LMFDB - Embedded newform 405.2.j.b.379.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,2,Mod(109,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("405.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	405.j (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$3.23394128186$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
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:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			379.2
Root			$-1.18614 + 1.26217i$ of defining polynomial
Character	$\chi$	$=$	405.379
Dual form			405.2.j.b.109.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.686141 + 0.396143i) q^{2} +(-0.686141 - 1.18843i) q^{4} +(-2.18614 + 0.469882i) q^{5} +(3.00000 + 1.73205i) q^{7} -2.67181i q^{8} +O(q^{10})$	\(q+(0.686141 + 0.396143i) q^{2} +(-0.686141 - 1.18843i) q^{4} +(-2.18614 + 0.469882i) q^{5} +(3.00000 + 1.73205i) q^{7} -2.67181i q^{8} +(-1.68614 - 0.543620i) q^{10} +(2.18614 - 3.78651i) q^{11} +(5.05842 - 2.92048i) q^{13} +(1.37228 + 2.37686i) q^{14} +(-0.313859 + 0.543620i) q^{16} -0.792287i q^{17} +0.372281 q^{19} +(2.05842 + 2.27567i) q^{20} +(3.00000 - 1.73205i) q^{22} +(1.37228 - 0.792287i) q^{23} +(4.55842 - 2.05446i) q^{25} +4.62772 q^{26} -4.75372i q^{28} +(-2.87228 + 4.97494i) q^{29} +(-3.18614 - 5.51856i) q^{31} +(-5.05842 + 2.92048i) q^{32} +(0.313859 - 0.543620i) q^{34} +(-7.37228 - 2.37686i) q^{35} +2.37686i q^{37} +(0.255437 + 0.147477i) q^{38} +(1.25544 + 5.84096i) q^{40} +(-2.18614 - 3.78651i) q^{41} +(3.00000 + 1.73205i) q^{43} -6.00000 q^{44} +1.25544 q^{46} +(1.62772 + 0.939764i) q^{47} +(2.50000 + 4.33013i) q^{49} +(3.94158 + 0.396143i) q^{50} +(-6.94158 - 4.00772i) q^{52} +11.9769i q^{53} +(-3.00000 + 9.30506i) q^{55} +(4.62772 - 8.01544i) q^{56} +(-3.94158 + 2.27567i) q^{58} +(0.813859 + 1.40965i) q^{59} +(-4.68614 + 8.11663i) q^{61} -5.04868i q^{62} -3.37228 q^{64} +(-9.68614 + 8.76144i) q^{65} +(-10.1168 + 5.84096i) q^{67} +(-0.941578 + 0.543620i) q^{68} +(-4.11684 - 4.55134i) q^{70} +13.1168 q^{71} -2.37686i q^{73} +(-0.941578 + 1.63086i) q^{74} +(-0.255437 - 0.442430i) q^{76} +(13.1168 - 7.57301i) q^{77} +(3.37228 - 5.84096i) q^{79} +(0.430703 - 1.33591i) q^{80} -3.46410i q^{82} +(-10.3723 - 5.98844i) q^{83} +(0.372281 + 1.73205i) q^{85} +(1.37228 + 2.37686i) q^{86} +(-10.1168 - 5.84096i) q^{88} +3.00000 q^{89} +20.2337 q^{91} +(-1.88316 - 1.08724i) q^{92} +(0.744563 + 1.28962i) q^{94} +(-0.813859 + 0.174928i) q^{95} +(-1.11684 - 0.644810i) q^{97} +3.96143i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{2} + 3 q^{4} - 3 q^{5} + 12 q^{7}+O(q^{10}) $ 4 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 + 12 * q^7	$ 4 q - 3 q^{2} + 3 q^{4} - 3 q^{5} + 12 q^{7} - q^{10} + 3 q^{11} + 3 q^{13} - 6 q^{14} - 7 q^{16} - 10 q^{19} - 9 q^{20} + 12 q^{22} - 6 q^{23} + q^{25} + 30 q^{26} - 7 q^{31} - 3 q^{32} + 7 q^{34} - 18 q^{35} + 24 q^{38} + 28 q^{40} - 3 q^{41} + 12 q^{43} - 24 q^{44} + 28 q^{46} + 18 q^{47} + 10 q^{49} + 33 q^{50} - 45 q^{52} - 12 q^{55} + 30 q^{56} - 33 q^{58} + 9 q^{59} - 13 q^{61} - 2 q^{64} - 33 q^{65} - 6 q^{67} - 21 q^{68} + 18 q^{70} + 18 q^{71} - 21 q^{74} - 24 q^{76} + 18 q^{77} + 2 q^{79} - 27 q^{80} - 30 q^{83} - 10 q^{85} - 6 q^{86} - 6 q^{88} + 12 q^{89} + 12 q^{91} - 42 q^{92} - 20 q^{94} - 9 q^{95} + 30 q^{97}+O(q^{100}) $ 4 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 + 12 * q^7 - q^10 + 3 * q^11 + 3 * q^13 - 6 * q^14 - 7 * q^16 - 10 * q^19 - 9 * q^20 + 12 * q^22 - 6 * q^23 + q^25 + 30 * q^26 - 7 * q^31 - 3 * q^32 + 7 * q^34 - 18 * q^35 + 24 * q^38 + 28 * q^40 - 3 * q^41 + 12 * q^43 - 24 * q^44 + 28 * q^46 + 18 * q^47 + 10 * q^49 + 33 * q^50 - 45 * q^52 - 12 * q^55 + 30 * q^56 - 33 * q^58 + 9 * q^59 - 13 * q^61 - 2 * q^64 - 33 * q^65 - 6 * q^67 - 21 * q^68 + 18 * q^70 + 18 * q^71 - 21 * q^74 - 24 * q^76 + 18 * q^77 + 2 * q^79 - 27 * q^80 - 30 * q^83 - 10 * q^85 - 6 * q^86 - 6 * q^88 + 12 * q^89 + 12 * q^91 - 42 * q^92 - 20 * q^94 - 9 * q^95 + 30 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	0.686141	+	0.396143i	0.485175	+	0.280116i	0.722570	−	0.691297i	$-0.242960\pi$
$2$	0.686141	+	0.396143i	0.485175	+	0.280116i	−0.237396	+	0.971413i	$0.576294\pi$
$3$		0			0
$3$		0			0
$4$	−0.686141	−	1.18843i	−0.343070	−	0.594215i
$5$	−2.18614	+	0.469882i	−0.977672	+	0.210138i
$5$	−2.18614	+	0.469882i	−0.977672	+	0.210138i
$6$		0			0
$7$	3.00000	+	1.73205i	1.13389	+	0.654654i	0.944911	−	0.327327i	$-0.106148\pi$
$7$	3.00000	+	1.73205i	1.13389	+	0.654654i	0.188982	+	0.981981i	$0.439481\pi$
$8$		−	2.67181i		−	0.944629i
$9$		0			0
$10$	−1.68614	−	0.543620i	−0.533204	−	0.171908i
$11$	2.18614	−	3.78651i	0.659146	−	1.14167i	−0.321691	−	0.946845i	$-0.604251\pi$
$11$	2.18614	−	3.78651i	0.659146	−	1.14167i	0.980837	−	0.194830i	$-0.0624155\pi$
$12$		0			0
$13$	5.05842	−	2.92048i	1.40295	−	0.809996i	0.408259	−	0.912866i	$-0.366136\pi$
$13$	5.05842	−	2.92048i	1.40295	−	0.809996i	0.994695	+	0.102870i	$0.0328027\pi$
$14$	1.37228	+	2.37686i	0.366758	+	0.635243i
$15$		0			0
$16$	−0.313859	+	0.543620i	−0.0784648	+	0.135905i
$17$		−	0.792287i		−	0.192158i	−0.995374	−	0.0960789i	$-0.969370\pi$
$17$		−	0.792287i		−	0.192158i	0.995374	−	0.0960789i	$-0.0306301\pi$
$18$		0			0
$19$	0.372281			0.0854072			0.0427036	−	0.999088i	$-0.486403\pi$
$19$	0.372281			0.0854072			0.0427036	+	0.999088i	$0.486403\pi$
$20$	2.05842	+	2.27567i	0.460277	+	0.508856i
$21$		0			0
$22$	3.00000	−	1.73205i	0.639602	−	0.369274i
$23$	1.37228	−	0.792287i	0.286140	−	0.165203i	−0.350060	−	0.936727i	$-0.613839\pi$
$23$	1.37228	−	0.792287i	0.286140	−	0.165203i	0.636200	+	0.771524i	$0.280505\pi$
$24$		0			0
$25$	4.55842	−	2.05446i	0.911684	−	0.410891i
$26$	4.62772			0.907570
$27$		0			0
$28$		−	4.75372i		−	0.898369i
$29$	−2.87228	+	4.97494i	−0.533369	+	0.923823i	0.465871	+	0.884853i	$0.345741\pi$
$29$	−2.87228	+	4.97494i	−0.533369	+	0.923823i	−0.999240	+	0.0389701i	$0.987592\pi$
$30$		0			0
$31$	−3.18614	−	5.51856i	−0.572248	−	0.991162i	−0.996335	−	0.0855407i	$-0.972738\pi$
$31$	−3.18614	−	5.51856i	−0.572248	−	0.991162i	0.424087	−	0.905621i	$-0.360595\pi$
