LMFDB - Embedded newform 1224.1.bu.a.53.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1224,1,Mod(53,1224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1224, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 4, 4, 7]))

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

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

chi := DirichletCharacter("1224.53");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1224 = 2^{3} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1224.bu (of order $8$, degree $4$, 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:	$0.610855575463$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{8}$
Projective field:	Galois closure of 8.0.153158089019904.1

Embedding invariants

Embedding label			53.1
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1224.53
Dual form			1224.1.bu.a.485.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{2} -1.00000i q^{4} +(0.292893 - 0.707107i) q^{7} +(-0.707107 - 0.707107i) q^{8} +O(q^{10})$	$q+(0.707107 - 0.707107i) q^{2} -1.00000i q^{4} +(0.292893 - 0.707107i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(-0.292893 - 0.707107i) q^{14} -1.00000 q^{16} -1.00000i q^{17} +(-0.707107 + 1.70711i) q^{23} +(0.707107 - 0.707107i) q^{25} +(-0.707107 - 0.292893i) q^{28} +(0.707107 - 0.292893i) q^{31} +(-0.707107 + 0.707107i) q^{32} +(-0.707107 - 0.707107i) q^{34} +(0.707107 + 0.292893i) q^{41} +(0.707107 + 1.70711i) q^{46} -1.41421 q^{47} +(0.292893 + 0.292893i) q^{49} -1.00000i q^{50} +(-0.707107 + 0.292893i) q^{56} +(0.292893 - 0.707107i) q^{62} +1.00000i q^{64} -1.00000 q^{68} +(0.292893 + 0.707107i) q^{71} +(0.707107 + 1.70711i) q^{73} +(-1.70711 - 0.707107i) q^{79} +(0.707107 - 0.292893i) q^{82} +1.41421 q^{89} +(1.70711 + 0.707107i) q^{92} +(-1.00000 + 1.00000i) q^{94} +(0.707107 + 1.70711i) q^{97} +0.414214 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^7	$ 4 q + 4 q^{7} - 4 q^{14} - 4 q^{16} + 4 q^{49} + 4 q^{62} - 4 q^{68} + 4 q^{71} - 4 q^{79} + 4 q^{92} - 4 q^{94} - 4 q^{98}+O(q^{100}) $ 4 * q + 4 * q^7 - 4 * q^14 - 4 * q^16 + 4 * q^49 + 4 * q^62 - 4 * q^68 + 4 * q^71 - 4 * q^79 + 4 * q^92 - 4 * q^94 - 4 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$.

$n$	$137$	$613$	$649$	$919$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{7}{8}\right)$	$1$

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 $

$2$	0.707107	−	0.707107i	0.707107	−	0.707107i
$2$	0.707107	−	0.707107i	0.707107	−	0.707107i
$3$		0			0
$3$		0			0
$4$		−	1.00000i		−	1.00000i
$5$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$5$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$6$		0			0
$7$	0.292893	−	0.707107i	0.292893	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$7$	0.292893	−	0.707107i	0.292893	−	0.707107i	1.00000			$0$
$8$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$9$		0			0
$10$		0			0
$11$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$11$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$15$		0			0
$16$	−1.00000			−1.00000
$17$		−	1.00000i		−	1.00000i
$17$		−	1.00000i		−	1.00000i
$18$		0			0
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.707107	+	1.70711i	−0.707107	+	1.70711i			1.00000i	$0.5\pi$
$23$	−0.707107	+	1.70711i	−0.707107	+	1.70711i	−0.707107	+	0.707107i	$0.750000\pi$
$24$		0			0
$25$	0.707107	−	0.707107i	0.707107	−	0.707107i
$26$		0			0
$27$		0			0
$28$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$29$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$29$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$30$		0			0
$31$	0.707107	−	0.292893i	0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$31$	0.707107	−	0.292893i	0.707107	−	0.292893i	0.707107	+	0.707107i	$0.250000\pi$
$32$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$33$		0			0
$34$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$35$		0			0
$36$		0			0
$37$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$37$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.707107	+	0.292893i	0.707107	+	0.292893i	0.707107	−	0.707107i	$-0.250000\pi$
$41$	0.707107	+	0.292893i	0.707107	+	0.292893i			1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$		0			0
$46$	0.707107	+	1.70711i	0.707107	+	1.70711i
$47$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$47$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$	0.292893	+	0.292893i	0.292893	+	0.292893i
$50$		−	1.00000i		−	1.00000i
$51$		0			0
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$		0			0
$55$		0			0
$56$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$57$		0			0
$58$		0			0
$59$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$59$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$60$		0			0
$61$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$61$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$62$	0.292893	−	0.707107i	0.292893	−	0.707107i
$63$		0			0
$64$			1.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	−1.00000			−1.00000
$69$		0			0
$70$		0			0
$71$	0.292893	+	0.707107i	0.292893	+	0.707107i	1.00000			$0$
$71$	0.292893	+	0.707107i	0.292893	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$72$		0			0
