LMFDB - Embedded newform 1397.1.l.a.1142.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1397,1,Mod(126,1397)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1397, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([4, 5]))

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

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

chi := DirichletCharacter("1397.126");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1397 = 11 \cdot 127 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1397.l (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:	$0.697193822648$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$5$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{50})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{15} + x^{10} - x^{5} + 1 $ x^20 - x^15 + x^10 - x^5 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{25}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{25} - \cdots)$

Embedding invariants

Embedding label			1142.2
Root			$-0.0627905 + 0.998027i$ of defining polynomial
Character	$\chi$	$=$	1397.1142
Dual form			1397.1.l.a.126.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.393950 + 1.21245i) q^{2} +(-0.505828 - 0.367505i) q^{4} +(-0.386520 + 0.280823i) q^{8} +(0.309017 - 0.951057i) q^{9} +O(q^{10})$	\(q+(-0.393950 + 1.21245i) q^{2} +(-0.505828 - 0.367505i) q^{4} +(-0.386520 + 0.280823i) q^{8} +(0.309017 - 0.951057i) q^{9} +(0.728969 - 0.684547i) q^{11} +(0.450527 - 1.38658i) q^{13} +(-0.381424 - 1.17390i) q^{16} +(0.598617 + 1.84235i) q^{17} +(1.03137 + 0.749337i) q^{18} +(1.50441 - 1.09302i) q^{19} +(0.542804 + 1.15352i) q^{22} +(-0.809017 + 0.587785i) q^{25} +(1.50368 + 1.09249i) q^{26} +(-0.393950 + 1.21245i) q^{31} +1.09580 q^{32} -2.46959 q^{34} +(-0.505828 + 0.367505i) q^{36} +(-1.56720 - 1.13864i) q^{37} +(0.732570 + 2.25462i) q^{38} +(1.60528 - 1.16630i) q^{41} +(-0.620307 + 0.0783630i) q^{44} +(-1.17950 + 0.856954i) q^{47} +(0.309017 + 0.951057i) q^{49} +(-0.393950 - 1.21245i) q^{50} +(-0.737465 + 0.535800i) q^{52} +(0.0388067 + 0.119435i) q^{61} +(-1.31484 - 0.955291i) q^{62} +(-0.0502653 + 0.154701i) q^{64} +(0.374278 - 1.15191i) q^{68} +(0.331159 + 1.01920i) q^{71} +(0.147638 + 0.454382i) q^{72} +(0.688925 + 0.500534i) q^{73} +(1.99794 - 1.45159i) q^{74} -1.16266 q^{76} +(-0.115808 + 0.356420i) q^{79} +(-0.809017 - 0.587785i) q^{81} +(0.781687 + 2.40578i) q^{82} +(-0.0895243 + 0.469303i) q^{88} +(-0.574354 - 1.76768i) q^{94} -1.27485 q^{98} +(-0.425779 - 0.904827i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 5 q^{4} - 5 q^{9}+O(q^{10}) $ 20 * q - 5 * q^4 - 5 * q^9	$ 20 q - 5 q^{4} - 5 q^{9} - 5 q^{16} - 5 q^{25} - 10 q^{32} - 10 q^{34} - 5 q^{36} + 15 q^{38} - 5 q^{44} - 5 q^{49} + 15 q^{52} - 10 q^{62} - 5 q^{64} + 15 q^{74} - 5 q^{81} - 5 q^{88}+O(q^{100}) $ 20 * q - 5 * q^4 - 5 * q^9 - 5 * q^16 - 5 * q^25 - 10 * q^32 - 10 * q^34 - 5 * q^36 + 15 * q^38 - 5 * q^44 - 5 * q^49 + 15 * q^52 - 10 * q^62 - 5 * q^64 + 15 * q^74 - 5 * q^81 - 5 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1397\mathbb{Z}\right)^\times$.

