LMFDB - Embedded newform 495.2.c.a.199.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [495,2,Mod(199,495)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [4,0,0,-6,3,0,0,0,0,-10,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$3.95259490005$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.3
Root			$1.68614 - 0.396143i$ of defining polynomial
Character	$\chi$	$=$	495.199
Dual form			495.2.c.a.199.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.792287i q^{2} +1.37228 q^{4} +(2.18614 - 0.469882i) q^{5} -3.46410i q^{7} +2.67181i q^{8} +(0.372281 + 1.73205i) q^{10} +1.00000 q^{11} +2.74456 q^{14} +0.627719 q^{16} -5.04868i q^{17} -4.00000 q^{19} +(3.00000 - 0.644810i) q^{20} +0.792287i q^{22} +2.52434i q^{23} +(4.55842 - 2.05446i) q^{25} -4.75372i q^{28} -2.74456 q^{29} -2.37228 q^{31} +5.84096i q^{32} +4.00000 q^{34} +(-1.62772 - 7.57301i) q^{35} +11.0371i q^{37} -3.16915i q^{38} +(1.25544 + 5.84096i) q^{40} +2.74456 q^{41} +3.46410i q^{43} +1.37228 q^{44} -2.00000 q^{46} +6.63325i q^{47} -5.00000 q^{49} +(1.62772 + 3.61158i) q^{50} +3.16915i q^{53} +(2.18614 - 0.469882i) q^{55} +9.25544 q^{56} -2.17448i q^{58} -1.62772 q^{59} +10.7446 q^{61} -1.87953i q^{62} -3.37228 q^{64} +0.644810i q^{67} -6.92820i q^{68} +(6.00000 - 1.28962i) q^{70} -7.11684 q^{71} -6.92820i q^{73} -8.74456 q^{74} -5.48913 q^{76} -3.46410i q^{77} -12.7446 q^{79} +(1.37228 - 0.294954i) q^{80} +2.17448i q^{82} +6.63325i q^{83} +(-2.37228 - 11.0371i) q^{85} -2.74456 q^{86} +2.67181i q^{88} -4.37228 q^{89} +3.46410i q^{92} -5.25544 q^{94} +(-8.74456 + 1.87953i) q^{95} +4.10891i q^{97} -3.96143i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{4} + 3 q^{5} - 10 q^{10} + 4 q^{11} - 12 q^{14} + 14 q^{16} - 16 q^{19} + 12 q^{20} + q^{25} + 12 q^{29} + 2 q^{31} + 16 q^{34} - 18 q^{35} + 28 q^{40} - 12 q^{41} - 6 q^{44} - 8 q^{46} - 20 q^{49}+ \cdots - 12 q^{95}+O(q^{100}) $ 4 * q - 6 * q^4 + 3 * q^5 - 10 * q^10 + 4 * q^11 - 12 * q^14 + 14 * q^16 - 16 * q^19 + 12 * q^20 + q^25 + 12 * q^29 + 2 * q^31 + 16 * q^34 - 18 * q^35 + 28 * q^40 - 12 * q^41 - 6 * q^44 - 8 * q^46 - 20 * q^49 + 18 * q^50 + 3 * q^55 + 60 * q^56 - 18 * q^59 + 20 * q^61 - 2 * q^64 + 24 * q^70 + 6 * q^71 - 12 * q^74 + 24 * q^76 - 28 * q^79 - 6 * q^80 + 2 * q^85 + 12 * q^86 - 6 * q^89 - 44 * q^94 - 12 * 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 $

$2$			0.792287i			0.560232i	0.959966	+	0.280116i	$0.0903729\pi$
$2$			0.792287i			0.560232i	−0.959966	+	0.280116i	$0.909627\pi$
$3$		0			0
$3$		0			0
$4$	1.37228			0.686141
$5$	2.18614	−	0.469882i	0.977672	−	0.210138i
$5$	2.18614	−	0.469882i	0.977672	−	0.210138i
$6$		0			0
$7$		−	3.46410i		−	1.30931i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$		−	3.46410i		−	1.30931i	0.755929	−	0.654654i	$-0.227186\pi$
$8$			2.67181i			0.944629i
$9$		0			0
$10$	0.372281	+	1.73205i	0.117726	+	0.547723i
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	2.74456			0.733515
$15$		0			0
$16$	0.627719			0.156930
$17$		−	5.04868i		−	1.22448i	−0.790671	−	0.612242i	$-0.790268\pi$
$17$		−	5.04868i		−	1.22448i	0.790671	−	0.612242i	$-0.209732\pi$
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$	3.00000	−	0.644810i	0.670820	−	0.144184i
$21$		0			0
$22$			0.792287i			0.168916i
$23$			2.52434i			0.526361i	0.964747	+	0.263180i	$0.0847714\pi$
$23$			2.52434i			0.526361i	−0.964747	+	0.263180i	$0.915229\pi$
$24$		0			0
$25$	4.55842	−	2.05446i	0.911684	−	0.410891i
$26$		0			0
$27$		0			0
$28$		−	4.75372i		−	0.898369i
$29$	−2.74456			−0.509652			−0.254826	−	0.966987i	$-0.582018\pi$
$29$	−2.74456			−0.509652			−0.254826	+	0.966987i	$0.582018\pi$
$30$		0			0
$31$	−2.37228			−0.426074			−0.213037	−	0.977044i	$-0.568336\pi$
$31$	−2.37228			−0.426074			−0.213037	+	0.977044i	$0.568336\pi$
$32$			5.84096i			1.03255i
$33$		0			0
$34$	4.00000			0.685994
$35$	−1.62772	−	7.57301i	−0.275135	−	1.28007i
$36$		0			0
$37$			11.0371i			1.81449i	0.420602	+	0.907245i	$0.361819\pi$
$37$			11.0371i			1.81449i	−0.420602	+	0.907245i	$0.638181\pi$
$38$		−	3.16915i		−	0.514104i
$39$		0			0
$40$	1.25544	+	5.84096i	0.198502	+	0.923537i
$41$	2.74456			0.428629			0.214314	−	0.976765i	$-0.431248\pi$
$41$	2.74456			0.428629			0.214314	+	0.976765i	$0.431248\pi$
$42$		0			0
$43$			3.46410i			0.528271i	0.964486	+	0.264135i	$0.0850865\pi$
$43$			3.46410i			0.528271i	−0.964486	+	0.264135i	$0.914913\pi$
$44$	1.37228			0.206879
$45$		0			0
$46$	−2.00000			−0.294884
$47$			6.63325i			0.967559i	0.875190	+	0.483779i	$0.160736\pi$
$47$			6.63325i			0.967559i	−0.875190	+	0.483779i	$0.839264\pi$
$48$		0			0
$49$	−5.00000			−0.714286
$50$	1.62772	+	3.61158i	0.230194	+	0.510754i
$51$		0			0
$52$		0			0
$53$			3.16915i			0.435316i	0.976025	+	0.217658i	$0.0698417\pi$
$53$			3.16915i			0.435316i	−0.976025	+	0.217658i	$0.930158\pi$
$54$		0			0
$55$	2.18614	−	0.469882i	0.294779	−	0.0633589i
$56$	9.25544			1.23681
$57$		0			0
$58$		−	2.17448i		−	0.285523i
$59$	−1.62772			−0.211911			−0.105955	−	0.994371i	$-0.533790\pi$
$59$	−1.62772			−0.211911			−0.105955	+	0.994371i	$0.533790\pi$
$60$		0			0
$61$	10.7446			1.37570			0.687850	−	0.725853i	$-0.258555\pi$
$61$	10.7446			1.37570			0.687850	+	0.725853i	$0.258555\pi$
$62$		−	1.87953i		−	0.238700i
$63$		0			0
$64$	−3.37228			−0.421535
$65$		0			0
$66$		0			0
$67$			0.644810i			0.0787761i	0.999224	+	0.0393880i	$0.0125408\pi$
$67$			0.644810i			0.0787761i	−0.999224	+	0.0393880i	$0.987459\pi$
$68$		−	6.92820i		−	0.840168i
$69$		0			0
$70$	6.00000	−	1.28962i	0.717137	−	0.154139i
$71$	−7.11684			−0.844614			−0.422307	−	0.906453i	$-0.638780\pi$
$71$	−7.11684			−0.844614			−0.422307	+	0.906453i	$0.638780\pi$
$72$		0			0
$73$		−	6.92820i		−	0.810885i	−0.914121	−	0.405442i	$-0.867117\pi$
$73$		−	6.92820i		−	0.810885i	0.914121	−	0.405442i	$-0.132883\pi$
$74$	−8.74456			−1.01653
$75$		0			0
$76$	−5.48913			−0.629646
$77$		−	3.46410i		−	0.394771i
$78$		0			0
$79$	−12.7446			−1.43388			−0.716938	−	0.697137i	$-0.754457\pi$
$79$	−12.7446			−1.43388			−0.716938	+	0.697137i	$0.754457\pi$
