LMFDB - Embedded newform 3528.1.dh.a.1189.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3528,1,Mod(181,3528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3528, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([0, 7, 0, 3]))

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

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

chi := DirichletCharacter("3528.181");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3528.dh (of order $14$, degree $6$, 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:	$1.76070136457$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{14})$
Coefficient field:	$\Q(\zeta_{28})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{14}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{14} + \cdots)$

Embedding invariants

Embedding label			1189.2
Root			$0.974928 - 0.222521i$ of defining polynomial
Character	$\chi$	$=$	3528.1189
Dual form			3528.1.dh.a.181.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.974928 - 0.222521i) q^{2} +(0.900969 - 0.433884i) q^{4} +(-1.21572 - 1.52446i) q^{5} +(0.623490 - 0.781831i) q^{7} +(0.781831 - 0.623490i) q^{8} +O(q^{10})$	$q+(0.974928 - 0.222521i) q^{2} +(0.900969 - 0.433884i) q^{4} +(-1.21572 - 1.52446i) q^{5} +(0.623490 - 0.781831i) q^{7} +(0.781831 - 0.623490i) q^{8} +(-1.52446 - 1.21572i) q^{10} +(-1.75676 + 0.400969i) q^{11} +(0.433884 - 0.900969i) q^{14} +(0.623490 - 0.781831i) q^{16} +(-1.75676 - 0.846011i) q^{20} +(-1.62349 + 0.781831i) q^{22} +(-0.623490 + 2.73169i) q^{25} +(0.222521 - 0.974928i) q^{28} +(0.193096 - 0.400969i) q^{29} -1.56366i q^{31} +(0.433884 - 0.900969i) q^{32} -1.94986 q^{35} +(-1.90097 - 0.433884i) q^{40} +(-1.40881 + 1.12349i) q^{44} +(-0.222521 - 0.974928i) q^{49} +2.80194i q^{50} +(0.867767 + 1.80194i) q^{53} +(2.74698 + 2.19064i) q^{55} -1.00000i q^{56} +(0.0990311 - 0.433884i) q^{58} +(-0.347948 - 1.52446i) q^{62} +(0.222521 - 0.974928i) q^{64} +(-1.90097 + 0.433884i) q^{70} +(-1.52446 - 0.347948i) q^{73} +(-0.781831 + 1.62349i) q^{77} -1.24698 q^{79} -1.94986 q^{80} +(0.433884 - 1.90097i) q^{83} +(-1.12349 + 1.40881i) q^{88} +0.867767i q^{97} +(-0.433884 - 0.900969i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{4} - 2 q^{7}+O(q^{10}) $ 12 * q + 2 * q^4 - 2 * q^7	$ 12 q + 2 q^{4} - 2 q^{7} - 2 q^{16} - 10 q^{22} + 2 q^{25} + 2 q^{28} - 14 q^{40} - 2 q^{49} + 14 q^{55} + 10 q^{58} + 2 q^{64} - 14 q^{70} + 4 q^{79} - 4 q^{88}+O(q^{100}) $ 12 * q + 2 * q^4 - 2 * q^7 - 2 * q^16 - 10 * q^22 + 2 * q^25 + 2 * q^28 - 14 * q^40 - 2 * q^49 + 14 * q^55 + 10 * q^58 + 2 * q^64 - 14 * q^70 + 4 * q^79 - 4 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$1$	$e\left(\frac{11}{14}\right)$	$-1$	$1$

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 $

$2$	0.974928	−	0.222521i	0.974928	−	0.222521i
$2$	0.974928	−	0.222521i	0.974928	−	0.222521i
$3$		0			0
$3$		0			0
$4$	0.900969	−	0.433884i	0.900969	−	0.433884i
$5$	−1.21572	−	1.52446i	−1.21572	−	1.52446i	−0.781831	−	0.623490i	$-0.785714\pi$
$5$	−1.21572	−	1.52446i	−1.21572	−	1.52446i	−0.433884	−	0.900969i	$-0.642857\pi$
$6$		0			0
$7$	0.623490	−	0.781831i	0.623490	−	0.781831i
$7$	0.623490	−	0.781831i	0.623490	−	0.781831i
$8$	0.781831	−	0.623490i	0.781831	−	0.623490i
$9$		0			0
$10$	−1.52446	−	1.21572i	−1.52446	−	1.21572i
$11$	−1.75676	+	0.400969i	−1.75676	+	0.400969i	−0.974928	−	0.222521i	$-0.928571\pi$
$11$	−1.75676	+	0.400969i	−1.75676	+	0.400969i	−0.781831	+	0.623490i	$0.785714\pi$
$12$		0			0
$13$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$13$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$14$	0.433884	−	0.900969i	0.433884	−	0.900969i
$15$		0			0
$16$	0.623490	−	0.781831i	0.623490	−	0.781831i
$17$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$17$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−1.75676	−	0.846011i	−1.75676	−	0.846011i
$21$		0			0
$22$	−1.62349	+	0.781831i	−1.62349	+	0.781831i
$23$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$23$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$24$		0			0
$25$	−0.623490	+	2.73169i	−0.623490	+	2.73169i
$26$		0			0
$27$		0			0
$28$	0.222521	−	0.974928i	0.222521	−	0.974928i
$29$	0.193096	−	0.400969i	0.193096	−	0.400969i	−0.781831	−	0.623490i	$-0.785714\pi$
$29$	0.193096	−	0.400969i	0.193096	−	0.400969i	0.974928	+	0.222521i	$0.0714286\pi$
$30$		0			0
$31$		−	1.56366i		−	1.56366i	−0.623490	−	0.781831i	$-0.714286\pi$
$31$		−	1.56366i		−	1.56366i	0.623490	−	0.781831i	$-0.285714\pi$
$32$	0.433884	−	0.900969i	0.433884	−	0.900969i
$33$		0			0
$34$		0			0
$35$	−1.94986			−1.94986
$36$		0			0
$37$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$37$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$38$		0			0
$39$		0			0
$40$	−1.90097	−	0.433884i	−1.90097	−	0.433884i
$41$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$41$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$42$		0			0
$43$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$43$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$44$	−1.40881	+	1.12349i	−1.40881	+	1.12349i
$45$		0			0
$46$		0			0
$47$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$47$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$48$		0			0