$n$	$255$	$892$
$\chi(n)$	$e\left(\frac{3}{5}\right)$	$-1$

Coefficient data

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

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

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

$2$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	0.535827	+	0.844328i	$0.320000\pi$
$2$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	−0.929776	+	0.368125i	$0.880000\pi$
$3$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$3$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$4$	−0.505828	−	0.367505i	−0.505828	−	0.367505i
$5$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$5$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$6$		0			0
$7$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$7$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$8$	−0.386520	+	0.280823i	−0.386520	+	0.280823i
$9$	0.309017	−	0.951057i	0.309017	−	0.951057i
$10$		0			0
$11$	0.728969	−	0.684547i	0.728969	−	0.684547i
$11$	0.728969	−	0.684547i	0.728969	−	0.684547i
$12$		0			0
$13$	0.450527	−	1.38658i	0.450527	−	1.38658i	−0.425779	−	0.904827i	$-0.640000\pi$
$13$	0.450527	−	1.38658i	0.450527	−	1.38658i	0.876307	−	0.481754i	$-0.160000\pi$
$14$		0			0
$15$		0			0
$16$	−0.381424	−	1.17390i	−0.381424	−	1.17390i
$17$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.535827	+	0.844328i	$0.320000\pi$
$17$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.0627905	+	0.998027i	$0.480000\pi$
$18$	1.03137	+	0.749337i	1.03137	+	0.749337i
$19$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.535827	−	0.844328i	$-0.320000\pi$
$19$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.968583	−	0.248690i	$-0.0800000\pi$
$20$		0			0
$21$		0			0
$22$	0.542804	+	1.15352i	0.542804	+	1.15352i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$26$	1.50368	+	1.09249i	1.50368	+	1.09249i
$27$		0			0
$28$		0			0
$29$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$29$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$30$		0			0
$31$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	0.535827	+	0.844328i	$0.320000\pi$
$31$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	−0.929776	+	0.368125i	$0.880000\pi$
$32$	1.09580			1.09580
$33$		0			0
$34$	−2.46959			−2.46959
$35$		0			0
$36$	−0.505828	+	0.367505i	−0.505828	+	0.367505i
$37$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.929776	−	0.368125i	$-0.880000\pi$
$37$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.637424	−	0.770513i	$-0.720000\pi$
$38$	0.732570	+	2.25462i	0.732570	+	2.25462i
$39$		0			0
$40$		0			0
$41$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.728969	−	0.684547i	$-0.240000\pi$
$41$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.876307	−	0.481754i	$-0.160000\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	−0.620307	+	0.0783630i	−0.620307	+	0.0783630i
$45$		0			0
$46$		0			0
$47$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.992115	−	0.125333i	$-0.960000\pi$
$47$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.187381	+	0.982287i	$0.560000\pi$
$48$		0			0
$49$	0.309017	+	0.951057i	0.309017	+	0.951057i
$50$	−0.393950	−	1.21245i	−0.393950	−	1.21245i
$51$		0			0
$52$	−0.737465	+	0.535800i	−0.737465	+	0.535800i
$53$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$53$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$59$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$60$		0			0
$61$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	0.968583	−	0.248690i	$-0.0800000\pi$
$61$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	−0.929776	+	0.368125i	$0.880000\pi$
$62$	−1.31484	−	0.955291i	−1.31484	−	0.955291i
$63$		0			0
$64$	−0.0502653	+	0.154701i	−0.0502653	+	0.154701i
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	0.374278	−	1.15191i	0.374278	−	1.15191i
$69$		0			0
$70$		0			0
$71$	0.331159	+	1.01920i	0.331159	+	1.01920i	0.968583	+	0.248690i	$0.0800000\pi$
$71$	0.331159	+	1.01920i	0.331159	+	1.01920i	−0.637424	+	0.770513i	$0.720000\pi$
$72$	0.147638	+	0.454382i	0.147638	+	0.454382i
$73$	0.688925	+	0.500534i	0.688925	+	0.500534i	0.876307	−	0.481754i	$-0.160000\pi$
$73$	0.688925	+	0.500534i	0.688925	+	0.500534i	−0.187381	+	0.982287i	$0.560000\pi$
$74$	1.99794	−	1.45159i	1.99794	−	1.45159i
$75$		0			0
$76$	−1.16266			−1.16266
$77$		0			0
$78$		0			0
$79$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	−0.992115	−	0.125333i	$-0.960000\pi$
$79$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	0.876307	+	0.481754i	$0.160000\pi$
$80$		0			0
$81$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$82$	0.781687	+	2.40578i	0.781687	+	2.40578i
$83$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$83$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−0.0895243	+	0.469303i	−0.0895243	+	0.469303i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	−0.574354	−	1.76768i	−0.574354	−	1.76768i
$95$		0			0
$96$		0			0
$97$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$97$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$98$	−1.27485			−1.27485
$99$	−0.425779	−	0.904827i	−0.425779	−	0.904827i
$100$	0.625237			0.625237
$101$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$101$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$102$		0			0
$103$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.992115	−	0.125333i	$-0.960000\pi$
