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Properties

Label 3380.1.bq.a
Level $3380$
Weight $1$
Character orbit 3380.bq
Analytic conductor $1.687$
Analytic rank $0$
Dimension $12$
Projective image $D_{26}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3380,1,Mod(259,3380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3380, base_ring=CyclotomicField(26))
 
chi = DirichletCharacter(H, H._module([13, 13, 17]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3380.259");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3380.bq (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: \(1.68683974270\)
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + \zeta_{26}^{10} q^{2} - \zeta_{26}^{7} q^{4} - \zeta_{26}^{2} q^{5} + \zeta_{26}^{4} q^{8} - \zeta_{26}^{10} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{26}^{10} q^{2} - \zeta_{26}^{7} q^{4} - \zeta_{26}^{2} q^{5} + \zeta_{26}^{4} q^{8} - \zeta_{26}^{10} q^{9} - \zeta_{26}^{12} q^{10} + \zeta_{26}^{9} q^{13} - \zeta_{26} q^{16} + ( - \zeta_{26}^{11} - \zeta_{26}^{10}) q^{17} + \zeta_{26}^{7} q^{18} + \zeta_{26}^{9} q^{20} + \zeta_{26}^{4} q^{25} - \zeta_{26}^{6} q^{26} + (\zeta_{26}^{5} - \zeta_{26}^{2}) q^{29} - \zeta_{26}^{11} q^{32} + (\zeta_{26}^{8} + \zeta_{26}^{7}) q^{34} - \zeta_{26}^{4} q^{36} + ( - \zeta_{26}^{3} - \zeta_{26}) q^{37} - \zeta_{26}^{6} q^{40} + ( - \zeta_{26}^{8} + \zeta_{26}^{2}) q^{41} + \zeta_{26}^{12} q^{45} + \zeta_{26}^{8} q^{49} - \zeta_{26} q^{50} + \zeta_{26}^{3} q^{52} + ( - \zeta_{26}^{8} + 1) q^{53} + ( - \zeta_{26}^{12} - \zeta_{26}^{2}) q^{58} + ( - \zeta_{26}^{4} + \zeta_{26}) q^{61} + \zeta_{26}^{8} q^{64} - \zeta_{26}^{11} q^{65} + ( - \zeta_{26}^{5} - \zeta_{26}^{4}) q^{68} + \zeta_{26} q^{72} - \zeta_{26}^{3} q^{73} + ( - \zeta_{26}^{11} + 1) q^{74} + \zeta_{26}^{3} q^{80} - \zeta_{26}^{7} q^{81} + (\zeta_{26}^{12} + \zeta_{26}^{5}) q^{82} + (\zeta_{26}^{12} - 1) q^{85} + (\zeta_{26}^{12} + \zeta_{26}) q^{89} - \zeta_{26}^{9} q^{90} + ( - \zeta_{26}^{12} + \zeta_{26}^{3}) q^{97} - \zeta_{26}^{5} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} - q^{4} + q^{5} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{2} - q^{4} + q^{5} - q^{8} + q^{9} + q^{10} + q^{13} - q^{16} + q^{18} + q^{20} - q^{25} + q^{26} + 2 q^{29} - q^{32} + q^{36} - 2 q^{37} + q^{40} - q^{45} - q^{49} - q^{50} + q^{52} + 13 q^{53} + 2 q^{58} + 2 q^{61} - q^{64} - q^{65} + q^{72} - 2 q^{73} + 11 q^{74} + q^{80} - q^{81} - 13 q^{85} - q^{90} + 2 q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1691\) \(1861\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{26}^{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
259.1
0.354605 − 0.935016i
0.748511 + 0.663123i
−0.885456 + 0.464723i
−0.120537 − 0.992709i
0.970942 + 0.239316i
−0.568065 + 0.822984i
−0.568065 − 0.822984i
0.970942 − 0.239316i
−0.120537 + 0.992709i
−0.885456 − 0.464723i
0.748511 − 0.663123i
0.354605 + 0.935016i
0.885456 + 0.464723i 0 0.568065 + 0.822984i 0.748511 + 0.663123i 0 0 0.120537 + 0.992709i −0.885456 − 0.464723i 0.354605 + 0.935016i
519.1 0.568065 + 0.822984i 0 −0.354605 + 0.935016i −0.120537 − 0.992709i 0 0 −0.970942 + 0.239316i −0.568065 − 0.822984i 0.748511 − 0.663123i
