Properties

Label 1.76.a.a
Level $1$
Weight $76$
Character orbit 1.a
Self dual yes
Analytic conductor $35.623$
Analytic rank $0$
Dimension $6$
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] = [1,76,Mod(1,1)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 76, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 76);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 76 \)
Character orbit: \([\chi]\) \(=\) 1.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: \(35.6228392822\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3 x^{5} + \cdots - 67\!\cdots\!50 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{58}\cdot 3^{26}\cdot 5^{7}\cdot 7^{3}\cdot 11\cdot 13\cdot 19 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 9513470340) q^{2} + ( - \beta_{2} + 60703 \beta_1 - 13\!\cdots\!40) q^{3}+ \cdots + ( - 1665408 \beta_{5} - 1671114570 \beta_{4} + \cdots + 35\!\cdots\!97) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 9513470340) q^{2} + ( - \beta_{2} + 60703 \beta_1 - 13\!\cdots\!40) q^{3}+ \cdots + ( - 12\!\cdots\!92 \beta_{5} + \cdots - 31\!\cdots\!56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 57080822040 q^{2} - 78\!\cdots\!40 q^{3}+ \cdots + 21\!\cdots\!82 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 57080822040 q^{2} - 78\!\cdots\!40 q^{3}+ \cdots - 18\!\cdots\!36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3 x^{5} + \cdots - 67\!\cdots\!50 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 72\nu - 36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 251418152991 \nu^{5} + \cdots + 14\!\cdots\!38 ) / 10\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 20\!\cdots\!81 \nu^{5} + \cdots - 47\!\cdots\!06 ) / 53\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 22\!\cdots\!83 \nu^{5} + \cdots + 20\!\cdots\!30 ) / 26\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 45\!\cdots\!49 \nu^{5} + \cdots + 24\!\cdots\!66 ) / 26\!\cdots\!48 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 36 ) / 72 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 16382\beta_{2} + 28859343499\beta _1 + 66455170111740773427552 ) / 5184 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2781 \beta_{5} + 5496670 \beta_{4} + 13763505040 \beta_{3} + 227096891305632 \beta_{2} + \cdots + 23\!\cdots\!88 ) / 46656 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 19416283325423 \beta_{5} + \cdots + 34\!\cdots\!24 ) / 139968 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 51\!\cdots\!48 \beta_{5} + \cdots + 66\!\cdots\!64 ) / 23328 \) 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
−4.40594e9
−3.89081e9
−1.60397e9
1.46671e9
3.19848e9
5.23553e9
−3.26741e11 −1.51453e18 6.89808e22 −4.33451e25 4.94860e29 4.24638e31 −1.01949e34 1.68554e36 1.41626e37
1.2 −2.89652e11 1.03577e18 4.61192e22 −2.47463e26 −3.00013e29 −6.83079e31 −2.41577e33 4.64556e35 7.16781e37
1.3 −1.25000e11 1.35148e17 −2.21540e22 1.80927e26 −1.68935e28 2.76229e31 7.49160e33 −5.90002e35 −2.26158e37
1.4 9.60898e10 −6.47242e17 −2.85457e22 −1.49650e26 −6.21934e28 −3.16100e31 −6.37312e33 −1.89344e35 −1.43799e37
1.5 2.20777e11 1.08881e18 1.09637e22 −2.30711e25 2.40385e29 1.96027e31 −5.92020e33 5.77242e35 −5.09358e36
1.6 3.67444e11 −8.83049e17 9.72365e22 2.43619e26 −3.24471e29 1.21530e31 2.18473e34 1.71508e35 8.95165e37
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

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 1.76.a.a 6
3.b odd 2 1 9.76.a.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.76.a.a 6 1.a even 1 1 trivial
9.76.a.c 6 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{76}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 57080822040 T^{5} + \cdots - 92\!\cdots\!16 \) Copy content Toggle raw display
$3$ \( T^{6} + \cdots - 13\!\cdots\!64 \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 60\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots - 76\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 87\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 35\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots - 14\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots - 18\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 52\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 83\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 41\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 57\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 66\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 36\!\cdots\!24 \) Copy content Toggle raw display
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