$103$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.425779	−	0.904827i	$-0.640000\pi$
$104$	0.215246	+	0.662460i	0.215246	+	0.662460i
$105$		0			0
$106$		0			0
$107$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	−0.929776	−	0.368125i	$-0.880000\pi$
$107$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	0.0627905	+	0.998027i	$0.480000\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.688925	−	0.500534i	0.688925	−	0.500534i	−0.187381	−	0.982287i	$-0.560000\pi$
$113$	0.688925	−	0.500534i	0.688925	−	0.500534i	0.876307	+	0.481754i	$0.160000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.17950	−	0.856954i	−1.17950	−	0.856954i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.0627905	−	0.998027i	0.0627905	−	0.998027i
$122$	−0.160097			−0.160097
$123$		0			0
$124$	0.644853	−	0.468513i	0.644853	−	0.468513i
$125$		0			0
$126$		0			0
$127$	0.309017	+	0.951057i	0.309017	+	0.951057i
$127$	0.309017	+	0.951057i	0.309017	+	0.951057i
$128$	0.718755	+	0.522206i	0.718755	+	0.522206i
$129$		0			0
$130$		0			0
$131$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$131$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.748754	−	0.544002i	−0.748754	−	0.544002i
$137$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$137$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$138$		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			0
$142$	−1.36620			−1.36620
$143$	−0.620759	−	1.31918i	−0.620759	−	1.31918i
$144$	−1.23432			−1.23432
$145$		0			0
$146$	−0.878275	+	0.638104i	−0.878275	+	0.638104i
$147$		0			0
$148$	0.374278	+	1.15191i	0.374278	+	1.15191i
$149$	0.331159	+	1.01920i	0.331159	+	1.01920i	0.968583	+	0.248690i	$0.0800000\pi$
$149$	0.331159	+	1.01920i	0.331159	+	1.01920i	−0.637424	+	0.770513i	$0.720000\pi$
$150$		0			0
$151$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$151$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$152$	−0.274540	+	0.844947i	−0.274540	+	0.844947i
$153$	1.93717			1.93717
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.728969	−	0.684547i	$-0.240000\pi$
$157$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.876307	−	0.481754i	$-0.160000\pi$
$158$	−0.386520	−	0.280823i	−0.386520	−	0.280823i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	1.03137	−	0.749337i	1.03137	−	0.749337i
$163$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$163$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$164$	−1.24061			−1.24061
$165$		0			0
$166$		0			0
$167$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$167$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$168$		0			0
$169$	−0.910614	−	0.661600i	−0.910614	−	0.661600i
$170$		0			0
$171$	−0.574633	−	1.76854i	−0.574633	−	1.76854i
$172$		0			0
$173$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$173$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$174$		0			0
$175$		0			0
$176$	−1.08164	−	0.594636i	−1.08164	−	0.594636i
$177$		0			0
$178$		0			0
$179$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	−0.637424	−	0.770513i	$-0.720000\pi$
$179$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	0.535827	+	0.844328i	$0.320000\pi$
$180$		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
$187$	1.69755	+	0.933237i	1.69755	+	0.933237i
$188$	0.911557			0.911557
$189$		0			0
$190$		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			0
$193$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$193$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$194$		0			0
$195$		0			0
$196$	0.193209	−	0.594636i	0.193209	−	0.594636i
$197$	−0.374763			−0.374763			−0.187381	−	0.982287i	$-0.560000\pi$
$197$	−0.374763			−0.374763			−0.187381	+	0.982287i	$0.560000\pi$
$198$	1.26480	−	0.159781i	1.26480	−	0.159781i
$199$	−1.98423			−1.98423			−0.992115	−	0.125333i	$-0.960000\pi$
$199$	−1.98423			−1.98423			−0.992115	+	0.125333i	$0.960000\pi$
$200$	0.147638	−	0.454382i	0.147638	−	0.454382i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	1.80760	−	1.31330i	1.80760	−	1.31330i
$207$		0			0
$208$	−1.79955			−1.79955
$209$	0.348445	−	1.82662i	0.348445	−	1.82662i
$210$		0			0
$211$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$211$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$212$		0			0
$213$		0			0
$214$	−0.422178	−	1.29933i	−0.422178	−	1.29933i
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.82427			2.82427
$222$		0			0
$223$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$223$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$224$		0			0
$225$	0.309017	+	0.951057i	0.309017	+	0.951057i
$226$	0.335471	+	1.03247i	0.335471	+	1.03247i
$227$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$227$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$228$		0			0
$229$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$229$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$233$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$234$	1.50368	−	1.09249i	1.50368	−	1.09249i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$239$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	1.18532	+	0.469303i	1.18532	+	0.469303i
$243$		0			0
$244$	0.0242634	−	0.0746750i	0.0242634	−	0.0746750i
$245$		0			0
$246$		0			0
$247$	−0.837780	−	2.57842i	−0.837780	−	2.57842i
$248$	−0.188216	−	0.579268i	−0.188216	−	0.579268i
$249$		0			0
$250$		0			0
$251$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$251$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$252$		0			0
