LMFDB - Embedded newform 1925.1.dn.b.1168.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1925,1,Mod(118,1925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1925, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([15, 10, 6]))

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

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

chi := DirichletCharacter("1925.118");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1925 = 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1925.dn (of order $20$, degree $8$, 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:	$0.960700149319$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{20})$
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:	$D_{10}$
Projective field:	Galois closure of 10.0.17691976269034375.2

Embedding invariants

Embedding label			1168.2
Root			$0.453990 - 0.891007i$ of defining polynomial
Character	$\chi$	$=$	1925.1168
Dual form			1925.1.dn.b.1007.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.183900 - 1.16110i) q^{3} +(-0.587785 + 0.809017i) q^{4} +(-0.987688 + 0.156434i) q^{7} +(-0.363271 - 0.118034i) q^{9} +O(q^{10})$	\(q+(0.183900 - 1.16110i) q^{3} +(-0.587785 + 0.809017i) q^{4} +(-0.987688 + 0.156434i) q^{7} +(-0.363271 - 0.118034i) q^{9} +(-0.309017 + 0.951057i) q^{11} +(0.831254 + 0.831254i) q^{12} +(1.44168 - 0.734572i) q^{13} +(-0.309017 - 0.951057i) q^{16} +(0.550672 + 0.280582i) q^{17} +1.17557i q^{21} +(0.329843 - 0.647354i) q^{27} +(0.453990 - 0.891007i) q^{28} +(1.53884 + 1.11803i) q^{29} +(1.04744 + 0.533698i) q^{33} +(0.309017 - 0.224514i) q^{36} +(-0.587785 - 1.80902i) q^{39} +(-0.587785 - 0.809017i) q^{44} +(1.87869 + 0.297556i) q^{47} +(-1.16110 + 0.183900i) q^{48} +(0.951057 - 0.309017i) q^{49} +(0.427051 - 0.587785i) q^{51} +(-0.253116 + 1.59811i) q^{52} +(0.377263 + 0.0597526i) q^{63} +(0.951057 + 0.309017i) q^{64} +(-0.550672 + 0.280582i) q^{68} +(-0.500000 - 1.53884i) q^{71} +(0.0966818 + 0.610425i) q^{73} +(0.156434 - 0.987688i) q^{77} +(-0.363271 + 1.11803i) q^{79} +(-1.00000 - 0.726543i) q^{81} +(0.734572 - 1.44168i) q^{83} +(-0.951057 - 0.690983i) q^{84} +(1.58114 - 1.58114i) q^{87} +(-1.30902 + 0.951057i) q^{91} +(-1.69480 + 0.863541i) q^{97} +(0.224514 - 0.309017i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 4 q^{11} + 4 q^{16} - 4 q^{36} - 20 q^{51} - 8 q^{71} - 16 q^{81} - 12 q^{91}+O(q^{100}) $ 16 * q + 4 * q^11 + 4 * q^16 - 4 * q^36 - 20 * q^51 - 8 * q^71 - 16 * q^81 - 12 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times$.

$n$	$276$	$1002$	$1751$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$	$e\left(\frac{1}{10}\right)$

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			0		−0.453990	−	0.891007i	$-0.650000\pi$
$2$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$3$	0.183900	−	1.16110i	0.183900	−	1.16110i	−0.707107	−	0.707107i	$-0.750000\pi$
$3$	0.183900	−	1.16110i	0.183900	−	1.16110i	0.891007	−	0.453990i	$-0.150000\pi$
$4$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.987688	+	0.156434i	−0.987688	+	0.156434i
$7$	−0.987688	+	0.156434i	−0.987688	+	0.156434i
$8$		0			0
$9$	−0.363271	−	0.118034i	−0.363271	−	0.118034i
$10$		0			0
$11$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$11$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$12$	0.831254	+	0.831254i	0.831254	+	0.831254i
$13$	1.44168	−	0.734572i	1.44168	−	0.734572i	0.453990	−	0.891007i	$-0.350000\pi$
$13$	1.44168	−	0.734572i	1.44168	−	0.734572i	0.987688	+	0.156434i	$0.0500000\pi$
$14$		0			0
$15$		0			0
$16$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$17$	0.550672	+	0.280582i	0.550672	+	0.280582i	0.707107	−	0.707107i	$-0.250000\pi$
$17$	0.550672	+	0.280582i	0.550672	+	0.280582i	−0.156434	+	0.987688i	$0.550000\pi$
$18$		0			0
$19$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$19$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$20$		0			0
$21$			1.17557i			1.17557i
$22$		0			0
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	0.329843	−	0.647354i	0.329843	−	0.647354i
$28$	0.453990	−	0.891007i	0.453990	−	0.891007i
$29$	1.53884	+	1.11803i	1.53884	+	1.11803i	0.951057	+	0.309017i	$0.100000\pi$
$29$	1.53884	+	1.11803i	1.53884	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$30$		0			0
$31$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$31$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$32$		0			0
$33$	1.04744	+	0.533698i	1.04744	+	0.533698i
$34$		0			0
$35$		0			0
$36$	0.309017	−	0.224514i	0.309017	−	0.224514i
$37$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$37$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$38$		0			0
$39$	−0.587785	−	1.80902i	−0.587785	−	1.80902i
$40$		0			0