$80$	1.37228	−	0.294954i	0.153426	−	0.0329768i
$81$		0			0
$82$			2.17448i			0.240131i
$83$			6.63325i			0.728094i	0.931381	+	0.364047i	$0.118605\pi$
$83$			6.63325i			0.728094i	−0.931381	+	0.364047i	$0.881395\pi$
$84$		0			0
$85$	−2.37228	−	11.0371i	−0.257310	−	1.19714i
$86$	−2.74456			−0.295954
$87$		0			0
$88$			2.67181i			0.284816i
$89$	−4.37228			−0.463461			−0.231730	−	0.972780i	$-0.574439\pi$
$89$	−4.37228			−0.463461			−0.231730	+	0.972780i	$0.574439\pi$
$90$		0			0
$91$		0			0
$92$			3.46410i			0.361158i
$93$		0			0
$94$	−5.25544			−0.542057
$95$	−8.74456	+	1.87953i	−0.897173	+	0.192835i
$96$		0			0
$97$			4.10891i			0.417197i	0.978001	+	0.208598i	$0.0668902\pi$
$97$			4.10891i			0.417197i	−0.978001	+	0.208598i	$0.933110\pi$
$98$		−	3.96143i		−	0.400165i
$99$		0			0
$100$	6.25544	−	2.81929i	0.625544	−	0.281929i
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$		−	10.3923i		−	1.02398i	−0.858990	−	0.511992i	$-0.828908\pi$
$103$		−	10.3923i		−	1.02398i	0.858990	−	0.511992i	$-0.171092\pi$
$104$		0			0
$105$		0			0
$106$	−2.51087			−0.243878
$107$			6.63325i			0.641260i	0.947204	+	0.320630i	$0.103895\pi$
$107$			6.63325i			0.641260i	−0.947204	+	0.320630i	$0.896105\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$	0.372281	+	1.73205i	0.0354956	+	0.165145i
$111$		0			0
$112$		−	2.17448i		−	0.205469i
$113$		−	16.0858i		−	1.51322i	−0.653864	−	0.756612i	$-0.726853\pi$
$113$		−	16.0858i		−	1.51322i	0.653864	−	0.756612i	$-0.273147\pi$
$114$		0			0
$115$	1.18614	+	5.51856i	0.110608	+	0.514608i
$116$	−3.76631			−0.349693
$117$		0			0
$118$		−	1.28962i		−	0.118719i
$119$	−17.4891			−1.60323
$120$		0			0
$121$	1.00000			0.0909091
$122$			8.51278i			0.770711i
$123$		0			0
$124$	−3.25544			−0.292347
$125$	9.00000	−	6.63325i	0.804984	−	0.593296i
$126$		0			0
$127$		−	11.6819i		−	1.03660i	−0.855198	−	0.518302i	$-0.826564\pi$
$127$		−	11.6819i		−	1.03660i	0.855198	−	0.518302i	$-0.173436\pi$
$128$			9.01011i			0.796389i
$129$		0			0
$130$		0			0
$131$	−8.74456			−0.764016			−0.382008	−	0.924159i	$-0.624767\pi$
$131$	−8.74456			−0.764016			−0.382008	+	0.924159i	$0.624767\pi$
$132$		0			0
$133$			13.8564i			1.20150i
$134$	−0.510875			−0.0441329
$135$		0			0
$136$	13.4891			1.15668
$137$		−	2.22938i		−	0.190469i	−0.995455	−	0.0952346i	$-0.969640\pi$
$137$		−	2.22938i		−	0.190469i	0.995455	−	0.0952346i	$-0.0303601\pi$
$138$		0			0
$139$	−18.2337			−1.54656			−0.773281	−	0.634064i	$-0.781386\pi$
$139$	−18.2337			−1.54656			−0.773281	+	0.634064i	$0.781386\pi$
$140$	−2.23369	−	10.3923i	−0.188781	−	0.878310i
$141$		0			0
$142$		−	5.63858i		−	0.473179i
$143$		0			0
$144$		0			0
$145$	−6.00000	+	1.28962i	−0.498273	+	0.107097i
$146$	5.48913			0.454283
$147$		0			0
$148$			15.1460i			1.24500i
$149$	11.4891			0.941226			0.470613	−	0.882340i	$-0.344033\pi$
$149$	11.4891			0.941226			0.470613	+	0.882340i	$0.344033\pi$
$150$		0			0
$151$	22.2337			1.80935			0.904676	−	0.426100i	$-0.140113\pi$
$151$	22.2337			1.80935			0.904676	+	0.426100i	$0.140113\pi$
$152$		−	10.6873i		−	0.866851i
$153$		0			0
$154$	2.74456			0.221163
$155$	−5.18614	+	1.11469i	−0.416561	+	0.0895342i
$156$		0			0
$157$		−	5.39853i		−	0.430850i	−0.976520	−	0.215425i	$-0.930886\pi$
$157$		−	5.39853i		−	0.430850i	0.976520	−	0.215425i	$-0.0691136\pi$
$158$		−	10.0974i		−	0.803302i
$159$		0			0
$160$	2.74456	+	12.7692i	0.216977	+	1.00949i
$161$	8.74456			0.689168
$162$		0			0
$163$		−	3.46410i		−	0.271329i	−0.990755	−	0.135665i	$-0.956683\pi$
$163$		−	3.46410i		−	0.271329i	0.990755	−	0.135665i	$-0.0433170\pi$
$164$	3.76631			0.294100
$165$		0			0
$166$	−5.25544			−0.407901
$167$		−	22.3692i		−	1.73098i	−0.500927	−	0.865490i	$-0.667007\pi$
$167$		−	22.3692i		−	1.73098i	0.500927	−	0.865490i	$-0.332993\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$	8.74456	−	1.87953i	0.670677	−	0.144153i
$171$		0			0
$172$			4.75372i			0.362468i
$173$			1.87953i			0.142898i	0.997444	+	0.0714489i	$0.0227623\pi$
$173$			1.87953i			0.142898i	−0.997444	+	0.0714489i	$0.977238\pi$
$174$		0			0
$175$	−7.11684	−	15.7908i	−0.537983	−	1.19368i
$176$	0.627719			0.0473161
$177$		0			0
$178$		−	3.46410i		−	0.259645i
$179$	−15.8614			−1.18554			−0.592769	−	0.805373i	$-0.701965\pi$
$179$	−15.8614			−1.18554			−0.592769	+	0.805373i	$0.701965\pi$
$180$		0			0
$181$	6.88316			0.511621			0.255810	−	0.966727i	$-0.417658\pi$
$181$	6.88316			0.511621			0.255810	+	0.966727i	$0.417658\pi$
$182$		0			0
$183$		0			0
$184$	−6.74456			−0.497216
$185$	5.18614	+	24.1287i	0.381293	+	1.77398i
$186$		0			0
$187$		−	5.04868i		−	0.369196i
$188$			9.10268i			0.663881i
$189$		0			0
$190$	−1.48913	−	6.92820i	−0.108033	−	0.502625i
$191$	13.6277			0.986067			0.493034	−	0.870010i	$-0.335888\pi$
$191$	13.6277			0.986067			0.493034	+	0.870010i	$0.335888\pi$
$192$		0			0
$193$			23.3639i			1.68177i	0.541216	+	0.840883i	$0.317964\pi$
$193$			23.3639i			1.68177i	−0.541216	+	0.840883i	$0.682036\pi$
$194$	−3.25544			−0.233727
$195$		0			0
$196$	−6.86141			−0.490100
$197$			1.87953i			0.133911i	0.997756	+	0.0669554i	$0.0213285\pi$
$197$			1.87953i			0.133911i	−0.997756	+	0.0669554i	$0.978671\pi$
$198$		0			0
$199$	8.00000			0.567105			0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000			0.567105			0.283552	+	0.958957i	$0.408487\pi$
$200$	5.48913	+	12.1793i	0.388140	+	0.861204i
$201$		0			0
$202$		−	4.75372i		−	0.334471i
$203$			9.50744i			0.667292i
$204$		0			0
$205$	6.00000	−	1.28962i	0.419058	−	0.0900710i
$206$	8.23369			0.573668
$207$		0			0
$208$		0			0
$209$	−4.00000			−0.276686
$210$		0			0
$211$	−21.4891			−1.47937			−0.739686	−	0.672952i	$-0.765026\pi$
$211$	−21.4891			−1.47937			−0.739686	+	0.672952i	$0.765026\pi$
$212$			4.34896i			0.298688i
$213$		0			0
$214$	−5.25544			−0.359254
$215$	1.62772	+	7.57301i	0.111009	+	0.516475i
$216$		0			0
$217$			8.21782i			0.557862i
$218$		−	7.92287i		−	0.536604i
$219$		0			0
$220$	3.00000	−	0.644810i	0.202260	−	0.0434731i
$221$		0			0
$222$		0			0
