LMFDB - Embedded newform 693.1.bp.a.62.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,1,Mod(62,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 5, 7]))

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

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

chi := DirichletCharacter("693.62");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	693.bp (of order $10$, 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:	$0.345852053755$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{40})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{20}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{20} - \cdots)$

Embedding invariants

Embedding label			62.2
Root			$0.156434 - 0.987688i$ of defining polynomial
Character	$\chi$	$=$	693.62
Dual form			693.1.bp.a.503.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.734572 - 0.533698i) q^{2} +(-0.0542543 - 0.166977i) q^{4} +(0.951057 - 0.309017i) q^{7} +(-0.329843 + 1.01515i) q^{8} +O(q^{10})$	$q+(-0.734572 - 0.533698i) q^{2} +(-0.0542543 - 0.166977i) q^{4} +(0.951057 - 0.309017i) q^{7} +(-0.329843 + 1.01515i) q^{8} +(0.987688 + 0.156434i) q^{11} +(-0.863541 - 0.280582i) q^{14} +(0.642040 - 0.466469i) q^{16} +(-0.642040 - 0.642040i) q^{22} -1.78201i q^{23} +(-0.309017 + 0.951057i) q^{25} +(-0.103198 - 0.142040i) q^{28} +(-0.0966818 - 0.297556i) q^{29} +0.346818 q^{32} +(-0.587785 - 1.80902i) q^{37} +1.61803i q^{43} +(-0.0274653 - 0.173409i) q^{44} +(-0.951057 + 1.30902i) q^{46} +(0.809017 - 0.587785i) q^{49} +(0.734572 - 0.533698i) q^{50} +(0.183900 - 0.253116i) q^{53} +1.06740i q^{56} +(-0.0877853 + 0.270175i) q^{58} +(-0.896802 - 0.651565i) q^{64} -1.61803 q^{67} +(1.16110 + 1.59811i) q^{71} +(-0.533698 + 1.64255i) q^{74} +(0.987688 - 0.156434i) q^{77} +(-0.690983 + 0.951057i) q^{79} +(0.863541 - 1.18856i) q^{86} +(-0.484587 + 0.951057i) q^{88} +(-0.297556 + 0.0966818i) q^{92} -0.907981 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{4}+O(q^{10}) $ 16 * q - 4 * q^4	$ 16 q - 4 q^{4} + 4 q^{16} - 4 q^{22} + 4 q^{25} - 20 q^{28} + 4 q^{49} + 8 q^{58} + 4 q^{64} - 8 q^{67} - 20 q^{79} + 20 q^{88}+O(q^{100}) $ 16 * q - 4 * q^4 + 4 * q^16 - 4 * q^22 + 4 * q^25 - 20 * q^28 + 4 * q^49 + 8 * q^58 + 4 * q^64 - 8 * q^67 - 20 * q^79 + 20 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/693\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$442$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{7}{10}\right)$

Coefficient data

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

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

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

$2$	−0.734572	−	0.533698i	−0.734572	−	0.533698i	0.156434	−	0.987688i	$-0.450000\pi$
$2$	−0.734572	−	0.533698i	−0.734572	−	0.533698i	−0.891007	+	0.453990i	$0.850000\pi$
$3$		0			0
$3$		0			0
$4$	−0.0542543	−	0.166977i	−0.0542543	−	0.166977i
$5$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$5$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$6$		0			0
$7$	0.951057	−	0.309017i	0.951057	−	0.309017i
$7$	0.951057	−	0.309017i	0.951057	−	0.309017i
$8$	−0.329843	+	1.01515i	−0.329843	+	1.01515i
$9$		0			0
$10$		0			0
$11$	0.987688	+	0.156434i	0.987688	+	0.156434i
$11$	0.987688	+	0.156434i	0.987688	+	0.156434i
$12$		0			0
$13$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$13$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$14$	−0.863541	−	0.280582i	−0.863541	−	0.280582i
$15$		0			0
$16$	0.642040	−	0.466469i	0.642040	−	0.466469i
$17$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$17$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$18$		0			0
$19$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$19$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$20$		0			0
$21$		0			0
$22$	−0.642040	−	0.642040i	−0.642040	−	0.642040i
$23$		−	1.78201i		−	1.78201i	−0.453990	−	0.891007i	$-0.650000\pi$
$23$		−	1.78201i		−	1.78201i	0.453990	−	0.891007i	$-0.350000\pi$
$24$		0			0
$25$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$26$		0			0
$27$		0			0
$28$	−0.103198	−	0.142040i	−0.103198	−	0.142040i
$29$	−0.0966818	−	0.297556i	−0.0966818	−	0.297556i	0.891007	−	0.453990i	$-0.150000\pi$
$29$	−0.0966818	−	0.297556i	−0.0966818	−	0.297556i	−0.987688	+	0.156434i	$0.950000\pi$
$30$		0			0
$31$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$31$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$32$	0.346818			0.346818
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.587785	−	1.80902i	−0.587785	−	1.80902i	−0.587785	−	0.809017i	$-0.700000\pi$
$37$	−0.587785	−	1.80902i	−0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$41$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$42$		0			0
$43$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$43$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$44$	−0.0274653	−	0.173409i	−0.0274653	−	0.173409i
