Properties

Label 1.122.a.a
Level $1$
Weight $122$
Character orbit 1.a
Self dual yes
Analytic conductor $92.717$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1,122,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, 122, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 122);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 122 \)
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: \(92.7173263878\)
Analytic rank: \(1\)
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: \( x^{9} - 2 x^{8} + \cdots + 32\!\cdots\!74 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{145}\cdot 3^{53}\cdot 5^{20}\cdot 7^{8}\cdot 11^{6}\cdot 13^{2}\cdot 17^{2} \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 26\!\cdots\!25) q^{2}+ \cdots + (\beta_{8} - 111 \beta_{7} + 89884 \beta_{6} + \cdots + 88\!\cdots\!41) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 26\!\cdots\!25) q^{2}+ \cdots + ( - 53\!\cdots\!72 \beta_{8} + \cdots - 48\!\cdots\!05) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 23\!\cdots\!28 q^{2}+ \cdots + 79\!\cdots\!17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 23\!\cdots\!28 q^{2}+ \cdots - 44\!\cdots\!96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 2 x^{8} + \cdots + 32\!\cdots\!74 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 48\nu - 11 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 69\!\cdots\!11 \nu^{8} + \cdots + 15\!\cdots\!02 ) / 44\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 79\!\cdots\!87 \nu^{8} + \cdots - 53\!\cdots\!90 ) / 14\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14\!\cdots\!31 \nu^{8} + \cdots - 15\!\cdots\!06 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 55\!\cdots\!91 \nu^{8} + \cdots - 47\!\cdots\!66 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 33\!\cdots\!11 \nu^{8} + \cdots + 11\!\cdots\!86 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 95\!\cdots\!97 \nu^{8} + \cdots + 31\!\cdots\!78 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 50\!\cdots\!01 \nu^{8} + \cdots - 26\!\cdots\!74 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 11 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 3414351\beta_{2} + 189828498139459931\beta _1 + 3530888858815588277459233691606826100 ) / 2304 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 100 \beta_{6} - 8716729 \beta_{5} + 611104015289 \beta_{4} + \cdots + 67\!\cdots\!03 ) / 110592 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 33743922161664 \beta_{8} + \cdots + 14\!\cdots\!99 ) / 331776 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 97\!\cdots\!60 \beta_{8} + \cdots + 88\!\cdots\!39 ) / 248832 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 86\!\cdots\!80 \beta_{8} + \cdots + 23\!\cdots\!35 ) / 165888 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 78\!\cdots\!60 \beta_{8} + \cdots + 54\!\cdots\!07 ) / 331776 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 75\!\cdots\!56 \beta_{8} + \cdots + 17\!\cdots\!21 ) / 331776 \) 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
−6.03020e16
−3.77468e16
−3.52210e16
−1.87667e16
−1.78332e14
1.54621e16
2.68569e16
4.65892e16
6.33066e16
−3.15505e18 6.48457e28 7.29589e36 −2.23954e41 −2.04592e47 1.98659e51 −1.46313e55 −1.18606e57 7.06586e59
1.2 −2.07240e18 −2.64580e28 1.63640e36 −5.16214e41 5.48316e46 −2.39892e51 2.11812e54 −4.69101e57 1.06980e60
1.3 −1.95116e18 −1.11766e29 1.14858e36 9.20370e41 2.18074e47 2.38466e51 2.94601e54 7.10063e57 −1.79579e60
1.4 −1.16136e18 1.09371e29 −1.30970e36 1.47325e42 −1.27019e47 −1.67990e50 4.60846e54 6.57089e57 −1.71098e60
1.5 −2.69116e17 3.14043e28 −2.58603e36 −3.76295e42 −8.45141e45 1.39208e51 1.41138e54 −4.40480e57 1.01267e60
1.6 4.81626e17 −2.96062e28 −2.42649e36 2.39673e42 −1.42591e46 2.39069e50 −2.44905e54 −4.51451e57 1.15433e60
1.7 1.02857e18 −1.22395e29 −1.60049e36 −2.11004e42 −1.25893e47 −1.42065e51 −4.38064e54 9.58953e57 −2.17033e60
1.8 1.97572e18 8.85385e28 1.24503e36 −2.55522e41 1.74928e47 −5.36128e50 −2.79253e54 2.44803e57 −5.04842e59
1.9 2.77816e18 −4.92324e28 5.05972e36 1.85701e41 −1.36775e47 7.17028e50 6.67109e54 −2.96721e57 5.15907e59
\(n\): 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 1.122.a.a 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.122.a.a 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace is the entire newspace \(S_{122}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + \cdots + 10\!\cdots\!08 \) Copy content Toggle raw display
$3$ \( T^{9} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$5$ \( T^{9} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots - 34\!\cdots\!72 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots - 16\!\cdots\!52 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots - 75\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots + 53\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots - 33\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{9} + \cdots - 79\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots - 44\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots + 10\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots - 16\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots + 11\!\cdots\!28 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots - 12\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots + 25\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots + 35\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots + 20\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots + 26\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots + 11\!\cdots\!08 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots + 34\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots - 28\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots - 17\!\cdots\!12 \) Copy content Toggle raw display
show more
show less