LMFDB - Embedded newform 936.2.g.a.469.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [936,2,Mod(469,936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(936, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("936.469");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 936 = 2^{3} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	936.g (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$7.47399762919$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			469.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	936.469
Dual form			936.2.g.a.469.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +3.00000i q^{5} +3.00000 q^{7} +(-2.00000 + 2.00000i) q^{8} +O(q^{10})$	\(q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +3.00000i q^{5} +3.00000 q^{7} +(-2.00000 + 2.00000i) q^{8} +(-3.00000 + 3.00000i) q^{10} -1.00000i q^{13} +(3.00000 + 3.00000i) q^{14} -4.00000 q^{16} +7.00000 q^{17} +4.00000i q^{19} -6.00000 q^{20} -4.00000 q^{23} -4.00000 q^{25} +(1.00000 - 1.00000i) q^{26} +6.00000i q^{28} -4.00000i q^{29} -8.00000 q^{31} +(-4.00000 - 4.00000i) q^{32} +(7.00000 + 7.00000i) q^{34} +9.00000i q^{35} -7.00000i q^{37} +(-4.00000 + 4.00000i) q^{38} +(-6.00000 - 6.00000i) q^{40} -2.00000 q^{41} +1.00000i q^{43} +(-4.00000 - 4.00000i) q^{46} +7.00000 q^{47} +2.00000 q^{49} +(-4.00000 - 4.00000i) q^{50} +2.00000 q^{52} +4.00000i q^{53} +(-6.00000 + 6.00000i) q^{56} +(4.00000 - 4.00000i) q^{58} -14.0000i q^{59} +10.0000i q^{61} +(-8.00000 - 8.00000i) q^{62} -8.00000i q^{64} +3.00000 q^{65} -2.00000i q^{67} +14.0000i q^{68} +(-9.00000 + 9.00000i) q^{70} +3.00000 q^{71} +14.0000 q^{73} +(7.00000 - 7.00000i) q^{74} -8.00000 q^{76} -10.0000 q^{79} -12.0000i q^{80} +(-2.00000 - 2.00000i) q^{82} +14.0000i q^{83} +21.0000i q^{85} +(-1.00000 + 1.00000i) q^{86} -3.00000i q^{91} -8.00000i q^{92} +(7.00000 + 7.00000i) q^{94} -12.0000 q^{95} +8.00000 q^{97} +(2.00000 + 2.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 6 q^{7} - 4 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 6 * q^7 - 4 * q^8	$ 2 q + 2 q^{2} + 6 q^{7} - 4 q^{8} - 6 q^{10} + 6 q^{14} - 8 q^{16} + 14 q^{17} - 12 q^{20} - 8 q^{23} - 8 q^{25} + 2 q^{26} - 16 q^{31} - 8 q^{32} + 14 q^{34} - 8 q^{38} - 12 q^{40} - 4 q^{41} - 8 q^{46} + 14 q^{47} + 4 q^{49} - 8 q^{50} + 4 q^{52} - 12 q^{56} + 8 q^{58} - 16 q^{62} + 6 q^{65} - 18 q^{70} + 6 q^{71} + 28 q^{73} + 14 q^{74} - 16 q^{76} - 20 q^{79} - 4 q^{82} - 2 q^{86} + 14 q^{94} - 24 q^{95} + 16 q^{97} + 4 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 6 * q^7 - 4 * q^8 - 6 * q^10 + 6 * q^14 - 8 * q^16 + 14 * q^17 - 12 * q^20 - 8 * q^23 - 8 * q^25 + 2 * q^26 - 16 * q^31 - 8 * q^32 + 14 * q^34 - 8 * q^38 - 12 * q^40 - 4 * q^41 - 8 * q^46 + 14 * q^47 + 4 * q^49 - 8 * q^50 + 4 * q^52 - 12 * q^56 + 8 * q^58 - 16 * q^62 + 6 * q^65 - 18 * q^70 + 6 * q^71 + 28 * q^73 + 14 * q^74 - 16 * q^76 - 20 * q^79 - 4 * q^82 - 2 * q^86 + 14 * q^94 - 24 * q^95 + 16 * q^97 + 4 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/936\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$1$	$1$	$-1$	$1$

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 $

$2$	1.00000	+	1.00000i	0.707107	+	0.707107i
$2$	1.00000	+	1.00000i	0.707107	+	0.707107i
$3$		0			0
$3$		0			0
$4$			2.00000i			1.00000i
$5$			3.00000i			1.34164i	0.741620	+	0.670820i	$0.234058\pi$
$5$			3.00000i			1.34164i	−0.741620	+	0.670820i	$0.765942\pi$
$6$		0			0
$7$	3.00000			1.13389			0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000			1.13389			0.566947	+	0.823754i	$0.308125\pi$
$8$	−2.00000	+	2.00000i	−0.707107	+	0.707107i
$9$		0			0
$10$	−3.00000	+	3.00000i	−0.948683	+	0.948683i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$	3.00000	+	3.00000i	0.801784	+	0.801784i
$15$		0			0
$16$	−4.00000			−1.00000
$17$	7.00000			1.69775			0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000			1.69775			0.848875	+	0.528594i	$0.177281\pi$
$18$		0			0
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	−6.00000			−1.34164
$21$		0			0
$22$		0			0
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$	1.00000	−	1.00000i	0.196116	−	0.196116i
$27$		0			0
$28$			6.00000i			1.13389i
$29$		−	4.00000i		−	0.742781i	−0.928477	−	0.371391i	$-0.878881\pi$
$29$		−	4.00000i		−	0.742781i	0.928477	−	0.371391i	$-0.121119\pi$
$30$		0			0
$31$	−8.00000			−1.43684			−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000			−1.43684			−0.718421	+	0.695608i	$0.755135\pi$
$32$	−4.00000	−	4.00000i	−0.707107	−	0.707107i
$33$		0			0
$34$	7.00000	+	7.00000i	1.20049	+	1.20049i
$35$			9.00000i			1.52128i
$36$		0			0
$37$		−	7.00000i		−	1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$		−	7.00000i		−	1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$	−4.00000	+	4.00000i	−0.648886	+	0.648886i
$39$		0			0
$40$	−6.00000	−	6.00000i	−0.948683	−	0.948683i
