LMFDB - Embedded newform 495.2.c.e.199.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,2,Mod(199,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("495.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$495 = 3^{2} \cdot 5 \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	495.c (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:	$3.95259490005$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.3
Root			$1.45161 + 1.45161i$ of defining polynomial
Character	$\chi$	$=$	495.199
Dual form			495.2.c.e.199.4

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.311108i q^{2} +1.90321 q^{4} +(-0.311108 - 2.21432i) q^{5} -0.903212i q^{7} -1.21432i q^{8} +(-0.688892 + 0.0967881i) q^{10} -1.00000 q^{11} -2.90321i q^{13} -0.280996 q^{14} +3.42864 q^{16} -2.28100i q^{17} -2.42864 q^{19} +(-0.592104 - 4.21432i) q^{20} +0.311108i q^{22} +4.00000i q^{23} +(-4.80642 + 1.37778i) q^{25} -0.903212 q^{26} -1.71900i q^{28} +7.05086 q^{29} -2.62222 q^{31} -3.49532i q^{32} -0.709636 q^{34} +(-2.00000 + 0.280996i) q^{35} +5.80642i q^{37} +0.755569i q^{38} +(-2.68889 + 0.377784i) q^{40} +10.6637 q^{41} -10.7096i q^{43} -1.90321 q^{44} +1.24443 q^{46} -0.949145i q^{47} +6.18421 q^{49} +(0.428639 + 1.49532i) q^{50} -5.52543i q^{52} +0.815792i q^{53} +(0.311108 + 2.21432i) q^{55} -1.09679 q^{56} -2.19358i q^{58} -1.67307 q^{59} -7.24443 q^{61} +0.815792i q^{62} +5.76986 q^{64} +(-6.42864 + 0.903212i) q^{65} +12.8573i q^{67} -4.34122i q^{68} +(0.0874201 + 0.622216i) q^{70} -9.28592 q^{71} +5.65878i q^{73} +1.80642 q^{74} -4.62222 q^{76} +0.903212i q^{77} +16.5303 q^{79} +(-1.06668 - 7.59210i) q^{80} -3.31756i q^{82} +7.76049i q^{83} +(-5.05086 + 0.709636i) q^{85} -3.33185 q^{86} +1.21432i q^{88} +6.13335 q^{89} -2.62222 q^{91} +7.61285i q^{92} -0.295286 q^{94} +(0.755569 + 5.37778i) q^{95} -12.4701i q^{97} -1.92396i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 2 q^{4} - 2 q^{5} - 4 q^{10} - 6 q^{11} + 12 q^{14} - 6 q^{16} + 12 q^{19} + 10 q^{20} - 2 q^{25} + 8 q^{26} + 16 q^{29} - 16 q^{31} + 36 q^{34} - 12 q^{35} - 16 q^{40} - 16 q^{41} + 2 q^{44} + 8 q^{46}+ \cdots + 4 q^{95}+O(q^{100})$ 6 * q - 2 * q^4 - 2 * q^5 - 4 * q^10 - 6 * q^11 + 12 * q^14 - 6 * q^16 + 12 * q^19 + 10 * q^20 - 2 * q^25 + 8 * q^26 + 16 * q^29 - 16 * q^31 + 36 * q^34 - 12 * q^35 - 16 * q^40 - 16 * q^41 + 2 * q^44 + 8 * q^46 + 10 * q^49 - 24 * q^50 + 2 * q^55 - 20 * q^56 + 16 * q^59 - 44 * q^61 + 22 * q^64 - 12 * q^65 - 40 * q^70 + 24 * q^71 - 16 * q^74 - 28 * q^76 + 20 * q^79 - 6 * q^80 - 4 * q^85 + 20 * q^86 + 36 * q^89 - 16 * q^91 + 24 * q^94 + 4 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/495\mathbb{Z}\right)^\times$ .

$n$	$46$	$56$	$397$
$\chi(n)$	$1$	$1$	$-1$

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

)

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

$2$		−	0.311108i		−	0.219986i	−0.993932	−	0.109993i	$-0.964917\pi$
$2$		−	0.311108i		−	0.219986i	0.993932	−	0.109993i	$-0.0350829\pi$
$3$		0			0
$3$		0			0
$4$	1.90321			0.951606
$5$	−0.311108	−	2.21432i	−0.139132	−	0.990274i
$5$	−0.311108	−	2.21432i	−0.139132	−	0.990274i
$6$		0			0
$7$		−	0.903212i		−	0.341382i	−0.985325	−	0.170691i	$-0.945400\pi$
$7$		−	0.903212i		−	0.341382i	0.985325	−	0.170691i	$-0.0546000\pi$
$8$		−	1.21432i		−	0.429327i
$9$		0			0
$10$	−0.688892	+	0.0967881i	−0.217847	+	0.0306071i
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$		0			0
$13$		−	2.90321i		−	0.805206i	−0.915375	−	0.402603i	$-0.868106\pi$
$13$		−	2.90321i		−	0.805206i	0.915375	−	0.402603i	$-0.131894\pi$
$14$	−0.280996			−0.0750994
$15$		0			0
$16$	3.42864			0.857160
$17$		−	2.28100i		−	0.553223i	−0.960982	−	0.276611i	$-0.910789\pi$
$17$		−	2.28100i		−	0.553223i	0.960982	−	0.276611i	$-0.0892115\pi$
$18$		0			0
$19$	−2.42864			−0.557168			−0.278584	−	0.960412i	$-0.589865\pi$
$19$	−2.42864			−0.557168			−0.278584	+	0.960412i	$0.589865\pi$
$20$	−0.592104	−	4.21432i	−0.132399	−	0.942351i
$21$		0			0
$22$			0.311108i			0.0663284i
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$	−4.80642	+	1.37778i	−0.961285	+	0.275557i
$26$	−0.903212			−0.177134
$27$		0			0
$28$		−	1.71900i		−	0.324861i
$29$	7.05086			1.30931			0.654655	−	0.755927i	$-0.272814\pi$
$29$	7.05086			1.30931			0.654655	+	0.755927i	$0.272814\pi$
$30$		0			0
$31$	−2.62222			−0.470964			−0.235482	−	0.971879i	$-0.575667\pi$
$31$	−2.62222			−0.470964			−0.235482	+	0.971879i	$0.575667\pi$
$32$		−	3.49532i		−	0.617890i
$33$		0			0
$34$	−0.709636			−0.121702
$35$	−2.00000	+	0.280996i	−0.338062	+	0.0474970i
$36$		0			0
$37$			5.80642i			0.954570i	0.878749	+	0.477285i	$0.158379\pi$
$37$			5.80642i			0.954570i	−0.878749	+	0.477285i	$0.841621\pi$
$38$			0.755569i			0.122569i
$39$		0			0
$40$	−2.68889	+	0.377784i	−0.425151	+	0.0597330i
$41$	10.6637			1.66539			0.832695	−	0.553731i	$-0.186797\pi$
$41$	10.6637			1.66539			0.832695	+	0.553731i	$0.186797\pi$
$42$		0			0
$43$		−	10.7096i		−	1.63320i	−0.577201	−	0.816602i	$-0.695855\pi$
$43$		−	10.7096i		−	1.63320i	0.577201	−	0.816602i	$-0.304145\pi$
$44$	−1.90321			−0.286920
$45$		0			0
$46$	1.24443			0.183481
$47$		−	0.949145i		−	0.138447i	−0.997601	−	0.0692235i	$-0.977948\pi$
$47$		−	0.949145i		−	0.138447i	0.997601	−	0.0692235i	$-0.0220522\pi$
$48$		0			0
$49$	6.18421			0.883458
$50$	0.428639	+	1.49532i	0.0606188	+	0.211470i
$51$		0			0
$52$		−	5.52543i		−	0.766239i
$53$			0.815792i			0.112058i	0.998429	+	0.0560288i	$0.0178439\pi$
$53$			0.815792i			0.112058i	−0.998429	+	0.0560288i	$0.982156\pi$
$54$		0			0
$55$	0.311108	+	2.21432i	0.0419498	+	0.298579i
$56$	−1.09679			−0.146564
$57$		0			0
$58$		−	2.19358i		−	0.288031i
$59$	−1.67307			−0.217815			−0.108908	−	0.994052i	$-0.534735\pi$
$59$	−1.67307			−0.217815			−0.108908	+	0.994052i	$0.534735\pi$
$60$		0			0
$61$	−7.24443			−0.927554			−0.463777	−	0.885952i	$-0.653506\pi$
$61$	−7.24443			−0.927554			−0.463777	+	0.885952i	$0.653506\pi$