$32$	−5.05842	+	2.92048i	−0.894211	+	0.516273i
$33$		0			0
$34$	0.313859	−	0.543620i	0.0538264	−	0.0932301i
$35$	−7.37228	−	2.37686i	−1.24614	−	0.401763i
$36$		0			0
$37$			2.37686i			0.390754i	0.980728	+	0.195377i	$0.0625930\pi$
$37$			2.37686i			0.390754i	−0.980728	+	0.195377i	$0.937407\pi$
$38$	0.255437	+	0.147477i	0.0414374	+	0.0239239i
$39$		0			0
$40$	1.25544	+	5.84096i	0.198502	+	0.923537i
$41$	−2.18614	−	3.78651i	−0.341418	−	0.591353i	0.643278	−	0.765632i	$-0.277574\pi$
$41$	−2.18614	−	3.78651i	−0.341418	−	0.591353i	−0.984696	+	0.174279i	$0.944240\pi$
$42$		0			0
$43$	3.00000	+	1.73205i	0.457496	+	0.264135i	0.710991	−	0.703201i	$-0.248247\pi$
$43$	3.00000	+	1.73205i	0.457496	+	0.264135i	−0.253495	+	0.967337i	$0.581580\pi$
$44$	−6.00000			−0.904534
$45$		0			0
$46$	1.25544			0.185104
$47$	1.62772	+	0.939764i	0.237427	+	0.137079i	0.613994	−	0.789311i	$-0.289562\pi$
$47$	1.62772	+	0.939764i	0.237427	+	0.137079i	−0.376566	+	0.926390i	$0.622895\pi$
$48$		0			0
$49$	2.50000	+	4.33013i	0.357143	+	0.618590i
$50$	3.94158	+	0.396143i	0.557423	+	0.0560232i
$51$		0			0
$52$	−6.94158	−	4.00772i	−0.962624	−	0.555771i
$53$			11.9769i			1.64515i	0.568656	+	0.822575i	$0.307464\pi$
$53$			11.9769i			1.64515i	−0.568656	+	0.822575i	$0.692536\pi$
$54$		0			0
$55$	−3.00000	+	9.30506i	−0.404520	+	1.25469i
$56$	4.62772	−	8.01544i	0.618405	−	1.07111i
$57$		0			0
$58$	−3.94158	+	2.27567i	−0.517555	+	0.298810i
$59$	0.813859	+	1.40965i	0.105955	+	0.183520i	0.914128	−	0.405425i	$-0.132877\pi$
$59$	0.813859	+	1.40965i	0.105955	+	0.183520i	−0.808173	+	0.588946i	$0.799543\pi$
$60$		0			0
$61$	−4.68614	+	8.11663i	−0.599999	+	1.03923i	0.392822	+	0.919615i	$0.371499\pi$
$61$	−4.68614	+	8.11663i	−0.599999	+	1.03923i	−0.992820	+	0.119614i	$0.961834\pi$
$62$		−	5.04868i		−	0.641182i
$63$		0			0
$64$	−3.37228			−0.421535
$65$	−9.68614	+	8.76144i	−1.20142	+	1.08672i
$66$		0			0
$67$	−10.1168	+	5.84096i	−1.23597	+	0.713587i	−0.968268	−	0.249915i	$-0.919597\pi$
$67$	−10.1168	+	5.84096i	−1.23597	+	0.713587i	−0.267701	+	0.963502i	$0.586264\pi$
$68$	−0.941578	+	0.543620i	−0.114183	+	0.0659236i
$69$		0			0
$70$	−4.11684	−	4.55134i	−0.492057	−	0.543989i
$71$	13.1168			1.55668			0.778341	−	0.627841i	$-0.216061\pi$
$71$	13.1168			1.55668			0.778341	+	0.627841i	$0.216061\pi$
$72$		0			0
$73$		−	2.37686i		−	0.278191i	−0.990279	−	0.139095i	$-0.955581\pi$
$73$		−	2.37686i		−	0.278191i	0.990279	−	0.139095i	$-0.0444194\pi$
$74$	−0.941578	+	1.63086i	−0.109456	+	0.189584i
$75$		0			0
$76$	−0.255437	−	0.442430i	−0.0293007	−	0.0507503i
$77$	13.1168	−	7.57301i	1.49480	−	0.863025i
$78$		0			0
$79$	3.37228	−	5.84096i	0.379411	−	0.657160i	−0.611565	−	0.791194i	$-0.709460\pi$
$79$	3.37228	−	5.84096i	0.379411	−	0.657160i	0.990977	+	0.134034i	$0.0427932\pi$
$80$	0.430703	−	1.33591i	0.0481541	−	0.149359i
$81$		0			0
$82$		−	3.46410i		−	0.382546i
$83$	−10.3723	−	5.98844i	−1.13851	−	0.657317i	−0.192446	−	0.981308i	$-0.561642\pi$
$83$	−10.3723	−	5.98844i	−1.13851	−	0.657317i	−0.946060	+	0.323991i	$0.894975\pi$
$84$		0			0
$85$	0.372281	+	1.73205i	0.0403796	+	0.187867i
$86$	1.37228	+	2.37686i	0.147977	+	0.256304i
$87$		0			0
$88$	−10.1168	−	5.84096i	−1.07846	−	0.622649i
$89$	3.00000			0.317999			0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000			0.317999			0.159000	+	0.987279i	$0.449173\pi$
$90$		0			0
$91$	20.2337			2.12107
$92$	−1.88316	−	1.08724i	−0.196333	−	0.113353i
$93$		0			0
$94$	0.744563	+	1.28962i	0.0767958	+	0.133014i
$95$	−0.813859	+	0.174928i	−0.0835002	+	0.0179473i
$96$		0			0
$97$	−1.11684	−	0.644810i	−0.113398	−	0.0654706i	0.442228	−	0.896903i	$-0.354188\pi$
$97$	−1.11684	−	0.644810i	−0.113398	−	0.0654706i	−0.555626	+	0.831432i	$0.687522\pi$
$98$			3.96143i			0.400165i
$99$		0			0
$100$	−5.56930	−	4.00772i	−0.556930	−	0.400772i
$101$	8.18614	−	14.1788i	0.814551	−	1.41084i	−0.0950981	−	0.995468i	$-0.530316\pi$
$101$	8.18614	−	14.1788i	0.814551	−	1.41084i	0.909650	−	0.415377i	$-0.136350\pi$
$102$		0			0
$103$	−6.00000	+	3.46410i	−0.591198	+	0.341328i	−0.765571	−	0.643352i	$-0.777543\pi$
$103$	−6.00000	+	3.46410i	−0.591198	+	0.341328i	0.174373	+	0.984680i	$0.444210\pi$
$104$	−7.80298	−	13.5152i	−0.765146	−	1.32527i
$105$		0			0
$106$	−4.74456	+	8.21782i	−0.460833	+	0.798186i
$107$			13.2665i			1.28252i	0.767323	+	0.641260i	$0.221588\pi$
$107$			13.2665i			1.28252i	−0.767323	+	0.641260i	$0.778412\pi$
$108$		0			0
$109$	−9.74456			−0.933360			−0.466680	−	0.884426i	$-0.654550\pi$
$109$	−9.74456			−0.933360			−0.466680	+	0.884426i	$0.654550\pi$
$110$	−5.74456	+	5.19615i	−0.547723	+	0.495434i
$111$		0			0
$112$	−1.88316	+	1.08724i	−0.177942	+	0.102735i
$113$	5.31386	−	3.06796i	0.499886	−	0.288609i	−0.228781	−	0.973478i	$-0.573474\pi$
$113$	5.31386	−	3.06796i	0.499886	−	0.288609i	0.728666	+	0.684869i	$0.240140\pi$
$114$		0			0
$115$	−2.62772	+	2.37686i	−0.245036	+	0.221643i
$116$	7.88316			0.731933
$117$		0			0
$118$			1.28962i			0.118719i
$119$	1.37228	−	2.37686i	0.125797	−	0.217886i
$120$		0			0
$121$	−4.05842	−	7.02939i	−0.368947	−	0.639036i
$122$	−6.43070	+	3.71277i	−0.582209	+	0.336138i
$123$		0			0
$124$	−4.37228	+	7.57301i	−0.392642	+	0.680077i
$125$	−9.00000	+	6.63325i	−0.804984	+	0.593296i
$126$		0			0
$127$			10.3923i			0.922168i	0.887357	+	0.461084i	$0.152539\pi$
$127$			10.3923i			0.922168i	−0.887357	+	0.461084i	$0.847461\pi$
$128$	7.80298	+	4.50506i	0.689693	+	0.398194i
$129$		0			0
$130$	−10.1168	+	2.17448i	−0.887306	+	0.190715i
$131$	−2.18614	−	3.78651i	−0.191004	−	0.330829i	0.754579	−	0.656209i	$-0.227841\pi$
$131$	−2.18614	−	3.78651i	−0.191004	−	0.330829i	−0.945583	+	0.325380i	$0.894508\pi$
$132$		0			0
$133$	1.11684	+	0.644810i	0.0968427	+	0.0559121i
$134$	−9.25544			−0.799548
$135$		0			0
$136$	−2.11684			−0.181518
$137$	−12.4307	−	7.17687i	−1.06203	−	0.613161i	−0.136034	−	0.990704i	$-0.543436\pi$
$137$	−12.4307	−	7.17687i	−1.06203	−	0.613161i	−0.925992	+	0.377543i	$0.876769\pi$
$138$		0			0
$139$	5.55842	+	9.62747i	0.471459	+	0.816591i	0.999467	−	0.0326483i	$-0.0103941\pi$