$223$			7.57301i			0.507126i	0.967319	+	0.253563i	$0.0816026\pi$
$223$			7.57301i			0.507126i	−0.967319	+	0.253563i	$0.918397\pi$
$224$	20.2337			1.35192
$225$		0			0
$226$	12.7446			0.847756
$227$		−	9.80240i		−	0.650608i	−0.945609	−	0.325304i	$-0.894533\pi$
$227$		−	9.80240i		−	0.650608i	0.945609	−	0.325304i	$-0.105467\pi$
$228$		0			0
$229$	−20.3723			−1.34624			−0.673119	−	0.739534i	$-0.735046\pi$
$229$	−20.3723			−1.34624			−0.673119	+	0.739534i	$0.735046\pi$
$230$	−4.37228	+	0.939764i	−0.288300	+	0.0619662i
$231$		0			0
$232$		−	7.33296i		−	0.481433i
$233$			17.0256i			1.11538i	0.830049	+	0.557691i	$0.188312\pi$
$233$			17.0256i			1.11538i	−0.830049	+	0.557691i	$0.811688\pi$
$234$		0			0
$235$	3.11684	+	14.5012i	0.203320	+	0.945955i
$236$	−2.23369			−0.145401
$237$		0			0
$238$		−	13.8564i		−	0.898177i
$239$	−3.25544			−0.210577			−0.105288	−	0.994442i	$-0.533577\pi$
$239$	−3.25544			−0.210577			−0.105288	+	0.994442i	$0.533577\pi$
$240$		0			0
$241$	5.25544			0.338532			0.169266	−	0.985570i	$-0.445860\pi$
$241$	5.25544			0.338532			0.169266	+	0.985570i	$0.445860\pi$
$242$			0.792287i			0.0509301i
$243$		0			0
$244$	14.7446			0.943924
$245$	−10.9307	+	2.34941i	−0.698337	+	0.150098i
$246$		0			0
$247$		0			0
$248$		−	6.33830i		−	0.402482i
$249$		0			0
$250$	5.25544	+	7.13058i	0.332383	+	0.450978i
$251$	4.88316			0.308222			0.154111	−	0.988054i	$-0.450749\pi$
$251$	4.88316			0.308222			0.154111	+	0.988054i	$0.450749\pi$
$252$		0			0
$253$			2.52434i			0.158704i
$254$	9.25544			0.580738
$255$		0			0
$256$	−13.8832			−0.867697
$257$			23.9538i			1.49419i	0.664715	+	0.747097i	$0.268553\pi$
$257$			23.9538i			1.49419i	−0.664715	+	0.747097i	$0.731447\pi$
$258$		0			0
$259$	38.2337			2.37573
$260$		0			0
$261$		0			0
$262$		−	6.92820i		−	0.428026i
$263$		−	14.1514i		−	0.872610i	−0.899799	−	0.436305i	$-0.856287\pi$
$263$		−	14.1514i		−	0.872610i	0.899799	−	0.436305i	$-0.143713\pi$
$264$		0			0
$265$	1.48913	+	6.92820i	0.0914762	+	0.425596i
$266$	−10.9783			−0.673120
$267$		0			0
$268$			0.884861i			0.0540515i
$269$	11.4891			0.700504			0.350252	−	0.936655i	$-0.386096\pi$
$269$	11.4891			0.700504			0.350252	+	0.936655i	$0.386096\pi$
$270$		0			0
$271$	−9.48913			−0.576423			−0.288212	−	0.957567i	$-0.593061\pi$
$271$	−9.48913			−0.576423			−0.288212	+	0.957567i	$0.593061\pi$
$272$		−	3.16915i		−	0.192158i
$273$		0			0
$274$	1.76631			0.106707
$275$	4.55842	−	2.05446i	0.274883	−	0.123888i
$276$		0			0
$277$		−	8.21782i		−	0.493761i	−0.969046	−	0.246881i	$-0.920594\pi$
$277$		−	8.21782i		−	0.493761i	0.969046	−	0.246881i	$-0.0794055\pi$
$278$		−	14.4463i		−	0.866432i
$279$		0			0
$280$	20.2337	−	4.34896i	1.20919	−	0.259900i
$281$	23.4891			1.40124			0.700622	−	0.713533i	$-0.252906\pi$
$281$	23.4891			1.40124			0.700622	+	0.713533i	$0.252906\pi$
$282$		0			0
$283$		−	4.75372i		−	0.282579i	−0.989968	−	0.141290i	$-0.954875\pi$
$283$		−	4.75372i		−	0.282579i	0.989968	−	0.141290i	$-0.0451249\pi$
$284$	−9.76631			−0.579524
$285$		0			0
$286$		0			0
$287$		−	9.50744i		−	0.561207i
$288$		0			0
$289$	−8.48913			−0.499360
$290$	−1.02175	−	4.75372i	−0.0599992	−	0.279148i
$291$		0			0
$292$		−	9.50744i		−	0.556381i
$293$			10.0974i			0.589894i	0.955514	+	0.294947i	$0.0953019\pi$
$293$			10.0974i			0.589894i	−0.955514	+	0.294947i	$0.904698\pi$
$294$		0			0
$295$	−3.55842	+	0.764836i	−0.207179	+	0.0445304i
$296$	−29.4891			−1.71402
$297$		0			0
$298$			9.10268i			0.527304i
$299$		0			0
$300$		0			0
$301$	12.0000			0.691669
$302$			17.6155i			1.01366i
$303$		0			0
$304$	−2.51087			−0.144009
$305$	23.4891	−	5.04868i	1.34498	−	0.289086i
$306$		0			0
$307$			28.1176i			1.60475i	0.596817	+	0.802377i	$0.296432\pi$
$307$			28.1176i			1.60475i	−0.596817	+	0.802377i	$0.703568\pi$
$308$		−	4.75372i		−	0.270868i
$309$		0			0
$310$	−0.883156	−	4.10891i	−0.0501599	−	0.233371i
$311$	17.4891			0.991717			0.495859	−	0.868403i	$-0.334853\pi$
$311$	17.4891			0.991717			0.495859	+	0.868403i	$0.334853\pi$
$312$		0			0
$313$		−	31.8217i		−	1.79867i	−0.437260	−	0.899335i	$-0.644051\pi$
$313$		−	31.8217i		−	1.79867i	0.437260	−	0.899335i	$-0.355949\pi$
$314$	4.27719			0.241376
$315$		0			0
$316$	−17.4891			−0.983840
$317$		−	3.51900i		−	0.197647i	−0.995105	−	0.0988235i	$-0.968492\pi$
$317$		−	3.51900i		−	0.197647i	0.995105	−	0.0988235i	$-0.0315079\pi$
$318$		0			0
$319$	−2.74456			−0.153666
$320$	−7.37228	+	1.58457i	−0.412123	+	0.0885804i
$321$		0			0
$322$			6.92820i			0.386094i
$323$			20.1947i			1.12366i
$324$		0			0
$325$		0			0
$326$	2.74456			0.152007
$327$		0			0
$328$			7.33296i			0.404895i
$329$	22.9783			1.26683
$330$		0			0
$331$	3.11684			0.171317			0.0856586	−	0.996325i	$-0.472701\pi$
$331$	3.11684			0.171317			0.0856586	+	0.996325i	$0.472701\pi$
$332$			9.10268i			0.499575i
$333$		0			0
$334$	17.7228			0.969749
$335$	0.302985	+	1.40965i	0.0165538	+	0.0770172i
$336$		0			0
$337$			12.5668i			0.684556i	0.939599	+	0.342278i	$0.111199\pi$
$337$			12.5668i			0.684556i	−0.939599	+	0.342278i	$0.888801\pi$
$338$			10.2997i			0.560232i
$339$		0			0
$340$	−3.25544	−	15.1460i	−0.176551	−	0.821409i
$341$	−2.37228			−0.128466
$342$		0			0
$343$		−	6.92820i		−	0.374088i
$344$	−9.25544			−0.499020
$345$		0			0
$346$	−1.48913			−0.0800559
$347$		−	29.2974i		−	1.57277i	−0.617739	−	0.786383i	$-0.711951\pi$
$347$		−	29.2974i		−	1.57277i	0.617739	−	0.786383i	$-0.288049\pi$
$348$		0			0
$349$	7.48913			0.400884			0.200442	−	0.979706i	$-0.435762\pi$
$349$	7.48913			0.400884			0.200442	+	0.979706i	$0.435762\pi$
$350$	12.5109	−	5.63858i	0.668734	−	0.301395i
$351$		0			0
$352$			5.84096i			0.311324i
$353$		−	21.7244i		−	1.15627i	−0.815941	−	0.578136i	$-0.803780\pi$
$353$		−	21.7244i		−	1.15627i	0.815941	−	0.578136i	$-0.196220\pi$
$354$		0			0
$355$	−15.5584	+	3.34408i	−0.825755	+	0.177485i
$356$	−6.00000			−0.317999
$357$		0			0
$358$		−	12.5668i		−	0.664175i
$359$	6.51087			0.343631			0.171815	−	0.985129i	$-0.445037\pi$
$359$	6.51087			0.343631			0.171815	+	0.985129i	$0.445037\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$			5.45343i			0.286626i