$49$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$50$			2.80194i			2.80194i
$51$		0			0
$52$		0			0
$53$	0.867767	+	1.80194i	0.867767	+	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$53$	0.867767	+	1.80194i	0.867767	+	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$54$		0			0
$55$	2.74698	+	2.19064i	2.74698	+	2.19064i
$56$		−	1.00000i		−	1.00000i
$57$		0			0
$58$	0.0990311	−	0.433884i	0.0990311	−	0.433884i
$59$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$59$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$60$		0			0
$61$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$61$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$62$	−0.347948	−	1.52446i	−0.347948	−	1.52446i
$63$		0			0
$64$	0.222521	−	0.974928i	0.222521	−	0.974928i
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$	−1.90097	+	0.433884i	−1.90097	+	0.433884i
$71$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$71$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$72$		0			0
$73$	−1.52446	−	0.347948i	−1.52446	−	0.347948i	−0.623490	−	0.781831i	$-0.714286\pi$
$73$	−1.52446	−	0.347948i	−1.52446	−	0.347948i	−0.900969	+	0.433884i	$0.857143\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.781831	+	1.62349i	−0.781831	+	1.62349i
$78$		0			0
$79$	−1.24698			−1.24698			−0.623490	−	0.781831i	$-0.714286\pi$
$79$	−1.24698			−1.24698			−0.623490	+	0.781831i	$0.714286\pi$
$80$	−1.94986			−1.94986
$81$		0			0
$82$		0			0
$83$	0.433884	−	1.90097i	0.433884	−	1.90097i		−	1.00000i	$-0.5\pi$
$83$	0.433884	−	1.90097i	0.433884	−	1.90097i	0.433884	−	0.900969i	$-0.357143\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−1.12349	+	1.40881i	−1.12349	+	1.40881i
$89$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$89$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$			0.867767i			0.867767i	0.900969	+	0.433884i	$0.142857\pi$
$97$			0.867767i			0.867767i	−0.900969	+	0.433884i	$0.857143\pi$
$98$	−0.433884	−	0.900969i	−0.433884	−	0.900969i
$99$		0			0
$100$	0.623490	+	2.73169i	0.623490	+	2.73169i
$101$	0.541044	+	0.678448i	0.541044	+	0.678448i	0.974928	−	0.222521i	$-0.0714286\pi$
$101$	0.541044	+	0.678448i	0.541044	+	0.678448i	−0.433884	+	0.900969i	$0.642857\pi$
$102$		0			0
$103$	0.678448	−	0.541044i	0.678448	−	0.541044i	−0.222521	−	0.974928i	$-0.571429\pi$
$103$	0.678448	−	0.541044i	0.678448	−	0.541044i	0.900969	+	0.433884i	$0.142857\pi$
$104$		0			0
$105$		0			0
$106$	1.24698	+	1.56366i	1.24698	+	1.56366i
$107$	1.21572	+	0.277479i	1.21572	+	0.277479i	0.781831	−	0.623490i	$-0.214286\pi$
$107$	1.21572	+	0.277479i	1.21572	+	0.277479i	0.433884	+	0.900969i	$0.357143\pi$
$108$		0			0
$109$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$109$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$110$	3.16557	+	1.52446i	3.16557	+	1.52446i
$111$		0			0
$112$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$113$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$113$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$114$		0			0
$115$		0			0
$116$		−	0.445042i		−	0.445042i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	2.02446	−	0.974928i	2.02446	−	0.974928i
$122$		0			0
$123$		0			0
$124$	−0.678448	−	1.40881i	−0.678448	−	1.40881i
$125$	3.16557	−	1.52446i	3.16557	−	1.52446i
$126$		0			0
$127$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.222521	−	0.974928i	$-0.571429\pi$
$127$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.900969	+	0.433884i	$0.857143\pi$
$128$		−	1.00000i		−	1.00000i
$129$		0			0
$130$		0			0
$131$	0.974928	−	1.22252i	0.974928	−	1.22252i		−	1.00000i	$-0.5\pi$
$131$	0.974928	−	1.22252i	0.974928	−	1.22252i	0.974928	−	0.222521i	$-0.0714286\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$137$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$138$		0			0
$139$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$139$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$140$	−1.75676	+	0.846011i	−1.75676	+	0.846011i
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−0.846011	+	0.193096i	−0.846011	+	0.193096i
$146$	−1.56366			−1.56366
$147$		0			0
$148$		0			0
$149$	1.21572	−	0.277479i	1.21572	−	0.277479i	0.433884	−	0.900969i	$-0.357143\pi$
$149$	1.21572	−	0.277479i	1.21572	−	0.277479i	0.781831	+	0.623490i	$0.214286\pi$
$150$		0			0
$151$	1.80194	−	0.867767i	1.80194	−	0.867767i	0.900969	−	0.433884i	$-0.142857\pi$
$151$	1.80194	−	0.867767i	1.80194	−	0.867767i	0.900969	−	0.433884i	$-0.142857\pi$
$152$		0			0
$153$		0			0
$154$	−0.400969	+	1.75676i	−0.400969	+	1.75676i
$155$	−2.38374	+	1.90097i	−2.38374	+	1.90097i
$156$		0			0
$157$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$157$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$158$	−1.21572	+	0.277479i	−1.21572	+	0.277479i
$159$		0			0
$160$	−1.90097	+	0.433884i	−1.90097	+	0.433884i
$161$		0			0
$162$		0			0
$163$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$163$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$164$		0			0
$165$		0			0
$166$		−	1.94986i		−	1.94986i
$167$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$167$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$168$		0			0
