Properties

Label 9.46.a.c
Level $9$
Weight $46$
Character orbit 9.a
Self dual yes
Analytic conductor $115.430$
Analytic rank $0$
Dimension $4$
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] = [9,46,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, 46, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 46);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 46 \)
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: \(115.430153467\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 379806423667x^{2} - 17154201580335707x + 14524158011475498861774 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{13}\cdot 5\cdot 7\cdot 11 \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1200829) q^{2} + (\beta_{2} - 1182173 \beta_1 + 27786259442267) q^{4} + ( - \beta_{3} + 71 \beta_{2} - 377854944 \beta_1 - 149822847872851) q^{5} + ( - 13275 \beta_{3} + 37309 \beta_{2} + \cdots + 38\!\cdots\!17) q^{7}+ \cdots + ( - 267200 \beta_{3} + 1969462 \beta_{2} + \cdots + 63\!\cdots\!06) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1200829) q^{2} + (\beta_{2} - 1182173 \beta_1 + 27786259442267) q^{4} + ( - \beta_{3} + 71 \beta_{2} - 377854944 \beta_1 - 149822847872851) q^{5} + ( - 13275 \beta_{3} + 37309 \beta_{2} + \cdots + 38\!\cdots\!17) q^{7}+ \cdots + (16\!\cdots\!00 \beta_{3} + \cdots + 56\!\cdots\!73) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4803318 q^{2} + 111145040133412 q^{4} - 599290635781656 q^{5} + 15\!\cdots\!24 q^{7}+ \cdots + 25\!\cdots\!84 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4803318 q^{2} + 111145040133412 q^{4} - 599290635781656 q^{5} + 15\!\cdots\!24 q^{7}+ \cdots + 22\!\cdots\!98 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 379806423667x^{2} - 17154201580335707x + 14524158011475498861774 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 18\nu - 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 324\nu^{2} - 21950910\nu - 61528635146408 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 729\nu^{3} - 66138120\nu^{2} - 216642793886163\nu + 3180628200133744474 ) / 33400 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 5 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 1219495\beta _1 + 61528641243883 ) / 324 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 66800\beta_{3} + 408260\beta_{2} + 24569292571607\beta _1 + 18758547031067899167 ) / 1458 \) 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
607176.
181379.
−240747.
−547807.
−9.72833e6 0 5.94561e13 −1.42612e15 0 −2.77214e17 −2.36123e20 0 1.38738e22
1.2 −2.06399e6 0 −3.09243e13 −4.26171e15 0 1.48233e19 1.36448e20 0 8.79615e21
1.3 5.53428e6 0 −4.55610e12 −2.40990e15 0 −1.32831e19 −2.19935e20 0 −1.33371e22
1.4 1.10614e7 0 8.71694e13 7.49845e15 0 1.42547e19 5.75025e20 0 8.29431e22
\(n\): 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 9.46.a.c 4
3.b odd 2 1 3.46.a.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.46.a.b 4 3.b odd 2 1
9.46.a.c 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 4803318T_{2}^{3} - 114405332339808T_{2}^{2} + 388659694534929432576T_{2} + 1229183068335614137177473024 \) acting on \(S_{46}^{\mathrm{new}}(\Gamma_0(9))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4803318 T^{3} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 599290635781656 T^{3} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots - 23\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 53\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 37\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots - 79\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots - 69\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 29\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 91\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 40\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 48\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 53\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 58\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 89\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 44\!\cdots\!84 \) Copy content Toggle raw display
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