$45$		0			0
$46$	−0.951057	+	1.30902i	−0.951057	+	1.30902i
$47$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$47$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$48$		0			0
$49$	0.809017	−	0.587785i	0.809017	−	0.587785i
$50$	0.734572	−	0.533698i	0.734572	−	0.533698i
$51$		0			0
$52$		0			0
$53$	0.183900	−	0.253116i	0.183900	−	0.253116i	−0.707107	−	0.707107i	$-0.750000\pi$
$53$	0.183900	−	0.253116i	0.183900	−	0.253116i	0.891007	+	0.453990i	$0.150000\pi$
$54$		0			0
$55$		0			0
$56$			1.06740i			1.06740i
$57$		0			0
$58$	−0.0877853	+	0.270175i	−0.0877853	+	0.270175i
$59$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$59$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$60$		0			0
$61$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$61$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$62$		0			0
$63$		0			0
$64$	−0.896802	−	0.651565i	−0.896802	−	0.651565i
$65$		0			0
$66$		0			0
$67$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$67$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.16110	+	1.59811i	1.16110	+	1.59811i	0.707107	+	0.707107i	$0.250000\pi$
$71$	1.16110	+	1.59811i	1.16110	+	1.59811i	0.453990	+	0.891007i	$0.350000\pi$
$72$		0			0
$73$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$73$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$74$	−0.533698	+	1.64255i	−0.533698	+	1.64255i
$75$		0			0
$76$		0			0
$77$	0.987688	−	0.156434i	0.987688	−	0.156434i
$78$		0			0
$79$	−0.690983	+	0.951057i	−0.690983	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$79$	−0.690983	+	0.951057i	−0.690983	+	0.951057i	−1.00000			$1.00000\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$83$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$84$		0			0
$85$		0			0
$86$	0.863541	−	1.18856i	0.863541	−	1.18856i
$87$		0			0
$88$	−0.484587	+	0.951057i	−0.484587	+	0.951057i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	−0.297556	+	0.0966818i	−0.297556	+	0.0966818i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$97$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$98$	−0.907981			−0.907981
$99$		0			0
$100$	0.175571			0.175571
$101$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$101$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$102$		0			0
$103$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$103$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$104$		0			0
$105$		0			0
$106$	−0.270175	+	0.0877853i	−0.270175	+	0.0877853i
$107$	−0.550672	+	1.69480i	−0.550672	+	1.69480i	0.156434	+	0.987688i	$0.450000\pi$
$107$	−0.550672	+	1.69480i	−0.550672	+	1.69480i	−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$109$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$110$		0			0
$111$		0			0
$112$	0.466469	−	0.642040i	0.466469	−	0.642040i
$113$	−1.87869	−	0.610425i	−1.87869	−	0.610425i	−0.987688	−	0.156434i	$-0.950000\pi$
$113$	−1.87869	−	0.610425i	−1.87869	−	0.610425i	−0.891007	−	0.453990i	$-0.850000\pi$
$114$		0			0
$115$		0			0
$116$	−0.0444398	+	0.0322874i	−0.0444398	+	0.0322874i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.951057	+	0.309017i	0.951057	+	0.309017i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	1.11803	+	1.53884i	1.11803	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$127$	1.11803	+	1.53884i	1.11803	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$128$	0.203854	+	0.627399i	0.203854	+	0.627399i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$	1.18856	+	0.863541i	1.18856	+	0.863541i
$135$		0			0
$136$		0			0
$137$	0.533698	+	0.734572i	0.533698	+	0.734572i	0.987688	−	0.156434i	$-0.0500000\pi$
$137$	0.533698	+	0.734572i	0.533698	+	0.734572i	−0.453990	+	0.891007i	$0.650000\pi$
$138$		0			0
$139$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$139$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$140$		0			0
$141$		0			0
$142$		−	1.79360i		−	1.79360i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−0.270175	+	0.196294i	−0.270175	+	0.196294i
$149$	−1.14412	+	0.831254i	−1.14412	+	0.831254i	−0.987688	−	0.156434i	$-0.950000\pi$
$149$	−1.14412	+	0.831254i	−1.14412	+	0.831254i	−0.156434	+	0.987688i	$0.550000\pi$
$150$		0			0
$151$	0.587785	+	0.190983i	0.587785	+	0.190983i	0.587785	−	0.809017i	$-0.300000\pi$
$151$	0.587785	+	0.190983i	0.587785	+	0.190983i			1.00000i	$0.5\pi$
$152$		0			0
$153$		0			0
$154$	−0.809017	−	0.412215i	−0.809017	−	0.412215i
$155$		0			0
$156$		0			0
$157$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$157$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$158$	1.01515	−	0.329843i	1.01515	−	0.329843i
$159$		0			0
$160$		0			0
$161$	−0.550672	−	1.69480i	−0.550672	−	1.69480i
$162$		0			0
