LMFDB - Embedded newform 2385.1.q.b.847.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2385,1,Mod(847,2385)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2385, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 2]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2385.847");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2385 = 3^{2} \cdot 5 \cdot 53 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2385.q (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:	$1.19027005513$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{40})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{20}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{20} + \cdots)$

Embedding invariants

Embedding label			847.5
Root			$-0.156434 - 0.987688i$ of defining polynomial
Character	$\chi$	$=$	2385.847
Dual form			2385.1.q.b.2278.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.437016 - 0.437016i) q^{2} +0.618034i q^{4} +(-0.987688 + 0.156434i) q^{5} +(-0.642040 - 0.642040i) q^{7} +(0.707107 + 0.707107i) q^{8} +O(q^{10})$	$q+(0.437016 - 0.437016i) q^{2} +0.618034i q^{4} +(-0.987688 + 0.156434i) q^{5} +(-0.642040 - 0.642040i) q^{7} +(0.707107 + 0.707107i) q^{8} +(-0.363271 + 0.500000i) q^{10} +(1.39680 - 1.39680i) q^{13} -0.561163 q^{14} +(-0.0966818 - 0.610425i) q^{20} +(1.34500 + 1.34500i) q^{23} +(0.951057 - 0.309017i) q^{25} -1.22085i q^{26} +(0.396802 - 0.396802i) q^{28} +(-0.707107 + 0.707107i) q^{32} +(0.734572 + 0.533698i) q^{35} +(1.26007 + 1.26007i) q^{37} +(-0.809017 - 0.587785i) q^{40} +0.907981i q^{41} +(0.221232 - 0.221232i) q^{43} +1.17557 q^{46} -0.175571i q^{49} +(0.280582 - 0.550672i) q^{50} +(0.863271 + 0.863271i) q^{52} +(0.707107 + 0.707107i) q^{53} -0.907981i q^{56} +0.618034i q^{64} +(-1.16110 + 1.59811i) q^{65} +(0.554254 - 0.0877853i) q^{70} -1.97538i q^{71} +1.10134 q^{74} +(0.396802 + 0.396802i) q^{82} +(-0.831254 - 0.831254i) q^{83} -0.193364i q^{86} -1.79360 q^{91} +(-0.831254 + 0.831254i) q^{92} +(0.221232 + 0.221232i) q^{97} +(-0.0767271 - 0.0767271i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{7}+O(q^{10}) $ 16 * q - 4 * q^7	$ 16 q - 4 q^{7} + 4 q^{13} - 12 q^{28} - 4 q^{37} - 4 q^{40} + 4 q^{43} + 8 q^{52} + 12 q^{70} - 12 q^{82} + 8 q^{91} + 4 q^{97}+O(q^{100}) $ 16 * q - 4 * q^7 + 4 * q^13 - 12 * q^28 - 4 * q^37 - 4 * q^40 + 4 * q^43 + 8 * q^52 + 12 * q^70 - 12 * q^82 + 8 * q^91 + 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2385\mathbb{Z}\right)^\times$.

$n$	$1432$	$1486$	$1856$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$1$

Coefficient data

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

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

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

$2$	0.437016	−	0.437016i	0.437016	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$2$	0.437016	−	0.437016i	0.437016	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$3$		0			0
$3$		0			0
$4$			0.618034i			0.618034i
$5$	−0.987688	+	0.156434i	−0.987688	+	0.156434i
$5$	−0.987688	+	0.156434i	−0.987688	+	0.156434i
$6$		0			0
$7$	−0.642040	−	0.642040i	−0.642040	−	0.642040i	0.309017	−	0.951057i	$-0.400000\pi$
$7$	−0.642040	−	0.642040i	−0.642040	−	0.642040i	−0.951057	+	0.309017i	$0.900000\pi$
$8$	0.707107	+	0.707107i	0.707107	+	0.707107i
$9$		0			0
$10$	−0.363271	+	0.500000i	−0.363271	+	0.500000i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.587785	−	0.809017i	$-0.300000\pi$
$13$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.809017	−	0.587785i	$-0.200000\pi$
$14$	−0.561163			−0.561163
$15$		0			0
$16$		0			0
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−0.0966818	−	0.610425i	−0.0966818	−	0.610425i
$21$		0			0
$22$		0			0
$23$	1.34500	+	1.34500i	1.34500	+	1.34500i	0.891007	+	0.453990i	$0.150000\pi$
$23$	1.34500	+	1.34500i	1.34500	+	1.34500i	0.453990	+	0.891007i	$0.350000\pi$
$24$		0			0
$25$	0.951057	−	0.309017i	0.951057	−	0.309017i
$26$		−	1.22085i		−	1.22085i
$27$		0			0
$28$	0.396802	−	0.396802i	0.396802	−	0.396802i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$33$		0			0