$139$	5.55842	+	9.62747i	0.471459	+	0.816591i	−0.528008	+	0.849240i	$0.677061\pi$
$140$	2.23369	+	10.3923i	0.188781	+	0.878310i
$141$		0			0
$142$	9.00000	+	5.19615i	0.755263	+	0.436051i
$143$		−	25.5383i		−	2.13562i
$144$		0			0
$145$	3.94158	−	12.2255i	0.327330	−	1.01528i
$146$	0.941578	−	1.63086i	0.0779256	−	0.134971i
$147$		0			0
$148$	2.82473	−	1.63086i	0.232192	−	0.134056i
$149$	5.31386	+	9.20387i	0.435328	+	0.754011i	0.997322	−	0.0731305i	$-0.0232990\pi$
$149$	5.31386	+	9.20387i	0.435328	+	0.754011i	−0.561994	+	0.827141i	$0.689966\pi$
$150$		0			0
$151$	−0.441578	+	0.764836i	−0.0359351	+	0.0622414i	−0.883434	−	0.468556i	$-0.844774\pi$
$151$	−0.441578	+	0.764836i	−0.0359351	+	0.0622414i	0.847499	+	0.530798i	$0.178108\pi$
$152$		−	0.994667i		−	0.0806781i
$153$		0			0
$154$	12.0000			0.966988
$155$	9.55842	+	10.5672i	0.767751	+	0.848781i
$156$		0			0
$157$	9.94158	−	5.73977i	0.793424	−	0.458084i	−0.0477424	−	0.998860i	$-0.515203\pi$
$157$	9.94158	−	5.73977i	0.793424	−	0.458084i	0.841167	+	0.540776i	$0.181869\pi$
$158$	4.62772	−	2.67181i	0.368162	−	0.212558i
$159$		0			0
$160$	9.68614	−	8.76144i	0.765757	−	0.692653i
$161$	5.48913			0.432604
$162$		0			0
$163$			15.1460i			1.18633i	0.805082	+	0.593164i	$0.202122\pi$
$163$			15.1460i			1.18633i	−0.805082	+	0.593164i	$0.797878\pi$
$164$	−3.00000	+	5.19615i	−0.234261	+	0.405751i
$165$		0			0
$166$	−4.74456	−	8.21782i	−0.368249	−	0.637827i
$167$	−11.7446	+	6.78073i	−0.908822	+	0.524708i	−0.880052	−	0.474878i	$-0.842492\pi$
$167$	−11.7446	+	6.78073i	−0.908822	+	0.524708i	−0.0287697	+	0.999586i	$0.509159\pi$
$168$		0			0
$169$	10.5584	−	18.2877i	0.812186	−	1.40675i
$170$	−0.430703	+	1.33591i	−0.0330334	+	0.102459i
$171$		0			0
$172$		−	4.75372i		−	0.362468i
$173$	9.68614	+	5.59230i	0.736424	+	0.425174i	0.820768	−	0.571262i	$-0.193546\pi$
$173$	9.68614	+	5.59230i	0.736424	+	0.425174i	−0.0843439	+	0.996437i	$0.526879\pi$
$174$		0			0
$175$	17.2337	+	1.73205i	1.30274	+	0.130931i
$176$	1.37228	+	2.37686i	0.103440	+	0.179163i
$177$		0			0
$178$	2.05842	+	1.18843i	0.154285	+	0.0890766i
$179$	4.88316			0.364984			0.182492	−	0.983207i	$-0.441584\pi$
$179$	4.88316			0.364984			0.182492	+	0.983207i	$0.441584\pi$
$180$		0			0
$181$	−13.8614			−1.03031			−0.515155	−	0.857097i	$-0.672266\pi$
$181$	−13.8614			−1.03031			−0.515155	+	0.857097i	$0.672266\pi$
$182$	13.8832	+	8.01544i	1.02909	+	0.594144i
$183$		0			0
$184$	−2.11684	−	3.66648i	−0.156056	−	0.270297i
$185$	−1.11684	−	5.19615i	−0.0821120	−	0.382029i
$186$		0			0
$187$	−3.00000	−	1.73205i	−0.219382	−	0.126660i
$188$		−	2.57924i		−	0.188110i
$189$		0			0
$190$	−0.627719	−	0.202380i	−0.0455395	−	0.0146822i
$191$	−3.81386	+	6.60580i	−0.275961	+	0.477979i	−0.970377	−	0.241595i	$-0.922329\pi$
$191$	−3.81386	+	6.60580i	−0.275961	+	0.477979i	0.694416	+	0.719574i	$0.255663\pi$
$192$		0			0
$193$	9.94158	−	5.73977i	0.715610	−	0.413158i	−0.0975245	−	0.995233i	$-0.531092\pi$
$193$	9.94158	−	5.73977i	0.715610	−	0.413158i	0.813135	+	0.582075i	$0.197759\pi$
$194$	−0.510875	−	0.884861i	−0.0366787	−	0.0635293i
$195$		0			0
$196$	3.43070	−	5.94215i	0.245050	−	0.424439i
$197$		−	6.43087i		−	0.458181i	−0.973405	−	0.229090i	$-0.926425\pi$
$197$		−	6.43087i		−	0.458181i	0.973405	−	0.229090i	$-0.0735751\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$	−5.48913	−	12.1793i	−0.388140	−	0.861204i
$201$		0			0
$202$	11.2337	−	6.48577i	0.790400	−	0.456337i
$203$	−17.2337	+	9.94987i	−1.20957	+	0.698344i
$204$		0			0
$205$	6.55842	+	7.25061i	0.458060	+	0.506404i
$206$	−5.48913			−0.382445
$207$		0			0
$208$			3.66648i			0.254225i
$209$	0.813859	−	1.40965i	0.0562958	−	0.0975072i
$210$		0			0
$211$	0.930703	+	1.61203i	0.0640723	+	0.110976i	0.896282	−	0.443484i	$-0.146258\pi$
$211$	0.930703	+	1.61203i	0.0640723	+	0.110976i	−0.832210	+	0.554461i	$0.812925\pi$
$212$	14.2337	−	8.21782i	0.977574	−	0.564402i
$213$		0			0
$214$	−5.25544	+	9.10268i	−0.359254	+	0.622247i
$215$	−7.37228	−	2.37686i	−0.502785	−	0.162101i
$216$		0			0
$217$		−	22.0742i		−	1.49850i
$218$	−6.68614	−	3.86025i	−0.452843	−	0.261449i
$219$		0			0
$220$	13.1168	−	2.81929i	0.884337	−	0.190077i
$221$	−2.31386	−	4.00772i	−0.155647	−	0.269589i
$222$		0			0
$223$	16.1168	+	9.30506i	1.07926	+	0.623113i	0.930698	−	0.365789i	$-0.119201\pi$
$223$	16.1168	+	9.30506i	1.07926	+	0.623113i	0.148566	+	0.988902i	$0.452534\pi$
$224$	−20.2337			−1.35192
$225$		0			0
$226$	4.86141			0.323376
$227$	11.7446	+	6.78073i	0.779514	+	0.450053i	0.836258	−	0.548336i	$-0.184739\pi$
$227$	11.7446	+	6.78073i	0.779514	+	0.450053i	−0.0567440	+	0.998389i	$0.518072\pi$
$228$		0			0
$229$	−8.80298	−	15.2472i	−0.581718	−	1.00756i	−0.995276	−	0.0970868i	$-0.969048\pi$
$229$	−8.80298	−	15.2472i	−0.581718	−	1.00756i	0.413558	−	0.910478i	$-0.364286\pi$
$230$	−2.74456	+	0.589907i	−0.180971	+	0.0388973i
$231$		0			0
$232$	13.2921	+	7.67420i	0.872670	+	0.503836i
$233$			6.13592i			0.401977i	0.979594	+	0.200989i	$0.0644154\pi$
$233$			6.13592i			0.401977i	−0.979594	+	0.200989i	$0.935585\pi$
$234$		0			0
$235$	−4.00000	−	1.28962i	−0.260931	−	0.0841256i
$236$	1.11684	−	1.93443i	0.0727004	−	0.125921i
$237$		0			0
$238$	1.88316	−	1.08724i	0.122067	−	0.0704753i
$239$	11.4891	+	19.8997i	0.743170	+	1.28721i	0.951045	+	0.309052i	$0.100012\pi$
$239$	11.4891	+	19.8997i	0.743170	+	1.28721i	−0.207875	+	0.978155i	$0.566655\pi$
$240$		0			0
$241$	0.755437	−	1.30846i	0.0486620	−	0.0842851i	−0.840668	−	0.541550i	$-0.817838\pi$
$241$	0.755437	−	1.30846i	0.0486620	−	0.0842851i	0.889330	+	0.457265i	$0.151171\pi$
$242$		−	6.43087i		−	0.413392i
$243$		0			0
$244$	12.8614			0.823367
$245$	−7.50000	−	8.29156i	−0.479157	−	0.529728i
$246$		0			0
$247$	1.88316	−	1.08724i	0.119822	−	0.0691795i
$248$	−14.7446	+	8.51278i	−0.936281	+	0.540562i
$249$		0			0
$250$	−8.80298	+	0.986051i	−0.556750	+	0.0623633i
$251$	−20.2337			−1.27714			−0.638570	−	0.769564i	$-0.720474\pi$
$251$	−20.2337			−1.27714			−0.638570	+	0.769564i	$0.720474\pi$
$252$		0			0
$253$		−	6.92820i		−	0.435572i
$254$	−4.11684	+	7.13058i	−0.258314	+	0.447413i
$255$		0			0