$73$	0.707107	+	1.70711i	0.707107	+	1.70711i	0.707107	+	0.707107i	$0.250000\pi$
$73$	0.707107	+	1.70711i	0.707107	+	1.70711i			1.00000i	$0.5\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$79$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−1.00000			$\pi$
$80$		0			0
$81$		0			0
$82$	0.707107	−	0.292893i	0.707107	−	0.292893i
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$89$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$90$		0			0
$91$		0			0
$92$	1.70711	+	0.707107i	1.70711	+	0.707107i
$93$		0			0
$94$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$95$		0			0
$96$		0			0
$97$	0.707107	+	1.70711i	0.707107	+	1.70711i	0.707107	+	0.707107i	$0.250000\pi$
$97$	0.707107	+	1.70711i	0.707107	+	1.70711i			1.00000i	$0.5\pi$
$98$	0.414214			0.414214
$99$		0			0
$100$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$103$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$107$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$108$		0			0
$109$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$109$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$110$		0			0
$111$		0			0
$112$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$113$	−0.292893	+	0.707107i	−0.292893	+	0.707107i	0.707107	+	0.707107i	$0.250000\pi$
$113$	−0.292893	+	0.707107i	−0.292893	+	0.707107i	−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$120$		0			0
$121$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$122$		0			0
$123$		0			0
$124$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$125$		0			0
$126$		0			0
$127$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$127$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$128$	0.707107	+	0.707107i	0.707107	+	0.707107i
$129$		0			0
$130$		0			0
$131$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$131$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$137$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$137$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$138$		0			0
$139$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$139$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$140$		0			0
$141$		0			0
$142$	0.707107	+	0.292893i	0.707107	+	0.292893i
$143$		0			0
$144$		0			0
$145$		0			0
$146$	1.70711	+	0.707107i	1.70711	+	0.707107i
$147$		0			0
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$151$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$159$		0			0
$160$		0			0
$161$	1.00000	+	1.00000i	1.00000	+	1.00000i
$162$		0			0
$163$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$163$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$164$	0.292893	−	0.707107i	0.292893	−	0.707107i
$165$		0			0
$166$		0			0
$167$	0.707107	+	1.70711i	0.707107	+	1.70711i	0.707107	+	0.707107i	$0.250000\pi$
$167$	0.707107	+	1.70711i	0.707107	+	1.70711i			1.00000i	$0.5\pi$
$168$		0			0
$169$	−1.00000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$173$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$174$		0			0
$175$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$176$		0			0
$177$		0			0
$178$	1.00000	−	1.00000i	1.00000	−	1.00000i
$179$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$179$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$180$		0			0
$181$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$181$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$182$		0			0
$183$		0			0
$184$	1.70711	−	0.707107i	1.70711	−	0.707107i
$185$		0			0
$186$		0			0
$187$		0			0
$188$			1.41421i			1.41421i
$189$		0			0
$190$		0			0
$191$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$191$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$192$		0			0
$193$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$193$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−1.00000			$\pi$
$194$	1.70711	+	0.707107i	1.70711	+	0.707107i
$195$		0			0
$196$	0.292893	−	0.292893i	0.292893	−	0.292893i
$197$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$197$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$198$		0			0
$199$	0.292893	+	0.707107i	0.292893	+	0.707107i	1.00000			$0$
$199$	0.292893	+	0.707107i	0.292893	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$200$	−1.00000			−1.00000
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	1.00000	−	1.00000i	1.00000	−	1.00000i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$211$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	0.585786i		−	0.585786i
$218$		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			$\pi$
$224$	0.292893	+	0.707107i	0.292893	+	0.707107i
$225$		0			0
$226$	0.292893	+	0.707107i	0.292893	+	0.707107i
$227$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$227$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$228$		0			0
$229$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$229$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.707107	+	0.292893i	−0.707107	+	0.292893i	−0.707107	−	0.707107i	$-0.750000\pi$
$233$	−0.707107	+	0.292893i	−0.707107	+	0.292893i			1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$239$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$239$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$240$		0			0
