LMFDB - Embedded newform 3400.1.y.a.251.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3400,1,Mod(251,3400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3400, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("3400.251");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3400 = 2^{3} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3400.y (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:	$1.69682104295$
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 136)
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.0.314432.1
Artin image:	$C_4^2:C_2^2$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{16} - \cdots)$

Embedding invariants

Embedding label			251.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3400.251
Dual form			3400.1.y.a.3251.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +(1.00000 + 1.00000i) q^{3} -1.00000 q^{4} +(-1.00000 + 1.00000i) q^{6} -1.00000i q^{8} +1.00000i q^{9} +(1.00000 - 1.00000i) q^{11} +(-1.00000 - 1.00000i) q^{12} +1.00000 q^{16} +1.00000i q^{17} -1.00000 q^{18} +2.00000i q^{19} +(1.00000 + 1.00000i) q^{22} +(1.00000 - 1.00000i) q^{24} +1.00000i q^{32} +2.00000 q^{33} -1.00000 q^{34} -1.00000i q^{36} -2.00000 q^{38} +(-1.00000 + 1.00000i) q^{41} +(-1.00000 + 1.00000i) q^{44} +(1.00000 + 1.00000i) q^{48} -1.00000i q^{49} +(-1.00000 + 1.00000i) q^{51} +(-2.00000 + 2.00000i) q^{57} -1.00000 q^{64} +2.00000i q^{66} -1.00000i q^{68} +1.00000 q^{72} +(1.00000 + 1.00000i) q^{73} -2.00000i q^{76} +1.00000 q^{81} +(-1.00000 - 1.00000i) q^{82} +(-1.00000 - 1.00000i) q^{88} +(-1.00000 + 1.00000i) q^{96} +(-1.00000 - 1.00000i) q^{97} +1.00000 q^{98} +(1.00000 + 1.00000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{4} - 2 q^{6} + 2 q^{11} - 2 q^{12} + 2 q^{16} - 2 q^{18} + 2 q^{22} + 2 q^{24} + 4 q^{33} - 2 q^{34} - 4 q^{38} - 2 q^{41} - 2 q^{44} + 2 q^{48} - 2 q^{51} - 4 q^{57} - 2 q^{64} + 2 q^{72}+ \cdots + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^4 - 2 * q^6 + 2 * q^11 - 2 * q^12 + 2 * q^16 - 2 * q^18 + 2 * q^22 + 2 * q^24 + 4 * q^33 - 2 * q^34 - 4 * q^38 - 2 * q^41 - 2 * q^44 + 2 * q^48 - 2 * q^51 - 4 * q^57 - 2 * q^64 + 2 * q^72 + 2 * q^73 + 2 * q^81 - 2 * q^82 - 2 * q^88 - 2 * q^96 - 2 * q^97 + 2 * q^98 + 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3400\mathbb{Z}\right)^\times$.

$n$	$1601$	$1701$	$2177$	$2551$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$			1.00000i			1.00000i
$2$			1.00000i			1.00000i
$3$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$3$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$4$	−1.00000			−1.00000
$5$		0			0
$5$		0			0
$6$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$		−	1.00000i		−	1.00000i
$9$			1.00000i			1.00000i
$10$		0			0
$11$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$11$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$12$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			1.00000
$17$			1.00000i			1.00000i
$17$			1.00000i			1.00000i
$18$	−1.00000			−1.00000
$19$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$19$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$	1.00000	+	1.00000i	1.00000	+	1.00000i
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$	1.00000	−	1.00000i	1.00000	−	1.00000i
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$29$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$30$		0			0
$31$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$31$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$32$			1.00000i			1.00000i
$33$	2.00000			2.00000
$34$	−1.00000			−1.00000
$35$		0			0
$36$		−	1.00000i		−	1.00000i
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$	−2.00000			−2.00000
$39$		0			0
$40$		0			0
$41$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$41$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	1.00000	+	1.00000i	1.00000	+	1.00000i
$49$		−	1.00000i		−	1.00000i
$50$		0			0
$51$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−2.00000	+	2.00000i	−2.00000	+	2.00000i
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$61$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$62$		0			0
$63$		0			0
$64$	−1.00000			−1.00000
$65$		0			0
$66$			2.00000i			2.00000i
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		−	1.00000i		−	1.00000i
$69$		0			0
$70$		0			0
$71$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$71$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$72$	1.00000			1.00000
$73$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$73$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$74$		0			0
$75$		0			0
$76$		−	2.00000i		−	2.00000i
$77$		0			0
$78$		0			0
