Properties

Label 9.74.a.a
Level $9$
Weight $74$
Character orbit 9.a
Self dual yes
Analytic conductor $303.736$
Analytic rank $0$
Dimension $5$
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] = [9,74,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, 74, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 74);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 74 \)
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: \(303.735576363\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{39}\cdot 3^{22}\cdot 5^{6}\cdot 7^{2} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 18417866698) q^{2} + (\beta_{3} - 2 \beta_{2} + 26620058826 \beta_1 + 17\!\cdots\!62) q^{4}+ \cdots + ( - 15102976 \beta_{4} + 41549502192 \beta_{3} + \cdots + 76\!\cdots\!32) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 18417866698) q^{2} + (\beta_{3} - 2 \beta_{2} + 26620058826 \beta_1 + 17\!\cdots\!62) q^{4}+ \cdots + (66\!\cdots\!12 \beta_{4} + \cdots + 15\!\cdots\!34) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 92089333488 q^{2} + 89\!\cdots\!60 q^{4}+ \cdots + 38\!\cdots\!80 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 92089333488 q^{2} + 89\!\cdots\!60 q^{4}+ \cdots + 76\!\cdots\!16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} + \cdots + 17\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 48\nu - 10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 24381 \nu^{4} + 9422283614913 \nu^{3} + \cdots - 11\!\cdots\!56 ) / 69\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 24381 \nu^{4} + 9422283614913 \nu^{3} + \cdots - 43\!\cdots\!56 ) / 34\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 66468519 \nu^{4} + \cdots - 23\!\cdots\!20 ) / 34\!\cdots\!12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 10 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 2\beta_{2} - 10215674550\beta _1 + 9283737247896553731050 ) / 2304 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 943936 \beta_{4} - 856506117 \beta_{3} + 6859809162 \beta_{2} + \cdots - 59\!\cdots\!58 ) / 6912 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 10\!\cdots\!84 \beta_{4} + \cdots + 71\!\cdots\!62 ) / 20736 \) 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
−3.13124e9
−6.50494e8
−2.60629e8
1.10442e9
2.93794e9
−1.31882e11 0 7.94807e21 7.75470e24 0 1.05974e30 1.97383e32 0 −1.02270e36
1.2 −1.28059e10 0 −9.28074e21 −4.05981e25 0 −7.12492e30 2.39796e32 0 5.19894e35
1.3 5.90767e9 0 −9.40983e21 −1.69056e25 0 −2.69951e30 −1.11387e32 0 −9.98727e34
1.4 7.14302e10 0 −4.34246e21 5.37334e25 0 1.02465e31 −9.84822e32 0 3.83819e36
1.5 1.59439e11 0 1.59761e22 −2.70839e25 0 −5.83802e30 1.04135e33 0 −4.31824e36
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 9.74.a.a 5
3.b odd 2 1 1.74.a.a 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.74.a.a 5 3.b odd 2 1
9.74.a.a 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} - 92089333488 T_{2}^{4} + \cdots - 11\!\cdots\!68 \) acting on \(S_{74}^{\mathrm{new}}(\Gamma_0(9))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 92089333488 T^{4} + \cdots - 11\!\cdots\!68 \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( T^{5} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{5} + \cdots + 12\!\cdots\!68 \) Copy content Toggle raw display
$11$ \( T^{5} + \cdots + 24\!\cdots\!32 \) Copy content Toggle raw display
$13$ \( T^{5} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{5} + \cdots + 89\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{5} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{5} + \cdots - 36\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{5} + \cdots - 75\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots + 18\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots + 56\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots + 15\!\cdots\!32 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots + 41\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots - 49\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots + 33\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots + 19\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots + 10\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots + 72\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots + 28\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots + 33\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots + 75\!\cdots\!68 \) Copy content Toggle raw display
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