$163$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$163$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$167$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$168$		0			0
$169$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$170$		0			0
$171$		0			0
$172$	0.270175	−	0.0877853i	0.270175	−	0.0877853i
$173$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$173$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$174$		0			0
$175$			1.00000i			1.00000i
$176$	0.707107	−	0.360289i	0.707107	−	0.360289i
$177$		0			0
$178$		0			0
$179$	0.863541	+	0.280582i	0.863541	+	0.280582i	0.707107	−	0.707107i	$-0.250000\pi$
$179$	0.863541	+	0.280582i	0.863541	+	0.280582i	0.156434	+	0.987688i	$0.450000\pi$
$180$		0			0
$181$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$181$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$182$		0			0
$183$		0			0
$184$	1.80902	+	0.587785i	1.80902	+	0.587785i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.297556	−	0.0966818i	0.297556	−	0.0966818i	−0.156434	−	0.987688i	$-0.550000\pi$
$191$	0.297556	−	0.0966818i	0.297556	−	0.0966818i	0.453990	+	0.891007i	$0.350000\pi$
$192$		0			0
$193$	−0.951057	−	1.30902i	−0.951057	−	1.30902i	−0.951057	−	0.309017i	$-0.900000\pi$
$193$	−0.951057	−	1.30902i	−0.951057	−	1.30902i		−	1.00000i	$-0.5\pi$
$194$		0			0
$195$		0			0
$196$	−0.142040	−	0.103198i	−0.142040	−	0.103198i
$197$	−1.78201			−1.78201			−0.891007	−	0.453990i	$-0.850000\pi$
$197$	−1.78201			−1.78201			−0.891007	+	0.453990i	$0.850000\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−0.863541	−	0.627399i	−0.863541	−	0.627399i
$201$		0			0
$202$		0			0
$203$	−0.183900	−	0.253116i	−0.183900	−	0.253116i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.363271	+	0.500000i	−0.363271	+	0.500000i	−0.951057	−	0.309017i	$-0.900000\pi$
$211$	−0.363271	+	0.500000i	−0.363271	+	0.500000i	0.587785	+	0.809017i	$0.300000\pi$
$212$	−0.0522421	−	0.0169745i	−0.0522421	−	0.0169745i
$213$		0			0
$214$	1.30902	−	0.951057i	1.30902	−	0.951057i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.329843	−	0.453990i	0.329843	−	0.453990i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$223$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$224$	0.329843	−	0.107173i	0.329843	−	0.107173i
$225$		0			0
$226$	1.05425	+	1.45106i	1.05425	+	1.45106i
$227$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$227$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$228$		0			0
$229$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$229$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$230$		0			0
$231$		0			0
$232$	0.333955			0.333955
$233$	−1.14412	−	0.831254i	−1.14412	−	0.831254i	−0.156434	−	0.987688i	$-0.550000\pi$
$233$	−1.14412	−	0.831254i	−1.14412	−	0.831254i	−0.987688	+	0.156434i	$0.950000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.610425	−	1.87869i	0.610425	−	1.87869i	0.156434	−	0.987688i	$-0.450000\pi$
$239$	0.610425	−	1.87869i	0.610425	−	1.87869i	0.453990	−	0.891007i	$-0.350000\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	−0.533698	−	0.734572i	−0.533698	−	0.734572i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$251$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$252$		0			0
$253$	0.278768	−	1.76007i	0.278768	−	1.76007i
$254$		−	1.72708i		−	1.72708i
$255$		0			0
$256$	−0.157452	+	0.484587i	−0.157452	+	0.484587i
$257$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$257$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$258$		0			0
$259$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.312869			−0.312869			−0.156434	−	0.987688i	$-0.550000\pi$
$263$	−0.312869			−0.312869			−0.156434	+	0.987688i	$0.550000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.0877853	+	0.270175i	0.0877853	+	0.270175i
$269$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$269$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$270$		0			0
$271$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$271$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$272$		0			0
$273$		0			0
$274$		−	0.824429i		−	0.824429i
$275$	−0.453990	+	0.891007i	−0.453990	+	0.891007i
$276$		0			0
$277$	1.11803	−	1.53884i	1.11803	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$277$	1.11803	−	1.53884i	1.11803	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.59811	−	1.16110i	1.59811	−	1.16110i	0.707107	−	0.707107i	$-0.250000\pi$
$281$	1.59811	−	1.16110i	1.59811	−	1.16110i	0.891007	−	0.453990i	$-0.150000\pi$
$282$		0			0
$283$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$283$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$284$	0.203854	−	0.280582i	0.203854	−	0.280582i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.309017	−	0.951057i	0.309017	−	0.951057i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$293$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$294$		0			0