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$			1.00000i			0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$			1.00000i			0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$		0			0
$45$		0			0
$46$	−4.00000	−	4.00000i	−0.589768	−	0.589768i
$47$	7.00000			1.02105			0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000			1.02105			0.510527	+	0.859861i	$0.329450\pi$
$48$		0			0
$49$	2.00000			0.285714
$50$	−4.00000	−	4.00000i	−0.565685	−	0.565685i
$51$		0			0
$52$	2.00000			0.277350
$53$			4.00000i			0.549442i	0.961524	+	0.274721i	$0.0885855\pi$
$53$			4.00000i			0.549442i	−0.961524	+	0.274721i	$0.911414\pi$
$54$		0			0
$55$		0			0
$56$	−6.00000	+	6.00000i	−0.801784	+	0.801784i
$57$		0			0
$58$	4.00000	−	4.00000i	0.525226	−	0.525226i
$59$		−	14.0000i		−	1.82264i	−0.411693	−	0.911322i	$-0.635063\pi$
$59$		−	14.0000i		−	1.82264i	0.411693	−	0.911322i	$-0.364937\pi$
$60$		0			0
$61$			10.0000i			1.28037i	0.768221	+	0.640184i	$0.221142\pi$
$61$			10.0000i			1.28037i	−0.768221	+	0.640184i	$0.778858\pi$
$62$	−8.00000	−	8.00000i	−1.01600	−	1.01600i
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$	3.00000			0.372104
$66$		0			0
$67$		−	2.00000i		−	0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$		−	2.00000i		−	0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$			14.0000i			1.69775i
$69$		0			0
$70$	−9.00000	+	9.00000i	−1.07571	+	1.07571i
$71$	3.00000			0.356034			0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000			0.356034			0.178017	+	0.984027i	$0.443032\pi$
$72$		0			0
$73$	14.0000			1.63858			0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000			1.63858			0.819288	+	0.573382i	$0.194369\pi$
$74$	7.00000	−	7.00000i	0.813733	−	0.813733i
$75$		0			0
$76$	−8.00000			−0.917663
$77$		0			0
$78$		0			0
$79$	−10.0000			−1.12509			−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000			−1.12509			−0.562544	+	0.826767i	$0.690177\pi$
$80$		−	12.0000i		−	1.34164i
$81$		0			0
$82$	−2.00000	−	2.00000i	−0.220863	−	0.220863i
$83$			14.0000i			1.53670i	0.640030	+	0.768350i	$0.278922\pi$
$83$			14.0000i			1.53670i	−0.640030	+	0.768350i	$0.721078\pi$
$84$		0			0
$85$			21.0000i			2.27777i
$86$	−1.00000	+	1.00000i	−0.107833	+	0.107833i
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		−	3.00000i		−	0.314485i
$92$		−	8.00000i		−	0.834058i
$93$		0			0
$94$	7.00000	+	7.00000i	0.721995	+	0.721995i
$95$	−12.0000			−1.23117
$96$		0			0
$97$	8.00000			0.812277			0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000			0.812277			0.406138	+	0.913812i	$0.366875\pi$
$98$	2.00000	+	2.00000i	0.202031	+	0.202031i
$99$		0			0
$100$		−	8.00000i		−	0.800000i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−6.00000			−0.591198			−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000			−0.591198			−0.295599	+	0.955312i	$0.595519\pi$
$104$	2.00000	+	2.00000i	0.196116	+	0.196116i
$105$		0			0
$106$	−4.00000	+	4.00000i	−0.388514	+	0.388514i
$107$		−	8.00000i		−	0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$		−	8.00000i		−	0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$		0			0
$109$		−	1.00000i		−	0.0957826i	−0.998853	−	0.0478913i	$-0.984750\pi$
$109$		−	1.00000i		−	0.0957826i	0.998853	−	0.0478913i	$-0.0152501\pi$
$110$		0			0
$111$		0			0
$112$	−12.0000			−1.13389
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$		−	12.0000i		−	1.11901i
$116$	8.00000			0.742781
$117$		0			0
$118$	14.0000	−	14.0000i	1.28880	−	1.28880i
$119$	21.0000			1.92507
$120$		0			0
$121$	11.0000			1.00000
$122$	−10.0000	+	10.0000i	−0.905357	+	0.905357i
$123$		0			0
$124$		−	16.0000i		−	1.43684i
$125$			3.00000i			0.268328i
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$	8.00000	−	8.00000i	0.707107	−	0.707107i
$129$		0			0
$130$	3.00000	+	3.00000i	0.263117	+	0.263117i
$131$		−	15.0000i		−	1.31056i	−0.755388	−	0.655278i	$-0.772551\pi$
$131$		−	15.0000i		−	1.31056i	0.755388	−	0.655278i	$-0.227449\pi$
$132$		0			0
$133$			12.0000i			1.04053i
$134$	2.00000	−	2.00000i	0.172774	−	0.172774i
$135$		0			0
$136$	−14.0000	+	14.0000i	−1.20049	+	1.20049i
$137$	12.0000			1.02523			0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000			1.02523			0.512615	+	0.858619i	$0.328677\pi$
$138$		0			0
$139$		−	11.0000i		−	0.933008i	−0.884519	−	0.466504i	$-0.845513\pi$
$139$		−	11.0000i		−	0.933008i	0.884519	−	0.466504i	$-0.154487\pi$
$140$	−18.0000			−1.52128
$141$		0			0
$142$	3.00000	+	3.00000i	0.251754	+	0.251754i
$143$		0			0
$144$		0			0
$145$	12.0000			0.996546
$146$	14.0000	+	14.0000i	1.15865	+	1.15865i
$147$		0			0
$148$	14.0000			1.15079
$149$			6.00000i			0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$			6.00000i			0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$		0			0
$151$	17.0000			1.38344			0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000			1.38344			0.691720	+	0.722166i	$0.256853\pi$
$152$	−8.00000	−	8.00000i	−0.648886	−	0.648886i
$153$		0			0