$241$	0.707107	−	0.292893i	0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$241$	0.707107	−	0.292893i	0.707107	−	0.292893i	0.707107	+	0.707107i	$0.250000\pi$
$242$	−1.00000			−1.00000
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$			2.00000i			2.00000i
$255$		0			0
$256$	1.00000			1.00000
$257$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$257$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$263$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$269$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$270$		0			0
$271$	−2.00000			−2.00000			−1.00000			$\pi$
$271$	−2.00000			−2.00000			−1.00000			$\pi$
$272$			1.00000i			1.00000i
$273$		0			0
$274$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$275$		0			0
$276$		0			0
$277$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$277$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$281$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$282$		0			0
$283$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$283$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$284$	0.707107	−	0.292893i	0.707107	−	0.292893i
$285$		0			0
$286$		0			0
$287$	0.414214	−	0.414214i	0.414214	−	0.414214i
$288$		0			0
$289$	−1.00000			−1.00000
$290$		0			0
$291$		0			0
$292$	1.70711	−	0.707107i	1.70711	−	0.707107i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.70711	−	0.707107i	1.70711	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$311$	1.70711	−	0.707107i	1.70711	−	0.707107i	1.00000			$0$
$312$		0			0
$313$	0.707107	−	1.70711i	0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$313$	0.707107	−	1.70711i	0.707107	−	1.70711i	0.707107	−	0.707107i	$-0.250000\pi$
$314$		0			0
$315$		0			0
$316$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$317$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$317$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$	1.41421			1.41421
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$329$	−0.414214	+	1.00000i	−0.414214	+	1.00000i
$330$		0			0
$331$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$331$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$332$		0			0
$333$		0			0
$334$	1.70711	+	0.707107i	1.70711	+	0.707107i
$335$		0			0
$336$		0			0
$337$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$337$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−1.00000			$\pi$
$338$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	1.00000	−	0.414214i	1.00000	−	0.414214i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$347$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$348$		0			0
$349$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$349$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$350$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$351$		0			0
$352$		0			0
$353$	−2.00000			−2.00000			−1.00000			$\pi$
$353$	−2.00000			−2.00000			−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		−	1.41421i		−	1.41421i
$357$		0			0
$358$		0			0
$359$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$359$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$360$		0			0
$361$			1.00000i			1.00000i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.707107	+	1.70711i	−0.707107	+	1.70711i			1.00000i	$0.5\pi$
$367$	−0.707107	+	1.70711i	−0.707107	+	1.70711i	−0.707107	+	0.707107i	$0.750000\pi$
$368$	0.707107	−	1.70711i	0.707107	−	1.70711i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	1.00000	+	1.00000i	1.00000	+	1.00000i
$377$		0			0
$378$		0			0
$379$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$379$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$380$		0			0
$381$		0			0
$382$	1.00000	−	1.00000i	1.00000	−	1.00000i
$383$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$383$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$387$		0			0
$388$	1.70711	−	0.707107i	1.70711	−	0.707107i
$389$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$389$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$390$		0			0
$391$	1.70711	+	0.707107i	1.70711	+	0.707107i
$392$		−	0.414214i		−	0.414214i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$397$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$398$	0.707107	+	0.292893i	0.707107	+	0.292893i
$399$		0			0
$400$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$401$	0.707107	+	0.292893i	0.707107	+	0.292893i	0.707107	−	0.707107i	$-0.250000\pi$
$401$	0.707107	+	0.292893i	0.707107	+	0.292893i			1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$409$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$410$		0			0
$411$		0			0
$412$		−	1.41421i		−	1.41421i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$419$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.292893	−	0.707107i	0.292893	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$431$	0.292893	−	0.707107i	0.292893	−	0.707107i	1.00000			$0$
$432$		0			0
$433$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$433$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$434$	−0.414214	−	0.414214i	−0.414214	−	0.414214i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	1.70711	−	0.707107i	1.70711	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$439$	1.70711	−	0.707107i	1.70711	−	0.707107i	1.00000			$0$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$	−1.41421			−1.41421