$253$		0			0
$254$	−1.27485			−1.27485
$255$		0			0
$256$	−1.04790	+	0.761344i	−1.04790	+	0.761344i
$257$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$257$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	0.637424	−	1.96179i	0.637424	−	1.96179i
$263$	0.125581			0.125581			0.0627905	−	0.998027i	$-0.480000\pi$
$263$	0.125581			0.125581			0.0627905	+	0.998027i	$0.480000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	−0.637424	−	0.770513i	$-0.720000\pi$
$269$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	0.0627905	−	0.998027i	$-0.480000\pi$
$270$		0			0
$271$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$271$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$272$	1.93442	−	1.40544i	1.93442	−	1.40544i
$273$		0			0
$274$		0			0
$275$	−0.187381	+	0.982287i	−0.187381	+	0.982287i
$276$		0			0
$277$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$277$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$278$		0			0
$279$	1.03137	+	0.749337i	1.03137	+	0.749337i
$280$		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			0		0.809017	−	0.587785i	$-0.200000\pi$
$283$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$284$	0.207053	−	0.637244i	0.207053	−	0.637244i
$285$		0			0
$286$	1.84399	−	0.232950i	1.84399	−	0.232950i
$287$		0			0
$288$	0.338621	−	1.04217i	0.338621	−	1.04217i
$289$	−2.22691	+	1.61795i	−2.22691	+	1.61795i
$290$		0			0
$291$		0			0
$292$	−0.164529	−	0.506367i	−0.164529	−	0.506367i
$293$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$293$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$294$		0			0
$295$		0			0
$296$	0.925511			0.925511
$297$		0			0
$298$	−1.36620			−1.36620
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−1.85692	−	1.34913i	−1.85692	−	1.34913i
$305$		0			0
$306$	−0.763146	+	2.34872i	−0.763146	+	2.34872i
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		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		−0.309017	−	0.951057i	$-0.600000\pi$
$313$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$314$	0.781687	+	2.40578i	0.781687	+	2.40578i
$315$		0			0
$316$	0.189565	−	0.137727i	0.189565	−	0.137727i
$317$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$317$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.91429	+	2.11736i	2.91429	+	2.11736i
$324$	0.193209	+	0.594636i	0.193209	+	0.594636i
$325$	0.450527	+	1.38658i	0.450527	+	1.38658i
$326$	0.637424	+	0.463116i	0.637424	+	0.463116i
$327$		0			0
$328$	−0.292947	+	0.901598i	−0.292947	+	0.901598i
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	−1.56720	+	1.13864i	−1.56720	+	1.13864i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$337$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$338$	1.16089	−	0.843439i	1.16089	−	0.843439i
$339$		0			0
$340$		0			0
$341$	0.542804	+	1.15352i	0.542804	+	1.15352i
$342$	2.37065			2.37065
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$347$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$348$		0			0
$349$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$349$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$350$		0			0
$351$		0			0
$352$	0.798803	−	0.750126i	0.798803	−	0.750126i
$353$	−1.85955			−1.85955			−0.929776	−	0.368125i	$-0.880000\pi$
$353$	−1.85955			−1.85955			−0.929776	+	0.368125i	$0.880000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−0.0494726	−	0.152261i	−0.0494726	−	0.152261i
$359$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$359$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$360$		0			0
$361$	0.759544	−	2.33764i	0.759544	−	2.33764i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	0.535827	−	0.844328i	$-0.320000\pi$
$367$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	−0.637424	+	0.770513i	$0.720000\pi$
$368$		0			0
$369$	−0.613161	−	1.88711i	−0.613161	−	1.88711i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$	−1.80026	+	1.69055i	−1.80026	+	1.69055i
$375$		0			0
$376$	0.215246	−	0.662460i	0.215246	−	0.662460i
$377$		0			0
$378$		0			0
$379$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$379$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$380$		0			0
$381$		0			0
$382$	−1.66880	+	1.21245i	−1.66880	+	1.21245i
$383$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$383$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.187381	−	0.982287i	$-0.560000\pi$
$389$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.992115	+	0.125333i	$0.960000\pi$
$390$		0			0
$391$		0			0
$392$	−0.386520	−	0.280823i	−0.386520	−	0.280823i
$393$		0			0
$394$	0.147638	−	0.454382i	0.147638	−	0.454382i
$395$		0			0
$396$	−0.117158	+	0.614163i	−0.117158	+	0.614163i
$397$	0.125581			0.125581			0.0627905	−	0.998027i	$-0.480000\pi$
$397$	0.125581			0.125581			0.0627905	+	0.998027i	$0.480000\pi$
$398$	0.781687	−	2.40578i	0.781687	−	2.40578i
$399$		0			0
$400$	0.998582	+	0.725513i	0.998582	+	0.725513i
$401$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$401$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$402$		0			0
$403$	1.50368	+	1.09249i	1.50368	+	1.09249i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.92189	+	0.242791i	−1.92189	+	0.242791i
$408$		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
$412$	0.338621	+	1.04217i	0.338621	+	1.04217i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	0.493688	−	1.51941i	0.493688	−	1.51941i
$417$		0			0