$295$		0			0
$296$	2.03031			2.03031
$297$		0			0
$298$	1.28408			1.28408
$299$		0			0
$300$		0			0
$301$	0.500000	+	1.53884i	0.500000	+	1.53884i
$302$	−0.329843	−	0.453990i	−0.329843	−	0.453990i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	−0.0797073	−	0.156434i	−0.0797073	−	0.156434i
$309$		0			0
$310$		0			0
$311$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$311$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$312$		0			0
$313$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$313$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$314$		0			0
$315$		0			0
$316$	0.196294	+	0.0637797i	0.196294	+	0.0637797i
$317$	−1.16110	+	1.59811i	−1.16110	+	1.59811i	−0.453990	+	0.891007i	$0.650000\pi$
$317$	−1.16110	+	1.59811i	−1.16110	+	1.59811i	−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$	−0.0489435	−	0.309017i	−0.0489435	−	0.309017i
$320$		0			0
$321$		0			0
$322$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$323$		0			0
$324$		0			0
$325$		0			0
$326$	0.173409	+	0.533698i	0.173409	+	0.533698i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$331$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.587785	+	0.190983i	−0.587785	+	0.190983i	−0.587785	−	0.809017i	$-0.700000\pi$
$337$	−0.587785	+	0.190983i	−0.587785	+	0.190983i			1.00000i	$0.5\pi$
$338$	−0.280582	+	0.863541i	−0.280582	+	0.863541i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.587785	−	0.809017i	0.587785	−	0.809017i
$344$	−1.64255	−	0.533698i	−1.64255	−	0.533698i
$345$		0			0
$346$		0			0
$347$	−1.59811	+	1.16110i	−1.59811	+	1.16110i	−0.707107	+	0.707107i	$0.750000\pi$
$347$	−1.59811	+	1.16110i	−1.59811	+	1.16110i	−0.891007	+	0.453990i	$0.850000\pi$
$348$		0			0
$349$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$349$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$350$	0.533698	−	0.734572i	0.533698	−	0.734572i
$351$		0			0
$352$	0.342548	+	0.0542543i	0.342548	+	0.0542543i
$353$		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
$358$	−0.484587	−	0.666977i	−0.484587	−	0.666977i
$359$	0.550672	+	1.69480i	0.550672	+	1.69480i	0.707107	+	0.707107i	$0.250000\pi$
$359$	0.550672	+	1.69480i	0.550672	+	1.69480i	−0.156434	+	0.987688i	$0.550000\pi$
$360$		0			0
$361$	0.809017	+	0.587785i	0.809017	+	0.587785i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$367$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$368$	−0.831254	−	1.14412i	−0.831254	−	1.14412i
$369$		0			0
$370$		0			0
$371$	0.0966818	−	0.297556i	0.0966818	−	0.297556i
$372$		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
$379$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.587785	+	0.809017i	$0.700000\pi$
$379$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$380$		0			0
$381$		0			0
$382$	−0.270175	−	0.0877853i	−0.270175	−	0.0877853i
$383$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$383$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$384$		0			0
$385$		0			0
$386$			1.46914i			1.46914i
$387$		0			0
$388$		0			0
$389$	0.863541	−	0.280582i	0.863541	−	0.280582i	0.156434	−	0.987688i	$-0.450000\pi$
$389$	0.863541	−	0.280582i	0.863541	−	0.280582i	0.707107	+	0.707107i	$0.250000\pi$
$390$		0			0
$391$		0			0
$392$	0.329843	+	1.01515i	0.329843	+	1.01515i
$393$		0			0
$394$	1.30902	+	0.951057i	1.30902	+	0.951057i
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	0.245237	+	0.754763i	0.245237	+	0.754763i
$401$	1.04744	+	1.44168i	1.04744	+	1.44168i	0.891007	+	0.453990i	$0.150000\pi$
$401$	1.04744	+	1.44168i	1.04744	+	1.44168i	0.156434	+	0.987688i	$0.450000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$			0.284079i			0.284079i
$407$	−0.297556	−	1.87869i	−0.297556	−	1.87869i
$408$		0			0
$409$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$409$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$421$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$422$	0.533698	−	0.173409i	0.533698	−	0.173409i
$423$		0			0
$424$	0.196294	+	0.270175i	0.196294	+	0.270175i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	0.312869			0.312869
$429$		0			0
$430$		0			0
$431$	−1.44168	−	1.04744i	−1.44168	−	1.04744i	−0.987688	−	0.156434i	$-0.950000\pi$
$431$	−1.44168	−	1.04744i	−1.44168	−	1.04744i	−0.453990	−	0.891007i	$-0.650000\pi$
$432$		0			0
$433$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$433$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$434$		0			0
$435$		0			0
$436$	0.103198	−	0.0335310i	0.103198	−	0.0335310i
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.87869	+	0.610425i	1.87869	+	0.610425i	0.987688	+	0.156434i	$0.0500000\pi$
$443$	1.87869	+	0.610425i	1.87869	+	0.610425i	0.891007	+	0.453990i	$0.150000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−1.05425	−	0.342548i	−1.05425	−	0.342548i
