LMFDB - Embedded newform 810.2.c.b.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,2,Mod(649,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("810.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	810.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$6.46788256372$
Analytic rank:	$1$
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 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	810.649
Dual form			810.2.c.b.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000i q^{8} +(2.00000 - 1.00000i) q^{10} -2.00000 q^{11} +6.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -6.00000 q^{19} +(1.00000 + 2.00000i) q^{20} -2.00000i q^{22} -1.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} -6.00000 q^{26} +1.00000i q^{28} -9.00000 q^{29} -2.00000 q^{31} +1.00000i q^{32} +2.00000 q^{34} +(-2.00000 + 1.00000i) q^{35} +2.00000i q^{37} -6.00000i q^{38} +(-2.00000 + 1.00000i) q^{40} -11.0000 q^{41} +4.00000i q^{43} +2.00000 q^{44} +1.00000 q^{46} -7.00000i q^{47} +6.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} -6.00000i q^{52} +(2.00000 + 4.00000i) q^{55} -1.00000 q^{56} -9.00000i q^{58} +4.00000 q^{59} -7.00000 q^{61} -2.00000i q^{62} -1.00000 q^{64} +(12.0000 - 6.00000i) q^{65} -11.0000i q^{67} +2.00000i q^{68} +(-1.00000 - 2.00000i) q^{70} -6.00000 q^{71} +4.00000i q^{73} -2.00000 q^{74} +6.00000 q^{76} +2.00000i q^{77} +12.0000 q^{79} +(-1.00000 - 2.00000i) q^{80} -11.0000i q^{82} +11.0000i q^{83} +(-4.00000 + 2.00000i) q^{85} -4.00000 q^{86} +2.00000i q^{88} -1.00000 q^{89} +6.00000 q^{91} +1.00000i q^{92} +7.00000 q^{94} +(6.00000 + 12.0000i) q^{95} -8.00000i q^{97} +6.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{5} + 4 q^{10} - 4 q^{11} + 2 q^{14} + 2 q^{16} - 12 q^{19} + 2 q^{20} - 6 q^{25} - 12 q^{26} - 18 q^{29} - 4 q^{31} + 4 q^{34} - 4 q^{35} - 4 q^{40} - 22 q^{41} + 4 q^{44} + 2 q^{46}+ \cdots + 12 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^5 + 4 * q^10 - 4 * q^11 + 2 * q^14 + 2 * q^16 - 12 * q^19 + 2 * q^20 - 6 * q^25 - 12 * q^26 - 18 * q^29 - 4 * q^31 + 4 * q^34 - 4 * q^35 - 4 * q^40 - 22 * q^41 + 4 * q^44 + 2 * q^46 + 12 * q^49 - 8 * q^50 + 4 * q^55 - 2 * q^56 + 8 * q^59 - 14 * q^61 - 2 * q^64 + 24 * q^65 - 2 * q^70 - 12 * q^71 - 4 * q^74 + 12 * q^76 + 24 * q^79 - 2 * q^80 - 8 * q^85 - 8 * q^86 - 2 * q^89 + 12 * q^91 + 14 * q^94 + 12 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$-1$	$1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$6$		0			0
$7$		−	1.00000i		−	0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$		−	1.00000i		−	0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	2.00000	−	1.00000i	0.632456	−	0.316228i
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$			6.00000i			1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$			6.00000i			1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	1.00000			0.267261
$15$		0			0
$16$	1.00000			0.250000
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$		0			0
$19$	−6.00000			−1.37649			−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000			−1.37649			−0.688247	+	0.725476i	$0.741620\pi$
$20$	1.00000	+	2.00000i	0.223607	+	0.447214i
$21$		0			0
$22$		−	2.00000i		−	0.426401i
$23$		−	1.00000i		−	0.208514i	−0.994550	−	0.104257i	$-0.966753\pi$
$23$		−	1.00000i		−	0.208514i	0.994550	−	0.104257i	$-0.0332465\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$	−6.00000			−1.17670
$27$		0			0
$28$			1.00000i			0.188982i
$29$	−9.00000			−1.67126			−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000			−1.67126			−0.835629	+	0.549294i	$0.814897\pi$
$30$		0			0
$31$	−2.00000			−0.359211			−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000			−0.359211			−0.179605	+	0.983739i	$0.557482\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	2.00000			0.342997
$35$	−2.00000	+	1.00000i	−0.338062	+	0.169031i
$36$		0			0
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		−	6.00000i		−	0.973329i
$39$		0			0
$40$	−2.00000	+	1.00000i	−0.316228	+	0.158114i
$41$	−11.0000			−1.71791			−0.858956	−	0.512050i	$-0.828886\pi$
$41$	−11.0000			−1.71791			−0.858956	+	0.512050i	$0.828886\pi$
$42$		0			0
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	2.00000			0.301511
$45$		0			0
$46$	1.00000			0.147442
$47$		−	7.00000i		−	1.02105i	−0.859861	−	0.510527i	$-0.829450\pi$
$47$		−	7.00000i		−	1.02105i	0.859861	−	0.510527i	$-0.170550\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$	−4.00000	−	3.00000i	−0.565685	−	0.424264i
$51$		0			0
$52$		−	6.00000i		−	0.832050i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$	2.00000	+	4.00000i	0.269680	+	0.539360i
$56$	−1.00000			−0.133631
$57$		0			0
$58$		−	9.00000i		−	1.18176i