$62$			0.815792i			0.103606i
$63$		0			0
$64$	5.76986			0.721232
$65$	−6.42864	+	0.903212i	−0.797375	+	0.112030i
$66$		0			0
$67$			12.8573i			1.57077i	0.619010	+	0.785383i	$0.287534\pi$
$67$			12.8573i			1.57077i	−0.619010	+	0.785383i	$0.712466\pi$
$68$		−	4.34122i		−	0.526450i
$69$		0			0
$70$	0.0874201	+	0.622216i	0.0104487	+	0.0743690i
$71$	−9.28592			−1.10204			−0.551018	−	0.834493i	$-0.685760\pi$
$71$	−9.28592			−1.10204			−0.551018	+	0.834493i	$0.685760\pi$
$72$		0			0
$73$			5.65878i			0.662310i	0.943576	+	0.331155i	$0.107438\pi$
$73$			5.65878i			0.662310i	−0.943576	+	0.331155i	$0.892562\pi$
$74$	1.80642			0.209993
$75$		0			0
$76$	−4.62222			−0.530204
$77$			0.903212i			0.102931i
$78$		0			0
$79$	16.5303			1.85981			0.929905	−	0.367800i	$-0.119889\pi$
$79$	16.5303			1.85981			0.929905	+	0.367800i	$0.119889\pi$
$80$	−1.06668	−	7.59210i	−0.119258	−	0.848823i
$81$		0			0
$82$		−	3.31756i		−	0.366363i
$83$			7.76049i			0.851825i	0.904764	+	0.425912i	$0.140047\pi$
$83$			7.76049i			0.851825i	−0.904764	+	0.425912i	$0.859953\pi$
$84$		0			0
$85$	−5.05086	+	0.709636i	−0.547842	+	0.0769708i
$86$	−3.33185			−0.359283
$87$		0			0
$88$			1.21432i			0.129447i
$89$	6.13335			0.650134			0.325067	−	0.945691i	$-0.394613\pi$
$89$	6.13335			0.650134			0.325067	+	0.945691i	$0.394613\pi$
$90$		0			0
$91$	−2.62222			−0.274883
$92$			7.61285i			0.793694i
$93$		0			0
$94$	−0.295286			−0.0304565
$95$	0.755569	+	5.37778i	0.0775197	+	0.551749i
$96$		0			0
$97$		−	12.4701i		−	1.26615i	−0.774091	−	0.633075i	$-0.781793\pi$
$97$		−	12.4701i		−	1.26615i	0.774091	−	0.633075i	$-0.218207\pi$
$98$		−	1.92396i		−	0.194349i
$99$		0			0
$100$	−9.14764	+	2.62222i	−0.914764	+	0.262222i
$101$	−16.1748			−1.60946			−0.804728	−	0.593643i	$-0.797689\pi$
$101$	−16.1748			−1.60946			−0.804728	+	0.593643i	$0.797689\pi$
$102$		0			0
$103$			17.1526i			1.69009i	0.534693	+	0.845046i	$0.320427\pi$
$103$			17.1526i			1.69009i	−0.534693	+	0.845046i	$0.679573\pi$
$104$	−3.52543			−0.345697
$105$		0			0
$106$	0.253799			0.0246512
$107$			13.5669i			1.31156i	0.754951	+	0.655782i	$0.227661\pi$
$107$			13.5669i			1.31156i	−0.754951	+	0.655782i	$0.772339\pi$
$108$		0			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$	0.688892	−	0.0967881i	0.0656833	−	0.00922838i
$111$		0			0
$112$		−	3.09679i		−	0.292619i
$113$			14.2351i			1.33912i	0.742757	+	0.669561i	$0.233518\pi$
$113$			14.2351i			1.33912i	−0.742757	+	0.669561i	$0.766482\pi$
$114$		0			0
$115$	8.85728	−	1.24443i	0.825946	−	0.116044i
$116$	13.4193			1.24595
$117$		0			0
$118$			0.520505i			0.0479164i
$119$	−2.06022			−0.188860
$120$		0			0
$121$	1.00000			0.0909091
$122$			2.25380i			0.204049i
$123$		0			0
$124$	−4.99063			−0.448172
$125$	4.54617	+	10.2143i	0.406622	+	0.913597i
$126$		0			0
$127$			11.0049i			0.976529i	0.872696	+	0.488264i	$0.162370\pi$
$127$			11.0049i			0.976529i	−0.872696	+	0.488264i	$0.837630\pi$
$128$		−	8.78568i		−	0.776552i
$129$		0			0
$130$	0.280996	+	2.00000i	0.0246450	+	0.175412i
$131$	−1.24443			−0.108726			−0.0543632	−	0.998521i	$-0.517313\pi$
$131$	−1.24443			−0.108726			−0.0543632	+	0.998521i	$0.517313\pi$
$132$		0			0
$133$			2.19358i			0.190207i
$134$	4.00000			0.345547
$135$		0			0
$136$	−2.76986			−0.237513
$137$			4.42864i			0.378364i	0.981942	+	0.189182i	$0.0605836\pi$
$137$			4.42864i			0.378364i	−0.981942	+	0.189182i	$0.939416\pi$
$138$		0			0
$139$	−0.917502			−0.0778215			−0.0389108	−	0.999243i	$-0.512389\pi$
$139$	−0.917502			−0.0778215			−0.0389108	+	0.999243i	$0.512389\pi$
$140$	−3.80642	+	0.534795i	−0.321702	+	0.0451985i
$141$		0			0
$142$			2.88892i			0.242433i
$143$			2.90321i			0.242779i
$144$		0			0
$145$	−2.19358	−	15.6128i	−0.182167	−	1.29658i
$146$	1.76049			0.145699
$147$		0			0
$148$			11.0509i			0.908375i
$149$	2.19358			0.179705			0.0898524	−	0.995955i	$-0.471360\pi$
$149$	2.19358			0.179705			0.0898524	+	0.995955i	$0.471360\pi$
$150$		0			0
$151$	−10.4286			−0.848671			−0.424335	−	0.905505i	$-0.639492\pi$
$151$	−10.4286			−0.848671			−0.424335	+	0.905505i	$0.639492\pi$
$152$			2.94914i			0.239207i
$153$		0			0
$154$	0.280996			0.0226433
$155$	0.815792	+	5.80642i	0.0655260	+	0.466383i
$156$		0			0
$157$			5.80642i			0.463403i	0.972787	+	0.231702i	$0.0744293\pi$
$157$			5.80642i			0.463403i	−0.972787	+	0.231702i	$0.925571\pi$
$158$		−	5.14272i		−	0.409133i
$159$		0			0
$160$	−7.73975	+	1.08742i	−0.611881	+	0.0859681i
$161$	3.61285			0.284732
$162$		0			0
$163$		−	11.0509i		−	0.865570i	−0.901497	−	0.432785i	$-0.857531\pi$
$163$		−	11.0509i		−	0.865570i	0.901497	−	0.432785i	$-0.142469\pi$
$164$	20.2953			1.58480
$165$		0			0
$166$	2.41435			0.187390
$167$		−	13.9541i		−	1.07980i	−0.841730	−	0.539899i	$-0.818462\pi$
$167$		−	13.9541i		−	1.07980i	0.841730	−	0.539899i	$-0.181538\pi$
$168$		0			0
$169$	4.57136			0.351643
$170$	0.220773	+	1.57136i	0.0169325	+	0.120518i
$171$		0			0
$172$		−	20.3827i		−	1.55417i
$173$		−	18.8430i		−	1.43261i	−0.697789	−	0.716303i	$-0.745833\pi$
$173$		−	18.8430i		−	1.43261i	0.697789	−	0.716303i	$-0.254167\pi$
$174$		0			0
$175$	1.24443	+	4.34122i	0.0940702	+	0.328165i
$176$	−3.42864			−0.258443
$177$		0			0
$178$		−	1.90813i		−	0.143021i
$179$	4.85728			0.363050			0.181525	−	0.983386i	$-0.441897\pi$
$179$	4.85728			0.363050			0.181525	+	0.983386i	$0.441897\pi$
$180$		0			0
$181$	−16.7971			−1.24852			−0.624258	−	0.781219i	$-0.714598\pi$
$181$	−16.7971			−1.24852			−0.624258	+	0.781219i	$0.714598\pi$
$182$			0.815792i			0.0604705i
$183$		0			0
$184$	4.85728			0.358083
$185$	12.8573	−	1.80642i	0.945286	−	0.132811i
$186$		0			0
$187$			2.28100i			0.166803i
$188$		−	1.80642i		−	0.131747i
$189$		0			0
$190$	1.67307	−	0.235063i	0.121377	−	0.0170533i
$191$	−16.8573			−1.21975			−0.609875	−	0.792498i	$-0.708780\pi$
$191$	−16.8573			−1.21975			−0.609875	+	0.792498i	$0.708780\pi$
$192$		0			0
$193$			24.6178i			1.77203i	0.463661	+	0.886013i	$0.346536\pi$
