002i			1.98521i	−0.121387	+	0.992605i	$0.538734\pi$
$524$		0			0
$525$		0			0
$526$	5.67376	−	3.01756i	0.247388	−	0.131572i
$527$		−	10.8895i		−	0.474356i
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$		−	44.4972i		−	1.93101i
$532$		0			0
$533$		0			0
$534$		0			0
$535$			43.0421i			1.86087i
$536$		0			0
$537$		0			0
$538$	9.29692	+	17.4805i	0.400819	+	0.753637i
$539$		−	8.38398i		−	0.361123i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$	5.96556	−	7.11714i	0.255771	−	0.305145i
$545$		0			0
$546$		0			0
$547$			8.35948i			0.357425i	0.983901	+	0.178713i	$0.0571933\pi$
$547$			8.35948i			0.357425i	−0.983901	+	0.178713i	$0.942807\pi$
$548$		0			0
$549$		0			0
$550$	−20.7058	+	11.0123i	−0.882898	+	0.469565i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	8.97871	+	16.8822i	0.381469	+	0.717255i
$555$		0			0
$556$		0			0
$557$	41.3376			1.75153			0.875764	−	0.482739i	$-0.160358\pi$
$557$	41.3376			1.75153			0.875764	+	0.482739i	$0.160358\pi$
$558$	−24.8469	+	13.2147i	−1.05186	+	0.559424i
$559$			90.9299i			3.84593i
$560$	−25.5938	+	10.3532i	−1.08153	+	0.437502i
$561$		0			0
$562$		0			0
$563$			17.7917i			0.749829i	0.927059	+	0.374915i	$0.122328\pi$
$563$			17.7917i			0.749829i	−0.927059	+	0.374915i	$0.877672\pi$
$564$		0			0
$565$		0			0
$566$	33.5623	−	17.8500i	1.41073	−	0.750290i
$567$			27.7805i			1.16667i
$568$	−4.37194	+	41.7239i	−0.183442	+	1.75070i
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$	−38.2480	−	25.7868i	−1.59923	−	1.07820i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−23.4787	−	4.97494i	−0.978280	−	0.207289i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−9.49943	−	17.8612i	−0.395124	−	0.742930i
$579$		0			0
$580$		0			0
$581$	−2.24922			−0.0933135
$582$		0			0
$583$		0			0
$584$	−34.5066	−	3.61568i	−1.42789	−	0.149618i
$585$	46.6501			1.92874
$586$	0.257354	+	0.483889i	0.0106312	+	0.0199893i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$		0			0
$590$	41.4116	−	22.0246i	1.70489	−	0.906737i
$591$		0			0
$592$		0			0
$593$	36.0250			1.47937			0.739686	−	0.672953i	$-0.234974\pi$
$593$	36.0250			1.47937			0.739686	+	0.672953i	$0.234974\pi$
$594$		0			0
$595$		−	11.3309i		−	0.464523i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			19.8997i			0.813082i	0.913633	+	0.406541i	$0.133265\pi$
$599$			19.8997i			0.813082i	−0.913633	+	0.406541i	$0.866735\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	−26.8021	−	50.3946i	−1.09237	−	2.05393i
$603$		0			0
$604$		0			0
$605$	24.5967			1.00000
$606$		0			0
$607$		−	24.5218i		−	0.995308i	−0.867376	−	0.497654i	$-0.834195\pi$
$607$		−	24.5218i		−	0.995308i	0.867376	−	0.497654i	$-0.165805\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	5.50630	−	8.16716i	0.222579	−	0.330138i
$613$	−16.8041			−0.678713			−0.339357	−	0.940658i	$-0.610209\pi$
$613$	−16.8041			−0.678713			−0.339357	+	0.940658i	$0.610209\pi$
$614$	−41.2705	+	21.9495i	−1.66554	+	0.885811i
$615$		0			0
$616$	28.7984	+	3.01756i	1.16032	+	0.121581i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		−	46.4327i		−	1.86629i	−0.359501	−	0.933145i	$-0.617053\pi$
$619$		−	46.4327i		−	1.86629i	0.359501	−	0.933145i	$-0.382947\pi$
$620$	−24.5967	−	16.5831i	−0.987829	−	0.665994i
$621$		0			0
$622$	18.5198	−	9.84968i	0.742577	−	0.394936i
$623$		−	41.4127i		−	1.65917i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$	−25.8541	+	13.7504i	−1.03005	+	0.547829i