$169$	0.900969	−	0.433884i	0.900969	−	0.433884i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.75676	−	0.846011i	1.75676	−	0.846011i	0.781831	−	0.623490i	$-0.214286\pi$
$173$	1.75676	−	0.846011i	1.75676	−	0.846011i	0.974928	−	0.222521i	$-0.0714286\pi$
$174$		0			0
$175$	1.74698	+	2.19064i	1.74698	+	2.19064i
$176$	−0.781831	+	1.62349i	−0.781831	+	1.62349i
$177$		0			0
$178$		0			0
$179$	−0.193096	+	0.400969i	−0.193096	+	0.400969i	−0.974928	−	0.222521i	$-0.928571\pi$
$179$	−0.193096	+	0.400969i	−0.193096	+	0.400969i	0.781831	+	0.623490i	$0.214286\pi$
$180$		0			0
$181$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$181$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$191$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$192$		0			0
$193$	0.777479	+	0.974928i	0.777479	+	0.974928i	1.00000			$0$
$193$	0.777479	+	0.974928i	0.777479	+	0.974928i	−0.222521	+	0.974928i	$0.571429\pi$
$194$	0.193096	+	0.846011i	0.193096	+	0.846011i
$195$		0			0
$196$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$197$		−	1.80194i		−	1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$197$		−	1.80194i		−	1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$198$		0			0
$199$	−1.22252	+	0.974928i	−1.22252	+	0.974928i	−0.222521	+	0.974928i	$0.571429\pi$
$199$	−1.22252	+	0.974928i	−1.22252	+	0.974928i	−1.00000			$\pi$
$200$	1.21572	+	2.52446i	1.21572	+	2.52446i
$201$		0			0
$202$	0.678448	+	0.541044i	0.678448	+	0.541044i
$203$	−0.193096	−	0.400969i	−0.193096	−	0.400969i
$204$		0			0
$205$		0			0
$206$	0.541044	−	0.678448i	0.541044	−	0.678448i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$211$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$212$	1.56366	+	1.24698i	1.56366	+	1.24698i
$213$		0			0
$214$	1.24698			1.24698
$215$		0			0
$216$		0			0
$217$	−1.22252	−	0.974928i	−1.22252	−	0.974928i
$218$		0			0
$219$		0			0
$220$	3.42543	+	0.781831i	3.42543	+	0.781831i
$221$		0			0
$222$		0			0
$223$	0.376510	+	0.781831i	0.376510	+	0.781831i	1.00000			$0$
$223$	0.376510	+	0.781831i	0.376510	+	0.781831i	−0.623490	+	0.781831i	$0.714286\pi$
$224$	−0.433884	−	0.900969i	−0.433884	−	0.900969i
$225$		0			0
$226$		0			0
$227$	−0.867767			−0.867767			−0.433884	−	0.900969i	$-0.642857\pi$
$227$	−0.867767			−0.867767			−0.433884	+	0.900969i	$0.642857\pi$
$228$		0			0
$229$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$229$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$230$		0			0
$231$		0			0
$232$	−0.0990311	−	0.433884i	−0.0990311	−	0.433884i
$233$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$233$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$239$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$240$		0			0
$241$	0.846011	+	1.75676i	0.846011	+	1.75676i	0.623490	+	0.781831i	$0.285714\pi$
$241$	0.846011	+	1.75676i	0.846011	+	1.75676i	0.222521	+	0.974928i	$0.428571\pi$
$242$	1.75676	−	1.40097i	1.75676	−	1.40097i
$243$		0			0
$244$		0			0
$245$	−1.21572	+	1.52446i	−1.21572	+	1.52446i
$246$		0			0
$247$		0			0
$248$	−0.974928	−	1.22252i	−0.974928	−	1.22252i
$249$		0			0
$250$	2.74698	−	2.19064i	2.74698	−	2.19064i
$251$	−0.974928	+	1.22252i	−0.974928	+	1.22252i			1.00000i	$0.5\pi$
$251$	−0.974928	+	1.22252i	−0.974928	+	1.22252i	−0.974928	+	0.222521i	$0.928571\pi$
$252$		0			0
$253$		0			0
$254$	−1.21572	−	0.277479i	−1.21572	−	0.277479i
$255$		0			0
$256$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$257$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$257$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	0.678448	−	1.40881i	0.678448	−	1.40881i
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$	1.69202	−	3.51352i	1.69202	−	3.51352i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.433884	−	1.90097i	0.433884	−	1.90097i		−	1.00000i	$-0.5\pi$
$269$	0.433884	−	1.90097i	0.433884	−	1.90097i	0.433884	−	0.900969i	$-0.357143\pi$
$270$		0			0
$271$	0.376510	+	0.781831i	0.376510	+	0.781831i	1.00000			$0$
$271$	0.376510	+	0.781831i	0.376510	+	0.781831i	−0.623490	+	0.781831i	$0.714286\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	5.04892i		−	5.04892i
$276$		0			0
$277$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$277$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$278$		0			0
$279$		0			0
$280$	−1.52446	+	1.21572i	−1.52446	+	1.21572i
$281$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$281$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$282$		0			0
$283$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$283$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.623490	+	0.781831i	0.623490	+	0.781831i
$290$	−0.781831	+	0.376510i	−0.781831	+	0.376510i
$291$		0			0
$292$	−1.52446	+	0.347948i	−1.52446	+	0.347948i
$293$	−0.867767			−0.867767			−0.433884	−	0.900969i	$-0.642857\pi$
$293$	−0.867767			−0.867767			−0.433884	+	0.900969i	$0.642857\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$	1.12349	−	0.541044i	1.12349	−	0.541044i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	1.56366	−	1.24698i	1.56366	−	1.24698i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$307$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$308$			1.80194i			1.80194i
$309$		0			0
$310$	−1.90097	+	2.38374i	−1.90097	+	2.38374i