$193$			24.6178i			1.77203i	−0.463661	+	0.886013i	$0.653464\pi$
$194$	−3.87955			−0.278536
$195$		0			0
$196$	11.7699			0.840704
$197$		−	14.8716i		−	1.05956i	−0.848137	−	0.529778i	$-0.822275\pi$
$197$		−	14.8716i		−	1.05956i	0.848137	−	0.529778i	$-0.177725\pi$
$198$		0			0
$199$	11.2257			0.795768			0.397884	−	0.917436i	$-0.369745\pi$
$199$	11.2257			0.795768			0.397884	+	0.917436i	$0.369745\pi$
$200$	1.67307	+	5.83654i	0.118304	+	0.412705i
$201$		0			0
$202$			5.03212i			0.354059i
$203$		−	6.36842i		−	0.446975i
$204$		0			0
$205$	−3.31756	−	23.6128i	−0.231709	−	1.64919i
$206$	5.33630			0.371797
$207$		0			0
$208$		−	9.95407i		−	0.690190i
$209$	2.42864			0.167993
$210$		0			0
$211$	−11.9398			−0.821968			−0.410984	−	0.911643i	$-0.634815\pi$
$211$	−11.9398			−0.821968			−0.410984	+	0.911643i	$0.634815\pi$
$212$			1.55262i			0.106635i
$213$		0			0
$214$	4.22077			0.288526
$215$	−23.7146	+	3.33185i	−1.61732	+	0.227230i
$216$		0			0
$217$			2.36842i			0.160779i
$218$		−	3.11108i		−	0.210709i
$219$		0			0
$220$	0.592104	+	4.21432i	0.0399197	+	0.284129i
$221$	−6.62222			−0.445458
$222$		0			0
$223$		−	21.8064i		−	1.46027i	−0.683305	−	0.730133i	$-0.739458\pi$
$223$		−	21.8064i		−	1.46027i	0.683305	−	0.730133i	$-0.260542\pi$
$224$	−3.15701			−0.210937
$225$		0			0
$226$	4.42864			0.294589
$227$		−	3.19850i		−	0.212292i	−0.994351	−	0.106146i	$-0.966149\pi$
$227$		−	3.19850i		−	0.212292i	0.994351	−	0.106146i	$-0.0338511\pi$
$228$		0			0
$229$	−7.12399			−0.470766			−0.235383	−	0.971903i	$-0.575634\pi$
$229$	−7.12399			−0.470766			−0.235383	+	0.971903i	$0.575634\pi$
$230$	−0.387152	−	2.75557i	−0.0255281	−	0.181697i
$231$		0			0
$232$		−	8.56199i		−	0.562122i
$233$		−	19.5254i		−	1.27915i	−0.768727	−	0.639577i	$-0.779110\pi$
$233$		−	19.5254i		−	1.27915i	0.768727	−	0.639577i	$-0.220890\pi$
$234$		0			0
$235$	−2.10171	+	0.295286i	−0.137100	+	0.0192624i
$236$	−3.18421			−0.207274
$237$		0			0
$238$			0.640951i			0.0415467i
$239$	−21.9813			−1.42185			−0.710925	−	0.703268i	$-0.751723\pi$
$239$	−21.9813			−1.42185			−0.710925	+	0.703268i	$0.751723\pi$
$240$		0			0
$241$	5.34614			0.344375			0.172188	−	0.985064i	$-0.444916\pi$
$241$	5.34614			0.344375			0.172188	+	0.985064i	$0.444916\pi$
$242$		−	0.311108i		−	0.0199988i
$243$		0			0
$244$	−13.7877			−0.882666
$245$	−1.92396	−	13.6938i	−0.122917	−	0.874866i
$246$		0			0
$247$			7.05086i			0.448635i
$248$			3.18421i			0.202197i
$249$		0			0
$250$	3.17775	−	1.41435i	0.200979	−	0.0894513i
$251$	23.7748			1.50065			0.750325	−	0.661069i	$-0.229897\pi$
$251$	23.7748			1.50065			0.750325	+	0.661069i	$0.229897\pi$
$252$		0			0
$253$		−	4.00000i		−	0.251478i
$254$	3.42372			0.214823
$255$		0			0
$256$	8.80642			0.550401
$257$			8.13335i			0.507345i	0.967290	+	0.253672i	$0.0816385\pi$
$257$			8.13335i			0.507345i	−0.967290	+	0.253672i	$0.918362\pi$
$258$		0			0
$259$	5.24443			0.325873
$260$	−12.2351	+	1.71900i	−0.758786	+	0.106608i
$261$		0			0
$262$			0.387152i			0.0239183i
$263$			22.9032i			1.41227i	0.708076	+	0.706136i	$0.249563\pi$
$263$			22.9032i			1.41227i	−0.708076	+	0.706136i	$0.750437\pi$
$264$		0			0
$265$	1.80642	−	0.253799i	0.110968	−	0.0155908i
$266$	0.682439			0.0418430
$267$		0			0
$268$			24.4701i			1.49475i
$269$	11.8350			0.721593			0.360796	−	0.932645i	$-0.382505\pi$
$269$	11.8350			0.721593			0.360796	+	0.932645i	$0.382505\pi$
$270$		0			0
$271$	14.8988			0.905036			0.452518	−	0.891755i	$-0.350526\pi$
$271$	14.8988			0.905036			0.452518	+	0.891755i	$0.350526\pi$
$272$		−	7.82071i		−	0.474200i
$273$		0			0
$274$	1.37778			0.0832350
$275$	4.80642	−	1.37778i	0.289838	−	0.0830835i
$276$		0			0
$277$		−	27.6686i		−	1.66245i	−0.555939	−	0.831223i	$-0.687641\pi$
$277$		−	27.6686i		−	1.66245i	0.555939	−	0.831223i	$-0.312359\pi$
$278$			0.285442i			0.0171197i
$279$		0			0
$280$	0.341219	+	2.42864i	0.0203918	+	0.145139i
$281$	−9.80642			−0.585002			−0.292501	−	0.956265i	$-0.594488\pi$
$281$	−9.80642			−0.585002			−0.292501	+	0.956265i	$0.594488\pi$
$282$		0			0
$283$			19.0049i			1.12973i	0.825185	+	0.564863i	$0.191071\pi$
$283$			19.0049i			1.12973i	−0.825185	+	0.564863i	$0.808929\pi$
$284$	−17.6731			−1.04870
$285$		0			0
$286$	0.903212			0.0534080
$287$		−	9.63158i		−	0.568534i
$288$		0			0
$289$	11.7971			0.693944
$290$	−4.85728	+	0.682439i	−0.285229	+	0.0400742i
$291$		0			0
$292$			10.7699i			0.630258i
$293$		−	30.7511i		−	1.79650i	−0.439485	−	0.898250i	$-0.644839\pi$
$293$		−	30.7511i		−	1.79650i	0.439485	−	0.898250i	$-0.355161\pi$
$294$		0			0
$295$	0.520505	+	3.70471i	0.0303050	+	0.215697i
$296$	7.05086			0.409823
$297$		0			0
$298$		−	0.682439i		−	0.0395326i
$299$	11.6128			0.671588
$300$		0			0
$301$	−9.67307			−0.557547
$302$			3.24443i			0.186696i
$303$		0			0
$304$	−8.32693			−0.477582
$305$	2.25380	+	16.0415i	0.129052	+	0.918533i
$306$		0			0
$307$		−	13.4938i		−	0.770131i	−0.922889	−	0.385065i	$-0.874179\pi$
$307$		−	13.4938i		−	0.770131i	0.922889	−	0.385065i	$-0.125821\pi$
$308$			1.71900i			0.0979493i
$309$		0			0
$310$	1.80642	−	0.253799i	0.102598	−	0.0144148i
$311$	17.5526			0.995318			0.497659	−	0.867373i	$-0.334193\pi$
$311$	17.5526			0.995318			0.497659	+	0.867373i	$0.334193\pi$
$312$		0			0
$313$		−	14.3970i		−	0.813766i	−0.913480	−	0.406883i	$-0.866616\pi$
$313$		−	14.3970i		−	0.813766i	0.913480	−	0.406883i	$-0.133384\pi$
$314$	1.80642			0.101942
$315$		0			0
$316$	31.4608			1.76981
$317$		−	29.4608i		−	1.65468i	−0.561701	−	0.827341i	$-0.689853\pi$
$317$		−	29.4608i		−	1.65468i	0.561701	−	0.827341i	$-0.310147\pi$
$318$		0			0
$319$	−7.05086			−0.394772
$320$	−1.79505	−	12.7763i	−0.100346	−	0.714218i
$321$		0			0
$322$		−	1.12399i		−	0.0626372i
$323$			5.53972i			0.308238i
$324$		0			0
$325$	4.00000	+	13.9541i	0.221880	+	0.774032i
$326$	−3.43801			−0.190414
$327$		0			0
$328$		−	12.9491i		−	0.714997i
$329$	−0.857279			−0.0472633
$330$		0			0