$34$		0			0
$35$	0.734572	+	0.533698i	0.734572	+	0.533698i
$36$		0			0
$37$	1.26007	+	1.26007i	1.26007	+	1.26007i	0.951057	+	0.309017i	$0.100000\pi$
$37$	1.26007	+	1.26007i	1.26007	+	1.26007i	0.309017	+	0.951057i	$0.400000\pi$
$38$		0			0
$39$		0			0
$40$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$41$			0.907981i			0.907981i	0.891007	+	0.453990i	$0.150000\pi$
$41$			0.907981i			0.907981i	−0.891007	+	0.453990i	$0.850000\pi$
$42$		0			0
$43$	0.221232	−	0.221232i	0.221232	−	0.221232i	−0.587785	−	0.809017i	$-0.700000\pi$
$43$	0.221232	−	0.221232i	0.221232	−	0.221232i	0.809017	+	0.587785i	$0.200000\pi$
$44$		0			0
$45$		0			0
$46$	1.17557			1.17557
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$		−	0.175571i		−	0.175571i
$50$	0.280582	−	0.550672i	0.280582	−	0.550672i
$51$		0			0
$52$	0.863271	+	0.863271i	0.863271	+	0.863271i
$53$	0.707107	+	0.707107i	0.707107	+	0.707107i
$53$	0.707107	+	0.707107i	0.707107	+	0.707107i
$54$		0			0
$55$		0			0
$56$		−	0.907981i		−	0.907981i
$57$		0			0
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$			0.618034i			0.618034i
$65$	−1.16110	+	1.59811i	−1.16110	+	1.59811i
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$		0			0
$69$		0			0
$70$	0.554254	−	0.0877853i	0.554254	−	0.0877853i
$71$		−	1.97538i		−	1.97538i	−0.156434	−	0.987688i	$-0.550000\pi$
$71$		−	1.97538i		−	1.97538i	0.156434	−	0.987688i	$-0.450000\pi$
$72$		0			0
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$	1.10134			1.10134
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$		0			0
$82$	0.396802	+	0.396802i	0.396802	+	0.396802i
$83$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	0.156434	−	0.987688i	$-0.450000\pi$
$83$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	−0.987688	+	0.156434i	$0.950000\pi$
$84$		0			0
$85$		0			0
$86$		−	0.193364i		−	0.193364i
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	−1.79360			−1.79360
$92$	−0.831254	+	0.831254i	−0.831254	+	0.831254i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.221232	+	0.221232i	0.221232	+	0.221232i	0.809017	−	0.587785i	$-0.200000\pi$
$97$	0.221232	+	0.221232i	0.221232	+	0.221232i	−0.587785	+	0.809017i	$0.700000\pi$
$98$	−0.0767271	−	0.0767271i	−0.0767271	−	0.0767271i
$99$		0			0
$100$	0.190983	+	0.587785i	0.190983	+	0.587785i
$101$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$101$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$102$		0			0
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$	1.97538			1.97538
$105$		0			0
$106$	0.618034			0.618034
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$		0			0
$115$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.00000			−1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.891007	+	0.453990i	−0.891007	+	0.453990i
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$	−0.437016	−	0.437016i	−0.437016	−	0.437016i
$129$		0			0
$130$	0.190983	+	1.20582i	0.190983	+	1.20582i
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.831254	−	0.831254i	0.831254	−	0.831254i	−0.156434	−	0.987688i	$-0.550000\pi$
$137$	0.831254	−	0.831254i	0.831254	−	0.831254i	0.987688	+	0.156434i	$0.0500000\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$	−0.329843	+	0.453990i	−0.329843	+	0.453990i
$141$		0			0
$142$	−0.863271	−	0.863271i	−0.863271	−	0.863271i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−0.778768	+	0.778768i	−0.778768	+	0.778768i
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$		0			0
$160$	0.587785	−	0.809017i	0.587785	−	0.809017i
$161$		−	1.72708i		−	1.72708i
$162$		0			0
$163$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$163$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$164$	−0.561163			−0.561163
$165$		0			0
$166$	−0.726543			−0.726543
$167$	−1.14412	+	1.14412i	−1.14412	+	1.14412i	−0.156434	+	0.987688i	$0.550000\pi$
$167$	−1.14412	+	1.14412i	−1.14412	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$168$		0			0
$169$		−	2.90211i		−	2.90211i
$170$		0			0
$171$		0			0
$172$	0.136729	+	0.136729i	0.136729	+	0.136729i
