LMFDB - Embedded newform 450.3.g.e.343.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,3,Mod(307,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("450.307");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	450.g (of order $4$, degree $2$, 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:	$12.2616118962$
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 50)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			343.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	450.343
Dual form			450.3.g.e.307.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-3.00000 + 3.00000i) q^{7} +(-2.00000 - 2.00000i) q^{8} +O(q^{10})$	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-3.00000 + 3.00000i) q^{7} +(-2.00000 - 2.00000i) q^{8} -12.0000 q^{11} +(-12.0000 - 12.0000i) q^{13} +6.00000i q^{14} -4.00000 q^{16} +(-12.0000 + 12.0000i) q^{17} -20.0000i q^{19} +(-12.0000 + 12.0000i) q^{22} +(-3.00000 - 3.00000i) q^{23} -24.0000 q^{26} +(6.00000 + 6.00000i) q^{28} -30.0000i q^{29} -8.00000 q^{31} +(-4.00000 + 4.00000i) q^{32} +24.0000i q^{34} +(-48.0000 + 48.0000i) q^{37} +(-20.0000 - 20.0000i) q^{38} +48.0000 q^{41} +(-27.0000 - 27.0000i) q^{43} +24.0000i q^{44} -6.00000 q^{46} +(-27.0000 + 27.0000i) q^{47} +31.0000i q^{49} +(-24.0000 + 24.0000i) q^{52} +(12.0000 + 12.0000i) q^{53} +12.0000 q^{56} +(-30.0000 - 30.0000i) q^{58} -60.0000i q^{59} +32.0000 q^{61} +(-8.00000 + 8.00000i) q^{62} +8.00000i q^{64} +(-3.00000 + 3.00000i) q^{67} +(24.0000 + 24.0000i) q^{68} +48.0000 q^{71} +(-12.0000 - 12.0000i) q^{73} +96.0000i q^{74} -40.0000 q^{76} +(36.0000 - 36.0000i) q^{77} -40.0000i q^{79} +(48.0000 - 48.0000i) q^{82} +(-93.0000 - 93.0000i) q^{83} -54.0000 q^{86} +(24.0000 + 24.0000i) q^{88} +30.0000i q^{89} +72.0000 q^{91} +(-6.00000 + 6.00000i) q^{92} +54.0000i q^{94} +(12.0000 - 12.0000i) q^{97} +(31.0000 + 31.0000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 24 q^{11} - 24 q^{13} - 8 q^{16} - 24 q^{17} - 24 q^{22} - 6 q^{23} - 48 q^{26} + 12 q^{28} - 16 q^{31} - 8 q^{32} - 96 q^{37} - 40 q^{38} + 96 q^{41} - 54 q^{43} - 12 q^{46} - 54 q^{47} - 48 q^{52} + 24 q^{53} + 24 q^{56} - 60 q^{58} + 64 q^{61} - 16 q^{62} - 6 q^{67} + 48 q^{68} + 96 q^{71} - 24 q^{73} - 80 q^{76} + 72 q^{77} + 96 q^{82} - 186 q^{83} - 108 q^{86} + 48 q^{88} + 144 q^{91} - 12 q^{92} + 24 q^{97} + 62 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 6 * q^7 - 4 * q^8 - 24 * q^11 - 24 * q^13 - 8 * q^16 - 24 * q^17 - 24 * q^22 - 6 * q^23 - 48 * q^26 + 12 * q^28 - 16 * q^31 - 8 * q^32 - 96 * q^37 - 40 * q^38 + 96 * q^41 - 54 * q^43 - 12 * q^46 - 54 * q^47 - 48 * q^52 + 24 * q^53 + 24 * q^56 - 60 * q^58 + 64 * q^61 - 16 * q^62 - 6 * q^67 + 48 * q^68 + 96 * q^71 - 24 * q^73 - 80 * q^76 + 72 * q^77 + 96 * q^82 - 186 * q^83 - 108 * q^86 + 48 * q^88 + 144 * q^91 - 12 * q^92 + 24 * q^97 + 62 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	1.00000	−	1.00000i	0.500000	−	0.500000i
$2$	1.00000	−	1.00000i	0.500000	−	0.500000i
$3$		0			0
$3$		0			0
$4$		−	2.00000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−3.00000	+	3.00000i	−0.428571	+	0.428571i	−0.888142	−	0.459570i	$-0.848004\pi$
$7$	−3.00000	+	3.00000i	−0.428571	+	0.428571i	0.459570	+	0.888142i	$0.348004\pi$
$8$	−2.00000	−	2.00000i	−0.250000	−	0.250000i
$9$		0			0
$10$		0			0
$11$	−12.0000			−1.09091			−0.545455	−	0.838140i	$-0.683643\pi$
$11$	−12.0000			−1.09091			−0.545455	+	0.838140i	$0.683643\pi$
$12$		0			0
$13$	−12.0000	−	12.0000i	−0.923077	−	0.923077i	0.0741688	−	0.997246i	$-0.476370\pi$
$13$	−12.0000	−	12.0000i	−0.923077	−	0.923077i	−0.997246	+	0.0741688i	$0.976370\pi$
$14$			6.00000i			0.428571i
$15$		0			0
$16$	−4.00000			−0.250000
$17$	−12.0000	+	12.0000i	−0.705882	+	0.705882i	−0.965667	−	0.259784i	$-0.916349\pi$
$17$	−12.0000	+	12.0000i	−0.705882	+	0.705882i	0.259784	+	0.965667i	$0.416349\pi$
$18$		0			0
$19$		−	20.0000i		−	1.05263i	−0.850289	−	0.526316i	$-0.823573\pi$
$19$		−	20.0000i		−	1.05263i	0.850289	−	0.526316i	$-0.176427\pi$
$20$		0			0
$21$		0			0
$22$	−12.0000	+	12.0000i	−0.545455	+	0.545455i
$23$	−3.00000	−	3.00000i	−0.130435	−	0.130435i	0.638875	−	0.769310i	$-0.279400\pi$
$23$	−3.00000	−	3.00000i	−0.130435	−	0.130435i	−0.769310	+	0.638875i	$0.779400\pi$
$24$		0			0
$25$		0			0
$26$	−24.0000			−0.923077
$27$		0			0
$28$	6.00000	+	6.00000i	0.214286	+	0.214286i
$29$		−	30.0000i		−	1.03448i	−0.855840	−	0.517241i	$-0.826959\pi$
$29$		−	30.0000i		−	1.03448i	0.855840	−	0.517241i	$-0.173041\pi$
$30$		0			0
$31$	−8.00000			−0.258065			−0.129032	−	0.991640i	$-0.541187\pi$
$31$	−8.00000			−0.258065			−0.129032	+	0.991640i	$0.541187\pi$
$32$	−4.00000	+	4.00000i	−0.125000	+	0.125000i
$33$		0			0
$34$			24.0000i			0.705882i
$35$		0			0
$36$		0			0
$37$	−48.0000	+	48.0000i	−1.29730	+	1.29730i	−0.367126	+	0.930171i	$0.619658\pi$
$37$	−48.0000	+	48.0000i	−1.29730	+	1.29730i	−0.930171	+	0.367126i	$0.880342\pi$
$38$	−20.0000	−	20.0000i	−0.526316	−	0.526316i
$39$		0			0
$40$		0			0
$41$	48.0000			1.17073			0.585366	−	0.810769i	$-0.300951\pi$
$41$	48.0000			1.17073			0.585366	+	0.810769i	$0.300951\pi$
$42$		0			0