$331$	−2.62222			−0.144130			−0.0720650	−	0.997400i	$-0.522959\pi$
$331$	−2.62222			−0.144130			−0.0720650	+	0.997400i	$0.522959\pi$
$332$			14.7699i			0.810601i
$333$		0			0
$334$	−4.34122			−0.237541
$335$	28.4701	−	4.00000i	1.55549	−	0.218543i
$336$		0			0
$337$		−	5.00492i		−	0.272635i	−0.990665	−	0.136318i	$-0.956473\pi$
$337$		−	5.00492i		−	0.272635i	0.990665	−	0.136318i	$-0.0435268\pi$
$338$		−	1.42219i		−	0.0773567i
$339$		0			0
$340$	−9.61285	+	1.35059i	−0.521330	+	0.0732459i
$341$	2.62222			0.142001
$342$		0			0
$343$		−	11.9081i		−	0.642979i
$344$	−13.0049			−0.701178
$345$		0			0
$346$	−5.86220			−0.315154
$347$			22.8113i			1.22458i	0.790634	+	0.612289i	$0.209751\pi$
$347$			22.8113i			1.22458i	−0.790634	+	0.612289i	$0.790249\pi$
$348$		0			0
$349$	−21.2257			−1.13619			−0.568093	−	0.822965i	$-0.692319\pi$
$349$	−21.2257			−1.13619			−0.568093	+	0.822965i	$0.692319\pi$
$350$	1.35059	−	0.387152i	0.0721919	−	0.0206942i
$351$		0			0
$352$			3.49532i			0.186301i
$353$			7.18421i			0.382377i	0.981553	+	0.191188i	$0.0612341\pi$
$353$			7.18421i			0.382377i	−0.981553	+	0.191188i	$0.938766\pi$
$354$		0			0
$355$	2.88892	+	20.5620i	0.153328	+	1.09132i
$356$	11.6731			0.618672
$357$		0			0
$358$		−	1.51114i		−	0.0798661i
$359$	−14.1017			−0.744260			−0.372130	−	0.928181i	$-0.621372\pi$
$359$	−14.1017			−0.744260			−0.372130	+	0.928181i	$0.621372\pi$
$360$		0			0
$361$	−13.1017			−0.689564
$362$			5.22570i			0.274656i
$363$		0			0
$364$	−4.99063			−0.261580
$365$	12.5303	−	1.76049i	0.655868	−	0.0921483i
$366$		0			0
$367$		−	3.90813i		−	0.204003i	−0.994784	−	0.102001i	$-0.967475\pi$
$367$		−	3.90813i		−	0.204003i	0.994784	−	0.102001i	$-0.0325246\pi$
$368$			13.7146i			0.714921i
$369$		0			0
$370$	−0.561993	−	4.00000i	−0.0292166	−	0.207950i
$371$	0.736833			0.0382545
$372$		0			0
$373$		−	12.9763i		−	0.671890i	−0.941882	−	0.335945i	$-0.890944\pi$
$373$		−	12.9763i		−	0.671890i	0.941882	−	0.335945i	$-0.109056\pi$
$374$	0.709636			0.0366944
$375$		0			0
$376$	−1.15257			−0.0594390
$377$		−	20.4701i		−	1.05427i
$378$		0			0
$379$	−36.0830			−1.85346			−0.926729	−	0.375731i	$-0.877392\pi$
$379$	−36.0830			−1.85346			−0.926729	+	0.375731i	$0.877392\pi$
$380$	1.43801	+	10.2351i	0.0737682	+	0.525048i
$381$		0			0
$382$			5.24443i			0.268328i
$383$			20.2953i			1.03704i	0.855065	+	0.518520i	$0.173517\pi$
$383$			20.2953i			1.03704i	−0.855065	+	0.518520i	$0.826483\pi$
$384$		0			0
$385$	2.00000	−	0.280996i	0.101929	−	0.0143209i
$386$	7.65878			0.389822
$387$		0			0
$388$		−	23.7333i		−	1.20488i
$389$	30.4701			1.54490			0.772448	−	0.635078i	$-0.219032\pi$
$389$	30.4701			1.54490			0.772448	+	0.635078i	$0.219032\pi$
$390$		0			0
$391$	9.12399			0.461420
$392$		−	7.50961i		−	0.379292i
$393$		0			0
$394$	−4.62666			−0.233088
$395$	−5.14272	−	36.6035i	−0.258758	−	1.84172i
$396$		0			0
$397$			4.97773i			0.249825i	0.992168	+	0.124912i	$0.0398650\pi$
$397$			4.97773i			0.249825i	−0.992168	+	0.124912i	$0.960135\pi$
$398$		−	3.49240i		−	0.175058i
$399$		0			0
$400$	−16.4795	+	4.72393i	−0.823975	+	0.236196i
$401$	1.86665			0.0932159			0.0466079	−	0.998913i	$-0.485159\pi$
$401$	1.86665			0.0932159			0.0466079	+	0.998913i	$0.485159\pi$
$402$		0			0
$403$			7.61285i			0.379223i
$404$	−30.7841			−1.53157
$405$		0			0
$406$	−1.98126			−0.0983285
$407$		−	5.80642i		−	0.287814i
$408$		0			0
$409$	3.63158			0.179570			0.0897851	−	0.995961i	$-0.471382\pi$
$409$	3.63158			0.179570			0.0897851	+	0.995961i	$0.471382\pi$
$410$	−7.34614	+	1.03212i	−0.362800	+	0.0509727i
$411$		0			0
$412$			32.6450i			1.60830i
$413$			1.51114i			0.0743582i
$414$		0			0
$415$	17.1842	−	2.41435i	0.843540	−	0.118516i
$416$	−10.1476			−0.497529
$417$		0			0
$418$		−	0.755569i		−	0.0369561i
$419$	4.85728			0.237294			0.118647	−	0.992937i	$-0.462144\pi$
$419$	4.85728			0.237294			0.118647	+	0.992937i	$0.462144\pi$
$420$		0			0
$421$	22.6321			1.10302			0.551510	−	0.834169i	$-0.314052\pi$
$421$	22.6321			1.10302			0.551510	+	0.834169i	$0.314052\pi$
$422$			3.71456i			0.180822i
$423$		0			0
$424$	0.990632			0.0481093
$425$	3.14272	+	10.9634i	0.152444	+	0.531805i
$426$		0			0
$427$			6.54326i			0.316650i
$428$			25.8207i			1.24809i
$429$		0			0
$430$	1.03657	+	7.37778i	0.0499876	+	0.355788i
$431$	−1.24443			−0.0599421			−0.0299711	−	0.999551i	$-0.509542\pi$
$431$	−1.24443			−0.0599421			−0.0299711	+	0.999551i	$0.509542\pi$
$432$		0			0
$433$			16.0000i			0.768911i	0.923144	+	0.384455i	$0.125611\pi$
$433$			16.0000i			0.768911i	−0.923144	+	0.384455i	$0.874389\pi$
$434$	0.736833			0.0353691
$435$		0			0
$436$	19.0321			0.911473
$437$		−	9.71456i		−	0.464710i
$438$		0			0
$439$	2.42864			0.115913			0.0579563	−	0.998319i	$-0.481542\pi$
$439$	2.42864			0.115913			0.0579563	+	0.998319i	$0.481542\pi$
$440$	2.68889	−	0.377784i	0.128188	−	0.0180102i
$441$		0			0
$442$			2.06022i			0.0979948i
$443$			31.0509i			1.47527i	0.675199	+	0.737635i	$0.264058\pi$
$443$			31.0509i			1.47527i	−0.675199	+	0.737635i	$0.735942\pi$
$444$		0			0
$445$	−1.90813	−	13.5812i	−0.0904542	−	0.643811i
$446$	−6.78415			−0.321239
$447$		0			0
$448$		−	5.21141i		−	0.246216i
$449$	−37.3590			−1.76308			−0.881541	−	0.472107i	$-0.843494\pi$
$449$	−37.3590			−1.76308			−0.881541	+	0.472107i	$0.843494\pi$
$450$		0			0
$451$	−10.6637			−0.502134
$452$			27.0923i			1.27432i
$453$		0			0
$454$	−0.995078			−0.0467013
$455$	0.815792	+	5.80642i	0.0382449	+	0.272209i
$456$		0			0
$457$			8.73822i			0.408757i	0.978892	+	0.204378i	$0.0655172\pi$
$457$			8.73822i			0.408757i	−0.978892	+	0.204378i	$0.934483\pi$
$458$			2.21633i			0.103562i
$459$		0			0
$460$	16.8573	−	2.36842i	0.785975	−	0.110428i
$461$	−31.7877			−1.48050			−0.740250	−	0.672332i	$-0.765293\pi$
$461$	−31.7877			−1.48050			−0.740250	+	0.672332i	$0.765293\pi$
$462$		0			0
$463$		−	12.0919i		−	0.561957i	−0.959714	−	0.280978i	$-0.909341\pi$