$449$	−1.04744	+	1.44168i	−1.04744	+	1.44168i	−0.156434	+	0.987688i	$0.550000\pi$
$449$	−1.04744	+	1.44168i	−1.04744	+	1.44168i	−0.891007	+	0.453990i	$0.850000\pi$
$450$		0			0
$451$		0			0
$452$			0.346818i			0.346818i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.363271	−	0.500000i	−0.363271	−	0.500000i	0.587785	−	0.809017i	$-0.300000\pi$
$457$	−0.363271	−	0.500000i	−0.363271	−	0.500000i	−0.951057	+	0.309017i	$0.900000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$463$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$464$	−0.200874	−	0.145944i	−0.200874	−	0.145944i
$465$		0			0
$466$	0.396802	+	1.22123i	0.396802	+	1.22123i
$467$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$467$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$468$		0			0
$469$	−1.53884	+	0.500000i	−1.53884	+	0.500000i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.253116	+	1.59811i	−0.253116	+	1.59811i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−1.45106	+	1.05425i	−1.45106	+	1.05425i
$479$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$479$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		−	0.175571i		−	0.175571i
$485$		0			0
$486$		0			0
$487$	0.363271	−	1.11803i	0.363271	−	1.11803i	−0.587785	−	0.809017i	$-0.700000\pi$
$487$	0.363271	−	1.11803i	0.363271	−	1.11803i	0.951057	−	0.309017i	$-0.100000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.437016	+	1.34500i	0.437016	+	1.34500i	0.891007	+	0.453990i	$0.150000\pi$
$491$	0.437016	+	1.34500i	0.437016	+	1.34500i	−0.453990	+	0.891007i	$0.650000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.59811	+	1.16110i	1.59811	+	1.16110i
$498$		0			0
$499$	0.363271	+	1.11803i	0.363271	+	1.11803i	0.951057	+	0.309017i	$0.100000\pi$
$499$	0.363271	+	1.11803i	0.363271	+	1.11803i	−0.587785	+	0.809017i	$0.700000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$503$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$504$		0			0
$505$		0			0
$506$	−1.14412	+	1.14412i	−1.14412	+	1.14412i
$507$		0			0
$508$	0.196294	−	0.270175i	0.196294	−	0.270175i
$509$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$509$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$510$		0			0
$511$		0			0
$512$	0.907981	−	0.659687i	0.907981	−	0.659687i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$			1.72708i			1.72708i
$519$		0			0
$520$		0			0
$521$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$521$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$522$		0			0
$523$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$523$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$524$		0			0
$525$		0			0
$526$	0.229825	+	0.166977i	0.229825	+	0.166977i
$527$		0			0
$528$		0			0
$529$	−2.17557			−2.17557
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	0.533698	−	1.64255i	0.533698	−	1.64255i
$537$		0			0
$538$		0			0
$539$	0.891007	−	0.453990i	0.891007	−	0.453990i
$540$		0			0
$541$	−1.11803	+	1.53884i	−1.11803	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$541$	−1.11803	+	1.53884i	−1.11803	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$547$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$548$	0.0937016	−	0.128969i	0.0937016	−	0.128969i
$549$		0			0
$550$	0.809017	−	0.412215i	0.809017	−	0.412215i
$551$		0			0
$552$		0			0
$553$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$554$	−1.64255	+	0.533698i	−1.64255	+	0.533698i
$555$		0			0
$556$		0			0
$557$	−0.610425	−	1.87869i	−0.610425	−	1.87869i	−0.453990	−	0.891007i	$-0.650000\pi$
$557$	−0.610425	−	1.87869i	−0.610425	−	1.87869i	−0.156434	−	0.987688i	$-0.550000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−1.79360			−1.79360
$563$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$563$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−2.00531	+	0.651565i	−2.00531	+	0.651565i
$569$	0.437016	−	1.34500i	0.437016	−	1.34500i	−0.453990	−	0.891007i	$-0.650000\pi$
$569$	0.437016	−	1.34500i	0.437016	−	1.34500i	0.891007	−	0.453990i	$-0.150000\pi$
$570$		0			0
$571$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$571$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.69480	+	0.550672i	1.69480	+	0.550672i
$576$		0			0
$577$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$577$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$578$	−0.734572	+	0.533698i	−0.734572	+	0.533698i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	0.221232	−	0.221232i	0.221232	−	0.221232i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$587$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.22123	−	0.887277i	−1.22123	−	0.887277i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$	0.200874	+	0.145944i	0.200874	+	0.145944i
$597$		0			0
$598$		0			0
$599$	−0.533698	−	0.734572i	−0.533698	−	0.734572i	0.453990	−	0.891007i	$-0.350000\pi$
