LMFDB - Embedded newform 275.2.e.c.43.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [275,2,Mod(32,275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 2])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("275.32"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	275.e (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.19588605559$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.1499238400.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 59x^{4} - 66x^{3} + 54x^{2} - 26x + 5 $ x^8 - 4x^7 + 16x^6 - 34x^5 + 59x^4 - 66x^3 + 54x^2 - 26*x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			43.4
Root			$0.500000 - 2.41267i$ of defining polynomial
Character	$\chi$	$=$	275.43
Dual form			275.2.e.c.32.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.91267 + 1.91267i) q^{2} +5.31662i q^{4} +(0.605599 + 0.605599i) q^{7} +(-6.34361 + 6.34361i) q^{8} -3.00000i q^{9} -3.31662 q^{11} +(4.43094 - 4.43094i) q^{13} +2.31662i q^{14} -13.6332 q^{16} +(3.21974 + 3.21974i) q^{17} +(5.73801 - 5.73801i) q^{18} +(-6.34361 - 6.34361i) q^{22} +16.9499 q^{26} +(-3.21974 + 3.21974i) q^{28} -6.63325 q^{31} +(-13.3887 - 13.3887i) q^{32} +12.3166i q^{34} +15.9499 q^{36} +(7.04509 - 7.04509i) q^{43} -17.6332i q^{44} -6.26650i q^{49} +(23.5577 + 23.5577i) q^{52} -7.68338 q^{56} +4.00000i q^{59} +(-12.6872 - 12.6872i) q^{62} +(1.81680 - 1.81680i) q^{63} -23.9499i q^{64} +(-17.1182 + 17.1182i) q^{68} -8.00000 q^{71} +(19.0308 + 19.0308i) q^{72} +(-12.0816 + 12.0816i) q^{73} +(-2.00855 - 2.00855i) q^{77} -9.00000 q^{81} +(-9.46748 + 9.46748i) q^{83} +26.9499 q^{86} +(21.0394 - 21.0394i) q^{88} -13.2665i q^{89} +5.36675 q^{91} +(11.9858 - 11.9858i) q^{98} +9.94987i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 56 q^{16} + 56 q^{26} + 48 q^{36} - 88 q^{56} - 64 q^{71} - 72 q^{81} + 136 q^{86} + 96 q^{91}+O(q^{100}) $ 8 * q - 56 * q^16 + 56 * q^26 + 48 * q^36 - 88 * q^56 - 64 * q^71 - 72 * q^81 + 136 * q^86 + 96 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$.

$n$	$101$	$177$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	1.91267	+	1.91267i	1.35246	+	1.35246i	0.882898	+	0.469565i	$0.155589\pi$
$2$	1.91267	+	1.91267i	1.35246	+	1.35246i	0.469565	+	0.882898i	$0.344411\pi$
$3$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$			5.31662i			2.65831i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.605599	+	0.605599i	0.228895	+	0.228895i	0.812231	−	0.583336i	$-0.198253\pi$
$7$	0.605599	+	0.605599i	0.228895	+	0.228895i	−0.583336	+	0.812231i	$0.698253\pi$
$8$	−6.34361	+	6.34361i	−2.24281	+	2.24281i
$9$		−	3.00000i		−	1.00000i
$10$		0			0
$11$	−3.31662			−1.00000
$11$	−3.31662			−1.00000
$12$		0			0
$13$	4.43094	−	4.43094i	1.22892	−	1.22892i	0.264550	−	0.964372i	$-0.414776\pi$
$13$	4.43094	−	4.43094i	1.22892	−	1.22892i	0.964372	−	0.264550i	$-0.0852236\pi$
$14$			2.31662i			0.619144i
$15$		0			0
$16$	−13.6332			−3.40831
$17$	3.21974	+	3.21974i	0.780903	+	0.780903i	0.979983	−	0.199081i	$-0.0637955\pi$
$17$	3.21974	+	3.21974i	0.780903	+	0.780903i	−0.199081	+	0.979983i	$0.563796\pi$
$18$	5.73801	−	5.73801i	1.35246	−	1.35246i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$	−6.34361	−	6.34361i	−1.35246	−	1.35246i
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$		0			0
$25$		0			0
$26$	16.9499			3.32414
$27$		0			0
$28$	−3.21974	+	3.21974i	−0.608474	+	0.608474i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−6.63325			−1.19137			−0.595683	−	0.803219i	$-0.703119\pi$
$31$	−6.63325			−1.19137			−0.595683	+	0.803219i	$0.703119\pi$
$32$	−13.3887	−	13.3887i	−2.36681	−	2.36681i
$33$		0			0
$34$			12.3166i			2.11228i
$35$		0			0
$36$	15.9499			2.65831
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	7.04509	−	7.04509i	1.07437	−	1.07437i	0.0773627	−	0.997003i	$-0.475350\pi$
$43$	7.04509	−	7.04509i	1.07437	−	1.07437i	0.997003	−	0.0773627i	$-0.0246499\pi$
$44$		−	17.6332i		−	2.65831i
$45$		0			0
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$		0			0
$49$		−	6.26650i		−	0.895214i
$50$		0			0
$51$		0			0
$52$	23.5577	+	23.5577i	3.26686	+	3.26686i
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$		0			0
