Properties

Label 9.24.a.b
Level $9$
Weight $24$
Character orbit 9.a
Self dual yes
Analytic conductor $30.168$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9,24,Mod(1,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 24, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 24);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 9.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: \(30.1683633611\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{144169}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 36042 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: no (minimal twist has level 1)
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 \(\beta = 12\sqrt{144169}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 540) q^{2} + (1080 \beta + 12663328) q^{4} + (15040 \beta - 36534510) q^{5} + ( - 985824 \beta - 679592200) q^{7} + ( - 4857920 \beta - 24729511680) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 540) q^{2} + (1080 \beta + 12663328) q^{4} + (15040 \beta - 36534510) q^{5} + ( - 985824 \beta - 679592200) q^{7} + ( - 4857920 \beta - 24729511680) q^{8} + (28412910 \beta - 292506818040) q^{10} + ( - 38671600 \beta - 428400984132) q^{11} + (1268350272 \beta + 2188054661030) q^{13} + (1211937160 \beta + 20833017264864) q^{14} + (18293091840 \beta + 7978293200896) q^{16} + (23522231424 \beta - 127014073798770) q^{17} + (137218594320 \beta + 2130300489980) q^{19} + (150999182320 \beta - 125434193734080) q^{20} + (449283648132 \beta + 10\!\cdots\!80) q^{22}+ \cdots + (60\!\cdots\!07 \beta - 24\!\cdots\!20) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1080 q^{2} + 25326656 q^{4} - 73069020 q^{5} - 1359184400 q^{7} - 49459023360 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 1080 q^{2} + 25326656 q^{4} - 73069020 q^{5} - 1359184400 q^{7} - 49459023360 q^{8} - 585013636080 q^{10} - 856801968264 q^{11} + 4376109322060 q^{13} + 41666034529728 q^{14} + 15956586401792 q^{16} - 254028147597540 q^{17} + 4260600979960 q^{19} - 250868387468160 q^{20} + 20\!\cdots\!60 q^{22}+ \cdots - 48\!\cdots\!40 q^{98}+O(q^{100}) \) 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
190.348
−189.348
−5096.35 0 1.75842e7 3.19930e7 0 −5.17135e9 −4.68639e10 0 −1.63048e11
1.2 4016.35 0 7.74247e6 −1.05062e8 0 3.81217e9 −2.59512e9 0 −4.21966e11
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

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 9.24.a.b 2
3.b odd 2 1 1.24.a.a 2
12.b even 2 1 16.24.a.b 2
15.d odd 2 1 25.24.a.a 2
15.e even 4 2 25.24.b.a 4
21.c even 2 1 49.24.a.b 2
24.f even 2 1 64.24.a.g 2
24.h odd 2 1 64.24.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.24.a.a 2 3.b odd 2 1
9.24.a.b 2 1.a even 1 1 trivial
16.24.a.b 2 12.b even 2 1
25.24.a.a 2 15.d odd 2 1
25.24.b.a 4 15.e even 4 2
49.24.a.b 2 21.c even 2 1
64.24.a.d 2 24.h odd 2 1
64.24.a.g 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 1080T_{2} - 20468736 \) acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(9))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1080 T - 20468736 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 73069020 T - 33\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} + 1359184400 T - 19\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{2} + 856801968264 T + 15\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( T^{2} - 4376109322060 T - 28\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{2} + 254028147597540 T + 46\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{2} - 4260600979960 T - 39\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 85\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 42\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 51\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 43\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 21\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 48\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 28\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 13\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 40\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 40\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 11\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 93\!\cdots\!36 \) Copy content Toggle raw display
show more
show less