$43$	−27.0000	−	27.0000i	−0.627907	−	0.627907i	0.319634	−	0.947541i	$-0.396440\pi$
$43$	−27.0000	−	27.0000i	−0.627907	−	0.627907i	−0.947541	+	0.319634i	$0.896440\pi$
$44$			24.0000i			0.545455i
$45$		0			0
$46$	−6.00000			−0.130435
$47$	−27.0000	+	27.0000i	−0.574468	+	0.574468i	−0.933374	−	0.358906i	$-0.883150\pi$
$47$	−27.0000	+	27.0000i	−0.574468	+	0.574468i	0.358906	+	0.933374i	$0.383150\pi$
$48$		0			0
$49$			31.0000i			0.632653i
$50$		0			0
$51$		0			0
$52$	−24.0000	+	24.0000i	−0.461538	+	0.461538i
$53$	12.0000	+	12.0000i	0.226415	+	0.226415i	0.811193	−	0.584778i	$-0.198818\pi$
$53$	12.0000	+	12.0000i	0.226415	+	0.226415i	−0.584778	+	0.811193i	$0.698818\pi$
$54$		0			0
$55$		0			0
$56$	12.0000			0.214286
$57$		0			0
$58$	−30.0000	−	30.0000i	−0.517241	−	0.517241i
$59$		−	60.0000i		−	1.01695i	−0.861077	−	0.508475i	$-0.830210\pi$
$59$		−	60.0000i		−	1.01695i	0.861077	−	0.508475i	$-0.169790\pi$
$60$		0			0
$61$	32.0000			0.524590			0.262295	−	0.964988i	$-0.415521\pi$
$61$	32.0000			0.524590			0.262295	+	0.964988i	$0.415521\pi$
$62$	−8.00000	+	8.00000i	−0.129032	+	0.129032i
$63$		0			0
$64$			8.00000i			0.125000i
$65$		0			0
$66$		0			0
$67$	−3.00000	+	3.00000i	−0.0447761	+	0.0447761i	−0.729140	−	0.684364i	$-0.760080\pi$
$67$	−3.00000	+	3.00000i	−0.0447761	+	0.0447761i	0.684364	+	0.729140i	$0.260080\pi$
$68$	24.0000	+	24.0000i	0.352941	+	0.352941i
$69$		0			0
$70$		0			0
$71$	48.0000			0.676056			0.338028	−	0.941136i	$-0.390240\pi$
$71$	48.0000			0.676056			0.338028	+	0.941136i	$0.390240\pi$
$72$		0			0
$73$	−12.0000	−	12.0000i	−0.164384	−	0.164384i	0.620122	−	0.784505i	$-0.287083\pi$
$73$	−12.0000	−	12.0000i	−0.164384	−	0.164384i	−0.784505	+	0.620122i	$0.787083\pi$
$74$			96.0000i			1.29730i
$75$		0			0
$76$	−40.0000			−0.526316
$77$	36.0000	−	36.0000i	0.467532	−	0.467532i
$78$		0			0
$79$		−	40.0000i		−	0.506329i	−0.967423	−	0.253165i	$-0.918529\pi$
$79$		−	40.0000i		−	0.506329i	0.967423	−	0.253165i	$-0.0814714\pi$
$80$		0			0
$81$		0			0
$82$	48.0000	−	48.0000i	0.585366	−	0.585366i
$83$	−93.0000	−	93.0000i	−1.12048	−	1.12048i	−0.991669	−	0.128813i	$-0.958883\pi$
$83$	−93.0000	−	93.0000i	−1.12048	−	1.12048i	−0.128813	−	0.991669i	$-0.541117\pi$
$84$		0			0
$85$		0			0
$86$	−54.0000			−0.627907
$87$		0			0
$88$	24.0000	+	24.0000i	0.272727	+	0.272727i
$89$			30.0000i			0.337079i	0.985695	+	0.168539i	$0.0539050\pi$
$89$			30.0000i			0.337079i	−0.985695	+	0.168539i	$0.946095\pi$
$90$		0			0
$91$	72.0000			0.791209
$92$	−6.00000	+	6.00000i	−0.0652174	+	0.0652174i
$93$		0			0
$94$			54.0000i			0.574468i
$95$		0			0
$96$		0			0
$97$	12.0000	−	12.0000i	0.123711	−	0.123711i	−0.642540	−	0.766252i	$-0.722120\pi$
$97$	12.0000	−	12.0000i	0.123711	−	0.123711i	0.766252	+	0.642540i	$0.222120\pi$
$98$	31.0000	+	31.0000i	0.316327	+	0.316327i
$99$		0			0
$100$		0			0
$101$	78.0000			0.772277			0.386139	−	0.922441i	$-0.373809\pi$
$101$	78.0000			0.772277			0.386139	+	0.922441i	$0.373809\pi$
$102$		0			0
$103$	93.0000	+	93.0000i	0.902913	+	0.902913i	0.995687	−	0.0927745i	$-0.0295736\pi$
$103$	93.0000	+	93.0000i	0.902913	+	0.902913i	−0.0927745	+	0.995687i	$0.529574\pi$
$104$			48.0000i			0.461538i
$105$		0			0
$106$	24.0000			0.226415
$107$	−27.0000	+	27.0000i	−0.252336	+	0.252336i	−0.821928	−	0.569591i	$-0.807101\pi$
$107$	−27.0000	+	27.0000i	−0.252336	+	0.252336i	0.569591	+	0.821928i	$0.307101\pi$
$108$		0			0
$109$		−	160.000i		−	1.46789i	−0.679209	−	0.733945i	$-0.737677\pi$
$109$		−	160.000i		−	1.46789i	0.679209	−	0.733945i	$-0.262323\pi$
$110$		0			0
$111$		0			0
$112$	12.0000	−	12.0000i	0.107143	−	0.107143i
$113$	72.0000	+	72.0000i	0.637168	+	0.637168i	0.949856	−	0.312688i	$-0.101229\pi$
$113$	72.0000	+	72.0000i	0.637168	+	0.637168i	−0.312688	+	0.949856i	$0.601229\pi$
$114$		0			0
$115$		0			0
$116$	−60.0000			−0.517241
$117$		0			0
$118$	−60.0000	−	60.0000i	−0.508475	−	0.508475i
$119$		−	72.0000i		−	0.605042i
$120$		0			0
$121$	23.0000			0.190083
$122$	32.0000	−	32.0000i	0.262295	−	0.262295i
$123$		0			0
$124$			16.0000i			0.129032i
$125$		0			0
$126$		0			0
$127$	117.000	−	117.000i	0.921260	−	0.921260i	−0.0758587	−	0.997119i	$-0.524170\pi$
$127$	117.000	−	117.000i	0.921260	−	0.921260i	0.997119	+	0.0758587i	$0.0241698\pi$
$128$	8.00000	+	8.00000i	0.0625000	+	0.0625000i
$129$		0			0
$130$		0			0
$131$	−132.000			−1.00763			−0.503817	−	0.863811i	$-0.668071\pi$
$131$	−132.000			−1.00763			−0.503817	+	0.863811i	$0.668071\pi$
$132$		0			0
$133$	60.0000	+	60.0000i	0.451128	+	0.451128i
$134$			6.00000i			0.0447761i
$135$		0			0
$136$	48.0000			0.352941
$137$	168.000	−	168.000i	1.22628	−	1.22628i	0.260916	−	0.965362i	$-0.415975\pi$
$137$	168.000	−	168.000i	1.22628	−	1.22628i	0.965362	−	0.260916i	$-0.0840245\pi$
$138$		0			0
$139$			100.000i			0.719424i	0.933063	+	0.359712i	$0.117125\pi$
$139$			100.000i			0.719424i	−0.933063	+	0.359712i	$0.882875\pi$
$140$		0			0
$141$		0			0
$142$	48.0000	−	48.0000i	0.338028	−	0.338028i
$143$	144.000	+	144.000i	1.00699	+	1.00699i
$144$		0			0
$145$		0			0
$146$	−24.0000			−0.164384
$147$		0			0
$148$	96.0000	+	96.0000i	0.648649	+	0.648649i
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	−248.000			−1.64238			−0.821192	−	0.570652i	$-0.806691\pi$