$55$		0			0
$56$	−7.68338			−1.02673
$57$		0			0
$58$		0			0
$59$			4.00000i			0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$			4.00000i			0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	−12.6872	−	12.6872i	−1.61128	−	1.61128i
$63$	1.81680	−	1.81680i	0.228895	−	0.228895i
$64$		−	23.9499i		−	2.99373i
$65$		0			0
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$	−17.1182	+	17.1182i	−2.07588	+	2.07588i
$69$		0			0
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$	19.0308	+	19.0308i	2.24281	+	2.24281i
$73$	−12.0816	+	12.0816i	−1.41405	+	1.41405i	−0.696193	+	0.717855i	$0.745124\pi$
$73$	−12.0816	+	12.0816i	−1.41405	+	1.41405i	−0.717855	+	0.696193i	$0.754876\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.00855	−	2.00855i	−0.228895	−	0.228895i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$	−9.46748	+	9.46748i	−1.03919	+	1.03919i	−0.0399913	+	0.999200i	$0.512733\pi$
$83$	−9.46748	+	9.46748i	−1.03919	+	1.03919i	−0.999200	+	0.0399913i	$0.987267\pi$
$84$		0			0
$85$		0			0
$86$	26.9499			2.90608
$87$		0			0
$88$	21.0394	−	21.0394i	2.24281	−	2.24281i
$89$		−	13.2665i		−	1.40625i	−0.711068	−	0.703123i	$-0.751788\pi$
$89$		−	13.2665i		−	1.40625i	0.711068	−	0.703123i	$-0.248212\pi$
$90$		0			0
$91$	5.36675			0.562588
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$97$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$98$	11.9858	−	11.9858i	1.21074	−	1.21074i
$99$			9.94987i			1.00000i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$			56.2164i			5.51247i
$105$		0			0
$106$		0			0
$107$	5.83389	+	5.83389i	0.563983	+	0.563983i	0.930436	−	0.366453i	$-0.119428\pi$
$107$	5.83389	+	5.83389i	0.563983	+	0.563983i	−0.366453	+	0.930436i	$0.619428\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$	−8.25629	−	8.25629i	−0.780146	−	0.780146i
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−13.2928	−	13.2928i	−1.22892	−	1.22892i
$118$	−7.65069	+	7.65069i	−0.704303	+	0.704303i
$119$			3.89975i			0.357489i
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$		−	35.2665i		−	3.16703i
$125$		0			0
$126$	6.94987			0.619144
$127$	−15.9070	−	15.9070i	−1.41152	−	1.41152i	−0.749416	−	0.662100i	$-0.769666\pi$
$127$	−15.9070	−	15.9070i	−1.41152	−	1.41152i	−0.662100	−	0.749416i	$-0.730334\pi$
$128$	19.0308	−	19.0308i	1.68210	−	1.68210i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−40.8496			−3.50283
$137$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$137$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	−15.3014	−	15.3014i	−1.28406	−	1.28406i
$143$	−14.6958	+	14.6958i	−1.22892	+	1.22892i
$144$			40.8997i			3.40831i
$145$		0			0
$146$	−46.2164			−3.82489
$147$		0			0
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	9.65923	−	9.65923i	0.780903	−	0.780903i
$154$		−	7.68338i		−	0.619144i
$155$		0			0
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−17.2140	−	17.2140i	−1.35246	−	1.35246i
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$		0			0
$165$		0			0
$166$	−36.2164			−2.81094
$167$	17.1182	+	17.1182i	1.32464	+	1.32464i	0.909970	+	0.414673i	$0.136104\pi$
$167$	17.1182	+	17.1182i	1.32464	+	1.32464i	0.414673	+	0.909970i	$0.363896\pi$
$168$		0			0
$169$		−	26.2665i		−	2.02050i
$170$		0			0
$171$		0			0
$172$	37.4561	+	37.4561i	2.85600	+	2.85600i
$173$	−6.85334	+	6.85334i	−0.521050	+	0.521050i	−0.917888	−	0.396839i	$-0.870107\pi$
$173$	−6.85334	+	6.85334i	−0.521050	+	0.521050i	0.396839	+	0.917888i	$0.370107\pi$
$174$		0			0
$175$		0			0
$176$	45.2164			3.40831
$177$		0			0
$178$	25.3745	−	25.3745i	1.90190	−	1.90190i
$179$			19.8997i			1.48738i	0.668526	+	0.743689i	$0.266925\pi$
$179$			19.8997i			1.48738i	−0.668526	+	0.743689i	$0.733075\pi$
$180$		0			0
$181$	26.5330			1.97218			0.986091	−	0.166206i	$-0.0531515\pi$
$181$	26.5330			1.97218			0.986091	+	0.166206i	$0.0531515\pi$
$182$	10.2648	+	10.2648i	0.760880	+	0.760880i
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−10.6787	−	10.6787i	−0.780903	−	0.780903i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.63325			−0.479965			−0.239983	−	0.970777i	$-0.577142\pi$