$41$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$41$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$45$		0			0
$46$		0			0
$47$	1.87869	+	0.297556i	1.87869	+	0.297556i	0.987688	−	0.156434i	$-0.0500000\pi$
$47$	1.87869	+	0.297556i	1.87869	+	0.297556i	0.891007	+	0.453990i	$0.150000\pi$
$48$	−1.16110	+	0.183900i	−1.16110	+	0.183900i
$49$	0.951057	−	0.309017i	0.951057	−	0.309017i
$50$		0			0
$51$	0.427051	−	0.587785i	0.427051	−	0.587785i
$52$	−0.253116	+	1.59811i	−0.253116	+	1.59811i
$53$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$53$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$59$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$60$		0			0
$61$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$61$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$62$		0			0
$63$	0.377263	+	0.0597526i	0.377263	+	0.0597526i
$64$	0.951057	+	0.309017i	0.951057	+	0.309017i
$65$		0			0
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$	−0.550672	+	0.280582i	−0.550672	+	0.280582i
$69$		0			0
$70$		0			0
$71$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$71$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$72$		0			0
$73$	0.0966818	+	0.610425i	0.0966818	+	0.610425i	0.987688	+	0.156434i	$0.0500000\pi$
$73$	0.0966818	+	0.610425i	0.0966818	+	0.610425i	−0.891007	+	0.453990i	$0.850000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.156434	−	0.987688i	0.156434	−	0.987688i
$78$		0			0
$79$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$79$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$80$		0			0
$81$	−1.00000	−	0.726543i	−1.00000	−	0.726543i
$82$		0			0
$83$	0.734572	−	1.44168i	0.734572	−	1.44168i	−0.156434	−	0.987688i	$-0.550000\pi$
$83$	0.734572	−	1.44168i	0.734572	−	1.44168i	0.891007	−	0.453990i	$-0.150000\pi$
$84$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$85$		0			0
$86$		0			0
$87$	1.58114	−	1.58114i	1.58114	−	1.58114i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.69480	+	0.863541i	−1.69480	+	0.863541i	−0.707107	+	0.707107i	$0.750000\pi$
$97$	−1.69480	+	0.863541i	−1.69480	+	0.863541i	−0.987688	+	0.156434i	$0.950000\pi$
$98$		0			0
$99$	0.224514	−	0.309017i	0.224514	−	0.309017i
$100$		0			0
$101$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$101$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$102$		0			0
$103$	−1.87869	+	0.297556i	−1.87869	+	0.297556i	−0.987688	−	0.156434i	$-0.950000\pi$
$103$	−1.87869	+	0.297556i	−1.87869	+	0.297556i	−0.891007	+	0.453990i	$0.850000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$107$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$108$	0.329843	+	0.647354i	0.329843	+	0.647354i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$	0.453990	+	0.891007i	0.453990	+	0.891007i
$113$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$113$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$114$		0			0
$115$		0			0
$116$	−1.80902	+	0.587785i	−1.80902	+	0.587785i
$117$	−0.610425	+	0.0966818i	−0.610425	+	0.0966818i
$118$		0			0
$119$	−0.587785	−	0.190983i	−0.587785	−	0.190983i
$120$		0			0
$121$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$127$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	−1.04744	+	0.533698i	−1.04744	+	0.533698i
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$137$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$138$		0			0
$139$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$139$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$140$		0			0
$141$	0.690983	−	2.12663i	0.690983	−	2.12663i
$142$		0			0
$143$	0.253116	+	1.59811i	0.253116	+	1.59811i
$144$			0.381966i			0.381966i
$145$		0			0
$146$		0			0
$147$	−0.183900	−	1.16110i	−0.183900	−	1.16110i
$148$		0			0
$149$	0.363271	+	1.11803i	0.363271	+	1.11803i	0.951057	+	0.309017i	$0.100000\pi$
$149$	0.363271	+	1.11803i	0.363271	+	1.11803i	−0.587785	+	0.809017i	$0.700000\pi$
$150$		0			0
$151$	1.11803	+	1.53884i	1.11803	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$151$	1.11803	+	1.53884i	1.11803	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$152$		0			0
$153$	−0.166925	−	0.166925i	−0.166925	−	0.166925i
$154$		0			0
$155$		0			0
$156$	1.80902	+	0.587785i	1.80902	+	0.587785i
$157$	−1.87869	−	0.297556i	−1.87869	−	0.297556i	−0.891007	−	0.453990i	$-0.850000\pi$
$157$	−1.87869	−	0.297556i	−1.87869	−	0.297556i	−0.987688	+	0.156434i	$0.950000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$163$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−0.907981	−	1.78201i	−0.907981	−	1.78201i	−0.453990	−	0.891007i	$-0.650000\pi$
$167$	−0.907981	−	1.78201i	−0.907981	−	1.78201i	−0.453990	−	0.891007i	$-0.650000\pi$
$168$		0			0