$447$		0			0
$448$	0.707107	+	0.292893i	0.707107	+	0.292893i
$449$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$449$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$	0.707107	+	0.292893i	0.707107	+	0.292893i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$457$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$461$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$462$		0			0
$463$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$463$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$477$		0			0
$478$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$479$	−0.707107	−	1.70711i	−0.707107	−	1.70711i	−0.707107	−	0.707107i	$-0.750000\pi$
$479$	−0.707107	−	1.70711i	−0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$480$		0			0
$481$		0			0
$482$	0.292893	−	0.707107i	0.292893	−	0.707107i
$483$		0			0
$484$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$485$		0			0
$486$		0			0
$487$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$487$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$491$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$497$	0.585786			0.585786
$498$		0			0
$499$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$499$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$503$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	1.41421	+	1.41421i	1.41421	+	1.41421i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	1.41421			1.41421
$512$	0.707107	−	0.707107i	0.707107	−	0.707107i
$513$		0			0
$514$		−	1.41421i		−	1.41421i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.707107	+	1.70711i	−0.707107	+	1.70711i			1.00000i	$0.5\pi$
$521$	−0.707107	+	1.70711i	−0.707107	+	1.70711i	−0.707107	+	0.707107i	$0.750000\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$	−1.41421			−1.41421
$527$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$528$		0			0
$529$	−1.70711	−	1.70711i	−1.70711	−	1.70711i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$541$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$542$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$543$		0			0
$544$	0.707107	+	0.707107i	0.707107	+	0.707107i
$545$		0			0
$546$		0			0
$547$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$547$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$548$	−2.00000			−2.00000
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−1.41421			−1.41421
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	0.292893	−	0.707107i	0.292893	−	0.707107i
$569$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$569$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$570$		0			0
$571$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$571$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$572$		0			0
$573$		0			0
$574$		−	0.585786i		−	0.585786i
$575$	0.707107	+	1.70711i	0.707107	+	1.70711i
$576$		0			0
$577$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$577$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$578$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.707107	−	1.70711i	0.707107	−	1.70711i
$585$		0			0
$586$		0			0
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$		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
$598$		0			0
$599$	2.00000			2.00000			1.00000			$0$
$599$	2.00000			2.00000			1.00000			$0$
$600$		0			0
$601$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$601$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−0.707107	−	1.70711i	−0.707107	−	1.70711i	−0.707107	−	0.707107i	$-0.750000\pi$
$607$	−0.707107	−	1.70711i	−0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$617$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−1.00000			$\pi$
$618$		0			0
$619$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$619$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$620$		0			0
$621$		0			0
$622$	0.707107	−	1.70711i	0.707107	−	1.70711i
$623$	0.414214	−	1.00000i	0.414214	−	1.00000i
$624$		0			0
$625$		−	1.00000i		−	1.00000i
$626$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$631$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$632$	0.707107	+	1.70711i	0.707107	+	1.70711i
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.707107	+	0.292893i	−0.707107	+	0.292893i	−0.707107	−	0.707107i	$-0.750000\pi$
$641$	−0.707107	+	0.292893i	−0.707107	+	0.292893i			1.00000i	$0.5\pi$
$642$		0			0
$643$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$643$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$644$	1.00000	−	1.00000i	1.00000	−	1.00000i
$645$		0			0
$646$		0			0
$647$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$647$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$653$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$654$		0			0
$655$		0			0
$656$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$657$		0			0
$658$	0.414214	+	1.00000i	0.414214	+	1.00000i