$173$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.891007	−	0.453990i	$-0.850000\pi$
$173$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.453990	−	0.891007i	$-0.650000\pi$
$174$		0			0
$175$	−0.809017	−	0.412215i	−0.809017	−	0.412215i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−0.907981			−0.907981			−0.453990	−	0.891007i	$-0.650000\pi$
$179$	−0.907981			−0.907981			−0.453990	+	0.891007i	$0.650000\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$	−0.783834	+	0.783834i	−0.783834	+	0.783834i
$183$		0			0
$184$			1.90211i			1.90211i
$185$	−1.44168	−	1.04744i	−1.44168	−	1.04744i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$			1.78201i			1.78201i	0.453990	+	0.891007i	$0.350000\pi$
$191$			1.78201i			1.78201i	−0.453990	+	0.891007i	$0.650000\pi$
$192$		0			0
$193$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$193$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$194$	0.193364			0.193364
$195$		0			0
$196$	0.108509			0.108509
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$199$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$200$	0.891007	+	0.453990i	0.891007	+	0.453990i
$201$		0			0
$202$	−0.618034	−	0.618034i	−0.618034	−	0.618034i
$203$		0			0
$204$		0			0
$205$	−0.142040	−	0.896802i	−0.142040	−	0.896802i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$211$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$212$	−0.437016	+	0.437016i	−0.437016	+	0.437016i
$213$		0			0
$214$		0			0
$215$	−0.183900	+	0.253116i	−0.183900	+	0.253116i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$223$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$224$	0.907981			0.907981
$225$		0			0
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$229$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$230$	−1.16110	+	0.183900i	−1.16110	+	0.183900i
$231$		0			0
$232$		0			0
$233$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$233$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$239$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$240$		0			0
$241$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$241$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$242$	−0.437016	+	0.437016i	−0.437016	+	0.437016i
$243$		0			0
$244$		0			0
$245$	0.0274653	+	0.173409i	0.0274653	+	0.173409i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$251$		−	1.78201i		−	1.78201i	−0.453990	−	0.891007i	$-0.650000\pi$
$251$		−	1.78201i		−	1.78201i	0.453990	−	0.891007i	$-0.350000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−1.00000			−1.00000
$257$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.453990	−	0.891007i	$-0.350000\pi$
$257$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.891007	−	0.453990i	$-0.150000\pi$
$258$		0			0
$259$		−	1.61803i		−	1.61803i
$260$	−0.987688	−	0.717598i	−0.987688	−	0.717598i
$261$		0			0
$262$		0			0
$263$	0.831254	+	0.831254i	0.831254	+	0.831254i	0.987688	−	0.156434i	$-0.0500000\pi$
$263$	0.831254	+	0.831254i	0.831254	+	0.831254i	−0.156434	+	0.987688i	$0.550000\pi$
$264$		0			0
$265$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$271$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$272$		0			0
$273$		0			0
$274$		−	0.726543i		−	0.726543i
$275$		0			0
$276$		0			0
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$		0			0
$279$		0			0
$280$	0.142040	+	0.896802i	0.142040	+	0.896802i
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$	1.22085			1.22085
$285$		0			0
$286$		0			0
$287$	0.582960	−	0.582960i	0.582960	−	0.582960i
$288$		0			0
$289$		−	1.00000i		−	1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$		0			0
$296$			1.78201i			1.78201i
$297$		0			0
$298$		0			0
$299$	3.75739			3.75739
$300$		0			0
$301$	−0.284079			−0.284079
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−1.39680	−	1.39680i	−1.39680	−	1.39680i	−0.809017	−	0.587785i	$-0.800000\pi$
$307$	−1.39680	−	1.39680i	−1.39680	−	1.39680i	−0.587785	−	0.809017i	$-0.700000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$	−0.0966818	−	0.610425i	−0.0966818	−	0.610425i
$321$		0			0