$59$	4.00000			0.520756			0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000			0.520756			0.260378	+	0.965507i	$0.416153\pi$
$60$		0			0
$61$	−7.00000			−0.896258			−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000			−0.896258			−0.448129	+	0.893969i	$0.647910\pi$
$62$		−	2.00000i		−	0.254000i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	12.0000	−	6.00000i	1.48842	−	0.744208i
$66$		0			0
$67$		−	11.0000i		−	1.34386i	−0.740613	−	0.671932i	$-0.765465\pi$
$67$		−	11.0000i		−	1.34386i	0.740613	−	0.671932i	$-0.234535\pi$
$68$			2.00000i			0.242536i
$69$		0			0
$70$	−1.00000	−	2.00000i	−0.119523	−	0.239046i
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	−2.00000			−0.232495
$75$		0			0
$76$	6.00000			0.688247
$77$			2.00000i			0.227921i
$78$		0			0
$79$	12.0000			1.35011			0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000			1.35011			0.675053	+	0.737769i	$0.264121\pi$
$80$	−1.00000	−	2.00000i	−0.111803	−	0.223607i
$81$		0			0
$82$		−	11.0000i		−	1.21475i
$83$			11.0000i			1.20741i	0.797209	+	0.603703i	$0.206309\pi$
$83$			11.0000i			1.20741i	−0.797209	+	0.603703i	$0.793691\pi$
$84$		0			0
$85$	−4.00000	+	2.00000i	−0.433861	+	0.216930i
$86$	−4.00000			−0.431331
$87$		0			0
$88$			2.00000i			0.213201i
$89$	−1.00000			−0.106000			−0.0529999	−	0.998595i	$-0.516878\pi$
$89$	−1.00000			−0.106000			−0.0529999	+	0.998595i	$0.516878\pi$
$90$		0			0
$91$	6.00000			0.628971
$92$			1.00000i			0.104257i
$93$		0			0
$94$	7.00000			0.721995
$95$	6.00000	+	12.0000i	0.615587	+	1.23117i
$96$		0			0
$97$		−	8.00000i		−	0.812277i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$		−	8.00000i		−	0.812277i	0.913812	−	0.406138i	$-0.133125\pi$
$98$			6.00000i			0.606092i
$99$		0			0
$100$	3.00000	−	4.00000i	0.300000	−	0.400000i
$101$	2.00000			0.199007			0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000			0.199007			0.0995037	+	0.995037i	$0.468274\pi$
$102$		0			0
$103$			8.00000i			0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$			8.00000i			0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	6.00000			0.588348
$105$		0			0
$106$		0			0
$107$			3.00000i			0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$			3.00000i			0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$		0			0
$109$	−7.00000			−0.670478			−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000			−0.670478			−0.335239	+	0.942133i	$0.608817\pi$
$110$	−4.00000	+	2.00000i	−0.381385	+	0.190693i
$111$		0			0
$112$		−	1.00000i		−	0.0944911i
$113$		−	12.0000i		−	1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$		−	12.0000i		−	1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$		0			0
$115$	−2.00000	+	1.00000i	−0.186501	+	0.0932505i
$116$	9.00000			0.835629
$117$		0			0
$118$			4.00000i			0.368230i
$119$	−2.00000			−0.183340
$120$		0			0
$121$	−7.00000			−0.636364
$122$		−	7.00000i		−	0.633750i
$123$		0			0
$124$	2.00000			0.179605
$125$	11.0000	+	2.00000i	0.983870	+	0.178885i
$126$		0			0
$127$		−	19.0000i		−	1.68598i	−0.537931	−	0.842989i	$-0.680794\pi$
$127$		−	19.0000i		−	1.68598i	0.537931	−	0.842989i	$-0.319206\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	6.00000	+	12.0000i	0.526235	+	1.05247i
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$			6.00000i			0.520266i
$134$	11.0000			0.950255
$135$		0			0
$136$	−2.00000			−0.171499
$137$			12.0000i			1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$			12.0000i			1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$		0			0
$139$	−16.0000			−1.35710			−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000			−1.35710			−0.678551	+	0.734553i	$0.737392\pi$
$140$	2.00000	−	1.00000i	0.169031	−	0.0845154i
$141$		0			0
$142$		−	6.00000i		−	0.503509i
$143$		−	12.0000i		−	1.00349i
$144$		0			0
$145$	9.00000	+	18.0000i	0.747409	+	1.49482i
$146$	−4.00000			−0.331042
$147$		0			0
$148$		−	2.00000i		−	0.164399i
$149$	1.00000			0.0819232			0.0409616	−	0.999161i	$-0.486958\pi$
$149$	1.00000			0.0819232			0.0409616	+	0.999161i	$0.486958\pi$
$150$		0			0
$151$	10.0000			0.813788			0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000			0.813788			0.406894	+	0.913475i	$0.366612\pi$
$152$			6.00000i			0.486664i
$153$		0			0
$154$	−2.00000			−0.161165
$155$	2.00000	+	4.00000i	0.160644	+	0.321288i
$156$		0			0
$157$			4.00000i			0.319235i	0.987179	+	0.159617i	$0.0510260\pi$
$157$			4.00000i			0.319235i	−0.987179	+	0.159617i	$0.948974\pi$
$158$			12.0000i			0.954669i
$159$		0			0
$160$	2.00000	−	1.00000i	0.158114	−	0.0790569i
$161$	−1.00000			−0.0788110
$162$		0			0
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	11.0000			0.858956
$165$		0			0
$166$	−11.0000			−0.853766