$191$	−6.63325			−0.479965			−0.239983	+	0.970777i	$0.577142\pi$
$192$		0			0
$193$	−0.797347	+	0.797347i	−0.0573943	+	0.0573943i	−0.735221	−	0.677827i	$-0.762922\pi$
$193$	−0.797347	+	0.797347i	−0.0573943	+	0.0573943i	0.677827	+	0.735221i	$0.262922\pi$
$194$		0			0
$195$		0			0
$196$	33.3166			2.37976
$197$	19.7323	+	19.7323i	1.40587	+	1.40587i	0.779635	+	0.626234i	$0.215405\pi$
$197$	19.7323	+	19.7323i	1.40587	+	1.40587i	0.626234	+	0.779635i	$0.284595\pi$
$198$	−19.0308	+	19.0308i	−1.35246	+	1.35246i
$199$			24.0000i			1.70131i	0.525720	+	0.850657i	$0.323796\pi$
$199$			24.0000i			1.70131i	−0.525720	+	0.850657i	$0.676204\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−60.4081	+	60.4081i	−4.18855	+	4.18855i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$			22.3166i			1.52553i
$215$		0			0
$216$		0			0
$217$	−4.01709	−	4.01709i	−0.272698	−	0.272698i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	28.5330			1.91934
$222$		0			0
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$		−	16.2164i		−	1.08350i
$225$		0			0
$226$		0			0
$227$	−4.62269	−	4.62269i	−0.306819	−	0.306819i	0.536855	−	0.843674i	$-0.319612\pi$
$227$	−4.62269	−	4.62269i	−0.306819	−	0.306819i	−0.843674	+	0.536855i	$0.819612\pi$
$228$		0			0
$229$		−	6.00000i		−	0.396491i	−0.980152	−	0.198246i	$-0.936476\pi$
$229$		−	6.00000i		−	0.396491i	0.980152	−	0.198246i	$-0.0635244\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	20.9435	−	20.9435i	1.37206	−	1.37206i	0.514662	−	0.857393i	$-0.327917\pi$
$233$	20.9435	−	20.9435i	1.37206	−	1.37206i	0.857393	−	0.514662i	$-0.172083\pi$
$234$		−	50.8496i		−	3.32414i
$235$		0			0
$236$	−21.2665			−1.38433
$237$		0			0
$238$	−7.45894	+	7.45894i	−0.483491	+	0.483491i
$239$		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			$\pi$
$242$	21.0394	+	21.0394i	1.35246	+	1.35246i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$	42.0788	−	42.0788i	2.67200	−	2.67200i
$249$		0			0
$250$		0			0
$251$	−28.0000			−1.76734			−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000			−1.76734			−0.883672	+	0.468106i	$0.844936\pi$
$252$	9.65923	+	9.65923i	0.608474	+	0.608474i
$253$		0			0
$254$		−	60.8496i		−	3.81804i
$255$		0			0
$256$	24.8997			1.55623
$257$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$257$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	18.3294	−	18.3294i	1.13024	−	1.13024i	0.140100	−	0.990137i	$-0.455258\pi$
$263$	18.3294	−	18.3294i	1.13024	−	1.13024i	0.990137	−	0.140100i	$-0.0447424\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			14.0000i			0.853595i	0.904347	+	0.426798i	$0.140358\pi$
$269$			14.0000i			0.853595i	−0.904347	+	0.426798i	$0.859642\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−43.8956	−	43.8956i	−2.66156	−	2.66156i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	8.44803	+	8.44803i	0.507593	+	0.507593i	0.913787	−	0.406194i	$-0.133144\pi$
$277$	8.44803	+	8.44803i	0.507593	+	0.507593i	−0.406194	+	0.913787i	$0.633144\pi$
$278$		0			0
$279$			19.8997i			1.19137i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	23.5577	−	23.5577i	1.40036	−	1.40036i	0.601438	−	0.798920i	$-0.294595\pi$
$283$	23.5577	−	23.5577i	1.40036	−	1.40036i	0.798920	−	0.601438i	$-0.205405\pi$
$284$		−	42.5330i		−	2.52387i
$285$		0			0
$286$	−56.2164			−3.32414
$287$		0			0
$288$	−40.1661	+	40.1661i	−2.36681	+	2.36681i
$289$			3.73350i			0.219618i
$290$		0			0
$291$		0			0
$292$	−64.2335	−	64.2335i	−3.75898	−	3.75898i
$293$	−17.3099	+	17.3099i	−1.01126	+	1.01126i	−0.0113203	+	0.999936i	$0.503603\pi$
$293$	−17.3099	+	17.3099i	−1.01126	+	1.01126i	−0.999936	+	0.0113203i	$0.996397\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	8.53300			0.491834
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$	36.9499			2.11228
$307$	22.3465	+	22.3465i	1.27538	+	1.27538i	0.943225	+	0.332155i	$0.107776\pi$
$307$	22.3465	+	22.3465i	1.27538	+	1.27538i	0.332155	+	0.943225i	$0.392224\pi$
$308$	10.6787	−	10.6787i	0.608474	−	0.608474i
$309$		0			0
$310$		0			0
$311$	32.0000			1.81455			0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000			1.81455			0.907277	+	0.420534i	$0.138157\pi$
$312$		0			0