$463$		−	12.0919i		−	0.561957i	0.959714	−	0.280978i	$-0.0906589\pi$
$464$	24.1748			1.12229
$465$		0			0
$466$	−6.07451			−0.281396
$467$			15.3461i			0.710135i	0.934841	+	0.355067i	$0.115542\pi$
$467$			15.3461i			0.710135i	−0.934841	+	0.355067i	$0.884458\pi$
$468$		0			0
$469$	11.6128			0.536231
$470$	0.0918659	+	0.653858i	0.00423746	+	0.0301602i
$471$		0			0
$472$			2.03164i			0.0935139i
$473$			10.7096i			0.492430i
$474$		0			0
$475$	11.6731	−	3.34614i	0.535597	−	0.153532i
$476$	−3.92104			−0.179721
$477$		0			0
$478$			6.83854i			0.312788i
$479$	5.89829			0.269500			0.134750	−	0.990880i	$-0.456977\pi$
$479$	5.89829			0.269500			0.134750	+	0.990880i	$0.456977\pi$
$480$		0			0
$481$	16.8573			0.768626
$482$		−	1.66323i		−	0.0757579i
$483$		0			0
$484$	1.90321			0.0865096
$485$	−27.6128	+	3.87955i	−1.25383	+	0.176161i
$486$		0			0
$487$			31.3461i			1.42043i	0.703985	+	0.710215i	$0.251402\pi$
$487$			31.3461i			1.42043i	−0.703985	+	0.710215i	$0.748598\pi$
$488$			8.79706i			0.398224i
$489$		0			0
$490$	−4.26025	+	0.598558i	−0.192459	+	0.0270401i
$491$	8.00000			0.361035			0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000			0.361035			0.180517	+	0.983572i	$0.442223\pi$
$492$		0			0
$493$		−	16.0830i		−	0.724341i
$494$	2.19358			0.0986937
$495$		0			0
$496$	−8.99063			−0.403691
$497$			8.38715i			0.376215i
$498$		0			0
$499$	−15.1427			−0.677881			−0.338941	−	0.940808i	$-0.610069\pi$
$499$	−15.1427			−0.677881			−0.338941	+	0.940808i	$0.610069\pi$
$500$	8.65233	+	19.4400i	0.386944	+	0.869384i
$501$		0			0
$502$		−	7.39652i		−	0.330123i
$503$			26.0370i			1.16093i	0.814284	+	0.580467i	$0.197130\pi$
$503$			26.0370i			1.16093i	−0.814284	+	0.580467i	$0.802870\pi$
$504$		0			0
$505$	5.03212	+	35.8163i	0.223926	+	1.59380i
$506$	−1.24443			−0.0553217
$507$		0			0
$508$			20.9447i			0.929271i
$509$	−24.5718			−1.08913			−0.544564	−	0.838719i	$-0.683305\pi$
$509$	−24.5718			−1.08913			−0.544564	+	0.838719i	$0.683305\pi$
$510$		0			0
$511$	5.11108			0.226101
$512$		−	20.3111i		−	0.897633i
$513$		0			0
$514$	2.53035			0.111609
$515$	37.9813	−	5.33630i	1.67365	−	0.235145i
$516$		0			0
$517$			0.949145i			0.0417433i
$518$		−	1.63158i		−	0.0716877i
$519$		0			0
$520$	1.09679	+	7.80642i	0.0480973	+	0.342334i
$521$	4.88892			0.214188			0.107094	−	0.994249i	$-0.465845\pi$
$521$	4.88892			0.214188			0.107094	+	0.994249i	$0.465845\pi$
$522$		0			0
$523$			4.22077i			0.184562i	0.995733	+	0.0922808i	$0.0294157\pi$
$523$			4.22077i			0.184562i	−0.995733	+	0.0922808i	$0.970584\pi$
$524$	−2.36842			−0.103465
$525$		0			0
$526$	7.12537			0.310681
$527$			5.98126i			0.260548i
$528$		0			0
$529$	7.00000			0.304348
$530$	−0.0789589	−	0.561993i	−0.00342976	−	0.0244114i
$531$		0			0
$532$			4.17484i			0.181002i
$533$		−	30.9590i		−	1.34098i
$534$		0			0
$535$	30.0415	−	4.22077i	1.29881	−	0.182480i
$536$	15.6128			0.674372
$537$		0			0
$538$		−	3.68196i		−	0.158741i
$539$	−6.18421			−0.266373
$540$		0			0
$541$	−13.6128			−0.585262			−0.292631	−	0.956225i	$-0.594531\pi$
$541$	−13.6128			−0.585262			−0.292631	+	0.956225i	$0.594531\pi$
$542$		−	4.63512i		−	0.199096i
$543$		0			0
$544$	−7.97280			−0.341831
$545$	−3.11108	−	22.1432i	−0.133264	−	0.948510i
$546$		0			0
$547$		−	7.48394i		−	0.319990i	−0.987118	−	0.159995i	$-0.948852\pi$
$547$		−	7.48394i		−	0.319990i	0.987118	−	0.159995i	$-0.0511478\pi$
$548$			8.42864i			0.360054i
$549$		0			0
$550$	−0.428639	−	1.49532i	−0.0182772	−	0.0637605i
$551$	−17.1240			−0.729506
$552$		0			0
$553$		−	14.9304i		−	0.634906i
$554$	−8.60793			−0.365716
$555$		0			0
$556$	−1.74620			−0.0740554
$557$			32.2908i			1.36821i	0.729385	+	0.684103i	$0.239806\pi$
$557$			32.2908i			1.36821i	−0.729385	+	0.684103i	$0.760194\pi$
$558$		0			0
$559$	−31.0923			−1.31507
$560$	−6.85728	+	0.963435i	−0.289773	+	0.0407126i
$561$		0			0
$562$			3.05086i			0.128693i
$563$		−	7.49378i		−	0.315825i	−0.987453	−	0.157913i	$-0.949524\pi$
$563$		−	7.49378i		−	0.315825i	0.987453	−	0.157913i	$-0.0504765\pi$
$564$		0			0
$565$	31.5210	−	4.42864i	1.32610	−	0.186314i
$566$	5.91258			0.248524
$567$		0			0
$568$			11.2761i			0.473134i
$569$	−12.9491			−0.542856			−0.271428	−	0.962459i	$-0.587496\pi$
$569$	−12.9491			−0.542856			−0.271428	+	0.962459i	$0.587496\pi$
$570$		0			0
$571$	−15.2859			−0.639696			−0.319848	−	0.947469i	$-0.603632\pi$
$571$	−15.2859			−0.639696			−0.319848	+	0.947469i	$0.603632\pi$
$572$			5.52543i			0.231030i
$573$		0			0
$574$	−2.99646			−0.125070
$575$	−5.51114	−	19.2257i	−0.229830	−	0.801767i
$576$		0			0
$577$		−	28.4415i		−	1.18404i	−0.805924	−	0.592019i	$-0.798331\pi$
$577$		−	28.4415i		−	1.18404i	0.805924	−	0.592019i	$-0.201669\pi$
$578$		−	3.67016i		−	0.152658i
$579$		0			0
$580$	−4.17484	−	29.7146i	−0.173351	−	1.23383i
$581$	7.00937			0.290798
$582$		0			0
$583$		−	0.815792i		−	0.0337866i
$584$	6.87157			0.284348
$585$		0			0
$586$	−9.56691			−0.395206
$587$		−	8.47013i		−	0.349600i	−0.984604	−	0.174800i	$-0.944072\pi$
$587$		−	8.47013i		−	0.349600i	0.984604	−	0.174800i	$-0.0559278\pi$
$588$		0			0
$589$	6.36842			0.262406
$590$	1.15257	−	0.161933i	0.0474504	−	0.00666669i
$591$		0			0
$592$			19.9081i			0.818219i
$593$		−	26.5763i		−	1.09136i	−0.837995	−	0.545679i	$-0.816272\pi$
$593$		−	26.5763i		−	1.09136i	0.837995	−	0.545679i	$-0.183728\pi$
$594$		0			0
$595$	0.640951	+	4.56199i	0.0262764	+	0.187023i
$596$	4.17484			0.171008
$597$		0			0
$598$		−	3.61285i		−	0.147740i
$599$	−8.77430			−0.358508			−0.179254	−	0.983803i	$-0.557368\pi$
$599$	−8.77430			−0.358508			−0.179254	+	0.983803i	$0.557368\pi$
$600$		0			0
$601$	−41.8163			−1.70572			−0.852861	−	0.522139i	$-0.825134\pi$
$601$	−41.8163			−1.70572			−0.852861	+	0.522139i	$0.825134\pi$
$602$			3.00937i			0.122653i
$603$		0			0
$604$	−19.8479			−0.807600
$605$	−0.311108	−	2.21432i	−0.0126483	−	0.0900249i
$606$		0			0
$607$		−	29.9353i		−	1.21504i	−0.794305	−	0.607519i	$-0.792165\pi$