$169$	0.951057	−	1.30902i	0.951057	−	1.30902i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.59811	+	0.253116i	1.59811	+	0.253116i	0.891007	−	0.453990i	$-0.150000\pi$
$173$	1.59811	+	0.253116i	1.59811	+	0.253116i	0.707107	+	0.707107i	$0.250000\pi$
$174$		0			0
$175$		0			0
$176$	1.00000			1.00000
$177$		0			0
$178$		0			0
$179$	−0.363271	−	0.500000i	−0.363271	−	0.500000i	0.587785	−	0.809017i	$-0.300000\pi$
$179$	−0.363271	−	0.500000i	−0.363271	−	0.500000i	−0.951057	+	0.309017i	$0.900000\pi$
$180$		0			0
$181$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$181$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−0.437016	+	0.437016i	−0.437016	+	0.437016i
$188$	−1.34500	+	1.34500i	−1.34500	+	1.34500i
$189$	−0.224514	+	0.690983i	−0.224514	+	0.690983i
$190$		0			0
$191$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$191$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$192$	0.533698	−	1.04744i	0.533698	−	1.04744i
$193$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$193$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$194$		0			0
$195$		0			0
$196$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.69480	−	0.863541i	−1.69480	−	0.863541i
$204$	0.224514	+	0.690983i	0.224514	+	0.690983i
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−1.14412	−	1.14412i	−1.14412	−	1.14412i
$209$		0			0
$210$		0			0
$211$	1.11803	+	0.363271i	1.11803	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$211$	1.11803	+	0.363271i	1.11803	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$212$		0			0
$213$	−1.87869	+	0.297556i	−1.87869	+	0.297556i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	0.726543			0.726543
$220$		0			0
$221$	1.00000			1.00000
$222$		0			0
$223$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$223$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.59811	+	0.253116i	−1.59811	+	0.253116i	−0.891007	−	0.453990i	$-0.850000\pi$
$227$	−1.59811	+	0.253116i	−1.59811	+	0.253116i	−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$229$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$230$		0			0
$231$	−1.11803	−	0.363271i	−1.11803	−	0.363271i
$232$		0			0
$233$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$233$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.23134	+	0.627399i	1.23134	+	0.627399i
$238$		0			0
$239$	−0.951057	+	0.690983i	−0.951057	+	0.690983i	−0.951057	−	0.309017i	$-0.900000\pi$
$239$	−0.951057	+	0.690983i	−0.951057	+	0.690983i			1.00000i	$0.5\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$	−0.513743	+	0.513743i	−0.513743	+	0.513743i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$250$		0			0
$251$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$251$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$252$	−0.270091	+	0.270091i	−0.270091	+	0.270091i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$257$	−0.297556	−	1.87869i	−0.297556	−	1.87869i	−0.453990	−	0.891007i	$-0.650000\pi$
$257$	−0.297556	−	1.87869i	−0.297556	−	1.87869i	0.156434	−	0.987688i	$-0.450000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−0.427051	−	0.587785i	−0.427051	−	0.587785i
$262$		0			0
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$269$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$270$		0			0
$271$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$271$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$272$	0.0966818	−	0.610425i	0.0966818	−	0.610425i
$273$	0.863541	+	1.69480i	0.863541	+	1.69480i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$277$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$281$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$282$		0			0
$283$	−0.610425	−	0.0966818i	−0.610425	−	0.0966818i	−0.156434	−	0.987688i	$-0.550000\pi$
$283$	−0.610425	−	0.0966818i	−0.610425	−	0.0966818i	−0.453990	+	0.891007i	$0.650000\pi$
$284$	1.53884	+	0.500000i	1.53884	+	0.500000i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.363271	−	0.500000i	−0.363271	−	0.500000i
$290$		0			0
$291$	0.690983	+	2.12663i	0.690983	+	2.12663i
$292$	−0.550672	−	0.280582i	−0.550672	−	0.280582i
$293$	0.312869	+	1.97538i	0.312869	+	1.97538i	0.156434	+	0.987688i	$0.450000\pi$
$293$	0.312869	+	1.97538i	0.312869	+	1.97538i	0.156434	+	0.987688i	$0.450000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	0.513743	+	0.513743i	0.513743	+	0.513743i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$307$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$308$	0.707107	+	0.707107i	0.707107	+	0.707107i
$309$			2.23607i			2.23607i
$310$		0			0
$311$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$311$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$312$		0			0
$313$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$313$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$314$		0			0