$79$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$79$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$80$		0			0
$81$	1.00000			1.00000
$82$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$97$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$97$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$98$	1.00000			1.00000
$99$	1.00000	+	1.00000i	1.00000	+	1.00000i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$107$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$108$		0			0
$109$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$109$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$113$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$114$	−2.00000	−	2.00000i	−2.00000	−	2.00000i
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	1.00000i		−	1.00000i
$122$		0			0
$123$	−2.00000			−2.00000
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		−	1.00000i		−	1.00000i
$129$		0			0
$130$		0			0
$131$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$131$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$132$	−2.00000			−2.00000
$133$		0			0
$134$		0			0
$135$		0			0
$136$	1.00000			1.00000
$137$	2.00000			2.00000			1.00000			$0$
$137$	2.00000			2.00000			1.00000			$0$
$138$		0			0
$139$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$139$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$			1.00000i			1.00000i
$145$		0			0
$146$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$147$	1.00000	−	1.00000i	1.00000	−	1.00000i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	2.00000			2.00000
$153$	−1.00000			−1.00000
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$			1.00000i			1.00000i
$163$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$163$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$164$	1.00000	−	1.00000i	1.00000	−	1.00000i
$165$		0			0
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$	1.00000			1.00000
$170$		0			0
$171$	−2.00000			−2.00000
$172$		0			0
$173$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$173$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$174$		0			0
$175$		0			0
$176$	1.00000	−	1.00000i	1.00000	−	1.00000i
$177$		0			0
$178$		0			0
$179$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$179$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$180$		0			0
$181$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$181$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	1.00000	+	1.00000i	1.00000	+	1.00000i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$193$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$193$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$194$	1.00000	−	1.00000i	1.00000	−	1.00000i
$195$		0			0
$196$			1.00000i			1.00000i
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$199$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$199$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$	1.00000	−	1.00000i	1.00000	−	1.00000i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.00000	+	2.00000i	2.00000	+	2.00000i
$210$		0			0
$211$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$211$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$	1.00000	−	1.00000i	1.00000	−	1.00000i
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$			2.00000i			2.00000i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$	1.00000	+	1.00000i	1.00000	+	1.00000i
$227$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$227$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$228$	2.00000	−	2.00000i	2.00000	−	2.00000i
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$233$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$241$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$242$	1.00000			1.00000
$243$	1.00000	+	1.00000i	1.00000	+	1.00000i
$244$		0			0
$245$		0			0
$246$		−	2.00000i		−	2.00000i
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1.00000			1.00000
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	1.00000	−	1.00000i	1.00000	−	1.00000i
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		−	2.00000i		−	2.00000i
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$269$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$			1.00000i			1.00000i
$273$		0			0
$274$			2.00000i			2.00000i
$275$		0			0
$276$		0			0
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$	1.00000	−	1.00000i	1.00000	−	1.00000i
$279$		0			0
$280$		0			0
$281$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$281$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$282$		0			0
$283$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$283$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−1.00000			−1.00000
$289$	−1.00000			−1.00000
$290$		0			0
