Properties

Label 304.7.e.a
Level $304$
Weight $7$
Character orbit 304.e
Self dual yes
Analytic conductor $69.936$
Analytic rank $0$
Dimension $1$
CM discriminant -19
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [304,7,Mod(113,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.113");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 304.e (of order \(2\), degree \(1\), not 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: yes
Analytic conductor: \(69.9364414204\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 54 q^{5} - 610 q^{7} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 54 q^{5} - 610 q^{7} + 729 q^{9} + 1062 q^{11} - 9630 q^{17} + 6859 q^{19} - 20610 q^{23} - 12709 q^{25} + 32940 q^{35} + 142630 q^{43} - 39366 q^{45} + 75150 q^{47} + 254451 q^{49} - 57348 q^{55} - 57062 q^{61} - 444690 q^{63} + 384050 q^{73} - 647820 q^{77} + 531441 q^{81} + 1131030 q^{83} + 520020 q^{85} - 370386 q^{95} + 774198 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
0
0 0 0 −54.0000 0 −610.000 0 729.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.7.e.a 1
4.b odd 2 1 19.7.b.a 1
12.b even 2 1 171.7.c.a 1
19.b odd 2 1 CM 304.7.e.a 1
76.d even 2 1 19.7.b.a 1
228.b odd 2 1 171.7.c.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.7.b.a 1 4.b odd 2 1
19.7.b.a 1 76.d even 2 1
171.7.c.a 1 12.b even 2 1
171.7.c.a 1 228.b odd 2 1
304.7.e.a 1 1.a even 1 1 trivial
304.7.e.a 1 19.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(304, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5} + 54 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 54 \) Copy content Toggle raw display
$7$ \( T + 610 \) Copy content Toggle raw display
$11$ \( T - 1062 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 9630 \) Copy content Toggle raw display
$19$ \( T - 6859 \) Copy content Toggle raw display
$23$ \( T + 20610 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 142630 \) Copy content Toggle raw display
$47$ \( T - 75150 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 57062 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 384050 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T - 1131030 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
show more
show less