$313$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		−	47.8496i		−	2.65831i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−6.63325			−0.364596			−0.182298	−	0.983243i	$-0.558354\pi$
$331$	−6.63325			−0.364596			−0.182298	+	0.983243i	$0.558354\pi$
$332$	−50.3351	−	50.3351i	−2.76250	−	2.76250i
$333$		0			0
$334$			65.4829i			3.58306i
$335$		0			0
$336$		0			0
$337$	−8.06454	−	8.06454i	−0.439303	−	0.439303i	0.452474	−	0.891778i	$-0.350541\pi$
$337$	−8.06454	−	8.06454i	−0.439303	−	0.439303i	−0.891778	+	0.452474i	$0.850541\pi$
$338$	50.2392	−	50.2392i	2.73265	−	2.73265i
$339$		0			0
$340$		0			0
$341$	22.0000			1.19137
$342$		0			0
$343$	8.03418	−	8.03418i	0.433805	−	0.433805i
$344$			89.3826i			4.81919i
$345$		0			0
$346$	−26.2164			−1.40940
$347$	11.8899	+	11.8899i	0.638282	+	0.638282i	0.950132	−	0.311849i	$-0.100948\pi$
$347$	11.8899	+	11.8899i	0.638282	+	0.638282i	−0.311849	+	0.950132i	$0.600948\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	44.4053	+	44.4053i	2.36681	+	2.36681i
$353$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$353$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$		0			0
$356$	70.5330			3.73824
$357$		0			0
$358$	−38.0617	+	38.0617i	−2.01162	+	2.01162i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$	50.7489	+	50.7489i	2.66730	+	2.66730i
$363$		0			0
$364$			28.5330i			1.49554i
$365$		0			0
$366$		0			0
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	15.7152	−	15.7152i	0.813703	−	0.813703i	−0.171484	−	0.985187i	$-0.554856\pi$
$373$	15.7152	−	15.7152i	0.813703	−	0.813703i	0.985187	+	0.171484i	$0.0548560\pi$
$374$		−	40.8496i		−	2.11228i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	36.0000i		−	1.84920i	−0.380945	−	0.924598i	$-0.624401\pi$
$379$		−	36.0000i		−	1.84920i	0.380945	−	0.924598i	$-0.375599\pi$
$380$		0			0
$381$		0			0
$382$	−12.6872	−	12.6872i	−0.649135	−	0.649135i
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$	−3.05013			−0.155247
$387$	−21.1353	−	21.1353i	−1.07437	−	1.07437i
$388$		0			0
$389$		−	26.0000i		−	1.31825i	−0.752032	−	0.659126i	$-0.770926\pi$
$389$		−	26.0000i		−	1.31825i	0.752032	−	0.659126i	$-0.229074\pi$
$390$		0			0
$391$		0			0
$392$	39.7523	+	39.7523i	2.00779	+	2.00779i
$393$		0			0
$394$			75.4829i			3.80277i
$395$		0			0
$396$	−52.8997			−2.65831
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$	−45.9041	+	45.9041i	−2.30097	+	2.30097i
$399$		0			0
$400$		0			0
$401$	−39.7995			−1.98749			−0.993746	−	0.111664i	$-0.964382\pi$
$401$	−39.7995			−1.98749			−0.993746	+	0.111664i	$0.964382\pi$
$402$		0			0
$403$	−29.3915	+	29.3915i	−1.46410	+	1.46410i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.42240	+	2.42240i	−0.119198	+	0.119198i
$414$		0			0
$415$		0			0
$416$	−118.649			−5.81725
$417$		0			0
$418$		0			0
$419$			19.8997i			0.972166i	0.873913	+	0.486083i	$0.161575\pi$
$419$			19.8997i			0.972166i	−0.873913	+	0.486083i	$0.838425\pi$
$420$		0			0
$421$	26.5330			1.29314			0.646570	−	0.762855i	$-0.276203\pi$
$421$	26.5330			1.29314			0.646570	+	0.762855i	$0.276203\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−31.0166	+	31.0166i	−1.49924	+	1.49924i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$		−	15.3668i		−	0.737628i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−18.7995			−0.895214
$442$	54.5743	+	54.5743i	2.59583	+	2.59583i
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	14.5040	−	14.5040i	0.685251	−	0.685251i
$449$		−	13.2665i		−	0.626085i	−0.949739	−	0.313042i	$-0.898652\pi$
$449$		−	13.2665i		−	0.626085i	0.949739	−	0.313042i	$-0.101348\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		−	17.6834i		−	0.829922i
$455$		0			0
$456$		0			0
$457$	−7.23683	−	7.23683i	−0.338525	−	0.338525i	0.517287	−	0.855812i	$-0.326942\pi$
$457$	−7.23683	−	7.23683i	−0.338525	−	0.338525i	−0.855812	+	0.517287i	$0.826942\pi$
$458$	11.4760	−	11.4760i	0.536240	−	0.536240i
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$	80.1161			3.71131
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$	70.6730	−	70.6730i	3.26686	−	3.26686i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−25.3745	−	25.3745i	−1.16795	−	1.16795i
