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Properties

Label 507.1.o.a
Level $507$
Weight $1$
Character orbit 507.o
Analytic conductor $0.253$
Analytic rank $0$
Dimension $12$
Projective image $D_{13}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,1,Mod(14,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(26))
 
chi = DirichletCharacter(H, H._module([13, 18]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.14");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 507.o (of order \(26\), degree \(12\), 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: \(0.253025961405\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{26})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{13}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{13} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{26}^{7} q^{3} + \zeta_{26}^{2} q^{4} + (\zeta_{26}^{10} - \zeta_{26}^{9}) q^{7} - \zeta_{26} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{26}^{7} q^{3} + \zeta_{26}^{2} q^{4} + (\zeta_{26}^{10} - \zeta_{26}^{9}) q^{7} - \zeta_{26} q^{9} - \zeta_{26}^{9} q^{12} + q^{13} + \zeta_{26}^{4} q^{16} + (\zeta_{26}^{12} - \zeta_{26}) q^{19} + (\zeta_{26}^{4} - \zeta_{26}^{3}) q^{21} - \zeta_{26}^{5} q^{25} + \zeta_{26}^{8} q^{27} + (\zeta_{26}^{12} - \zeta_{26}^{11}) q^{28} + (\zeta_{26}^{10} + \zeta_{26}^{6}) q^{31} - \zeta_{26}^{3} q^{36} + ( - \zeta_{26}^{11} - \zeta_{26}^{5}) q^{37} - \zeta_{26}^{7} q^{39} + (\zeta_{26}^{8} + \zeta_{26}^{2}) q^{43} - \zeta_{26}^{11} q^{48} + ( - \zeta_{26}^{7} + \cdots - \zeta_{26}^{5}) q^{49} + \cdots + \zeta_{26}^{4} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{3} - q^{4} - 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{3} - q^{4} - 2 q^{7} - q^{9} - q^{12} + 12 q^{13} - q^{16} - 2 q^{19} - 2 q^{21} - q^{25} - q^{27} - 2 q^{28} - 2 q^{31} - q^{36} - 2 q^{37} - q^{39} - 2 q^{43} - q^{48} - 3 q^{49} - q^{52} - 2 q^{57} - 2 q^{61} - 2 q^{63} - q^{64} - 2 q^{67} - 2 q^{73} - q^{75} - 2 q^{76} - 2 q^{79} - q^{81} - 2 q^{84} - 2 q^{91} + 11 q^{93} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(\zeta_{26}^{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1
−0.568065 + 0.822984i
0.748511 − 0.663123i
−0.885456 + 0.464723i
0.970942 − 0.239316i
0.970942 + 0.239316i
−0.885456 − 0.464723i
0.748511 + 0.663123i
−0.568065 − 0.822984i
0.354605 + 0.935016i
−0.120537 − 0.992709i
−0.120537 + 0.992709i
0.354605 − 0.935016i
0 0.885456 − 0.464723i −0.354605 − 0.935016i 0 0 −1.71945 − 0.423807i 0 0.568065 − 0.822984i 0
53.1 0 −0.354605 − 0.935016i 0.120537 − 0.992709i 0 0 −0.402877 − 0.583668i 0 −0.748511 + 0.663123i 0
92.1 0 −0.970942 + 0.239316i 0.568065 − 0.822984i 0 0 −0.234068 + 1.92773i 0 0.885456 − 0.464723i 0
131.1 0 0.120537 + 0.992709i 0.885456 − 0.464723i 0 0 −0.180446 + 0.159861i 0 −0.970942 + 0.239316i 0
209.1 0 0.120537 − 0.992709i 0.885456 + 0.464723i 0 0 −0.180446 − 0.159861i 0 −0.970942 − 0.239316i 0
248.1 0 −0.970942 − 0.239316i 0.568065 + 0.822984i 0 0 −0.234068 − 1.92773i 0 0.885456 + 0.464723i 0
287.1 0 −0.354605 + 0.935016i 0.120537 + 0.992709i 0 0 −0.402877 + 0.583668i 0 −0.748511 − 0.663123i 0
326.1 0 0.885456 + 0.464723i −0.354605 + 0.935016i 0 0 −1.71945 + 0.423807i 0 0.568065 + 0.822984i 0
365.1 0 0.568065 − 0.822984i −0.748511 + 0.663123i 0 0 1.00599 + 0.527986i 0 −0.354605 − 0.935016i 0
404.1 0 −0.748511 − 0.663123i −0.970942 + 0.239316i 0 0 0.530851 + 1.39974i 0 0.120537 + 0.992709i 0
443.1 0 −0.748511 + 0.663123i −0.970942 − 0.239316i 0 0 0.530851 − 1.39974i 0 0.120537 − 0.992709i 0
482.1 0 0.568065 + 0.822984i −0.748511 − 0.663123i 0 0 1.00599 − 0.527986i 0 −0.354605 + 0.935016i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
169.g even 13 1 inner
507.o odd 26 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.1.o.a ✓ 12
3.b odd 2 1 CM 507.1.o.a ✓ 12
169.g even 13 1 inner 507.1.o.a ✓ 12
507.o odd 26 1 inner 507.1.o.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.1.o.a ✓ 12 1.a even 1 1 trivial
507.1.o.a ✓ 12 3.b odd 2 1 CM
507.1.o.a ✓ 12 169.g even 13 1 inner
507.1.o.a ✓ 12 507.o odd 26 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(507, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( (T - 1)^{12} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( (T^{6} + T^{5} - 5 T^{4} + \cdots - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} + 2 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
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