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$661$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	1.70711	−	0.707107i	1.70711	−	0.707107i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.292893	−	0.707107i	0.292893	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$673$	0.292893	−	0.707107i	0.292893	−	0.707107i	1.00000			$0$
$674$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$675$		0			0
$676$			1.00000i			1.00000i
$677$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$677$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$678$		0			0
$679$	1.41421			1.41421
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$683$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$684$		0			0
$685$		0			0
$686$	0.414214	−	1.00000i	0.414214	−	1.00000i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$691$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	0.292893	−	0.707107i	0.292893	−	0.707107i
$698$		0			0
$699$		0			0
$700$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$707$		0			0
$708$		0			0
$709$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$709$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$710$		0			0
$711$		0			0
$712$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$713$			1.41421i			1.41421i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.70711	−	0.707107i	1.70711	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$719$	1.70711	−	0.707107i	1.70711	−	0.707107i	1.00000			$0$
$720$		0			0
$721$	0.414214	−	1.00000i	0.414214	−	1.00000i
$722$	0.707107	+	0.707107i	0.707107	+	0.707107i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$727$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$	0.707107	+	1.70711i	0.707107	+	1.70711i
$735$		0			0
$736$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$737$		0			0
$738$		0			0
$739$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$739$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.70711	−	0.707107i	1.70711	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$743$	1.70711	−	0.707107i	1.70711	−	0.707107i	1.00000			$0$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.70711	−	0.707107i	1.70711	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$751$	1.70711	−	0.707107i	1.70711	−	0.707107i	1.00000			$0$
$752$	1.41421			1.41421
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.00000			2.00000			1.00000			$0$
$761$	2.00000			2.00000			1.00000			$0$
$762$		0			0
$763$		0			0
$764$		−	1.41421i		−	1.41421i
$765$		0			0
$766$	−1.41421			−1.41421
$767$		0			0
$768$		0			0
$769$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$769$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$770$		0			0
$771$		0			0
$772$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$		0			0
$775$	0.292893	−	0.707107i	0.292893	−	0.707107i
$776$	0.707107	−	1.70711i	0.707107	−	1.70711i
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	1.70711	−	0.707107i	1.70711	−	0.707107i
$783$		0			0
$784$	−0.292893	−	0.292893i	−0.292893	−	0.292893i
$785$		0			0
$786$		0			0
$787$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$787$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	0.414214	+	0.414214i	0.414214	+	0.414214i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	0.707107	−	0.292893i	0.707107	−	0.292893i
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$			1.41421i			1.41421i
$800$			1.00000i			1.00000i
$801$		0			0
$802$	0.707107	−	0.292893i	0.707107	−	0.292893i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$809$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$810$		0			0
$811$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$811$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$819$		0			0
$820$		0			0
$821$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$821$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$822$		0			0
$823$	0.292893	−	0.707107i	0.292893	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$823$	0.292893	−	0.707107i	0.292893	−	0.707107i	1.00000			$0$
$824$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$825$		0			0
$826$		0			0
$827$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$827$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.292893	−	0.292893i	0.292893	−	0.292893i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−0.292893	+	0.707107i	−0.292893	+	0.707107i	0.707107	+	0.707107i	$0.250000\pi$
$839$	−0.292893	+	0.707107i	−0.292893	+	0.707107i	−1.00000			$\pi$
$840$		0			0
$841$	0.707107	−	0.707107i	0.707107	−	0.707107i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$848$		0			0
$849$		0			0
$850$	−1.00000			−1.00000
$851$		0			0
$852$		0			0
$853$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$853$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$857$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$859$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$860$		0			0
$861$		0			0
$862$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		−	1.41421i		−	1.41421i
$867$		0			0
$868$	−0.585786			−0.585786
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$877$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$878$	0.707107	−	1.70711i	0.707107	−	1.70711i
$879$		0			0
$880$		0			0
$881$	0.707107	+	1.70711i	0.707107	+	1.70711i	0.707107	+	0.707107i	$0.250000\pi$