$473$	−23.3659	+	23.3659i	−1.07437	+	1.07437i
$474$		0			0
$475$		0			0
$476$	−20.7335			−0.950318
$477$		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
$484$			58.4829i			2.65831i
$485$		0			0
$486$		0			0
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	90.4327			4.06055
$497$	−4.84479	−	4.84479i	−0.217319	−	0.217319i
$498$		0			0
$499$			4.00000i			0.179065i	0.995984	+	0.0895323i	$0.0285372\pi$
$499$			4.00000i			0.179065i	−0.995984	+	0.0895323i	$0.971463\pi$
$500$		0			0
$501$		0			0
$502$	−53.5548	−	53.5548i	−2.39027	−	2.39027i
$503$	−31.2083	+	31.2083i	−1.39151	+	1.39151i	−0.569565	+	0.821946i	$0.692888\pi$
$503$	−31.2083	+	31.2083i	−1.39151	+	1.39151i	−0.821946	+	0.569565i	$0.807112\pi$
$504$			23.0501i			1.02673i
$505$		0			0
$506$		0			0
$507$		0			0
$508$	84.5714	−	84.5714i	3.75225	−	3.75225i
$509$			34.0000i			1.50702i	0.657434	+	0.753512i	$0.271642\pi$
$509$			34.0000i			1.50702i	−0.657434	+	0.753512i	$0.728358\pi$
$510$		0			0
$511$	−14.6332			−0.647337
$512$	9.56336	+	9.56336i	0.422645	+	0.422645i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−39.7995			−1.74365			−0.871824	−	0.489820i	$-0.837063\pi$
$521$	−39.7995			−1.74365			−0.871824	+	0.489820i	$0.837063\pi$
$522$		0			0
$523$	−19.9241	+	19.9241i	−0.871218	+	0.871218i	−0.992605	−	0.121387i	$-0.961266\pi$
$523$	−19.9241	+	19.9241i	−0.871218	+	0.871218i	0.121387	+	0.992605i	$0.461266\pi$
$524$		0			0
$525$		0			0
$526$	70.1161			3.05721
$527$	−21.3574	−	21.3574i	−0.930341	−	0.930341i
$528$		0			0
$529$		−	23.0000i		−	1.00000i
$530$		0			0
$531$	12.0000			0.520756
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−26.7774	+	26.7774i	−1.15446	+	1.15446i
$539$			20.7836i			0.895214i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		−	86.2164i		−	3.69650i
$545$		0			0
$546$		0			0
$547$	−27.1913	−	27.1913i	−1.16261	−	1.16261i	−0.983901	−	0.178713i	$-0.942807\pi$
$547$	−27.1913	−	27.1913i	−1.16261	−	1.16261i	−0.178713	−	0.983901i	$-0.557193\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$			32.3166i			1.37300i
$555$		0			0
$556$		0			0
$557$	9.27574	+	9.27574i	0.393026	+	0.393026i	0.875764	−	0.482739i	$-0.160358\pi$
$557$	9.27574	+	9.27574i	0.393026	+	0.393026i	−0.482739	+	0.875764i	$0.660358\pi$
$558$	−38.0617	+	38.0617i	−1.61128	+	1.61128i
$559$		−	62.4327i		−	2.64062i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	13.1011	−	13.1011i	0.552145	−	0.552145i	−0.374915	−	0.927059i	$-0.622328\pi$
$563$	13.1011	−	13.1011i	0.552145	−	0.552145i	0.927059	+	0.374915i	$0.122328\pi$
$564$		0			0
$565$		0			0
$566$	90.1161			3.78786
$567$	−5.45039	−	5.45039i	−0.228895	−	0.228895i
$568$	50.7489	−	50.7489i	2.12938	−	2.12938i
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$	−78.1319	−	78.1319i	−3.26686	−	3.26686i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−71.8496			−2.99373
$577$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$577$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$578$	−7.14096	+	7.14096i	−0.297025	+	0.297025i
$579$		0			0
$580$		0			0
$581$	−11.4670			−0.475731
$582$		0			0
$583$		0			0
$584$		−	153.282i		−	6.34287i
$585$		0			0
$586$	−66.2164			−2.73537
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.62505	+	1.62505i	−0.0667328	+	0.0667328i	−0.739686	−	0.672953i	$-0.765026\pi$
$593$	−1.62505	+	1.62505i	−0.0667328	+	0.0667328i	0.672953	+	0.739686i	$0.265026\pi$
$594$		0			0
$595$		0			0
$596$		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			$\pi$
$602$	16.3208	+	16.3208i	0.665187	+	0.665187i
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	33.6307	+	33.6307i	1.36503	+	1.36503i	0.867376	+	0.497654i	$0.165805\pi$
$607$	33.6307	+	33.6307i	1.36503	+	1.36503i	0.497654	+	0.867376i	$0.334195\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	51.3545	+	51.3545i	2.07588	+	2.07588i
$613$	14.8875	−	14.8875i	0.601301	−	0.601301i	−0.339357	−	0.940658i	$-0.610209\pi$
$613$	14.8875	−	14.8875i	0.601301	−	0.601301i	0.940658	+	0.339357i	$0.110209\pi$
$614$			85.4829i			3.44981i
$615$		0			0
$616$	25.4829			1.02673
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$		0			0