$154$		0			0
$155$		−	24.0000i		−	1.92773i
$156$		0			0
$157$		−	2.00000i		−	0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$		−	2.00000i		−	0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	−10.0000	−	10.0000i	−0.795557	−	0.795557i
$159$		0			0
$160$	12.0000	−	12.0000i	0.948683	−	0.948683i
$161$	−12.0000			−0.945732
$162$		0			0
$163$		−	24.0000i		−	1.87983i	−0.341415	−	0.939913i	$-0.610906\pi$
$163$		−	24.0000i		−	1.87983i	0.341415	−	0.939913i	$-0.389094\pi$
$164$		−	4.00000i		−	0.312348i
$165$		0			0
$166$	−14.0000	+	14.0000i	−1.08661	+	1.08661i
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$	−21.0000	+	21.0000i	−1.61063	+	1.61063i
$171$		0			0
$172$	−2.00000			−0.152499
$173$		−	6.00000i		−	0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$		−	6.00000i		−	0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$		0			0
$175$	−12.0000			−0.907115
$176$		0			0
$177$		0			0
$178$		0			0
$179$			21.0000i			1.56961i	0.619740	+	0.784807i	$0.287238\pi$
$179$			21.0000i			1.56961i	−0.619740	+	0.784807i	$0.712762\pi$
$180$		0			0
$181$		−	10.0000i		−	0.743294i	−0.928374	−	0.371647i	$-0.878793\pi$
$181$		−	10.0000i		−	0.743294i	0.928374	−	0.371647i	$-0.121207\pi$
$182$	3.00000	−	3.00000i	0.222375	−	0.222375i
$183$		0			0
$184$	8.00000	−	8.00000i	0.589768	−	0.589768i
$185$	21.0000			1.54395
$186$		0			0
$187$		0			0
$188$			14.0000i			1.02105i
$189$		0			0
$190$	−12.0000	−	12.0000i	−0.870572	−	0.870572i
$191$	−12.0000			−0.868290			−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000			−0.868290			−0.434145	+	0.900843i	$0.642949\pi$
$192$		0			0
$193$	−6.00000			−0.431889			−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000			−0.431889			−0.215945	+	0.976406i	$0.569283\pi$
$194$	8.00000	+	8.00000i	0.574367	+	0.574367i
$195$		0			0
$196$			4.00000i			0.285714i
$197$			17.0000i			1.21120i	0.795769	+	0.605600i	$0.207067\pi$
$197$			17.0000i			1.21120i	−0.795769	+	0.605600i	$0.792933\pi$
$198$		0			0
$199$	−10.0000			−0.708881			−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000			−0.708881			−0.354441	+	0.935079i	$0.615329\pi$
$200$	8.00000	−	8.00000i	0.565685	−	0.565685i
$201$		0			0
$202$		0			0
$203$		−	12.0000i		−	0.842235i
$204$		0			0
$205$		−	6.00000i		−	0.419058i
$206$	−6.00000	−	6.00000i	−0.418040	−	0.418040i
$207$		0			0
$208$			4.00000i			0.277350i
$209$		0			0
$210$		0			0
$211$			15.0000i			1.03264i	0.856395	+	0.516321i	$0.172699\pi$
$211$			15.0000i			1.03264i	−0.856395	+	0.516321i	$0.827301\pi$
$212$	−8.00000			−0.549442
$213$		0			0
$214$	8.00000	−	8.00000i	0.546869	−	0.546869i
$215$	−3.00000			−0.204598
$216$		0			0
$217$	−24.0000			−1.62923
$218$	1.00000	−	1.00000i	0.0677285	−	0.0677285i
$219$		0			0
$220$		0			0
$221$		−	7.00000i		−	0.470871i
$222$		0			0
$223$	−1.00000			−0.0669650			−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000			−0.0669650			−0.0334825	+	0.999439i	$0.510660\pi$
$224$	−12.0000	−	12.0000i	−0.801784	−	0.801784i
$225$		0			0
$226$	6.00000	+	6.00000i	0.399114	+	0.399114i
$227$		−	18.0000i		−	1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$		−	18.0000i		−	1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$		0			0
$229$		−	21.0000i		−	1.38772i	−0.720110	−	0.693860i	$-0.755909\pi$
$229$		−	21.0000i		−	1.38772i	0.720110	−	0.693860i	$-0.244091\pi$
$230$	12.0000	−	12.0000i	0.791257	−	0.791257i
$231$		0			0
$232$	8.00000	+	8.00000i	0.525226	+	0.525226i
$233$	−19.0000			−1.24473			−0.622366	−	0.782727i	$-0.713828\pi$
$233$	−19.0000			−1.24473			−0.622366	+	0.782727i	$0.713828\pi$
$234$		0			0
$235$			21.0000i			1.36989i
$236$	28.0000			1.82264
$237$		0			0
$238$	21.0000	+	21.0000i	1.36123	+	1.36123i
$239$	−5.00000			−0.323423			−0.161712	−	0.986838i	$-0.551701\pi$
$239$	−5.00000			−0.323423			−0.161712	+	0.986838i	$0.551701\pi$
$240$		0			0
$241$	−8.00000			−0.515325			−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000			−0.515325			−0.257663	+	0.966235i	$0.582952\pi$
$242$	11.0000	+	11.0000i	0.707107	+	0.707107i
$243$		0			0
$244$	−20.0000			−1.28037
$245$			6.00000i			0.383326i
$246$		0			0
$247$	4.00000			0.254514
$248$	16.0000	−	16.0000i	1.01600	−	1.01600i
$249$		0			0
$250$	−3.00000	+	3.00000i	−0.189737	+	0.189737i
$251$			20.0000i			1.26239i	0.775625	+	0.631194i	$0.217435\pi$
$251$			20.0000i			1.26239i	−0.775625	+	0.631194i	$0.782565\pi$
$252$		0			0
$253$		0			0
$254$	8.00000	+	8.00000i	0.501965	+	0.501965i
$255$		0			0
$256$	16.0000			1.00000
$257$	7.00000			0.436648			0.218324	−	0.975876i	$-0.429941\pi$
$257$	7.00000			0.436648			0.218324	+	0.975876i	$0.429941\pi$
$258$		0			0
$259$		−	21.0000i		−	1.30488i
$260$			6.00000i			0.372104i
$261$		0			0
$262$	15.0000	−	15.0000i	0.926703	−	0.926703i
$263$	−14.0000			−0.863277			−0.431638	−	0.902047i	$-0.642064\pi$
$263$	−14.0000			−0.863277			−0.431638	+	0.902047i	$0.642064\pi$
$264$		0			0
$265$	−12.0000			−0.737154