$363$		0			0
$364$		0			0
$365$	−3.25544	−	15.1460i	−0.170397	−	0.792779i
$366$		0			0
$367$			24.0087i			1.25324i	0.779324	+	0.626621i	$0.215563\pi$
$367$			24.0087i			1.25324i	−0.779324	+	0.626621i	$0.784437\pi$
$368$			1.58457i			0.0826016i
$369$		0			0
$370$	−19.1168	+	4.10891i	−0.993837	+	0.213612i
$371$	10.9783			0.569962
$372$		0			0
$373$			8.21782i			0.425503i	0.977106	+	0.212751i	$0.0682424\pi$
$373$			8.21782i			0.425503i	−0.977106	+	0.212751i	$0.931758\pi$
$374$	4.00000			0.206835
$375$		0			0
$376$	−17.7228			−0.913984
$377$		0			0
$378$		0			0
$379$	6.37228			0.327322			0.163661	−	0.986517i	$-0.447670\pi$
$379$	6.37228			0.327322			0.163661	+	0.986517i	$0.447670\pi$
$380$	−12.0000	+	2.57924i	−0.615587	+	0.132312i
$381$		0			0
$382$			10.7971i			0.552426i
$383$		−	5.69349i		−	0.290924i	−0.989364	−	0.145462i	$-0.953533\pi$
$383$		−	5.69349i		−	0.290924i	0.989364	−	0.145462i	$-0.0464668\pi$
$384$		0			0
$385$	−1.62772	−	7.57301i	−0.0829562	−	0.385957i
$386$	−18.5109			−0.942179
$387$		0			0
$388$			5.63858i			0.286256i
$389$	−9.86141			−0.499993			−0.249997	−	0.968247i	$-0.580429\pi$
$389$	−9.86141			−0.499993			−0.249997	+	0.968247i	$0.580429\pi$
$390$		0			0
$391$	12.7446			0.644520
$392$		−	13.3591i		−	0.674735i
$393$		0			0
$394$	−1.48913			−0.0750210
$395$	−27.8614	+	5.98844i	−1.40186	+	0.301311i
$396$		0			0
$397$		−	23.3639i		−	1.17260i	−0.810095	−	0.586299i	$-0.800584\pi$
$397$		−	23.3639i		−	1.17260i	0.810095	−	0.586299i	$-0.199416\pi$
$398$			6.33830i			0.317710i
$399$		0			0
$400$	2.86141	−	1.28962i	0.143070	−	0.0644810i
$401$	−11.4891			−0.573740			−0.286870	−	0.957970i	$-0.592615\pi$
$401$	−11.4891			−0.573740			−0.286870	+	0.957970i	$0.592615\pi$
$402$		0			0
$403$		0			0
$404$	−8.23369			−0.409641
$405$		0			0
$406$	−7.53262			−0.373838
$407$			11.0371i			0.547089i
$408$		0			0
$409$	−4.51087			−0.223048			−0.111524	−	0.993762i	$-0.535573\pi$
$409$	−4.51087			−0.223048			−0.111524	+	0.993762i	$0.535573\pi$
$410$	1.02175	+	4.75372i	0.0504606	+	0.234770i
$411$		0			0
$412$		−	14.2612i		−	0.702597i
$413$			5.63858i			0.277457i
$414$		0			0
$415$	3.11684	+	14.5012i	0.153000	+	0.711837i
$416$		0			0
$417$		0			0
$418$		−	3.16915i		−	0.155008i
$419$	22.9783			1.12256			0.561280	−	0.827626i	$-0.310309\pi$
$419$	22.9783			1.12256			0.561280	+	0.827626i	$0.310309\pi$
$420$		0			0
$421$	31.4891			1.53469			0.767343	−	0.641237i	$-0.221578\pi$
$421$	31.4891			1.53469			0.767343	+	0.641237i	$0.221578\pi$
$422$		−	17.0256i		−	0.828791i
$423$		0			0
$424$	−8.46738			−0.411212
$425$	−10.3723	−	23.0140i	−0.503130	−	1.11634i
$426$		0			0
$427$		−	37.2203i		−	1.80121i
$428$			9.10268i			0.439995i
$429$		0			0
$430$	−6.00000	+	1.28962i	−0.289346	+	0.0621910i
$431$	−31.7228			−1.52803			−0.764017	−	0.645196i	$-0.776776\pi$
$431$	−31.7228			−1.52803			−0.764017	+	0.645196i	$0.776776\pi$
$432$		0			0
$433$		−	20.5446i		−	0.987308i	−0.869658	−	0.493654i	$-0.835661\pi$
$433$		−	20.5446i		−	0.987308i	0.869658	−	0.493654i	$-0.164339\pi$
$434$	−6.51087			−0.312532
$435$		0			0
$436$	−13.7228			−0.657204
$437$		−	10.0974i		−	0.483022i
$438$		0			0
$439$	1.48913			0.0710721			0.0355360	−	0.999368i	$-0.488686\pi$
$439$	1.48913			0.0710721			0.0355360	+	0.999368i	$0.488686\pi$
$440$	1.25544	+	5.84096i	0.0598506	+	0.278457i
$441$		0			0
$442$		0			0
$443$			15.0911i			0.717001i	0.933530	+	0.358500i	$0.116712\pi$
$443$			15.0911i			0.717001i	−0.933530	+	0.358500i	$0.883288\pi$
$444$		0			0
$445$	−9.55842	+	2.05446i	−0.453113	+	0.0973905i
$446$	−6.00000			−0.284108
$447$		0			0
$448$			11.6819i			0.551919i
$449$	−21.8614			−1.03170			−0.515852	−	0.856678i	$-0.672524\pi$
$449$	−21.8614			−1.03170			−0.515852	+	0.856678i	$0.672524\pi$
$450$		0			0
$451$	2.74456			0.129236
$452$		−	22.0742i		−	1.03828i
$453$		0			0
$454$	7.76631			0.364491
$455$		0			0
$456$		0			0
$457$			20.7846i			0.972263i	0.873886	+	0.486132i	$0.161592\pi$
$457$			20.7846i			0.972263i	−0.873886	+	0.486132i	$0.838408\pi$
$458$		−	16.1407i		−	0.754205i
$459$		0			0
$460$	1.62772	+	7.57301i	0.0758928	+	0.353094i
$461$	32.2337			1.50127			0.750636	−	0.660716i	$-0.229747\pi$
$461$	32.2337			1.50127			0.750636	+	0.660716i	$0.229747\pi$
$462$		0			0
$463$		−	20.1398i		−	0.935976i	−0.883735	−	0.467988i	$-0.844979\pi$
$463$		−	20.1398i		−	0.935976i	0.883735	−	0.467988i	$-0.155021\pi$
$464$	−1.72281			−0.0799796
$465$		0			0
$466$	−13.4891			−0.624872
$467$		−	4.40387i		−	0.203787i	−0.994795	−	0.101893i	$-0.967510\pi$
$467$		−	4.40387i		−	0.203787i	0.994795	−	0.101893i	$-0.0324900\pi$
$468$		0			0
$469$	2.23369			0.103142
$470$	−11.4891	+	2.46943i	−0.529954	+	0.113907i
$471$		0			0
$472$		−	4.34896i		−	0.200177i
$473$			3.46410i			0.159280i
$474$		0			0
$475$	−18.2337	+	8.21782i	−0.836619	+	0.377060i
$476$	−24.0000			−1.10004
$477$		0			0
$478$		−	2.57924i		−	0.117972i
$479$	17.4891			0.799099			0.399549	−	0.916712i	$-0.369167\pi$
$479$	17.4891			0.799099			0.399549	+	0.916712i	$0.369167\pi$
$480$		0			0
$481$		0			0
$482$			4.16381i			0.189657i
$483$		0			0
$484$	1.37228			0.0623764
$485$	1.93070	+	8.98266i	0.0876687	+	0.407882i
$486$		0			0
$487$		−	22.7190i		−	1.02950i	−0.857341	−	0.514749i	$-0.827885\pi$
$487$		−	22.7190i		−	1.02950i	0.857341	−	0.514749i	$-0.172115\pi$
$488$			28.7075i			1.29953i
$489$		0			0
$490$	−1.86141	−	8.66025i	−0.0840898	−	0.391230i
$491$	−29.4891			−1.33083			−0.665413	−	0.746476i	$-0.731744\pi$
$491$	−29.4891			−1.33083			−0.665413	+	0.746476i	$0.731744\pi$
$492$		0			0
$493$			13.8564i			0.624061i
$494$		0			0
$495$		0			0
$496$	−1.48913			−0.0668637
$497$			24.6535i			1.10586i
$498$		0			0
$499$	20.0000			0.895323			0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000			0.895323			0.447661	+	0.894203i	$0.352257\pi$
$500$	12.3505	−	9.10268i	0.552333	−	0.407084i
$501$		0			0
$502$			3.86886i			0.172676i
$503$		−	0.294954i		−	0.0131513i	−0.999978	−	0.00657567i	$-0.997907\pi$