$631$		−	14.8324i		−	0.590468i	−0.955425	−	0.295234i	$-0.904602\pi$
$631$		−	14.8324i		−	0.590468i	0.955425	−	0.295234i	$-0.0953977\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			37.7694i			1.49883i
$636$		0			0
$637$	−17.5792			−0.696515
$638$		0			0
$639$			44.4972i			1.76028i
$640$	−6.99119	−	24.3130i	−0.276351	−	0.961057i
$641$	−31.3050			−1.23647			−0.618236	−	0.785993i	$-0.712152\pi$
$641$	−31.3050			−1.23647			−0.618236	+	0.785993i	$0.712152\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−25.3172	−	2.65280i	−0.994555	−	0.104212i
$649$	−49.1935			−1.93101
$650$	23.0902	+	43.4152i	0.905671	+	1.70288i
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−36.8001			−1.43571
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−42.0000			−1.63361			−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000			−1.63361			−0.816805	+	0.576913i	$0.804257\pi$
$662$	−8.28232	+	4.40491i	−0.321901	+	0.171202i
$663$		0			0
$664$	0.214782	−	2.04979i	0.00833515	−	0.0795472i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−17.7730	−	11.9826i	−0.687658	−	0.463619i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−19.6084			−0.755850			−0.377925	−	0.925836i	$-0.623362\pi$
$673$	−19.6084			−0.755850			−0.377925	+	0.925836i	$0.623362\pi$
$674$	−21.7426	−	40.8815i	−0.837495	−	1.57470i
$675$		0			0
$676$	−39.5344	+	58.6391i	−1.52056	+	2.25535i
$677$	51.9626			1.99709			0.998543	−	0.0539677i	$-0.0171868\pi$
$677$	51.9626			1.99709			0.998543	+	0.0539677i	$0.0171868\pi$
$678$		0			0
$679$		0			0
$680$	10.3262	+	1.08201i	0.395993	+	0.0414931i
$681$		0			0
$682$	14.6094	+	27.4693i	0.559424	+	1.05186i
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$	−17.2361	+	9.16693i	−0.658076	+	0.349995i
$687$		0			0
$688$	48.4855	−	19.6134i	1.84849	−	0.747752i
$689$		0			0
$690$		0			0
$691$		−	44.4972i		−	1.69275i	−0.532585	−	0.846376i	$-0.678779\pi$
$691$		−	44.4972i		−	1.69275i	0.532585	−	0.846376i	$-0.321221\pi$
$692$	−26.9959	+	40.0414i	−1.02623	+	1.52215i
$693$	30.7125			1.16667
$694$	14.5066	−	7.71526i	0.550662	−	0.292867i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−25.5938	−	17.2553i	−0.967353	−	0.652189i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	−5.50000	+	25.9567i	−0.207289	+	0.978280i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−4.47214			−0.167955			−0.0839773	−	0.996468i	$-0.526762\pi$
$709$	−4.47214			−0.167955			−0.0839773	+	0.996468i	$0.526762\pi$
$710$	−41.4116	+	22.0246i	−1.55415	+	0.826567i
$711$		0			0
$712$	37.7407	+	3.95457i	1.41439	+	0.148204i
$713$		0			0
$714$		0			0
$715$		−	51.5736i		−	1.92874i
$716$	33.0000	+	22.2486i	1.23327	+	0.831469i
$717$		0			0
$718$		0			0
$719$			46.4327i			1.73165i	0.500348	+	0.865825i	$0.333206\pi$
$719$			46.4327i			1.73165i	−0.500348	+	0.865825i	$0.666794\pi$
$720$	−10.0623	−	24.8747i	−0.375000	−	0.927025i
$721$		0			0
$722$	−12.6172	−	23.7235i	−0.469565	−	0.882898i
$723$		0			0
$724$	5.00000	−	7.41620i	0.185824	−	0.275621i
$725$		0			0
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$	6.32713	−	60.3834i	0.234499	−	2.23796i
$729$	−27.0000			−1.00000
$730$	−18.2148	−	34.2483i	−0.674159	−	1.26758i
$731$			21.4656i			0.793936i
$732$		0			0
$733$	45.3960			1.67674			0.838369	−	0.545103i	$-0.183509\pi$
$733$	45.3960			1.67674			0.838369	+	0.545103i	$0.183509\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			18.3483i			0.673135i	0.941659	+	0.336567i	$0.109266\pi$