$291$		−	2.00000i		−	2.00000i
$292$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	1.00000	+	1.00000i	1.00000	+	1.00000i
$295$		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
$304$			2.00000i			2.00000i
$305$		0			0
$306$		−	1.00000i		−	1.00000i
$307$	−2.00000			−2.00000			−1.00000			$\pi$
$307$	−2.00000			−2.00000			−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$311$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$312$		0			0
$313$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$313$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$314$		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			0
$321$		−	2.00000i		−	2.00000i
$322$		0			0
$323$	−2.00000			−2.00000
$324$	−1.00000			−1.00000
$325$		0			0
$326$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$327$		0			0
$328$	1.00000	+	1.00000i	1.00000	+	1.00000i
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$337$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$338$			1.00000i			1.00000i
$339$	2.00000			2.00000
$340$		0			0
$341$		0			0
$342$		−	2.00000i		−	2.00000i
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$347$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	1.00000	+	1.00000i	1.00000	+	1.00000i
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	2.00000			2.00000
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−3.00000			−3.00000
$362$		0			0
$363$	1.00000	−	1.00000i	1.00000	−	1.00000i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$		0			0
$369$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$379$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	1.00000	−	1.00000i	1.00000	−	1.00000i
$385$		0			0
$386$	1.00000	+	1.00000i	1.00000	+	1.00000i
$387$		0			0
$388$	1.00000	+	1.00000i	1.00000	+	1.00000i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	−1.00000			−1.00000
$393$		−	2.00000i		−	2.00000i
$394$		0			0
$395$		0			0
$396$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$401$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	1.00000	+	1.00000i	1.00000	+	1.00000i
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$	2.00000	+	2.00000i	2.00000	+	2.00000i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	2.00000i		−	2.00000i
$418$	−2.00000	+	2.00000i	−2.00000	+	2.00000i
$419$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$419$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	1.00000	+	1.00000i	1.00000	+	1.00000i
$429$		0			0
$430$		0			0
$431$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$431$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$432$		0			0
$433$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$433$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−2.00000			−2.00000
$439$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$439$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$440$		0			0
$441$	1.00000			1.00000
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$449$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$450$		0			0
$451$			2.00000i			2.00000i
$452$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$453$		0			0
$454$	1.00000	+	1.00000i	1.00000	+	1.00000i
$455$		0			0
$456$	2.00000	+	2.00000i	2.00000	+	2.00000i
$457$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$457$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	1.00000	−	1.00000i	1.00000	−	1.00000i
$467$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$467$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$468$		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			0
$479$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$479$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$480$		0			0
$481$		0			0
$482$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$483$		0			0
$484$			1.00000i			1.00000i
$485$		0			0
$486$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$		0			0
$489$	−2.00000			−2.00000
$490$		0			0
$491$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$491$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$492$	2.00000			2.00000
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$499$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	1.00000	+	1.00000i	1.00000	+	1.00000i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$			1.00000i			1.00000i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$521$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$	1.00000	+	1.00000i	1.00000	+	1.00000i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	2.00000			2.00000
$529$		−	1.00000i		−	1.00000i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	2.00000	−	2.00000i	2.00000	−	2.00000i
$538$		0			0
$539$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$540$		0			0