$503$		−	0.294954i		−	0.0131513i	0.999978	−	0.00657567i	$-0.00209311\pi$
$504$		0			0
$505$	−13.1168	+	2.81929i	−0.583692	+	0.125457i
$506$	−2.00000			−0.0889108
$507$		0			0
$508$		−	16.0309i		−	0.711256i
$509$	28.3723			1.25758			0.628790	−	0.777575i	$-0.283551\pi$
$509$	28.3723			1.25758			0.628790	+	0.777575i	$0.283551\pi$
$510$		0			0
$511$	−24.0000			−1.06170
$512$			7.02078i			0.310277i
$513$		0			0
$514$	−18.9783			−0.837095
$515$	−4.88316	−	22.7190i	−0.215178	−	1.00112i
$516$		0			0
$517$			6.63325i			0.291730i
$518$			30.2921i			1.33096i
$519$		0			0
$520$		0			0
$521$	−18.6060			−0.815142			−0.407571	−	0.913173i	$-0.633624\pi$
$521$	−18.6060			−0.815142			−0.407571	+	0.913173i	$0.633624\pi$
$522$		0			0
$523$			9.10268i			0.398033i	0.979996	+	0.199016i	$0.0637747\pi$
$523$			9.10268i			0.398033i	−0.979996	+	0.199016i	$0.936225\pi$
$524$	−12.0000			−0.524222
$525$		0			0
$526$	11.2119			0.488864
$527$			11.9769i			0.521721i
$528$		0			0
$529$	16.6277			0.722944
$530$	−5.48913	+	1.17981i	−0.238432	+	0.0512479i
$531$		0			0
$532$			19.0149i			0.824400i
$533$		0			0
$534$		0			0
$535$	3.11684	+	14.5012i	0.134753	+	0.626942i
$536$	−1.72281			−0.0744142
$537$		0			0
$538$			9.10268i			0.392445i
$539$	−5.00000			−0.215365
$540$		0			0
$541$	−0.233688			−0.0100470			−0.00502351	−	0.999987i	$-0.501599\pi$
$541$	−0.233688			−0.0100470			−0.00502351	+	0.999987i	$0.501599\pi$
$542$		−	7.51811i		−	0.322930i
$543$		0			0
$544$	29.4891			1.26434
$545$	−21.8614	+	4.69882i	−0.936440	+	0.201275i
$546$		0			0
$547$			9.10268i			0.389203i	0.980882	+	0.194601i	$0.0623413\pi$
$547$			9.10268i			0.389203i	−0.980882	+	0.194601i	$0.937659\pi$
$548$		−	3.05934i		−	0.130689i
$549$		0			0
$550$	1.62772	+	3.61158i	0.0694062	+	0.153998i
$551$	10.9783			0.467689
$552$		0			0
$553$			44.1485i			1.87738i
$554$	6.51087			0.276621
$555$		0			0
$556$	−25.0217			−1.06116
$557$			32.1716i			1.36315i	0.731747	+	0.681577i	$0.238705\pi$
$557$			32.1716i			1.36315i	−0.731747	+	0.681577i	$0.761295\pi$
$558$		0			0
$559$		0			0
$560$	−1.02175	−	4.75372i	−0.0431768	−	0.200881i
$561$		0			0
$562$			18.6101i			0.785021i
$563$			12.2718i			0.517196i	0.965985	+	0.258598i	$0.0832605\pi$
$563$			12.2718i			0.517196i	−0.965985	+	0.258598i	$0.916739\pi$
$564$		0			0
$565$	−7.55842	−	35.1658i	−0.317985	−	1.47944i
$566$	3.76631			0.158310
$567$		0			0
$568$		−	19.0149i		−	0.797847i
$569$	−38.7446			−1.62426			−0.812128	−	0.583479i	$-0.801691\pi$
$569$	−38.7446			−1.62426			−0.812128	+	0.583479i	$0.801691\pi$
$570$		0			0
$571$	−21.4891			−0.899292			−0.449646	−	0.893207i	$-0.648450\pi$
$571$	−21.4891			−0.899292			−0.449646	+	0.893207i	$0.648450\pi$
$572$		0			0
$573$		0			0
$574$	7.53262			0.314406
$575$	5.18614	+	11.5070i	0.216277	+	0.479875i
$576$		0			0
$577$		−	31.8217i		−	1.32476i	−0.749170	−	0.662378i	$-0.769547\pi$
$577$		−	31.8217i		−	1.32476i	0.749170	−	0.662378i	$-0.230453\pi$
$578$		−	6.72582i		−	0.279757i
$579$		0			0
$580$	−8.23369	+	1.76972i	−0.341885	+	0.0734837i
$581$	22.9783			0.953298
$582$		0			0
$583$			3.16915i			0.131253i
$584$	18.5109			0.765985
$585$		0			0
$586$	−8.00000			−0.330477
$587$		−	28.0078i		−	1.15600i	−0.816035	−	0.578002i	$-0.803833\pi$
$587$		−	28.0078i		−	1.15600i	0.816035	−	0.578002i	$-0.196167\pi$
$588$		0			0
$589$	9.48913			0.390993
$590$	−0.605969	−	2.81929i	−0.0249474	−	0.116068i
$591$		0			0
$592$			6.92820i			0.284747i
$593$		−	43.5586i		−	1.78874i	−0.447333	−	0.894368i	$-0.647626\pi$
$593$		−	43.5586i		−	1.78874i	0.447333	−	0.894368i	$-0.352374\pi$
$594$		0			0
$595$	−38.2337	+	8.21782i	−1.56743	+	0.336898i
$596$	15.7663			0.645813
$597$		0			0
$598$		0			0
$599$	−34.9783			−1.42917			−0.714586	−	0.699547i	$-0.753385\pi$
$599$	−34.9783			−1.42917			−0.714586	+	0.699547i	$0.753385\pi$
$600$		0			0
$601$	30.4674			1.24279			0.621395	−	0.783497i	$-0.286566\pi$
$601$	30.4674			1.24279			0.621395	+	0.783497i	$0.286566\pi$
$602$			9.50744i			0.387494i
$603$		0			0
$604$	30.5109			1.24147
$605$	2.18614	−	0.469882i	0.0888793	−	0.0191034i
$606$		0			0
$607$		−	3.46410i		−	0.140604i	−0.997526	−	0.0703018i	$-0.977604\pi$
$607$		−	3.46410i		−	0.140604i	0.997526	−	0.0703018i	$-0.0223962\pi$
$608$		−	23.3639i		−	0.947529i
$609$		0			0
$610$	4.00000	+	18.6101i	0.161955	+	0.753502i
$611$		0			0
$612$		0			0
$613$		−	44.1485i		−	1.78314i	−0.452883	−	0.891570i	$-0.649605\pi$
$613$		−	44.1485i		−	1.78314i	0.452883	−	0.891570i	$-0.350395\pi$
$614$	−22.2772			−0.899034
$615$		0			0
$616$	9.25544			0.372912
$617$		−	3.75906i		−	0.151334i	−0.997133	−	0.0756669i	$-0.975891\pi$
$617$		−	3.75906i		−	0.151334i	0.997133	−	0.0756669i	$-0.0241086\pi$
$618$		0			0
$619$	3.11684			0.125277			0.0626383	−	0.998036i	$-0.480049\pi$
$619$	3.11684			0.125277			0.0626383	+	0.998036i	$0.480049\pi$
$620$	−7.11684	+	1.52967i	−0.285819	+	0.0614331i
$621$		0			0
$622$			13.8564i			0.555591i
$623$			15.1460i			0.606813i
$624$		0			0
$625$	16.5584	−	18.7302i	0.662337	−	0.749206i
$626$	25.2119			1.00767
$627$		0			0
$628$		−	7.40830i		−	0.295624i
$629$	55.7228			2.22181
$630$		0			0
$631$	−16.6060			−0.661073			−0.330537	−	0.943793i	$-0.607230\pi$
$631$	−16.6060			−0.661073			−0.330537	+	0.943793i	$0.607230\pi$
$632$		−	34.0511i		−	1.35448i
$633$		0			0
$634$	2.78806			0.110728
$635$	−5.48913	−	25.5383i	−0.217829	−	1.01346i
$636$		0			0
$637$		0			0
$638$		−	2.17448i		−	0.0860885i
$639$		0			0
$640$	4.23369	+	19.6974i	0.167351	+	0.778607i
$641$	19.6277			0.775248			0.387624	−	0.921818i	$-0.373296\pi$
$641$	19.6277			0.775248			0.387624	+	0.921818i	$0.373296\pi$
$642$		0			0
$643$			39.1547i			1.54411i	0.635556	+	0.772055i	$0.280771\pi$
$643$			39.1547i			1.54411i	−0.635556	+	0.772055i	$0.719229\pi$
$644$	12.0000			0.472866
$645$		0			0
$646$	−16.0000			−0.629512
$647$			41.0342i			1.61322i	0.591083	+	0.806611i	$0.298701\pi$
$647$			41.0342i			1.61322i	−0.591083	+	0.806611i	$0.701299\pi$
$648$		0			0
$649$	−1.62772			−0.0638935
$650$		0			0
$651$		0			0
$652$		−	4.75372i		−	0.186170i
