Properties

Label 371.1.n.a
Level $371$
Weight $1$
Character orbit 371.n
Analytic conductor $0.185$
Analytic rank $0$
Dimension $12$
Projective image $D_{13}$
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(13,371)]
 
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, 12]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("371.13");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 371 = 7 \cdot 53 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 371.n (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: \(D_{13}\)
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.

\(f(q)\) \(=\) \( q + (\zeta_{26}^{6} - \zeta_{26}) q^{2} + (\zeta_{26}^{12} - \zeta_{26}^{7} + \zeta_{26}^{2}) q^{4} + \zeta_{26}^{10} q^{7} + (\zeta_{26}^{8} - \zeta_{26}^{5} - \zeta_{26}^{3} - 1) q^{8} + \zeta_{26}^{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{26}^{6} - \zeta_{26}) q^{2} + (\zeta_{26}^{12} - \zeta_{26}^{7} + \zeta_{26}^{2}) q^{4} + \zeta_{26}^{10} q^{7} + (\zeta_{26}^{8} - \zeta_{26}^{5} - \zeta_{26}^{3} - 1) q^{8} + \zeta_{26}^{2} q^{9} + (\zeta_{26}^{10} + \zeta_{26}^{6}) q^{11} + ( - \zeta_{26}^{11} - \zeta_{26}^{3}) q^{14} + ( - \zeta_{26}^{11} - \zeta_{26}^{9} - \zeta_{26}^{6} + \zeta_{26}^{4} + \zeta_{26}) q^{16} + (\zeta_{26}^{8} - \zeta_{26}^{3}) q^{18} + (\zeta_{26}^{12} - \zeta_{26}^{11} - \zeta_{26}^{7} - \zeta_{26}^{3}) q^{22} + (\zeta_{26}^{8} - \zeta_{26}^{5}) q^{23} + \zeta_{26}^{4} q^{25} + (\zeta_{26}^{12} - \zeta_{26}^{9} + \zeta_{26}^{4}) q^{28} + ( - \zeta_{26}^{9} - \zeta_{26}) q^{29} + (\zeta_{26}^{12} + \zeta_{26}^{10} - \zeta_{26}^{7} - \zeta_{26}^{5} + \zeta_{26}^{4} - \zeta_{26}^{2}) q^{32} + ( - \zeta_{26}^{9} + \zeta_{26}^{4} - \zeta_{26}) q^{36} + (\zeta_{26}^{8} - \zeta_{26}^{7}) q^{37} + ( - \zeta_{26}^{11} + 1) q^{43} + (\zeta_{26}^{12} - \zeta_{26}^{9} + \zeta_{26}^{8} - \zeta_{26}^{5} + \zeta_{26}^{4} + 1) q^{44} + ( - \zeta_{26}^{11} - \zeta_{26}^{9} + \zeta_{26}^{6} - \zeta_{26}) q^{46} - \zeta_{26}^{7} q^{49} + (\zeta_{26}^{10} - \zeta_{26}^{5}) q^{50} - \zeta_{26}^{11} q^{53} + (\zeta_{26}^{10} - \zeta_{26}^{5} + \zeta_{26}^{2} + 1) q^{56} + (\zeta_{26}^{10} - \zeta_{26}^{7} + \zeta_{26}^{2}) q^{58} + \zeta_{26}^{12} q^{63} + ( - \zeta_{26}^{11} + \zeta_{26}^{10} + \zeta_{26}^{8} + \zeta_{26}^{6} + \zeta_{26}^{5} + \zeta_{26}^{3} - 1) q^{64} + ( - \zeta_{26}^{9} - \zeta_{26}^{5}) q^{67} + ( - \zeta_{26}^{9} + \zeta_{26}^{2}) q^{71} + (\zeta_{26}^{10} - \zeta_{26}^{7} - \zeta_{26}^{5} + \zeta_{26}^{2}) q^{72} + ( - \zeta_{26}^{9} + \zeta_{26}^{8} - \zeta_{26} + 1) q^{74} + ( - \zeta_{26}^{7} - \zeta_{26}^{3}) q^{77} + (\zeta_{26}^{6} + 1) q^{79} + \zeta_{26}^{4} q^{81} + (\zeta_{26}^{12} + \zeta_{26}^{6} + \zeta_{26}^{4} - \zeta_{26}) q^{86} + ( - \zeta_{26}^{11} + \zeta_{26}^{10} - \zeta_{26}^{9} + \zeta_{26}^{6} - \zeta_{26}^{5} + \zeta_{26}^{2} - \zeta_{26} + 1) q^{88} + (\zeta_{26}^{12} + \zeta_{26}^{10} - \zeta_{26}^{7} + \zeta_{26}^{4} + \zeta_{26}^{2}) q^{92} + (\zeta_{26}^{8} + 1) q^{98} + (\zeta_{26}^{12} + \zeta_{26}^{8}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} - 3 q^{4} - q^{7} + 9 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} - 3 q^{4} - q^{7} + 9 q^{8} - q^{9} - 2 q^{11} - 2 q^{14} - 5 q^{16} - 2 q^{18} - 4 q^{22} - 2 q^{23} - q^{25} - 3 q^{28} - 2 q^{29} - 6 q^{32} - 3 q^{36} - 2 q^{37} + 11 q^{43} + 7 q^{44} - 4 q^{46} - q^{49} - 2 q^{50} - q^{53} + 9 q^{56} - 4 q^{58} - q^{63} + 6 q^{64} - 2 q^{67} - 2 q^{71} - 4 q^{72} + 9 q^{74} - 2 q^{77} + 11 q^{79} - q^{81} - 4 q^{86} + 5 q^{88} - 6 q^{92} + 11 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(213\) \(267\)
\(\chi(n)\) \(-1\) \(-\zeta_{26}^{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
−0.885456 + 0.464723i