$266$	−12.0000	+	12.0000i	−0.735767	+	0.735767i
$267$		0			0
$268$	4.00000			0.244339
$269$			6.00000i			0.365826i	0.983129	+	0.182913i	$0.0585527\pi$
$269$			6.00000i			0.365826i	−0.983129	+	0.182913i	$0.941447\pi$
$270$		0			0
$271$	−3.00000			−0.182237			−0.0911185	−	0.995840i	$-0.529044\pi$
$271$	−3.00000			−0.182237			−0.0911185	+	0.995840i	$0.529044\pi$
$272$	−28.0000			−1.69775
$273$		0			0
$274$	12.0000	+	12.0000i	0.724947	+	0.724947i
$275$		0			0
$276$		0			0
$277$			18.0000i			1.08152i	0.841178	+	0.540758i	$0.181862\pi$
$277$			18.0000i			1.08152i	−0.841178	+	0.540758i	$0.818138\pi$
$278$	11.0000	−	11.0000i	0.659736	−	0.659736i
$279$		0			0
$280$	−18.0000	−	18.0000i	−1.07571	−	1.07571i
$281$	8.00000			0.477240			0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000			0.477240			0.238620	+	0.971113i	$0.423305\pi$
$282$		0			0
$283$		−	4.00000i		−	0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$		−	4.00000i		−	0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$			6.00000i			0.356034i
$285$		0			0
$286$		0			0
$287$	−6.00000			−0.354169
$288$		0			0
$289$	32.0000			1.88235
$290$	12.0000	+	12.0000i	0.704664	+	0.704664i
$291$		0			0
$292$			28.0000i			1.63858i
$293$		−	1.00000i		−	0.0584206i	−0.999573	−	0.0292103i	$-0.990701\pi$
$293$		−	1.00000i		−	0.0584206i	0.999573	−	0.0292103i	$-0.00929925\pi$
$294$		0			0
$295$	42.0000			2.44533
$296$	14.0000	+	14.0000i	0.813733	+	0.813733i
$297$		0			0
$298$	−6.00000	+	6.00000i	−0.347571	+	0.347571i
$299$			4.00000i			0.231326i
$300$		0			0
$301$			3.00000i			0.172917i
$302$	17.0000	+	17.0000i	0.978240	+	0.978240i
$303$		0			0
$304$		−	16.0000i		−	0.917663i
$305$	−30.0000			−1.71780
$306$		0			0
$307$		−	2.00000i		−	0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$		−	2.00000i		−	0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$		0			0
$309$		0			0
$310$	24.0000	−	24.0000i	1.36311	−	1.36311i
$311$	−32.0000			−1.81455			−0.907277	−	0.420534i	$-0.861843\pi$
$311$	−32.0000			−1.81455			−0.907277	+	0.420534i	$0.861843\pi$
$312$		0			0
$313$	29.0000			1.63918			0.819588	−	0.572953i	$-0.194202\pi$
$313$	29.0000			1.63918			0.819588	+	0.572953i	$0.194202\pi$
$314$	2.00000	−	2.00000i	0.112867	−	0.112867i
$315$		0			0
$316$		−	20.0000i		−	1.12509i
$317$		−	18.0000i		−	1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$		−	18.0000i		−	1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$		0			0
$319$		0			0
$320$	24.0000			1.34164
$321$		0			0
$322$	−12.0000	−	12.0000i	−0.668734	−	0.668734i
$323$			28.0000i			1.55796i
$324$		0			0
$325$			4.00000i			0.221880i
$326$	24.0000	−	24.0000i	1.32924	−	1.32924i
$327$		0			0
$328$	4.00000	−	4.00000i	0.220863	−	0.220863i
$329$	21.0000			1.15777
$330$		0			0
$331$		−	10.0000i		−	0.549650i	−0.961494	−	0.274825i	$-0.911380\pi$
$331$		−	10.0000i		−	0.549650i	0.961494	−	0.274825i	$-0.0886199\pi$
$332$	−28.0000			−1.53670
$333$		0			0
$334$	12.0000	+	12.0000i	0.656611	+	0.656611i
$335$	6.00000			0.327815
$336$		0			0
$337$	−17.0000			−0.926049			−0.463025	−	0.886345i	$-0.653236\pi$
$337$	−17.0000			−0.926049			−0.463025	+	0.886345i	$0.653236\pi$
$338$	−1.00000	−	1.00000i	−0.0543928	−	0.0543928i
$339$		0			0
$340$	−42.0000			−2.27777
$341$		0			0
$342$		0			0
$343$	−15.0000			−0.809924
$344$	−2.00000	−	2.00000i	−0.107833	−	0.107833i
$345$		0			0
$346$	6.00000	−	6.00000i	0.322562	−	0.322562i
$347$			7.00000i			0.375780i	0.982190	+	0.187890i	$0.0601648\pi$
$347$			7.00000i			0.375780i	−0.982190	+	0.187890i	$0.939835\pi$
$348$		0			0
$349$			19.0000i			1.01705i	0.861048	+	0.508523i	$0.169808\pi$
$349$			19.0000i			1.01705i	−0.861048	+	0.508523i	$0.830192\pi$
$350$	−12.0000	−	12.0000i	−0.641427	−	0.641427i
$351$		0			0
$352$		0			0
$353$	6.00000			0.319348			0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000			0.319348			0.159674	+	0.987170i	$0.448956\pi$
$354$		0			0
$355$			9.00000i			0.477670i
$356$		0			0
$357$		0			0
$358$	−21.0000	+	21.0000i	−1.10988	+	1.10988i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$	10.0000	−	10.0000i	0.525588	−	0.525588i
$363$		0			0
$364$	6.00000			0.314485
$365$			42.0000i			2.19838i
$366$		0			0
$367$	−22.0000			−1.14839			−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000			−1.14839			−0.574195	+	0.818718i	$0.694685\pi$
$368$	16.0000			0.834058
$369$		0			0
$370$	21.0000	+	21.0000i	1.09174	+	1.09174i
$371$			12.0000i			0.623009i
$372$		0			0
$373$			36.0000i			1.86401i	0.362446	+	0.932005i	$0.381942\pi$
$373$			36.0000i			1.86401i	−0.362446	+	0.932005i	$0.618058\pi$
$374$		0			0
$375$		0			0
$376$	−14.0000	+	14.0000i	−0.721995	+	0.721995i
$377$	−4.00000			−0.206010
$378$		0			0
$379$			4.00000i			0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$			4.00000i			0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$		−	24.0000i		−	1.23117i
$381$		0			0
$382$	−12.0000	−	12.0000i	−0.613973	−	0.613973i