$607$		−	29.9353i		−	1.21504i	0.794305	−	0.607519i	$-0.207835\pi$
$608$			8.48886i			0.344269i
$609$		0			0
$610$	4.99063	−	0.701175i	0.202065	−	0.0283897i
$611$	−2.75557			−0.111478
$612$		0			0
$613$			26.1289i			1.05534i	0.849450	+	0.527668i	$0.176934\pi$
$613$			26.1289i			1.05534i	−0.849450	+	0.527668i	$0.823066\pi$
$614$	−4.19802			−0.169418
$615$		0			0
$616$	1.09679			0.0441909
$617$		−	3.66323i		−	0.147476i	−0.997278	−	0.0737380i	$-0.976507\pi$
$617$		−	3.66323i		−	0.147476i	0.997278	−	0.0737380i	$-0.0234928\pi$
$618$		0			0
$619$	−43.2958			−1.74020			−0.870102	−	0.492872i	$-0.835947\pi$
$619$	−43.2958			−1.74020			−0.870102	+	0.492872i	$0.835947\pi$
$620$	1.55262	+	11.0509i	0.0623549	+	0.443813i
$621$		0			0
$622$		−	5.46076i		−	0.218956i
$623$		−	5.53972i		−	0.221944i
$624$		0			0
$625$	21.2034	−	13.2444i	0.848137	−	0.529777i
$626$	−4.47902			−0.179018
$627$		0			0
$628$			11.0509i			0.440977i
$629$	13.2444			0.528090
$630$		0			0
$631$	8.97773			0.357398			0.178699	−	0.983904i	$-0.442811\pi$
$631$	8.97773			0.357398			0.178699	+	0.983904i	$0.442811\pi$
$632$		−	20.0731i		−	0.798466i
$633$		0			0
$634$	−9.16547			−0.364007
$635$	24.3684	−	3.42372i	0.967031	−	0.135866i
$636$		0			0
$637$		−	17.9541i		−	0.711366i
$638$			2.19358i			0.0868445i
$639$		0			0
$640$	−19.4543	+	2.73329i	−0.768999	+	0.108043i
$641$	−9.21279			−0.363883			−0.181942	−	0.983309i	$-0.558238\pi$
$641$	−9.21279			−0.363883			−0.181942	+	0.983309i	$0.558238\pi$
$642$		0			0
$643$		−	16.3783i		−	0.645896i	−0.946417	−	0.322948i	$-0.895326\pi$
$643$		−	16.3783i		−	0.645896i	0.946417	−	0.322948i	$-0.104674\pi$
$644$	6.87601			0.270953
$645$		0			0
$646$	1.72345			0.0678082
$647$			9.80642i			0.385530i	0.981245	+	0.192765i	$0.0617456\pi$
$647$			9.80642i			0.385530i	−0.981245	+	0.192765i	$0.938254\pi$
$648$		0			0
$649$	1.67307			0.0656738
$650$	4.34122	−	1.24443i	0.170277	−	0.0488106i
$651$		0			0
$652$		−	21.0321i		−	0.823681i
$653$		−	33.0736i		−	1.29427i	−0.762375	−	0.647135i	$-0.775967\pi$
$653$		−	33.0736i		−	1.29427i	0.762375	−	0.647135i	$-0.224033\pi$
$654$		0			0
$655$	0.387152	+	2.75557i	0.0151273	+	0.107669i
$656$	36.5620			1.42751
$657$		0			0
$658$			0.266706i			0.0103973i
$659$	34.1017			1.32841			0.664207	−	0.747549i	$-0.268769\pi$
$659$	34.1017			1.32841			0.664207	+	0.747549i	$0.268769\pi$
$660$		0			0
$661$	−5.40943			−0.210402			−0.105201	−	0.994451i	$-0.533549\pi$
$661$	−5.40943			−0.210402			−0.105201	+	0.994451i	$0.533549\pi$
$662$			0.815792i			0.0317066i
$663$		0			0
$664$	9.42372			0.365711
$665$	4.85728	−	0.682439i	0.188357	−	0.0264638i
$666$		0			0
$667$			28.2034i			1.09204i
$668$		−	26.5575i		−	1.02754i
$669$		0			0
$670$	−1.24443	−	8.85728i	−0.0480766	−	0.342187i
$671$	7.24443			0.279668
$672$		0			0
$673$			24.1476i			0.930823i	0.885094	+	0.465412i	$0.154094\pi$
$673$			24.1476i			0.930823i	−0.885094	+	0.465412i	$0.845906\pi$
$674$	−1.55707			−0.0599761
$675$		0			0
$676$	8.70027			0.334626
$677$			26.2810i			1.01006i	0.863102	+	0.505030i	$0.168519\pi$
$677$			26.2810i			1.01006i	−0.863102	+	0.505030i	$0.831481\pi$
$678$		0			0
$679$	−11.2632			−0.432241
$680$	0.861725	+	6.13335i	0.0330456	+	0.235203i
$681$		0			0
$682$		−	0.815792i		−	0.0312383i
$683$		−	15.3176i		−	0.586110i	−0.956096	−	0.293055i	$-0.905328\pi$
$683$		−	15.3176i		−	0.586110i	0.956096	−	0.293055i	$-0.0946719\pi$
$684$		0			0
$685$	9.80642	−	1.37778i	0.374684	−	0.0526424i
$686$	−3.70471			−0.141447
$687$		0			0
$688$		−	36.7195i		−	1.39992i
$689$	2.36842			0.0902295
$690$		0			0
$691$	15.0223			0.571474			0.285737	−	0.958308i	$-0.407762\pi$
$691$	15.0223			0.571474			0.285737	+	0.958308i	$0.407762\pi$
$692$		−	35.8622i		−	1.36328i
$693$		0			0
$694$	7.09679			0.269390
$695$	0.285442	+	2.03164i	0.0108274	+	0.0770646i
$696$		0			0
$697$		−	24.3239i		−	0.921332i
$698$			6.60348i			0.249945i
$699$		0			0
$700$	2.36842	+	8.26226i	0.0895177	+	0.312284i
$701$	19.9081			0.751920			0.375960	−	0.926636i	$-0.377313\pi$
$701$	19.9081			0.751920			0.375960	+	0.926636i	$0.377313\pi$
$702$		0			0
$703$		−	14.1017i		−	0.531856i
$704$	−5.76986			−0.217460
$705$		0			0
$706$	2.23506			0.0841177
$707$			14.6093i			0.549440i
$708$		0			0
$709$	13.5081			0.507306			0.253653	−	0.967295i	$-0.418368\pi$
$709$	13.5081			0.507306			0.253653	+	0.967295i	$0.418368\pi$
$710$	6.39700	−	0.898766i	0.240075	−	0.0337301i
$711$		0			0
$712$		−	7.44785i		−	0.279120i
$713$		−	10.4889i		−	0.392811i
$714$		0			0
$715$	6.42864	−	0.903212i	0.240417	−	0.0337782i
$716$	9.24443			0.345481
$717$		0			0
$718$			4.38715i			0.163727i
$719$	−16.0830			−0.599794			−0.299897	−	0.953972i	$-0.596952\pi$
$719$	−16.0830			−0.599794			−0.299897	+	0.953972i	$0.596952\pi$
$720$		0			0
$721$	15.4924			0.576967
$722$			4.07604i			0.151695i
$723$		0			0
$724$	−31.9684			−1.18809
$725$	−33.8894	+	9.71456i	−1.25862	+	0.360790i
$726$		0			0
$727$			23.6128i			0.875752i	0.899035	+	0.437876i	$0.144269\pi$
$727$			23.6128i			0.875752i	−0.899035	+	0.437876i	$0.855731\pi$
$728$			3.18421i			0.118015i
$729$		0			0
$730$	−0.547702	−	3.89829i	−0.0202714	−	0.144282i
$731$	−24.4286			−0.903526
$732$		0			0
$733$			30.0459i			1.10977i	0.831926	+	0.554886i	$0.187238\pi$
$733$			30.0459i			1.10977i	−0.831926	+	0.554886i	$0.812762\pi$
$734$	−1.21585			−0.0448779
$735$		0			0
$736$	13.9813			0.515356
$737$		−	12.8573i		−	0.473604i
$738$		0			0
$739$	24.4099			0.897933			0.448966	−	0.893549i	$-0.351792\pi$
$739$	24.4099			0.897933			0.448966	+	0.893549i	$0.351792\pi$
$740$	24.4701	−	3.43801i	0.899540	−	0.126384i
$741$		0			0
$742$		−	0.229234i		−	0.00841546i
$743$		−	33.1798i		−	1.21725i	−0.793459	−	0.608624i	$-0.791722\pi$
$743$		−	33.1798i		−	1.21725i	0.793459	−	0.608624i	$-0.208278\pi$
$744$		0			0
$745$	−0.682439	−	4.85728i	−0.0250026	−	0.177957i
$746$	−4.03704			−0.147807
$747$		0			0
$748$			4.34122i			0.158731i
$749$	12.2538			0.447744