$619$		−	46.4327i		−	1.86629i	−0.359501	−	0.933145i	$-0.617053\pi$
$619$		−	46.4327i		−	1.86629i	0.359501	−	0.933145i	$-0.382947\pi$
$620$		0			0
$621$		0			0
$622$	61.2055	+	61.2055i	2.45412	+	2.45412i
$623$	8.03418	−	8.03418i	0.321883	−	0.321883i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−48.0000			−1.91085			−0.955425	−	0.295234i	$-0.904602\pi$
$631$	−48.0000			−1.91085			−0.955425	+	0.295234i	$0.904602\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−27.7665	−	27.7665i	−1.10015	−	1.10015i
$638$		0			0
$639$			24.0000i			0.949425i
$640$		0			0
$641$	−39.7995			−1.57199			−0.785993	−	0.618236i	$-0.787848\pi$
$641$	−39.7995			−1.57199			−0.785993	+	0.618236i	$0.787848\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$	57.0925	−	57.0925i	2.24281	−	2.24281i
$649$		−	13.2665i		−	0.520756i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$653$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	36.2449	+	36.2449i	1.41405	+	1.41405i
$658$		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	$-0.195743\pi$
$661$	42.0000			1.63361			0.816805	+	0.576913i	$0.195743\pi$
$662$	−12.6872	−	12.6872i	−0.493103	−	0.493103i
$663$		0			0
$664$		−	120.116i		−	4.66141i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−91.0109	+	91.0109i	−3.52132	+	3.52132i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−33.8225	+	33.8225i	−1.30376	+	1.30376i	−0.377925	+	0.925836i	$0.623362\pi$
$673$	−33.8225	+	33.8225i	−1.30376	+	1.30376i	−0.925836	+	0.377925i	$0.876638\pi$
$674$		−	30.8496i		−	1.18828i
$675$		0			0
$676$	139.649			5.37112
$677$	−24.5771	−	24.5771i	−0.944575	−	0.944575i	0.0539677	−	0.998543i	$-0.482813\pi$
$677$	−24.5771	−	24.5771i	−0.944575	−	0.944575i	−0.998543	+	0.0539677i	$0.982813\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	42.0788	+	42.0788i	1.61128	+	1.61128i
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$		0			0
$686$	30.7335			1.17341
$687$		0			0
$688$	−96.0474	+	96.0474i	−3.66177	+	3.66177i
$689$		0			0
$690$		0			0
$691$	−28.0000			−1.06517			−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000			−1.06517			−0.532585	+	0.846376i	$0.678779\pi$
$692$	−36.4366	−	36.4366i	−1.38511	−	1.38511i
$693$	−6.02564	+	6.02564i	−0.228895	+	0.228895i
$694$			45.4829i			1.72651i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$			79.4327i			2.99373i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			53.0660i			1.99294i	0.0839773	+	0.996468i	$0.473238\pi$
$709$			53.0660i			1.99294i	−0.0839773	+	0.996468i	$0.526762\pi$
$710$		0			0
$711$		0			0
$712$	84.1575	+	84.1575i	3.15394	+	3.15394i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−105.799			−3.95391
$717$		0			0
$718$		0			0
$719$		−	46.4327i		−	1.73165i	−0.500348	−	0.865825i	$-0.666794\pi$
$719$		−	46.4327i		−	1.73165i	0.500348	−	0.865825i	$-0.333206\pi$
$720$		0			0
$721$		0			0
$722$	−36.3408	−	36.3408i	−1.35246	−	1.35246i
$723$		0			0
$724$			141.066i			5.24268i
$725$		0			0
$726$		0			0
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$	−34.0446	+	34.0446i	−1.26178	+	1.26178i
$729$			27.0000i			1.00000i
$730$		0			0
$731$	45.3668			1.67795
$732$		0			0
$733$	37.4561	−	37.4561i	1.38347	−	1.38347i	0.545103	−	0.838369i	$-0.316491\pi$
$733$	37.4561	−	37.4561i	1.38347	−	1.38347i	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
$743$	34.8419	−	34.8419i	1.27823	−	1.27823i	0.336567	−	0.941659i	$-0.390734\pi$
$743$	34.8419	−	34.8419i	1.27823	−	1.27823i	0.941659	−	0.336567i	$-0.109266\pi$
$744$		0			0
$745$		0			0
$746$	60.1161			2.20101
$747$	28.4025	+	28.4025i	1.03919	+	1.03919i
$748$	56.7745	−	56.7745i	2.07588	−	2.07588i
$749$			7.06600i			0.258186i
$750$		0			0
$751$	32.0000			1.16770			0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000			1.16770			0.583848	+	0.811863i	$0.301546\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$	68.8562	−	68.8562i	2.50097	−	2.50097i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		−	35.2665i		−	1.27590i
$765$		0			0
$766$		0			0
$767$	17.7238	+	17.7238i	0.639968	+	0.639968i
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	−4.23919	−	4.23919i	−0.152572	−	0.152572i