$151$	−248.000			−1.64238			−0.821192	+	0.570652i	$0.806691\pi$
$152$	−40.0000	+	40.0000i	−0.263158	+	0.263158i
$153$		0			0
$154$		−	72.0000i		−	0.467532i
$155$		0			0
$156$		0			0
$157$	72.0000	−	72.0000i	0.458599	−	0.458599i	−0.439597	−	0.898195i	$-0.644879\pi$
$157$	72.0000	−	72.0000i	0.458599	−	0.458599i	0.898195	+	0.439597i	$0.144879\pi$
$158$	−40.0000	−	40.0000i	−0.253165	−	0.253165i
$159$		0			0
$160$		0			0
$161$	18.0000			0.111801
$162$		0			0
$163$	93.0000	+	93.0000i	0.570552	+	0.570552i	0.932283	−	0.361731i	$-0.117814\pi$
$163$	93.0000	+	93.0000i	0.570552	+	0.570552i	−0.361731	+	0.932283i	$0.617814\pi$
$164$		−	96.0000i		−	0.585366i
$165$		0			0
$166$	−186.000			−1.12048
$167$	3.00000	−	3.00000i	0.0179641	−	0.0179641i	−0.698068	−	0.716032i	$-0.745957\pi$
$167$	3.00000	−	3.00000i	0.0179641	−	0.0179641i	0.716032	+	0.698068i	$0.245957\pi$
$168$		0			0
$169$			119.000i			0.704142i
$170$		0			0
$171$		0			0
$172$	−54.0000	+	54.0000i	−0.313953	+	0.313953i
$173$	−168.000	−	168.000i	−0.971098	−	0.971098i	0.0284957	−	0.999594i	$-0.490928\pi$
$173$	−168.000	−	168.000i	−0.971098	−	0.971098i	−0.999594	+	0.0284957i	$0.990928\pi$
$174$		0			0
$175$		0			0
$176$	48.0000			0.272727
$177$		0			0
$178$	30.0000	+	30.0000i	0.168539	+	0.168539i
$179$		−	300.000i		−	1.67598i	−0.545687	−	0.837989i	$-0.683731\pi$
$179$		−	300.000i		−	1.67598i	0.545687	−	0.837989i	$-0.316269\pi$
$180$		0			0
$181$	142.000			0.784530			0.392265	−	0.919852i	$-0.371692\pi$
$181$	142.000			0.784530			0.392265	+	0.919852i	$0.371692\pi$
$182$	72.0000	−	72.0000i	0.395604	−	0.395604i
$183$		0			0
$184$			12.0000i			0.0652174i
$185$		0			0
$186$		0			0
$187$	144.000	−	144.000i	0.770053	−	0.770053i
$188$	54.0000	+	54.0000i	0.287234	+	0.287234i
$189$		0			0
$190$		0			0
$191$	−192.000			−1.00524			−0.502618	−	0.864509i	$-0.667630\pi$
$191$	−192.000			−1.00524			−0.502618	+	0.864509i	$0.667630\pi$
$192$		0			0
$193$	−132.000	−	132.000i	−0.683938	−	0.683938i	0.276947	−	0.960885i	$-0.410677\pi$
$193$	−132.000	−	132.000i	−0.683938	−	0.683938i	−0.960885	+	0.276947i	$0.910677\pi$
$194$		−	24.0000i		−	0.123711i
$195$		0			0
$196$	62.0000			0.316327
$197$	−132.000	+	132.000i	−0.670051	+	0.670051i	−0.957728	−	0.287677i	$-0.907117\pi$
$197$	−132.000	+	132.000i	−0.670051	+	0.670051i	0.287677	+	0.957728i	$0.407117\pi$
$198$		0			0
$199$		−	160.000i		−	0.804020i	−0.915635	−	0.402010i	$-0.868312\pi$
$199$		−	160.000i		−	0.804020i	0.915635	−	0.402010i	$-0.131688\pi$
$200$		0			0
$201$		0			0
$202$	78.0000	−	78.0000i	0.386139	−	0.386139i
$203$	90.0000	+	90.0000i	0.443350	+	0.443350i
$204$		0			0
$205$		0			0
$206$	186.000			0.902913
$207$		0			0
$208$	48.0000	+	48.0000i	0.230769	+	0.230769i
$209$			240.000i			1.14833i
$210$		0			0
$211$	−28.0000			−0.132701			−0.0663507	−	0.997796i	$-0.521136\pi$
$211$	−28.0000			−0.132701			−0.0663507	+	0.997796i	$0.521136\pi$
$212$	24.0000	−	24.0000i	0.113208	−	0.113208i
$213$		0			0
$214$			54.0000i			0.252336i
$215$		0			0
$216$		0			0
$217$	24.0000	−	24.0000i	0.110599	−	0.110599i
$218$	−160.000	−	160.000i	−0.733945	−	0.733945i
$219$		0			0
$220$		0			0
$221$	288.000			1.30317
$222$		0			0
$223$	−117.000	−	117.000i	−0.524664	−	0.524664i	0.394313	−	0.918976i	$-0.370983\pi$
$223$	−117.000	−	117.000i	−0.524664	−	0.524664i	−0.918976	+	0.394313i	$0.870983\pi$
$224$		−	24.0000i		−	0.107143i
$225$		0			0
$226$	144.000			0.637168
$227$	93.0000	−	93.0000i	0.409692	−	0.409692i	−0.471939	−	0.881631i	$-0.656446\pi$
$227$	93.0000	−	93.0000i	0.409692	−	0.409692i	0.881631	+	0.471939i	$0.156446\pi$
$228$		0			0
$229$			370.000i			1.61572i	0.589374	+	0.807860i	$0.299374\pi$
$229$			370.000i			1.61572i	−0.589374	+	0.807860i	$0.700626\pi$
$230$		0			0
$231$		0			0
$232$	−60.0000	+	60.0000i	−0.258621	+	0.258621i
$233$	252.000	+	252.000i	1.08155	+	1.08155i	0.996366	+	0.0851794i	$0.0271464\pi$
$233$	252.000	+	252.000i	1.08155	+	1.08155i	0.0851794	+	0.996366i	$0.472854\pi$
$234$		0			0
$235$		0			0
$236$	−120.000			−0.508475
$237$		0			0
$238$	−72.0000	−	72.0000i	−0.302521	−	0.302521i
$239$			360.000i			1.50628i	0.657862	+	0.753138i	$0.271461\pi$
$239$			360.000i			1.50628i	−0.657862	+	0.753138i	$0.728539\pi$
$240$		0			0
$241$	32.0000			0.132780			0.0663900	−	0.997794i	$-0.478852\pi$
$241$	32.0000			0.132780			0.0663900	+	0.997794i	$0.478852\pi$
$242$	23.0000	−	23.0000i	0.0950413	−	0.0950413i
$243$		0			0
$244$		−	64.0000i		−	0.262295i
$245$		0			0
$246$		0			0
$247$	−240.000	+	240.000i	−0.971660	+	0.971660i
$248$	16.0000	+	16.0000i	0.0645161	+	0.0645161i
$249$		0			0
$250$		0			0
$251$	−252.000			−1.00398			−0.501992	−	0.864872i	$-0.667399\pi$
$251$	−252.000			−1.00398			−0.501992	+	0.864872i	$0.667399\pi$
$252$		0			0
$253$	36.0000	+	36.0000i	0.142292	+	0.142292i
$254$		−	234.000i		−	0.921260i
$255$		0			0
$256$	16.0000			0.0625000
$257$	−192.000	+	192.000i	−0.747082	+	0.747082i	−0.973930	−	0.226848i	$-0.927158\pi$
$257$	−192.000	+	192.000i	−0.747082	+	0.747082i	0.226848	+	0.973930i	$0.427158\pi$
$258$		0			0
$259$		−	288.000i		−	1.11197i
$260$		0			0
$261$		0			0