$167$		−	3.00000i		−	0.232147i	−0.993241	−	0.116073i	$-0.962969\pi$
$167$		−	3.00000i		−	0.232147i	0.993241	−	0.116073i	$-0.0370308\pi$
$168$		0			0
$169$	−23.0000			−1.76923
$170$	−2.00000	−	4.00000i	−0.153393	−	0.306786i
$171$		0			0
$172$		−	4.00000i		−	0.304997i
$173$			4.00000i			0.304114i	0.988372	+	0.152057i	$0.0485898\pi$
$173$			4.00000i			0.304114i	−0.988372	+	0.152057i	$0.951410\pi$
$174$		0			0
$175$	4.00000	+	3.00000i	0.302372	+	0.226779i
$176$	−2.00000			−0.150756
$177$		0			0
$178$		−	1.00000i		−	0.0749532i
$179$	−2.00000			−0.149487			−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000			−0.149487			−0.0747435	+	0.997203i	$0.523814\pi$
$180$		0			0
$181$	−13.0000			−0.966282			−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000			−0.966282			−0.483141	+	0.875542i	$0.660504\pi$
$182$			6.00000i			0.444750i
$183$		0			0
$184$	−1.00000			−0.0737210
$185$	4.00000	−	2.00000i	0.294086	−	0.147043i
$186$		0			0
$187$			4.00000i			0.292509i
$188$			7.00000i			0.510527i
$189$		0			0
$190$	−12.0000	+	6.00000i	−0.870572	+	0.435286i
$191$	6.00000			0.434145			0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000			0.434145			0.217072	+	0.976156i	$0.430349\pi$
$192$		0			0
$193$			10.0000i			0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$			10.0000i			0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	8.00000			0.574367
$195$		0			0
$196$	−6.00000			−0.428571
$197$			8.00000i			0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$			8.00000i			0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$		0			0
$199$	18.0000			1.27599			0.637993	−	0.770042i	$-0.279765\pi$
$199$	18.0000			1.27599			0.637993	+	0.770042i	$0.279765\pi$
$200$	4.00000	+	3.00000i	0.282843	+	0.212132i
$201$		0			0
$202$			2.00000i			0.140720i
$203$			9.00000i			0.631676i
$204$		0			0
$205$	11.0000	+	22.0000i	0.768273	+	1.53655i
$206$	−8.00000			−0.557386
$207$		0			0
$208$			6.00000i			0.416025i
$209$	12.0000			0.830057
$210$		0			0
$211$	18.0000			1.23917			0.619586	−	0.784929i	$-0.287301\pi$
$211$	18.0000			1.23917			0.619586	+	0.784929i	$0.287301\pi$
$212$		0			0
$213$		0			0
$214$	−3.00000			−0.205076
$215$	8.00000	−	4.00000i	0.545595	−	0.272798i
$216$		0			0
$217$			2.00000i			0.135769i
$218$		−	7.00000i		−	0.474100i
$219$		0			0
$220$	−2.00000	−	4.00000i	−0.134840	−	0.269680i
$221$	12.0000			0.807207
$222$		0			0
$223$		−	23.0000i		−	1.54019i	−0.637927	−	0.770097i	$-0.720208\pi$
$223$		−	23.0000i		−	1.54019i	0.637927	−	0.770097i	$-0.279792\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$	12.0000			0.798228
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$		0			0
$229$	7.00000			0.462573			0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000			0.462573			0.231287	+	0.972886i	$0.425707\pi$
$230$	−1.00000	−	2.00000i	−0.0659380	−	0.131876i
$231$		0			0
$232$			9.00000i			0.590879i
$233$			10.0000i			0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$			10.0000i			0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$		0			0
$235$	−14.0000	+	7.00000i	−0.913259	+	0.456630i
$236$	−4.00000			−0.260378
$237$		0			0
$238$		−	2.00000i		−	0.129641i
$239$	−28.0000			−1.81117			−0.905585	−	0.424165i	$-0.860568\pi$
$239$	−28.0000			−1.81117			−0.905585	+	0.424165i	$0.860568\pi$
$240$		0			0
$241$	1.00000			0.0644157			0.0322078	−	0.999481i	$-0.489746\pi$
$241$	1.00000			0.0644157			0.0322078	+	0.999481i	$0.489746\pi$
$242$		−	7.00000i		−	0.449977i
$243$		0			0
$244$	7.00000			0.448129
$245$	−6.00000	−	12.0000i	−0.383326	−	0.766652i
$246$		0			0
$247$		−	36.0000i		−	2.29063i
$248$			2.00000i			0.127000i
$249$		0			0
$250$	−2.00000	+	11.0000i	−0.126491	+	0.695701i
$251$	18.0000			1.13615			0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000			1.13615			0.568075	+	0.822977i	$0.307688\pi$
$252$		0			0
$253$			2.00000i			0.125739i
$254$	19.0000			1.19217
$255$		0			0
$256$	1.00000			0.0625000
$257$			12.0000i			0.748539i	0.927320	+	0.374270i	$0.122107\pi$
$257$			12.0000i			0.748539i	−0.927320	+	0.374270i	$0.877893\pi$
$258$		0			0
$259$	2.00000			0.124274
$260$	−12.0000	+	6.00000i	−0.744208	+	0.372104i
$261$		0			0
$262$			12.0000i			0.741362i
$263$		−	16.0000i		−	0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$		−	16.0000i		−	0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$		0			0
$265$		0			0
$266$	−6.00000			−0.367884
$267$		0			0
$268$			11.0000i			0.671932i
$269$	−3.00000			−0.182913			−0.0914566	−	0.995809i	$-0.529152\pi$
$269$	−3.00000			−0.182913			−0.0914566	+	0.995809i	$0.529152\pi$
$270$		0			0
$271$	−14.0000			−0.850439			−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000			−0.850439			−0.425220	+	0.905090i	$0.639803\pi$