$743$			18.3483i			0.673135i	−0.941659	+	0.336567i	$0.890734\pi$
$744$		0			0
$745$		0			0
$746$	−25.2705	−	47.5148i	−0.925219	−	1.73964i
$747$		−	2.18603i		−	0.0799827i
$748$	−9.02913	−	6.08744i	−0.330138	−	0.222579i
$749$	59.4164			2.17103
$750$		0			0
$751$			44.4972i			1.62373i	0.583848	+	0.811863i	$0.301546\pi$
$751$			44.4972i			1.62373i	−0.583848	+	0.811863i	$0.698454\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−18.5198	+	9.84968i	−0.672670	+	0.357757i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$	11.0000	+	7.41620i	0.397966	+	0.268309i
$765$	11.0126			0.398161
$766$		0			0
$767$			103.147i			3.72443i
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$	15.2016	+	28.5828i	0.547829	+	1.03005i
$771$		0			0
$772$	−21.0563	+	31.2316i	−0.757834	+	1.12405i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$	48.9787	−	26.0491i	1.76050	−	0.936316i
$775$		−	33.1662i		−	1.19137i
$776$		0			0
$777$		0			0
$778$	−17.2657	−	32.4638i	−0.619005	−	1.16388i
$779$		0			0
$780$		0			0
$781$	49.1935			1.76028
$782$		0			0
$783$		0			0
$784$	3.79180	+	9.37357i	0.135421	+	0.334770i
$785$		0			0
$786$		0			0
$787$		−	46.8575i		−	1.67029i	−0.550030	−	0.835145i	$-0.685384\pi$
$787$		−	46.8575i		−	1.67029i	0.550030	−	0.835145i	$-0.314616\pi$
$788$	−19.6542	+	29.1519i	−0.700151	+	1.03849i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−2.93278	+	27.9893i	−0.104212	+	0.994555i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	24.5967	+	16.5831i	0.871809	+	0.587773i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	18.1693	−	21.6766i	0.642381	−	0.766385i
$801$	40.2492			1.42214
$802$	2.96979	+	5.58394i	0.104867	+	0.197176i
$803$			40.6841i			1.43571i
$804$		0			0
$805$		0			0
$806$	57.5967	−	30.6326i	2.02876	−	1.07899i
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	−13.3641	−	25.1277i	−0.469565	−	0.882898i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		−	64.3969i		−	2.25021i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$	−30.4033	−	57.1656i	−1.05786	−	1.98904i
$827$			53.0310i			1.84407i	0.387110	+	0.922034i	$0.373473\pi$
$827$			53.0310i			1.84407i	−0.387110	+	0.922034i	$0.626527\pi$
$828$		0			0
$829$	−22.3607			−0.776619			−0.388309	−	0.921529i	$-0.626941\pi$
$829$	−22.3607			−0.776619			−0.388309	+	0.921529i	$0.626941\pi$
$830$	2.03444	−	1.08201i	0.0706165	−	0.0375571i
$831$		0			0
$832$	54.4251	+	11.5322i	1.88685	+	0.399807i
$833$	−4.14989			−0.143785
$834$		0			0
$835$		−	23.9651i		−	0.829347i
$836$		0			0
$837$		0			0
$838$	24.8469	−	13.2147i	0.858324	−	0.456496i
$839$		−	14.8324i		−	0.512071i	−0.966667	−	0.256036i	$-0.917584\pi$
$839$		−	14.8324i		−	0.512071i	0.966667	−	0.256036i	$-0.0824164\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$	20.7885	+	39.0876i	0.716420	+	1.34705i
$843$		0			0
$844$		0			0
$845$	−79.0689			−2.72005
$846$		0			0
$847$		−	33.9540i		−	1.16667i
$848$		0			0
$849$		0			0
$850$	5.45085	+	10.2489i	0.186963	+	0.351536i
$851$		0			0
$852$		0			0
$853$	−55.2459			−1.89158			−0.945792	−	0.324772i	$-0.894712\pi$
$853$	−55.2459			−1.89158			−0.945792	+	0.324772i	$0.894712\pi$
$854$		0			0
$855$		0			0
$856$	−5.67376	+	54.1480i	−0.193925	+	1.85074i
$857$	25.4000			0.867647			0.433824	−	0.900998i	$-0.357164\pi$
$857$	25.4000			0.867647			0.433824	+	0.900998i	$0.357164\pi$
$858$		0			0
$859$		−	46.4327i		−	1.58426i	−0.610349	−	0.792132i	$-0.708971\pi$