$262$	−132.000	+	132.000i	−0.503817	+	0.503817i
$263$	−333.000	−	333.000i	−1.26616	−	1.26616i	−0.948056	−	0.318104i	$-0.896954\pi$
$263$	−333.000	−	333.000i	−1.26616	−	1.26616i	−0.318104	−	0.948056i	$-0.603046\pi$
$264$		0			0
$265$		0			0
$266$	120.000			0.451128
$267$		0			0
$268$	6.00000	+	6.00000i	0.0223881	+	0.0223881i
$269$			480.000i			1.78439i	0.451654	+	0.892193i	$0.350834\pi$
$269$			480.000i			1.78439i	−0.451654	+	0.892193i	$0.649166\pi$
$270$		0			0
$271$	−88.0000			−0.324723			−0.162362	−	0.986731i	$-0.551911\pi$
$271$	−88.0000			−0.324723			−0.162362	+	0.986731i	$0.551911\pi$
$272$	48.0000	−	48.0000i	0.176471	−	0.176471i
$273$		0			0
$274$		−	336.000i		−	1.22628i
$275$		0			0
$276$		0			0
$277$	−288.000	+	288.000i	−1.03971	+	1.03971i	−0.0405330	+	0.999178i	$0.512906\pi$
$277$	−288.000	+	288.000i	−1.03971	+	1.03971i	−0.999178	+	0.0405330i	$0.987094\pi$
$278$	100.000	+	100.000i	0.359712	+	0.359712i
$279$		0			0
$280$		0			0
$281$	288.000			1.02491			0.512456	−	0.858714i	$-0.328736\pi$
$281$	288.000			1.02491			0.512456	+	0.858714i	$0.328736\pi$
$282$		0			0
$283$	−117.000	−	117.000i	−0.413428	−	0.413428i	0.469503	−	0.882931i	$-0.344433\pi$
$283$	−117.000	−	117.000i	−0.413428	−	0.413428i	−0.882931	+	0.469503i	$0.844433\pi$
$284$		−	96.0000i		−	0.338028i
$285$		0			0
$286$	288.000			1.00699
$287$	−144.000	+	144.000i	−0.501742	+	0.501742i
$288$		0			0
$289$			1.00000i			0.00346021i
$290$		0			0
$291$		0			0
$292$	−24.0000	+	24.0000i	−0.0821918	+	0.0821918i
$293$	−168.000	−	168.000i	−0.573379	−	0.573379i	0.359692	−	0.933071i	$-0.382882\pi$
$293$	−168.000	−	168.000i	−0.573379	−	0.573379i	−0.933071	+	0.359692i	$0.882882\pi$
$294$		0			0
$295$		0			0
$296$	192.000			0.648649
$297$		0			0
$298$		0			0
$299$			72.0000i			0.240803i
$300$		0			0
$301$	162.000			0.538206
$302$	−248.000	+	248.000i	−0.821192	+	0.821192i
$303$		0			0
$304$			80.0000i			0.263158i
$305$		0			0
$306$		0			0
$307$	−243.000	+	243.000i	−0.791531	+	0.791531i	−0.981743	−	0.190212i	$-0.939082\pi$
$307$	−243.000	+	243.000i	−0.791531	+	0.791531i	0.190212	+	0.981743i	$0.439082\pi$
$308$	−72.0000	−	72.0000i	−0.233766	−	0.233766i
$309$		0			0
$310$		0			0
$311$	−552.000			−1.77492			−0.887460	−	0.460885i	$-0.847532\pi$
$311$	−552.000			−1.77492			−0.887460	+	0.460885i	$0.847532\pi$
$312$		0			0
$313$	48.0000	+	48.0000i	0.153355	+	0.153355i	0.779614	−	0.626260i	$-0.215415\pi$
$313$	48.0000	+	48.0000i	0.153355	+	0.153355i	−0.626260	+	0.779614i	$0.715415\pi$
$314$		−	144.000i		−	0.458599i
$315$		0			0
$316$	−80.0000			−0.253165
$317$	228.000	−	228.000i	0.719243	−	0.719243i	−0.249207	−	0.968450i	$-0.580170\pi$
$317$	228.000	−	228.000i	0.719243	−	0.719243i	0.968450	+	0.249207i	$0.0801700\pi$
$318$		0			0
$319$			360.000i			1.12853i
$320$		0			0
$321$		0			0
$322$	18.0000	−	18.0000i	0.0559006	−	0.0559006i
$323$	240.000	+	240.000i	0.743034	+	0.743034i
$324$		0			0
$325$		0			0
$326$	186.000			0.570552
$327$		0			0
$328$	−96.0000	−	96.0000i	−0.292683	−	0.292683i
$329$		−	162.000i		−	0.492401i
$330$		0			0
$331$	−148.000			−0.447130			−0.223565	−	0.974689i	$-0.571769\pi$
$331$	−148.000			−0.447130			−0.223565	+	0.974689i	$0.571769\pi$
$332$	−186.000	+	186.000i	−0.560241	+	0.560241i
$333$		0			0
$334$		−	6.00000i		−	0.0179641i
$335$		0			0
$336$		0			0
$337$	192.000	−	192.000i	0.569733	−	0.569733i	−0.362321	−	0.932054i	$-0.618015\pi$
$337$	192.000	−	192.000i	0.569733	−	0.569733i	0.932054	+	0.362321i	$0.118015\pi$
$338$	119.000	+	119.000i	0.352071	+	0.352071i
$339$		0			0
$340$		0			0
$341$	96.0000			0.281525
$342$		0			0
$343$	−240.000	−	240.000i	−0.699708	−	0.699708i
$344$			108.000i			0.313953i
$345$		0			0
$346$	−336.000			−0.971098
$347$	−117.000	+	117.000i	−0.337176	+	0.337176i	−0.855303	−	0.518128i	$-0.826629\pi$
$347$	−117.000	+	117.000i	−0.337176	+	0.337176i	0.518128	+	0.855303i	$0.326629\pi$
$348$		0			0
$349$			130.000i			0.372493i	0.982503	+	0.186246i	$0.0596323\pi$
$349$			130.000i			0.372493i	−0.982503	+	0.186246i	$0.940368\pi$
$350$		0			0
$351$		0			0
$352$	48.0000	−	48.0000i	0.136364	−	0.136364i
$353$	−288.000	−	288.000i	−0.815864	−	0.815864i	0.169642	−	0.985506i	$-0.445739\pi$
$353$	−288.000	−	288.000i	−0.815864	−	0.815864i	−0.985506	+	0.169642i	$0.945739\pi$
$354$		0			0
$355$		0			0
$356$	60.0000			0.168539
$357$		0			0
$358$	−300.000	−	300.000i	−0.837989	−	0.837989i
$359$		−	120.000i		−	0.334262i	−0.985935	−	0.167131i	$-0.946550\pi$
$359$		−	120.000i		−	0.334262i	0.985935	−	0.167131i	$-0.0534503\pi$
$360$		0			0
$361$	−39.0000			−0.108033
$362$	142.000	−	142.000i	0.392265	−	0.392265i
$363$		0			0
$364$		−	144.000i		−	0.395604i
$365$		0			0
$366$		0			0
$367$	−213.000	+	213.000i	−0.580381	+	0.580381i	−0.935008	−	0.354627i	$-0.884608\pi$
$367$	−213.000	+	213.000i	−0.580381	+	0.580381i	0.354627	+	0.935008i	$0.384608\pi$
$368$	12.0000	+	12.0000i	0.0326087	+	0.0326087i
$369$		0			0
$370$		0			0
$371$	−72.0000			−0.194070
$372$		0			0
$373$	168.000	+	168.000i	0.450402	+	0.450402i	0.895488	−	0.445086i	$-0.146827\pi$
$373$	168.000	+	168.000i	0.450402	+	0.450402i	−0.445086	+	0.895488i	$0.646827\pi$
$374$		−	288.000i		−	0.770053i