$272$		−	2.00000i		−	0.121268i
$273$		0			0
$274$	−12.0000			−0.724947
$275$	6.00000	−	8.00000i	0.361814	−	0.482418i
$276$		0			0
$277$		−	22.0000i		−	1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$		−	22.0000i		−	1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$		−	16.0000i		−	0.959616i
$279$		0			0
$280$	1.00000	+	2.00000i	0.0597614	+	0.119523i
$281$	−3.00000			−0.178965			−0.0894825	−	0.995988i	$-0.528521\pi$
$281$	−3.00000			−0.178965			−0.0894825	+	0.995988i	$0.528521\pi$
$282$		0			0
$283$		−	1.00000i		−	0.0594438i	−0.999558	−	0.0297219i	$-0.990538\pi$
$283$		−	1.00000i		−	0.0594438i	0.999558	−	0.0297219i	$-0.00946217\pi$
$284$	6.00000			0.356034
$285$		0			0
$286$	12.0000			0.709575
$287$			11.0000i			0.649309i
$288$		0			0
$289$	13.0000			0.764706
$290$	−18.0000	+	9.00000i	−1.05700	+	0.528498i
$291$		0			0
$292$		−	4.00000i		−	0.234082i
$293$			18.0000i			1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$			18.0000i			1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$		0			0
$295$	−4.00000	−	8.00000i	−0.232889	−	0.465778i
$296$	2.00000			0.116248
$297$		0			0
$298$			1.00000i			0.0579284i
$299$	6.00000			0.346989
$300$		0			0
$301$	4.00000			0.230556
$302$			10.0000i			0.575435i
$303$		0			0
$304$	−6.00000			−0.344124
$305$	7.00000	+	14.0000i	0.400819	+	0.801638i
$306$		0			0
$307$		−	9.00000i		−	0.513657i	−0.966457	−	0.256829i	$-0.917322\pi$
$307$		−	9.00000i		−	0.513657i	0.966457	−	0.256829i	$-0.0826776\pi$
$308$		−	2.00000i		−	0.113961i
$309$		0			0
$310$	−4.00000	+	2.00000i	−0.227185	+	0.113592i
$311$	6.00000			0.340229			0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000			0.340229			0.170114	+	0.985424i	$0.445586\pi$
$312$		0			0
$313$			22.0000i			1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$			22.0000i			1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$	−4.00000			−0.225733
$315$		0			0
$316$	−12.0000			−0.675053
$317$		−	2.00000i		−	0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$		−	2.00000i		−	0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$		0			0
$319$	18.0000			1.00781
$320$	1.00000	+	2.00000i	0.0559017	+	0.111803i
$321$		0			0
$322$		−	1.00000i		−	0.0557278i
$323$			12.0000i			0.667698i
$324$		0			0
$325$	−24.0000	−	18.0000i	−1.33128	−	0.998460i
$326$	4.00000			0.221540
$327$		0			0
$328$			11.0000i			0.607373i
$329$	−7.00000			−0.385922
$330$		0			0
$331$	−8.00000			−0.439720			−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000			−0.439720			−0.219860	+	0.975531i	$0.570560\pi$
$332$		−	11.0000i		−	0.603703i
$333$		0			0
$334$	3.00000			0.164153
$335$	−22.0000	+	11.0000i	−1.20199	+	0.600994i
$336$		0			0
$337$			8.00000i			0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$			8.00000i			0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$		−	23.0000i		−	1.25104i
$339$		0			0
$340$	4.00000	−	2.00000i	0.216930	−	0.108465i
$341$	4.00000			0.216612
$342$		0			0
$343$		−	13.0000i		−	0.701934i
$344$	4.00000			0.215666
$345$		0			0
$346$	−4.00000			−0.215041
$347$		−	12.0000i		−	0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$		−	12.0000i		−	0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$		0			0
$349$	11.0000			0.588817			0.294408	−	0.955680i	$-0.404877\pi$
$349$	11.0000			0.588817			0.294408	+	0.955680i	$0.404877\pi$
$350$	−3.00000	+	4.00000i	−0.160357	+	0.213809i
$351$		0			0
$352$		−	2.00000i		−	0.106600i
$353$			16.0000i			0.851594i	0.904819	+	0.425797i	$0.140006\pi$
$353$			16.0000i			0.851594i	−0.904819	+	0.425797i	$0.859994\pi$
$354$		0			0
$355$	6.00000	+	12.0000i	0.318447	+	0.636894i
$356$	1.00000			0.0529999
$357$		0			0
$358$		−	2.00000i		−	0.105703i
$359$	−30.0000			−1.58334			−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000			−1.58334			−0.791670	+	0.610949i	$0.790788\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		−	13.0000i		−	0.683265i
$363$		0			0
$364$	−6.00000			−0.314485
$365$	8.00000	−	4.00000i	0.418739	−	0.209370i
$366$		0			0
$367$		−	16.0000i		−	0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$		−	16.0000i		−	0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$		−	1.00000i		−	0.0521286i
$369$		0			0
$370$	2.00000	+	4.00000i	0.103975	+	0.207950i
$371$		0			0
$372$		0			0
$373$		−	12.0000i		−	0.621336i	−0.950518	−	0.310668i	$-0.899447\pi$
$373$		−	12.0000i		−	0.621336i	0.950518	−	0.310668i	$-0.100553\pi$
$374$	−4.00000			−0.206835
$375$		0			0
$376$	−7.00000			−0.360997
$377$		−	54.0000i		−	2.78114i
$378$		0			0
$379$	16.0000			0.821865			0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000			0.821865			0.410932	+	0.911666i	$0.365203\pi$
$380$	−6.00000	−	12.0000i	−0.307794	−	0.615587i