$256$	6.94158	+	12.0232i	0.433849	+	0.751448i
$257$	−5.56930	+	3.21543i	−0.347403	+	0.200573i	−0.663541	−	0.748140i	$-0.730947\pi$
$257$	−5.56930	+	3.21543i	−0.347403	+	0.200573i	0.316138	+	0.948713i	$0.397614\pi$
$258$		0			0
$259$	−4.11684	+	7.13058i	−0.255808	+	0.443073i
$260$	17.0584	+	5.49972i	1.05792	+	0.341078i
$261$		0			0
$262$		−	3.46410i		−	0.214013i
$263$	22.9783	+	13.2665i	1.41690	+	0.818047i	0.996025	−	0.0890715i	$-0.0283900\pi$
$263$	22.9783	+	13.2665i	1.41690	+	0.818047i	0.420874	+	0.907119i	$0.361723\pi$
$264$		0			0
$265$	−5.62772	−	26.1831i	−0.345708	−	1.60842i
$266$	0.510875	+	0.884861i	0.0313237	+	0.0542543i
$267$		0			0
$268$	13.8832	+	8.01544i	0.848049	+	0.489621i
$269$	29.2337			1.78241			0.891205	−	0.453601i	$-0.149861\pi$
$269$	29.2337			1.78241			0.891205	+	0.453601i	$0.149861\pi$
$270$		0			0
$271$	5.25544			0.319245			0.159623	−	0.987178i	$-0.448972\pi$
$271$	5.25544			0.319245			0.159623	+	0.987178i	$0.448972\pi$
$272$	0.430703	+	0.248667i	0.0261152	+	0.0150776i
$273$		0			0
$274$	−5.68614	−	9.84868i	−0.343512	−	0.594981i
$275$	2.18614	−	21.7518i	0.131829	−	1.31168i
$276$		0			0
$277$	−19.1168	−	11.0371i	−1.14862	−	0.663156i	−0.200069	−	0.979782i	$-0.564117\pi$
$277$	−19.1168	−	11.0371i	−1.14862	−	0.663156i	−0.948551	+	0.316626i	$0.897450\pi$
$278$			8.80773i			0.528253i
$279$		0			0
$280$	−6.35053	+	19.6974i	−0.379517	+	1.17714i
$281$	0.686141	−	1.18843i	0.0409317	−	0.0708958i	−0.844834	−	0.535029i	$-0.820301\pi$
$281$	0.686141	−	1.18843i	0.0409317	−	0.0708958i	0.885765	+	0.464133i	$0.153634\pi$
$282$		0			0
$283$	−24.0000	+	13.8564i	−1.42665	+	0.823678i	−0.996855	−	0.0792477i	$-0.974748\pi$
$283$	−24.0000	+	13.8564i	−1.42665	+	0.823678i	−0.429797	+	0.902926i	$0.641415\pi$
$284$	−9.00000	−	15.5885i	−0.534052	−	0.925005i
$285$		0			0
$286$	10.1168	−	17.5229i	0.598222	−	1.03615i
$287$		−	15.1460i		−	0.894042i
$288$		0			0
$289$	16.3723			0.963075
$290$	7.54755	−	6.82701i	0.443207	−	0.400896i
$291$		0			0
$292$	−2.82473	+	1.63086i	−0.165305	+	0.0954389i
$293$	−0.686141	+	0.396143i	−0.0400848	+	0.0231430i	−0.519908	−	0.854222i	$-0.674034\pi$
$293$	−0.686141	+	0.396143i	−0.0400848	+	0.0231430i	0.479824	+	0.877365i	$0.340701\pi$
$294$		0			0
$295$	−2.44158	−	2.69927i	−0.142154	−	0.157157i
$296$	6.35053			0.369117
$297$		0			0
$298$			8.42020i			0.487769i
$299$	4.62772	−	8.01544i	0.267628	−	0.463545i
$300$		0			0
$301$	6.00000	+	10.3923i	0.345834	+	0.599002i
$302$	−0.605969	+	0.349857i	−0.0348696	+	0.0201320i
$303$		0			0
$304$	−0.116844	+	0.202380i	−0.00670146	+	0.0116073i
$305$	6.43070	−	19.9460i	0.368221	−	1.14211i
$306$		0			0
$307$			15.1460i			0.864429i	0.901771	+	0.432215i	$0.142268\pi$
$307$			15.1460i			0.864429i	−0.901771	+	0.432215i	$0.857732\pi$
$308$	−18.0000	−	10.3923i	−1.02565	−	0.592157i
$309$		0			0
$310$	2.37228	+	11.0371i	0.134737	+	0.626866i
$311$	7.93070	+	13.7364i	0.449709	+	0.778919i	0.998367	−	0.0571282i	$-0.0181944\pi$
$311$	7.93070	+	13.7364i	0.449709	+	0.778919i	−0.548658	+	0.836047i	$0.684861\pi$
$312$		0			0
$313$	−21.1753	−	12.2255i	−1.19690	−	0.691029i	−0.237035	−	0.971501i	$-0.576176\pi$
$313$	−21.1753	−	12.2255i	−1.19690	−	0.691029i	−0.959862	+	0.280472i	$0.909509\pi$
$314$	9.09509			0.513266
$315$		0			0
$316$	−9.25544			−0.520659
$317$	−25.5475	−	14.7499i	−1.43489	−	0.828436i	−0.437405	−	0.899265i	$-0.644102\pi$
$317$	−25.5475	−	14.7499i	−1.43489	−	0.828436i	−0.997488	+	0.0708288i	$0.977436\pi$
$318$		0			0
$319$	12.5584	+	21.7518i	0.703137	+	1.21787i
$320$	7.37228	−	1.58457i	0.412123	−	0.0885804i
$321$		0			0
$322$	3.76631	+	2.17448i	0.209888	+	0.121179i
$323$		−	0.294954i		−	0.0164117i
$324$		0			0
$325$	17.0584	−	23.7051i	0.946231	−	1.31492i
$326$	−6.00000	+	10.3923i	−0.332309	+	0.575577i
$327$		0			0
$328$	−10.1168	+	5.84096i	−0.558609	+	0.322513i
$329$	3.25544	+	5.63858i	0.179478	+	0.310865i
$330$		0			0
$331$	−4.55842	+	7.89542i	−0.250554	+	0.433971i	−0.963678	−	0.267066i	$-0.913946\pi$
$331$	−4.55842	+	7.89542i	−0.250554	+	0.433971i	0.713125	+	0.701037i	$0.247279\pi$
$332$			16.4356i			0.902023i
$333$		0			0
$334$	−10.7446			−0.587916
$335$	19.3723	−	17.5229i	1.05842	−	0.957378i
$336$		0			0
$337$	−6.00000	+	3.46410i	−0.326841	+	0.188702i	−0.654438	−	0.756116i	$-0.727095\pi$
$337$	−6.00000	+	3.46410i	−0.326841	+	0.188702i	0.327597	+	0.944818i	$0.393761\pi$
$338$	14.4891	−	8.36530i	0.788105	−	0.455012i
$339$		0			0
$340$	1.80298	−	1.63086i	0.0977806	−	0.0884459i
$341$	−27.8614			−1.50878
$342$		0			0
$343$		−	6.92820i		−	0.374088i
$344$	4.62772	−	8.01544i	0.249510	−	0.432164i
$345$		0			0
$346$	4.43070	+	7.67420i	0.238196	+	0.412568i
$347$	2.48913	−	1.43710i	0.133623	−	0.0771474i	−0.431698	−	0.902018i	$-0.642085\pi$
$347$	2.48913	−	1.43710i	0.133623	−	0.0771474i	0.565321	+	0.824871i	$0.308752\pi$
$348$		0			0
$349$	−13.3030	+	23.0414i	−0.712092	+	1.23338i	0.251978	+	0.967733i	$0.418919\pi$
$349$	−13.3030	+	23.0414i	−0.712092	+	1.23338i	−0.964070	+	0.265647i	$0.914414\pi$
$350$	11.1386	+	8.01544i	0.595383	+	0.428443i
$351$		0			0
$352$			25.5383i			1.36120i
$353$	1.62772	+	0.939764i	0.0866347	+	0.0500186i	0.542692	−	0.839932i	$-0.317405\pi$
$353$	1.62772	+	0.939764i	0.0866347	+	0.0500186i	−0.456057	+	0.889951i	$0.650739\pi$
$354$		0			0
$355$	−28.6753	+	6.16337i	−1.52193	+	0.327118i
$356$	−2.05842	−	3.56529i	−0.109096	−	0.188960i
$357$		0			0
$358$	3.35053	+	1.93443i	0.177081	+	0.102238i
$359$	10.8832			0.574391			0.287196	−	0.957872i	$-0.407277\pi$
$359$	10.8832			0.574391			0.287196	+	0.957872i	$0.407277\pi$
$360$		0			0
$361$	−18.8614			−0.992706
$362$	−9.51087	−	5.49111i	−0.499880	−	0.288606i
$363$		0			0
$364$	−13.8832	−	24.0463i	−0.727675	−	1.26037i
$365$	1.11684	+	5.19615i	0.0584583	+	0.271979i
$366$		0			0
$367$	−14.2337	−	8.21782i	−0.742992	−	0.428967i	0.0801639	−	0.996782i	$-0.474456\pi$
$367$	−14.2337	−	8.21782i	−0.742992	−	0.428967i	−0.823156	+	0.567815i	$0.807789\pi$
$368$			0.994667i			0.0518506i
$369$		0			0
$370$	1.29211	−	4.00772i	0.0671736	−	0.208352i
$371$	−20.7446	+	35.9306i	−1.07700	+	1.86543i
$372$		0			0