$653$		−	25.5932i		−	1.00154i	−0.865580	−	0.500770i	$-0.833050\pi$
$653$		−	25.5932i		−	1.00154i	0.865580	−	0.500770i	$-0.166950\pi$
$654$		0			0
$655$	−19.1168	+	4.10891i	−0.746957	+	0.160548i
$656$	1.72281			0.0672646
$657$		0			0
$658$			18.2054i			0.709719i
$659$	32.7446			1.27555			0.637774	−	0.770224i	$-0.279856\pi$
$659$	32.7446			1.27555			0.637774	+	0.770224i	$0.279856\pi$
$660$		0			0
$661$	35.3505			1.37498			0.687488	−	0.726196i	$-0.258713\pi$
$661$	35.3505			1.37498			0.687488	+	0.726196i	$0.258713\pi$
$662$			2.46943i			0.0959773i
$663$		0			0
$664$	−17.7228			−0.687779
$665$	6.51087	+	30.2921i	0.252481	+	1.17468i
$666$		0			0
$667$		−	6.92820i		−	0.268261i
$668$		−	30.6968i		−	1.18770i
$669$		0			0
$670$	−1.11684	+	0.240051i	−0.0431474	+	0.00927397i
$671$	10.7446			0.414789
$672$		0			0
$673$		−	1.28962i		−	0.0497112i	−0.999691	−	0.0248556i	$-0.992087\pi$
$673$		−	1.28962i		−	0.0497112i	0.999691	−	0.0248556i	$-0.00791260\pi$
$674$	−9.95650			−0.383510
$675$		0			0
$676$	17.8397			0.686141
$677$			43.4487i			1.66987i	0.550348	+	0.834935i	$0.314495\pi$
$677$			43.4487i			1.66987i	−0.550348	+	0.834935i	$0.685505\pi$
$678$		0			0
$679$	14.2337			0.546239
$680$	29.4891	−	6.33830i	1.13086	−	0.243063i
$681$		0			0
$682$		−	1.87953i		−	0.0719708i
$683$		−	44.4434i		−	1.70058i	−0.526314	−	0.850290i	$-0.676427\pi$
$683$		−	44.4434i		−	1.70058i	0.526314	−	0.850290i	$-0.323573\pi$
$684$		0			0
$685$	−1.04755	−	4.87375i	−0.0400247	−	0.186216i
$686$	5.48913			0.209576
$687$		0			0
$688$			2.17448i			0.0829013i
$689$		0			0
$690$		0			0
$691$	16.1386			0.613941			0.306971	−	0.951719i	$-0.400685\pi$
$691$	16.1386			0.613941			0.306971	+	0.951719i	$0.400685\pi$
$692$			2.57924i			0.0980480i
$693$		0			0
$694$	23.2119			0.881113
$695$	−39.8614	+	8.56768i	−1.51203	+	0.324991i
$696$		0			0
$697$		−	13.8564i		−	0.524849i
$698$			5.93354i			0.224588i
$699$		0			0
$700$	−9.76631	−	21.6695i	−0.369132	−	0.819029i
$701$	35.4891			1.34041			0.670203	−	0.742178i	$-0.266207\pi$
$701$	35.4891			1.34041			0.670203	+	0.742178i	$0.266207\pi$
$702$		0			0
$703$		−	44.1485i		−	1.66509i
$704$	−3.37228			−0.127098
$705$		0			0
$706$	17.2119			0.647780
$707$			20.7846i			0.781686i
$708$		0			0
$709$	−41.1168			−1.54418			−0.772088	−	0.635516i	$-0.780787\pi$
$709$	−41.1168			−1.54418			−0.772088	+	0.635516i	$0.780787\pi$
$710$	−2.64947	−	12.3267i	−0.0994328	−	0.462614i
$711$		0			0
$712$		−	11.6819i		−	0.437799i
$713$		−	5.98844i		−	0.224269i
$714$		0			0
$715$		0			0
$716$	−21.7663			−0.813445
$717$		0			0
$718$			5.15848i			0.192513i
$719$	21.3505			0.796240			0.398120	−	0.917333i	$-0.369663\pi$
$719$	21.3505			0.796240			0.398120	+	0.917333i	$0.369663\pi$
$720$		0			0
$721$	−36.0000			−1.34071
$722$		−	2.37686i		−	0.0884576i
$723$		0			0
$724$	9.44563			0.351044
$725$	−12.5109	+	5.63858i	−0.464642	+	0.209412i
$726$		0			0
$727$			15.7908i			0.585650i	0.956166	+	0.292825i	$0.0945953\pi$
$727$			15.7908i			0.585650i	−0.956166	+	0.292825i	$0.905405\pi$
$728$		0			0
$729$		0			0
$730$	12.0000	−	2.57924i	0.444140	−	0.0954620i
$731$	17.4891			0.646859
$732$		0			0
$733$			30.2921i			1.11886i	0.828877	+	0.559431i	$0.188980\pi$
$733$			30.2921i			1.11886i	−0.828877	+	0.559431i	$0.811020\pi$
$734$	−19.0217			−0.702106
$735$		0			0
$736$	−14.7446			−0.543492
$737$			0.644810i			0.0237519i
$738$		0			0
$739$	−0.744563			−0.0273892			−0.0136946	−	0.999906i	$-0.504359\pi$
$739$	−0.744563			−0.0273892			−0.0136946	+	0.999906i	$0.504359\pi$
$740$	7.11684	+	33.1113i	0.261620	+	1.21720i
$741$		0			0
$742$			8.69793i			0.319311i
$743$			21.7793i			0.799004i	0.916732	+	0.399502i	$0.130817\pi$
$743$			21.7793i			0.799004i	−0.916732	+	0.399502i	$0.869183\pi$
$744$		0			0
$745$	25.1168	−	5.39853i	0.920210	−	0.197787i
$746$	−6.51087			−0.238380
$747$		0			0
$748$		−	6.92820i		−	0.253320i
$749$	22.9783			0.839607
$750$		0			0
$751$	21.6277			0.789207			0.394603	−	0.918852i	$-0.370882\pi$
$751$	21.6277			0.789207			0.394603	+	0.918852i	$0.370882\pi$
$752$			4.16381i			0.151839i
$753$		0			0
$754$		0			0
$755$	48.6060	−	10.4472i	1.76895	−	0.380213i
$756$		0			0
$757$			39.7995i			1.44654i	0.690567	+	0.723269i	$0.257361\pi$
$757$			39.7995i			1.44654i	−0.690567	+	0.723269i	$0.742639\pi$
$758$			5.04868i			0.183376i
$759$		0			0
$760$	−5.02175	−	23.3639i	−0.182158	−	0.847496i
$761$	−21.2554			−0.770509			−0.385255	−	0.922810i	$-0.625886\pi$
$761$	−21.2554			−0.770509			−0.385255	+	0.922810i	$0.625886\pi$
$762$		0			0
$763$			34.6410i			1.25409i
$764$	18.7011			0.676581
$765$		0			0
$766$	4.51087			0.162985
$767$		0			0
$768$		0			0
$769$	51.2119			1.84675			0.923375	−	0.383900i	$-0.125419\pi$
$769$	51.2119			1.84675			0.923375	+	0.383900i	$0.125419\pi$
$770$	6.00000	−	1.28962i	0.216225	−	0.0464747i
$771$		0			0
$772$			32.0618i			1.15393i
$773$			30.8820i			1.11075i	0.831601	+	0.555373i	$0.187425\pi$
$773$			30.8820i			1.11075i	−0.831601	+	0.555373i	$0.812575\pi$
$774$		0			0
$775$	−10.8139	+	4.87375i	−0.388445	+	0.175070i
$776$	−10.9783			−0.394096
$777$		0			0
$778$		−	7.81306i		−	0.280112i
$779$	−10.9783			−0.393337
$780$		0			0
$781$	−7.11684			−0.254661
$782$			10.0974i			0.361081i
$783$		0			0
$784$	−3.13859			−0.112093
$785$	−2.53667	−	11.8020i	−0.0905377	−	0.421230i
$786$		0			0
$787$		−	4.75372i		−	0.169452i	−0.996404	−	0.0847259i	$-0.972999\pi$
$787$		−	4.75372i		−	0.169452i	0.996404	−	0.0847259i	$-0.0270015\pi$
$788$			2.57924i			0.0918816i
$789$		0			0
$790$	−4.74456	−	22.0742i	−0.168804	−	0.785366i
$791$	−55.7228			−1.98128
$792$		0			0
$793$		0			0
$794$	18.5109			0.656926
$795$		0			0
$796$	10.9783			0.389114
$797$		−	31.2318i		−	1.10629i	−0.833086	−	0.553144i	$-0.813428\pi$
$797$		−	31.2318i		−	1.10629i	0.833086	−	0.553144i	$-0.186572\pi$
$798$		0			0
$799$	33.4891			1.18476
$800$	12.0000	+	26.6256i	0.424264	+	0.941356i
$801$		0			0
$802$		−	9.10268i		−	0.321427i
$803$		−	6.92820i		−	0.244491i
$804$		0			0
$805$	19.1168	−	4.10891i	0.673780	−	0.144820i
$806$		0			0