$381$		0			0
$382$			6.00000i			0.306987i
$383$		−	32.0000i		−	1.63512i	−0.575841	−	0.817562i	$-0.695325\pi$
$383$		−	32.0000i		−	1.63512i	0.575841	−	0.817562i	$-0.304675\pi$
$384$		0			0
$385$	4.00000	−	2.00000i	0.203859	−	0.101929i
$386$	−10.0000			−0.508987
$387$		0			0
$388$			8.00000i			0.406138i
$389$	−19.0000			−0.963338			−0.481669	−	0.876353i	$-0.659969\pi$
$389$	−19.0000			−0.963338			−0.481669	+	0.876353i	$0.659969\pi$
$390$		0			0
$391$	−2.00000			−0.101144
$392$		−	6.00000i		−	0.303046i
$393$		0			0
$394$	−8.00000			−0.403034
$395$	−12.0000	−	24.0000i	−0.603786	−	1.20757i
$396$		0			0
$397$		−	4.00000i		−	0.200754i	−0.994949	−	0.100377i	$-0.967995\pi$
$397$		−	4.00000i		−	0.200754i	0.994949	−	0.100377i	$-0.0320049\pi$
$398$			18.0000i			0.902258i
$399$		0			0
$400$	−3.00000	+	4.00000i	−0.150000	+	0.200000i
$401$	−10.0000			−0.499376			−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000			−0.499376			−0.249688	+	0.968326i	$0.580328\pi$
$402$		0			0
$403$		−	12.0000i		−	0.597763i
$404$	−2.00000			−0.0995037
$405$		0			0
$406$	−9.00000			−0.446663
$407$		−	4.00000i		−	0.198273i
$408$		0			0
$409$	−38.0000			−1.87898			−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000			−1.87898			−0.939490	+	0.342578i	$0.888700\pi$
$410$	−22.0000	+	11.0000i	−1.08650	+	0.543251i
$411$		0			0
$412$		−	8.00000i		−	0.394132i
$413$		−	4.00000i		−	0.196827i
$414$		0			0
$415$	22.0000	−	11.0000i	1.07994	−	0.539969i
$416$	−6.00000			−0.294174
$417$		0			0
$418$			12.0000i			0.586939i
$419$	−34.0000			−1.66101			−0.830504	−	0.557012i	$-0.811948\pi$
$419$	−34.0000			−1.66101			−0.830504	+	0.557012i	$0.811948\pi$
$420$		0			0
$421$	22.0000			1.07221			0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000			1.07221			0.536107	+	0.844150i	$0.319894\pi$
$422$			18.0000i			0.876226i
$423$		0			0
$424$		0			0
$425$	8.00000	+	6.00000i	0.388057	+	0.291043i
$426$		0			0
$427$			7.00000i			0.338754i
$428$		−	3.00000i		−	0.145010i
$429$		0			0
$430$	4.00000	+	8.00000i	0.192897	+	0.385794i
$431$	16.0000			0.770693			0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000			0.770693			0.385346	+	0.922772i	$0.374082\pi$
$432$		0			0
$433$		−	2.00000i		−	0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$		−	2.00000i		−	0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	−2.00000			−0.0960031
$435$		0			0
$436$	7.00000			0.335239
$437$			6.00000i			0.287019i
$438$		0			0
$439$	−24.0000			−1.14546			−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000			−1.14546			−0.572729	+	0.819745i	$0.694115\pi$
$440$	4.00000	−	2.00000i	0.190693	−	0.0953463i
$441$		0			0
$442$			12.0000i			0.570782i
$443$			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
$445$	1.00000	+	2.00000i	0.0474045	+	0.0948091i
$446$	23.0000			1.08908
$447$		0			0
$448$			1.00000i			0.0472456i
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$	22.0000			1.03594
$452$			12.0000i			0.564433i
$453$		0			0
$454$	8.00000			0.375459
$455$	−6.00000	−	12.0000i	−0.281284	−	0.562569i
$456$		0			0
$457$		−	10.0000i		−	0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$		−	10.0000i		−	0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$			7.00000i			0.327089i
$459$		0			0
$460$	2.00000	−	1.00000i	0.0932505	−	0.0466252i
$461$	−21.0000			−0.978068			−0.489034	−	0.872265i	$-0.662651\pi$
$461$	−21.0000			−0.978068			−0.489034	+	0.872265i	$0.662651\pi$
$462$		0			0
$463$		−	36.0000i		−	1.67306i	−0.547920	−	0.836531i	$-0.684580\pi$
$463$		−	36.0000i		−	1.67306i	0.547920	−	0.836531i	$-0.315420\pi$
$464$	−9.00000			−0.417815
$465$		0			0
$466$	−10.0000			−0.463241
$467$			36.0000i			1.66588i	0.553362	+	0.832941i	$0.313345\pi$
$467$			36.0000i			1.66588i	−0.553362	+	0.832941i	$0.686655\pi$
$468$		0			0
$469$	−11.0000			−0.507933
$470$	−7.00000	−	14.0000i	−0.322886	−	0.645772i
$471$		0			0
$472$		−	4.00000i		−	0.184115i
$473$		−	8.00000i		−	0.367840i
$474$		0			0
$475$	18.0000	−	24.0000i	0.825897	−	1.10120i
$476$	2.00000			0.0916698
$477$		0			0
$478$		−	28.0000i		−	1.28069i
$479$	−28.0000			−1.27935			−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000			−1.27935			−0.639676	+	0.768644i	$0.720932\pi$
$480$		0			0
$481$	−12.0000			−0.547153
$482$			1.00000i			0.0455488i
$483$		0			0
$484$	7.00000			0.318182
$485$	−16.0000	+	8.00000i	−0.726523	+	0.363261i
$486$		0			0
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$			7.00000i			0.316875i
$489$		0			0
$490$	12.0000	−	6.00000i	0.542105	−	0.271052i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$			18.0000i			0.810679i
$494$	36.0000			1.61972
$495$		0			0
$496$	−2.00000			−0.0898027
$497$			6.00000i			0.269137i
$498$		0			0
