Properties

Label 1.98.a.a
Level $1$
Weight $98$
Character orbit 1.a
Self dual yes
Analytic conductor $59.585$
Analytic rank $1$
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] = [1,98,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, 98, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 98);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 98 \)
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: \(59.5852992940\)
Analytic rank: \(1\)
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: \( x^{7} - x^{6} + \cdots - 60\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{83}\cdot 3^{30}\cdot 5^{10}\cdot 7^{8}\cdot 11^{2}\cdot 19 \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 2385320155001) q^{2} + ( - \beta_{2} - 23858416 \beta_1 + 14\!\cdots\!54) q^{3}+ \cdots + ( - 178740 \beta_{6} + 63549828 \beta_{5} + \cdots + 49\!\cdots\!03) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 2385320155001) q^{2} + ( - \beta_{2} - 23858416 \beta_1 + 14\!\cdots\!54) q^{3}+ \cdots + (13\!\cdots\!80 \beta_{6} + \cdots - 19\!\cdots\!22) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 16697241085008 q^{2} + 10\!\cdots\!96 q^{3}+ \cdots + 34\!\cdots\!51 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 16697241085008 q^{2} + 10\!\cdots\!96 q^{3}+ \cdots - 13\!\cdots\!28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} + \cdots - 60\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 48\nu - 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 39\!\cdots\!11 \nu^{6} + \cdots - 13\!\cdots\!80 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 16\!\cdots\!49 \nu^{6} + \cdots - 53\!\cdots\!24 ) / 26\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 92\!\cdots\!71 \nu^{6} + \cdots + 35\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 20\!\cdots\!73 \nu^{6} + \cdots + 51\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 28\!\cdots\!71 \nu^{6} + \cdots - 11\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 7 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 168236\beta_{2} + 31316295066249\beta _1 + 181744504410671833814939351084 ) / 2304 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 12 \beta_{6} + 1614 \beta_{5} - 34647766 \beta_{4} - 37912750497 \beta_{3} + \cdots + 35\!\cdots\!18 ) / 6912 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 83208391554236 \beta_{6} + \cdots + 22\!\cdots\!10 ) / 20736 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 21\!\cdots\!08 \beta_{6} + \cdots + 76\!\cdots\!38 ) / 6912 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 64\!\cdots\!72 \beta_{6} + \cdots + 10\!\cdots\!82 ) / 6912 \) 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
−1.21138e13
−1.07106e13
−2.29850e12
−1.00063e11
1.42059e12
1.16497e13
1.21527e13
−5.83847e14 −6.70202e22 1.82421e29 3.35065e33 3.91296e37 6.90378e40 −1.39919e43 −1.45964e46 −1.95627e48
1.2 −5.16494e14 2.18475e23 1.08310e29 −1.00097e34 −1.12841e38 −1.23831e41 2.59003e43 2.86433e46 5.16994e48
1.3 −1.12713e14 −2.23668e23 −1.45752e29 −2.02292e33 2.52104e37 −1.81030e41 3.42884e43 3.09393e46 2.28010e47
1.4 −7.18835e12 1.45423e23 −1.58405e29 1.38920e34 −1.04535e36 −1.18505e40 2.27771e42 2.05969e45 −9.98605e46
1.5 6.58029e13 3.71452e20 −1.54126e29 −8.33150e33 2.44426e34 1.07181e41 −2.05689e43 −1.90879e46 −5.48237e47
1.6 5.56799e14 −1.35917e23 1.51568e29 4.66064e33 −7.56782e37 2.73237e40 −3.83521e42 −6.14732e44 2.59504e48
1.7 5.80945e14 1.62851e23 1.79040e29 −5.19758e33 9.46076e37 −7.14729e40 1.19582e43 7.43249e45 −3.01951e48
\(n\): 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 1.98.a.a 7
3.b odd 2 1 9.98.a.a 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.98.a.a 7 1.a even 1 1 trivial
9.98.a.a 7 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{98}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} + 16697241085008 T^{6} + \cdots - 52\!\cdots\!12 \) Copy content Toggle raw display
$3$ \( T^{7} + \cdots + 39\!\cdots\!56 \) Copy content Toggle raw display
$5$ \( T^{7} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{7} + \cdots - 38\!\cdots\!52 \) Copy content Toggle raw display
$11$ \( T^{7} + \cdots - 14\!\cdots\!88 \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots - 15\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots - 17\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots + 66\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots - 75\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots + 32\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots + 14\!\cdots\!08 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots - 21\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots + 27\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots - 59\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots + 55\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots + 28\!\cdots\!12 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots + 19\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots - 18\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots - 50\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots - 38\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots - 61\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots + 28\!\cdots\!28 \) Copy content Toggle raw display
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