$859$		−	46.4327i		−	1.58426i	0.610349	−	0.792132i	$-0.291029\pi$
$860$	48.4855	+	32.6889i	1.65334	+	1.11468i
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	−53.9918			−1.83578
$866$		0			0
$867$		0			0
$868$	−22.8918	+	33.9540i	−0.776997	+	1.15247i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	34.5106i		−	1.16667i
$876$		0			0
$877$	−44.6209			−1.50674			−0.753370	−	0.657597i	$-0.771573\pi$
$877$	−44.6209			−1.50674			−0.753370	+	0.657597i	$0.771573\pi$
$878$		0			0
$879$		0			0
$880$	−27.5000	+	11.1243i	−0.927025	+	0.375000i
$881$	2.00000			0.0673817			0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000			0.0673817			0.0336909	+	0.999432i	$0.489274\pi$
$882$	5.03600	+	9.46892i	0.169571	+	0.318835i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$	−12.7639	+	18.9320i	−0.429297	+	0.636751i
$885$		0			0
$886$		0			0
$887$		−	13.9763i		−	0.469277i	−0.972083	−	0.234639i	$-0.924609\pi$
$887$		−	13.9763i		−	0.469277i	0.972083	−	0.234639i	$-0.0753906\pi$
$888$		0			0
$889$	52.1378			1.74864
$890$	19.9220	+	37.4582i	0.667786	+	1.25560i
$891$			29.8496i			1.00000i
$892$		0			0
$893$		0			0
$894$		0			0
$895$			44.4972i			1.48738i
$896$	−33.5623	+	9.65081i	−1.12124	+	0.322411i
$897$		0			0
$898$	26.7281	+	50.2554i	0.891929	+	1.67705i
$899$		0			0
$900$	16.7705	−	24.8747i	0.559017	−	0.829156i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	10.0000			0.332411
$906$		0			0
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$	−42.1584	−	28.4232i	−1.39908	−	0.943256i
$909$		0			0
$910$	59.9314	−	31.8742i	1.98671	−	1.05662i
$911$		−	59.6992i		−	1.97792i	−0.148168	−	0.988962i	$-0.547338\pi$
$911$		−	59.6992i		−	1.97792i	0.148168	−	0.988962i	$-0.452662\pi$
$912$		0			0
$913$	−2.41675			−0.0799827
$914$	−14.6869	−	27.6150i	−0.485800	−	0.913423i
$915$		0			0
$916$	−6.70820	+	9.94987i	−0.221645	+	0.328753i
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	103.147i		−	3.39513i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−14.0000			−0.459325			−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000			−0.459325			−0.229663	+	0.973270i	$0.573762\pi$
$930$		0			0
$931$		0			0
$932$	29.2646	−	43.4065i	0.958595	−	1.42183i
$933$		0			0
$934$		0			0
$935$		−	12.1749i		−	0.398161i
$936$	58.6869	+	6.14936i	1.91824	+	0.200998i
$937$	−60.5585			−1.97836			−0.989179	−	0.146712i	$-0.953131\pi$
$937$	−60.5585			−1.97836			−0.989179	+	0.146712i	$0.953131\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$	55.0000	−	22.2486i	1.79010	−	0.724130i
$945$		0			0
$946$	−28.7984	−	54.1480i	−0.936316	−	1.76050i
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$	85.3050			2.76912
$950$		0			0
$951$		0			0
$952$	1.49363	−	14.2546i	0.0484089	−	0.461994i
$953$	33.5168			1.08572			0.542858	−	0.839825i	$-0.317342\pi$
$953$	33.5168			1.08572			0.542858	+	0.839825i	$0.317342\pi$
$954$		0			0
$955$			14.8324i			0.479965i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−13.0000			−0.419355
$962$		0			0
$963$			57.7471i			1.86087i
$964$		0			0
$965$	−42.1126			−1.35565
$966$		0			0
$967$		−	61.5625i		−	1.97972i	−0.142063	−	0.989858i	$-0.545374\pi$
$967$		−	61.5625i		−	1.97972i	0.142063	−	0.989858i	$-0.454626\pi$
$968$	30.9433	+	3.24231i	0.994555	+	0.104212i
$969$		0			0
$970$		0			0
$971$			59.6992i			1.91584i	0.287035	+	0.957920i	$0.407330\pi$