$541$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$541$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$542$		0			0
$543$		0			0
$544$	−1.00000			−1.00000
$545$		0			0
$546$		0			0
$547$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$547$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$548$	−2.00000			−2.00000
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	1.00000	+	1.00000i	1.00000	+	1.00000i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$			2.00000i			2.00000i
$562$	2.00000			2.00000
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$571$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		−	1.00000i		−	1.00000i
$577$	−2.00000			−2.00000			−1.00000			$\pi$
$577$	−2.00000			−2.00000			−1.00000			$\pi$
$578$		−	1.00000i		−	1.00000i
$579$	2.00000			2.00000
$580$		0			0
$581$		0			0
$582$	2.00000			2.00000
$583$		0			0
$584$	1.00000	−	1.00000i	1.00000	−	1.00000i
$585$		0			0
$586$		0			0
$587$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$587$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$588$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$601$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$	−2.00000			−2.00000
$609$		0			0
$610$		0			0
$611$		0			0
$612$	1.00000			1.00000
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		−	2.00000i		−	2.00000i
$615$		0			0
$616$		0			0
$617$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$617$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$618$		0			0
$619$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$619$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$627$			4.00000i			4.00000i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$	−2.00000			−2.00000
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$641$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$642$	2.00000			2.00000
$643$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$643$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		−	2.00000i		−	2.00000i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		−	1.00000i		−	1.00000i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	1.00000	−	1.00000i	1.00000	−	1.00000i
$653$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$653$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$		0			0
$656$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$657$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$658$		0			0
$659$	−2.00000			−2.00000			−1.00000			$\pi$
$659$	−2.00000			−2.00000			−1.00000			$\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$673$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$674$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$675$		0			0
$676$	−1.00000			−1.00000
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$			2.00000i			2.00000i
$679$		0			0
$680$		0			0
$681$	2.00000			2.00000
$682$		0			0
$683$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$683$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$684$	2.00000			2.00000
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$691$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$692$		0			0
$693$		0			0
$694$	1.00000	+	1.00000i	1.00000	+	1.00000i
$695$		0			0
$696$		0			0
$697$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$698$		0			0
$699$		−	2.00000i		−	2.00000i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$709$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$			2.00000i			2.00000i
$717$		0			0
$718$		0			0
$719$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$719$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$720$		0			0
$721$		0			0
$722$		−	3.00000i		−	3.00000i
$723$			2.00000i			2.00000i
$724$		0			0
$725$		0			0
$726$	1.00000	+	1.00000i	1.00000	+	1.00000i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$			1.00000i			1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	1.00000	−	1.00000i	1.00000	−	1.00000i
$739$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$739$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		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
$748$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$749$		0			0
$750$		0			0
$751$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$751$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	1.00000	−	1.00000i	1.00000	−	1.00000i
$759$		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
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	1.00000	+	1.00000i	1.00000	+	1.00000i
$769$	2.00000			2.00000			1.00000			$0$
$769$	2.00000			2.00000			1.00000			$0$
$770$		0			0
$771$		0			0
$772$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$777$		0			0
$778$		0			0