$599$	−0.533698	−	0.734572i	−0.533698	−	0.734572i	−0.987688	+	0.156434i	$0.950000\pi$
$600$		0			0
$601$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$601$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$602$	0.453990	−	1.39724i	0.453990	−	1.39724i
$603$		0			0
$604$		−	0.108509i		−	0.108509i
$605$		0			0
$606$		0			0
$607$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$607$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.11803	+	0.363271i	1.11803	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$613$	1.11803	+	0.363271i	1.11803	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$614$		0			0
$615$		0			0
$616$	−0.166977	+	1.05425i	−0.166977	+	1.05425i
$617$			0.312869i			0.312869i	0.987688	+	0.156434i	$0.0500000\pi$
$617$			0.312869i			0.312869i	−0.987688	+	0.156434i	$0.950000\pi$
$618$		0			0
$619$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$619$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	0.809017	−	0.587785i	$-0.200000\pi$
$631$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	−1.00000			$\pi$
$632$	−0.737552	−	1.01515i	−0.737552	−	1.01515i
$633$		0			0
$634$	1.70582	−	0.554254i	1.70582	−	0.554254i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−0.128969	+	0.253116i	−0.128969	+	0.253116i
$639$		0			0
$640$		0			0
$641$	0.297556	+	0.0966818i	0.297556	+	0.0966818i	0.453990	−	0.891007i	$-0.350000\pi$
$641$	0.297556	+	0.0966818i	0.297556	+	0.0966818i	−0.156434	+	0.987688i	$0.550000\pi$
$642$		0			0
$643$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$643$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$644$	−0.253116	+	0.183900i	−0.253116	+	0.183900i
$645$		0			0
$646$		0			0
$647$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$647$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−0.0335310	+	0.103198i	−0.0335310	+	0.103198i
$653$	1.69480	−	0.550672i	1.69480	−	0.550672i	0.707107	−	0.707107i	$-0.250000\pi$
$653$	1.69480	−	0.550672i	1.69480	−	0.550672i	0.987688	+	0.156434i	$0.0500000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$659$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	1.39724	+	1.01515i	1.39724	+	1.01515i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−0.530249	+	0.172288i	−0.530249	+	0.172288i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.951057	−	1.30902i	0.951057	−	1.30902i		−	1.00000i	$-0.5\pi$
$673$	0.951057	−	1.30902i	0.951057	−	1.30902i	0.951057	−	0.309017i	$-0.100000\pi$
$674$	0.533698	+	0.173409i	0.533698	+	0.173409i
$675$		0			0
$676$	−0.142040	+	0.103198i	−0.142040	+	0.103198i
$677$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$677$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	0.907981i		−	0.907981i	−0.891007	−	0.453990i	$-0.850000\pi$
$683$		−	0.907981i		−	0.907981i	0.891007	−	0.453990i	$-0.150000\pi$
$684$		0			0
$685$		0			0
$686$	−0.863541	+	0.280582i	−0.863541	+	0.280582i
$687$		0			0
$688$	0.754763	+	1.03884i	0.754763	+	1.03884i
$689$		0			0
$690$		0			0
$691$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$691$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$692$		0			0
$693$		0			0
$694$	1.79360			1.79360
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.166977	−	0.0542543i	0.166977	−	0.0542543i
$701$	0.550672	−	1.69480i	0.550672	−	1.69480i	−0.156434	−	0.987688i	$-0.550000\pi$
$701$	0.550672	−	1.69480i	0.550672	−	1.69480i	0.707107	−	0.707107i	$-0.250000\pi$
$702$		0			0
$703$		0			0
$704$	−0.783834	−	0.783834i	−0.783834	−	0.783834i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.951057	+	0.690983i	−0.951057	+	0.690983i	−0.951057	−	0.309017i	$-0.900000\pi$
$709$	−0.951057	+	0.690983i	−0.951057	+	0.690983i			1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		−	0.159415i		−	0.159415i
$717$		0			0
$718$	0.500000	−	1.53884i	0.500000	−	1.53884i
$719$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$719$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$720$		0			0
$721$		0			0
$722$	−0.280582	−	0.863541i	−0.280582	−	0.863541i
$723$		0			0
$724$		0			0
$725$	0.312869			0.312869
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$733$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$734$		0			0
$735$		0			0
$736$		−	0.618034i		−	0.618034i
$737$	−1.59811	−	0.253116i	−1.59811	−	0.253116i
$738$		0			0
$739$	0.690983	−	0.951057i	0.690983	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$739$	0.690983	−	0.951057i	0.690983	−	0.951057i	1.00000			$0$
$740$		0			0
$741$		0			0
$742$	−0.229825	+	0.166977i	−0.229825	+	0.166977i
$743$	−0.253116	+	0.183900i	−0.253116	+	0.183900i	−0.707107	−	0.707107i	$-0.750000\pi$
$743$	−0.253116	+	0.183900i	−0.253116	+	0.183900i	0.453990	+	0.891007i	$0.350000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$			1.78201i			1.78201i
