Properties

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

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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: \(\Q(\sqrt{530401}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 132600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 3)
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 \(\beta = 3\sqrt{530401}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 621) q^{2} + ( - 1242 \beta - 3229358) q^{4} + ( - 62720 \beta + 23404410) q^{5} + (3642624 \beta - 105981952) q^{7} + (10846684 \beta - 1285934508) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 621) q^{2} + ( - 1242 \beta - 3229358) q^{4} + ( - 62720 \beta + 23404410) q^{5} + (3642624 \beta - 105981952) q^{7} + (10846684 \beta - 1285934508) q^{8} + ( - 62353530 \beta + 313934895090) q^{10} + ( - 135883264 \beta - 734486183244) q^{11} + (1301405184 \beta + 5245827132374) q^{13} + (2368051456 \beta - 17454277502208) q^{14} + (18440376408 \beta - 25486575338360) q^{16} + (15197836800 \beta + 105444005760414) q^{17} + (142703202816 \beta - 453691224268972) q^{19} + (173477056540 \beta + 296274520879380) q^{20} + (650102676300 \beta + 192537652185252) q^{22} + (114772628992 \beta - 50\!\cdots\!56) q^{23}+ \cdots + ( - 36\!\cdots\!61 \beta + 26\!\cdots\!09) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 1242 q^{2} - 6458716 q^{4} + 46808820 q^{5} - 211963904 q^{7} - 2571869016 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 1242 q^{2} - 6458716 q^{4} + 46808820 q^{5} - 211963904 q^{7} - 2571869016 q^{8} + 627869790180 q^{10} - 1468972366488 q^{11} + 10491654264748 q^{13} - 34908555004416 q^{14} - 50973150676720 q^{16} + 210888011520828 q^{17} - 907382448537944 q^{19} + 592549041758760 q^{20} + 385075304370504 q^{22} - 10\!\cdots\!12 q^{23}+ \cdots + 52\!\cdots\!18 q^{98}+O(q^{100}) \) 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
364.643
−363.643
−1563.86 0 −5.94295e6 −1.13630e8 0 7.85264e9 2.24125e10 0 1.77701e11
1.2 2805.86 0 −515763. 1.60439e8 0 −8.06460e9 −2.49844e10 0 4.50169e11
\(n\): 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.c 2
3.b odd 2 1 3.24.a.b 2
12.b even 2 1 48.24.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.24.a.b 2 3.b odd 2 1
9.24.a.c 2 1.a even 1 1 trivial
48.24.a.g 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 1242T_{2} - 4387968 \) acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(9))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 1242 T - 4387968 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 46808820 T - 18\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} + 211963904 T - 63\!\cdots\!80 \) Copy content Toggle raw display
$11$ \( T^{2} + 1468972366488 T + 45\!\cdots\!72 \) Copy content Toggle raw display
$13$ \( T^{2} - 10491654264748 T + 19\!\cdots\!72 \) Copy content Toggle raw display
$17$ \( T^{2} - 210888011520828 T + 10\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{2} + 907382448537944 T + 10\!\cdots\!80 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 25\!\cdots\!60 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 48\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} + 478995036787364 T - 14\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 71\!\cdots\!20 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 99\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 26\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 38\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 43\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 47\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 66\!\cdots\!80 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 48\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 20\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
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