$418$	2.07741	+	1.14207i	2.07741	+	1.14207i
$419$	−1.85955			−1.85955			−0.929776	−	0.368125i	$-0.880000\pi$
$419$	−1.85955			−1.85955			−0.929776	+	0.368125i	$0.880000\pi$
$420$		0			0
$421$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$421$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$422$	−0.878275	−	0.638104i	−0.878275	−	0.638104i
$423$	0.450527	+	1.38658i	0.450527	+	1.38658i
$424$		0			0
$425$	−1.56720	−	1.13864i	−1.56720	−	1.13864i
$426$		0			0
$427$		0			0
$428$	0.670038			0.670038
$429$		0			0
$430$		0			0
$431$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.187381	+	0.982287i	$0.560000\pi$
$431$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.425779	+	0.904827i	$0.640000\pi$
$432$		0			0
$433$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$433$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	−0.929776	+	0.368125i	$0.880000\pi$
$434$		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
$441$	1.00000			1.00000
$442$	−1.11262	+	3.42429i	−1.11262	+	3.42429i
$443$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.637424	+	0.770513i	$0.720000\pi$
$443$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.929776	+	0.368125i	$0.880000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.187381	+	0.982287i	$0.560000\pi$
$449$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.425779	+	0.904827i	$0.640000\pi$
$450$	−1.27485			−1.27485
$451$	0.371808	−	1.94908i	0.371808	−	1.94908i
$452$	−0.532426			−0.532426
$453$		0			0
$454$	0.637424	−	0.463116i	0.637424	−	0.463116i
$455$		0			0
$456$		0			0
$457$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	0.876307	−	0.481754i	$-0.160000\pi$
$457$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	−0.992115	+	0.125333i	$0.960000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	1.93717			1.93717			0.968583	−	0.248690i	$-0.0800000\pi$
$463$	1.93717			1.93717			0.968583	+	0.248690i	$0.0800000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$467$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$468$	0.281687	+	0.866942i	0.281687	+	0.866942i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.574633	+	1.76854i	−0.574633	+	1.76854i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	−0.992115	−	0.125333i	$-0.960000\pi$
$479$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	0.728969	−	0.684547i	$-0.240000\pi$
$480$		0			0
$481$	−2.28488	+	1.66006i	−2.28488	+	1.66006i
$482$		0			0
$483$		0			0
$484$	−0.398541	+	0.481754i	−0.398541	+	0.481754i
$485$		0			0
$486$		0			0
$487$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$487$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$488$	−0.0485396	−	0.0352661i	−0.0485396	−	0.0352661i
$489$		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			0
$494$	3.45626			3.45626
$495$		0			0
$496$	1.57356			1.57356
$497$		0			0
$498$		0			0
$499$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$499$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$500$		0			0
$501$		0			0
$502$	−0.878275	−	0.638104i	−0.878275	−	0.638104i
$503$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.425779	+	0.904827i	$0.640000\pi$
$503$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.992115	+	0.125333i	$0.960000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	0.193209	−	0.594636i	0.193209	−	0.594636i
$509$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$509$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$510$		0			0
$511$		0			0
$512$	−0.235733	−	0.725513i	−0.235733	−	0.725513i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−0.273190	+	1.43211i	−0.273190	+	1.43211i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.968583	−	0.248690i	$-0.0800000\pi$
$521$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.0627905	+	0.998027i	$0.480000\pi$
$522$		0			0
$523$	0.541587	+	1.66683i	0.541587	+	1.66683i	0.728969	+	0.684547i	$0.240000\pi$
$523$	0.541587	+	1.66683i	0.541587	+	1.66683i	−0.187381	+	0.982287i	$0.560000\pi$
$524$	0.818446	+	0.594636i	0.818446	+	0.594636i
$525$		0			0
$526$	−0.0494726	+	0.152261i	−0.0494726	+	0.152261i
$527$	−2.46959			−2.46959
$528$		0			0
$529$	1.00000			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.893950	−	2.75129i	−0.893950	−	2.75129i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	2.37065			2.37065
$539$	0.876307	+	0.481754i	0.876307	+	0.481754i
$540$		0			0
$541$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$541$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$542$	0.637424	−	0.463116i	0.637424	−	0.463116i
$543$		0			0
$544$	0.655964	+	2.01885i	0.655964	+	2.01885i
$545$		0			0
$546$		0			0
$547$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$547$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$548$		0			0
$549$	0.125581			0.125581
$550$	−1.11716	−	0.614163i	−1.11716	−	0.614163i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$557$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$558$	−1.31484	+	0.955291i	−1.31484	+	0.955291i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$563$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−0.414216	−	0.300945i	−0.414216	−	0.300945i
$569$	0.303189	−	0.220280i	0.303189	−	0.220280i	−0.425779	−	0.904827i	$-0.640000\pi$
$569$	0.303189	−	0.220280i	0.303189	−	0.220280i	0.728969	+	0.684547i	$0.240000\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$	−0.170809	+	0.895411i	−0.170809	+	0.895411i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.131596	+	0.0956103i	0.131596	+	0.0956103i