$881$	0.707107	+	1.70711i	0.707107	+	1.70711i			1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.707107	+	1.70711i	0.707107	+	1.70711i	0.707107	+	0.707107i	$0.250000\pi$
$887$	0.707107	+	1.70711i	0.707107	+	1.70711i			1.00000i	$0.5\pi$
$888$		0			0
$889$	0.585786	+	1.41421i	0.585786	+	1.41421i
$890$		0			0
$891$		0			0
$892$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$893$		0			0
$894$		0			0
$895$		0			0
$896$	0.707107	−	0.292893i	0.707107	−	0.292893i
$897$		0			0
$898$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	0.707107	−	0.292893i	0.707107	−	0.292893i
$905$		0			0
$906$		0			0
$907$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$907$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.707107	+	0.292893i	0.707107	+	0.292893i	0.707107	−	0.707107i	$-0.250000\pi$
$911$	0.707107	+	0.292893i	0.707107	+	0.292893i			1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$	1.41421			1.41421
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$	1.00000	+	1.00000i	1.00000	+	1.00000i
$927$		0			0
$928$		0			0
$929$	−0.707107	+	1.70711i	−0.707107	+	1.70711i			1.00000i	$0.5\pi$
$929$	−0.707107	+	1.70711i	−0.707107	+	1.70711i	−0.707107	+	0.707107i	$0.750000\pi$
$930$		0			0
$931$		0			0
$932$	0.292893	+	0.707107i	0.292893	+	0.707107i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$941$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$942$		0			0
$943$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$947$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$	0.292893	+	0.707107i	0.292893	+	0.707107i
$953$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$953$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$954$		0			0
$955$		0			0
$956$	−1.41421			−1.41421
$957$		0			0
$958$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$959$	−1.41421	−	0.585786i	−1.41421	−	0.585786i
$960$		0			0
$961$	−0.292893	+	0.292893i	−0.292893	+	0.292893i
$962$		0			0
$963$		0			0
$964$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$965$		0			0
$966$		0			0
$967$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$967$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$968$			1.00000i			1.00000i
$969$		0			0
$970$		0			0
$971$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$971$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$972$		0			0
$973$		0			0
$974$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$975$		0			0
$976$		0			0
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	0.292893	+	0.707107i	0.292893	+	0.707107i	1.00000			$0$
$983$	0.292893	+	0.707107i	0.292893	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$991$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	−1.00000			$\pi$
$992$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$993$		0			0
$994$	0.414214	−	0.414214i	0.414214	−	0.414214i
$995$		0			0
$996$		0			0
$997$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$997$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1224.1.bu.a.53.1	✓	4
3.2	odd	2		1224.1.bu.b.53.1	yes	4
8.5	even	2	RM	1224.1.bu.a.53.1	✓	4
17.9	even	8		1224.1.bu.b.485.1	yes	4
24.5	odd	2		1224.1.bu.b.53.1	yes	4
51.26	odd	8	inner	1224.1.bu.a.485.1	yes	4
136.77	even	8		1224.1.bu.b.485.1	yes	4
408.77	odd	8	inner	1224.1.bu.a.485.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1224.1.bu.a.53.1	✓	4	1.1	even	1	trivial
1224.1.bu.a.53.1	✓	4	8.5	even	2	RM
1224.1.bu.a.485.1	yes	4	51.26	odd	8	inner
1224.1.bu.a.485.1	yes	4	408.77	odd	8	inner
1224.1.bu.b.53.1	yes	4	3.2	odd	2
1224.1.bu.b.53.1	yes	4	24.5	odd	2
1224.1.bu.b.485.1	yes	4	17.9	even	8
1224.1.bu.b.485.1	yes	4	136.77	even	8

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:	\( N \)	\(=\)	\( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1224.bu (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{2} -1.00000i q^{4} +(0.292893 - 0.707107i) q^{7} +(-0.707107 - 0.707107i) q^{8} +O(q^{10})\)	\(q+(0.707107 - 0.707107i) q^{2} -1.00000i q^{4} +(0.292893 - 0.707107i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(-0.292893 - 0.707107i) q^{14} -1.00000 q^{16} -1.00000i q^{17} +(-0.707107 + 1.70711i) q^{23} +(0.707107 - 0.707107i) q^{25} +(-0.707107 - 0.292893i) q^{28} +(0.707107 - 0.292893i) q^{31} +(-0.707107 + 0.707107i) q^{32} +(-0.707107 - 0.707107i) q^{34} +(0.707107 + 0.292893i) q^{41} +(0.707107 + 1.70711i) q^{46} -1.41421 q^{47} +(0.292893 + 0.292893i) q^{49} -1.00000i q^{50} +(-0.707107 + 0.292893i) q^{56} +(0.292893 - 0.707107i) q^{62} +1.00000i q^{64} -1.00000 q^{68} +(0.292893 + 0.707107i) q^{71} +(0.707107 + 1.70711i) q^{73} +(-1.70711 - 0.707107i) q^{79} +(0.707107 - 0.292893i) q^{82} +1.41421 q^{89} +(1.70711 + 0.707107i) q^{92} +(-1.00000 + 1.00000i) q^{94} +(0.707107 + 1.70711i) q^{97} +0.414214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^7	\( 4 q + 4 q^{7} - 4 q^{14} - 4 q^{16} + 4 q^{49} + 4 q^{62} - 4 q^{68} + 4 q^{71} - 4 q^{79} + 4 q^{92} - 4 q^{94} - 4 q^{98}+O(q^{100}) \) 4 * q + 4 * q^7 - 4 * q^14 - 4 * q^16 + 4 * q^49 + 4 * q^62 - 4 * q^68 + 4 * q^71 - 4 * q^79 + 4 * q^92 - 4 * q^94 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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