Properties

Label 7800.2.a.bf
Level $7800$
Weight $2$
Character orbit 7800.a
Self dual yes
Analytic conductor $62.283$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1849.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 14x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q - q^{3} + (\beta_1 - 2) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + (\beta_1 - 2) q^{7} + q^{9} + (\beta_{2} + 1) q^{11} - q^{13} - \beta_1 q^{17} + ( - \beta_{2} + \beta_1 + 1) q^{19} + ( - \beta_1 + 2) q^{21} + (\beta_1 - 2) q^{23} - q^{27} + ( - \beta_{2} - \beta_1 - 1) q^{29} + ( - 2 \beta_1 + 4) q^{31} + ( - \beta_{2} - 1) q^{33} + (\beta_{2} - 2 \beta_1 - 1) q^{37} + q^{39} + ( - \beta_{2} + 2 \beta_1 + 1) q^{41} - 4 q^{43} - 2 \beta_1 q^{47} + (\beta_{2} - 2 \beta_1 + 6) q^{49} + \beta_1 q^{51} + ( - \beta_{2} - 2 \beta_1 + 5) q^{53} + (\beta_{2} - \beta_1 - 1) q^{57} + (\beta_{2} + 7) q^{61} + (\beta_1 - 2) q^{63} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{67} + ( - \beta_1 + 2) q^{69} + ( - \beta_{2} + 2 \beta_1 - 1) q^{71} + ( - \beta_{2} - \beta_1 - 9) q^{73} + ( - 3 \beta_{2} + 4 \beta_1 - 3) q^{77} + (\beta_{2} - 7) q^{79} + q^{81} + (2 \beta_{2} + 2 \beta_1 - 2) q^{83} + (\beta_{2} + \beta_1 + 1) q^{87} + ( - \beta_{2} - 4 \beta_1 + 5) q^{89} + ( - \beta_1 + 2) q^{91} + (2 \beta_1 - 4) q^{93} + ( - \beta_1 - 8) q^{97} + (\beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 5 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 5 q^{7} + 3 q^{9} + 3 q^{11} - 3 q^{13} - q^{17} + 4 q^{19} + 5 q^{21} - 5 q^{23} - 3 q^{27} - 4 q^{29} + 10 q^{31} - 3 q^{33} - 5 q^{37} + 3 q^{39} + 5 q^{41} - 12 q^{43} - 2 q^{47} + 16 q^{49} + q^{51} + 13 q^{53} - 4 q^{57} + 21 q^{61} - 5 q^{63} - 8 q^{67} + 5 q^{69} - q^{71} - 28 q^{73} - 5 q^{77} - 21 q^{79} + 3 q^{81} - 4 q^{83} + 4 q^{87} + 11 q^{89} + 5 q^{91} - 10 q^{93} - 25 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 14x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 9 \) 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
−2.88824
−0.615072
4.50331
0 −1.00000 0 0 0 −4.88824 0 1.00000 0
1.2 0 −1.00000 0 0 0 −2.61507 0 1.00000 0
1.3 0 −1.00000 0 0 0 2.50331 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.bf 3
5.b even 2 1 1560.2.a.p 3
15.d odd 2 1 4680.2.a.bl 3
20.d odd 2 1 3120.2.a.bh 3
60.h even 2 1 9360.2.a.db 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.a.p 3 5.b even 2 1
3120.2.a.bh 3 20.d odd 2 1
4680.2.a.bl 3 15.d odd 2 1
7800.2.a.bf 3 1.a even 1 1 trivial
9360.2.a.db 3 60.h even 2 1

Hecke kernels

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

\( T_{7}^{3} + 5T_{7}^{2} - 6T_{7} - 32 \) Copy content Toggle raw display
\( T_{11}^{3} - 3T_{11}^{2} - 40T_{11} + 128 \) Copy content Toggle raw display
\( T_{17}^{3} + T_{17}^{2} - 14T_{17} + 8 \) Copy content Toggle raw display
\( T_{19}^{3} - 4T_{19}^{2} - 52T_{19} + 176 \) 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
$7$ \( T^{3} + 5 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$11$ \( T^{3} - 3 T^{2} + \cdots + 128 \) Copy content Toggle raw display
$13$ \( (T + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + T^{2} - 14T + 8 \) Copy content Toggle raw display
$19$ \( T^{3} - 4 T^{2} + \cdots + 176 \) Copy content Toggle raw display
$23$ \( T^{3} + 5 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$29$ \( T^{3} + 4 T^{2} + \cdots - 176 \) Copy content Toggle raw display
$31$ \( T^{3} - 10 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$37$ \( T^{3} + 5 T^{2} + \cdots - 548 \) Copy content Toggle raw display
$41$ \( T^{3} - 5 T^{2} + \cdots + 548 \) Copy content Toggle raw display
$43$ \( (T + 4)^{3} \) Copy content Toggle raw display
$47$ \( T^{3} + 2 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$53$ \( T^{3} - 13 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$59$ \( T^{3} \) Copy content Toggle raw display
$61$ \( T^{3} - 21 T^{2} + \cdots + 44 \) Copy content Toggle raw display
$67$ \( T^{3} + 8 T^{2} + \cdots - 1408 \) Copy content Toggle raw display
$71$ \( T^{3} + T^{2} + \cdots + 352 \) Copy content Toggle raw display
$73$ \( T^{3} + 28 T^{2} + \cdots + 176 \) Copy content Toggle raw display
$79$ \( T^{3} + 21 T^{2} + \cdots + 128 \) Copy content Toggle raw display
$83$ \( T^{3} + 4 T^{2} + \cdots + 512 \) Copy content Toggle raw display
$89$ \( T^{3} - 11 T^{2} + \cdots + 2596 \) Copy content Toggle raw display
$97$ \( T^{3} + 25 T^{2} + \cdots + 472 \) Copy content Toggle raw display
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