779.1 0.120537 + 0.992709i 0 −0.970942 + 0.239316i −0.568065 + 0.822984i 0 0 −0.354605 − 0.935016i −0.120537 − 0.992709i −0.885456 − 0.464723i
1039.1 −0.354605 + 0.935016i 0 −0.748511 − 0.663123i 0.970942 − 0.239316i 0 0 0.885456 − 0.464723i 0.354605 − 0.935016i −0.120537 + 0.992709i
1299.1 −0.748511 + 0.663123i 0 0.120537 − 0.992709i −0.885456 − 0.464723i 0 0 0.568065 + 0.822984i 0.748511 − 0.663123i 0.970942 − 0.239316i
1559.1 −0.970942 + 0.239316i 0 0.885456 − 0.464723i 0.354605 + 0.935016i 0 0 −0.748511 + 0.663123i 0.970942 − 0.239316i −0.568065 − 0.822984i
1819.1 −0.970942 − 0.239316i 0 0.885456 + 0.464723i 0.354605 − 0.935016i 0 0 −0.748511 − 0.663123i 0.970942 + 0.239316i −0.568065 + 0.822984i
2079.1 −0.748511 − 0.663123i 0 0.120537 + 0.992709i −0.885456 + 0.464723i 0 0 0.568065 − 0.822984i 0.748511 + 0.663123i 0.970942 + 0.239316i
2339.1 −0.354605 − 0.935016i 0 −0.748511 + 0.663123i 0.970942 + 0.239316i 0 0 0.885456 + 0.464723i 0.354605 + 0.935016i −0.120537 − 0.992709i
2599.1 0.120537 − 0.992709i 0 −0.970942 − 0.239316i −0.568065 − 0.822984i 0 0 −0.354605 + 0.935016i −0.120537 + 0.992709i −0.885456 + 0.464723i
2859.1 0.568065 − 0.822984i 0 −0.354605 − 0.935016i −0.120537 + 0.992709i 0 0 −0.970942 − 0.239316i −0.568065 + 0.822984i 0.748511 + 0.663123i
3119.1 0.885456 − 0.464723i 0 0.568065 − 0.822984i 0.748511 − 0.663123i 0 0 0.120537 − 0.992709i −0.885456 + 0.464723i 0.354605 − 0.935016i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 259.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
845.v even 26 1 inner
3380.bq odd 26 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3380.1.bq.a ✓ 12
4.b odd 2 1 CM 3380.1.bq.a ✓ 12
5.b even 2 1 3380.1.bq.b yes 12
20.d odd 2 1 3380.1.bq.b yes 12
169.h even 26 1 3380.1.bq.b yes 12
676.p odd 26 1 3380.1.bq.b yes 12
845.v even 26 1 inner 3380.1.bq.a ✓ 12
3380.bq odd 26 1 inner 3380.1.bq.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3380.1.bq.a ✓ 12 1.a even 1 1 trivial
3380.1.bq.a ✓ 12 4.b odd 2 1 CM
3380.1.bq.a ✓ 12 845.v even 26 1 inner
3380.1.bq.a ✓ 12 3380.bq odd 26 1 inner
3380.1.bq.b yes 12 5.b even 2 1
3380.1.bq.b yes 12 20.d odd 2 1
3380.1.bq.b yes 12 169.h even 26 1
3380.1.bq.b yes 12 676.p odd 26 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{12} + 26T_{17}^{7} + 13T_{17}^{6} - 13T_{17}^{3} + 65T_{17}^{2} - 52T_{17} + 13 \) acting on \(S_{1}^{\mathrm{new}}(3380, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - T^{11} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} - T^{11} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{12} + 26 T^{7} + \cdots + 13 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} \) 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
$41$ \( T^{12} + 13 T^{8} + \cdots + 13 \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} - 13 T^{11} + \cdots + 13 \) 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
$67$ \( T^{12} \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} + 2 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} + 13 T^{10} + \cdots + 13 \) Copy content Toggle raw display
$97$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
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