$779$	−2.00000	−	2.00000i	−2.00000	−	2.00000i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		−	1.00000i		−	1.00000i
$785$		0			0
$786$	2.00000			2.00000
$787$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$787$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	1.00000	−	1.00000i	1.00000	−	1.00000i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$	1.00000	+	1.00000i	1.00000	+	1.00000i
$803$	2.00000			2.00000
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$809$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$810$		0			0
$811$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$811$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$821$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$822$	−2.00000	+	2.00000i	−2.00000	+	2.00000i
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$827$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.00000			1.00000
$834$	2.00000			2.00000
$835$		0			0
$836$	−2.00000	−	2.00000i	−2.00000	−	2.00000i
$837$		0			0
$838$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$839$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$839$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$840$		0			0
$841$			1.00000i			1.00000i
$842$		0			0
$843$	2.00000	−	2.00000i	2.00000	−	2.00000i
$844$	1.00000	−	1.00000i	1.00000	−	1.00000i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−2.00000			−2.00000
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$		0			0
$856$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$857$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$857$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$	−2.00000			−2.00000
$867$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1.00000	−	1.00000i	1.00000	−	1.00000i
$874$		0			0
$875$		0			0
$876$		−	2.00000i		−	2.00000i
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$881$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$882$			1.00000i			1.00000i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1.00000	−	1.00000i	1.00000	−	1.00000i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	1.00000	+	1.00000i	1.00000	+	1.00000i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	−2.00000			−2.00000
$903$		0			0
$904$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$905$		0			0
$906$		0			0
$907$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$907$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$908$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$909$		0			0
$910$		0			0
$911$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$911$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$912$	−2.00000	+	2.00000i	−2.00000	+	2.00000i
$913$		0			0
$914$	2.00000			2.00000
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−2.00000	−	2.00000i	−2.00000	−	2.00000i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$929$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$930$		0			0
$931$	2.00000			2.00000
$932$	1.00000	+	1.00000i	1.00000	+	1.00000i
$933$		0			0
$934$	−2.00000			−2.00000
$935$		0			0
$936$		0			0
$937$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$937$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$938$		0			0
$939$	−2.00000			−2.00000
$940$		0			0
$941$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$941$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$947$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$			1.00000i			1.00000i
$962$		0			0
$963$	1.00000	−	1.00000i	1.00000	−	1.00000i
$964$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	−1.00000			−1.00000
$969$	−2.00000	−	2.00000i	−2.00000	−	2.00000i
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		−	2.00000i		−	2.00000i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−2.00000			−2.00000
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$			2.00000i			2.00000i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$991$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$	1.00000	+	1.00000i	1.00000	+	1.00000i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3400.1.y.a.251.1		2
5.2	odd	4		3400.1.bc.a.2699.1		2
5.3	odd	4		3400.1.bc.b.2699.1		2
5.4	even	2		136.1.j.a.115.1	✓	2
8.3	odd	2	CM	3400.1.y.a.251.1		2
15.14	odd	2		1224.1.s.a.523.1		2
17.4	even	4	inner	3400.1.y.a.3251.1		2
20.19	odd	2		544.1.n.a.47.1		2
40.3	even	4		3400.1.bc.b.2699.1		2
40.19	odd	2		136.1.j.a.115.1	✓	2
40.27	even	4		3400.1.bc.a.2699.1		2
40.29	even	2		544.1.n.a.47.1		2
85.4	even	4		136.1.j.a.123.1	yes	2
85.9	even	8		2312.1.e.a.1155.1		2
85.14	odd	16		2312.1.p.e.1579.2		8
85.19	even	8		2312.1.f.b.579.1		2
85.24	odd	16		2312.1.p.e.1555.1		8
85.29	odd	16		2312.1.p.e.155.1		8
85.38	odd	4		3400.1.bc.a.2299.1		2