$315$		0			0
$316$	−0.690983	−	0.951057i	−0.690983	−	0.951057i
$317$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$317$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$318$		0			0
$319$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	1.17557	−	0.381966i	1.17557	−	0.381966i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.90211			−1.90211
$330$		0			0
$331$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$331$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$332$	0.734572	+	1.44168i	0.734572	+	1.44168i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	1.11803	−	0.363271i	1.11803	−	0.363271i
$337$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$337$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.891007	+	0.453990i	−0.891007	+	0.453990i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$347$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$348$	0.349798	+	2.20854i	0.349798	+	2.20854i
$349$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$349$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$350$		0			0
$351$		−	1.17557i		−	1.17557i
$352$		0			0
$353$	0.831254	−	0.831254i	0.831254	−	0.831254i	−0.156434	−	0.987688i	$-0.550000\pi$
$353$	0.831254	−	0.831254i	0.831254	−	0.831254i	0.987688	+	0.156434i	$0.0500000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−0.329843	+	0.647354i	−0.329843	+	0.647354i
$358$		0			0
$359$	0.951057	+	0.690983i	0.951057	+	0.690983i	0.951057	−	0.309017i	$-0.100000\pi$
$359$	0.951057	+	0.690983i	0.951057	+	0.690983i			1.00000i	$0.5\pi$
$360$		0			0
$361$	0.309017	−	0.951057i	0.309017	−	0.951057i
$362$		0			0
$363$	−0.831254	+	0.831254i	−0.831254	+	0.831254i
$364$		−	1.61803i		−	1.61803i
$365$		0			0
$366$		0			0
$367$	−0.183900	−	1.16110i	−0.183900	−	1.16110i	−0.891007	−	0.453990i	$-0.850000\pi$
$367$	−0.183900	−	1.16110i	−0.183900	−	1.16110i	0.707107	−	0.707107i	$-0.250000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.03979	+	0.481456i	3.03979	+	0.481456i
$378$		0			0
$379$	−0.587785	+	0.190983i	−0.587785	+	0.190983i	−0.587785	−	0.809017i	$-0.700000\pi$
$379$	−0.587785	+	0.190983i	−0.587785	+	0.190983i			1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.533698	−	1.04744i	−0.533698	−	1.04744i	−0.987688	−	0.156434i	$-0.950000\pi$
$383$	−0.533698	−	1.04744i	−0.533698	−	1.04744i	0.453990	−	0.891007i	$-0.350000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.297556	−	1.87869i	0.297556	−	1.87869i
$389$	0.951057	−	1.30902i	0.951057	−	1.30902i		−	1.00000i	$-0.5\pi$
$389$	0.951057	−	1.30902i	0.951057	−	1.30902i	0.951057	−	0.309017i	$-0.100000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	0.118034	+	0.363271i	0.118034	+	0.363271i
$397$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	0.156434	−	0.987688i	$-0.450000\pi$
$397$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	−0.987688	+	0.156434i	$0.950000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	0.809017	−	0.587785i	$-0.200000\pi$
$401$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$409$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$410$		0			0
$411$		0			0
$412$	0.863541	−	1.69480i	0.863541	−	1.69480i
$413$		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
$421$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$421$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$422$		0			0
$423$	−0.647354	−	0.329843i	−0.647354	−	0.329843i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	1.90211			1.90211
$430$		0			0
$431$	−1.80902	−	0.587785i	−1.80902	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$431$	−1.80902	−	0.587785i	−1.80902	−	0.587785i	−1.00000			$\pi$
$432$	−0.717598	−	0.113656i	−0.717598	−	0.113656i
$433$	−1.16110	+	0.183900i	−1.16110	+	0.183900i	−0.707107	−	0.707107i	$-0.750000\pi$
$433$	−1.16110	+	0.183900i	−1.16110	+	0.183900i	−0.453990	+	0.891007i	$0.650000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−0.381966			−0.381966
$442$		0			0
$443$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$443$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	1.36495	−	0.216187i	1.36495	−	0.216187i
$448$	−0.987688	−	0.156434i	−0.987688	−	0.156434i
$449$	−1.53884	−	0.500000i	−1.53884	−	0.500000i	−0.587785	−	0.809017i	$-0.700000\pi$
$449$	−1.53884	−	0.500000i	−1.53884	−	0.500000i	−0.951057	+	0.309017i	$0.900000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	1.99235	−	1.01515i	1.99235	−	1.01515i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$457$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$458$		0			0
$459$	0.363271	−	0.263932i	0.363271	−	0.263932i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$	0.587785	−	1.80902i	0.587785	−	1.80902i
$465$		0			0