$807$		0			0
$808$		−	16.0309i		−	0.563965i
$809$	−21.2554			−0.747301			−0.373651	−	0.927569i	$-0.621894\pi$
$809$	−21.2554			−0.747301			−0.373651	+	0.927569i	$0.621894\pi$
$810$		0			0
$811$	34.2337			1.20211			0.601054	−	0.799209i	$-0.294748\pi$
$811$	34.2337			1.20211			0.601054	+	0.799209i	$0.294748\pi$
$812$			13.0469i			0.457856i
$813$		0			0
$814$	−8.74456			−0.306497
$815$	−1.62772	−	7.57301i	−0.0570165	−	0.265271i
$816$		0			0
$817$		−	13.8564i		−	0.484774i
$818$		−	3.57391i		−	0.124959i
$819$		0			0
$820$	8.23369	−	1.76972i	0.287533	−	0.0618014i
$821$	18.0000			0.628204			0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000			0.628204			0.314102	+	0.949389i	$0.398297\pi$
$822$		0			0
$823$		−	33.5161i		−	1.16830i	−0.811646	−	0.584149i	$-0.801428\pi$
$823$		−	33.5161i		−	1.16830i	0.811646	−	0.584149i	$-0.198572\pi$
$824$	27.7663			0.967285
$825$		0			0
$826$	−4.46738			−0.155440
$827$		−	18.0202i		−	0.626624i	−0.949650	−	0.313312i	$-0.898561\pi$
$827$		−	18.0202i		−	0.626624i	0.949650	−	0.313312i	$-0.101439\pi$
$828$		0			0
$829$	−31.3505			−1.08885			−0.544424	−	0.838810i	$-0.683252\pi$
$829$	−31.3505			−1.08885			−0.544424	+	0.838810i	$0.683252\pi$
$830$	−11.4891	+	2.46943i	−0.398793	+	0.0857153i
$831$		0			0
$832$		0			0
$833$			25.2434i			0.874631i
$834$		0			0
$835$	−10.5109	−	48.9022i	−0.363744	−	1.69233i
$836$	−5.48913			−0.189845
$837$		0			0
$838$			18.2054i			0.628894i
$839$	7.11684			0.245701			0.122850	−	0.992425i	$-0.460796\pi$
$839$	7.11684			0.245701			0.122850	+	0.992425i	$0.460796\pi$
$840$		0			0
$841$	−21.4674			−0.740254
$842$			24.9484i			0.859779i
$843$		0			0
$844$	−29.4891			−1.01506
$845$	28.4198	−	6.10846i	0.977672	−	0.210138i
$846$		0			0
$847$		−	3.46410i		−	0.119028i
$848$			1.98933i			0.0683140i
$849$		0			0
$850$	18.2337	−	8.21782i	0.625410	−	0.281869i
$851$	−27.8614			−0.955077
$852$		0			0
$853$		−	24.6535i		−	0.844119i	−0.906568	−	0.422059i	$-0.861307\pi$
$853$		−	24.6535i		−	0.844119i	0.906568	−	0.422059i	$-0.138693\pi$
$854$	29.4891			1.00910
$855$		0			0
$856$	−17.7228			−0.605753
$857$		−	10.6873i		−	0.365070i	−0.983199	−	0.182535i	$-0.941570\pi$
$857$		−	10.6873i		−	0.365070i	0.983199	−	0.182535i	$-0.0584303\pi$
$858$		0			0
$859$	−11.1168			−0.379302			−0.189651	−	0.981852i	$-0.560736\pi$
$859$	−11.1168			−0.379302			−0.189651	+	0.981852i	$0.560736\pi$
$860$	2.23369	+	10.3923i	0.0761681	+	0.354375i
$861$		0			0
$862$		−	25.1336i		−	0.856053i
$863$		−	23.6588i		−	0.805355i	−0.915342	−	0.402678i	$-0.868080\pi$
$863$		−	23.6588i		−	0.805355i	0.915342	−	0.402678i	$-0.131920\pi$
$864$		0			0
$865$	0.883156	+	4.10891i	0.0300282	+	0.139707i
$866$	16.2772			0.553121
$867$		0			0
$868$			11.2772i			0.382772i
$869$	−12.7446			−0.432330
$870$		0			0
$871$		0			0
$872$		−	26.7181i		−	0.904791i
$873$		0			0
$874$	8.00000			0.270604
$875$	−22.9783	−	31.1769i	−0.776807	−	1.05397i
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$			1.17981i			0.0398168i
$879$		0			0
$880$	1.37228	−	0.294954i	0.0462596	−	0.00994289i
$881$	−21.8614			−0.736530			−0.368265	−	0.929721i	$-0.620048\pi$
$881$	−21.8614			−0.736530			−0.368265	+	0.929721i	$0.620048\pi$
$882$		0			0
$883$			24.2487i			0.816034i	0.912974	+	0.408017i	$0.133780\pi$
$883$			24.2487i			0.816034i	−0.912974	+	0.408017i	$0.866220\pi$
$884$		0			0
$885$		0			0
$886$	−11.9565			−0.401687
$887$		−	14.1514i		−	0.475156i	−0.971368	−	0.237578i	$-0.923646\pi$
$887$		−	14.1514i		−	0.475156i	0.971368	−	0.237578i	$-0.0763536\pi$
$888$		0			0
$889$	−40.4674			−1.35723
$890$	−1.62772	−	7.57301i	−0.0545613	−	0.253848i
$891$		0			0
$892$			10.3923i			0.347960i
$893$		−	26.5330i		−	0.887893i
$894$		0			0
$895$	−34.6753	+	7.45299i	−1.15907	+	0.249126i
$896$	31.2119			1.04272
$897$		0			0
$898$		−	17.3205i		−	0.577993i
$899$	6.51087			0.217150
$900$		0			0
$901$	16.0000			0.533037
$902$			2.17448i			0.0724023i
$903$		0			0
$904$	42.9783			1.42944
$905$	15.0475	−	3.23427i	0.500197	−	0.107511i
$906$		0			0
$907$		−	19.8997i		−	0.660760i	−0.943848	−	0.330380i	$-0.892823\pi$
$907$		−	19.8997i		−	0.660760i	0.943848	−	0.330380i	$-0.107177\pi$
$908$		−	13.4516i		−	0.446409i
$909$		0			0
$910$		0			0
$911$	30.5109			1.01087			0.505435	−	0.862865i	$-0.331332\pi$
$911$	30.5109			1.01087			0.505435	+	0.862865i	$0.331332\pi$
$912$		0			0
$913$			6.63325i			0.219529i
$914$	−16.4674			−0.544692
$915$		0			0
$916$	−27.9565			−0.923709
$917$			30.2921i			1.00033i
$918$		0			0
$919$	−6.23369			−0.205630			−0.102815	−	0.994700i	$-0.532785\pi$
$919$	−6.23369			−0.205630			−0.102815	+	0.994700i	$0.532785\pi$
$920$	−14.7446	+	3.16915i	−0.486114	+	0.104484i
$921$		0			0
$922$			25.5383i			0.841060i
$923$		0			0
$924$		0			0
$925$	22.6753	+	50.3118i	0.745558	+	1.65424i
$926$	15.9565			0.524363
$927$		0			0
$928$		−	16.0309i		−	0.526240i
$929$	−52.9783			−1.73816			−0.869080	−	0.494672i	$-0.835288\pi$
$929$	−52.9783			−1.73816			−0.869080	+	0.494672i	$0.835288\pi$
$930$		0			0
$931$	20.0000			0.655474
$932$			23.3639i			0.765308i
$933$		0			0
$934$	3.48913			0.114168
$935$	−2.37228	−	11.0371i	−0.0775819	−	0.360952i
$936$		0			0
$937$			25.9431i			0.847524i	0.905774	+	0.423762i	$0.139291\pi$
$937$			25.9431i			0.847524i	−0.905774	+	0.423762i	$0.860709\pi$
$938$			1.76972i			0.0577835i
$939$		0			0
$940$	4.27719	+	19.8997i	0.139506	+	0.649058i
$941$	−10.4674			−0.341227			−0.170613	−	0.985338i	$-0.554575\pi$
$941$	−10.4674			−0.341227			−0.170613	+	0.985338i	$0.554575\pi$
$942$		0			0
$943$			6.92820i			0.225613i
$944$	−1.02175			−0.0332551
$945$		0			0
$946$	−2.74456			−0.0892334
$947$			56.1802i			1.82561i	0.408393	+	0.912806i	$0.366089\pi$
$947$			56.1802i			1.82561i	−0.408393	+	0.912806i	$0.633911\pi$
$948$		0			0
$949$		0			0
$950$	−6.51087	−	14.4463i	−0.211241	−	0.468700i
$951$		0			0
$952$		−	46.7277i		−	1.51445i
$953$			41.6790i			1.35012i	0.737765	+	0.675058i	$0.235881\pi$
$953$			41.6790i			1.35012i	−0.737765	+	0.675058i	$0.764119\pi$
$954$		0			0