$499$	24.0000			1.07439			0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000			1.07439			0.537194	+	0.843459i	$0.319484\pi$
$500$	−11.0000	−	2.00000i	−0.491935	−	0.0894427i
$501$		0			0
$502$			18.0000i			0.803379i
$503$		−	27.0000i		−	1.20387i	−0.798545	−	0.601935i	$-0.794397\pi$
$503$		−	27.0000i		−	1.20387i	0.798545	−	0.601935i	$-0.205603\pi$
$504$		0			0
$505$	−2.00000	−	4.00000i	−0.0889988	−	0.177998i
$506$	−2.00000			−0.0889108
$507$		0			0
$508$			19.0000i			0.842989i
$509$	15.0000			0.664863			0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000			0.664863			0.332432	+	0.943127i	$0.392131\pi$
$510$		0			0
$511$	4.00000			0.176950
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−12.0000			−0.529297
$515$	16.0000	−	8.00000i	0.705044	−	0.352522i
$516$		0			0
$517$			14.0000i			0.615719i
$518$			2.00000i			0.0878750i
$519$		0			0
$520$	−6.00000	−	12.0000i	−0.263117	−	0.526235i
$521$	−37.0000			−1.62100			−0.810500	−	0.585739i	$-0.800804\pi$
$521$	−37.0000			−1.62100			−0.810500	+	0.585739i	$0.800804\pi$
$522$		0			0
$523$			29.0000i			1.26808i	0.773300	+	0.634041i	$0.218605\pi$
$523$			29.0000i			1.26808i	−0.773300	+	0.634041i	$0.781395\pi$
$524$	−12.0000			−0.524222
$525$		0			0
$526$	16.0000			0.697633
$527$			4.00000i			0.174243i
$528$		0			0
$529$	22.0000			0.956522
$530$		0			0
$531$		0			0
$532$		−	6.00000i		−	0.260133i
$533$		−	66.0000i		−	2.85878i
$534$		0			0
$535$	6.00000	−	3.00000i	0.259403	−	0.129701i
$536$	−11.0000			−0.475128
$537$		0			0
$538$		−	3.00000i		−	0.129339i
$539$	−12.0000			−0.516877
$540$		0			0
$541$	−17.0000			−0.730887			−0.365444	−	0.930834i	$-0.619083\pi$
$541$	−17.0000			−0.730887			−0.365444	+	0.930834i	$0.619083\pi$
$542$		−	14.0000i		−	0.601351i
$543$		0			0
$544$	2.00000			0.0857493
$545$	7.00000	+	14.0000i	0.299847	+	0.599694i
$546$		0			0
$547$			35.0000i			1.49649i	0.663421	+	0.748246i	$0.269104\pi$
$547$			35.0000i			1.49649i	−0.663421	+	0.748246i	$0.730896\pi$
$548$		−	12.0000i		−	0.512615i
$549$		0			0
$550$	8.00000	+	6.00000i	0.341121	+	0.255841i
$551$	54.0000			2.30048
$552$		0			0
$553$		−	12.0000i		−	0.510292i
$554$	22.0000			0.934690
$555$		0			0
$556$	16.0000			0.678551
$557$		−	24.0000i		−	1.01691i	−0.861088	−	0.508456i	$-0.830216\pi$
$557$		−	24.0000i		−	1.01691i	0.861088	−	0.508456i	$-0.169784\pi$
$558$		0			0
$559$	−24.0000			−1.01509
$560$	−2.00000	+	1.00000i	−0.0845154	+	0.0422577i
$561$		0			0
$562$		−	3.00000i		−	0.126547i
$563$			37.0000i			1.55936i	0.626176	+	0.779682i	$0.284619\pi$
$563$			37.0000i			1.55936i	−0.626176	+	0.779682i	$0.715381\pi$
$564$		0			0
$565$	−24.0000	+	12.0000i	−1.00969	+	0.504844i
$566$	1.00000			0.0420331
$567$		0			0
$568$			6.00000i			0.251754i
$569$	2.00000			0.0838444			0.0419222	−	0.999121i	$-0.486652\pi$
$569$	2.00000			0.0838444			0.0419222	+	0.999121i	$0.486652\pi$
$570$		0			0
$571$	−20.0000			−0.836974			−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000			−0.836974			−0.418487	+	0.908223i	$0.637439\pi$
$572$			12.0000i			0.501745i
$573$		0			0
$574$	−11.0000			−0.459131
$575$	4.00000	+	3.00000i	0.166812	+	0.125109i
$576$		0			0
$577$			32.0000i			1.33218i	0.745873	+	0.666089i	$0.232033\pi$
$577$			32.0000i			1.33218i	−0.745873	+	0.666089i	$0.767967\pi$
$578$			13.0000i			0.540729i
$579$		0			0
$580$	−9.00000	−	18.0000i	−0.373705	−	0.747409i
$581$	11.0000			0.456357
$582$		0			0
$583$		0			0
$584$	4.00000			0.165521
$585$		0			0
$586$	−18.0000			−0.743573
$587$			3.00000i			0.123823i	0.998082	+	0.0619116i	$0.0197197\pi$
$587$			3.00000i			0.123823i	−0.998082	+	0.0619116i	$0.980280\pi$
$588$		0			0
$589$	12.0000			0.494451
$590$	8.00000	−	4.00000i	0.329355	−	0.164677i
$591$		0			0
$592$			2.00000i			0.0821995i
$593$			30.0000i			1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$			30.0000i			1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$		0			0
$595$	2.00000	+	4.00000i	0.0819920	+	0.163984i
$596$	−1.00000			−0.0409616
$597$		0			0
$598$			6.00000i			0.245358i
$599$	12.0000			0.490307			0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000			0.490307			0.245153	+	0.969484i	$0.421162\pi$
$600$		0			0
$601$	−22.0000			−0.897399			−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000			−0.897399			−0.448699	+	0.893683i	$0.648113\pi$
$602$			4.00000i			0.163028i
$603$		0			0
$604$	−10.0000			−0.406894
$605$	7.00000	+	14.0000i	0.284590	+	0.569181i
$606$		0			0
$607$			1.00000i			0.0405887i	0.999794	+	0.0202944i	$0.00646034\pi$
$607$			1.00000i			0.0405887i	−0.999794	+	0.0202944i	$0.993540\pi$
$608$		−	6.00000i		−	0.243332i
$609$		0			0
$610$	−14.0000	+	7.00000i	−0.566843	+	0.283422i
$611$	42.0000			1.69914
$612$		0			0