$373$	7.11684	−	4.10891i	0.368496	−	0.212751i	−0.304305	−	0.952575i	$-0.598424\pi$
$373$	7.11684	−	4.10891i	0.368496	−	0.212751i	0.672801	+	0.739823i	$0.265091\pi$
$374$	−1.37228	−	2.37686i	−0.0709590	−	0.122905i
$375$		0			0
$376$	2.51087	−	4.34896i	0.129488	−	0.224281i
$377$			33.5538i			1.72811i
$378$		0			0
$379$	1.48913			0.0764912			0.0382456	−	0.999268i	$-0.487823\pi$
$379$	1.48913			0.0764912			0.0382456	+	0.999268i	$0.487823\pi$
$380$	0.766312	+	0.847190i	0.0393110	+	0.0434599i
$381$		0			0
$382$	−5.23369	+	3.02167i	−0.267779	+	0.154602i
$383$	9.60597	−	5.54601i	0.490842	−	0.283388i	−0.234082	−	0.972217i	$-0.575208\pi$
$383$	9.60597	−	5.54601i	0.490842	−	0.283388i	0.724924	+	0.688829i	$0.241875\pi$
$384$		0			0
$385$	−25.1168	+	22.7190i	−1.28007	+	1.15787i
$386$	9.09509			0.462928
$387$		0			0
$388$			1.76972i			0.0898440i
$389$	−0.255437	+	0.442430i	−0.0129512	+	0.0224321i	−0.872428	−	0.488742i	$-0.837456\pi$
$389$	−0.255437	+	0.442430i	−0.0129512	+	0.0224321i	0.859477	+	0.511174i	$0.170789\pi$
$390$		0			0
$391$	−0.627719	−	1.08724i	−0.0317451	−	0.0549841i
$392$	11.5693	−	6.67954i	0.584338	−	0.337368i
$393$		0			0
$394$	2.54755	−	4.41248i	0.128344	−	0.222298i
$395$	−4.62772	+	14.3537i	−0.232846	+	0.722215i
$396$		0			0
$397$		−	17.5229i		−	0.879449i	−0.898133	−	0.439724i	$-0.855076\pi$
$397$		−	17.5229i		−	0.879449i	0.898133	−	0.439724i	$-0.144924\pi$
$398$	−10.9783	−	6.33830i	−0.550290	−	0.317710i
$399$		0			0
$400$	−0.313859	+	3.12286i	−0.0156930	+	0.156143i
$401$	−12.6861	−	21.9730i	−0.633516	−	1.09728i	−0.986828	−	0.161776i	$-0.948278\pi$
$401$	−12.6861	−	21.9730i	−0.633516	−	1.09728i	0.353312	−	0.935506i	$-0.385055\pi$
$402$		0			0
$403$	−32.2337	−	18.6101i	−1.60567	−	0.927037i
$404$	−22.4674			−1.11779
$405$		0			0
$406$	−15.7663			−0.782469
$407$	9.00000	+	5.19615i	0.446113	+	0.257564i
$408$		0			0
$409$	11.4307	+	19.7986i	0.565212	+	0.978976i	0.997030	+	0.0770150i	$0.0245389\pi$
$409$	11.4307	+	19.7986i	0.565212	+	0.978976i	−0.431818	+	0.901961i	$0.642128\pi$
$410$	1.62772	+	7.57301i	0.0803873	+	0.374004i
$411$		0			0
$412$	8.23369	+	4.75372i	0.405645	+	0.234199i
$413$			5.63858i			0.277457i
$414$		0			0
$415$	25.4891	+	8.21782i	1.25121	+	0.403397i
$416$	−17.0584	+	29.5461i	−0.836358	+	1.44861i
$417$		0			0
$418$	1.11684	−	0.644810i	0.0546266	−	0.0315387i
$419$	−8.74456	−	15.1460i	−0.427200	−	0.739932i	0.569423	−	0.822045i	$-0.307167\pi$
$419$	−8.74456	−	15.1460i	−0.427200	−	0.739932i	−0.996623	+	0.0821127i	$0.973833\pi$
$420$		0			0
$421$	10.8723	−	18.8313i	0.529883	−	0.917784i	−0.469510	−	0.882927i	$-0.655569\pi$
$421$	10.8723	−	18.8313i	0.529883	−	0.917784i	0.999392	−	0.0348563i	$-0.0110973\pi$
$422$			1.47477i			0.0717906i
$423$		0			0
$424$	32.0000			1.55406
$425$	−1.62772	−	3.61158i	−0.0789560	−	0.175187i
$426$		0			0
$427$	−28.1168	+	16.2333i	−1.36067	+	0.785583i
$428$	15.7663	−	9.10268i	0.762093	−	0.439995i
$429$		0			0
$430$	−4.11684	−	4.55134i	−0.198532	−	0.219485i
$431$	−33.3505			−1.60644			−0.803219	−	0.595683i	$-0.796881\pi$
$431$	−33.3505			−1.60644			−0.803219	+	0.595683i	$0.796881\pi$
$432$		0			0
$433$		−	12.7692i		−	0.613647i	−0.951766	−	0.306823i	$-0.900734\pi$
$433$		−	12.7692i		−	0.613647i	0.951766	−	0.306823i	$-0.0992661\pi$
$434$	8.74456	−	15.1460i	0.419752	−	0.727033i
$435$		0			0
$436$	6.68614	+	11.5807i	0.320208	+	0.554617i
$437$	0.510875	−	0.294954i	0.0244385	−	0.0141095i
$438$		0			0
$439$	15.9307	−	27.5928i	0.760331	−	1.31693i	−0.182349	−	0.983234i	$-0.558370\pi$
$439$	15.9307	−	27.5928i	0.760331	−	1.31693i	0.942680	−	0.333698i	$-0.108297\pi$
$440$	24.8614	+	8.01544i	1.18522	+	0.382121i
$441$		0			0
$442$		−	3.66648i		−	0.174397i
$443$	−18.6060	−	10.7422i	−0.883996	−	0.510375i	−0.0120223	−	0.999928i	$-0.503827\pi$
$443$	−18.6060	−	10.7422i	−0.883996	−	0.510375i	−0.871974	+	0.489552i	$0.837160\pi$
$444$		0			0
$445$	−6.55842	+	1.40965i	−0.310899	+	0.0668236i
$446$	7.37228	+	12.7692i	0.349088	+	0.604638i
$447$		0			0
$448$	−10.1168	−	5.84096i	−0.477976	−	0.275960i
$449$	10.8832			0.513608			0.256804	−	0.966464i	$-0.417331\pi$
$449$	10.8832			0.513608			0.256804	+	0.966464i	$0.417331\pi$
$450$		0			0
$451$	−19.1168			−0.900177
$452$	−7.29211	−	4.21010i	−0.342992	−	0.198027i
$453$		0			0
$454$	5.37228	+	9.30506i	0.252134	+	0.436708i
$455$	−44.2337	+	9.50744i	−2.07371	+	0.445716i
$456$		0			0
$457$	18.1753	+	10.4935i	0.850203	+	0.490865i	0.860719	−	0.509080i	$-0.170014\pi$
$457$	18.1753	+	10.4935i	0.850203	+	0.490865i	−0.0105163	+	0.999945i	$0.503348\pi$
$458$		−	13.9490i		−	0.651793i
$459$		0			0
$460$	4.62772	+	1.49200i	0.215768	+	0.0695649i
$461$	−12.0475	+	20.8670i	−0.561110	+	0.971871i	0.436290	+	0.899806i	$0.356292\pi$
$461$	−12.0475	+	20.8670i	−0.561110	+	0.971871i	−0.997400	+	0.0720652i	$0.977041\pi$
$462$		0			0
$463$	12.0000	−	6.92820i	0.557687	−	0.321981i	−0.194529	−	0.980897i	$-0.562318\pi$
$463$	12.0000	−	6.92820i	0.557687	−	0.321981i	0.752217	+	0.658916i	$0.228985\pi$
$464$	−1.80298	−	3.12286i	−0.0837015	−	0.144975i
$465$		0			0
$466$	−2.43070	+	4.21010i	−0.112600	+	0.195029i
$467$		−	18.3152i		−	0.847525i	−0.905773	−	0.423763i	$-0.860709\pi$
$467$		−	18.3152i		−	0.847525i	0.905773	−	0.423763i	$-0.139291\pi$
$468$		0			0
$469$	−40.4674			−1.86861
$470$	−2.23369	−	2.46943i	−0.103032	−	0.113907i
$471$		0			0
$472$	3.76631	−	2.17448i	0.173359	−	0.100089i
$473$	13.1168	−	7.57301i	0.603113	−	0.348208i
$474$		0			0
$475$	1.69702	−	0.764836i	0.0778644	−	0.0350931i
$476$	−3.76631			−0.172629
$477$		0			0
$478$			18.2054i			0.832694i
$479$	9.30298	−	16.1132i	0.425064	−	0.736233i	−0.571362	−	0.820698i	$-0.693585\pi$
$479$	9.30298	−	16.1132i	0.425064	−	0.736233i	0.996426	+	0.0844652i	$0.0269182\pi$
$480$		0			0
$481$	6.94158	+	12.0232i	0.316509	+	0.548209i
$482$	1.03667	−	0.598523i	0.0472191	−	0.0272620i
$483$		0			0
$484$	−5.56930	+	9.64630i	−0.253150	+	0.438468i
$485$	2.74456	+	0.884861i	0.124624	+	0.0401795i
$486$		0			0
$487$			9.50744i			0.430823i	0.976523	+	0.215412i	$0.0691093\pi$
$487$			9.50744i			0.430823i	−0.976523	+	0.215412i	$0.930891\pi$
