Properties

Label 1.78.a.a
Level $1$
Weight $78$
Character orbit 1.a
Self dual yes
Analytic conductor $37.548$
Analytic rank $1$
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] = [1,78,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, 78, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 78);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 78 \)
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: \(37.5479417817\)
Analytic rank: \(1\)
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: \( x^{6} + \cdots - 44\!\cdots\!16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{64}\cdot 3^{20}\cdot 5^{8}\cdot 7^{3}\cdot 11^{2}\cdot 13^{2}\cdot 19 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 44120315520) q^{2} + ( - \beta_{2} + 328682 \beta_1 + 24\!\cdots\!80) q^{3}+ \cdots + (3311328 \beta_{5} + 1712740242 \beta_{4} + \cdots - 81\!\cdots\!07) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 44120315520) q^{2} + ( - \beta_{2} + 328682 \beta_1 + 24\!\cdots\!80) q^{3}+ \cdots + (20\!\cdots\!52 \beta_{5} + \cdots + 37\!\cdots\!96) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 264721893120 q^{2} + 14\!\cdots\!80 q^{3}+ \cdots - 48\!\cdots\!42 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 264721893120 q^{2} + 14\!\cdots\!80 q^{3}+ \cdots + 22\!\cdots\!76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + \cdots - 44\!\cdots\!16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 192\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 8104264459 \nu^{5} + \cdots + 16\!\cdots\!60 ) / 18\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 15406206736559 \nu^{5} + \cdots - 27\!\cdots\!76 ) / 26\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 93\!\cdots\!27 \nu^{5} + \cdots + 30\!\cdots\!64 ) / 78\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 75\!\cdots\!65 \nu^{5} + \cdots + 27\!\cdots\!64 ) / 67\!\cdots\!76 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 192 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 13307\beta_{2} + 87863724858\beta _1 + 218250578218837967990784 ) / 36864 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4131 \beta_{5} - 4639708 \beta_{4} + 6739710962 \beta_{3} + 107241347382062 \beta_{2} + \cdots + 11\!\cdots\!00 ) / 442368 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 543893760081 \beta_{5} - 25654290487476 \beta_{4} + \cdots + 23\!\cdots\!04 ) / 36864 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 72\!\cdots\!99 \beta_{5} + \cdots + 26\!\cdots\!00 ) / 442368 \) 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
3.81583e9
1.98644e9
7.14285e8
−1.47555e9
−1.61977e9
−3.42124e9
−6.88520e11 1.25704e18 3.22944e23 7.49066e26 −8.65494e29 4.37098e32 −1.18307e35 −3.89426e36 −5.15747e38
1.2 −3.37277e11 −2.07041e18 −3.73599e22 −2.41864e26 6.98302e29 −4.13826e32 6.35685e34 −1.18781e36 8.15750e37
1.3 −9.30224e10 3.16609e18 −1.42463e23 −6.28590e26 −2.94518e29 2.92216e32 2.73094e34 4.54975e36 5.84729e37
1.4 3.27426e11 −3.10225e18 −4.39082e22 −2.60453e25 −1.01576e30 6.63145e32 −6.38558e34 4.14958e36 −8.52790e36
1.5 3.55117e11 1.22969e18 −2.50080e22 1.38608e27 4.36684e29 −5.00002e32 −6.25444e34 −3.96226e36 4.92219e38
1.6 7.00999e11 9.61452e17 3.40284e23 −1.50068e27 6.73977e29 −2.07773e32 1.32607e35 −4.55001e36 −1.05197e39
\(n\): 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 1.78.a.a 6
3.b odd 2 1 9.78.a.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.78.a.a 6 1.a even 1 1 trivial
9.78.a.a 6 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{78}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 264721893120 T^{5} + \cdots - 17\!\cdots\!04 \) Copy content Toggle raw display
$3$ \( T^{6} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots - 36\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots - 41\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 24\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 11\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 31\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots - 47\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots - 87\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 31\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 45\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots - 16\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots - 10\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 49\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 68\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots - 81\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 79\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 90\!\cdots\!64 \) Copy content Toggle raw display
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