$577$	0.331159	+	1.01920i	0.331159	+	1.01920i	0.968583	+	0.248690i	$0.0800000\pi$
$577$	0.331159	+	1.01920i	0.331159	+	1.01920i	−0.637424	+	0.770513i	$0.720000\pi$
$578$	−1.08439	−	3.33741i	−1.08439	−	3.33741i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.406845			−0.406845
$585$		0			0
$586$		0			0
$587$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.968583	+	0.248690i	$0.0800000\pi$
$587$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.535827	+	0.844328i	$0.320000\pi$
$588$		0			0
$589$	0.732570	+	2.25462i	0.732570	+	2.25462i
$590$		0			0
$591$		0			0
$592$	−0.738883	+	2.27405i	−0.738883	+	2.27405i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$	0.207053	−	0.637244i	0.207053	−	0.637244i
$597$		0			0
$598$		0			0
$599$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$599$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$600$		0			0
$601$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$601$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	0.331159	−	1.01920i	0.331159	−	1.01920i	−0.637424	−	0.770513i	$-0.720000\pi$
$607$	0.331159	−	1.01920i	0.331159	−	1.01920i	0.968583	−	0.248690i	$-0.0800000\pi$
$608$	1.64853	−	1.19773i	1.64853	−	1.19773i
$609$		0			0
$610$		0			0
$611$	0.656841	+	2.02155i	0.656841	+	2.02155i
$612$	−0.979872	−	0.711919i	−0.979872	−	0.711919i
$613$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$613$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$619$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.309017	−	0.951057i	0.309017	−	0.951057i
$626$		0			0
$627$		0			0
$628$	−1.24061			−1.24061
$629$	1.15962	−	3.56895i	1.15962	−	3.56895i
$630$		0			0
$631$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$631$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$632$	−0.0553291	−	0.170285i	−0.0553291	−	0.170285i
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.45794			1.45794
$638$		0			0
$639$	1.07165			1.07165
$640$		0			0
$641$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$641$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$642$		0			0
$643$	0.541587	+	1.66683i	0.541587	+	1.66683i	0.728969	+	0.684547i	$0.240000\pi$
$643$	0.541587	+	1.66683i	0.541587	+	1.66683i	−0.187381	+	0.982287i	$0.560000\pi$
$644$		0			0
$645$		0			0
$646$	−3.71528	+	2.69931i	−3.71528	+	2.69931i
$647$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$647$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$648$	0.477765			0.477765
$649$		0			0
$650$	−1.85865			−1.85865
$651$		0			0
$652$	−0.312619	+	0.227131i	−0.312619	+	0.227131i
$653$	0.688925	+	0.500534i	0.688925	+	0.500534i	0.876307	−	0.481754i	$-0.160000\pi$
$653$	0.688925	+	0.500534i	0.688925	+	0.500534i	−0.187381	+	0.982287i	$0.560000\pi$
$654$		0			0
$655$		0			0
$656$	−1.98142	−	1.43958i	−1.98142	−	1.43958i
$657$	0.688925	−	0.500534i	0.688925	−	0.500534i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−0.851559			−0.851559			−0.425779	−	0.904827i	$-0.640000\pi$
$661$	−0.851559			−0.851559			−0.425779	+	0.904827i	$0.640000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.763146	−	2.34872i	−0.763146	−	2.34872i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	0.110048	+	0.0604991i	0.110048	+	0.0604991i
$672$		0			0
$673$	0.331159	−	1.01920i	0.331159	−	1.01920i	−0.637424	−	0.770513i	$-0.720000\pi$
$673$	0.331159	−	1.01920i	0.331159	−	1.01920i	0.968583	−	0.248690i	$-0.0800000\pi$
$674$		0			0
$675$		0			0
$676$	0.217472	+	0.669311i	0.217472	+	0.669311i
$677$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$677$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	−1.61242	+	0.203696i	−1.61242	+	0.203696i
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	−0.359282	+	1.10576i	−0.359282	+	1.10576i
$685$		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	$0.600000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	3.10969	+	2.25932i	3.10969	+	2.25932i
$698$		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
$703$	−3.60226			−3.60226
$704$	0.0692581	+	0.147181i	0.0692581	+	0.147181i
$705$		0			0
$706$	0.732570	−	2.25462i	0.732570	−	2.25462i
$707$		0			0
$708$		0			0
$709$	0.450527	+	1.38658i	0.450527	+	1.38658i	0.876307	+	0.481754i	$0.160000\pi$
$709$	0.450527	+	1.38658i	0.450527	+	1.38658i	−0.425779	+	0.904827i	$0.640000\pi$
$710$		0			0
$711$	0.303189	+	0.220280i	0.303189	+	0.220280i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.0785180			0.0785180
$717$		0			0
$718$		0			0
$719$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.929776	−	0.368125i	$-0.880000\pi$
$719$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.637424	−	0.770513i	$-0.720000\pi$
$720$		0			0
$721$		0			0
$722$	2.53505	+	1.84182i	2.53505	+	1.84182i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$733$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$734$	0.129521	−	0.0941025i	0.129521	−	0.0941025i
$735$		0			0
$736$		0			0
$737$		0			0
$738$	2.52959			2.52959
$739$	0.541587	−	1.66683i	0.541587	−	1.66683i	−0.187381	−	0.982287i	$-0.560000\pi$
$739$	0.541587	−	1.66683i	0.541587	−	1.66683i	0.728969	−	0.684547i	$-0.240000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$743$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$	−0.515699	−	1.09592i	−0.515699	−	1.09592i
$749$		0			0
$750$		0			0