$750$		0			0
$751$	−22.5718			−0.823658			−0.411829	−	0.911261i	$-0.635110\pi$
$751$	−22.5718			−0.823658			−0.411829	+	0.911261i	$0.635110\pi$
$752$		−	3.25428i		−	0.118671i
$753$		0			0
$754$	−6.36842			−0.231924
$755$	3.24443	+	23.0923i	0.118077	+	0.840416i
$756$		0			0
$757$		−	4.94914i		−	0.179880i	−0.995947	−	0.0899399i	$-0.971333\pi$
$757$		−	4.94914i		−	0.179880i	0.995947	−	0.0899399i	$-0.0286675\pi$
$758$			11.2257i			0.407736i
$759$		0			0
$760$	6.53035	−	0.917502i	0.236881	−	0.0332813i
$761$	−14.6637			−0.531559			−0.265779	−	0.964034i	$-0.585629\pi$
$761$	−14.6637			−0.531559			−0.265779	+	0.964034i	$0.585629\pi$
$762$		0			0
$763$		−	9.03212i		−	0.326985i
$764$	−32.0830			−1.16072
$765$		0			0
$766$	6.31402			0.228135
$767$			4.85728i			0.175386i
$768$		0			0
$769$	44.5718			1.60730			0.803651	−	0.595101i	$-0.202888\pi$
$769$	44.5718			1.60730			0.803651	+	0.595101i	$0.202888\pi$
$770$	−0.0874201	−	0.622216i	−0.00315040	−	0.0224231i
$771$		0			0
$772$			46.8528i			1.68627i
$773$		−	17.3145i		−	0.622759i	−0.950286	−	0.311380i	$-0.899209\pi$
$773$		−	17.3145i		−	0.622759i	0.950286	−	0.311380i	$-0.100791\pi$
$774$		0			0
$775$	12.6035	−	3.61285i	0.452730	−	0.129777i
$776$	−15.1427			−0.543592
$777$		0			0
$778$		−	9.47949i		−	0.339856i
$779$	−25.8983			−0.927903
$780$		0			0
$781$	9.28592			0.332276
$782$		−	2.83854i		−	0.101506i
$783$		0			0
$784$	21.2034			0.757265
$785$	12.8573	−	1.80642i	0.458896	−	0.0644740i
$786$		0			0
$787$		−	36.5161i		−	1.30166i	−0.759225	−	0.650828i	$-0.774422\pi$
$787$		−	36.5161i		−	1.30166i	0.759225	−	0.650828i	$-0.225578\pi$
$788$		−	28.3037i		−	1.00828i
$789$		0			0
$790$	−11.3876	+	1.59994i	−0.405154	+	0.0569233i
$791$	12.8573			0.457152
$792$		0			0
$793$			21.0321i			0.746872i
$794$	1.54861			0.0549581
$795$		0			0
$796$	21.3649			0.757258
$797$		−	14.3180i		−	0.507171i	−0.967313	−	0.253585i	$-0.918390\pi$
$797$		−	14.3180i		−	0.507171i	0.967313	−	0.253585i	$-0.0816099\pi$
$798$		0			0
$799$	−2.16500			−0.0765921
$800$	4.81579	+	16.8000i	0.170264	+	0.593969i
$801$		0			0
$802$		−	0.580728i		−	0.0205062i
$803$		−	5.65878i		−	0.199694i
$804$		0			0
$805$	−1.12399	−	8.00000i	−0.0396153	−	0.281963i
$806$	2.36842			0.0834239
$807$		0			0
$808$			19.6414i			0.690983i
$809$	32.0544			1.12697			0.563486	−	0.826125i	$-0.309460\pi$
$809$	32.0544			1.12697			0.563486	+	0.826125i	$0.309460\pi$
$810$		0			0
$811$	−8.44738			−0.296627			−0.148314	−	0.988940i	$-0.547385\pi$
$811$	−8.44738			−0.296627			−0.148314	+	0.988940i	$0.547385\pi$
$812$		−	12.1204i		−	0.425344i
$813$		0			0
$814$	−1.80642			−0.0633151
$815$	−24.4701	+	3.43801i	−0.857151	+	0.120428i
$816$		0			0
$817$			26.0098i			0.909969i
$818$		−	1.12981i		−	0.0395030i
$819$		0			0
$820$	−6.31402	−	44.9403i	−0.220495	−	1.56938i
$821$	−17.2159			−0.600837			−0.300419	−	0.953807i	$-0.597126\pi$
$821$	−17.2159			−0.600837			−0.300419	+	0.953807i	$0.597126\pi$
$822$		0			0
$823$			12.7654i			0.444974i	0.974936	+	0.222487i	$0.0714175\pi$
$823$			12.7654i			0.444974i	−0.974936	+	0.222487i	$0.928582\pi$
$824$	20.8287			0.725602
$825$		0			0
$826$	0.470127			0.0163578
$827$			8.70964i			0.302864i	0.988468	+	0.151432i	$0.0483884\pi$
$827$			8.70964i			0.302864i	−0.988468	+	0.151432i	$0.951612\pi$
$828$		0			0
$829$	8.32693			0.289206			0.144603	−	0.989490i	$-0.453809\pi$
$829$	8.32693			0.289206			0.144603	+	0.989490i	$0.453809\pi$
$830$	−0.751123	−	5.34614i	−0.0260719	−	0.185567i
$831$		0			0
$832$		−	16.7511i		−	0.580741i
$833$		−	14.1062i		−	0.488749i
$834$		0			0
$835$	−30.8988	+	4.34122i	−1.06930	+	0.150234i
$836$	4.62222			0.159863
$837$		0			0
$838$		−	1.51114i		−	0.0522014i
$839$	12.8988			0.445315			0.222657	−	0.974897i	$-0.428527\pi$
$839$	12.8988			0.445315			0.222657	+	0.974897i	$0.428527\pi$
$840$		0			0
$841$	20.7146			0.714295
$842$		−	7.04101i		−	0.242649i
$843$		0			0
$844$	−22.7239			−0.782190
$845$	−1.42219	−	10.1225i	−0.0489247	−	0.348223i
$846$		0			0
$847$		−	0.903212i		−	0.0310347i
$848$			2.79706i			0.0960513i
$849$		0			0
$850$	3.41081	−	0.977725i	0.116990	−	0.0335357i
$851$	−23.2257			−0.796167
$852$		0			0
$853$		−	19.6686i		−	0.673441i	−0.941605	−	0.336720i	$-0.890682\pi$
$853$		−	19.6686i		−	0.673441i	0.941605	−	0.336720i	$-0.109318\pi$
$854$	2.03566			0.0696588
$855$		0			0
$856$	16.4746			0.563089
$857$			31.8207i			1.08697i	0.839417	+	0.543487i	$0.182896\pi$
$857$			31.8207i			1.08697i	−0.839417	+	0.543487i	$0.817104\pi$
$858$		0			0
$859$	−27.8292			−0.949519			−0.474760	−	0.880116i	$-0.657465\pi$
$859$	−27.8292			−0.949519			−0.474760	+	0.880116i	$0.657465\pi$
$860$	−45.1338	+	6.34122i	−1.53905	+	0.216234i
$861$		0			0
$862$			0.387152i			0.0131865i
$863$			4.82870i			0.164371i	0.996617	+	0.0821854i	$0.0261900\pi$
$863$			4.82870i			0.164371i	−0.996617	+	0.0821854i	$0.973810\pi$
$864$		0			0
$865$	−41.7244	+	5.86220i	−1.41867	+	0.199321i
$866$	4.97773			0.169150
$867$		0			0
$868$			4.50760i			0.152998i
$869$	−16.5303			−0.560754
$870$		0			0
$871$	37.3274			1.26479
$872$		−	12.1432i		−	0.411221i
$873$		0			0
$874$	−3.02227			−0.102230
$875$	9.22570	−	4.10616i	0.311885	−	0.138813i
$876$		0			0
$877$		−	21.9826i		−	0.742301i	−0.928573	−	0.371151i	$-0.878963\pi$
$877$		−	21.9826i		−	0.742301i	0.928573	−	0.371151i	$-0.121037\pi$
$878$		−	0.755569i		−	0.0254992i
$879$		0			0
$880$	1.06668	+	7.59210i	0.0359577	+	0.255930i
$881$	12.1017			0.407717			0.203858	−	0.979000i	$-0.434652\pi$
$881$	12.1017			0.407717			0.203858	+	0.979000i	$0.434652\pi$
$882$		0			0
$883$			8.73683i			0.294018i	0.989135	+	0.147009i	$0.0469646\pi$
$883$			8.73683i			0.294018i	−0.989135	+	0.147009i	$0.953035\pi$
$884$	−12.6035			−0.423901
$885$		0			0
$886$	9.66016			0.324540
$887$		−	19.8524i		−	0.666577i	−0.942825	−	0.333288i	$-0.891842\pi$
$887$		−	19.8524i		−	0.666577i	0.942825	−	0.333288i	$-0.108158\pi$
$888$		0			0
$889$	9.93978			0.333369