$311$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$311$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$312$		0			0
$313$			1.56366i			1.56366i	0.623490	+	0.781831i	$0.285714\pi$
$313$			1.56366i			1.56366i	−0.623490	+	0.781831i	$0.714286\pi$
$314$		0			0
$315$		0			0
$316$	−1.12349	+	0.541044i	−1.12349	+	0.541044i
$317$	0.781831	+	1.62349i	0.781831	+	1.62349i	0.781831	+	0.623490i	$0.214286\pi$
$317$	0.781831	+	1.62349i	0.781831	+	1.62349i			1.00000i	$0.5\pi$
$318$		0			0
$319$	−0.178448	+	0.781831i	−0.178448	+	0.781831i
$320$	−1.75676	+	0.846011i	−1.75676	+	0.846011i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$331$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$332$	−0.433884	−	1.90097i	−0.433884	−	1.90097i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.24698	−	1.56366i	1.24698	−	1.56366i	0.623490	−	0.781831i	$-0.285714\pi$
$337$	1.24698	−	1.56366i	1.24698	−	1.56366i	0.623490	−	0.781831i	$-0.285714\pi$
$338$	0.781831	−	0.623490i	0.781831	−	0.623490i
$339$		0			0
$340$		0			0
$341$	0.626980	+	2.74698i	0.626980	+	2.74698i
$342$		0			0
$343$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$344$		0			0
$345$		0			0
$346$	1.52446	−	1.21572i	1.52446	−	1.21572i
$347$	−0.193096	−	0.400969i	−0.193096	−	0.400969i	0.781831	−	0.623490i	$-0.214286\pi$
$347$	−0.193096	−	0.400969i	−0.193096	−	0.400969i	−0.974928	+	0.222521i	$0.928571\pi$
$348$		0			0
$349$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$349$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$350$	2.19064	+	1.74698i	2.19064	+	1.74698i
$351$		0			0
$352$	−0.400969	+	1.75676i	−0.400969	+	1.75676i
$353$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$353$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−0.0990311	+	0.433884i	−0.0990311	+	0.433884i
$359$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$359$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$360$		0			0
$361$	−1.00000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.32288	+	2.74698i	1.32288	+	2.74698i
$366$		0			0
$367$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$367$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.94986	+	0.445042i	1.94986	+	0.445042i
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$379$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$383$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$384$		0			0
$385$	3.42543	−	0.781831i	3.42543	−	0.781831i
$386$	0.974928	+	0.777479i	0.974928	+	0.777479i
$387$		0			0
$388$	0.376510	+	0.781831i	0.376510	+	0.781831i
$389$	−0.347948	+	0.277479i	−0.347948	+	0.277479i	−0.781831	−	0.623490i	$-0.785714\pi$
$389$	−0.347948	+	0.277479i	−0.347948	+	0.277479i	0.433884	+	0.900969i	$0.357143\pi$
$390$		0			0
$391$		0			0
$392$	−0.781831	−	0.623490i	−0.781831	−	0.623490i
$393$		0			0
$394$	−0.400969	−	1.75676i	−0.400969	−	1.75676i
$395$	1.51597	+	1.90097i	1.51597	+	1.90097i
$396$		0			0
$397$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$397$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$398$	−0.974928	+	1.22252i	−0.974928	+	1.22252i
$399$		0			0
$400$	1.74698	+	2.19064i	1.74698	+	2.19064i
$401$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$401$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$402$		0			0
$403$		0			0
$404$	0.781831	+	0.376510i	0.781831	+	0.376510i
$405$		0			0
$406$	−0.277479	−	0.347948i	−0.277479	−	0.347948i
$407$		0			0
$408$		0			0
$409$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$409$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$410$		0			0
$411$		0			0
$412$	0.376510	−	0.781831i	0.376510	−	0.781831i
$413$		0			0
$414$		0			0
$415$	−3.42543	+	1.64960i	−3.42543	+	1.64960i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1.75676	+	0.846011i	−1.75676	+	0.846011i	−0.781831	+	0.623490i	$0.785714\pi$
$419$	−1.75676	+	0.846011i	−1.75676	+	0.846011i	−0.974928	+	0.222521i	$0.928571\pi$
$420$		0			0
$421$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$421$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$422$		0			0
$423$		0			0
$424$	1.80194	+	0.867767i	1.80194	+	0.867767i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	1.21572	−	0.277479i	1.21572	−	0.277479i
$429$		0			0
$430$		0			0
$431$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$431$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$432$		0			0
$433$	−1.52446	+	1.21572i	−1.52446	+	1.21572i	−0.623490	+	0.781831i	$0.714286\pi$
$433$	−1.52446	+	1.21572i	−1.52446	+	1.21572i	−0.900969	+	0.433884i	$0.857143\pi$
$434$	−1.40881	−	0.678448i	−1.40881	−	0.678448i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−0.846011	+	0.193096i	−0.846011	+	0.193096i	−0.623490	−	0.781831i	$-0.714286\pi$
$439$	−0.846011	+	0.193096i	−0.846011	+	0.193096i	−0.222521	+	0.974928i	$0.571429\pi$
$440$	3.51352			3.51352
$441$		0			0
$442$		0			0
$443$	−1.75676	+	0.400969i	−1.75676	+	0.400969i	−0.974928	−	0.222521i	$-0.928571\pi$
$443$	−1.75676	+	0.400969i	−1.75676	+	0.400969i	−0.781831	+	0.623490i	$0.785714\pi$
$444$		0			0
$445$		0			0
$446$	0.541044	+	0.678448i	0.541044	+	0.678448i
$447$		0			0