$613$			34.0000i			1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$			34.0000i			1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$	9.00000			0.363210
$615$		0			0
$616$	2.00000			0.0805823
$617$		−	32.0000i		−	1.28827i	−0.764911	−	0.644136i	$-0.777217\pi$
$617$		−	32.0000i		−	1.28827i	0.764911	−	0.644136i	$-0.222783\pi$
$618$		0			0
$619$	10.0000			0.401934			0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000			0.401934			0.200967	+	0.979598i	$0.435592\pi$
$620$	−2.00000	−	4.00000i	−0.0803219	−	0.160644i
$621$		0			0
$622$			6.00000i			0.240578i
$623$			1.00000i			0.0400642i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	−22.0000			−0.879297
$627$		0			0
$628$		−	4.00000i		−	0.159617i
$629$	4.00000			0.159490
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$		−	12.0000i		−	0.477334i
$633$		0			0
$634$	2.00000			0.0794301
$635$	−38.0000	+	19.0000i	−1.50798	+	0.753992i
$636$		0			0
$637$			36.0000i			1.42637i
$638$			18.0000i			0.712627i
$639$		0			0
$640$	−2.00000	+	1.00000i	−0.0790569	+	0.0395285i
$641$	13.0000			0.513469			0.256735	−	0.966482i	$-0.417353\pi$
$641$	13.0000			0.513469			0.256735	+	0.966482i	$0.417353\pi$
$642$		0			0
$643$			33.0000i			1.30139i	0.759338	+	0.650696i	$0.225523\pi$
$643$			33.0000i			1.30139i	−0.759338	+	0.650696i	$0.774477\pi$
$644$	1.00000			0.0394055
$645$		0			0
$646$	−12.0000			−0.472134
$647$			33.0000i			1.29736i	0.761060	+	0.648682i	$0.224679\pi$
$647$			33.0000i			1.29736i	−0.761060	+	0.648682i	$0.775321\pi$
$648$		0			0
$649$	−8.00000			−0.314027
$650$	18.0000	−	24.0000i	0.706018	−	0.941357i
$651$		0			0
$652$			4.00000i			0.156652i
$653$			26.0000i			1.01746i	0.860927	+	0.508729i	$0.169885\pi$
$653$			26.0000i			1.01746i	−0.860927	+	0.508729i	$0.830115\pi$
$654$		0			0
$655$	−12.0000	−	24.0000i	−0.468879	−	0.937758i
$656$	−11.0000			−0.429478
$657$		0			0
$658$		−	7.00000i		−	0.272888i
$659$	−20.0000			−0.779089			−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000			−0.779089			−0.389545	+	0.921008i	$0.627368\pi$
$660$		0			0
$661$	−10.0000			−0.388955			−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000			−0.388955			−0.194477	+	0.980907i	$0.562301\pi$
$662$		−	8.00000i		−	0.310929i
$663$		0			0
$664$	11.0000			0.426883
$665$	12.0000	−	6.00000i	0.465340	−	0.232670i
$666$		0			0
$667$			9.00000i			0.348481i
$668$			3.00000i			0.116073i
$669$		0			0
$670$	−11.0000	−	22.0000i	−0.424967	−	0.849934i
$671$	14.0000			0.540464
$672$		0			0
$673$		−	6.00000i		−	0.231283i	−0.993291	−	0.115642i	$-0.963108\pi$
$673$		−	6.00000i		−	0.231283i	0.993291	−	0.115642i	$-0.0368924\pi$
$674$	−8.00000			−0.308148
$675$		0			0
$676$	23.0000			0.884615
$677$			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
$679$	−8.00000			−0.307012
$680$	2.00000	+	4.00000i	0.0766965	+	0.153393i
$681$		0			0
$682$			4.00000i			0.153168i
$683$		−	4.00000i		−	0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$		−	4.00000i		−	0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$		0			0
$685$	24.0000	−	12.0000i	0.916993	−	0.458496i
$686$	13.0000			0.496342
$687$		0			0
$688$			4.00000i			0.152499i
$689$		0			0
$690$		0			0
$691$	−44.0000			−1.67384			−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000			−1.67384			−0.836919	+	0.547326i	$0.815646\pi$
$692$		−	4.00000i		−	0.152057i
$693$		0			0
$694$	12.0000			0.455514
$695$	16.0000	+	32.0000i	0.606915	+	1.21383i
$696$		0			0
$697$			22.0000i			0.833309i
$698$			11.0000i			0.416356i
$699$		0			0
$700$	−4.00000	−	3.00000i	−0.151186	−	0.113389i
$701$	13.0000			0.491003			0.245502	−	0.969396i	$-0.421047\pi$
$701$	13.0000			0.491003			0.245502	+	0.969396i	$0.421047\pi$
$702$		0			0
$703$		−	12.0000i		−	0.452589i
$704$	2.00000			0.0753778
$705$		0			0
$706$	−16.0000			−0.602168
$707$		−	2.00000i		−	0.0752177i
$708$		0			0
$709$	27.0000			1.01401			0.507003	−	0.861944i	$-0.330753\pi$
$709$	27.0000			1.01401			0.507003	+	0.861944i	$0.330753\pi$
$710$	−12.0000	+	6.00000i	−0.450352	+	0.225176i
$711$		0			0
$712$			1.00000i			0.0374766i
$713$			2.00000i			0.0749006i
$714$		0			0
$715$	−24.0000	+	12.0000i	−0.897549	+	0.448775i
$716$	2.00000			0.0747435
$717$		0			0
$718$		−	30.0000i		−	1.11959i
$719$	44.0000			1.64092			0.820462	−	0.571702i	$-0.193717\pi$
$719$	44.0000			1.64092			0.820462	+	0.571702i	$0.193717\pi$
$720$		0			0
$721$	8.00000			0.297936
$722$			17.0000i			0.632674i
$723$		0			0
$724$	13.0000			0.483141
$725$	27.0000	−	36.0000i	1.00275	−	1.33701i
$726$		0			0
$727$			21.0000i			0.778847i	0.921059	+	0.389423i	$0.127326\pi$
$727$			21.0000i			0.778847i	−0.921059	+	0.389423i	$0.872674\pi$
$728$		−	6.00000i		−	0.222375i
$729$		0			0
