Properties

Label 7800.2.a.bj
Level $7800$
Weight $2$
Character orbit 7800.a
Self dual yes
Analytic conductor $62.283$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7800,2,Mod(1,7800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7800.a (trivial)

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: \(62.2833135766\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1560)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{9} + ( - \beta_1 - 2) q^{11} - q^{13} + ( - \beta_{2} - \beta_1 + 1) q^{17} + ( - \beta_{2} - \beta_1 - 1) q^{19} + ( - \beta_{2} - \beta_1 + 3) q^{23} - q^{27} + ( - 2 \beta_{2} - 4) q^{29} + (2 \beta_1 + 4) q^{31} + (\beta_1 + 2) q^{33} + ( - 2 \beta_1 - 2) q^{37} + q^{39} + (\beta_{2} - 2 \beta_1 - 3) q^{41} + (\beta_{2} + \beta_1 - 1) q^{43} + ( - \beta_{2} - 2 \beta_1 + 5) q^{47} - 7 q^{49} + (\beta_{2} + \beta_1 - 1) q^{51} + (2 \beta_{2} + 4) q^{53} + (\beta_{2} + \beta_1 + 1) q^{57} + (3 \beta_1 - 2) q^{59} + (\beta_{2} - \beta_1 + 1) q^{61} + (2 \beta_{2} - 2 \beta_1 + 2) q^{67} + (\beta_{2} + \beta_1 - 3) q^{69} + (\beta_{2} + 3) q^{71} + (4 \beta_1 + 2) q^{73} + (\beta_{2} + \beta_1 + 7) q^{79} + q^{81} + ( - \beta_{2} + 2 \beta_1 + 1) q^{83} + (2 \beta_{2} + 4) q^{87} + (\beta_{2} - 2 \beta_1 - 7) q^{89} + ( - 2 \beta_1 - 4) q^{93} + 6 q^{97} + ( - \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 3 q^{9} - 6 q^{11} - 3 q^{13} + 4 q^{17} - 2 q^{19} + 10 q^{23} - 3 q^{27} - 10 q^{29} + 12 q^{31} + 6 q^{33} - 6 q^{37} + 3 q^{39} - 10 q^{41} - 4 q^{43} + 16 q^{47} - 21 q^{49} - 4 q^{51} + 10 q^{53} + 2 q^{57} - 6 q^{59} + 2 q^{61} + 4 q^{67} - 10 q^{69} + 8 q^{71} + 6 q^{73} + 20 q^{79} + 3 q^{81} + 4 q^{83} + 10 q^{87} - 22 q^{89} - 12 q^{93} + 18 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 3\nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 3\beta _1 + 9 ) / 2 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.12489
−1.76156
−0.363328
0 −1.00000 0 0 0 0 0 1.00000 0
1.2 0 −1.00000 0 0 0 0 0 1.00000 0
1.3 0 −1.00000 0 0 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( +1 \)
\(5\) \( -1 \)
\(13\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7800.2.a.bj 3
5.b even 2 1 7800.2.a.bp 3
5.c odd 4 2 1560.2.l.c 6
15.e even 4 2 4680.2.l.e 6
20.e even 4 2 3120.2.l.m 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.l.c 6 5.c odd 4 2
3120.2.l.m 6 20.e even 4 2
4680.2.l.e 6 15.e even 4 2
7800.2.a.bj 3 1.a even 1 1 trivial
7800.2.a.bp 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{3} + 6T_{11}^{2} + 2T_{11} - 20 \) Copy content Toggle raw display
\( T_{17}^{3} - 4T_{17}^{2} - 20T_{17} + 64 \) Copy content Toggle raw display
\( T_{19}^{3} + 2T_{19}^{2} - 24T_{19} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 6 T^{2} + \cdots - 20 \) Copy content Toggle raw display
$13$ \( (T + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 4 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( T^{3} + 2 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{3} - 10 T^{2} + \cdots + 80 \) Copy content Toggle raw display
$29$ \( T^{3} + 10 T^{2} + \cdots - 472 \) Copy content Toggle raw display
$31$ \( T^{3} - 12 T^{2} + \cdots + 160 \) Copy content Toggle raw display
$37$ \( T^{3} + 6 T^{2} + \cdots - 136 \) Copy content Toggle raw display
$41$ \( T^{3} + 10 T^{2} + \cdots - 332 \) Copy content Toggle raw display
$43$ \( T^{3} + 4 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$47$ \( T^{3} - 16 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{3} - 10 T^{2} + \cdots + 472 \) Copy content Toggle raw display
$59$ \( T^{3} + 6 T^{2} + \cdots + 44 \) Copy content Toggle raw display
$61$ \( T^{3} - 2 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$67$ \( T^{3} - 4 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$71$ \( T^{3} - 8 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$73$ \( T^{3} - 6 T^{2} + \cdots + 824 \) Copy content Toggle raw display
$79$ \( T^{3} - 20 T^{2} + \cdots - 160 \) Copy content Toggle raw display
$83$ \( T^{3} - 4 T^{2} + \cdots + 232 \) Copy content Toggle raw display
$89$ \( T^{3} + 22 T^{2} + \cdots - 244 \) Copy content Toggle raw display
$97$ \( (T - 6)^{3} \) Copy content Toggle raw display
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