$890$	−4.22522	+	0.593635i	−0.141630	+	0.0198987i
$891$		0			0
$892$		−	41.5022i		−	1.38960i
$893$			2.30513i			0.0771383i
$894$		0			0
$895$	−1.51114	−	10.7556i	−0.0505118	−	0.359519i
$896$	−7.93533			−0.265101
$897$		0			0
$898$			11.6227i			0.387854i
$899$	−18.4889			−0.616638
$900$		0			0
$901$	1.86082			0.0619928
$902$			3.31756i			0.110463i
$903$		0			0
$904$	17.2859			0.574921
$905$	5.22570	+	37.1941i	0.173708	+	1.23637i
$906$		0			0
$907$		−	32.8287i		−	1.09006i	−0.838417	−	0.545030i	$-0.816518\pi$
$907$		−	32.8287i		−	1.09006i	0.838417	−	0.545030i	$-0.183482\pi$
$908$		−	6.08742i		−	0.202018i
$909$		0			0
$910$	1.80642	−	0.253799i	0.0598824	−	0.00841336i
$911$	16.3497			0.541689			0.270845	−	0.962623i	$-0.412697\pi$
$911$	16.3497			0.541689			0.270845	+	0.962623i	$0.412697\pi$
$912$		0			0
$913$		−	7.76049i		−	0.256835i
$914$	2.71853			0.0899209
$915$		0			0
$916$	−13.5585			−0.447984
$917$			1.12399i			0.0371173i
$918$		0			0
$919$	−20.0228			−0.660490			−0.330245	−	0.943895i	$-0.607131\pi$
$919$	−20.0228			−0.660490			−0.330245	+	0.943895i	$0.607131\pi$
$920$	−1.51114	−	10.7556i	−0.0498207	−	0.354601i
$921$		0			0
$922$			9.88940i			0.325690i
$923$			26.9590i			0.887366i
$924$		0			0
$925$	−8.00000	−	27.9081i	−0.263038	−	0.917614i
$926$	−3.76187			−0.123623
$927$		0			0
$928$		−	24.6450i		−	0.809011i
$929$	−43.5308			−1.42820			−0.714100	−	0.700044i	$-0.753164\pi$
$929$	−43.5308			−1.42820			−0.714100	+	0.700044i	$0.753164\pi$
$930$		0			0
$931$	−15.0192			−0.492235
$932$		−	37.1610i		−	1.21725i
$933$		0			0
$934$	4.77430			0.156220
$935$	5.05086	−	0.709636i	0.165181	−	0.0232076i
$936$		0			0
$937$			43.4563i			1.41966i	0.704375	+	0.709828i	$0.251227\pi$
$937$			43.4563i			1.41966i	−0.704375	+	0.709828i	$0.748773\pi$
$938$		−	3.61285i		−	0.117964i
$939$		0			0
$940$	−4.00000	+	0.561993i	−0.130466	+	0.0183302i
$941$	23.7244			0.773393			0.386697	−	0.922207i	$-0.373616\pi$
$941$	23.7244			0.773393			0.386697	+	0.922207i	$0.373616\pi$
$942$		0			0
$943$			42.6548i			1.38903i
$944$	−5.73636			−0.186703
$945$		0			0
$946$	3.33185			0.108328
$947$		−	11.7047i		−	0.380352i	−0.981750	−	0.190176i	$-0.939094\pi$
$947$		−	11.7047i		−	0.380352i	0.981750	−	0.190176i	$-0.0609059\pi$
$948$		0			0
$949$	16.4286			0.533296
$950$	−1.04101	−	3.63158i	−0.0337748	−	0.117824i
$951$		0			0
$952$			2.50177i			0.0810828i
$953$			46.1258i			1.49416i	0.664733	+	0.747081i	$0.268545\pi$
$953$			46.1258i			1.49416i	−0.664733	+	0.747081i	$0.731455\pi$
$954$		0			0
$955$	5.24443	+	37.3274i	0.169706	+	1.20789i
$956$	−41.8350			−1.35304
$957$		0			0
$958$		−	1.83500i		−	0.0592863i
$959$	4.00000			0.129167
$960$		0			0
$961$	−24.1240			−0.778193
$962$		−	5.24443i		−	0.169087i
$963$		0			0
$964$	10.1748			0.327710
$965$	54.5116	−	7.65878i	1.75479	−	0.246545i
$966$		0			0
$967$			17.0495i			0.548274i	0.961691	+	0.274137i	$0.0883922\pi$
$967$			17.0495i			0.548274i	−0.961691	+	0.274137i	$0.911608\pi$
$968$		−	1.21432i		−	0.0390297i
$969$		0			0
$970$	1.20696	+	8.59057i	0.0387531	+	0.275827i
$971$	58.1847			1.86724			0.933618	−	0.358271i	$-0.116634\pi$
$971$	58.1847			1.86724			0.933618	+	0.358271i	$0.116634\pi$
$972$		0			0
$973$			0.828699i			0.0265669i
$974$	9.75203			0.312475
$975$		0			0
$976$	−24.8385			−0.795062
$977$			51.7373i			1.65522i	0.561301	+	0.827612i	$0.310301\pi$
$977$			51.7373i			1.65522i	−0.561301	+	0.827612i	$0.689699\pi$
$978$		0			0
$979$	−6.13335			−0.196023
$980$	−3.66170	−	26.0622i	−0.116969	−	0.832527i
$981$		0			0
$982$		−	2.48886i		−	0.0794228i
$983$		−	26.3970i		−	0.841933i	−0.907076	−	0.420967i	$-0.861691\pi$
$983$		−	26.3970i		−	0.841933i	0.907076	−	0.420967i	$-0.138309\pi$
$984$		0			0
$985$	−32.9304	+	4.62666i	−1.04925	+	0.147418i
$986$	−5.00354			−0.159345
$987$		0			0
$988$			13.4193i			0.426924i
$989$	42.8385			1.36219
$990$		0			0
$991$	−23.0923			−0.733552			−0.366776	−	0.930309i	$-0.619539\pi$
$991$	−23.0923			−0.733552			−0.366776	+	0.930309i	$0.619539\pi$
$992$			9.16547i			0.291004i
$993$		0			0
$994$	2.60931			0.0827622
$995$	−3.49240	−	24.8573i	−0.110717	−	0.788029i
$996$		0			0
$997$			12.9131i			0.408961i	0.978871	+	0.204480i	$0.0655504\pi$
$997$			12.9131i			0.408961i	−0.978871	+	0.204480i	$0.934450\pi$
$998$			4.71102i			0.149125i
$999$		0			0

Display

a_p

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up to: 50 250 1000 (See

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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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.2.c.e.199.3		6
3.2	odd	2		165.2.c.b.34.4	yes	6
5.2	odd	4		2475.2.a.bc.1.2		3
5.3	odd	4		2475.2.a.ba.1.2		3
5.4	even	2	inner	495.2.c.e.199.4		6
12.11	even	2		2640.2.d.h.529.5		6
15.2	even	4		825.2.a.j.1.2		3
15.8	even	4		825.2.a.l.1.2		3
15.14	odd	2		165.2.c.b.34.3	✓	6
33.32	even	2		1815.2.c.e.364.3		6
60.59	even	2		2640.2.d.h.529.2		6
165.32	odd	4		9075.2.a.ch.1.2		3
165.98	odd	4		9075.2.a.cg.1.2		3
165.164	even	2		1815.2.c.e.364.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.c.b.34.3	✓	6	15.14	odd	2
165.2.c.b.34.4	yes	6	3.2	odd	2
495.2.c.e.199.3		6	1.1	even	1	trivial
495.2.c.e.199.4		6	5.4	even	2	inner
825.2.a.j.1.2		3	15.2	even	4
825.2.a.l.1.2		3	15.8	even	4
1815.2.c.e.364.3		6	33.32	even	2
1815.2.c.e.364.4		6	165.164	even	2
2475.2.a.ba.1.2		3	5.3	odd	4
2475.2.a.bc.1.2		3	5.2	odd	4
2640.2.d.h.529.2		6	60.59	even	2
2640.2.d.h.529.5		6	12.11	even	2
9075.2.a.cg.1.2		3	165.98	odd	4
9075.2.a.ch.1.2		3	165.32	odd	4

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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Newspace parameters

Newform invariants

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$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	495.2.c.e.199.3

Level	$495$
Weight	$2$
Character	495.199
Analytic conductor	$3.953$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$