$466$		0			0
$467$	−0.533698	+	1.04744i	−0.533698	+	1.04744i	0.453990	+	0.891007i	$0.350000\pi$
$467$	−0.533698	+	1.04744i	−0.533698	+	1.04744i	−0.987688	+	0.156434i	$0.950000\pi$
$468$	0.280582	−	0.550672i	0.280582	−	0.550672i
$469$		0			0
$470$		0			0
$471$	−0.690983	+	2.12663i	−0.690983	+	2.12663i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$	0.500000	−	0.363271i	0.500000	−	0.363271i
$477$		0			0
$478$		0			0
$479$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$479$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.951057	−	0.309017i	0.951057	−	0.309017i
$485$		0			0
$486$		0			0
$487$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$487$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$491$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$492$		0			0
$493$	0.533698	+	1.04744i	0.533698	+	1.04744i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.734572	+	1.44168i	0.734572	+	1.44168i
$498$		0			0
$499$	−0.363271	+	0.500000i	−0.363271	+	0.500000i	−0.951057	−	0.309017i	$-0.900000\pi$
$499$	−0.363271	+	0.500000i	−0.363271	+	0.500000i	0.587785	+	0.809017i	$0.300000\pi$
$500$		0			0
$501$	−2.23607	+	0.726543i	−2.23607	+	0.726543i
$502$		0			0
$503$	1.59811	+	0.253116i	1.59811	+	0.253116i	0.891007	−	0.453990i	$-0.150000\pi$
$503$	1.59811	+	0.253116i	1.59811	+	0.253116i	0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.34500	−	1.34500i	−1.34500	−	1.34500i
$508$		0			0
$509$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$509$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$510$		0			0
$511$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−0.863541	+	1.69480i	−0.863541	+	1.69480i
$518$		0			0
$519$	0.587785	−	1.80902i	0.587785	−	1.80902i
$520$		0			0
$521$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$521$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$522$		0			0
$523$	0.280582	−	0.550672i	0.280582	−	0.550672i	−0.707107	−	0.707107i	$-0.750000\pi$
$523$	0.280582	−	0.550672i	0.280582	−	0.550672i	0.987688	+	0.156434i	$0.0500000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	0.183900	−	1.16110i	0.183900	−	1.16110i
$529$		−	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
$537$	−0.647354	+	0.329843i	−0.647354	+	0.329843i
$538$		0			0
$539$			1.00000i			1.00000i
$540$		0			0
$541$	−1.11803	−	0.363271i	−1.11803	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$541$	−1.11803	−	0.363271i	−1.11803	−	0.363271i	−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		0.156434	−	0.987688i	$-0.450000\pi$
$547$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.183900	−	1.16110i	0.183900	−	1.16110i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$557$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	0.427051	+	0.587785i	0.427051	+	0.587785i
$562$		0			0
$563$	−0.550672	+	0.280582i	−0.550672	+	0.280582i	−0.707107	−	0.707107i	$-0.750000\pi$
$563$	−0.550672	+	0.280582i	−0.550672	+	0.280582i	0.156434	+	0.987688i	$0.450000\pi$
$564$	1.31433	+	1.80902i	1.31433	+	1.80902i
$565$		0			0
$566$		0			0
$567$	1.10134	+	0.561163i	1.10134	+	0.561163i
$568$		0			0
$569$	1.53884	−	1.11803i	1.53884	−	1.11803i	0.587785	−	0.809017i	$-0.300000\pi$
$569$	1.53884	−	1.11803i	1.53884	−	1.11803i	0.951057	−	0.309017i	$-0.100000\pi$
$570$		0			0
$571$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$571$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$572$	−1.44168	−	0.734572i	−1.44168	−	0.734572i
$573$	1.34500	−	1.34500i	1.34500	−	1.34500i
$574$		0			0
$575$		0			0
$576$	−0.309017	−	0.224514i	−0.309017	−	0.224514i
$577$	−0.863541	+	1.69480i	−0.863541	+	1.69480i	−0.156434	+	0.987688i	$0.550000\pi$
$577$	−0.863541	+	1.69480i	−0.863541	+	1.69480i	−0.707107	+	0.707107i	$0.750000\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	0.183900	+	1.16110i	0.183900	+	1.16110i	0.891007	+	0.453990i	$0.150000\pi$
$587$	0.183900	+	1.16110i	0.183900	+	1.16110i	−0.707107	+	0.707107i	$0.750000\pi$
$588$	1.04744	+	0.533698i	1.04744	+	0.533698i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$593$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$594$		0			0
$595$		0			0
$596$	−1.11803	−	0.363271i	−1.11803	−	0.363271i
$597$		0			0
$598$		0			0
$599$	−0.587785	+	0.190983i	−0.587785	+	0.190983i	−0.587785	−	0.809017i	$-0.700000\pi$
$599$	−0.587785	+	0.190983i	−0.587785	+	0.190983i			1.00000i	$0.5\pi$
$600$		0			0
$601$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$601$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$602$		0			0
$603$		0			0
$604$	−1.90211			−1.90211
$605$		0			0
$606$		0			0
$607$	−0.280582	−	0.550672i	−0.280582	−	0.550672i	0.707107	−	0.707107i	$-0.250000\pi$
$607$	−0.280582	−	0.550672i	−0.280582	−	0.550672i	−0.987688	+	0.156434i	$0.950000\pi$