$750$		0			0
$751$	0.587785	−	1.80902i	0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$751$	0.587785	−	1.80902i	0.587785	−	1.80902i	0.587785	−	0.809017i	$-0.300000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.951057	+	0.690983i	0.951057	+	0.690983i	0.951057	−	0.309017i	$-0.100000\pi$
$757$	0.951057	+	0.690983i	0.951057	+	0.690983i			1.00000i	$0.5\pi$
$758$	1.72708			1.72708
$759$		0			0
$760$		0			0
$761$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$761$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$762$		0			0
$763$	0.190983	+	0.587785i	0.190983	+	0.587785i
$764$	−0.0322874	−	0.0444398i	−0.0322874	−	0.0444398i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	−0.166977	+	0.229825i	−0.166977	+	0.229825i
$773$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$773$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−0.784079	−	0.254763i	−0.784079	−	0.254763i
$779$		0			0
$780$		0			0
$781$	0.896802	+	1.76007i	0.896802	+	1.76007i
$782$		0			0
$783$		0			0
$784$	0.245237	−	0.754763i	0.245237	−	0.754763i
$785$		0			0
$786$		0			0
$787$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$787$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$788$	0.0966818	+	0.297556i	0.0966818	+	0.297556i
$789$		0			0
$790$		0			0
$791$	−1.97538			−1.97538
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$797$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$798$		0			0
$799$		0			0
$800$	−0.107173	+	0.329843i	−0.107173	+	0.329843i
$801$		0			0
$802$		−	1.61803i		−	1.61803i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.44168	−	1.04744i	1.44168	−	1.04744i	0.453990	−	0.891007i	$-0.350000\pi$
$809$	1.44168	−	1.04744i	1.44168	−	1.04744i	0.987688	−	0.156434i	$-0.0500000\pi$
$810$		0			0
$811$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$811$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$812$	−0.0322874	+	0.0444398i	−0.0322874	+	0.0444398i
$813$		0			0
$814$	−0.784079	+	1.53884i	−0.784079	+	1.53884i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.437016	−	1.34500i	−0.437016	−	1.34500i	−0.891007	−	0.453990i	$-0.850000\pi$
$821$	−0.437016	−	1.34500i	−0.437016	−	1.34500i	0.453990	−	0.891007i	$-0.350000\pi$
$822$		0			0
$823$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$823$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.14412	+	0.831254i	1.14412	+	0.831254i	0.987688	−	0.156434i	$-0.0500000\pi$
$827$	1.14412	+	0.831254i	1.14412	+	0.831254i	0.156434	+	0.987688i	$0.450000\pi$
$828$		0			0
$829$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$829$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$839$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$840$		0			0
$841$	0.729825	−	0.530249i	0.729825	−	0.530249i
$842$	−0.453990	+	0.329843i	−0.453990	+	0.329843i
$843$		0			0
$844$	0.103198	+	0.0335310i	0.103198	+	0.0335310i
$845$		0			0
$846$		0			0
$847$	1.00000			1.00000
$848$		−	0.248294i		−	0.248294i
$849$		0			0
$850$		0			0
$851$	−3.22369	+	1.04744i	−3.22369	+	1.04744i
$852$		0			0
$853$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$853$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$854$		0			0
$855$		0			0
$856$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	0.500000	+	1.53884i	0.500000	+	1.53884i
$863$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$863$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.831254	+	0.831254i	−0.831254	+	0.831254i
$870$		0			0
$871$		0			0
$872$	−0.627399	−	0.203854i	−0.627399	−	0.203854i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.11803	−	0.363271i	−1.11803	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$877$	−1.11803	−	0.363271i	−1.11803	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$883$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$884$		0			0
$885$		0			0
$886$	−1.05425	−	1.45106i	−1.05425	−	1.45106i
$887$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$887$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$888$		0			0
$889$	1.53884	+	1.11803i	1.53884	+	1.11803i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	0.387754	+	0.533698i	0.387754	+	0.533698i
$897$		0			0
$898$	1.53884	−	0.500000i	1.53884	−	0.500000i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	1.23935	−	1.70582i	1.23935	−	1.70582i
$905$		0			0
$906$		0			0
$907$	0.951057	−	0.690983i	0.951057	−	0.690983i		−	1.00000i	$-0.5\pi$
$907$	0.951057	−	0.690983i	0.951057	−	0.690983i	0.951057	+	0.309017i	$0.100000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.831254	−	1.14412i	0.831254	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$911$	0.831254	−	1.14412i	0.831254	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$912$		0			0
$913$		0			0