$955$	29.7921	−	6.40342i	0.964050	−	0.207210i
$956$	−4.46738			−0.144485
$957$		0			0
$958$			13.8564i			0.447680i
$959$	−7.72281			−0.249383
$960$		0			0
$961$	−25.3723			−0.818461
$962$		0			0
$963$		0			0
$964$	7.21194			0.232281
$965$	10.9783	+	51.0767i	0.353402	+	1.64422i
$966$		0			0
$967$			46.3229i			1.48965i	0.667262	+	0.744823i	$0.267466\pi$
$967$			46.3229i			1.48965i	−0.667262	+	0.744823i	$0.732534\pi$
$968$			2.67181i			0.0858754i
$969$		0			0
$970$	−7.11684	+	1.52967i	−0.228508	+	0.0491148i
$971$	−54.0951			−1.73599			−0.867997	−	0.496569i	$-0.834593\pi$
$971$	−54.0951			−1.73599			−0.867997	+	0.496569i	$0.834593\pi$
$972$		0			0
$973$			63.1633i			2.02492i
$974$	18.0000			0.576757
$975$		0			0
$976$	6.74456			0.215888
$977$		−	27.3630i		−	0.875419i	−0.899117	−	0.437709i	$-0.855790\pi$
$977$		−	27.3630i		−	0.875419i	0.899117	−	0.437709i	$-0.144210\pi$
$978$		0			0
$979$	−4.37228			−0.139739
$980$	−15.0000	+	3.22405i	−0.479157	+	0.102989i
$981$		0			0
$982$		−	23.3639i		−	0.745570i
$983$			8.16292i			0.260357i	0.991491	+	0.130178i	$0.0415550\pi$
$983$			8.16292i			0.260357i	−0.991491	+	0.130178i	$0.958445\pi$
$984$		0			0
$985$	0.883156	+	4.10891i	0.0281397	+	0.130921i
$986$	−10.9783			−0.349619
$987$		0			0
$988$		0			0
$989$	−8.74456			−0.278061
$990$		0			0
$991$	−26.9783			−0.856992			−0.428496	−	0.903544i	$-0.640956\pi$
$991$	−26.9783			−0.856992			−0.428496	+	0.903544i	$0.640956\pi$
$992$		−	13.8564i		−	0.439941i
$993$		0			0
$994$	−19.5326			−0.619537
$995$	17.4891	−	3.75906i	0.554443	−	0.119170i
$996$		0			0
$997$		−	22.0742i		−	0.699098i	−0.936918	−	0.349549i	$-0.886335\pi$
$997$		−	22.0742i		−	0.699098i	0.936918	−	0.349549i	$-0.113665\pi$
$998$			15.8457i			0.501588i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.2.c.a.199.3		4
3.2	odd	2		55.2.b.a.34.2	✓	4
5.2	odd	4		2475.2.a.bi.1.2		4
5.3	odd	4		2475.2.a.bi.1.3		4
5.4	even	2	inner	495.2.c.a.199.2		4
12.11	even	2		880.2.b.h.529.1		4
15.2	even	4		275.2.a.h.1.3		4
15.8	even	4		275.2.a.h.1.2		4
15.14	odd	2		55.2.b.a.34.3	yes	4
33.2	even	10		605.2.j.j.444.3		16
33.5	odd	10		605.2.j.i.124.3		16
33.8	even	10		605.2.j.j.9.3		16
33.14	odd	10		605.2.j.i.9.2		16
33.17	even	10		605.2.j.j.124.2		16
33.20	odd	10		605.2.j.i.444.2		16
33.26	odd	10		605.2.j.i.269.3		16
33.29	even	10		605.2.j.j.269.2		16
33.32	even	2		605.2.b.c.364.3		4
60.23	odd	4		4400.2.a.cc.1.4		4
60.47	odd	4		4400.2.a.cc.1.1		4
60.59	even	2		880.2.b.h.529.4		4
165.14	odd	10		605.2.j.i.9.3		16
165.29	even	10		605.2.j.j.269.3		16
165.32	odd	4		3025.2.a.ba.1.2		4
165.59	odd	10		605.2.j.i.269.2		16
165.74	even	10		605.2.j.j.9.2		16
165.98	odd	4		3025.2.a.ba.1.3		4
165.104	odd	10		605.2.j.i.124.2		16
165.119	odd	10		605.2.j.i.444.3		16
165.134	even	10		605.2.j.j.444.2		16
165.149	even	10		605.2.j.j.124.3		16
165.164	even	2		605.2.b.c.364.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.b.a.34.2	✓	4	3.2	odd	2
55.2.b.a.34.3	yes	4	15.14	odd	2
275.2.a.h.1.2		4	15.8	even	4
275.2.a.h.1.3		4	15.2	even	4
495.2.c.a.199.2		4	5.4	even	2	inner
495.2.c.a.199.3		4	1.1	even	1	trivial
605.2.b.c.364.2		4	165.164	even	2
605.2.b.c.364.3		4	33.32	even	2
605.2.j.i.9.2		16	33.14	odd	10
605.2.j.i.9.3		16	165.14	odd	10
605.2.j.i.124.2		16	165.104	odd	10
605.2.j.i.124.3		16	33.5	odd	10
605.2.j.i.269.2		16	165.59	odd	10
605.2.j.i.269.3		16	33.26	odd	10
605.2.j.i.444.2		16	33.20	odd	10
605.2.j.i.444.3		16	165.119	odd	10
605.2.j.j.9.2		16	165.74	even	10
605.2.j.j.9.3		16	33.8	even	10
605.2.j.j.124.2		16	33.17	even	10
605.2.j.j.124.3		16	165.149	even	10
605.2.j.j.269.2		16	33.29	even	10
605.2.j.j.269.3		16	165.29	even	10
605.2.j.j.444.2		16	165.134	even	10
605.2.j.j.444.3		16	33.2	even	10
880.2.b.h.529.1		4	12.11	even	2
880.2.b.h.529.4		4	60.59	even	2
2475.2.a.bi.1.2		4	5.2	odd	4
2475.2.a.bi.1.3		4	5.3	odd	4
3025.2.a.ba.1.2		4	165.32	odd	4
3025.2.a.ba.1.3		4	165.98	odd	4
4400.2.a.cc.1.1		4	60.47	odd	4
4400.2.a.cc.1.4		4	60.23	odd	4

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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	495.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.792287i q^{2} +1.37228 q^{4} +(2.18614 - 0.469882i) q^{5} -3.46410i q^{7} +2.67181i q^{8} +(0.372281 + 1.73205i) q^{10} +1.00000 q^{11} +2.74456 q^{14} +0.627719 q^{16} -5.04868i q^{17} -4.00000 q^{19} +(3.00000 - 0.644810i) q^{20} +0.792287i q^{22} +2.52434i q^{23} +(4.55842 - 2.05446i) q^{25} -4.75372i q^{28} -2.74456 q^{29} -2.37228 q^{31} +5.84096i q^{32} +4.00000 q^{34} +(-1.62772 - 7.57301i) q^{35} +11.0371i q^{37} -3.16915i q^{38} +(1.25544 + 5.84096i) q^{40} +2.74456 q^{41} +3.46410i q^{43} +1.37228 q^{44} -2.00000 q^{46} +6.63325i q^{47} -5.00000 q^{49} +(1.62772 + 3.61158i) q^{50} +3.16915i q^{53} +(2.18614 - 0.469882i) q^{55} +9.25544 q^{56} -2.17448i q^{58} -1.62772 q^{59} +10.7446 q^{61} -1.87953i q^{62} -3.37228 q^{64} +0.644810i q^{67} -6.92820i q^{68} +(6.00000 - 1.28962i) q^{70} -7.11684 q^{71} -6.92820i q^{73} -8.74456 q^{74} -5.48913 q^{76} -3.46410i q^{77} -12.7446 q^{79} +(1.37228 - 0.294954i) q^{80} +2.17448i q^{82} +6.63325i q^{83} +(-2.37228 - 11.0371i) q^{85} -2.74456 q^{86} +2.67181i q^{88} -4.37228 q^{89} +3.46410i q^{92} -5.25544 q^{94} +(-8.74456 + 1.87953i) q^{95} +4.10891i q^{97} -3.96143i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{4} + 3 q^{5} - 10 q^{10} + 4 q^{11} - 12 q^{14} + 14 q^{16} - 16 q^{19} + 12 q^{20} + q^{25} + 12 q^{29} + 2 q^{31} + 16 q^{34} - 18 q^{35} + 28 q^{40} - 12 q^{41} - 6 q^{44} - 8 q^{46} - 20 q^{49}+ \cdots - 12 q^{95}+O(q^{100}) \) 4 * q - 6 * q^4 + 3 * q^5 - 10 * q^10 + 4 * q^11 - 12 * q^14 + 14 * q^16 - 16 * q^19 + 12 * q^20 + q^25 + 12 * q^29 + 2 * q^31 + 16 * q^34 - 18 * q^35 + 28 * q^40 - 12 * q^41 - 6 * q^44 - 8 * q^46 - 20 * q^49 + 18 * q^50 + 3 * q^55 + 60 * q^56 - 18 * q^59 + 20 * q^61 - 2 * q^64 + 24 * q^70 + 6 * q^71 - 12 * q^74 + 24 * q^76 - 28 * q^79 - 6 * q^80 + 2 * q^85 + 12 * q^86 - 6 * q^89 - 44 * q^94 - 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more