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Properties

Label 371.1.o.a
Level $371$
Weight $1$
Character orbit 371.o
Analytic conductor $0.185$
Analytic rank $0$
Dimension $12$
Projective image $D_{26}$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [371,1,Mod(6,371)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(371, base_ring=CyclotomicField(26))
 
chi = DirichletCharacter(H, H._module([13, 9]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("371.6");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 371 = 7 \cdot 53 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 371.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: \(0.185153119687\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{26}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{26} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{26}^{9} - \zeta_{26}^{8}) q^{2} + ( - \zeta_{26}^{5} + \cdots - \zeta_{26}^{3}) q^{4}+ \cdots + \zeta_{26}^{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{26}^{9} - \zeta_{26}^{8}) q^{2} + ( - \zeta_{26}^{5} + \cdots - \zeta_{26}^{3}) q^{4}+ \cdots + (\zeta_{26}^{12} - \zeta_{26}^{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{4} - q^{7} - 13 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{4} - q^{7} - 13 q^{8} + q^{9} + 2 q^{11} - q^{16} + q^{25} - q^{28} - 2 q^{29} + q^{36} + 2 q^{37} - 11 q^{43} + 15 q^{44} - q^{49} - q^{53} - 13 q^{56} + q^{63} + 12 q^{64} - 13 q^{74} + 2 q^{77} - 13 q^{79} - q^{81} + 13 q^{88} + 13 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/371\mathbb{Z}\right)^\times\).

\(n\) \(213\) \(267\)
\(\chi(n)\) \(-1\) \(-\zeta_{26}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
0.970942 − 0.239316i
0.970942 + 0.239316i
−0.120537 + 0.992709i
0.748511 + 0.663123i
−0.568065 + 0.822984i
−0.120537 − 0.992709i
0.748511 − 0.663123i
−0.568065 − 0.822984i
−0.885456 − 0.464723i
0.354605 + 0.935016i
−0.885456 + 0.464723i
0.354605 − 0.935016i
0.922670 + 1.75800i 0 −1.67118 + 2.42112i 0 0 0.885456 − 0.464723i −3.82734 − 0.464723i 0.748511 − 0.663123i 0
62.1 0.922670 − 1.75800i 0 −1.67118 − 2.42112i 0 0 0.885456 + 0.464723i −3.82734 + 0.464723i 0.748511 + 0.663123i 0
90.1 0.317391 − 1.28771i 0 −0.671996 − 0.352691i 0 0 −0.970942 − 0.239316i 0.212015 − 0.239316i 0.354605 − 0.935016i 0
146.1 −1.85640 + 0.225408i 0 2.42446 − 0.597576i 0 0 0.120537 + 0.992709i −2.61756 + 0.992709i −0.568065 + 0.822984i 0
188.1 −0.869047 + 0.329586i 0 −0.101894 + 0.0902706i 0 0 −0.354605 − 0.935016i 0.490734 − 0.935016i 0.970942 + 0.239316i 0
202.1 0.317391 + 1.28771i 0 −0.671996 + 0.352691i 0 0 −0.970942 + 0.239316i 0.212015 + 0.239316i 0.354605 + 0.935016i 0
216.1 −1.85640 − 0.225408i 0 2.42446 + 0.597576i 0 0 0.120537 − 0.992709i −2.61756 − 0.992709i −0.568065 − 0.822984i 0
223.1 −0.869047 − 0.329586i 0 −0.101894 − 0.0902706i 0 0 −0.354605 + 0.935016i 0.490734 + 0.935016i 0.970942 − 0.239316i 0
237.1 0.393906 − 0.271894i 0 −0.273369 + 0.720815i 0 0 0.568065 + 0.822984i 0.202847 + 0.822984i −0.120537 − 0.992709i 0
272.1 1.09148 + 1.23202i 0 −0.206022 + 1.69675i 0 0 −0.748511 + 0.663123i −0.960699 + 0.663123i −0.885456 − 0.464723i 0
335.1 0.393906 + 0.271894i 0 −0.273369 − 0.720815i 0 0 0.568065 − 0.822984i 0.202847 − 0.822984i −0.120537 + 0.992709i 0
356.1 1.09148 − 1.23202i 0 −0.206022 − 1.69675i 0 0 −0.748511 − 0.663123i −0.960699 − 0.663123i −0.885456 + 0.464723i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
53.e even 26 1 inner
371.o odd 26 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 371.1.o.a ✓ 12
3.b odd 2 1 3339.1.cm.a 12
7.b odd 2 1 CM 371.1.o.a ✓ 12
7.c even 3 2 2597.1.be.a 24
7.d odd 6 2 2597.1.be.a 24
21.c even 2 1 3339.1.cm.a 12
53.e even 26 1 inner 371.1.o.a ✓ 12
159.h odd 26 1 3339.1.cm.a 12
371.o odd 26 1 inner 371.1.o.a ✓ 12
371.t even 78 2 2597.1.be.a 24
371.u odd 78 2 2597.1.be.a 24
1113.bb even 26 1 3339.1.cm.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
371.1.o.a ✓ 12 1.a even 1 1 trivial
371.1.o.a ✓ 12 7.b odd 2 1 CM
371.1.o.a ✓ 12 53.e even 26 1 inner
371.1.o.a ✓ 12 371.o odd 26 1 inner
2597.1.be.a 24 7.c even 3 2
2597.1.be.a 24 7.d odd 6 2
2597.1.be.a 24 371.t even 78 2
2597.1.be.a 24 371.u odd 78 2
3339.1.cm.a 12 3.b odd 2 1
3339.1.cm.a 12 21.c even 2 1
3339.1.cm.a 12 159.h odd 26 1
3339.1.cm.a 12 1113.bb even 26 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(371, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 13 T^{9} + \cdots + 13 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} + 13 T^{10} + \cdots + 13 \) Copy content Toggle raw display
$29$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{12} \) 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
$43$ \( T^{12} + 11 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} + 13 T^{5} + \cdots + 13 \) Copy content Toggle raw display
$71$ \( T^{12} + 13 T^{9} + \cdots + 13 \) Copy content Toggle raw display
$73$ \( T^{12} \) Copy content Toggle raw display
$79$ \( T^{12} + 13 T^{11} + \cdots + 13 \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} \) Copy content Toggle raw display
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