Properties

Label 1.108.a.a
Level $1$
Weight $108$
Character orbit 1.a
Self dual yes
Analytic conductor $72.504$
Analytic rank $0$
Dimension $9$
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] = [1,108,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, 108, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 108);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 108 \)
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: \(72.5037502298\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4 x^{8} + \cdots + 27\!\cdots\!52 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{143}\cdot 3^{48}\cdot 5^{18}\cdot 7^{8}\cdot 11^{2}\cdot 13^{2} \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 607308845937277) q^{2} + (\beta_{2} + 62113396 \beta_1 + 17\!\cdots\!33) q^{3}+ \cdots + (\beta_{8} - 245 \beta_{7} - 63368 \beta_{6} + \cdots + 41\!\cdots\!23) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 607308845937277) q^{2} + (\beta_{2} + 62113396 \beta_1 + 17\!\cdots\!33) q^{3}+ \cdots + (85\!\cdots\!88 \beta_{8} + \cdots + 23\!\cdots\!75) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 54\!\cdots\!96 q^{2}+ \cdots + 36\!\cdots\!13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 54\!\cdots\!96 q^{2}+ \cdots + 21\!\cdots\!36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4 x^{8} + \cdots + 27\!\cdots\!52 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 24\nu - 11 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 12\!\cdots\!79 \nu^{8} + \cdots + 21\!\cdots\!68 ) / 18\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 47\!\cdots\!19 \nu^{8} + \cdots - 24\!\cdots\!20 ) / 10\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 32\!\cdots\!89 \nu^{8} + \cdots + 16\!\cdots\!36 ) / 96\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 59\!\cdots\!89 \nu^{8} + \cdots - 93\!\cdots\!36 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 50\!\cdots\!53 \nu^{8} + \cdots - 31\!\cdots\!28 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 26\!\cdots\!07 \nu^{8} + \cdots + 53\!\cdots\!68 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 45\!\cdots\!37 \nu^{8} + \cdots - 14\!\cdots\!12 ) / 76\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 11 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 664037\beta_{2} + 1454496237795986\beta _1 + 233205805914080463879155578823890 ) / 576 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 287 \beta_{6} + 326098 \beta_{5} - 11536093720 \beta_{4} + \cdots + 33\!\cdots\!45 ) / 13824 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 399171419712 \beta_{8} + \cdots + 11\!\cdots\!87 ) / 41472 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 10\!\cdots\!80 \beta_{8} + \cdots + 33\!\cdots\!05 ) / 62208 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 69\!\cdots\!00 \beta_{8} + \cdots + 46\!\cdots\!93 ) / 20736 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 10\!\cdots\!80 \beta_{8} + \cdots + 33\!\cdots\!45 ) / 41472 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 42\!\cdots\!96 \beta_{8} + \cdots + 25\!\cdots\!07 ) / 124416 \) 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
1.03210e15
6.73733e14
4.82720e14
3.47594e14
−2.01415e13
−3.22655e14
−5.32911e14
−7.86617e14
−8.73826e14
−2.41632e16 −3.68177e24 4.21601e32 3.92322e37 8.89632e40 −1.24740e45 −6.26652e48 −1.11358e51 −9.47976e53
1.2 −1.55623e16 1.35366e25 7.99254e31 −3.05298e37 −2.10660e41 6.30256e44 1.28130e48 −9.43892e50 4.75113e53
1.3 −1.09780e16 −5.69800e25 −4.17435e31 2.81503e36 6.25524e41 3.42403e44 2.23954e48 2.11959e51 −3.09033e52
1.4 −7.73495e15 6.57221e25 −1.02430e32 3.43777e37 −5.08357e41 9.22106e44 2.04736e48 3.19227e51 −2.65909e53
1.5 1.09070e15 6.73736e23 −1.61070e32 1.05761e37 7.34847e38 −2.03867e45 −3.52656e47 −1.12668e51 1.15354e52
1.6 8.35103e15 −1.73817e25 −9.25196e31 3.17576e36 −1.45155e41 2.61780e45 −2.12767e48 −8.25006e50 2.65209e52
1.7 1.33972e16 4.78035e25 1.72251e31 −4.75138e37 6.40432e41 −7.19244e44 −1.94305e48 1.15805e51 −6.36552e53
1.8 1.94861e16 −5.50693e25 2.17450e32 −2.12470e37 −1.07309e42 −2.04209e45 1.07545e48 1.90550e51 −4.14022e53
1.9 2.15791e16 2.13607e25 3.03400e32 3.38441e37 4.60946e41 5.98905e44 3.04570e48 −6.70851e50 7.30327e53
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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 1.108.a.a 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.108.a.a 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace is the entire newspace \(S_{108}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + \cdots - 16\!\cdots\!56 \) Copy content Toggle raw display
$3$ \( T^{9} + \cdots - 12\!\cdots\!52 \) Copy content Toggle raw display
$5$ \( T^{9} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots - 11\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots - 20\!\cdots\!92 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots - 74\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots + 28\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{9} + \cdots + 25\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots + 48\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots - 77\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots - 24\!\cdots\!32 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots + 29\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots - 14\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots + 12\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots + 12\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots - 13\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots - 75\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots - 51\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots - 26\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots - 76\!\cdots\!16 \) Copy content Toggle raw display
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