$448$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$449$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$449$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$	−0.846011	+	0.193096i	−0.846011	+	0.193096i
$455$		0			0
$456$		0			0
$457$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	−0.900969	−	0.433884i	$-0.857143\pi$
$457$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	0.623490	+	0.781831i	$0.285714\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.40881	+	0.678448i	1.40881	+	0.678448i	0.974928	−	0.222521i	$-0.0714286\pi$
$461$	1.40881	+	0.678448i	1.40881	+	0.678448i	0.433884	+	0.900969i	$0.357143\pi$
$462$		0			0
$463$	−0.400969	+	0.193096i	−0.400969	+	0.193096i	−0.623490	−	0.781831i	$-0.714286\pi$
$463$	−0.400969	+	0.193096i	−0.400969	+	0.193096i	0.222521	+	0.974928i	$0.428571\pi$
$464$	−0.193096	−	0.400969i	−0.193096	−	0.400969i
$465$		0			0
$466$		0			0
$467$	0.781831	−	0.376510i	0.781831	−	0.376510i		−	1.00000i	$-0.5\pi$
$467$	0.781831	−	0.376510i	0.781831	−	0.376510i	0.781831	+	0.623490i	$0.214286\pi$
$468$		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		−0.222521	−	0.974928i	$-0.571429\pi$
$479$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$480$		0			0
$481$		0			0
$482$	1.21572	+	1.52446i	1.21572	+	1.52446i
$483$		0			0
$484$	1.40097	−	1.75676i	1.40097	−	1.75676i
$485$	1.32288	−	1.05496i	1.32288	−	1.05496i
$486$		0			0
$487$	−0.777479	−	0.974928i	−0.777479	−	0.974928i	0.222521	−	0.974928i	$-0.428571\pi$
$487$	−0.777479	−	0.974928i	−0.777479	−	0.974928i	−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$	−0.846011	+	1.75676i	−0.846011	+	1.75676i
$491$		−	0.445042i		−	0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$491$		−	0.445042i		−	0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.22252	−	0.974928i	−1.22252	−	0.974928i
$497$		0			0
$498$		0			0
$499$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$499$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$500$	2.19064	−	2.74698i	2.19064	−	2.74698i
$501$		0			0
$502$	−0.678448	+	1.40881i	−0.678448	+	1.40881i
$503$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$503$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$504$		0			0
$505$	0.376510	−	1.64960i	0.376510	−	1.64960i
$506$		0			0
$507$		0			0
$508$	−1.24698			−1.24698
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	−1.22252	+	0.974928i	−1.22252	+	0.974928i
$512$	−0.433884	−	0.900969i	−0.433884	−	0.900969i
$513$		0			0
$514$		0			0
$515$	−1.64960	−	0.376510i	−1.64960	−	0.376510i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$523$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$524$	0.347948	−	1.52446i	0.347948	−	1.52446i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$530$	0.867767	−	3.80194i	0.867767	−	3.80194i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−1.05496	−	2.19064i	−1.05496	−	2.19064i
$536$		0			0
$537$		0			0
$538$		−	1.94986i		−	1.94986i
$539$	0.781831	+	1.62349i	0.781831	+	1.62349i
$540$		0			0
$541$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$541$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$542$	0.541044	+	0.678448i	0.541044	+	0.678448i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$547$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$548$		0			0
$549$		0			0
$550$	−1.12349	−	4.92233i	−1.12349	−	4.92233i
$551$		0			0
$552$		0			0
$553$	−0.777479	+	0.974928i	−0.777479	+	0.974928i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			0.445042i			0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$557$			0.445042i			0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$558$		0			0
$559$		0			0
$560$	−1.21572	+	1.52446i	−1.21572	+	1.52446i
$561$		0			0
$562$		0			0
$563$	−0.347948	+	1.52446i	−0.347948	+	1.52446i	0.433884	+	0.900969i	$0.357143\pi$
$563$	−0.347948	+	1.52446i	−0.347948	+	1.52446i	−0.781831	+	0.623490i	$0.785714\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$571$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.52446	−	0.347948i	1.52446	−	0.347948i	0.623490	−	0.781831i	$-0.285714\pi$
$577$	1.52446	−	0.347948i	1.52446	−	0.347948i	0.900969	+	0.433884i	$0.142857\pi$
$578$	0.781831	+	0.623490i	0.781831	+	0.623490i
$579$		0			0
$580$	−0.678448	+	0.541044i	−0.678448	+	0.541044i
$581$	−1.21572	−	1.52446i	−1.21572	−	1.52446i
$582$		0			0
$583$	−2.24698	−	2.81762i	−2.24698	−	2.81762i
$584$	−1.40881	+	0.678448i	−1.40881	+	0.678448i
$585$		0			0
$586$	−0.846011	+	0.193096i	−0.846011	+	0.193096i
$587$	0.867767			0.867767			0.433884	−	0.900969i	$-0.357143\pi$
$587$	0.867767			0.867767			0.433884	+	0.900969i	$0.357143\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$593$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$594$		0			0
$595$		0			0
$596$	0.974928	−	0.777479i	0.974928	−	0.777479i
$597$		0			0
$598$		0			0
$599$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$599$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$600$		0			0
$601$	−1.52446	+	0.347948i	−1.52446	+	0.347948i	−0.900969	−	0.433884i	$-0.857143\pi$
$601$	−1.52446	+	0.347948i	−1.52446	+	0.347948i	−0.623490	+	0.781831i	$0.714286\pi$
$602$		0			0
$603$		0			0
$604$	1.24698	−	1.56366i	1.24698	−	1.56366i
$605$	−3.94740	−	1.90097i	−3.94740	−	1.90097i
$606$		0			0