$375$		0			0
$376$	108.000			0.287234
$377$	−360.000	+	360.000i	−0.954907	+	0.954907i
$378$		0			0
$379$		−	20.0000i		−	0.0527704i	−0.999652	−	0.0263852i	$-0.991600\pi$
$379$		−	20.0000i		−	0.0527704i	0.999652	−	0.0263852i	$-0.00839965\pi$
$380$		0			0
$381$		0			0
$382$	−192.000	+	192.000i	−0.502618	+	0.502618i
$383$	−123.000	−	123.000i	−0.321149	−	0.321149i	0.528059	−	0.849208i	$-0.322920\pi$
$383$	−123.000	−	123.000i	−0.321149	−	0.321149i	−0.849208	+	0.528059i	$0.822920\pi$
$384$		0			0
$385$		0			0
$386$	−264.000			−0.683938
$387$		0			0
$388$	−24.0000	−	24.0000i	−0.0618557	−	0.0618557i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$	72.0000			0.184143
$392$	62.0000	−	62.0000i	0.158163	−	0.158163i
$393$		0			0
$394$			264.000i			0.670051i
$395$		0			0
$396$		0			0
$397$	−108.000	+	108.000i	−0.272040	+	0.272040i	−0.829921	−	0.557881i	$-0.811615\pi$
$397$	−108.000	+	108.000i	−0.272040	+	0.272040i	0.557881	+	0.829921i	$0.311615\pi$
$398$	−160.000	−	160.000i	−0.402010	−	0.402010i
$399$		0			0
$400$		0			0
$401$	18.0000			0.0448878			0.0224439	−	0.999748i	$-0.492855\pi$
$401$	18.0000			0.0448878			0.0224439	+	0.999748i	$0.492855\pi$
$402$		0			0
$403$	96.0000	+	96.0000i	0.238213	+	0.238213i
$404$		−	156.000i		−	0.386139i
$405$		0			0
$406$	180.000			0.443350
$407$	576.000	−	576.000i	1.41523	−	1.41523i
$408$		0			0
$409$		−	80.0000i		−	0.195599i	−0.995206	−	0.0977995i	$-0.968820\pi$
$409$		−	80.0000i		−	0.195599i	0.995206	−	0.0977995i	$-0.0311804\pi$
$410$		0			0
$411$		0			0
$412$	186.000	−	186.000i	0.451456	−	0.451456i
$413$	180.000	+	180.000i	0.435835	+	0.435835i
$414$		0			0
$415$		0			0
$416$	96.0000			0.230769
$417$		0			0
$418$	240.000	+	240.000i	0.574163	+	0.574163i
$419$		−	540.000i		−	1.28878i	−0.764696	−	0.644391i	$-0.777111\pi$
$419$		−	540.000i		−	1.28878i	0.764696	−	0.644391i	$-0.222889\pi$
$420$		0			0
$421$	−608.000			−1.44418			−0.722090	−	0.691799i	$-0.756818\pi$
$421$	−608.000			−1.44418			−0.722090	+	0.691799i	$0.756818\pi$
$422$	−28.0000	+	28.0000i	−0.0663507	+	0.0663507i
$423$		0			0
$424$		−	48.0000i		−	0.113208i
$425$		0			0
$426$		0			0
$427$	−96.0000	+	96.0000i	−0.224824	+	0.224824i
$428$	54.0000	+	54.0000i	0.126168	+	0.126168i
$429$		0			0
$430$		0			0
$431$	−312.000			−0.723898			−0.361949	−	0.932198i	$-0.617889\pi$
$431$	−312.000			−0.723898			−0.361949	+	0.932198i	$0.617889\pi$
$432$		0			0
$433$	−252.000	−	252.000i	−0.581986	−	0.581986i	0.353463	−	0.935449i	$-0.385004\pi$
$433$	−252.000	−	252.000i	−0.581986	−	0.581986i	−0.935449	+	0.353463i	$0.885004\pi$
$434$		−	48.0000i		−	0.110599i
$435$		0			0
$436$	−320.000			−0.733945
$437$	−60.0000	+	60.0000i	−0.137300	+	0.137300i
$438$		0			0
$439$		−	40.0000i		−	0.0911162i	−0.998962	−	0.0455581i	$-0.985493\pi$
$439$		−	40.0000i		−	0.0911162i	0.998962	−	0.0455581i	$-0.0145066\pi$
$440$		0			0
$441$		0			0
$442$	288.000	−	288.000i	0.651584	−	0.651584i
$443$	−213.000	−	213.000i	−0.480813	−	0.480813i	0.424578	−	0.905391i	$-0.360422\pi$
$443$	−213.000	−	213.000i	−0.480813	−	0.480813i	−0.905391	+	0.424578i	$0.860422\pi$
$444$		0			0
$445$		0			0
$446$	−234.000			−0.524664
$447$		0			0
$448$	−24.0000	−	24.0000i	−0.0535714	−	0.0535714i
$449$			480.000i			1.06904i	0.845155	+	0.534521i	$0.179508\pi$
$449$			480.000i			1.06904i	−0.845155	+	0.534521i	$0.820492\pi$
$450$		0			0
$451$	−576.000			−1.27716
$452$	144.000	−	144.000i	0.318584	−	0.318584i
$453$		0			0
$454$		−	186.000i		−	0.409692i
$455$		0			0
$456$		0			0
$457$	432.000	−	432.000i	0.945295	−	0.945295i	−0.0532840	−	0.998579i	$-0.516969\pi$
$457$	432.000	−	432.000i	0.945295	−	0.945295i	0.998579	+	0.0532840i	$0.0169689\pi$
$458$	370.000	+	370.000i	0.807860	+	0.807860i
$459$		0			0
$460$		0			0
$461$	−222.000			−0.481562			−0.240781	−	0.970579i	$-0.577404\pi$
$461$	−222.000			−0.481562			−0.240781	+	0.970579i	$0.577404\pi$
$462$		0			0
$463$	213.000	+	213.000i	0.460043	+	0.460043i	0.898670	−	0.438626i	$-0.144535\pi$
$463$	213.000	+	213.000i	0.460043	+	0.460043i	−0.438626	+	0.898670i	$0.644535\pi$
$464$			120.000i			0.258621i
$465$		0			0
$466$	504.000			1.08155
$467$	3.00000	−	3.00000i	0.00642398	−	0.00642398i	−0.703887	−	0.710311i	$-0.748554\pi$
$467$	3.00000	−	3.00000i	0.00642398	−	0.00642398i	0.710311	+	0.703887i	$0.248554\pi$
$468$		0			0
$469$		−	18.0000i		−	0.0383795i
$470$		0			0
$471$		0			0
$472$	−120.000	+	120.000i	−0.254237	+	0.254237i
$473$	324.000	+	324.000i	0.684989	+	0.684989i
$474$		0			0
$475$		0			0
$476$	−144.000			−0.302521
$477$		0			0
$478$	360.000	+	360.000i	0.753138	+	0.753138i
$479$		−	240.000i		−	0.501044i	−0.968111	−	0.250522i	$-0.919398\pi$
$479$		−	240.000i		−	0.501044i	0.968111	−	0.250522i	$-0.0806022\pi$
$480$		0			0
$481$	1152.00			2.39501
$482$	32.0000	−	32.0000i	0.0663900	−	0.0663900i
$483$		0			0
$484$		−	46.0000i		−	0.0950413i
$485$		0			0
$486$		0			0
$487$	627.000	−	627.000i	1.28747	−	1.28747i	0.351158	−	0.936316i	$-0.385788\pi$
$487$	627.000	−	627.000i	1.28747	−	1.28747i	0.936316	−	0.351158i	$-0.114212\pi$
$488$	−64.0000	−	64.0000i	−0.131148	−	0.131148i
$489$		0			0
$490$		0			0
$491$	588.000			1.19756			0.598778	−	0.800915i	$-0.295653\pi$
