Properties

Label 2496.2.m.b
Level $2496$
Weight $2$
Character orbit 2496.m
Analytic conductor $19.931$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2496,2,Mod(2209,2496)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2496, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2496.2209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2496.m (of order \(2\), degree \(1\), 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: \(19.9306603445\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1871773696.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 31x^{4} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{3} q^{5} + 2 \beta_{4} q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{3} q^{5} + 2 \beta_{4} q^{7} - q^{9} - \beta_{6} q^{11} - \beta_{5} q^{13} - \beta_{4} q^{15} + 2 q^{17} - 2 \beta_{3} q^{21} - 3 q^{25} - \beta_1 q^{27} - 2 \beta_{5} q^{29} + \beta_{7} q^{33} - 4 \beta_1 q^{35} + 6 \beta_{3} q^{37} - \beta_{2} q^{39} - \beta_{7} q^{41} + 2 \beta_1 q^{43} + \beta_{3} q^{45} - 5 \beta_{4} q^{47} - q^{49} + 2 \beta_1 q^{51} - 2 \beta_{5} q^{53} - 2 \beta_{2} q^{55} - \beta_{6} q^{59} - 2 \beta_{5} q^{61} - 2 \beta_{4} q^{63} - \beta_{7} q^{65} + 2 \beta_{6} q^{67} + 3 \beta_{4} q^{71} - 2 \beta_{7} q^{73} - 3 \beta_1 q^{75} - 4 \beta_{5} q^{77} - 2 \beta_{2} q^{79} + q^{81} + \beta_{6} q^{83} - 2 \beta_{3} q^{85} - 2 \beta_{2} q^{87} + \beta_{7} q^{89} + 2 \beta_{6} q^{91} + 2 \beta_{7} q^{97} + \beta_{6} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{9} + 16 q^{17} - 24 q^{25} - 8 q^{49} + 8 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 31x^{4} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 40\nu^{2} ) / 63 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{4} + 31 ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{7} + 9\nu^{5} - 97\nu^{3} + 171\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{7} + 9\nu^{5} + 97\nu^{3} + 171\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} - 22\nu^{2} ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10\nu^{7} - 9\nu^{5} + 337\nu^{3} - 549\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10\nu^{7} + 9\nu^{5} + 337\nu^{3} + 549\nu ) / 189 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 7\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} - 5\beta_{4} + 5\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{2} - 31 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -19\beta_{7} + 19\beta_{6} + 61\beta_{4} + 61\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -20\beta_{5} - 77\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -97\beta_{7} - 97\beta_{6} + 337\beta_{4} - 337\beta_{3} ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(769\) \(833\) \(1093\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2209.1
−1.62831 + 1.62831i
0.921201 0.921201i
−0.921201 + 0.921201i
1.62831 1.62831i
−1.62831 1.62831i
0.921201 + 0.921201i
−0.921201 0.921201i
1.62831 + 1.62831i
0 1.00000i 0 −1.41421 0 2.82843i 0 −1.00000 0
2209.2 0 1.00000i 0 −1.41421 0 2.82843i 0 −1.00000 0
2209.3 0 1.00000i 0 1.41421 0 2.82843i 0 −1.00000 0
2209.4 0 1.00000i 0 1.41421 0 2.82843i 0 −1.00000 0
2209.5 0 1.00000i 0 −1.41421 0 2.82843i 0 −1.00000 0
2209.6 0 1.00000i 0 −1.41421 0 2.82843i 0 −1.00000 0
2209.7 0 1.00000i 0 1.41421 0 2.82843i 0 −1.00000 0
2209.8 0 1.00000i 0 1.41421 0 2.82843i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2209.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
13.b even 2 1 inner
52.b odd 2 1 inner
104.e even 2 1 inner
104.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2496.2.m.b 8
4.b odd 2 1 inner 2496.2.m.b 8
8.b even 2 1 inner 2496.2.m.b 8
8.d odd 2 1 inner 2496.2.m.b 8
13.b even 2 1 inner 2496.2.m.b 8
52.b odd 2 1 inner 2496.2.m.b 8
104.e even 2 1 inner 2496.2.m.b 8
104.h odd 2 1 inner 2496.2.m.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2496.2.m.b 8 1.a even 1 1 trivial
2496.2.m.b 8 4.b odd 2 1 inner
2496.2.m.b 8 8.b even 2 1 inner
2496.2.m.b 8 8.d odd 2 1 inner
2496.2.m.b 8 13.b even 2 1 inner
2496.2.m.b 8 52.b odd 2 1 inner
2496.2.m.b 8 104.e even 2 1 inner
2496.2.m.b 8 104.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2496, [\chi])\):

\( T_{5}^{2} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 26)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 13)^{4} \) Copy content Toggle raw display
$17$ \( (T - 2)^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{2} + 52)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{2} - 72)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 26)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 50)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 52)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 26)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 52)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 104)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 104)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 52)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 26)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 26)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 104)^{4} \) Copy content Toggle raw display
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