$488$	21.6861	+	12.5205i	0.981685	+	0.566776i
$489$		0			0
$490$	−1.86141	−	8.66025i	−0.0840898	−	0.391230i
$491$	15.8139	+	27.3904i	0.713669	+	1.23611i	0.963470	+	0.267815i	$0.0863015\pi$
$491$	15.8139	+	27.3904i	0.713669	+	1.23611i	−0.249801	+	0.968297i	$0.580365\pi$
$492$		0			0
$493$	3.94158	+	2.27567i	0.177520	+	0.102491i
$494$	1.72281			0.0775130
$495$		0			0
$496$	4.00000			0.179605
$497$	39.3505	+	22.7190i	1.76511	+	1.01909i
$498$		0			0
$499$	−12.1861	−	21.1070i	−0.545527	−	0.944880i	−0.998574	−	0.0533931i	$-0.982996\pi$
$499$	−12.1861	−	21.1070i	−0.545527	−	0.944880i	0.453047	−	0.891487i	$-0.350337\pi$
$500$	14.0584	+	6.14453i	0.628712	+	0.274792i
$501$		0			0
$502$	−13.8832	−	8.01544i	−0.619636	−	0.357747i
$503$		−	3.16915i		−	0.141305i	−0.997501	−	0.0706527i	$-0.977492\pi$
$503$		−	3.16915i		−	0.141305i	0.997501	−	0.0706527i	$-0.0225082\pi$
$504$		0			0
$505$	−11.2337	+	34.8434i	−0.499893	+	1.55051i
$506$	2.74456	−	4.75372i	0.122011	−	0.211329i
$507$		0			0
$508$	12.3505	−	7.13058i	0.547966	−	0.316368i
$509$	0.255437	+	0.442430i	0.0113221	+	0.0196104i	0.871631	−	0.490163i	$-0.163063\pi$
$509$	0.255437	+	0.442430i	0.0113221	+	0.0196104i	−0.860309	+	0.509773i	$0.829729\pi$
$510$		0			0
$511$	4.11684	−	7.13058i	0.182118	−	0.315438i
$512$		−	7.02078i		−	0.310277i
$513$		0			0
$514$	−5.09509			−0.224735
$515$	11.4891	−	10.3923i	0.506271	−	0.457940i
$516$		0			0
$517$	7.11684	−	4.10891i	0.312998	−	0.180710i
$518$	−5.64947	+	3.26172i	−0.248223	+	0.143312i
$519$		0			0
$520$	23.4090	+	25.8796i	1.02655	+	1.13489i
$521$	−18.0000			−0.788594			−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000			−0.788594			−0.394297	+	0.918983i	$0.629012\pi$
$522$		0			0
$523$		−	4.75372i		−	0.207866i	−0.994584	−	0.103933i	$-0.966857\pi$
$523$		−	4.75372i		−	0.207866i	0.994584	−	0.103933i	$-0.0331427\pi$
$524$	−3.00000	+	5.19615i	−0.131056	+	0.226995i
$525$		0			0
$526$	10.5109	+	18.2054i	0.458296	+	0.793792i
$527$	−4.37228	+	2.52434i	−0.190460	+	0.109962i
$528$		0			0
$529$	−10.2446	+	17.7441i	−0.445416	+	0.771483i
$530$	6.51087	−	20.1947i	0.282814	−	0.877202i
$531$		0			0
$532$		−	1.76972i		−	0.0767272i
$533$	−22.1168	−	12.7692i	−0.957987	−	0.553094i
$534$		0			0
$535$	−6.23369	−	29.0024i	−0.269506	−	1.25388i
$536$	15.6060	+	27.0303i	0.674075	+	1.16753i
$537$		0			0
$538$	20.0584	+	11.5807i	0.864780	+	0.499281i
$539$	21.8614			0.941637
$540$		0			0
$541$	19.2337			0.826921			0.413460	−	0.910522i	$-0.364320\pi$
$541$	19.2337			0.826921			0.413460	+	0.910522i	$0.364320\pi$
$542$	3.60597	+	2.08191i	0.154890	+	0.0894256i
$543$		0			0
$544$	2.31386	+	4.00772i	0.0992059	+	0.171830i
$545$	21.3030	−	4.57879i	0.912520	−	0.196134i
$546$		0			0
$547$	−6.00000	−	3.46410i	−0.256541	−	0.148114i	0.366214	−	0.930531i	$-0.380654\pi$
$547$	−6.00000	−	3.46410i	−0.256541	−	0.148114i	−0.622756	+	0.782416i	$0.713987\pi$
$548$			19.6974i			0.841430i
$549$		0			0
$550$	10.1168	−	14.0588i	0.431384	−	0.599469i
$551$	−1.06930	+	1.85208i	−0.0455536	+	0.0789011i
$552$		0			0
$553$	20.2337	−	11.6819i	0.860424	−	0.496766i
$554$	−8.74456	−	15.1460i	−0.371521	−	0.643493i
$555$		0			0
$556$	7.62772	−	13.2116i	0.323487	−	0.560296i
$557$		−	25.0410i		−	1.06102i	−0.847678	−	0.530511i	$-0.822000\pi$
$557$		−	25.0410i		−	1.06102i	0.847678	−	0.530511i	$-0.178000\pi$
$558$		0			0
$559$	20.2337			0.855794
$560$	3.60597	−	3.26172i	0.152380	−	0.137833i
$561$		0			0
$562$	0.941578	−	0.543620i	0.0397181	−	0.0229312i
$563$	−27.8614	+	16.0858i	−1.17422	+	0.677935i	−0.954670	−	0.297666i	$-0.903792\pi$
$563$	−27.8614	+	16.0858i	−1.17422	+	0.677935i	−0.219548	+	0.975602i	$0.570458\pi$
$564$		0			0
$565$	−10.1753	+	9.20387i	−0.428077	+	0.387210i
$566$	−21.9565			−0.922901
$567$		0			0
$568$		−	35.0458i		−	1.47049i
$569$	10.2446	−	17.7441i	0.429474	−	0.743871i	−0.567352	−	0.823475i	$-0.692032\pi$
$569$	10.2446	−	17.7441i	0.429474	−	0.743871i	0.996827	+	0.0796038i	$0.0253655\pi$
$570$		0			0
$571$	−2.06930	−	3.58413i	−0.0865974	−	0.149991i	0.819473	−	0.573117i	$-0.194266\pi$
$571$	−2.06930	−	3.58413i	−0.0865974	−	0.149991i	−0.906071	+	0.423126i	$0.860933\pi$
$572$	−30.3505	+	17.5229i	−1.26902	+	0.732669i
$573$		0			0
$574$	6.00000	−	10.3923i	0.250435	−	0.433766i
$575$	4.62772	−	6.43087i	0.192989	−	0.268186i
$576$		0			0
$577$		−	18.4077i		−	0.766325i	−0.923681	−	0.383162i	$-0.874835\pi$
$577$		−	18.4077i		−	0.766325i	0.923681	−	0.383162i	$-0.125165\pi$
$578$	11.2337	+	6.48577i	0.467260	+	0.269773i
$579$		0			0
$580$	−17.2337	+	3.70415i	−0.715590	+	0.153807i
$581$	−20.7446	−	35.9306i	−0.860629	−	1.49065i
$582$		0			0
$583$	45.3505	+	26.1831i	1.87823	+	1.08439i
$584$	−6.35053			−0.262787
$585$		0			0
$586$	−0.627719			−0.0259308
$587$	−14.4891	−	8.36530i	−0.598030	−	0.345273i	0.170236	−	0.985403i	$-0.445547\pi$
$587$	−14.4891	−	8.36530i	−0.598030	−	0.345273i	−0.768266	+	0.640130i	$0.778880\pi$
$588$		0			0
$589$	−1.18614	−	2.05446i	−0.0488741	−	0.0846524i
$590$	−0.605969	−	2.81929i	−0.0249474	−	0.116068i
$591$		0			0
$592$	−1.29211	−	0.746000i	−0.0531054	−	0.0306604i
$593$		−	1.67715i		−	0.0688722i	−0.999407	−	0.0344361i	$-0.989036\pi$
$593$		−	1.67715i		−	0.0688722i	0.999407	−	0.0344361i	$-0.0109635\pi$
$594$		0			0
$595$	−1.88316	+	5.84096i	−0.0772019	+	0.239456i
$596$	7.29211	−	12.6303i	0.298696	−	0.517357i
$597$		0			0
$598$	6.35053	−	3.66648i	0.259693	−	0.149934i
$599$	0.813859	+	1.40965i	0.0332534	+	0.0575966i	0.882173	−	0.470925i	$-0.156080\pi$
$599$	0.813859	+	1.40965i	0.0332534	+	0.0575966i	−0.848920	+	0.528522i	$0.822746\pi$
$600$		0			0
$601$	8.98913	−	15.5696i	0.366674	−	0.635098i	−0.622369	−	0.782724i	$-0.713830\pi$
$601$	8.98913	−	15.5696i	0.366674	−	0.635098i	0.989043	+	0.147626i	$0.0471631\pi$
$602$			9.50744i			0.387494i
$603$		0			0
$604$	1.21194			0.0493131
$605$	12.1753	+	13.4603i	0.494995	+	0.547238i
$606$		0			0
$607$	37.4674	−	21.6318i	1.52075	−	0.878008i	0.521054	−	0.853524i	$-0.325539\pi$
$607$	37.4674	−	21.6318i	1.52075	−	0.878008i	0.999700	−	0.0244837i	$-0.00779419\pi$
$608$	−1.88316	+	1.08724i	−0.0763721	+	0.0440934i