$914$			0.561163i			0.561163i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	0.951057	+	1.30902i	0.951057	+	1.30902i	0.951057	+	0.309017i	$0.100000\pi$
$919$	0.951057	+	1.30902i	0.951057	+	1.30902i			1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.90211			1.90211
$926$	−1.18856	−	0.863541i	−1.18856	−	0.863541i
$927$		0			0
$928$	−0.0335310	−	0.103198i	−0.0335310	−	0.103198i
$929$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$929$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$930$		0			0
$931$		0			0
$932$	−0.0767271	+	0.236142i	−0.0767271	+	0.236142i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$937$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$938$	1.39724	+	0.453990i	1.39724	+	0.453990i
$939$		0			0
$940$		0			0
$941$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$941$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	1.03884	−	1.03884i	1.03884	−	1.03884i
$947$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$947$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.280582	+	0.863541i	0.280582	+	0.863541i	0.987688	+	0.156434i	$0.0500000\pi$
$953$	0.280582	+	0.863541i	0.280582	+	0.863541i	−0.707107	+	0.707107i	$0.750000\pi$
$954$		0			0
$955$		0			0
$956$	−0.346818			−0.346818
$957$		0			0
$958$		0			0
$959$	0.734572	+	0.533698i	0.734572	+	0.533698i
$960$		0			0
$961$	0.309017	+	0.951057i	0.309017	+	0.951057i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$967$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$968$	−0.627399	+	0.863541i	−0.627399	+	0.863541i
$969$		0			0
$970$		0			0
$971$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$971$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$972$		0			0
$973$		0			0
$974$	−0.863541	+	0.627399i	−0.863541	+	0.627399i
$975$		0			0
$976$		0			0
$977$	−0.533698	+	0.734572i	−0.533698	+	0.734572i	−0.987688	−	0.156434i	$-0.950000\pi$
$977$	−0.533698	+	0.734572i	−0.533698	+	0.734572i	0.453990	+	0.891007i	$0.350000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	0.396802	−	1.22123i	0.396802	−	1.22123i
$983$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$983$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.88336			2.88336
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$	−0.554254	−	1.70582i	−0.554254	−	1.70582i
$995$		0			0
$996$		0			0
$997$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$997$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$998$	0.329843	−	1.01515i	0.329843	−	1.01515i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.1.bp.a.62.2	✓	16
3.2	odd	2	inner	693.1.bp.a.62.3	yes	16
7.6	odd	2	CM	693.1.bp.a.62.2	✓	16
11.8	odd	10	inner	693.1.bp.a.503.3	yes	16
21.20	even	2	inner	693.1.bp.a.62.3	yes	16
33.8	even	10	inner	693.1.bp.a.503.2	yes	16
77.41	even	10	inner	693.1.bp.a.503.3	yes	16
231.41	odd	10	inner	693.1.bp.a.503.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.1.bp.a.62.2	✓	16	1.1	even	1	trivial
693.1.bp.a.62.2	✓	16	7.6	odd	2	CM
693.1.bp.a.62.3	yes	16	3.2	odd	2	inner
693.1.bp.a.62.3	yes	16	21.20	even	2	inner
693.1.bp.a.503.2	yes	16	33.8	even	10	inner
693.1.bp.a.503.2	yes	16	231.41	odd	10	inner
693.1.bp.a.503.3	yes	16	11.8	odd	10	inner
693.1.bp.a.503.3	yes	16	77.41	even	10	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	693.bp (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.734572 - 0.533698i) q^{2} +(-0.0542543 - 0.166977i) q^{4} +(0.951057 - 0.309017i) q^{7} +(-0.329843 + 1.01515i) q^{8} +O(q^{10})\)	\(q+(-0.734572 - 0.533698i) q^{2} +(-0.0542543 - 0.166977i) q^{4} +(0.951057 - 0.309017i) q^{7} +(-0.329843 + 1.01515i) q^{8} +(0.987688 + 0.156434i) q^{11} +(-0.863541 - 0.280582i) q^{14} +(0.642040 - 0.466469i) q^{16} +(-0.642040 - 0.642040i) q^{22} -1.78201i q^{23} +(-0.309017 + 0.951057i) q^{25} +(-0.103198 - 0.142040i) q^{28} +(-0.0966818 - 0.297556i) q^{29} +0.346818 q^{32} +(-0.587785 - 1.80902i) q^{37} +1.61803i q^{43} +(-0.0274653 - 0.173409i) q^{44} +(-0.951057 + 1.30902i) q^{46} +(0.809017 - 0.587785i) q^{49} +(0.734572 - 0.533698i) q^{50} +(0.183900 - 0.253116i) q^{53} +1.06740i q^{56} +(-0.0877853 + 0.270175i) q^{58} +(-0.896802 - 0.651565i) q^{64} -1.61803 q^{67} +(1.16110 + 1.59811i) q^{71} +(-0.533698 + 1.64255i) q^{74} +(0.987688 - 0.156434i) q^{77} +(-0.690983 + 0.951057i) q^{79} +(0.863541 - 1.18856i) q^{86} +(-0.484587 + 0.951057i) q^{88} +(-0.297556 + 0.0966818i) q^{92} -0.907981 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{4}+O(q^{10}) \) 16 * q - 4 * q^4	\( 16 q - 4 q^{4} + 4 q^{16} - 4 q^{22} + 4 q^{25} - 20 q^{28} + 4 q^{49} + 8 q^{58} + 4 q^{64} - 8 q^{67} - 20 q^{79} + 20 q^{88}+O(q^{100}) \) 16 * q - 4 * q^4 + 4 * q^16 - 4 * q^22 + 4 * q^25 - 20 * q^28 + 4 * q^49 + 8 * q^58 + 4 * q^64 - 8 * q^67 - 20 * q^79 + 20 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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