$322$	−0.754763	−	0.754763i	−0.754763	−	0.754763i
$323$		0			0
$324$		0			0
$325$	0.896802	−	1.76007i	0.896802	−	1.76007i
$326$			0.874032i			0.874032i
$327$		0			0
$328$	−0.642040	+	0.642040i	−0.642040	+	0.642040i
$329$		0			0
$330$		0			0
$331$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$331$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$332$	0.513743	−	0.513743i	0.513743	−	0.513743i
$333$		0			0
$334$			1.00000i			1.00000i
$335$		0			0
$336$		0			0
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$	−1.26827	−	1.26827i	−1.26827	−	1.26827i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.754763	+	0.754763i	−0.754763	+	0.754763i
$344$	0.312869			0.312869
$345$		0			0
$346$	−1.17557			−1.17557
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	−0.533698	+	0.173409i	−0.533698	+	0.173409i
$351$		0			0
$352$		0			0
$353$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$353$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$354$		0			0
$355$	0.309017	+	1.95106i	0.309017	+	1.95106i
$356$		0			0
$357$		0			0
$358$	−0.396802	+	0.396802i	−0.396802	+	0.396802i
$359$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$359$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$360$		0			0
$361$	−1.00000			−1.00000
$362$		0			0
$363$		0			0
$364$		−	1.10851i		−	1.10851i
$365$		0			0
$366$		0			0
$367$	1.26007	+	1.26007i	1.26007	+	1.26007i	0.951057	+	0.309017i	$0.100000\pi$
$367$	1.26007	+	1.26007i	1.26007	+	1.26007i	0.309017	+	0.951057i	$0.400000\pi$
$368$		0			0
$369$		0			0
$370$	−1.08779	+	0.172288i	−1.08779	+	0.172288i
$371$		−	0.907981i		−	0.907981i
$372$		0			0
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$	0.778768	+	0.778768i	0.778768	+	0.778768i
$383$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$383$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.136729	+	0.136729i	−0.136729	+	0.136729i
$389$	0.907981			0.907981			0.453990	−	0.891007i	$-0.350000\pi$
$389$	0.907981			0.907981			0.453990	+	0.891007i	$0.350000\pi$
$390$		0			0
$391$		0			0
$392$	0.124147	−	0.124147i	0.124147	−	0.124147i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$	0.513743	+	0.513743i	0.513743	+	0.513743i
$399$		0			0
$400$		0			0
$401$			0.312869i			0.312869i	0.987688	+	0.156434i	$0.0500000\pi$
$401$			0.312869i			0.312869i	−0.987688	+	0.156434i	$0.950000\pi$
$402$		0			0
$403$		0			0
$404$	0.874032			0.874032
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$409$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$410$	−0.453990	−	0.329843i	−0.453990	−	0.329843i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	0.951057	+	0.690983i	0.951057	+	0.690983i
$416$			1.97538i			1.97538i
$417$		0			0
$418$		0			0
$419$	0.312869			0.312869			0.156434	−	0.987688i	$-0.450000\pi$
$419$	0.312869			0.312869			0.156434	+	0.987688i	$0.450000\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$	0.831254	−	0.831254i	0.831254	−	0.831254i
$423$		0			0
$424$			1.00000i			1.00000i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$	0.0302487	+	0.190983i	0.0302487	+	0.190983i
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.309017	+	0.951057i	$0.600000\pi$
$433$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.951057	+	0.309017i	$0.900000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$439$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.14412	−	1.14412i	−1.14412	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$443$	−1.14412	−	1.14412i	−1.14412	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$444$		0			0
$445$		0			0
$446$		−	0.874032i		−	0.874032i
$447$		0			0
$448$	0.396802	−	0.396802i	0.396802	−	0.396802i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.77152	−	0.280582i	1.77152	−	0.280582i
$456$		0			0
$457$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$457$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$458$	0.831254	+	0.831254i	0.831254	+	0.831254i
$459$		0			0
$460$	0.690983	−	0.951057i	0.690983	−	0.951057i
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$	1.00000			1.00000