$609$		0			0
$610$	12.3139	−	11.1383i	0.498574	−	0.450977i
$611$	10.9783			0.444132
$612$		0			0
$613$		−	30.2921i		−	1.22348i	−0.791057	−	0.611742i	$-0.790469\pi$
$613$		−	30.2921i		−	1.22348i	0.791057	−	0.611742i	$-0.209531\pi$
$614$	−6.00000	+	10.3923i	−0.242140	+	0.419399i
$615$		0			0
$616$	−20.2337	−	35.0458i	−0.815239	−	1.41203i
$617$	−17.9198	+	10.3460i	−0.721425	+	0.416515i	−0.815277	−	0.579071i	$-0.803415\pi$
$617$	−17.9198	+	10.3460i	−0.721425	+	0.416515i	0.0938519	+	0.995586i	$0.470082\pi$
$618$		0			0
$619$	2.00000	−	3.46410i	0.0803868	−	0.139234i	−0.823029	−	0.567999i	$-0.807718\pi$
$619$	2.00000	−	3.46410i	0.0803868	−	0.139234i	0.903416	+	0.428765i	$0.141051\pi$
$620$	6.00000	−	18.6101i	0.240966	−	0.747401i
$621$		0			0
$622$			12.5668i			0.503882i
$623$	9.00000	+	5.19615i	0.360577	+	0.208179i
$624$		0			0
$625$	16.5584	−	18.7302i	0.662337	−	0.749206i
$626$	−9.68614	−	16.7769i	−0.387136	−	0.670539i
$627$		0			0
$628$	−13.6426	−	7.87658i	−0.544401	−	0.314310i
$629$	1.88316			0.0750863
$630$		0			0
$631$	6.37228			0.253677			0.126838	−	0.991923i	$-0.459517\pi$
$631$	6.37228			0.253677			0.126838	+	0.991923i	$0.459517\pi$
$632$	−15.6060	−	9.01011i	−0.620772	−	0.358403i
$633$		0			0
$634$	−11.6861	−	20.2410i	−0.464116	−	0.803872i
$635$	−4.88316	−	22.7190i	−0.193782	−	0.901578i
$636$		0			0
$637$	25.2921	+	14.6024i	1.00211	+	0.578568i
$638$			19.8997i			0.787839i
$639$		0			0
$640$	−19.1753	−	6.18220i	−0.757969	−	0.244373i
$641$	−3.98913	+	6.90937i	−0.157561	+	0.272904i	−0.933989	−	0.357303i	$-0.883696\pi$
$641$	−3.98913	+	6.90937i	−0.157561	+	0.272904i	0.776428	+	0.630206i	$0.217030\pi$
$642$		0			0
$643$	−1.11684	+	0.644810i	−0.0440440	+	0.0254288i	−0.521860	−	0.853031i	$-0.674762\pi$
$643$	−1.11684	+	0.644810i	−0.0440440	+	0.0254288i	0.477816	+	0.878460i	$0.341428\pi$
$644$	−3.76631	−	6.52344i	−0.148413	−	0.257060i
$645$		0			0
$646$	0.116844	−	0.202380i	0.00459716	−	0.00796252i
$647$		−	28.7075i		−	1.12861i	−0.825567	−	0.564304i	$-0.809145\pi$
$647$		−	28.7075i		−	1.12861i	0.825567	−	0.564304i	$-0.190855\pi$
$648$		0			0
$649$	7.11684			0.279361
$650$	21.0951	−	9.50744i	0.827418	−	0.372913i
$651$		0			0
$652$	18.0000	−	10.3923i	0.704934	−	0.406994i
$653$	5.48913	−	3.16915i	0.214806	−	0.124018i	−0.388737	−	0.921349i	$-0.627089\pi$
$653$	5.48913	−	3.16915i	0.214806	−	0.124018i	0.603543	+	0.797330i	$0.293755\pi$
$654$		0			0
$655$	6.55842	+	7.25061i	0.256259	+	0.283305i
$656$	2.74456			0.107157
$657$		0			0
$658$			5.15848i			0.201099i
$659$	10.6277	−	18.4077i	0.413997	−	0.717064i	−0.581325	−	0.813671i	$-0.697466\pi$
$659$	10.6277	−	18.4077i	0.413997	−	0.717064i	0.995323	+	0.0966070i	$0.0307990\pi$
$660$		0			0
$661$	7.61684	+	13.1928i	0.296261	+	0.513139i	0.975277	−	0.220984i	$-0.0709269\pi$
$661$	7.61684	+	13.1928i	0.296261	+	0.513139i	−0.679017	+	0.734123i	$0.737594\pi$
$662$	−6.25544	+	3.61158i	−0.243124	+	0.140368i
$663$		0			0
$664$	−16.0000	+	27.7128i	−0.620920	+	1.07547i
$665$	−2.74456	−	0.884861i	−0.106430	−	0.0343134i
$666$		0			0
$667$			9.10268i			0.352457i
$668$	16.1168	+	9.30506i	0.623579	+	0.360024i
$669$		0			0
$670$	20.2337	−	4.34896i	0.781696	−	0.168015i
$671$	20.4891	+	35.4882i	0.790974	+	1.37001i
$672$		0			0
$673$	9.17527	+	5.29734i	0.353681	+	0.204198i	0.666305	−	0.745679i	$-0.267875\pi$
$673$	9.17527	+	5.29734i	0.353681	+	0.204198i	−0.312625	+	0.949877i	$0.601208\pi$
$674$	−5.48913			−0.211433
$675$		0			0
$676$	−28.9783			−1.11455
$677$	22.9783	+	13.2665i	0.883126	+	0.509873i	0.871688	−	0.490062i	$-0.163026\pi$
$677$	22.9783	+	13.2665i	0.883126	+	0.509873i	0.0114381	+	0.999935i	$0.496359\pi$
$678$		0			0
$679$	−2.23369	−	3.86886i	−0.0857211	−	0.148473i
$680$	4.62772	−	0.994667i	0.177465	−	0.0381437i
$681$		0			0
$682$	−19.1168	−	11.0371i	−0.732022	−	0.422633i
$683$			27.1229i			1.03783i	0.854826	+	0.518915i	$0.173664\pi$
$683$			27.1229i			1.03783i	−0.854826	+	0.518915i	$0.826336\pi$
$684$		0			0
$685$	30.5475	+	9.84868i	1.16716	+	0.376299i
$686$	2.74456	−	4.75372i	0.104788	−	0.181498i
$687$		0			0
$688$	−1.88316	+	1.08724i	−0.0717947	+	0.0414507i
$689$	34.9783	+	60.5841i	1.33257	+	2.30807i
$690$		0			0
$691$	−22.8614	+	39.5971i	−0.869689	+	1.50635i	−0.00737407	+	0.999973i	$0.502347\pi$
$691$	−22.8614	+	39.5971i	−0.869689	+	1.50635i	−0.862315	+	0.506373i	$0.830986\pi$
$692$		−	15.3484i		−	0.583459i
$693$		0			0
$694$	2.27719			0.0864408
$695$	−16.6753	−	18.4352i	−0.632529	−	0.699287i
$696$		0			0
$697$	−3.00000	+	1.73205i	−0.113633	+	0.0656061i
$698$	−18.2554	+	10.5398i	−0.690978	+	0.398937i
$699$		0			0
$700$	−9.76631	−	21.6695i	−0.369132	−	0.819029i
$701$	17.2337			0.650907			0.325454	−	0.945558i	$-0.394483\pi$
$701$	17.2337			0.650907			0.325454	+	0.945558i	$0.394483\pi$
$702$		0			0
$703$			0.884861i			0.0333732i
$704$	−7.37228	+	12.7692i	−0.277853	+	0.481256i
$705$		0			0
$706$	0.744563	+	1.28962i	0.0280220	+	0.0485355i
$707$	49.1168	−	28.3576i	1.84723	−	1.06650i
$708$		0			0
$709$	26.1753	−	45.3369i	0.983033	−	1.70266i	0.332661	−	0.943046i	$-0.392053\pi$
$709$	26.1753	−	45.3369i	0.983033	−	1.70266i	0.650372	−	0.759616i	$-0.274613\pi$
$710$	−22.1168	−	7.13058i	−0.830030	−	0.267606i
$711$		0			0
$712$		−	8.01544i		−	0.300391i
$713$	−8.74456	−	5.04868i	−0.327486	−	0.189074i
$714$		0			0
$715$	12.0000	+	55.8304i	0.448775	+	2.08794i
$716$	−3.35053	−	5.80329i	−0.125215	−	0.216879i
$717$		0			0
$718$	7.46738	+	4.31129i	0.278680	+	0.160896i
$719$	−10.8832			−0.405873			−0.202937	−	0.979192i	$-0.565049\pi$
$719$	−10.8832			−0.405873			−0.202937	+	0.979192i	$0.565049\pi$
$720$		0			0
$721$	−24.0000			−0.893807
$722$	−12.9416	−	7.47182i	−0.481636	−	0.278072i
$723$		0			0
$724$	9.51087	+	16.4733i	0.353469	+	0.612226i
$725$	−2.87228	+	28.5788i	−0.106674	+	1.06139i
$726$		0			0
$727$	42.3505	+	24.4511i	1.57069	+	0.906841i	0.996084	+	0.0884137i	$0.0281798\pi$
$727$	42.3505	+	24.4511i	1.57069	+	0.906841i	0.574610	+	0.818427i	$0.305154\pi$
$728$		−	54.0607i		−	2.00362i
$729$		0			0
$730$	−1.29211	+	4.00772i	−0.0478231	+	0.148332i
$731$	1.37228	−	2.37686i	0.0507557	−	0.0879114i