−0.568065 0.822984i
0.354605 0.935016i
0.354605 + 0.935016i
0.970942 0.239316i
0.748511 0.663123i
0.970942 + 0.239316i
0.748511 + 0.663123i
−0.120537 0.992709i
−0.885456 0.464723i
−0.568065 + 0.822984i
−0.120537 + 0.992709i
−0.0854858 0.704039i 0 0.482579 0.118945i 0 0 0.120537 + 0.992709i −0.376485 0.992709i 0.568065 0.822984i 0
69.1 1.45352 + 0.358261i 0 1.09892 + 0.576756i 0 0 −0.970942 0.239316i 0.270132 + 0.239316i −0.354605 + 0.935016i 0
97.1 0.213460 + 0.112032i 0 −0.535051 0.775155i 0 0 0.885456 + 0.464723i −0.0564276 0.464723i −0.748511 0.663123i 0
153.1 0.213460 0.112032i 0 −0.535051 + 0.775155i 0 0 0.885456 0.464723i −0.0564276 + 0.464723i −0.748511 + 0.663123i 0
174.1 −0.850405 0.753393i 0 0.0350509 + 0.288670i 0 0 −0.748511 0.663123i −0.457721 + 0.663123i 0.885456 0.464723i 0
195.1 −1.10312 + 1.59814i 0 −0.982579 2.59085i 0 0 0.568065 0.822984i 3.33898 + 0.822984i 0.120537 0.992709i 0
258.1 −0.850405 + 0.753393i 0 0.0350509 0.288670i 0 0 −0.748511 + 0.663123i −0.457721 0.663123i 0.885456 + 0.464723i 0
293.1 −1.10312 1.59814i 0 −0.982579 + 2.59085i 0 0 0.568065 + 0.822984i 3.33898 0.822984i 0.120537 + 0.992709i 0
307.1 −0.627974 + 1.65583i 0 −1.59892 1.41652i 0 0 −0.354605 + 0.935016i 1.78152 0.935016i −0.970942 + 0.239316i 0
314.1 −0.0854858 + 0.704039i 0 0.482579 + 0.118945i 0 0 0.120537 0.992709i −0.376485 + 0.992709i 0.568065 + 0.822984i 0
328.1 1.45352 0.358261i 0 1.09892 0.576756i 0 0 −0.970942 + 0.239316i 0.270132 0.239316i −0.354605 0.935016i 0
342.1 −0.627974 1.65583i 0 −1.59892 + 1.41652i 0 0 −0.354605 0.935016i 1.78152 + 0.935016i −0.970942 0.239316i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.1
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.d even 13 1 inner
371.n 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.n.a 12
3.b odd 2 1 3339.1.ck.a 12
7.b odd 2 1 CM 371.1.n.a 12
7.c even 3 2 2597.1.bf.a 24
7.d odd 6 2 2597.1.bf.a 24
21.c even 2 1 3339.1.ck.a 12
53.d even 13 1 inner 371.1.n.a 12
159.j odd 26 1 3339.1.ck.a 12
371.n odd 26 1 inner 371.1.n.a 12
371.q even 39 2 2597.1.bf.a 24
371.v odd 78 2 2597.1.bf.a 24
1113.bd even 26 1 3339.1.ck.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
371.1.n.a 12 1.a even 1 1 trivial
371.1.n.a 12 7.b odd 2 1 CM
371.1.n.a 12 53.d even 13 1 inner
371.1.n.a 12 371.n odd 26 1 inner
2597.1.bf.a 24 7.c even 3 2
2597.1.bf.a 24 7.d odd 6 2
2597.1.bf.a 24 371.q even 39 2
2597.1.bf.a 24 371.v odd 78 2
3339.1.ck.a 12 3.b odd 2 1
3339.1.ck.a 12 21.c even 2 1
3339.1.ck.a 12 159.j odd 26 1
3339.1.ck.a 12 1113.bd even 26 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 2 T^{11} + 4 T^{10} - 5 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + T^{11} + T^{10} + T^{9} + T^{8} + T^{7} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} + 2 T^{11} + 4 T^{10} - 5 T^{9} + \cdots + 1 \) 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
$23$ \( (T^{6} + T^{5} - 5 T^{4} - 4 T^{3} + 6 T^{2} + \cdots - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 2 T^{11} + 4 T^{10} + 8 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( T^{12} + 2 T^{11} + 4 T^{10} + 8 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} - 11 T^{11} + 56 T^{10} - 174 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} + T^{11} + T^{10} + T^{9} + T^{8} + T^{7} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} + 2 T^{11} + 4 T^{10} + 8 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{12} + 2 T^{11} + 4 T^{10} - 5 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{12} \) Copy content Toggle raw display
$79$ \( T^{12} - 11 T^{11} + 56 T^{10} - 174 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} \) Copy content Toggle raw display
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