$608$		0			0
$609$	−1.31433	+	1.80902i	−1.31433	+	1.80902i
$610$		0			0
$611$	2.92705	−	0.951057i	2.92705	−	0.951057i
$612$	0.233162	−	0.0369292i	0.233162	−	0.0369292i
$613$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$613$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$		0			0
$619$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$619$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$625$		0			0
$626$		0			0
$627$		0			0
$628$	1.34500	−	1.34500i	1.34500	−	1.34500i
$629$		0			0
$630$		0			0
$631$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$631$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$632$		0			0
$633$	0.627399	−	1.23134i	0.627399	−	1.23134i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.14412	−	1.14412i	1.14412	−	1.14412i
$638$		0			0
$639$			0.618034i			0.618034i
$640$		0			0
$641$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$641$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$642$		0			0
$643$	−1.04744	−	0.533698i	−1.04744	−	0.533698i	−0.156434	−	0.987688i	$-0.550000\pi$
$643$	−1.04744	−	0.533698i	−1.04744	−	0.533698i	−0.891007	+	0.453990i	$0.850000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.69480	−	0.863541i	1.69480	−	0.863541i	0.707107	−	0.707107i	$-0.250000\pi$
$647$	1.69480	−	0.863541i	1.69480	−	0.863541i	0.987688	−	0.156434i	$-0.0500000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$653$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	0.0369292	−	0.233162i	0.0369292	−	0.233162i
$658$		0			0
$659$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$659$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$	0.183900	−	1.16110i	0.183900	−	1.16110i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	1.97538	+	0.312869i	1.97538	+	0.312869i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$673$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$674$		0			0
$675$		0			0
$676$	0.500000	+	1.53884i	0.500000	+	1.53884i
$677$	0.550672	+	0.280582i	0.550672	+	0.280582i	0.707107	−	0.707107i	$-0.250000\pi$
$677$	0.550672	+	0.280582i	0.550672	+	0.280582i	−0.156434	+	0.987688i	$0.550000\pi$
$678$		0			0
$679$	1.53884	−	1.11803i	1.53884	−	1.11803i
$680$		0			0
$681$			1.90211i			1.90211i
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$691$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$692$	−1.14412	+	1.14412i	−1.14412	+	1.14412i
$693$	−0.173409	+	0.340334i	−0.173409	+	0.340334i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$701$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$702$		0			0
$703$		0			0
$704$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.53884	+	0.500000i	−1.53884	+	0.500000i	−0.951057	−	0.309017i	$-0.900000\pi$
$709$	−1.53884	+	0.500000i	−1.53884	+	0.500000i	−0.587785	+	0.809017i	$0.700000\pi$
$710$		0			0
$711$	0.263932	−	0.363271i	0.263932	−	0.363271i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.618034			0.618034
$717$	0.627399	+	1.23134i	0.627399	+	1.23134i
$718$		0			0
$719$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$719$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$720$		0			0
$721$	1.80902	−	0.587785i	1.80902	−	0.587785i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	0.831254	+	0.831254i	0.831254	+	0.831254i	0.987688	−	0.156434i	$-0.0500000\pi$
$727$	0.831254	+	0.831254i	0.831254	+	0.831254i	−0.156434	+	0.987688i	$0.550000\pi$
$728$		0			0
$729$	−0.224514	−	0.309017i	−0.224514	−	0.309017i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−0.253116	−	1.59811i	−0.253116	−	1.59811i	−0.707107	−	0.707107i	$-0.750000\pi$
$733$	−0.253116	−	1.59811i	−0.253116	−	1.59811i	0.453990	−	0.891007i	$-0.350000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$739$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$743$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−0.437016	+	0.437016i	−0.437016	+	0.437016i
$748$	−0.0966818	−	0.610425i	−0.0966818	−	0.610425i
$749$		0			0
$750$		0			0
$751$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$751$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$752$	−0.297556	−	1.87869i	−0.297556	−	1.87869i
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.427051	−	0.587785i	−0.427051	−	0.587785i
$757$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$757$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$761$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$762$		0			0
$763$		0			0
$764$	−1.53884	+	0.500000i	−1.53884	+	0.500000i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.533698	+	1.04744i	0.533698	+	1.04744i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−2.23607			−2.23607
$772$		0			0
$773$	−0.183900	+	1.16110i	−0.183900	+	1.16110i	0.707107	+	0.707107i	$0.250000\pi$