$491$	588.000			1.19756			0.598778	+	0.800915i	$0.295653\pi$
$492$		0			0
$493$	360.000	+	360.000i	0.730223	+	0.730223i
$494$			480.000i			0.971660i
$495$		0			0
$496$	32.0000			0.0645161
$497$	−144.000	+	144.000i	−0.289738	+	0.289738i
$498$		0			0
$499$		−	460.000i		−	0.921844i	−0.887441	−	0.460922i	$-0.847519\pi$
$499$		−	460.000i		−	0.921844i	0.887441	−	0.460922i	$-0.152481\pi$
$500$		0			0
$501$		0			0
$502$	−252.000	+	252.000i	−0.501992	+	0.501992i
$503$	627.000	+	627.000i	1.24652	+	1.24652i	0.957246	+	0.289275i	$0.0934141\pi$
$503$	627.000	+	627.000i	1.24652	+	1.24652i	0.289275	+	0.957246i	$0.406586\pi$
$504$		0			0
$505$		0			0
$506$	72.0000			0.142292
$507$		0			0
$508$	−234.000	−	234.000i	−0.460630	−	0.460630i
$509$			450.000i			0.884086i	0.896994	+	0.442043i	$0.145746\pi$
$509$			450.000i			0.884086i	−0.896994	+	0.442043i	$0.854254\pi$
$510$		0			0
$511$	72.0000			0.140900
$512$	16.0000	−	16.0000i	0.0312500	−	0.0312500i
$513$		0			0
$514$			384.000i			0.747082i
$515$		0			0
$516$		0			0
$517$	324.000	−	324.000i	0.626692	−	0.626692i
$518$	−288.000	−	288.000i	−0.555985	−	0.555985i
$519$		0			0
$520$		0			0
$521$	558.000			1.07102			0.535509	−	0.844530i	$-0.320120\pi$
$521$	558.000			1.07102			0.535509	+	0.844530i	$0.320120\pi$
$522$		0			0
$523$	123.000	+	123.000i	0.235182	+	0.235182i	0.814851	−	0.579670i	$-0.196818\pi$
$523$	123.000	+	123.000i	0.235182	+	0.235182i	−0.579670	+	0.814851i	$0.696818\pi$
$524$			264.000i			0.503817i
$525$		0			0
$526$	−666.000			−1.26616
$527$	96.0000	−	96.0000i	0.182163	−	0.182163i
$528$		0			0
$529$		−	511.000i		−	0.965974i
$530$		0			0
$531$		0			0
$532$	120.000	−	120.000i	0.225564	−	0.225564i
$533$	−576.000	−	576.000i	−1.08068	−	1.08068i
$534$		0			0
$535$		0			0
$536$	12.0000			0.0223881
$537$		0			0
$538$	480.000	+	480.000i	0.892193	+	0.892193i
$539$		−	372.000i		−	0.690167i
$540$		0			0
$541$	542.000			1.00185			0.500924	−	0.865491i	$-0.332994\pi$
$541$	542.000			1.00185			0.500924	+	0.865491i	$0.332994\pi$
$542$	−88.0000	+	88.0000i	−0.162362	+	0.162362i
$543$		0			0
$544$		−	96.0000i		−	0.176471i
$545$		0			0
$546$		0			0
$547$	147.000	−	147.000i	0.268739	−	0.268739i	−0.559853	−	0.828592i	$-0.689142\pi$
$547$	147.000	−	147.000i	0.268739	−	0.268739i	0.828592	+	0.559853i	$0.189142\pi$
$548$	−336.000	−	336.000i	−0.613139	−	0.613139i
$549$		0			0
$550$		0			0
$551$	−600.000			−1.08893
$552$		0			0
$553$	120.000	+	120.000i	0.216998	+	0.216998i
$554$			576.000i			1.03971i
$555$		0			0
$556$	200.000			0.359712
$557$	288.000	−	288.000i	0.517056	−	0.517056i	−0.399624	−	0.916679i	$-0.630859\pi$
$557$	288.000	−	288.000i	0.517056	−	0.517056i	0.916679	+	0.399624i	$0.130859\pi$
$558$		0			0
$559$			648.000i			1.15921i
$560$		0			0
$561$		0			0
$562$	288.000	−	288.000i	0.512456	−	0.512456i
$563$	477.000	+	477.000i	0.847247	+	0.847247i	0.989789	−	0.142542i	$-0.0455276\pi$
$563$	477.000	+	477.000i	0.847247	+	0.847247i	−0.142542	+	0.989789i	$0.545528\pi$
$564$		0			0
$565$		0			0
$566$	−234.000			−0.413428
$567$		0			0
$568$	−96.0000	−	96.0000i	−0.169014	−	0.169014i
$569$		−	240.000i		−	0.421793i	−0.977508	−	0.210896i	$-0.932362\pi$
$569$		−	240.000i		−	0.421793i	0.977508	−	0.210896i	$-0.0676382\pi$
$570$		0			0
$571$	692.000			1.21191			0.605954	−	0.795499i	$-0.292791\pi$
$571$	692.000			1.21191			0.605954	+	0.795499i	$0.292791\pi$
$572$	288.000	−	288.000i	0.503497	−	0.503497i
$573$		0			0
$574$			288.000i			0.501742i
$575$		0			0
$576$		0			0
$577$	−168.000	+	168.000i	−0.291161	+	0.291161i	−0.837539	−	0.546378i	$-0.816006\pi$
$577$	−168.000	+	168.000i	−0.291161	+	0.291161i	0.546378	+	0.837539i	$0.316006\pi$
$578$	1.00000	+	1.00000i	0.00173010	+	0.00173010i
$579$		0			0
$580$		0			0
$581$	558.000			0.960413
$582$		0			0
$583$	−144.000	−	144.000i	−0.246998	−	0.246998i
$584$			48.0000i			0.0821918i
$585$		0			0
$586$	−336.000			−0.573379
$587$	213.000	−	213.000i	0.362862	−	0.362862i	−0.502004	−	0.864866i	$-0.667404\pi$
$587$	213.000	−	213.000i	0.362862	−	0.362862i	0.864866	+	0.502004i	$0.167404\pi$
$588$		0			0
$589$			160.000i			0.271647i
$590$		0			0
$591$		0			0
$592$	192.000	−	192.000i	0.324324	−	0.324324i
$593$	312.000	+	312.000i	0.526138	+	0.526138i	0.919419	−	0.393280i	$-0.128660\pi$
$593$	312.000	+	312.000i	0.526138	+	0.526138i	−0.393280	+	0.919419i	$0.628660\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	72.0000	+	72.0000i	0.120401	+	0.120401i
$599$			240.000i			0.400668i	0.979728	+	0.200334i	$0.0642027\pi$
$599$			240.000i			0.400668i	−0.979728	+	0.200334i	$0.935797\pi$
$600$		0			0
$601$	−608.000			−1.01165			−0.505824	−	0.862637i	$-0.668811\pi$
$601$	−608.000			−1.01165			−0.505824	+	0.862637i	$0.668811\pi$
$602$	162.000	−	162.000i	0.269103	−	0.269103i
$603$		0			0
$604$			496.000i			0.821192i
$605$		0			0
$606$		0			0
$607$	267.000	−	267.000i	0.439868	−	0.439868i	−0.452099	−	0.891968i	$-0.649325\pi$
$607$	267.000	−	267.000i	0.439868	−	0.439868i	0.891968	+	0.452099i	$0.149325\pi$
$608$	80.0000	+	80.0000i	0.131579	+	0.131579i
$609$		0			0
$610$		0			0
$611$	648.000			1.06056
$612$		0			0
$613$	228.000	+	228.000i	0.371941	+	0.371941i	0.868184	−	0.496243i	$-0.165287\pi$