$607$			0.867767i			0.867767i	0.900969	+	0.433884i	$0.142857\pi$
$607$			0.867767i			0.867767i	−0.900969	+	0.433884i	$0.857143\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$613$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$614$		0			0
$615$		0			0
$616$	0.400969	+	1.75676i	0.400969	+	1.75676i
$617$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$617$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$	−1.32288	+	2.74698i	−1.32288	+	2.74698i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−3.64795	−	1.75676i	−3.64795	−	1.75676i
$626$	0.347948	+	1.52446i	0.347948	+	1.52446i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	1.12349	−	1.40881i	1.12349	−	1.40881i	0.222521	−	0.974928i	$-0.428571\pi$
$631$	1.12349	−	1.40881i	1.12349	−	1.40881i	0.900969	−	0.433884i	$-0.142857\pi$
$632$	−0.974928	+	0.777479i	−0.974928	+	0.777479i
$633$		0			0
$634$	1.12349	+	1.40881i	1.12349	+	1.40881i
$635$	0.541044	+	2.37047i	0.541044	+	2.37047i
$636$		0			0
$637$		0			0
$638$			0.801938i			0.801938i
$639$		0			0
$640$	−1.52446	+	1.21572i	−1.52446	+	1.21572i
$641$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$641$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$642$		0			0
$643$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$643$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$647$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.347948	+	0.277479i	0.347948	+	0.277479i	0.781831	−	0.623490i	$-0.214286\pi$
$653$	0.347948	+	0.277479i	0.347948	+	0.277479i	−0.433884	+	0.900969i	$0.642857\pi$
$654$		0			0
$655$	−3.04892			−3.04892
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.781831	+	1.62349i	0.781831	+	1.62349i	0.781831	+	0.623490i	$0.214286\pi$
$659$	0.781831	+	1.62349i	0.781831	+	1.62349i			1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$661$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$662$		0			0
$663$		0			0
$664$	−0.846011	−	1.75676i	−0.846011	−	1.75676i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.400969	−	1.75676i	−0.400969	−	1.75676i	−0.623490	−	0.781831i	$-0.714286\pi$
$673$	−0.400969	−	1.75676i	−0.400969	−	1.75676i	0.222521	−	0.974928i	$-0.428571\pi$
$674$	0.867767	−	1.80194i	0.867767	−	1.80194i
$675$		0			0
$676$	0.623490	−	0.781831i	0.623490	−	0.781831i
$677$	−0.193096	+	0.846011i	−0.193096	+	0.846011i	0.781831	+	0.623490i	$0.214286\pi$
$677$	−0.193096	+	0.846011i	−0.193096	+	0.846011i	−0.974928	+	0.222521i	$0.928571\pi$
$678$		0			0
$679$	0.678448	+	0.541044i	0.678448	+	0.541044i
$680$		0			0
$681$		0			0
$682$	1.22252	+	2.53859i	1.22252	+	2.53859i
$683$	0.347948	−	0.277479i	0.347948	−	0.277479i	−0.433884	−	0.900969i	$-0.642857\pi$
$683$	0.347948	−	0.277479i	0.347948	−	0.277479i	0.781831	+	0.623490i	$0.214286\pi$
$684$		0			0
$685$		0			0
$686$	−0.974928	−	0.222521i	−0.974928	−	0.222521i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$691$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$692$	1.21572	−	1.52446i	1.21572	−	1.52446i
$693$		0			0
$694$	−0.277479	−	0.347948i	−0.277479	−	0.347948i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	2.52446	+	1.21572i	2.52446	+	1.21572i
$701$	−1.21572	−	0.277479i	−1.21572	−	0.277479i	−0.433884	−	0.900969i	$-0.642857\pi$
$701$	−1.21572	−	0.277479i	−1.21572	−	0.277479i	−0.781831	+	0.623490i	$0.785714\pi$
$702$		0			0
$703$		0			0
$704$			1.80194i			1.80194i
$705$		0			0
$706$		0			0
$707$	0.867767			0.867767
$708$		0			0
$709$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$709$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$			0.445042i			0.445042i
$717$		0			0
$718$		0			0
$719$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$719$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$720$		0			0
$721$		−	0.867767i		−	0.867767i
$722$	−0.974928	+	0.222521i	−0.974928	+	0.222521i
$723$		0			0
$724$		0			0
$725$	0.974928	+	0.777479i	0.974928	+	0.777479i
$726$		0			0
$727$	1.52446	−	1.21572i	1.52446	−	1.21572i	0.623490	−	0.781831i	$-0.285714\pi$
$727$	1.52446	−	1.21572i	1.52446	−	1.21572i	0.900969	−	0.433884i	$-0.142857\pi$
$728$		0			0
$729$		0			0
$730$	1.90097	+	2.38374i	1.90097	+	2.38374i
$731$		0			0
$732$		0			0
$733$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$733$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$739$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$740$		0			0
$741$		0			0
$742$	2.00000			2.00000
$743$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$743$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$744$		0			0
$745$	−1.90097	−	1.51597i	−1.90097	−	1.51597i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	0.974928	−	0.777479i	0.974928	−	0.777479i
$750$		0			0
$751$	1.12349	−	1.40881i	1.12349	−	1.40881i	0.222521	−	0.974928i	$-0.428571\pi$
$751$	1.12349	−	1.40881i	1.12349	−	1.40881i	0.900969	−	0.433884i	$-0.142857\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3.51352	−	1.69202i	−3.51352	−	1.69202i
$756$		0			0
$757$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$757$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$761$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.52446	+	0.347948i	1.52446	+	0.347948i	0.900969	−	0.433884i	$-0.142857\pi$
