Properties

Label 1.84.a.a
Level $1$
Weight $84$
Character orbit 1.a
Self dual yes
Analytic conductor $43.627$
Analytic rank $0$
Dimension $7$
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] = [1,84,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, 84, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 84);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 84 \)
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: \(43.6272128266\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{82}\cdot 3^{30}\cdot 5^{8}\cdot 7^{4}\cdot 17 \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 49635823059) q^{2} + ( - \beta_{2} - 2898083 \beta_1 + 13\!\cdots\!89) q^{3}+ \cdots + ( - 13860 \beta_{6} + 8089752 \beta_{5} + \cdots + 10\!\cdots\!89) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 49635823059) q^{2} + ( - \beta_{2} - 2898083 \beta_1 + 13\!\cdots\!89) q^{3}+ \cdots + (21\!\cdots\!20 \beta_{6} + \cdots - 26\!\cdots\!69) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 347450761416 q^{2} + 92\!\cdots\!72 q^{3}+ \cdots + 71\!\cdots\!19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 347450761416 q^{2} + 92\!\cdots\!72 q^{3}+ \cdots - 18\!\cdots\!92 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} + \cdots + 22\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 24\nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 33\!\cdots\!59 \nu^{6} + \cdots - 74\!\cdots\!48 ) / 10\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 18\!\cdots\!03 \nu^{6} + \cdots - 56\!\cdots\!72 ) / 10\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 66\!\cdots\!99 \nu^{6} + \cdots + 12\!\cdots\!00 ) / 70\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 28\!\cdots\!99 \nu^{6} + \cdots - 56\!\cdots\!00 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31\!\cdots\!43 \nu^{6} + \cdots - 61\!\cdots\!00 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 3 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 55717\beta_{2} - 598680817450\beta _1 + 14752386507570773831110328 ) / 576 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 21 \beta_{6} - 19917 \beta_{5} - 12811516 \beta_{4} - 30990098711 \beta_{3} + \cdots - 11\!\cdots\!93 ) / 1728 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1716728799877 \beta_{6} + \cdots + 61\!\cdots\!63 ) / 5184 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 12\!\cdots\!99 \beta_{6} + \cdots - 92\!\cdots\!03 ) / 1728 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 48\!\cdots\!99 \beta_{6} + \cdots + 11\!\cdots\!41 ) / 1728 \) 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
−2.54071e11
−1.58823e11
−7.80143e10
2.25782e10
1.15014e11
1.16386e11
2.36930e11
−6.04807e12 9.44466e19 2.69078e25 −3.42567e28 −5.71220e32 −1.33934e34 −1.04247e38 4.92932e39 2.07187e41
1.2 −3.76211e12 −6.68720e19 4.48205e24 1.95538e28 2.51580e32 1.66208e35 1.95229e37 4.81025e38 −7.35637e40
1.3 −1.82271e12 1.57108e19 −6.34915e24 −2.84572e28 −2.86362e31 −2.18145e35 2.92008e37 −3.74401e39 5.18692e40
1.4 5.91512e11 9.95960e19 −9.32152e24 1.29632e29 5.89123e31 1.88228e35 −1.12345e37 5.92853e39 7.66792e40
1.5 2.80997e12 −9.69898e19 −1.77546e24 1.10914e29 −2.72539e32 −5.50994e34 −3.21654e37 5.41619e39 3.11666e41
1.6 2.84290e12 5.71915e17 −1.58931e24 −1.75639e29 1.62590e30 1.13550e35 −3.20131e37 −3.99051e39 −4.99324e41
1.7 5.73595e12 4.63055e19 2.32297e25 7.41344e28 2.65606e32 −1.39412e35 7.77697e37 −1.84664e39 4.25231e41
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 1.84.a.a 7
3.b odd 2 1 9.84.a.c 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.84.a.a 7 1.a even 1 1 trivial
9.84.a.c 7 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{84}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - 347450761416 T^{6} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$3$ \( T^{7} + \cdots - 25\!\cdots\!92 \) Copy content Toggle raw display
$5$ \( T^{7} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{7} + \cdots - 79\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{7} + \cdots + 50\!\cdots\!32 \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots - 12\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots - 38\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots + 15\!\cdots\!28 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots - 76\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots + 34\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots - 11\!\cdots\!88 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots - 28\!\cdots\!52 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots - 50\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots - 46\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots - 97\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots + 22\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots + 45\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots - 24\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots - 45\!\cdots\!64 \) Copy content Toggle raw display
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