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$		−	80.8496i		−	2.90608i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	49.7295	−	49.7295i	1.78289	−	1.78289i
$779$		0			0
$780$		0			0
$781$	26.5330			0.949425
$782$		0			0
$783$		0			0
$784$			85.4327i			3.05117i
$785$		0			0
$786$		0			0
$787$	38.8590	+	38.8590i	1.38517	+	1.38517i	0.835145	+	0.550030i	$0.185384\pi$
$787$	38.8590	+	38.8590i	1.38517	+	1.38517i	0.550030	+	0.835145i	$0.314616\pi$
$788$	−104.909	+	104.909i	−3.73724	+	3.73724i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−63.1182	−	63.1182i	−2.24281	−	2.24281i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−127.599			−4.52263
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−39.7995			−1.40625
$802$	−76.1234	−	76.1234i	−2.68801	−	2.68801i
$803$	40.0702	−	40.0702i	1.41405	−	1.41405i
$804$		0			0
$805$		0			0
$806$	−112.433			−3.96027
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		−	16.1003i		−	0.562588i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$	−9.26650			−0.322423
$827$	−37.6478	−	37.6478i	−1.30914	−	1.30914i	−0.922034	−	0.387110i	$-0.873473\pi$
$827$	−37.6478	−	37.6478i	−1.30914	−	1.30914i	−0.387110	−	0.922034i	$-0.626527\pi$
$828$		0			0
$829$			53.0660i			1.84306i	0.388309	+	0.921529i	$0.373059\pi$
$829$			53.0660i			1.84306i	−0.388309	+	0.921529i	$0.626941\pi$
$830$		0			0
$831$		0			0
$832$	−106.121	−	106.121i	−3.67907	−	3.67907i
$833$	20.1765	−	20.1765i	0.699075	−	0.699075i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−38.0617	+	38.0617i	−1.31482	+	1.31482i
$839$		−	56.0000i		−	1.93333i	−0.256036	−	0.966667i	$-0.582416\pi$
$839$		−	56.0000i		−	1.93333i	0.256036	−	0.966667i	$-0.417584\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$	50.7489	+	50.7489i	1.74892	+	1.74892i
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	6.66159	+	6.66159i	0.228895	+	0.228895i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−18.1376	+	18.1376i	−0.621020	+	0.621020i	−0.945792	−	0.324772i	$-0.894712\pi$
$853$	−18.1376	+	18.1376i	−0.621020	+	0.621020i	0.324772	+	0.945792i	$0.394712\pi$
$854$		0			0
$855$		0			0
$856$	−74.0159			−2.52981
$857$	13.6763	+	13.6763i	0.467174	+	0.467174i	0.900998	−	0.433824i	$-0.142836\pi$
$857$	13.6763	+	13.6763i	0.467174	+	0.467174i	−0.433824	+	0.900998i	$0.642836\pi$
$858$		0			0
$859$		−	46.4327i		−	1.58426i	−0.610349	−	0.792132i	$-0.708971\pi$
$859$		−	46.4327i		−	1.58426i	0.610349	−	0.792132i	$-0.291029\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	21.3574	−	21.3574i	0.724916	−	0.724916i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−2.83625	−	2.83625i	−0.0957733	−	0.0957733i	0.657597	−	0.753370i	$-0.271573\pi$
$877$	−2.83625	−	2.83625i	−0.0957733	−	0.0957733i	−0.753370	+	0.657597i	$0.771573\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2.00000			0.0673817			0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000			0.0673817			0.0336909	+	0.999432i	$0.489274\pi$
$882$	−35.9573	−	35.9573i	−1.21074	−	1.21074i
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$			151.699i			5.10220i
$885$		0			0
$886$		0			0
$887$	−21.9630	−	21.9630i	−0.737444	−	0.737444i	0.234639	−	0.972083i	$-0.424609\pi$
$887$	−21.9630	−	21.9630i	−0.737444	−	0.737444i	−0.972083	+	0.234639i	$0.924609\pi$
$888$		0			0
$889$		−	19.2665i		−	0.646178i
$890$		0			0
$891$	29.8496			1.00000
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	23.0501			0.770051
$897$		0			0
$898$	25.3745	−	25.3745i	0.846757	−	0.846757i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$	24.5771	−	24.5771i	0.815620	−	0.815620i
$909$		0			0
$910$		0			0
$911$	59.6992			1.97792			0.988962	−	0.148168i	$-0.0473378\pi$
$911$	59.6992			1.97792			0.988962	+	0.148168i	$0.0473378\pi$
$912$		0			0
$913$	31.4001	−	31.4001i	1.03919	−	1.03919i
$914$		−	27.6834i		−	0.915685i
$915$		0			0
$916$	31.8997			1.05400
$917$		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
$923$	−35.4475	+	35.4475i	−1.16677	+	1.16677i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$			14.0000i			0.459325i	0.973270	+	0.229663i	$0.0737623\pi$