$383$	1.00000			0.0510976			0.0255488	−	0.999674i	$-0.491867\pi$
$383$	1.00000			0.0510976			0.0255488	+	0.999674i	$0.491867\pi$
$384$		0			0
$385$		0			0
$386$	−6.00000	−	6.00000i	−0.305392	−	0.305392i
$387$		0			0
$388$			16.0000i			0.812277i
$389$		−	34.0000i		−	1.72387i	−0.507020	−	0.861934i	$-0.669253\pi$
$389$		−	34.0000i		−	1.72387i	0.507020	−	0.861934i	$-0.330747\pi$
$390$		0			0
$391$	−28.0000			−1.41602
$392$	−4.00000	+	4.00000i	−0.202031	+	0.202031i
$393$		0			0
$394$	−17.0000	+	17.0000i	−0.856448	+	0.856448i
$395$		−	30.0000i		−	1.50946i
$396$		0			0
$397$		−	22.0000i		−	1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$		−	22.0000i		−	1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$	−10.0000	−	10.0000i	−0.501255	−	0.501255i
$399$		0			0
$400$	16.0000			0.800000
$401$	−32.0000			−1.59800			−0.799002	−	0.601329i	$-0.794638\pi$
$401$	−32.0000			−1.59800			−0.799002	+	0.601329i	$0.794638\pi$
$402$		0			0
$403$			8.00000i			0.398508i
$404$		0			0
$405$		0			0
$406$	12.0000	−	12.0000i	0.595550	−	0.595550i
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$	6.00000	−	6.00000i	0.296319	−	0.296319i
$411$		0			0
$412$		−	12.0000i		−	0.591198i
$413$		−	42.0000i		−	2.06668i
$414$		0			0
$415$	−42.0000			−2.06170
$416$	−4.00000	+	4.00000i	−0.196116	+	0.196116i
$417$		0			0
$418$		0			0
$419$			11.0000i			0.537385i	0.963226	+	0.268693i	$0.0865916\pi$
$419$			11.0000i			0.537385i	−0.963226	+	0.268693i	$0.913408\pi$
$420$		0			0
$421$		−	15.0000i		−	0.731055i	−0.930800	−	0.365528i	$-0.880889\pi$
$421$		−	15.0000i		−	0.731055i	0.930800	−	0.365528i	$-0.119111\pi$
$422$	−15.0000	+	15.0000i	−0.730189	+	0.730189i
$423$		0			0
$424$	−8.00000	−	8.00000i	−0.388514	−	0.388514i
$425$	−28.0000			−1.35820
$426$		0			0
$427$			30.0000i			1.45180i
$428$	16.0000			0.773389
$429$		0			0
$430$	−3.00000	−	3.00000i	−0.144673	−	0.144673i
$431$	3.00000			0.144505			0.0722525	−	0.997386i	$-0.476981\pi$
$431$	3.00000			0.144505			0.0722525	+	0.997386i	$0.476981\pi$
$432$		0			0
$433$	−1.00000			−0.0480569			−0.0240285	−	0.999711i	$-0.507649\pi$
$433$	−1.00000			−0.0480569			−0.0240285	+	0.999711i	$0.507649\pi$
$434$	−24.0000	−	24.0000i	−1.15204	−	1.15204i
$435$		0			0
$436$	2.00000			0.0957826
$437$		−	16.0000i		−	0.765384i
$438$		0			0
$439$	−10.0000			−0.477274			−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000			−0.477274			−0.238637	+	0.971109i	$0.576701\pi$
$440$		0			0
$441$		0			0
$442$	7.00000	−	7.00000i	0.332956	−	0.332956i
$443$			9.00000i			0.427603i	0.976877	+	0.213801i	$0.0685846\pi$
$443$			9.00000i			0.427603i	−0.976877	+	0.213801i	$0.931415\pi$
$444$		0			0
$445$		0			0
$446$	−1.00000	−	1.00000i	−0.0473514	−	0.0473514i
$447$		0			0
$448$		−	24.0000i		−	1.13389i
$449$	20.0000			0.943858			0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000			0.943858			0.471929	+	0.881636i	$0.343558\pi$
$450$		0			0
$451$		0			0
$452$			12.0000i			0.564433i
$453$		0			0
$454$	18.0000	−	18.0000i	0.844782	−	0.844782i
$455$	9.00000			0.421927
$456$		0			0
$457$	−32.0000			−1.49690			−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000			−1.49690			−0.748448	+	0.663193i	$0.769201\pi$
$458$	21.0000	−	21.0000i	0.981266	−	0.981266i
$459$		0			0
$460$	24.0000			1.11901
$461$			5.00000i			0.232873i	0.993198	+	0.116437i	$0.0371472\pi$
$461$			5.00000i			0.232873i	−0.993198	+	0.116437i	$0.962853\pi$
$462$		0			0
$463$	−16.0000			−0.743583			−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000			−0.743583			−0.371792	+	0.928316i	$0.621256\pi$
$464$			16.0000i			0.742781i
$465$		0			0
$466$	−19.0000	−	19.0000i	−0.880158	−	0.880158i
$467$			12.0000i			0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$			12.0000i			0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$		0			0
$469$		−	6.00000i		−	0.277054i
$470$	−21.0000	+	21.0000i	−0.968658	+	0.968658i
$471$		0			0
$472$	28.0000	+	28.0000i	1.28880	+	1.28880i
$473$		0			0
$474$		0			0
$475$		−	16.0000i		−	0.734130i
$476$			42.0000i			1.92507i
$477$		0			0
$478$	−5.00000	−	5.00000i	−0.228695	−	0.228695i
$479$	−5.00000			−0.228456			−0.114228	−	0.993455i	$-0.536439\pi$
$479$	−5.00000			−0.228456			−0.114228	+	0.993455i	$0.536439\pi$
$480$		0			0
$481$	−7.00000			−0.319173
$482$	−8.00000	−	8.00000i	−0.364390	−	0.364390i
$483$		0			0
$484$			22.0000i			1.00000i
$485$			24.0000i			1.08978i
$486$		0			0
$487$	8.00000			0.362515			0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000			0.362515			0.181257	+	0.983436i	$0.441983\pi$
$488$	−20.0000	−	20.0000i	−0.905357	−	0.905357i
$489$		0			0
$490$	−6.00000	+	6.00000i	−0.271052	+	0.271052i
$491$			15.0000i			0.676941i	0.940977	+	0.338470i	$0.109909\pi$
$491$			15.0000i			0.676941i	−0.940977	+	0.338470i	$0.890091\pi$
$492$		0			0
$493$		−	28.0000i		−	1.26106i
$494$	4.00000	+	4.00000i	0.179969	+	0.179969i
$495$		0			0
$496$	32.0000			1.43684
$497$	9.00000			0.403705