$751$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$751$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$752$	1.45587	+	1.05775i	1.45587	+	1.05775i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	0.535827	+	0.844328i	$0.320000\pi$
$757$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	−0.929776	+	0.368125i	$0.880000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$761$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$762$		0			0
$763$		0			0
$764$	−0.312619	−	0.962141i	−0.312619	−	0.962141i
$765$		0			0
$766$	0.637424	+	0.463116i	0.637424	+	0.463116i
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	−0.637424	−	0.770513i	$-0.720000\pi$
$773$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	0.535827	+	0.844328i	$0.320000\pi$
$774$		0			0
$775$	−0.393950	−	1.21245i	−0.393950	−	1.21245i
$776$		0			0
$777$		0			0
$778$	1.50368	−	1.09249i	1.50368	−	1.09249i
$779$	1.14020	−	3.50919i	1.14020	−	3.50919i
$780$		0			0
$781$	0.939097	+	0.516273i	0.939097	+	0.516273i
$782$		0			0
$783$		0			0
$784$	0.998582	−	0.725513i	0.998582	−	0.725513i
$785$		0			0
$786$		0			0
$787$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.425779	−	0.904827i	$-0.640000\pi$
$787$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.187381	−	0.982287i	$-0.560000\pi$
$788$	0.189565	+	0.137727i	0.189565	+	0.137727i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	0.418669	+	0.230165i	0.418669	+	0.230165i
$793$	0.183089			0.183089
$794$	−0.0494726	+	0.152261i	−0.0494726	+	0.152261i
$795$		0			0
$796$	1.00368	+	0.729215i	1.00368	+	0.729215i
$797$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$797$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$798$		0			0
$799$	−2.28488	−	1.66006i	−2.28488	−	1.66006i
$800$	−0.886520	+	0.644095i	−0.886520	+	0.644095i
$801$		0			0
$802$		0			0
$803$	0.844844	−	0.106729i	0.844844	−	0.106729i
$804$		0			0
$805$		0			0
$806$	−1.91696	+	1.39275i	−1.91696	+	1.39275i
$807$		0			0
$808$		0			0
$809$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	−0.637424	−	0.770513i	$-0.720000\pi$
$809$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	0.0627905	−	0.998027i	$-0.480000\pi$
$810$		0			0
$811$	0.303189	−	0.220280i	0.303189	−	0.220280i	−0.425779	−	0.904827i	$-0.640000\pi$
$811$	0.303189	−	0.220280i	0.303189	−	0.220280i	0.728969	+	0.684547i	$0.240000\pi$
$812$		0			0
$813$		0			0
$814$	0.462756	−	2.42585i	0.462756	−	2.42585i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$821$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$822$		0			0
$823$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	−0.929776	−	0.368125i	$-0.880000\pi$
$823$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	0.968583	+	0.248690i	$0.0800000\pi$
$824$	0.837338			0.837338
$825$		0			0
$826$		0			0
$827$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$827$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$828$		0			0
$829$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$829$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$830$		0			0
$831$		0			0
$832$	0.191859	+	0.139394i	0.191859	+	0.139394i
$833$	−1.56720	+	1.13864i	−1.56720	+	1.13864i
$834$		0			0
$835$		0			0
$836$	−0.847544	+	0.795897i	−0.847544	+	0.795897i
$837$		0			0
$838$	0.732570	−	2.25462i	0.732570	−	2.25462i
$839$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$839$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$840$		0			0
$841$	0.309017	+	0.951057i	0.309017	+	0.951057i
$842$		0			0
$843$		0			0
$844$	0.430742	−	0.312952i	0.430742	−	0.312952i
$845$		0			0
$846$	−1.85865			−1.85865
$847$		0			0
$848$		0			0
$849$		0			0
$850$	1.99794	−	1.45159i	1.99794	−	1.45159i
$851$		0			0
$852$		0			0
$853$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$853$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$854$		0			0
$855$		0			0
$856$	0.158216	−	0.486940i	0.158216	−	0.486940i
$857$		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
$862$	−2.04648	−	1.48686i	−2.04648	−	1.48686i
$863$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$863$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$864$		0			0
$865$		0			0
$866$	1.10528	−	0.803030i	1.10528	−	0.803030i
$867$		0			0
$868$		0			0
$869$	0.159566	+	0.339095i	0.159566	+	0.339095i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.535827	−	0.844328i	$-0.320000\pi$
$877$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.968583	−	0.248690i	$-0.0800000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.393950	+	1.21245i	−0.393950	+	1.21245i
$883$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	−0.929776	−	0.368125i	$-0.880000\pi$
$883$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	0.0627905	+	0.998027i	$0.480000\pi$
$884$	−1.42859	−	1.03793i	−1.42859	−	1.03793i
$885$		0			0
$886$	−0.763146	−	2.34872i	−0.763146	−	2.34872i
$887$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$887$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.992115	+	0.125333i	−0.992115	+	0.125333i
$892$		0			0
$893$	−0.837780	+	2.57842i	−0.837780	+	2.57842i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−2.04648	−	1.48686i	−2.04648	−	1.48686i
$899$		0			0
$900$	0.193209	−	0.594636i	0.193209	−	0.594636i
$901$		0			0
$902$	2.21670	+	1.21864i	2.21670	+	1.21864i
$903$		0			0
$904$	−0.125722	+	0.386933i	−0.125722	+	0.386933i
$905$		0			0
$906$		0			0
