Properties

Label 9360.2.a.cp
Level $9360$
Weight $2$
Character orbit 9360.a
Self dual yes
Analytic conductor $74.740$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9360,2,Mod(1,9360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9360, 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("9360.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9360.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: \(74.7399762919\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2340)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} - \beta q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} - \beta q^{7} + ( - \beta + 2) q^{11} + q^{13} + (\beta + 4) q^{17} + 4 q^{19} + ( - \beta - 4) q^{23} + q^{25} + ( - 2 \beta + 4) q^{29} + 2 \beta q^{31} - \beta q^{35} + ( - 3 \beta + 2) q^{37} + ( - \beta + 2) q^{41} + ( - 2 \beta - 4) q^{43} + (4 \beta - 2) q^{47} + (\beta + 1) q^{49} + (\beta + 4) q^{53} + ( - \beta + 2) q^{55} + ( - 2 \beta - 2) q^{59} + ( - \beta + 10) q^{61} + q^{65} + 4 q^{67} + ( - \beta + 2) q^{71} + (4 \beta - 6) q^{73} + ( - \beta + 8) q^{77} + 5 \beta q^{79} + (2 \beta - 10) q^{83} + (\beta + 4) q^{85} + ( - 3 \beta + 6) q^{89} - \beta q^{91} + 4 q^{95} + ( - \beta - 14) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - q^{7} + 3 q^{11} + 2 q^{13} + 9 q^{17} + 8 q^{19} - 9 q^{23} + 2 q^{25} + 6 q^{29} + 2 q^{31} - q^{35} + q^{37} + 3 q^{41} - 10 q^{43} + 3 q^{49} + 9 q^{53} + 3 q^{55} - 6 q^{59} + 19 q^{61} + 2 q^{65} + 8 q^{67} + 3 q^{71} - 8 q^{73} + 15 q^{77} + 5 q^{79} - 18 q^{83} + 9 q^{85} + 9 q^{89} - q^{91} + 8 q^{95} - 29 q^{97}+O(q^{100}) \) 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
3.37228
−2.37228
0 0 0 1.00000 0 −3.37228 0 0 0
1.2 0 0 0 1.00000 0 2.37228 0 0 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 9360.2.a.cp 2
3.b odd 2 1 9360.2.a.cg 2
4.b odd 2 1 2340.2.a.m yes 2
12.b even 2 1 2340.2.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2340.2.a.j 2 12.b even 2 1
2340.2.a.m yes 2 4.b odd 2 1
9360.2.a.cg 2 3.b odd 2 1
9360.2.a.cp 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9360))\):

\( T_{7}^{2} + T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} - 6 \) Copy content Toggle raw display
\( T_{17}^{2} - 9T_{17} + 12 \) Copy content Toggle raw display
\( T_{19} - 4 \) Copy content Toggle raw display
\( T_{31}^{2} - 2T_{31} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T - 8 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 9T + 12 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 9T + 12 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$31$ \( T^{2} - 2T - 32 \) Copy content Toggle raw display
$37$ \( T^{2} - T - 74 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$43$ \( T^{2} + 10T - 8 \) Copy content Toggle raw display
$47$ \( T^{2} - 132 \) Copy content Toggle raw display
$53$ \( T^{2} - 9T + 12 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T - 24 \) Copy content Toggle raw display
$61$ \( T^{2} - 19T + 82 \) Copy content Toggle raw display
$67$ \( (T - 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$73$ \( T^{2} + 8T - 116 \) Copy content Toggle raw display
$79$ \( T^{2} - 5T - 200 \) Copy content Toggle raw display
$83$ \( T^{2} + 18T + 48 \) Copy content Toggle raw display
$89$ \( T^{2} - 9T - 54 \) Copy content Toggle raw display
$97$ \( T^{2} + 29T + 202 \) Copy content Toggle raw display
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