$498$		0			0
$499$		−	6.00000i		−	0.268597i	−0.990941	−	0.134298i	$-0.957122\pi$
$499$		−	6.00000i		−	0.268597i	0.990941	−	0.134298i	$-0.0428781\pi$
$500$	−6.00000			−0.268328
$501$		0			0
$502$	−20.0000	+	20.0000i	−0.892644	+	0.892644i
$503$	−4.00000			−0.178351			−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000			−0.178351			−0.0891756	+	0.996016i	$0.528423\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$			16.0000i			0.709885i
$509$		−	14.0000i		−	0.620539i	−0.950649	−	0.310270i	$-0.899581\pi$
$509$		−	14.0000i		−	0.620539i	0.950649	−	0.310270i	$-0.100419\pi$
$510$		0			0
$511$	42.0000			1.85797
$512$	16.0000	+	16.0000i	0.707107	+	0.707107i
$513$		0			0
$514$	7.00000	+	7.00000i	0.308757	+	0.308757i
$515$		−	18.0000i		−	0.793175i
$516$		0			0
$517$		0			0
$518$	21.0000	−	21.0000i	0.922687	−	0.922687i
$519$		0			0
$520$	−6.00000	+	6.00000i	−0.263117	+	0.263117i
$521$	3.00000			0.131432			0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000			0.131432			0.0657162	+	0.997838i	$0.479067\pi$
$522$		0			0
$523$		−	44.0000i		−	1.92399i	−0.273075	−	0.961993i	$-0.588041\pi$
$523$		−	44.0000i		−	1.92399i	0.273075	−	0.961993i	$-0.411959\pi$
$524$	30.0000			1.31056
$525$		0			0
$526$	−14.0000	−	14.0000i	−0.610429	−	0.610429i
$527$	−56.0000			−2.43940
$528$		0			0
$529$	−7.00000			−0.304348
$530$	−12.0000	−	12.0000i	−0.521247	−	0.521247i
$531$		0			0
$532$	−24.0000			−1.04053
$533$			2.00000i			0.0866296i
$534$		0			0
$535$	24.0000			1.03761
$536$	4.00000	+	4.00000i	0.172774	+	0.172774i
$537$		0			0
$538$	−6.00000	+	6.00000i	−0.258678	+	0.258678i
$539$		0			0
$540$		0			0
$541$			25.0000i			1.07483i	0.843317	+	0.537417i	$0.180600\pi$
$541$			25.0000i			1.07483i	−0.843317	+	0.537417i	$0.819400\pi$
$542$	−3.00000	−	3.00000i	−0.128861	−	0.128861i
$543$		0			0
$544$	−28.0000	−	28.0000i	−1.20049	−	1.20049i
$545$	3.00000			0.128506
$546$		0			0
$547$			13.0000i			0.555840i	0.960604	+	0.277920i	$0.0896450\pi$
$547$			13.0000i			0.555840i	−0.960604	+	0.277920i	$0.910355\pi$
$548$			24.0000i			1.02523i
$549$		0			0
$550$		0			0
$551$	16.0000			0.681623
$552$		0			0
$553$	−30.0000			−1.27573
$554$	−18.0000	+	18.0000i	−0.764747	+	0.764747i
$555$		0			0
$556$	22.0000			0.933008
$557$		−	13.0000i		−	0.550828i	−0.961326	−	0.275414i	$-0.911185\pi$
$557$		−	13.0000i		−	0.550828i	0.961326	−	0.275414i	$-0.0888149\pi$
$558$		0			0
$559$	1.00000			0.0422955
$560$		−	36.0000i		−	1.52128i
$561$		0			0
$562$	8.00000	+	8.00000i	0.337460	+	0.337460i
$563$		−	31.0000i		−	1.30649i	−0.757145	−	0.653247i	$-0.773406\pi$
$563$		−	31.0000i		−	1.30649i	0.757145	−	0.653247i	$-0.226594\pi$
$564$		0			0
$565$			18.0000i			0.757266i
$566$	4.00000	−	4.00000i	0.168133	−	0.168133i
$567$		0			0
$568$	−6.00000	+	6.00000i	−0.251754	+	0.251754i
$569$	−25.0000			−1.04805			−0.524027	−	0.851701i	$-0.675571\pi$
$569$	−25.0000			−1.04805			−0.524027	+	0.851701i	$0.675571\pi$
$570$		0			0
$571$		−	5.00000i		−	0.209243i	−0.994512	−	0.104622i	$-0.966637\pi$
$571$		−	5.00000i		−	0.209243i	0.994512	−	0.104622i	$-0.0333632\pi$
$572$		0			0
$573$		0			0
$574$	−6.00000	−	6.00000i	−0.250435	−	0.250435i
$575$	16.0000			0.667246
$576$		0			0
$577$	−2.00000			−0.0832611			−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000			−0.0832611			−0.0416305	+	0.999133i	$0.513255\pi$
$578$	32.0000	+	32.0000i	1.33102	+	1.33102i
$579$		0			0
$580$			24.0000i			0.996546i
$581$			42.0000i			1.74245i
$582$		0			0
$583$		0			0
$584$	−28.0000	+	28.0000i	−1.15865	+	1.15865i
$585$		0			0
$586$	1.00000	−	1.00000i	0.0413096	−	0.0413096i
$587$		−	28.0000i		−	1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$		−	28.0000i		−	1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$		0			0
$589$		−	32.0000i		−	1.31854i
$590$	42.0000	+	42.0000i	1.72911	+	1.72911i
$591$		0			0
$592$			28.0000i			1.15079i
$593$	−24.0000			−0.985562			−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000			−0.985562			−0.492781	+	0.870153i	$0.664020\pi$
$594$		0			0
$595$			63.0000i			2.58275i
$596$	−12.0000			−0.491539
$597$		0			0
$598$	−4.00000	+	4.00000i	−0.163572	+	0.163572i
$599$	−10.0000			−0.408589			−0.204294	−	0.978909i	$-0.565490\pi$
$599$	−10.0000			−0.408589			−0.204294	+	0.978909i	$0.565490\pi$
$600$		0			0
$601$	27.0000			1.10135			0.550676	−	0.834719i	$-0.314370\pi$
$601$	27.0000			1.10135			0.550676	+	0.834719i	$0.314370\pi$
$602$	−3.00000	+	3.00000i	−0.122271	+	0.122271i
$603$		0			0
$604$			34.0000i			1.38344i
$605$			33.0000i			1.34164i
$606$		0			0
$607$	−22.0000			−0.892952			−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000			−0.892952			−0.446476	+	0.894795i	$0.647321\pi$
$608$	16.0000	−	16.0000i	0.648886	−	0.648886i
$609$		0			0
$610$	−30.0000	−	30.0000i	−1.21466	−	1.21466i
$611$		−	7.00000i		−	0.283190i
$612$		0			0
$613$		−	14.0000i		−	0.565455i	−0.959200	−	0.282727i	$-0.908761\pi$