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−0.618034	+	0.618034i	−0.618034	+	0.618034i
$479$	−1.78201			−1.78201			−0.891007	−	0.453990i	$-0.850000\pi$
$479$	−1.78201			−1.78201			−0.891007	+	0.453990i	$0.850000\pi$
$480$		0			0
$481$	3.52015			3.52015
$482$	−0.270091	+	0.270091i	−0.270091	+	0.270091i
$483$		0			0
$484$		−	0.618034i		−	0.618034i
$485$	−0.253116	−	0.183900i	−0.253116	−	0.183900i
$486$		0			0
$487$	−0.221232	−	0.221232i	−0.221232	−	0.221232i	0.587785	−	0.809017i	$-0.300000\pi$
$487$	−0.221232	−	0.221232i	−0.221232	−	0.221232i	−0.809017	+	0.587785i	$0.800000\pi$
$488$		0			0
$489$		0			0
$490$	0.0877853	+	0.0637797i	0.0877853	+	0.0637797i
$491$		−	0.312869i		−	0.312869i	−0.987688	−	0.156434i	$-0.950000\pi$
$491$		−	0.312869i		−	0.312869i	0.987688	−	0.156434i	$-0.0500000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.26827	+	1.26827i	−1.26827	+	1.26827i
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−0.280582	−	0.550672i	−0.280582	−	0.550672i
$501$		0			0
$502$	−0.778768	−	0.778768i	−0.778768	−	0.778768i
$503$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$503$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$504$		0			0
$505$	0.221232	+	1.39680i	0.221232	+	1.39680i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.312869			0.312869			0.156434	−	0.987688i	$-0.450000\pi$
$509$	0.312869			0.312869			0.156434	+	0.987688i	$0.450000\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		−	1.17557i		−	1.17557i
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$519$		0			0
$520$	−1.95106	+	0.309017i	−1.95106	+	0.309017i
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.309017	+	0.951057i	$0.600000\pi$
$523$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.951057	+	0.309017i	$0.900000\pi$
$524$		0			0
$525$		0			0
$526$	0.726543			0.726543
$527$		0			0
$528$		0			0
$529$			2.61803i			2.61803i
$530$	−0.610425	+	0.0966818i	−0.610425	+	0.0966818i
$531$		0			0
$532$		0			0
$533$	1.26827	+	1.26827i	1.26827	+	1.26827i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$541$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$542$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.39680	−	1.39680i	−1.39680	−	1.39680i	−0.809017	−	0.587785i	$-0.800000\pi$
$547$	−1.39680	−	1.39680i	−1.39680	−	1.39680i	−0.587785	−	0.809017i	$-0.700000\pi$
$548$	0.513743	+	0.513743i	0.513743	+	0.513743i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.34500	+	1.34500i	−1.34500	+	1.34500i	−0.453990	+	0.891007i	$0.650000\pi$
$557$	−1.34500	+	1.34500i	−1.34500	+	1.34500i	−0.891007	+	0.453990i	$0.850000\pi$
$558$		0			0
$559$		−	0.618034i		−	0.618034i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$563$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	1.39680	−	1.39680i	1.39680	−	1.39680i
$569$	−1.97538			−1.97538			−0.987688	−	0.156434i	$-0.950000\pi$
$569$	−1.97538			−1.97538			−0.987688	+	0.156434i	$0.950000\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		−	0.509525i		−	0.509525i
$575$	1.69480	+	0.863541i	1.69480	+	0.863541i
$576$		0			0
$577$	−0.642040	−	0.642040i	−0.642040	−	0.642040i	0.309017	−	0.951057i	$-0.400000\pi$
$577$	−0.642040	−	0.642040i	−0.642040	−	0.642040i	−0.951057	+	0.309017i	$0.900000\pi$
$578$	−0.437016	−	0.437016i	−0.437016	−	0.437016i
$579$		0			0
$580$		0			0
$581$			1.06740i			1.06740i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	1.64204	−	1.64204i	1.64204	−	1.64204i
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	−0.124147	+	0.124147i	−0.124147	+	0.124147i
$603$		0			0
$604$		0			0
$605$	0.987688	−	0.156434i	0.987688	−	0.156434i
$606$		0			0
$607$	−0.221232	−	0.221232i	−0.221232	−	0.221232i	0.587785	−	0.809017i	$-0.300000\pi$
$607$	−0.221232	−	0.221232i	−0.221232	−	0.221232i	−0.809017	+	0.587785i	$0.800000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$	−1.22085			−1.22085
$615$		0			0
$616$		0			0
$617$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$617$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$618$		0			0