$613$	228.000	+	228.000i	0.371941	+	0.371941i	−0.496243	+	0.868184i	$0.665287\pi$
$614$			486.000i			0.791531i
$615$		0			0
$616$	−144.000			−0.233766
$617$	348.000	−	348.000i	0.564019	−	0.564019i	−0.366427	−	0.930447i	$-0.619419\pi$
$617$	348.000	−	348.000i	0.564019	−	0.564019i	0.930447	+	0.366427i	$0.119419\pi$
$618$		0			0
$619$		−	940.000i		−	1.51858i	−0.650753	−	0.759289i	$-0.725547\pi$
$619$		−	940.000i		−	1.51858i	0.650753	−	0.759289i	$-0.274453\pi$
$620$		0			0
$621$		0			0
$622$	−552.000	+	552.000i	−0.887460	+	0.887460i
$623$	−90.0000	−	90.0000i	−0.144462	−	0.144462i
$624$		0			0
$625$		0			0
$626$	96.0000			0.153355
$627$		0			0
$628$	−144.000	−	144.000i	−0.229299	−	0.229299i
$629$		−	1152.00i		−	1.83148i
$630$		0			0
$631$	−808.000			−1.28051			−0.640254	−	0.768164i	$-0.721171\pi$
$631$	−808.000			−1.28051			−0.640254	+	0.768164i	$0.721171\pi$
$632$	−80.0000	+	80.0000i	−0.126582	+	0.126582i
$633$		0			0
$634$		−	456.000i		−	0.719243i
$635$		0			0
$636$		0			0
$637$	372.000	−	372.000i	0.583987	−	0.583987i
$638$	360.000	+	360.000i	0.564263	+	0.564263i
$639$		0			0
$640$		0			0
$641$	768.000			1.19813			0.599064	−	0.800701i	$-0.295540\pi$
$641$	768.000			1.19813			0.599064	+	0.800701i	$0.295540\pi$
$642$		0			0
$643$	−477.000	−	477.000i	−0.741835	−	0.741835i	0.231096	−	0.972931i	$-0.425769\pi$
$643$	−477.000	−	477.000i	−0.741835	−	0.741835i	−0.972931	+	0.231096i	$0.925769\pi$
$644$		−	36.0000i		−	0.0559006i
$645$		0			0
$646$	480.000			0.743034
$647$	−627.000	+	627.000i	−0.969088	+	0.969088i	−0.999536	−	0.0304482i	$-0.990307\pi$
$647$	−627.000	+	627.000i	−0.969088	+	0.969088i	0.0304482	+	0.999536i	$0.490307\pi$
$648$		0			0
$649$			720.000i			1.10940i
$650$		0			0
$651$		0			0
$652$	186.000	−	186.000i	0.285276	−	0.285276i
$653$	12.0000	+	12.0000i	0.0183767	+	0.0183767i	0.716235	−	0.697859i	$-0.245864\pi$
$653$	12.0000	+	12.0000i	0.0183767	+	0.0183767i	−0.697859	+	0.716235i	$0.745864\pi$
$654$		0			0
$655$		0			0
$656$	−192.000			−0.292683
$657$		0			0
$658$	−162.000	−	162.000i	−0.246201	−	0.246201i
$659$		−	540.000i		−	0.819423i	−0.912215	−	0.409712i	$-0.865629\pi$
$659$		−	540.000i		−	0.819423i	0.912215	−	0.409712i	$-0.134371\pi$
$660$		0			0
$661$	352.000			0.532526			0.266263	−	0.963900i	$-0.414211\pi$
$661$	352.000			0.532526			0.266263	+	0.963900i	$0.414211\pi$
$662$	−148.000	+	148.000i	−0.223565	+	0.223565i
$663$		0			0
$664$			372.000i			0.560241i
$665$		0			0
$666$		0			0
$667$	−90.0000	+	90.0000i	−0.134933	+	0.134933i
$668$	−6.00000	−	6.00000i	−0.00898204	−	0.00898204i
$669$		0			0
$670$		0			0
$671$	−384.000			−0.572280
$672$		0			0
$673$	−732.000	−	732.000i	−1.08767	−	1.08767i	−0.995768	−	0.0918988i	$-0.970706\pi$
$673$	−732.000	−	732.000i	−1.08767	−	1.08767i	−0.0918988	−	0.995768i	$-0.529294\pi$
$674$		−	384.000i		−	0.569733i
$675$		0			0
$676$	238.000			0.352071
$677$	108.000	−	108.000i	0.159527	−	0.159527i	−0.622830	−	0.782357i	$-0.714017\pi$
$677$	108.000	−	108.000i	0.159527	−	0.159527i	0.782357	+	0.622830i	$0.214017\pi$
$678$		0			0
$679$			72.0000i			0.106038i
$680$		0			0
$681$		0			0
$682$	96.0000	−	96.0000i	0.140762	−	0.140762i
$683$	−933.000	−	933.000i	−1.36603	−	1.36603i	−0.866016	−	0.500016i	$-0.833327\pi$
$683$	−933.000	−	933.000i	−1.36603	−	1.36603i	−0.500016	−	0.866016i	$-0.666673\pi$
$684$		0			0
$685$		0			0
$686$	−480.000			−0.699708
$687$		0			0
$688$	108.000	+	108.000i	0.156977	+	0.156977i
$689$		−	288.000i		−	0.417997i
$690$		0			0
$691$	−68.0000			−0.0984081			−0.0492041	−	0.998789i	$-0.515668\pi$
$691$	−68.0000			−0.0984081			−0.0492041	+	0.998789i	$0.515668\pi$
$692$	−336.000	+	336.000i	−0.485549	+	0.485549i
$693$		0			0
$694$			234.000i			0.337176i
$695$		0			0
$696$		0			0
$697$	−576.000	+	576.000i	−0.826399	+	0.826399i
$698$	130.000	+	130.000i	0.186246	+	0.186246i
$699$		0			0
$700$		0			0
$701$	−192.000			−0.273894			−0.136947	−	0.990578i	$-0.543729\pi$
$701$	−192.000			−0.273894			−0.136947	+	0.990578i	$0.543729\pi$
$702$		0			0
$703$	960.000	+	960.000i	1.36558	+	1.36558i
$704$		−	96.0000i		−	0.136364i
$705$		0			0
$706$	−576.000			−0.815864
$707$	−234.000	+	234.000i	−0.330976	+	0.330976i
$708$		0			0
$709$		−	50.0000i		−	0.0705219i	−0.999378	−	0.0352609i	$-0.988774\pi$
$709$		−	50.0000i		−	0.0705219i	0.999378	−	0.0352609i	$-0.0112262\pi$
$710$		0			0
$711$		0			0
$712$	60.0000	−	60.0000i	0.0842697	−	0.0842697i
$713$	24.0000	+	24.0000i	0.0336606	+	0.0336606i
$714$		0			0
$715$		0			0
$716$	−600.000			−0.837989
$717$		0			0
$718$	−120.000	−	120.000i	−0.167131	−	0.167131i
$719$			840.000i			1.16829i	0.811650	+	0.584145i	$0.198570\pi$
$719$			840.000i			1.16829i	−0.811650	+	0.584145i	$0.801430\pi$
$720$		0			0
$721$	−558.000			−0.773925
$722$	−39.0000	+	39.0000i	−0.0540166	+	0.0540166i
$723$		0			0
$724$		−	284.000i		−	0.392265i
$725$		0			0
$726$		0			0
$727$	−963.000	+	963.000i	−1.32462	+	1.32462i	−0.414633	+	0.909989i	$0.636090\pi$
$727$	−963.000	+	963.000i	−1.32462	+	1.32462i	−0.909989	+	0.414633i	$0.863910\pi$
$728$	−144.000	−	144.000i	−0.197802	−	0.197802i
$729$		0			0
$730$		0			0
$731$	648.000			0.886457
$732$		0			0