$773$	−0.183900	+	1.16110i	−0.183900	+	1.16110i	−0.891007	+	0.453990i	$0.850000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	1.61803			1.61803
$782$		0			0
$783$	1.23134	−	0.627399i	1.23134	−	0.627399i
$784$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$785$		0			0
$786$		0			0
$787$	1.44168	+	0.734572i	1.44168	+	0.734572i	0.987688	−	0.156434i	$-0.0500000\pi$
$787$	1.44168	+	0.734572i	1.44168	+	0.734572i	0.453990	+	0.891007i	$0.350000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.533698	+	1.04744i	−0.533698	+	1.04744i	0.453990	+	0.891007i	$0.350000\pi$
$797$	−0.533698	+	1.04744i	−0.533698	+	1.04744i	−0.987688	+	0.156434i	$0.950000\pi$
$798$		0			0
$799$	0.951057	+	0.690983i	0.951057	+	0.690983i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.610425	−	0.0966818i	−0.610425	−	0.0966818i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.363271	−	1.11803i	−0.363271	−	1.11803i	−0.951057	−	0.309017i	$-0.900000\pi$
$809$	−0.363271	−	1.11803i	−0.363271	−	1.11803i	0.587785	−	0.809017i	$-0.300000\pi$
$810$		0			0
$811$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$811$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$812$	1.69480	−	0.863541i	1.69480	−	0.863541i
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−0.690983	−	0.224514i	−0.690983	−	0.224514i
$817$		0			0
$818$		0			0
$819$	0.587785	−	0.190983i	0.587785	−	0.190983i
$820$		0			0
$821$	−1.11803	+	1.53884i	−1.11803	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$821$	−1.11803	+	1.53884i	−1.11803	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$822$		0			0
$823$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$823$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$827$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$828$		0			0
$829$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$829$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$830$		0			0
$831$		0			0
$832$	1.59811	−	0.253116i	1.59811	−	0.253116i
$833$	0.610425	+	0.0966818i	0.610425	+	0.0966818i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$839$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$840$		0			0
$841$	0.809017	+	2.48990i	0.809017	+	2.48990i
$842$		0			0
$843$		0			0
$844$	−0.951057	+	0.690983i	−0.951057	+	0.690983i
$845$		0			0
$846$		0			0
$847$	0.891007	+	0.453990i	0.891007	+	0.453990i
$848$		0			0
$849$	−0.224514	+	0.690983i	−0.224514	+	0.690983i
$850$		0			0
$851$		0			0
$852$	0.863541	−	1.69480i	0.863541	−	1.69480i
$853$	−0.734572	+	1.44168i	−0.734572	+	1.44168i	0.156434	+	0.987688i	$0.450000\pi$
$853$	−0.734572	+	1.44168i	−0.734572	+	1.44168i	−0.891007	+	0.453990i	$0.850000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$857$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$863$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.647354	+	0.329843i	−0.647354	+	0.329843i
$868$		0			0
$869$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.717598	−	0.113656i	0.717598	−	0.113656i
$874$		0			0
$875$		0			0
$876$	−0.427051	+	0.587785i	−0.427051	+	0.587785i
$877$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$877$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$878$		0			0
$879$	2.35114			2.35114
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$883$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$884$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$885$		0			0
$886$		0			0
$887$	0.610425	−	0.0966818i	0.610425	−	0.0966818i	0.156434	−	0.987688i	$-0.450000\pi$
$887$	0.610425	−	0.0966818i	0.610425	−	0.0966818i	0.453990	+	0.891007i	$0.350000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1.00000	−	0.726543i	1.00000	−	0.726543i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$907$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$908$	0.734572	−	1.44168i	0.734572	−	1.44168i
$909$		0			0
$910$		0			0
$911$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$911$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$912$		0			0
$913$	1.14412	+	1.14412i	1.14412	+	1.14412i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	0.587785	+	1.80902i	0.587785	+	1.80902i	0.587785	+	0.809017i	$0.300000\pi$
$919$	0.587785	+	1.80902i	0.587785	+	1.80902i			1.00000i	$0.5\pi$
$920$		0			0
$921$	0.427051	+	0.587785i	0.427051	+	0.587785i
$922$		0			0
$923$	−1.85123	−	1.85123i	−1.85123	−	1.85123i
$924$	0.951057	−	0.690983i	0.951057	−	0.690983i
$925$		0			0
$926$		0			0
$927$	0.717598	+	0.113656i	0.717598	+	0.113656i
$928$		0			0
$929$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$929$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.734572	−	1.44168i	−0.734572	−	1.44168i	−0.891007	−	0.453990i	$-0.850000\pi$