$971$			59.6992i			1.91584i	−0.287035	+	0.957920i	$0.592670\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		−	44.4972i		−	1.42214i
$980$	−6.31966	+	9.37357i	−0.201874	+	0.299428i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	−39.3084			−1.25247
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−27.7797	+	14.7745i	−0.882898	+	0.469565i
$991$		−	6.63325i		−	0.210712i	−0.994435	−	0.105356i	$-0.966402\pi$
$991$		−	6.63325i		−	0.210712i	0.994435	−	0.105356i	$-0.0335982\pi$
$992$	−28.7573	−	24.1043i	−0.913046	−	0.765312i
$993$		0			0
$994$	30.4033	+	57.1656i	0.964333	+	1.81318i
$995$			33.1662i			1.05144i
$996$		0			0
$997$	−24.9210			−0.789255			−0.394627	−	0.918841i	$-0.629126\pi$
$997$	−24.9210			−0.789255			−0.394627	+	0.918841i	$0.629126\pi$
$998$	55.5595	−	29.5490i	1.75870	−	0.935359i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	220.2.g.a.219.6	yes	8
4.3	odd	2	inner	220.2.g.a.219.5	yes	8
5.4	even	2	inner	220.2.g.a.219.3	✓	8
11.10	odd	2	inner	220.2.g.a.219.3	✓	8
20.19	odd	2	inner	220.2.g.a.219.4	yes	8
44.43	even	2	inner	220.2.g.a.219.4	yes	8
55.54	odd	2	CM	220.2.g.a.219.6	yes	8
220.219	even	2	inner	220.2.g.a.219.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.2.g.a.219.3	✓	8	5.4	even	2	inner
220.2.g.a.219.3	✓	8	11.10	odd	2	inner
220.2.g.a.219.4	yes	8	20.19	odd	2	inner
220.2.g.a.219.4	yes	8	44.43	even	2	inner
220.2.g.a.219.5	yes	8	4.3	odd	2	inner
220.2.g.a.219.5	yes	8	220.219	even	2	inner
220.2.g.a.219.6	yes	8	1.1	even	1	trivial
220.2.g.a.219.6	yes	8	55.54	odd	2	CM

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:	\( N \)	\(=\)	\( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	220.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.664066 + 1.24861i) q^{2} +(-1.11803 + 1.65831i) q^{4} -2.23607 q^{5} +3.08672i q^{7} +(-2.81303 - 0.294756i) q^{8} -3.00000 q^{9} +(-1.48490 - 2.79197i) q^{10} +3.31662i q^{11} +6.95418 q^{13} +(-3.85410 + 2.04979i) q^{14} +(-1.50000 - 3.70810i) q^{16} +1.64166 q^{17} +(-1.99220 - 3.74582i) q^{18} +(2.50000 - 3.70810i) q^{20} +(-4.14116 + 2.20246i) q^{22} +5.00000 q^{25} +(4.61803 + 8.68304i) q^{26} +(-5.11875 - 3.45106i) q^{28} -6.63325i q^{31} +(3.63386 - 4.33533i) q^{32} +(1.09017 + 2.04979i) q^{34} -6.90212i q^{35} +(3.35410 - 4.97494i) q^{36} +(6.29012 + 0.659094i) q^{40} +13.0756i q^{43} +(-5.50000 - 3.70810i) q^{44} +6.70820 q^{45} -2.52786 q^{49} +(3.32033 + 6.24303i) q^{50} +(-7.77501 + 11.5322i) q^{52} -7.41620i q^{55} +(0.909830 - 8.68304i) q^{56} +14.8324i q^{59} +(8.28232 - 4.40491i) q^{62} -9.26017i q^{63} +(7.82624 + 1.65831i) q^{64} -15.5500 q^{65} +(-1.83543 + 2.72239i) q^{68} +(8.61803 - 4.58346i) q^{70} -14.8324i q^{71} +(8.43908 + 0.884268i) q^{72} +12.2667 q^{73} -10.2375 q^{77} +(3.35410 + 8.29156i) q^{80} +9.00000 q^{81} +0.728677i q^{83} -3.67086 q^{85} +(-16.3262 + 8.68304i) q^{86} +(0.977595 - 9.32975i) q^{88} -13.4164 q^{89} +(4.45469 + 8.37590i) q^{90} +21.4656i q^{91} +(-1.67867 - 3.15631i) q^{98} -9.94987i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9} - 4 q^{14} - 12 q^{16} + 20 q^{20} + 40 q^{25} + 28 q^{26} - 36 q^{34} - 44 q^{44} - 56 q^{49} + 52 q^{56} + 60 q^{70} + 72 q^{81} - 68 q^{86}+O(q^{100}) \) 8 * q - 24 * q^9 - 4 * q^14 - 12 * q^16 + 20 * q^20 + 40 * q^25 + 28 * q^26 - 36 * q^34 - 44 * q^44 - 56 * q^49 + 52 * q^56 + 60 * q^70 + 72 * q^81 - 68 * q^86

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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