$730$	4.00000	+	8.00000i	0.148047	+	0.296093i
$731$	8.00000			0.295891
$732$		0			0
$733$		−	4.00000i		−	0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$		−	4.00000i		−	0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$	16.0000			0.590571
$735$		0			0
$736$	1.00000			0.0368605
$737$			22.0000i			0.810380i
$738$		0			0
$739$	−40.0000			−1.47142			−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000			−1.47142			−0.735712	+	0.677295i	$0.763152\pi$
$740$	−4.00000	+	2.00000i	−0.147043	+	0.0735215i
$741$		0			0
$742$		0			0
$743$			15.0000i			0.550297i	0.961402	+	0.275148i	$0.0887270\pi$
$743$			15.0000i			0.550297i	−0.961402	+	0.275148i	$0.911273\pi$
$744$		0			0
$745$	−1.00000	−	2.00000i	−0.0366372	−	0.0732743i
$746$	12.0000			0.439351
$747$		0			0
$748$		−	4.00000i		−	0.146254i
$749$	3.00000			0.109618
$750$		0			0
$751$	26.0000			0.948753			0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000			0.948753			0.474377	+	0.880322i	$0.342673\pi$
$752$		−	7.00000i		−	0.255264i
$753$		0			0
$754$	54.0000			1.96656
$755$	−10.0000	−	20.0000i	−0.363937	−	0.727875i
$756$		0			0
$757$		−	10.0000i		−	0.363456i	−0.983349	−	0.181728i	$-0.941831\pi$
$757$		−	10.0000i		−	0.363456i	0.983349	−	0.181728i	$-0.0581691\pi$
$758$			16.0000i			0.581146i
$759$		0			0
$760$	12.0000	−	6.00000i	0.435286	−	0.217643i
$761$	−9.00000			−0.326250			−0.163125	−	0.986605i	$-0.552157\pi$
$761$	−9.00000			−0.326250			−0.163125	+	0.986605i	$0.552157\pi$
$762$		0			0
$763$			7.00000i			0.253417i
$764$	−6.00000			−0.217072
$765$		0			0
$766$	32.0000			1.15621
$767$			24.0000i			0.866590i
$768$		0			0
$769$	15.0000			0.540914			0.270457	−	0.962732i	$-0.412825\pi$
$769$	15.0000			0.540914			0.270457	+	0.962732i	$0.412825\pi$
$770$	2.00000	+	4.00000i	0.0720750	+	0.144150i
$771$		0			0
$772$		−	10.0000i		−	0.359908i
$773$		−	12.0000i		−	0.431610i	−0.976436	−	0.215805i	$-0.930762\pi$
$773$		−	12.0000i		−	0.431610i	0.976436	−	0.215805i	$-0.0692376\pi$
$774$		0			0
$775$	6.00000	−	8.00000i	0.215526	−	0.287368i
$776$	−8.00000			−0.287183
$777$		0			0
$778$		−	19.0000i		−	0.681183i
$779$	66.0000			2.36470
$780$		0			0
$781$	12.0000			0.429394
$782$		−	2.00000i		−	0.0715199i
$783$		0			0
$784$	6.00000			0.214286
$785$	8.00000	−	4.00000i	0.285532	−	0.142766i
$786$		0			0
$787$			44.0000i			1.56843i	0.620489	+	0.784215i	$0.286934\pi$
$787$			44.0000i			1.56843i	−0.620489	+	0.784215i	$0.713066\pi$
$788$		−	8.00000i		−	0.284988i
$789$		0			0
$790$	24.0000	−	12.0000i	0.853882	−	0.426941i
$791$	−12.0000			−0.426671
$792$		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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	810.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000i q^{8} +(2.00000 - 1.00000i) q^{10} -2.00000 q^{11} +6.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -6.00000 q^{19} +(1.00000 + 2.00000i) q^{20} -2.00000i q^{22} -1.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} -6.00000 q^{26} +1.00000i q^{28} -9.00000 q^{29} -2.00000 q^{31} +1.00000i q^{32} +2.00000 q^{34} +(-2.00000 + 1.00000i) q^{35} +2.00000i q^{37} -6.00000i q^{38} +(-2.00000 + 1.00000i) q^{40} -11.0000 q^{41} +4.00000i q^{43} +2.00000 q^{44} +1.00000 q^{46} -7.00000i q^{47} +6.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} -6.00000i q^{52} +(2.00000 + 4.00000i) q^{55} -1.00000 q^{56} -9.00000i q^{58} +4.00000 q^{59} -7.00000 q^{61} -2.00000i q^{62} -1.00000 q^{64} +(12.0000 - 6.00000i) q^{65} -11.0000i q^{67} +2.00000i q^{68} +(-1.00000 - 2.00000i) q^{70} -6.00000 q^{71} +4.00000i q^{73} -2.00000 q^{74} +6.00000 q^{76} +2.00000i q^{77} +12.0000 q^{79} +(-1.00000 - 2.00000i) q^{80} -11.0000i q^{82} +11.0000i q^{83} +(-4.00000 + 2.00000i) q^{85} -4.00000 q^{86} +2.00000i q^{88} -1.00000 q^{89} +6.00000 q^{91} +1.00000i q^{92} +7.00000 q^{94} +(6.00000 + 12.0000i) q^{95} -8.00000i q^{97} +6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{5} + 4 q^{10} - 4 q^{11} + 2 q^{14} + 2 q^{16} - 12 q^{19} + 2 q^{20} - 6 q^{25} - 12 q^{26} - 18 q^{29} - 4 q^{31} + 4 q^{34} - 4 q^{35} - 4 q^{40} - 22 q^{41} + 4 q^{44} + 2 q^{46}+ \cdots + 12 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^5 + 4 * q^10 - 4 * q^11 + 2 * q^14 + 2 * q^16 - 12 * q^19 + 2 * q^20 - 6 * q^25 - 12 * q^26 - 18 * q^29 - 4 * q^31 + 4 * q^34 - 4 * q^35 - 4 * q^40 - 22 * q^41 + 4 * q^44 + 2 * q^46 + 12 * q^49 - 8 * q^50 + 4 * q^55 - 2 * q^56 + 8 * q^59 - 14 * q^61 - 2 * q^64 + 24 * q^65 - 2 * q^70 - 12 * q^71 - 4 * q^74 + 12 * q^76 + 24 * q^79 - 2 * q^80 - 8 * q^85 - 8 * q^86 - 2 * q^89 + 12 * q^91 + 14 * q^94 + 12 * q^95

Introduction

L-functions

Modular forms

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