$769$	1.52446	+	0.347948i	1.52446	+	0.347948i	0.623490	+	0.781831i	$0.285714\pi$
$770$	3.16557	−	1.52446i	3.16557	−	1.52446i
$771$		0			0
$772$	1.12349	+	0.541044i	1.12349	+	0.541044i
$773$	−0.193096	−	0.846011i	−0.193096	−	0.846011i	−0.974928	−	0.222521i	$-0.928571\pi$
$773$	−0.193096	−	0.846011i	−0.193096	−	0.846011i	0.781831	−	0.623490i	$-0.214286\pi$
$774$		0			0
$775$	4.27144	+	0.974928i	4.27144	+	0.974928i
$776$	0.541044	+	0.678448i	0.541044	+	0.678448i
$777$		0			0
$778$	−0.277479	+	0.347948i	−0.277479	+	0.347948i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$785$		0			0
$786$		0			0
$787$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$787$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$788$	−0.781831	−	1.62349i	−0.781831	−	1.62349i
$789$		0			0
$790$	1.90097	+	1.51597i	1.90097	+	1.51597i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−0.678448	+	1.40881i	−0.678448	+	1.40881i
$797$	−0.193096	−	0.846011i	−0.193096	−	0.846011i	−0.974928	−	0.222521i	$-0.928571\pi$
$797$	−0.193096	−	0.846011i	−0.193096	−	0.846011i	0.781831	−	0.623490i	$-0.214286\pi$
$798$		0			0
$799$		0			0
$800$	2.19064	+	1.74698i	2.19064	+	1.74698i
$801$		0			0
$802$		0			0
$803$	2.81762			2.81762
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.846011	+	0.193096i	0.846011	+	0.193096i
$809$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$809$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$810$		0			0
$811$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$811$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$812$	−0.347948	−	0.277479i	−0.347948	−	0.277479i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.541044	+	1.12349i	−0.541044	+	1.12349i	0.433884	+	0.900969i	$0.357143\pi$
$821$	−0.541044	+	1.12349i	−0.541044	+	1.12349i	−0.974928	+	0.222521i	$0.928571\pi$
$822$		0			0
$823$	−0.777479	+	0.974928i	−0.777479	+	0.974928i	0.222521	+	0.974928i	$0.428571\pi$
$823$	−0.777479	+	0.974928i	−0.777479	+	0.974928i	−1.00000			$\pi$
$824$	0.193096	−	0.846011i	0.193096	−	0.846011i
$825$		0			0
$826$		0			0
$827$	0.347948	+	0.277479i	0.347948	+	0.277479i	0.781831	−	0.623490i	$-0.214286\pi$
$827$	0.347948	+	0.277479i	0.347948	+	0.277479i	−0.433884	+	0.900969i	$0.642857\pi$
$828$		0			0
$829$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$829$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$830$	−2.97247	+	2.37047i	−2.97247	+	2.37047i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−1.52446	+	1.21572i	−1.52446	+	1.21572i
$839$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$839$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$840$		0			0
$841$	0.500000	+	0.626980i	0.500000	+	0.626980i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.75676	−	0.846011i	−1.75676	−	0.846011i
$846$		0			0
$847$	0.500000	−	2.19064i	0.500000	−	2.19064i
$848$	1.94986	+	0.445042i	1.94986	+	0.445042i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$853$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$854$		0			0
$855$		0			0
$856$	1.12349	−	0.541044i	1.12349	−	0.541044i
$857$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$857$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$858$		0			0
$859$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$859$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	−3.42543	−	1.64960i	−3.42543	−	1.64960i
$866$	−1.21572	+	1.52446i	−1.21572	+	1.52446i
$867$		0			0
$868$	−1.52446	−	0.347948i	−1.52446	−	0.347948i
$869$	2.19064	−	0.500000i	2.19064	−	0.500000i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.781831	−	3.42543i	0.781831	−	3.42543i
$876$		0			0
$877$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$877$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$878$	−0.781831	+	0.376510i	−0.781831	+	0.376510i
$879$		0			0
$880$	3.42543	−	0.781831i	3.42543	−	0.781831i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	−1.62349	+	0.781831i	−1.62349	+	0.781831i
$887$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$887$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$888$		0			0
$889$	−1.12349	+	0.541044i	−1.12349	+	0.541044i
$890$		0			0
$891$		0			0
$892$	0.678448	+	0.541044i	0.678448	+	0.541044i
$893$		0			0
$894$		0			0
$895$	0.846011	−	0.193096i	0.846011	−	0.193096i
$896$	−0.781831	−	0.623490i	−0.781831	−	0.623490i
$897$		0			0
$898$		0			0
$899$	−0.626980	−	0.301938i	−0.626980	−	0.301938i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$907$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$908$	−0.781831	+	0.376510i	−0.781831	+	0.376510i
$909$		0			0
$910$		0			0
$911$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$911$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$912$		0			0
$913$			3.51352i			3.51352i
$914$	−0.193096	+	0.400969i	−0.193096	+	0.400969i
$915$		0			0
$916$		0			0
$917$	−0.347948	−	1.52446i	−0.347948	−	1.52446i
$918$		0			0
$919$	1.62349	+	0.781831i	1.62349	+	0.781831i	1.00000			$0$
$919$	1.62349	+	0.781831i	1.62349	+	0.781831i	0.623490	+	0.781831i	$0.285714\pi$
$920$		0			0
$921$		0			0
$922$	1.52446	+	0.347948i	1.52446	+	0.347948i
$923$		0			0
$924$		0			0
$925$		0			0
$926$	−0.347948	+	0.277479i	−0.347948	+	0.277479i