85.39	odd	16		2312.1.p.e.155.2		8
85.44	odd	16		2312.1.p.e.1555.2		8
85.49	even	8		2312.1.f.b.579.2		2
85.54	odd	16		2312.1.p.e.1579.1		8
85.59	even	8		2312.1.e.a.1155.2		2
85.64	even	4		2312.1.j.b.1483.1		2
85.72	odd	4		3400.1.bc.b.2299.1		2
85.74	odd	16		2312.1.p.e.179.1		8
85.79	odd	16		2312.1.p.e.179.2		8
85.84	even	2		2312.1.j.b.251.1		2
120.59	even	2		1224.1.s.a.523.1		2
136.123	odd	4	inner	3400.1.y.a.3251.1		2
255.89	odd	4		1224.1.s.a.667.1		2
340.259	odd	4		544.1.n.a.463.1		2
680.19	odd	8		2312.1.f.b.579.1		2
680.59	odd	8		2312.1.e.a.1155.2		2
680.99	even	16		2312.1.p.e.1579.2		8
680.123	even	4		3400.1.bc.a.2299.1		2
680.139	even	16		2312.1.p.e.1579.1		8
680.179	odd	8		2312.1.e.a.1155.1		2
680.219	odd	8		2312.1.f.b.579.2		2
680.259	odd	4		136.1.j.a.123.1	yes	2
680.299	even	16		2312.1.p.e.1555.2		8
680.339	odd	2		2312.1.j.b.251.1		2
680.379	even	16		2312.1.p.e.155.2		8
680.419	even	16		2312.1.p.e.179.2		8
680.429	even	4		544.1.n.a.463.1		2
680.499	even	16		2312.1.p.e.179.1		8
680.539	even	16		2312.1.p.e.155.1		8
680.619	even	16		2312.1.p.e.1555.1		8
680.659	odd	4		2312.1.j.b.1483.1		2
680.667	even	4		3400.1.bc.b.2299.1		2
2040.1619	even	4		1224.1.s.a.667.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.1.j.a.115.1	✓	2	5.4	even	2
136.1.j.a.115.1	✓	2	40.19	odd	2
136.1.j.a.123.1	yes	2	85.4	even	4
136.1.j.a.123.1	yes	2	680.259	odd	4
544.1.n.a.47.1		2	20.19	odd	2
544.1.n.a.47.1		2	40.29	even	2
544.1.n.a.463.1		2	340.259	odd	4
544.1.n.a.463.1		2	680.429	even	4
1224.1.s.a.523.1		2	15.14	odd	2
1224.1.s.a.523.1		2	120.59	even	2
1224.1.s.a.667.1		2	255.89	odd	4
1224.1.s.a.667.1		2	2040.1619	even	4
2312.1.e.a.1155.1		2	85.9	even	8
2312.1.e.a.1155.1		2	680.179	odd	8
2312.1.e.a.1155.2		2	85.59	even	8
2312.1.e.a.1155.2		2	680.59	odd	8
2312.1.f.b.579.1		2	85.19	even	8
2312.1.f.b.579.1		2	680.19	odd	8
2312.1.f.b.579.2		2	85.49	even	8
2312.1.f.b.579.2		2	680.219	odd	8
2312.1.j.b.251.1		2	85.84	even	2
2312.1.j.b.251.1		2	680.339	odd	2
2312.1.j.b.1483.1		2	85.64	even	4
2312.1.j.b.1483.1		2	680.659	odd	4
2312.1.p.e.155.1		8	85.29	odd	16
2312.1.p.e.155.1		8	680.539	even	16
2312.1.p.e.155.2		8	85.39	odd	16
2312.1.p.e.155.2		8	680.379	even	16
2312.1.p.e.179.1		8	85.74	odd	16
2312.1.p.e.179.1		8	680.499	even	16
2312.1.p.e.179.2		8	85.79	odd	16
2312.1.p.e.179.2		8	680.419	even	16
2312.1.p.e.1555.1		8	85.24	odd	16
2312.1.p.e.1555.1		8	680.619	even	16
2312.1.p.e.1555.2		8	85.44	odd	16
2312.1.p.e.1555.2		8	680.299	even	16
2312.1.p.e.1579.1		8	85.54	odd	16
2312.1.p.e.1579.1		8	680.139	even	16
2312.1.p.e.1579.2		8	85.14	odd	16
2312.1.p.e.1579.2		8	680.99	even	16
3400.1.y.a.251.1		2	1.1	even	1	trivial
3400.1.y.a.251.1		2	8.3	odd	2	CM
3400.1.y.a.3251.1		2	17.4	even	4	inner
3400.1.y.a.3251.1		2	136.123	odd	4	inner
3400.1.bc.a.2299.1		2	85.38	odd	4
3400.1.bc.a.2299.1		2	680.123	even	4
3400.1.bc.a.2699.1		2	5.2	odd	4
3400.1.bc.a.2699.1		2	40.27	even	4
3400.1.bc.b.2299.1		2	85.72	odd	4
3400.1.bc.b.2299.1		2	680.667	even	4
3400.1.bc.b.2699.1		2	5.3	odd	4
3400.1.bc.b.2699.1		2	40.3	even	4

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:	\( N \)	\(=\)	\( 3400 = 2^{3} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3400.y (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +(1.00000 + 1.00000i) q^{3} -1.00000 q^{4} +(-1.00000 + 1.00000i) q^{6} -1.00000i q^{8} +1.00000i q^{9} +(1.00000 - 1.00000i) q^{11} +(-1.00000 - 1.00000i) q^{12} +1.00000 q^{16} +1.00000i q^{17} -1.00000 q^{18} +2.00000i q^{19} +(1.00000 + 1.00000i) q^{22} +(1.00000 - 1.00000i) q^{24} +1.00000i q^{32} +2.00000 q^{33} -1.00000 q^{34} -1.00000i q^{36} -2.00000 q^{38} +(-1.00000 + 1.00000i) q^{41} +(-1.00000 + 1.00000i) q^{44} +(1.00000 + 1.00000i) q^{48} -1.00000i q^{49} +(-1.00000 + 1.00000i) q^{51} +(-2.00000 + 2.00000i) q^{57} -1.00000 q^{64} +2.00000i q^{66} -1.00000i q^{68} +1.00000 q^{72} +(1.00000 + 1.00000i) q^{73} -2.00000i q^{76} +1.00000 q^{81} +(-1.00000 - 1.00000i) q^{82} +(-1.00000 - 1.00000i) q^{88} +(-1.00000 + 1.00000i) q^{96} +(-1.00000 - 1.00000i) q^{97} +1.00000 q^{98} +(1.00000 + 1.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{4} - 2 q^{6} + 2 q^{11} - 2 q^{12} + 2 q^{16} - 2 q^{18} + 2 q^{22} + 2 q^{24} + 4 q^{33} - 2 q^{34} - 4 q^{38} - 2 q^{41} - 2 q^{44} + 2 q^{48} - 2 q^{51} - 4 q^{57} - 2 q^{64} + 2 q^{72}+ \cdots + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^4 - 2 * q^6 + 2 * q^11 - 2 * q^12 + 2 * q^16 - 2 * q^18 + 2 * q^22 + 2 * q^24 + 4 * q^33 - 2 * q^34 - 4 * q^38 - 2 * q^41 - 2 * q^44 + 2 * q^48 - 2 * q^51 - 4 * q^57 - 2 * q^64 + 2 * q^72 + 2 * q^73 + 2 * q^81 - 2 * q^82 - 2 * q^88 - 2 * q^96 - 2 * q^97 + 2 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more