$613$		−	14.0000i		−	0.565455i	0.959200	−	0.282727i	$-0.0912392\pi$
$614$	2.00000	−	2.00000i	0.0807134	−	0.0807134i
$615$		0			0
$616$		0			0
$617$	12.0000			0.483102			0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000			0.483102			0.241551	+	0.970388i	$0.422344\pi$
$618$		0			0
$619$		−	6.00000i		−	0.241160i	−0.992704	−	0.120580i	$-0.961525\pi$
$619$		−	6.00000i		−	0.241160i	0.992704	−	0.120580i	$-0.0384755\pi$
$620$	48.0000			1.92773
$621$		0			0
$622$	−32.0000	−	32.0000i	−1.28308	−	1.28308i
$623$		0			0
$624$		0			0
$625$	−29.0000			−1.16000
$626$	29.0000	+	29.0000i	1.15907	+	1.15907i
$627$		0			0
$628$	4.00000			0.159617
$629$		−	49.0000i		−	1.95376i
$630$		0			0
$631$	−23.0000			−0.915616			−0.457808	−	0.889051i	$-0.651365\pi$
$631$	−23.0000			−0.915616			−0.457808	+	0.889051i	$0.651365\pi$
$632$	20.0000	−	20.0000i	0.795557	−	0.795557i
$633$		0			0
$634$	18.0000	−	18.0000i	0.714871	−	0.714871i
$635$			24.0000i			0.952411i
$636$		0			0
$637$		−	2.00000i		−	0.0792429i
$638$		0			0
$639$		0			0
$640$	24.0000	+	24.0000i	0.948683	+	0.948683i
$641$	−2.00000			−0.0789953			−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000			−0.0789953			−0.0394976	+	0.999220i	$0.512576\pi$
$642$		0			0
$643$			26.0000i			1.02534i	0.858586	+	0.512670i	$0.171344\pi$
$643$			26.0000i			1.02534i	−0.858586	+	0.512670i	$0.828656\pi$
$644$		−	24.0000i		−	0.945732i
$645$		0			0
$646$	−28.0000	+	28.0000i	−1.10165	+	1.10165i
$647$	−18.0000			−0.707653			−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000			−0.707653			−0.353827	+	0.935311i	$0.615120\pi$
$648$		0			0
$649$		0			0
$650$	−4.00000	+	4.00000i	−0.156893	+	0.156893i
$651$		0			0
$652$	48.0000			1.87983
$653$			44.0000i			1.72185i	0.508729	+	0.860927i	$0.330115\pi$
$653$			44.0000i			1.72185i	−0.508729	+	0.860927i	$0.669885\pi$
$654$		0			0
$655$	45.0000			1.75830
$656$	8.00000			0.312348
$657$		0			0
$658$	21.0000	+	21.0000i	0.818665	+	0.818665i
$659$		−	4.00000i		−	0.155818i	−0.996960	−	0.0779089i	$-0.975176\pi$
$659$		−	4.00000i		−	0.155818i	0.996960	−	0.0779089i	$-0.0248243\pi$
$660$		0			0
$661$		−	30.0000i		−	1.16686i	−0.812162	−	0.583432i	$-0.801709\pi$
$661$		−	30.0000i		−	1.16686i	0.812162	−	0.583432i	$-0.198291\pi$
$662$	10.0000	−	10.0000i	0.388661	−	0.388661i
$663$		0			0
$664$	−28.0000	−	28.0000i	−1.08661	−	1.08661i
$665$	−36.0000			−1.39602
$666$		0			0
$667$			16.0000i			0.619522i
$668$			24.0000i			0.928588i
$669$		0			0
$670$	6.00000	+	6.00000i	0.231800	+	0.231800i
$671$		0			0
$672$		0			0
$673$	29.0000			1.11787			0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000			1.11787			0.558934	+	0.829212i	$0.311211\pi$
$674$	−17.0000	−	17.0000i	−0.654816	−	0.654816i
$675$		0			0
$676$		−	2.00000i		−	0.0769231i
$677$			22.0000i			0.845529i	0.906240	+	0.422764i	$0.138940\pi$
$677$			22.0000i			0.845529i	−0.906240	+	0.422764i	$0.861060\pi$
$678$		0			0
$679$	24.0000			0.921035
$680$	−42.0000	−	42.0000i	−1.61063	−	1.61063i
$681$		0			0
$682$		0			0
$683$		−	26.0000i		−	0.994862i	−0.867503	−	0.497431i	$-0.834277\pi$
$683$		−	26.0000i		−	0.994862i	0.867503	−	0.497431i	$-0.165723\pi$
$684$		0			0
$685$			36.0000i			1.37549i
$686$	−15.0000	−	15.0000i	−0.572703	−	0.572703i
$687$		0			0
$688$		−	4.00000i		−	0.152499i
$689$	4.00000			0.152388
$690$		0			0
$691$			20.0000i			0.760836i	0.924815	+	0.380418i	$0.124220\pi$
$691$			20.0000i			0.760836i	−0.924815	+	0.380418i	$0.875780\pi$
$692$	12.0000			0.456172
$693$		0			0
$694$	−7.00000	+	7.00000i	−0.265716	+	0.265716i
$695$	33.0000			1.25176
$696$		0			0
$697$	−14.0000			−0.530288
$698$	−19.0000	+	19.0000i	−0.719161	+	0.719161i
$699$		0			0
$700$		−	24.0000i		−	0.907115i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	28.0000			1.05604
$704$		0			0
$705$		0			0
$706$	6.00000	+	6.00000i	0.225813	+	0.225813i
$707$		0			0
$708$		0			0
$709$		−	6.00000i		−	0.225335i	−0.993633	−	0.112667i	$-0.964061\pi$
$709$		−	6.00000i		−	0.225335i	0.993633	−	0.112667i	$-0.0359394\pi$
$710$	−9.00000	+	9.00000i	−0.337764	+	0.337764i
$711$		0			0
$712$		0			0
$713$	32.0000			1.19841
$714$		0			0
$715$		0			0
$716$	−42.0000			−1.56961
$717$		0			0
$718$		0			0
$719$	30.0000			1.11881			0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000			1.11881			0.559406	+	0.828894i	$0.311029\pi$
$720$		0			0
$721$	−18.0000			−0.670355
$722$	3.00000	+	3.00000i	0.111648	+	0.111648i
$723$		0			0
$724$	20.0000			0.743294
$725$			16.0000i			0.594225i
$726$		0			0
$727$	8.00000			0.296704			0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000			0.296704			0.148352	+	0.988935i	$0.452603\pi$
$728$	6.00000	+	6.00000i	0.222375	+	0.222375i
$729$		0			0
$730$	−42.0000	+	42.0000i	−1.55449	+	1.55449i
$731$			7.00000i			0.258904i
$732$		0			0
$733$			1.00000i			0.0369358i	0.999829	+	0.0184679i	$0.00587886\pi$
$733$			1.00000i			0.0369358i	−0.999829	+	0.0184679i	$0.994121\pi$
$734$	−22.0000	−	22.0000i	−0.812035	−	0.812035i