$732$		0			0
$733$	−14.2337	+	8.21782i	−0.525733	+	0.303532i	−0.739277	−	0.673401i	$-0.764833\pi$
$733$	−14.2337	+	8.21782i	−0.525733	+	0.303532i	0.213544	+	0.976933i	$0.431499\pi$
$734$	−6.51087	−	11.2772i	−0.240321	−	0.416248i
$735$		0			0
$736$	−4.62772	+	8.01544i	−0.170580	+	0.295453i
$737$			51.0767i			1.88143i
$738$		0			0
$739$	−5.62772			−0.207019			−0.103509	−	0.994628i	$-0.533007\pi$
$739$	−5.62772			−0.207019			−0.103509	+	0.994628i	$0.533007\pi$
$740$	−5.40895	+	4.89258i	−0.198837	+	0.179855i
$741$		0			0
$742$	−28.4674	+	16.4356i	−1.04507	+	0.603372i
$743$	8.13859	−	4.69882i	0.298576	−	0.172383i	−0.343227	−	0.939252i	$-0.611520\pi$
$743$	8.13859	−	4.69882i	0.298576	−	0.172383i	0.641803	+	0.766870i	$0.278187\pi$
$744$		0			0
$745$	−15.9416	−	17.6241i	−0.584054	−	0.645696i
$746$	6.51087			0.238380
$747$		0			0
$748$			4.75372i			0.173813i
$749$	−22.9783	+	39.7995i	−0.839607	+	1.45424i
$750$		0			0
$751$	−10.8614	−	18.8125i	−0.396338	−	0.686478i	0.596933	−	0.802291i	$-0.296386\pi$
$751$	−10.8614	−	18.8125i	−0.396338	−	0.686478i	−0.993271	+	0.115813i	$0.963053\pi$
$752$	−1.02175	+	0.589907i	−0.0372594	+	0.0215117i
$753$		0			0
$754$	−13.2921	+	23.0226i	−0.484070	+	0.838434i
$755$	0.605969	−	1.87953i	0.0220535	−	0.0684030i
$756$		0			0
$757$			41.5692i			1.51086i	0.655230	+	0.755429i	$0.272572\pi$
$757$			41.5692i			1.51086i	−0.655230	+	0.755429i	$0.727428\pi$
$758$	1.02175	+	0.589907i	0.0371116	+	0.0214264i
$759$		0			0
$760$	0.467376	+	2.17448i	0.0169535	+	0.0788767i
$761$	−16.2446	−	28.1364i	−0.588865	−	1.01994i	−0.994381	−	0.105856i	$-0.966242\pi$
$761$	−16.2446	−	28.1364i	−0.588865	−	1.01994i	0.405517	−	0.914088i	$-0.367092\pi$
$762$		0			0
$763$	−29.2337	−	16.8781i	−1.05833	−	0.611027i
$764$	10.4674			0.378696
$765$		0			0
$766$	8.78806			0.317526
$767$	8.23369	+	4.75372i	0.297301	+	0.171647i
$768$		0			0
$769$	−8.24456	−	14.2800i	−0.297307	−	0.514950i	0.678212	−	0.734866i	$-0.262755\pi$
$769$	−8.24456	−	14.2800i	−0.297307	−	0.514950i	−0.975519	+	0.219916i	$0.929422\pi$
$770$	−26.2337	+	5.63858i	−0.945396	+	0.203200i
$771$		0			0
$772$	−13.6426	−	7.87658i	−0.491009	−	0.283484i
$773$		−	21.5769i		−	0.776067i	−0.921645	−	0.388034i	$-0.873154\pi$
$773$		−	21.5769i		−	0.776067i	0.921645	−	0.388034i	$-0.126846\pi$
$774$		0			0
$775$	−25.8614	−	18.6101i	−0.928969	−	0.668496i
$776$	−1.72281	+	2.98400i	−0.0618454	+	0.107119i
$777$		0			0
$778$	−0.350532	+	0.202380i	−0.0125672	+	0.00725566i
$779$	−0.813859	−	1.40965i	−0.0291595	−	0.0505058i
$780$		0			0

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.j (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.686141 + 0.396143i) q^{2} +(-0.686141 - 1.18843i) q^{4} +(-2.18614 + 0.469882i) q^{5} +(3.00000 + 1.73205i) q^{7} -2.67181i q^{8} +O(q^{10})\)	\(q+(0.686141 + 0.396143i) q^{2} +(-0.686141 - 1.18843i) q^{4} +(-2.18614 + 0.469882i) q^{5} +(3.00000 + 1.73205i) q^{7} -2.67181i q^{8} +(-1.68614 - 0.543620i) q^{10} +(2.18614 - 3.78651i) q^{11} +(5.05842 - 2.92048i) q^{13} +(1.37228 + 2.37686i) q^{14} +(-0.313859 + 0.543620i) q^{16} -0.792287i q^{17} +0.372281 q^{19} +(2.05842 + 2.27567i) q^{20} +(3.00000 - 1.73205i) q^{22} +(1.37228 - 0.792287i) q^{23} +(4.55842 - 2.05446i) q^{25} +4.62772 q^{26} -4.75372i q^{28} +(-2.87228 + 4.97494i) q^{29} +(-3.18614 - 5.51856i) q^{31} +(-5.05842 + 2.92048i) q^{32} +(0.313859 - 0.543620i) q^{34} +(-7.37228 - 2.37686i) q^{35} +2.37686i q^{37} +(0.255437 + 0.147477i) q^{38} +(1.25544 + 5.84096i) q^{40} +(-2.18614 - 3.78651i) q^{41} +(3.00000 + 1.73205i) q^{43} -6.00000 q^{44} +1.25544 q^{46} +(1.62772 + 0.939764i) q^{47} +(2.50000 + 4.33013i) q^{49} +(3.94158 + 0.396143i) q^{50} +(-6.94158 - 4.00772i) q^{52} +11.9769i q^{53} +(-3.00000 + 9.30506i) q^{55} +(4.62772 - 8.01544i) q^{56} +(-3.94158 + 2.27567i) q^{58} +(0.813859 + 1.40965i) q^{59} +(-4.68614 + 8.11663i) q^{61} -5.04868i q^{62} -3.37228 q^{64} +(-9.68614 + 8.76144i) q^{65} +(-10.1168 + 5.84096i) q^{67} +(-0.941578 + 0.543620i) q^{68} +(-4.11684 - 4.55134i) q^{70} +13.1168 q^{71} -2.37686i q^{73} +(-0.941578 + 1.63086i) q^{74} +(-0.255437 - 0.442430i) q^{76} +(13.1168 - 7.57301i) q^{77} +(3.37228 - 5.84096i) q^{79} +(0.430703 - 1.33591i) q^{80} -3.46410i q^{82} +(-10.3723 - 5.98844i) q^{83} +(0.372281 + 1.73205i) q^{85} +(1.37228 + 2.37686i) q^{86} +(-10.1168 - 5.84096i) q^{88} +3.00000 q^{89} +20.2337 q^{91} +(-1.88316 - 1.08724i) q^{92} +(0.744563 + 1.28962i) q^{94} +(-0.813859 + 0.174928i) q^{95} +(-1.11684 - 0.644810i) q^{97} +3.96143i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{2} + 3 q^{4} - 3 q^{5} + 12 q^{7}+O(q^{10}) \) 4 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 + 12 * q^7	\( 4 q - 3 q^{2} + 3 q^{4} - 3 q^{5} + 12 q^{7} - q^{10} + 3 q^{11} + 3 q^{13} - 6 q^{14} - 7 q^{16} - 10 q^{19} - 9 q^{20} + 12 q^{22} - 6 q^{23} + q^{25} + 30 q^{26} - 7 q^{31} - 3 q^{32} + 7 q^{34} - 18 q^{35} + 24 q^{38} + 28 q^{40} - 3 q^{41} + 12 q^{43} - 24 q^{44} + 28 q^{46} + 18 q^{47} + 10 q^{49} + 33 q^{50} - 45 q^{52} - 12 q^{55} + 30 q^{56} - 33 q^{58} + 9 q^{59} - 13 q^{61} - 2 q^{64} - 33 q^{65} - 6 q^{67} - 21 q^{68} + 18 q^{70} + 18 q^{71} - 21 q^{74} - 24 q^{76} + 18 q^{77} + 2 q^{79} - 27 q^{80} - 30 q^{83} - 10 q^{85} - 6 q^{86} - 6 q^{88} + 12 q^{89} + 12 q^{91} - 42 q^{92} - 20 q^{94} - 9 q^{95} + 30 q^{97}+O(q^{100}) \) 4 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 + 12 * q^7 - q^10 + 3 * q^11 + 3 * q^13 - 6 * q^14 - 7 * q^16 - 10 * q^19 - 9 * q^20 + 12 * q^22 - 6 * q^23 + q^25 + 30 * q^26 - 7 * q^31 - 3 * q^32 + 7 * q^34 - 18 * q^35 + 24 * q^38 + 28 * q^40 - 3 * q^41 + 12 * q^43 - 24 * q^44 + 28 * q^46 + 18 * q^47 + 10 * q^49 + 33 * q^50 - 45 * q^52 - 12 * q^55 + 30 * q^56 - 33 * q^58 + 9 * q^59 - 13 * q^61 - 2 * q^64 - 33 * q^65 - 6 * q^67 - 21 * q^68 + 18 * q^70 + 18 * q^71 - 21 * q^74 - 24 * q^76 + 18 * q^77 + 2 * q^79 - 27 * q^80 - 30 * q^83 - 10 * q^85 - 6 * q^86 - 6 * q^88 + 12 * q^89 + 12 * q^91 - 42 * q^92 - 20 * q^94 - 9 * q^95 + 30 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more