$619$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$619$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.809017	−	0.587785i	0.809017	−	0.587785i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.245237	−	0.245237i	−0.245237	−	0.245237i
$638$		0			0
$639$		0			0
$640$	0.500000	+	0.363271i	0.500000	+	0.363271i
$641$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$641$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$642$		0			0
$643$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$643$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$644$	1.06740			1.06740
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$		0			0
$649$		0			0
$650$	−0.377263	−	1.16110i	−0.377263	−	1.16110i
$651$		0			0
$652$	−0.618034	−	0.618034i	−0.618034	−	0.618034i
$653$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$653$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−0.312869			−0.312869			−0.156434	−	0.987688i	$-0.550000\pi$
$659$	−0.312869			−0.312869			−0.156434	+	0.987688i	$0.550000\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$	−0.831254	+	0.831254i	−0.831254	+	0.831254i
$663$		0			0
$664$		−	1.17557i		−	1.17557i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.221232	+	0.221232i	−0.221232	+	0.221232i	−0.809017	−	0.587785i	$-0.800000\pi$
$673$	−0.221232	+	0.221232i	−0.221232	+	0.221232i	0.587785	+	0.809017i	$0.300000\pi$
$674$		0			0
$675$		0			0
$676$	1.79360			1.79360
$677$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$677$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$678$		0			0
$679$		−	0.284079i		−	0.284079i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$	−0.690983	+	0.951057i	−0.690983	+	0.951057i
$686$			0.659687i			0.659687i
$687$		0			0
$688$		0			0
$689$	1.97538			1.97538
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	0.831254	−	0.831254i	0.831254	−	0.831254i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.254763	−	0.500000i	0.254763	−	0.500000i
$701$		−	1.97538i		−	1.97538i	−0.156434	−	0.987688i	$-0.550000\pi$
$701$		−	1.97538i		−	1.97538i	0.156434	−	0.987688i	$-0.450000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	1.23607			1.23607
$707$	−0.907981	+	0.907981i	−0.907981	+	0.907981i
$708$		0			0
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$	0.987688	+	0.717598i	0.987688	+	0.717598i
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		−	0.561163i		−	0.561163i
$717$		0			0
$718$	−0.618034	+	0.618034i	−0.618034	+	0.618034i
$719$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$719$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$720$		0			0
$721$		0			0
$722$	−0.437016	+	0.437016i	−0.437016	+	0.437016i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	1.39680	+	1.39680i	1.39680	+	1.39680i	0.809017	+	0.587785i	$0.200000\pi$
$727$	1.39680	+	1.39680i	1.39680	+	1.39680i	0.587785	+	0.809017i	$0.300000\pi$
$728$	−1.26827	−	1.26827i	−1.26827	−	1.26827i
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	0.642040	−	0.642040i	0.642040	−	0.642040i	−0.309017	−	0.951057i	$-0.600000\pi$
$733$	0.642040	−	0.642040i	0.642040	−	0.642040i	0.951057	+	0.309017i	$0.100000\pi$
$734$	1.10134			1.10134
$735$		0			0
$736$	−1.90211			−1.90211
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	0.647354	−	0.891007i	0.647354	−	0.891007i
$741$		0			0
$742$	−0.396802	−	0.396802i	−0.396802	−	0.396802i
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$751$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$757$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1.78201i		−	1.78201i	−0.453990	−	0.891007i	$-0.650000\pi$
$761$		−	1.78201i		−	1.78201i	0.453990	−	0.891007i	$-0.350000\pi$
$762$		0			0
$763$		0			0
$764$	−1.10134			−1.10134
$765$		0			0
$766$	1.00000			1.00000
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.437016	+	0.437016i	0.437016	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$773$	0.437016	+	0.437016i	0.437016	+	0.437016i	−0.453990	+	0.891007i	$0.650000\pi$
$774$		0			0
$775$		0			0
$776$			0.312869i			0.312869i
$777$		0			0
$778$	0.396802	−	0.396802i	0.396802	−	0.396802i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−0.726543			−0.726543