$733$	−72.0000	−	72.0000i	−0.0982265	−	0.0982265i	0.656286	−	0.754512i	$-0.272127\pi$
$733$	−72.0000	−	72.0000i	−0.0982265	−	0.0982265i	−0.754512	+	0.656286i	$0.772127\pi$
$734$			426.000i			0.580381i
$735$		0			0
$736$	24.0000			0.0326087
$737$	36.0000	−	36.0000i	0.0488467	−	0.0488467i
$738$		0			0
$739$		−	20.0000i		−	0.0270636i	−0.999908	−	0.0135318i	$-0.995693\pi$
$739$		−	20.0000i		−	0.0270636i	0.999908	−	0.0135318i	$-0.00430744\pi$
$740$		0			0
$741$		0			0
$742$	−72.0000	+	72.0000i	−0.0970350	+	0.0970350i
$743$	−243.000	−	243.000i	−0.327052	−	0.327052i	0.524412	−	0.851465i	$-0.324285\pi$
$743$	−243.000	−	243.000i	−0.327052	−	0.327052i	−0.851465	+	0.524412i	$0.824285\pi$
$744$		0			0
$745$		0			0
$746$	336.000			0.450402
$747$		0			0
$748$	−288.000	−	288.000i	−0.385027	−	0.385027i
$749$		−	162.000i		−	0.216288i
$750$		0			0
$751$	1072.00			1.42743			0.713715	−	0.700436i	$-0.247011\pi$
$751$	1072.00			1.42743			0.713715	+	0.700436i	$0.247011\pi$
$752$	108.000	−	108.000i	0.143617	−	0.143617i
$753$		0			0
$754$			720.000i			0.954907i
$755$		0			0
$756$		0			0
$757$	−408.000	+	408.000i	−0.538970	+	0.538970i	−0.923226	−	0.384257i	$-0.874458\pi$
$757$	−408.000	+	408.000i	−0.538970	+	0.538970i	0.384257	+	0.923226i	$0.374458\pi$
$758$	−20.0000	−	20.0000i	−0.0263852	−	0.0263852i
$759$		0			0
$760$		0			0
$761$	−1362.00			−1.78975			−0.894875	−	0.446317i	$-0.852736\pi$
$761$	−1362.00			−1.78975			−0.894875	+	0.446317i	$0.852736\pi$
$762$		0			0
$763$	480.000	+	480.000i	0.629096	+	0.629096i
$764$			384.000i			0.502618i
$765$		0			0
$766$	−246.000			−0.321149
$767$	−720.000	+	720.000i	−0.938722	+	0.938722i
$768$		0			0
$769$		−	370.000i		−	0.481144i	−0.970631	−	0.240572i	$-0.922665\pi$
$769$		−	370.000i		−	0.481144i	0.970631	−	0.240572i	$-0.0773351\pi$
$770$		0			0
$771$		0			0
$772$	−264.000	+	264.000i	−0.341969	+	0.341969i
$773$	132.000	+	132.000i	0.170763	+	0.170763i	0.787315	−	0.616551i	$-0.211471\pi$
$773$	132.000	+	132.000i	0.170763	+	0.170763i	−0.616551	+	0.787315i	$0.711471\pi$
$774$		0			0
$775$		0			0
$776$	−48.0000			−0.0618557
$777$		0			0
$778$		0			0
$779$		−	960.000i		−	1.23235i
$780$		0			0
$781$	−576.000			−0.737516
$782$	72.0000	−	72.0000i	0.0920716	−	0.0920716i
$783$		0			0
$784$		−	124.000i		−	0.158163i
$785$		0			0
$786$		0			0
$787$	−93.0000	+	93.0000i	−0.118170	+	0.118170i	−0.763719	−	0.645549i	$-0.776629\pi$
$787$	−93.0000	+	93.0000i	−0.118170	+	0.118170i	0.645549	+	0.763719i	$0.276629\pi$
$788$	264.000	+	264.000i	0.335025

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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	450.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-3.00000 + 3.00000i) q^{7} +(-2.00000 - 2.00000i) q^{8} +O(q^{10})\)	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-3.00000 + 3.00000i) q^{7} +(-2.00000 - 2.00000i) q^{8} -12.0000 q^{11} +(-12.0000 - 12.0000i) q^{13} +6.00000i q^{14} -4.00000 q^{16} +(-12.0000 + 12.0000i) q^{17} -20.0000i q^{19} +(-12.0000 + 12.0000i) q^{22} +(-3.00000 - 3.00000i) q^{23} -24.0000 q^{26} +(6.00000 + 6.00000i) q^{28} -30.0000i q^{29} -8.00000 q^{31} +(-4.00000 + 4.00000i) q^{32} +24.0000i q^{34} +(-48.0000 + 48.0000i) q^{37} +(-20.0000 - 20.0000i) q^{38} +48.0000 q^{41} +(-27.0000 - 27.0000i) q^{43} +24.0000i q^{44} -6.00000 q^{46} +(-27.0000 + 27.0000i) q^{47} +31.0000i q^{49} +(-24.0000 + 24.0000i) q^{52} +(12.0000 + 12.0000i) q^{53} +12.0000 q^{56} +(-30.0000 - 30.0000i) q^{58} -60.0000i q^{59} +32.0000 q^{61} +(-8.00000 + 8.00000i) q^{62} +8.00000i q^{64} +(-3.00000 + 3.00000i) q^{67} +(24.0000 + 24.0000i) q^{68} +48.0000 q^{71} +(-12.0000 - 12.0000i) q^{73} +96.0000i q^{74} -40.0000 q^{76} +(36.0000 - 36.0000i) q^{77} -40.0000i q^{79} +(48.0000 - 48.0000i) q^{82} +(-93.0000 - 93.0000i) q^{83} -54.0000 q^{86} +(24.0000 + 24.0000i) q^{88} +30.0000i q^{89} +72.0000 q^{91} +(-6.00000 + 6.00000i) q^{92} +54.0000i q^{94} +(12.0000 - 12.0000i) q^{97} +(31.0000 + 31.0000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 6 q^{7} - 4 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 - 6 * q^7 - 4 * q^8	\( 2 q + 2 q^{2} - 6 q^{7} - 4 q^{8} - 24 q^{11} - 24 q^{13} - 8 q^{16} - 24 q^{17} - 24 q^{22} - 6 q^{23} - 48 q^{26} + 12 q^{28} - 16 q^{31} - 8 q^{32} - 96 q^{37} - 40 q^{38} + 96 q^{41} - 54 q^{43} - 12 q^{46} - 54 q^{47} - 48 q^{52} + 24 q^{53} + 24 q^{56} - 60 q^{58} + 64 q^{61} - 16 q^{62} - 6 q^{67} + 48 q^{68} + 96 q^{71} - 24 q^{73} - 80 q^{76} + 72 q^{77} + 96 q^{82} - 186 q^{83} - 108 q^{86} + 48 q^{88} + 144 q^{91} - 12 q^{92} + 24 q^{97} + 62 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 6 * q^7 - 4 * q^8 - 24 * q^11 - 24 * q^13 - 8 * q^16 - 24 * q^17 - 24 * q^22 - 6 * q^23 - 48 * q^26 + 12 * q^28 - 16 * q^31 - 8 * q^32 - 96 * q^37 - 40 * q^38 + 96 * q^41 - 54 * q^43 - 12 * q^46 - 54 * q^47 - 48 * q^52 + 24 * q^53 + 24 * q^56 - 60 * q^58 + 64 * q^61 - 16 * q^62 - 6 * q^67 + 48 * q^68 + 96 * q^71 - 24 * q^73 - 80 * q^76 + 72 * q^77 + 96 * q^82 - 186 * q^83 - 108 * q^86 + 48 * q^88 + 144 * q^91 - 12 * q^92 + 24 * q^97 + 62 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more