$735$		0			0
$736$	16.0000	+	16.0000i	0.589768	+	0.589768i
$737$		0			0
$738$		0			0
$739$			34.0000i			1.25071i	0.780340	+	0.625355i	$0.215046\pi$
$739$			34.0000i			1.25071i	−0.780340	+	0.625355i	$0.784954\pi$
$740$			42.0000i			1.54395i
$741$		0			0
$742$	−12.0000	+	12.0000i	−0.440534	+	0.440534i
$743$	21.0000			0.770415			0.385208	−	0.922830i	$-0.374130\pi$
$743$	21.0000			0.770415			0.385208	+	0.922830i	$0.374130\pi$
$744$		0			0
$745$	−18.0000			−0.659469
$746$	−36.0000	+	36.0000i	−1.31805	+	1.31805i
$747$		0			0
$748$		0			0
$749$		−	24.0000i		−	0.876941i
$750$		0			0
$751$	2.00000			0.0729810			0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000			0.0729810			0.0364905	+	0.999334i	$0.488382\pi$
$752$	−28.0000			−1.02105
$753$		0			0
$754$	−4.00000	−	4.00000i	−0.145671	−	0.145671i
$755$			51.0000i			1.85608i
$756$		0			0
$757$		−	32.0000i		−	1.16306i	−0.813525	−	0.581530i	$-0.802454\pi$
$757$		−	32.0000i		−	1.16306i	0.813525	−	0.581530i	$-0.197546\pi$
$758$	−4.00000	+	4.00000i	−0.145287	+	0.145287i
$759$		0			0
$760$	24.0000	−	24.0000i	0.870572	−	0.870572i
$761$	−22.0000			−0.797499			−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000			−0.797499			−0.398750	+	0.917060i	$0.630556\pi$
$762$		0			0
$763$		−	3.00000i		−	0.108607i
$764$		−	24.0000i		−	0.868290i
$765$		0			0
$766$	1.00000	+	1.00000i	0.0361315	+	0.0361315i
$767$	−14.0000			−0.505511
$768$		0			0
$769$	−20.0000			−0.721218			−0.360609	−	0.932717i	$-0.617431\pi$
$769$	−20.0000			−0.721218			−0.360609	+	0.932717i	$0.617431\pi$
$770$		0			0
$771$		0			0
$772$		−	12.0000i		−	0.431889i
$773$		−	21.0000i		−	0.755318i	−0.925945	−	0.377659i	$-0.876729\pi$
$773$		−	21.0000i		−	0.755318i	0.925945	−	0.377659i	$-0.123271\pi$
$774$		0			0
$775$	32.0000			1.14947
$776$	−16.0000	+	16.0000i	−0.574367	+	0.574367i
$777$		0			0
$778$	34.0000	−	34.0000i	1.21896	−	1.21896i
$779$		−	8.00000i		−	0.286630i
$780$		0			0
$781$		0			0
$782$	−28.0000	−	28.0000i	−1.00128	−	1.00128i
$783$		0			0
$784$	−8.00000			−0.285714
$785$	6.00000			0.214149
$786$		0			0
$787$		−	22.0000i		−	0.784215i	−0.919919	−	0.392108i	$-0.871746\pi$
$787$		−	22.0000i		−	0.784215i	0.919919	−	0.392108i	$-0.128254\pi$
$788$	−34.0000			−1.21120
$789$		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:	\( N \)	\(=\)	\( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	936.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +3.00000i q^{5} +3.00000 q^{7} +(-2.00000 + 2.00000i) q^{8} +O(q^{10})\)	\(q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +3.00000i q^{5} +3.00000 q^{7} +(-2.00000 + 2.00000i) q^{8} +(-3.00000 + 3.00000i) q^{10} -1.00000i q^{13} +(3.00000 + 3.00000i) q^{14} -4.00000 q^{16} +7.00000 q^{17} +4.00000i q^{19} -6.00000 q^{20} -4.00000 q^{23} -4.00000 q^{25} +(1.00000 - 1.00000i) q^{26} +6.00000i q^{28} -4.00000i q^{29} -8.00000 q^{31} +(-4.00000 - 4.00000i) q^{32} +(7.00000 + 7.00000i) q^{34} +9.00000i q^{35} -7.00000i q^{37} +(-4.00000 + 4.00000i) q^{38} +(-6.00000 - 6.00000i) q^{40} -2.00000 q^{41} +1.00000i q^{43} +(-4.00000 - 4.00000i) q^{46} +7.00000 q^{47} +2.00000 q^{49} +(-4.00000 - 4.00000i) q^{50} +2.00000 q^{52} +4.00000i q^{53} +(-6.00000 + 6.00000i) q^{56} +(4.00000 - 4.00000i) q^{58} -14.0000i q^{59} +10.0000i q^{61} +(-8.00000 - 8.00000i) q^{62} -8.00000i q^{64} +3.00000 q^{65} -2.00000i q^{67} +14.0000i q^{68} +(-9.00000 + 9.00000i) q^{70} +3.00000 q^{71} +14.0000 q^{73} +(7.00000 - 7.00000i) q^{74} -8.00000 q^{76} -10.0000 q^{79} -12.0000i q^{80} +(-2.00000 - 2.00000i) q^{82} +14.0000i q^{83} +21.0000i q^{85} +(-1.00000 + 1.00000i) q^{86} -3.00000i q^{91} -8.00000i q^{92} +(7.00000 + 7.00000i) q^{94} -12.0000 q^{95} +8.00000 q^{97} +(2.00000 + 2.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 6 q^{7} - 4 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 6 * q^7 - 4 * q^8	\( 2 q + 2 q^{2} + 6 q^{7} - 4 q^{8} - 6 q^{10} + 6 q^{14} - 8 q^{16} + 14 q^{17} - 12 q^{20} - 8 q^{23} - 8 q^{25} + 2 q^{26} - 16 q^{31} - 8 q^{32} + 14 q^{34} - 8 q^{38} - 12 q^{40} - 4 q^{41} - 8 q^{46} + 14 q^{47} + 4 q^{49} - 8 q^{50} + 4 q^{52} - 12 q^{56} + 8 q^{58} - 16 q^{62} + 6 q^{65} - 18 q^{70} + 6 q^{71} + 28 q^{73} + 14 q^{74} - 16 q^{76} - 20 q^{79} - 4 q^{82} - 2 q^{86} + 14 q^{94} - 24 q^{95} + 16 q^{97} + 4 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 6 * q^7 - 4 * q^8 - 6 * q^10 + 6 * q^14 - 8 * q^16 + 14 * q^17 - 12 * q^20 - 8 * q^23 - 8 * q^25 + 2 * q^26 - 16 * q^31 - 8 * q^32 + 14 * q^34 - 8 * q^38 - 12 * q^40 - 4 * q^41 - 8 * q^46 + 14 * q^47 + 4 * q^49 - 8 * q^50 + 4 * q^52 - 12 * q^56 + 8 * q^58 - 16 * q^62 + 6 * q^65 - 18 * q^70 + 6 * q^71 + 28 * q^73 + 14 * q^74 - 16 * q^76 - 20 * q^79 - 4 * q^82 - 2 * q^86 + 14 * q^94 - 24 * q^95 + 16 * q^97 + 4 * q^98

Introduction

L-functions

Modular forms

Varieties

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