$797$	0.437016	−	0.437016i	0.437016	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$797$	0.437016	−	0.437016i	0.437016	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$798$		0			0
$799$		0			0
$800$	−0.453990	+	0.891007i	−0.453990	+	0.891007i
$801$		0			0
$802$	0.136729	+	0.136729i	0.136729	+	0.136729i
$803$		0			0
$804$		0			0
$805$	0.270175	+	1.70582i	0.270175	+	1.70582i
$806$		0			0
$807$		0			0
$808$	1.00000	−	1.00000i	1.00000	−	1.00000i
$809$	−1.78201			−1.78201			−0.891007	−	0.453990i	$-0.850000\pi$
$809$	−1.78201			−1.78201			−0.891007	+	0.453990i	$0.850000\pi$
$810$		0			0
$811$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$811$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.831254	−	1.14412i	0.831254	−	1.14412i
$816$		0			0
$817$		0			0
$818$	−0.270091	−	0.270091i	−0.270091	−	0.270091i
$819$		0			0
$820$	0.554254	−	0.0877853i	0.554254	−	0.0877853i
$821$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$821$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$822$		0			0
$823$	−1.39680	+	1.39680i	−1.39680	+	1.39680i	−0.587785	+	0.809017i	$0.700000\pi$
$823$	−1.39680	+	1.39680i	−1.39680	+	1.39680i	−0.809017	+	0.587785i	$0.800000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.831254	−	0.831254i	0.831254	−	0.831254i	−0.156434	−	0.987688i	$-0.550000\pi$
$827$	0.831254	−	0.831254i	0.831254	−	0.831254i	0.987688	+	0.156434i	$0.0500000\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$	0.717598	−	0.113656i	0.717598	−	0.113656i
$831$		0			0
$832$	0.863271	+	0.863271i	0.863271	+	0.863271i
$833$		0			0
$834$		0			0
$835$	0.951057	−	1.30902i	0.951057	−	1.30902i
$836$		0			0
$837$		0			0
$838$	0.136729	−	0.136729i	0.136729	−	0.136729i
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$		0			0
$843$		0			0
$844$			1.17557i			1.17557i
$845$	0.453990	+	2.86638i	0.453990	+	2.86638i
$846$		0			0
$847$	0.642040	+	0.642040i	0.642040	+	0.642040i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			3.38959i			3.38959i
$852$		0			0
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$859$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$860$	−0.156434	−	0.113656i	−0.156434	−	0.113656i
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$	1.53884	+	1.11803i	1.53884	+	1.11803i
$866$			1.10134i			1.10134i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.863541	+	0.280582i	0.863541	+	0.280582i
$876$		0			0
$877$	0.221232	+	0.221232i	0.221232	+	0.221232i	0.809017	−	0.587785i	$-0.200000\pi$
$877$	0.221232	+	0.221232i	0.221232	+	0.221232i	−0.587785	+	0.809017i	$0.700000\pi$
$878$	0.831254	+	0.831254i	0.831254	+	0.831254i
$879$		0			0
$880$		0			0
$881$			1.78201i			1.78201i	0.453990	+	0.891007i	$0.350000\pi$
$881$			1.78201i			1.78201i	−0.453990	+	0.891007i	$0.650000\pi$
$882$		0			0
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$		0			0
$885$		0			0
$886$	−1.00000			−1.00000
$887$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$887$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	0.618034	+	0.618034i	0.618034	+	0.618034i
$893$		0			0
$894$		0			0
$895$	0.896802	−	0.142040i	0.896802	−	0.142040i
$896$			0.561163i			0.561163i
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.26007	+	1.26007i	1.26007	+	1.26007i	0.951057	+	0.309017i	$0.100000\pi$
$907$	1.26007	+	1.26007i	1.26007	+	1.26007i	0.309017	+	0.951057i	$0.400000\pi$
$908$		0			0
$909$		0			0
$910$	0.651565	−	0.896802i	0.651565	−	0.896802i
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.17557			−1.17557
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$	−0.297556	−	1.87869i	−0.297556	−	1.87869i
$921$		0			0
$922$		0			0
$923$	−2.75921	−	2.75921i	−2.75921	−	2.75921i
$924$		0			0
$925$	1.58779	+	0.809017i	1.58779	+	0.809017i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.951057	−	0.309017i	$-0.900000\pi$
$937$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.309017	−	0.951057i	$-0.600000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$	−1.22123	+	1.22123i	−1.22123	+	1.22123i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$953$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$954$		0			0