$929$			14.0000i			0.459325i	−0.973270	+	0.229663i	$0.926238\pi$
$930$		0			0
$931$		0			0
$932$	111.349	+	111.349i	3.64735	+	3.64735i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	168.649			5.51247
$937$	25.7883	+	25.7883i	0.842467	+	0.842467i	0.989179	−	0.146712i	$-0.0468691\pi$
$937$	25.7883	+	25.7883i	0.842467	+	0.842467i	−0.146712	+	0.989179i	$0.546869\pi$
$938$		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
$944$		−	54.5330i		−	1.77490i
$945$		0			0
$946$	−89.3826			−2.90608
$947$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$947$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$948$		0			0
$949$			107.066i			3.47551i
$950$		0			0
$951$		0			0
$952$	−24.7385	−	24.7385i	−0.801779	−	0.801779i
$953$	42.6844	−	42.6844i	1.38268	−	1.38268i	0.542858	−	0.839825i	$-0.317342\pi$
$953$	42.6844	−	42.6844i	1.38268	−	1.38268i	0.839825	−	0.542858i	$-0.182658\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	13.0000			0.419355
$962$		0			0
$963$	17.5017	−	17.5017i	0.563983	−	0.563983i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−26.3635	−	26.3635i	−0.847795	−	0.847795i	0.142063	−	0.989858i	$-0.454626\pi$
$967$	−26.3635	−	26.3635i	−0.847795	−	0.847795i	−0.989858	+	0.142063i	$0.954626\pi$
$968$	−69.7798	+	69.7798i	−2.24281	+	2.24281i
$969$		0			0
$970$		0			0
$971$	59.6992			1.91584			0.957920	−	0.287035i	$-0.0926697\pi$
$971$	59.6992			1.91584			0.957920	+	0.287035i	$0.0926697\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$977$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$978$		0			0
$979$			44.0000i			1.40625i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−6.63325			−0.210712			−0.105356	−	0.994435i	$-0.533598\pi$
$991$	−6.63325			−0.210712			−0.105356	+	0.994435i	$0.533598\pi$
$992$	88.8106	+	88.8106i	2.81974	+	2.81974i
$993$		0			0
$994$		−	18.5330i		−	0.587831i
$995$		0			0
$996$		0			0
$997$	41.4732	+	41.4732i	1.31347	+	1.31347i	0.918841	+	0.394627i	$0.129126\pi$
$997$	41.4732	+	41.4732i	1.31347	+	1.31347i	0.394627	+	0.918841i	$0.370874\pi$
$998$	−7.65069	+	7.65069i	−0.242178	+	0.242178i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.2.e.c.43.4	yes	8
5.2	odd	4	inner	275.2.e.c.32.1	✓	8
5.3	odd	4	inner	275.2.e.c.32.4	yes	8
5.4	even	2	inner	275.2.e.c.43.1	yes	8
11.10	odd	2	inner	275.2.e.c.43.1	yes	8
55.32	even	4	inner	275.2.e.c.32.4	yes	8
55.43	even	4	inner	275.2.e.c.32.1	✓	8
55.54	odd	2	CM	275.2.e.c.43.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.e.c.32.1	✓	8	5.2	odd	4	inner
275.2.e.c.32.1	✓	8	55.43	even	4	inner
275.2.e.c.32.4	yes	8	5.3	odd	4	inner
275.2.e.c.32.4	yes	8	55.32	even	4	inner
275.2.e.c.43.1	yes	8	5.4	even	2	inner
275.2.e.c.43.1	yes	8	11.10	odd	2	inner
275.2.e.c.43.4	yes	8	1.1	even	1	trivial
275.2.e.c.43.4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	275.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.91267 + 1.91267i) q^{2} +5.31662i q^{4} +(0.605599 + 0.605599i) q^{7} +(-6.34361 + 6.34361i) q^{8} -3.00000i q^{9} -3.31662 q^{11} +(4.43094 - 4.43094i) q^{13} +2.31662i q^{14} -13.6332 q^{16} +(3.21974 + 3.21974i) q^{17} +(5.73801 - 5.73801i) q^{18} +(-6.34361 - 6.34361i) q^{22} +16.9499 q^{26} +(-3.21974 + 3.21974i) q^{28} -6.63325 q^{31} +(-13.3887 - 13.3887i) q^{32} +12.3166i q^{34} +15.9499 q^{36} +(7.04509 - 7.04509i) q^{43} -17.6332i q^{44} -6.26650i q^{49} +(23.5577 + 23.5577i) q^{52} -7.68338 q^{56} +4.00000i q^{59} +(-12.6872 - 12.6872i) q^{62} +(1.81680 - 1.81680i) q^{63} -23.9499i q^{64} +(-17.1182 + 17.1182i) q^{68} -8.00000 q^{71} +(19.0308 + 19.0308i) q^{72} +(-12.0816 + 12.0816i) q^{73} +(-2.00855 - 2.00855i) q^{77} -9.00000 q^{81} +(-9.46748 + 9.46748i) q^{83} +26.9499 q^{86} +(21.0394 - 21.0394i) q^{88} -13.2665i q^{89} +5.36675 q^{91} +(11.9858 - 11.9858i) q^{98} +9.94987i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 56 q^{16} + 56 q^{26} + 48 q^{36} - 88 q^{56} - 64 q^{71} - 72 q^{81} + 136 q^{86} + 96 q^{91}+O(q^{100}) \) 8 * q - 56 * q^16 + 56 * q^26 + 48 * q^36 - 88 * q^56 - 64 * q^71 - 72 * q^81 + 136 * q^86 + 96 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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