Properties

Label 6930.2.a.cj
Level $6930$
Weight $2$
Character orbit 6930.a
Self dual yes
Analytic conductor $55.336$
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] = [6930,2,Mod(1,6930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6930, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6930.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6930.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: \(55.3363286007\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} - q^{10} - q^{11} + \beta_1 q^{13} + q^{14} + q^{16} + (\beta_1 - 2) q^{17} + ( - \beta_{2} - \beta_1 - 2) q^{19} - q^{20} - q^{22} + (\beta_{2} - 2 \beta_1 - 2) q^{23} + q^{25} + \beta_1 q^{26} + q^{28} + ( - 2 \beta_1 - 2) q^{29} + (\beta_{2} + \beta_1) q^{31} + q^{32} + (\beta_1 - 2) q^{34} - q^{35} + ( - \beta_1 - 2) q^{37} + ( - \beta_{2} - \beta_1 - 2) q^{38} - q^{40} + (\beta_{2} + \beta_1 - 2) q^{41} + ( - \beta_{2} - \beta_1 - 2) q^{43} - q^{44} + (\beta_{2} - 2 \beta_1 - 2) q^{46} - 6 q^{47} + q^{49} + q^{50} + \beta_1 q^{52} + ( - \beta_{2} + \beta_1 - 2) q^{53} + q^{55} + q^{56} + ( - 2 \beta_1 - 2) q^{58} + ( - \beta_{2} - \beta_1 - 4) q^{59} - 4 q^{61} + (\beta_{2} + \beta_1) q^{62} + q^{64} - \beta_1 q^{65} + (2 \beta_{2} - \beta_1 + 4) q^{67} + (\beta_1 - 2) q^{68} - q^{70} + (4 \beta_1 - 2) q^{71} + ( - \beta_{2} - \beta_1 - 2) q^{73} + ( - \beta_1 - 2) q^{74} + ( - \beta_{2} - \beta_1 - 2) q^{76} - q^{77} + 4 \beta_1 q^{79} - q^{80} + (\beta_{2} + \beta_1 - 2) q^{82} + ( - \beta_{2} + 3 \beta_1) q^{83} + ( - \beta_1 + 2) q^{85} + ( - \beta_{2} - \beta_1 - 2) q^{86} - q^{88} + (\beta_{2} - 4 \beta_1 - 4) q^{89} + \beta_1 q^{91} + (\beta_{2} - 2 \beta_1 - 2) q^{92} - 6 q^{94} + (\beta_{2} + \beta_1 + 2) q^{95} + (\beta_{2} + 3 \beta_1 + 2) q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8} - 3 q^{10} - 3 q^{11} + 3 q^{14} + 3 q^{16} - 6 q^{17} - 6 q^{19} - 3 q^{20} - 3 q^{22} - 6 q^{23} + 3 q^{25} + 3 q^{28} - 6 q^{29} + 3 q^{32} - 6 q^{34} - 3 q^{35} - 6 q^{37} - 6 q^{38} - 3 q^{40} - 6 q^{41} - 6 q^{43} - 3 q^{44} - 6 q^{46} - 18 q^{47} + 3 q^{49} + 3 q^{50} - 6 q^{53} + 3 q^{55} + 3 q^{56} - 6 q^{58} - 12 q^{59} - 12 q^{61} + 3 q^{64} + 12 q^{67} - 6 q^{68} - 3 q^{70} - 6 q^{71} - 6 q^{73} - 6 q^{74} - 6 q^{76} - 3 q^{77} - 3 q^{80} - 6 q^{82} + 6 q^{85} - 6 q^{86} - 3 q^{88} - 12 q^{89} - 6 q^{92} - 18 q^{94} + 6 q^{95} + 6 q^{97} + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} - 2\nu - 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 8 ) / 4 \) 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
−1.53209
−0.347296
1.87939
1.00000 0 1.00000 −1.00000 0 1.00000 1.00000 0 −1.00000
1.2 1.00000 0 1.00000 −1.00000 0 1.00000 1.00000 0 −1.00000
1.3 1.00000 0 1.00000 −1.00000 0 1.00000 1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)
\(11\) \(1\)

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 6930.2.a.cj yes 3
3.b odd 2 1 6930.2.a.ci 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6930.2.a.ci 3 3.b odd 2 1
6930.2.a.cj yes 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6930))\):

\( T_{13}^{3} - 12T_{13} - 8 \) Copy content Toggle raw display
\( T_{17}^{3} + 6T_{17}^{2} - 24 \) Copy content Toggle raw display
\( T_{19}^{3} + 6T_{19}^{2} - 36T_{19} - 152 \) Copy content Toggle raw display
\( T_{23}^{3} + 6T_{23}^{2} - 72T_{23} - 456 \) Copy content Toggle raw display
\( T_{29}^{3} + 6T_{29}^{2} - 36T_{29} - 24 \) Copy content Toggle raw display
\( T_{31}^{3} - 48T_{31} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( (T + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 12T - 8 \) Copy content Toggle raw display
$17$ \( T^{3} + 6T^{2} - 24 \) Copy content Toggle raw display
$19$ \( T^{3} + 6 T^{2} - 36 T - 152 \) Copy content Toggle raw display
$23$ \( T^{3} + 6 T^{2} - 72 T - 456 \) Copy content Toggle raw display
$29$ \( T^{3} + 6 T^{2} - 36 T - 24 \) Copy content Toggle raw display
$31$ \( T^{3} - 48T + 64 \) Copy content Toggle raw display
$37$ \( T^{3} + 6T^{2} - 8 \) Copy content Toggle raw display
$41$ \( T^{3} + 6 T^{2} - 36 T - 24 \) Copy content Toggle raw display
$43$ \( T^{3} + 6 T^{2} - 36 T - 152 \) Copy content Toggle raw display
$47$ \( (T + 6)^{3} \) Copy content Toggle raw display
$53$ \( T^{3} + 6 T^{2} - 36 T - 24 \) Copy content Toggle raw display
$59$ \( T^{3} + 12T^{2} - 192 \) Copy content Toggle raw display
$61$ \( (T + 4)^{3} \) Copy content Toggle raw display
$67$ \( T^{3} - 12 T^{2} - 108 T + 712 \) Copy content Toggle raw display
$71$ \( T^{3} + 6 T^{2} - 180 T - 888 \) Copy content Toggle raw display
$73$ \( T^{3} + 6 T^{2} - 36 T - 152 \) Copy content Toggle raw display
$79$ \( T^{3} - 192T - 512 \) Copy content Toggle raw display
$83$ \( T^{3} - 144T + 576 \) Copy content Toggle raw display
$89$ \( T^{3} + 12 T^{2} - 180 T - 1704 \) Copy content Toggle raw display
$97$ \( T^{3} - 6 T^{2} - 132 T - 296 \) Copy content Toggle raw display
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