$955$	−0.278768	−	1.76007i	−0.278768	−	1.76007i
$956$		−	0.874032i		−	0.874032i
$957$		0			0
$958$	−0.778768	+	0.778768i	−0.778768	+	0.778768i
$959$	−1.06740			−1.06740
$960$		0			0
$961$	1.00000			1.00000
$962$	1.53836	−	1.53836i	1.53836	−	1.53836i
$963$		0			0
$964$		−	0.381966i		−	0.381966i
$965$		0			0
$966$		0			0
$967$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.951057	−	0.309017i	$-0.900000\pi$
$967$	−1.26007	−	1.26007i	−1.26007	−	1.26007i	−0.309017	−	0.951057i	$-0.600000\pi$
$968$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$969$		0			0
$970$	−0.190983	+	0.0302487i	−0.190983	+	0.0302487i
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$	−0.193364			−0.193364
$975$		0			0
$976$		0			0
$977$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.453990	−	0.891007i	$-0.350000\pi$
$977$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.891007	−	0.453990i	$-0.150000\pi$
$978$		0			0
$979$		0			0
$980$	−0.107173	+	0.0169745i	−0.107173	+	0.0169745i
$981$		0			0
$982$	−0.136729	−	0.136729i	−0.136729	−	0.136729i
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.595112			0.595112
$990$		0			0
$991$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$991$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$992$		0			0
$993$		0			0
$994$			1.10851i			1.10851i
$995$	−0.183900	−	1.16110i	−0.183900	−	1.16110i
$996$		0			0
$997$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$997$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2385.1.q.b.847.5	yes	16
3.2	odd	2	inner	2385.1.q.b.847.4	✓	16
5.3	odd	4	inner	2385.1.q.b.2278.4	yes	16
15.8	even	4	inner	2385.1.q.b.2278.5	yes	16
53.52	even	2	inner	2385.1.q.b.847.4	✓	16
159.158	odd	2	CM	2385.1.q.b.847.5	yes	16
265.158	odd	4	inner	2385.1.q.b.2278.5	yes	16
795.158	even	4	inner	2385.1.q.b.2278.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2385.1.q.b.847.4	✓	16	3.2	odd	2	inner
2385.1.q.b.847.4	✓	16	53.52	even	2	inner
2385.1.q.b.847.5	yes	16	1.1	even	1	trivial
2385.1.q.b.847.5	yes	16	159.158	odd	2	CM
2385.1.q.b.2278.4	yes	16	5.3	odd	4	inner
2385.1.q.b.2278.4	yes	16	795.158	even	4	inner
2385.1.q.b.2278.5	yes	16	15.8	even	4	inner
2385.1.q.b.2278.5	yes	16	265.158	odd	4	inner

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:	\( N \)	\(=\)	\( 2385 = 3^{2} \cdot 5 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2385.q (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.437016 - 0.437016i) q^{2} +0.618034i q^{4} +(-0.987688 + 0.156434i) q^{5} +(-0.642040 - 0.642040i) q^{7} +(0.707107 + 0.707107i) q^{8} +O(q^{10})\)	\(q+(0.437016 - 0.437016i) q^{2} +0.618034i q^{4} +(-0.987688 + 0.156434i) q^{5} +(-0.642040 - 0.642040i) q^{7} +(0.707107 + 0.707107i) q^{8} +(-0.363271 + 0.500000i) q^{10} +(1.39680 - 1.39680i) q^{13} -0.561163 q^{14} +(-0.0966818 - 0.610425i) q^{20} +(1.34500 + 1.34500i) q^{23} +(0.951057 - 0.309017i) q^{25} -1.22085i q^{26} +(0.396802 - 0.396802i) q^{28} +(-0.707107 + 0.707107i) q^{32} +(0.734572 + 0.533698i) q^{35} +(1.26007 + 1.26007i) q^{37} +(-0.809017 - 0.587785i) q^{40} +0.907981i q^{41} +(0.221232 - 0.221232i) q^{43} +1.17557 q^{46} -0.175571i q^{49} +(0.280582 - 0.550672i) q^{50} +(0.863271 + 0.863271i) q^{52} +(0.707107 + 0.707107i) q^{53} -0.907981i q^{56} +0.618034i q^{64} +(-1.16110 + 1.59811i) q^{65} +(0.554254 - 0.0877853i) q^{70} -1.97538i q^{71} +1.10134 q^{74} +(0.396802 + 0.396802i) q^{82} +(-0.831254 - 0.831254i) q^{83} -0.193364i q^{86} -1.79360 q^{91} +(-0.831254 + 0.831254i) q^{92} +(0.221232 + 0.221232i) q^{97} +(-0.0767271 - 0.0767271i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{7}+O(q^{10}) \) 16 * q - 4 * q^7	\( 16 q - 4 q^{7} + 4 q^{13} - 12 q^{28} - 4 q^{37} - 4 q^{40} + 4 q^{43} + 8 q^{52} + 12 q^{70} - 12 q^{82} + 8 q^{91} + 4 q^{97}+O(q^{100}) \) 16 * q - 4 * q^7 + 4 * q^13 - 12 * q^28 - 4 * q^37 - 4 * q^40 + 4 * q^43 + 8 * q^52 + 12 * q^70 - 12 * q^82 + 8 * q^91 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more