$937$	−0.734572	−	1.44168i	−0.734572	−	1.44168i	0.156434	−	0.987688i	$-0.450000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$941$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$947$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$948$	−1.23134	+	0.627399i	−1.23134	+	0.627399i
$949$	0.587785	+	0.809017i	0.587785	+	0.809017i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$953$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$954$		0			0
$955$		0			0
$956$		−	1.17557i		−	1.17557i
$957$	1.01515	+	1.99235i	1.01515	+	1.99235i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$971$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$972$	−0.113656	−	0.717598i	−0.113656	−	0.717598i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$977$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.87869	−	0.297556i	1.87869	−	0.297556i	0.891007	−	0.453990i	$-0.150000\pi$
$983$	1.87869	−	0.297556i	1.87869	−	0.297556i	0.987688	+	0.156434i	$0.0500000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−0.349798	+	2.20854i	−0.349798	+	2.20854i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$991$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$992$		0			0
$993$	−0.297556	+	1.87869i	−0.297556	+	1.87869i
$994$		0			0
$995$		0			0
$996$	1.80902	−	0.587785i	1.80902	−	0.587785i
$997$	−0.610425	+	0.0966818i	−0.610425	+	0.0966818i	−0.453990	−	0.891007i	$-0.650000\pi$
$997$	−0.610425	+	0.0966818i	−0.610425	+	0.0966818i	−0.156434	+	0.987688i	$0.550000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.1.dn.b.1168.2	yes	16
5.2	odd	4	inner	1925.1.dn.b.1707.1	yes	16
5.3	odd	4	inner	1925.1.dn.b.1707.2	yes	16
5.4	even	2	inner	1925.1.dn.b.1168.1	yes	16
7.6	odd	2	inner	1925.1.dn.b.1168.1	yes	16
11.6	odd	10	inner	1925.1.dn.b.468.2	yes	16
35.13	even	4	inner	1925.1.dn.b.1707.1	yes	16
35.27	even	4	inner	1925.1.dn.b.1707.2	yes	16
35.34	odd	2	CM	1925.1.dn.b.1168.2	yes	16
55.17	even	20	inner	1925.1.dn.b.1007.1	yes	16
55.28	even	20	inner	1925.1.dn.b.1007.2	yes	16
55.39	odd	10	inner	1925.1.dn.b.468.1	✓	16
77.6	even	10	inner	1925.1.dn.b.468.1	✓	16
385.83	odd	20	inner	1925.1.dn.b.1007.1	yes	16
385.237	odd	20	inner	1925.1.dn.b.1007.2	yes	16
385.314	even	10	inner	1925.1.dn.b.468.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1925.1.dn.b.468.1	✓	16	55.39	odd	10	inner
1925.1.dn.b.468.1	✓	16	77.6	even	10	inner
1925.1.dn.b.468.2	yes	16	11.6	odd	10	inner
1925.1.dn.b.468.2	yes	16	385.314	even	10	inner
1925.1.dn.b.1007.1	yes	16	55.17	even	20	inner
1925.1.dn.b.1007.1	yes	16	385.83	odd	20	inner
1925.1.dn.b.1007.2	yes	16	55.28	even	20	inner
1925.1.dn.b.1007.2	yes	16	385.237	odd	20	inner
1925.1.dn.b.1168.1	yes	16	5.4	even	2	inner
1925.1.dn.b.1168.1	yes	16	7.6	odd	2	inner
1925.1.dn.b.1168.2	yes	16	1.1	even	1	trivial
1925.1.dn.b.1168.2	yes	16	35.34	odd	2	CM
1925.1.dn.b.1707.1	yes	16	5.2	odd	4	inner
1925.1.dn.b.1707.1	yes	16	35.13	even	4	inner
1925.1.dn.b.1707.2	yes	16	5.3	odd	4	inner
1925.1.dn.b.1707.2	yes	16	35.27	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1925.dn (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(0.183900 - 1.16110i) q^{3} +(-0.587785 + 0.809017i) q^{4} +(-0.987688 + 0.156434i) q^{7} +(-0.363271 - 0.118034i) q^{9} +O(q^{10})\)	\(q+(0.183900 - 1.16110i) q^{3} +(-0.587785 + 0.809017i) q^{4} +(-0.987688 + 0.156434i) q^{7} +(-0.363271 - 0.118034i) q^{9} +(-0.309017 + 0.951057i) q^{11} +(0.831254 + 0.831254i) q^{12} +(1.44168 - 0.734572i) q^{13} +(-0.309017 - 0.951057i) q^{16} +(0.550672 + 0.280582i) q^{17} +1.17557i q^{21} +(0.329843 - 0.647354i) q^{27} +(0.453990 - 0.891007i) q^{28} +(1.53884 + 1.11803i) q^{29} +(1.04744 + 0.533698i) q^{33} +(0.309017 - 0.224514i) q^{36} +(-0.587785 - 1.80902i) q^{39} +(-0.587785 - 0.809017i) q^{44} +(1.87869 + 0.297556i) q^{47} +(-1.16110 + 0.183900i) q^{48} +(0.951057 - 0.309017i) q^{49} +(0.427051 - 0.587785i) q^{51} +(-0.253116 + 1.59811i) q^{52} +(0.377263 + 0.0597526i) q^{63} +(0.951057 + 0.309017i) q^{64} +(-0.550672 + 0.280582i) q^{68} +(-0.500000 - 1.53884i) q^{71} +(0.0966818 + 0.610425i) q^{73} +(0.156434 - 0.987688i) q^{77} +(-0.363271 + 1.11803i) q^{79} +(-1.00000 - 0.726543i) q^{81} +(0.734572 - 1.44168i) q^{83} +(-0.951057 - 0.690983i) q^{84} +(1.58114 - 1.58114i) q^{87} +(-1.30902 + 0.951057i) q^{91} +(-1.69480 + 0.863541i) q^{97} +(0.224514 - 0.309017i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 4 q^{11} + 4 q^{16} - 4 q^{36} - 20 q^{51} - 8 q^{71} - 16 q^{81} - 12 q^{91}+O(q^{100}) \) 16 * q + 4 * q^11 + 4 * q^16 - 4 * q^36 - 20 * q^51 - 8 * q^71 - 16 * q^81 - 12 * q^91

Introduction

L-functions

Modular forms

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Database

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