$927$		0			0
$928$	−0.277479	−	0.347948i	−0.277479	−	0.347948i
$929$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$929$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	0.678448	−	0.541044i	0.678448	−	0.541044i
$935$		0			0
$936$		0			0
$937$	1.22252	+	0.974928i	1.22252	+	0.974928i	1.00000			$0$
$937$	1.22252	+	0.974928i	1.22252	+	0.974928i	0.222521	+	0.974928i	$0.428571\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.541044	+	0.678448i	−0.541044	+	0.678448i	−0.974928	−	0.222521i	$-0.928571\pi$
$941$	−0.541044	+	0.678448i	−0.541044	+	0.678448i	0.433884	+	0.900969i	$0.357143\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.974928	+	0.777479i	0.974928	+	0.777479i	0.974928	−	0.222521i	$-0.0714286\pi$
$947$	0.974928	+	0.777479i	0.974928	+	0.777479i			1.00000i	$0.5\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$953$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1.44504			−1.44504
$962$		0			0
$963$		0			0
$964$	1.52446	+	1.21572i	1.52446	+	1.21572i
$965$	0.541044	−	2.37047i	0.541044	−	2.37047i
$966$		0			0
$967$	−0.277479	−	1.21572i	−0.277479	−	1.21572i	−0.900969	−	0.433884i	$-0.857143\pi$
$967$	−0.277479	−	1.21572i	−0.277479	−	1.21572i	0.623490	−	0.781831i	$-0.285714\pi$
$968$	0.974928	−	2.02446i	0.974928	−	2.02446i
$969$		0			0
$970$	1.05496	−	1.32288i	1.05496	−	1.32288i
$971$	−0.347948	+	1.52446i	−0.347948	+	1.52446i	0.433884	+	0.900969i	$0.357143\pi$
$971$	−0.347948	+	1.52446i	−0.347948	+	1.52446i	−0.781831	+	0.623490i	$0.785714\pi$
$972$		0			0
$973$		0			0
$974$	−0.974928	−	0.777479i	−0.974928	−	0.777479i
$975$		0			0
$976$		0			0
$977$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$977$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$978$		0			0
$979$		0			0
$980$	−0.433884	+	1.90097i	−0.433884	+	1.90097i
$981$		0			0
$982$	−0.0990311	−	0.433884i	−0.0990311	−	0.433884i
$983$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$983$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$984$		0			0
$985$	−2.74698	+	2.19064i	−2.74698	+	2.19064i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.400969	+	1.75676i	0.400969	+	1.75676i	0.623490	+	0.781831i	$0.285714\pi$
$991$	0.400969	+	1.75676i	0.400969	+	1.75676i	−0.222521	+	0.974928i	$0.571429\pi$
$992$	−1.40881	−	0.678448i	−1.40881	−	0.678448i
$993$		0			0
$994$		0			0
$995$	2.97247	+	0.678448i	2.97247	+	0.678448i
$996$		0			0
$997$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$997$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.1.dh.a.1189.2	yes	12
3.2	odd	2	inner	3528.1.dh.a.1189.1	yes	12
8.5	even	2	inner	3528.1.dh.a.1189.1	yes	12
24.5	odd	2	CM	3528.1.dh.a.1189.2	yes	12
49.34	odd	14	inner	3528.1.dh.a.181.1	✓	12
147.83	even	14	inner	3528.1.dh.a.181.2	yes	12
392.181	odd	14	inner	3528.1.dh.a.181.2	yes	12
1176.965	even	14	inner	3528.1.dh.a.181.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3528.1.dh.a.181.1	✓	12	49.34	odd	14	inner
3528.1.dh.a.181.1	✓	12	1176.965	even	14	inner
3528.1.dh.a.181.2	yes	12	147.83	even	14	inner
3528.1.dh.a.181.2	yes	12	392.181	odd	14	inner
3528.1.dh.a.1189.1	yes	12	3.2	odd	2	inner
3528.1.dh.a.1189.1	yes	12	8.5	even	2	inner
3528.1.dh.a.1189.2	yes	12	1.1	even	1	trivial
3528.1.dh.a.1189.2	yes	12	24.5	odd	2	CM

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:	\( N \)	\(=\)	\( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3528.dh (of order \(14\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(0.974928 - 0.222521i) q^{2} +(0.900969 - 0.433884i) q^{4} +(-1.21572 - 1.52446i) q^{5} +(0.623490 - 0.781831i) q^{7} +(0.781831 - 0.623490i) q^{8} +O(q^{10})\)	\(q+(0.974928 - 0.222521i) q^{2} +(0.900969 - 0.433884i) q^{4} +(-1.21572 - 1.52446i) q^{5} +(0.623490 - 0.781831i) q^{7} +(0.781831 - 0.623490i) q^{8} +(-1.52446 - 1.21572i) q^{10} +(-1.75676 + 0.400969i) q^{11} +(0.433884 - 0.900969i) q^{14} +(0.623490 - 0.781831i) q^{16} +(-1.75676 - 0.846011i) q^{20} +(-1.62349 + 0.781831i) q^{22} +(-0.623490 + 2.73169i) q^{25} +(0.222521 - 0.974928i) q^{28} +(0.193096 - 0.400969i) q^{29} -1.56366i q^{31} +(0.433884 - 0.900969i) q^{32} -1.94986 q^{35} +(-1.90097 - 0.433884i) q^{40} +(-1.40881 + 1.12349i) q^{44} +(-0.222521 - 0.974928i) q^{49} +2.80194i q^{50} +(0.867767 + 1.80194i) q^{53} +(2.74698 + 2.19064i) q^{55} -1.00000i q^{56} +(0.0990311 - 0.433884i) q^{58} +(-0.347948 - 1.52446i) q^{62} +(0.222521 - 0.974928i) q^{64} +(-1.90097 + 0.433884i) q^{70} +(-1.52446 - 0.347948i) q^{73} +(-0.781831 + 1.62349i) q^{77} -1.24698 q^{79} -1.94986 q^{80} +(0.433884 - 1.90097i) q^{83} +(-1.12349 + 1.40881i) q^{88} +0.867767i q^{97} +(-0.433884 - 0.900969i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{4} - 2 q^{7}+O(q^{10}) \) 12 * q + 2 * q^4 - 2 * q^7	\( 12 q + 2 q^{4} - 2 q^{7} - 2 q^{16} - 10 q^{22} + 2 q^{25} + 2 q^{28} - 14 q^{40} - 2 q^{49} + 14 q^{55} + 10 q^{58} + 2 q^{64} - 14 q^{70} + 4 q^{79} - 4 q^{88}+O(q^{100}) \) 12 * q + 2 * q^4 - 2 * q^7 - 2 * q^16 - 10 * q^22 + 2 * q^25 + 2 * q^28 - 14 * q^40 - 2 * q^49 + 14 * q^55 + 10 * q^58 + 2 * q^64 - 14 * q^70 + 4 * q^79 - 4 * q^88

Introduction

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Database

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