$907$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.425779	−	0.904827i	$-0.640000\pi$
$907$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.187381	−	0.982287i	$-0.560000\pi$
$908$	0.119410	+	0.367505i	0.119410	+	0.367505i
$909$		0			0
$910$		0			0
$911$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$911$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$912$		0			0
$913$		0			0
$914$	0.477765			0.477765
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.535827	+	0.844328i	$0.320000\pi$
$919$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.0627905	+	0.998027i	$0.480000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.56240			1.56240
$924$		0			0
$925$	1.93717			1.93717
$926$	−0.763146	+	2.34872i	−0.763146	+	2.34872i
$927$	−1.41789	+	1.03016i	−1.41789	+	1.03016i
$928$		0			0
$929$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$929$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$930$		0			0
$931$	1.50441	+	1.09302i	1.50441	+	1.09302i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	0.696552			0.696552
$937$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$937$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	−0.992115	−	0.125333i	$-0.960000\pi$
$941$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	0.728969	−	0.684547i	$-0.240000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$	1.00441	−	0.729747i	1.00441	−	0.729747i
$950$	−1.91789	−	1.39343i	−1.91789	−	1.39343i
$951$		0			0
$952$		0			0
$953$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.992115	−	0.125333i	$-0.960000\pi$
$953$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.425779	−	0.904827i	$-0.640000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$	1.08561			1.08561
$959$		0			0
$960$		0			0
$961$	−0.505828	−	0.367505i	−0.505828	−	0.367505i
$962$	−1.11262	−	3.42429i	−1.11262	−	3.42429i
$963$	0.331159	+	1.01920i	0.331159	+	1.01920i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	0.255999	+	0.403391i	0.255999	+	0.403391i
$969$		0			0
$970$		0			0
$971$	0.303189	−	0.220280i	0.303189	−	0.220280i	−0.425779	−	0.904827i	$-0.640000\pi$
$971$	0.303189	−	0.220280i	0.303189	−	0.220280i	0.728969	+	0.684547i	$0.240000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.125403	−	0.0911106i	0.125403	−	0.0911106i
$977$	0.541587	−	1.66683i	0.541587	−	1.66683i	−0.187381	−	0.982287i	$-0.560000\pi$
$977$	0.541587	−	1.66683i	0.541587	−	1.66683i	0.728969	−	0.684547i	$-0.240000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	0.303189	+	0.220280i	0.303189	+	0.220280i	0.728969	−	0.684547i	$-0.240000\pi$
$983$	0.303189	+	0.220280i	0.303189	+	0.220280i	−0.425779	+	0.904827i	$0.640000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−0.523811	+	1.61212i	−0.523811	+	1.61212i
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−0.431690	+	1.32860i	−0.431690	+	1.32860i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$997$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1397.1.l.a.1142.2	yes	20
11.5	even	5	inner	1397.1.l.a.126.2	✓	20
127.126	odd	2	CM	1397.1.l.a.1142.2	yes	20
1397.126	odd	10	inner	1397.1.l.a.126.2	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1397.1.l.a.126.2	✓	20	11.5	even	5	inner
1397.1.l.a.126.2	✓	20	1397.126	odd	10	inner
1397.1.l.a.1142.2	yes	20	1.1	even	1	trivial
1397.1.l.a.1142.2	yes	20	127.126	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:	\( N \)	\(=\)	\( 1397 = 11 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1397.l (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.393950 + 1.21245i) q^{2} +(-0.505828 - 0.367505i) q^{4} +(-0.386520 + 0.280823i) q^{8} +(0.309017 - 0.951057i) q^{9} +O(q^{10})\)	\(q+(-0.393950 + 1.21245i) q^{2} +(-0.505828 - 0.367505i) q^{4} +(-0.386520 + 0.280823i) q^{8} +(0.309017 - 0.951057i) q^{9} +(0.728969 - 0.684547i) q^{11} +(0.450527 - 1.38658i) q^{13} +(-0.381424 - 1.17390i) q^{16} +(0.598617 + 1.84235i) q^{17} +(1.03137 + 0.749337i) q^{18} +(1.50441 - 1.09302i) q^{19} +(0.542804 + 1.15352i) q^{22} +(-0.809017 + 0.587785i) q^{25} +(1.50368 + 1.09249i) q^{26} +(-0.393950 + 1.21245i) q^{31} +1.09580 q^{32} -2.46959 q^{34} +(-0.505828 + 0.367505i) q^{36} +(-1.56720 - 1.13864i) q^{37} +(0.732570 + 2.25462i) q^{38} +(1.60528 - 1.16630i) q^{41} +(-0.620307 + 0.0783630i) q^{44} +(-1.17950 + 0.856954i) q^{47} +(0.309017 + 0.951057i) q^{49} +(-0.393950 - 1.21245i) q^{50} +(-0.737465 + 0.535800i) q^{52} +(0.0388067 + 0.119435i) q^{61} +(-1.31484 - 0.955291i) q^{62} +(-0.0502653 + 0.154701i) q^{64} +(0.374278 - 1.15191i) q^{68} +(0.331159 + 1.01920i) q^{71} +(0.147638 + 0.454382i) q^{72} +(0.688925 + 0.500534i) q^{73} +(1.99794 - 1.45159i) q^{74} -1.16266 q^{76} +(-0.115808 + 0.356420i) q^{79} +(-0.809017 - 0.587785i) q^{81} +(0.781687 + 2.40578i) q^{82} +(-0.0895243 + 0.469303i) q^{88} +(-0.574354 - 1.76768i) q^{94} -1.27485 q^{98} +(-0.425779 - 0.904827i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{4} - 5 q^{9}+O(q^{10}) \) 20 * q - 5 * q^4 - 5 * q^9	\( 20 q - 5 q^{4} - 5 q^{9} - 5 q^{16} - 5 q^{25} - 10 q^{32} - 10 q^{34} - 5 q^{36} + 15 q^{38} - 5 q^{44} - 5 q^{49} + 15 q^{52} - 10 q^{62} - 5 q^{64} + 15 q^{74} - 5 q^{81} - 5 q^{88}+O(q^{100}) \) 20 * q - 5 * q^4 - 5 * q^9 - 5 * q^16 - 5 * q^25 - 10 * q^32 - 10 * q^34 - 5 * q^36 + 15 * q^38 - 5 * q^44 - 5 * q^49 + 15 * q^52 - 10 * q^62 - 5 * q^64 + 15 * q^74 - 5 * q^81 - 5 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more