Properties

Label 4842.2.a.n
Level $4842$
Weight $2$
Character orbit 4842.a
Self dual yes
Analytic conductor $38.664$
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] = [4842,2,Mod(1,4842)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4842, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4842.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4842 = 2 \cdot 3^{2} \cdot 269 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4842.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: \(38.6635646587\)
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: \( x^{7} - x^{6} - 15x^{5} + 16x^{4} + 49x^{3} - 53x^{2} - 44x + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 538)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( - \beta_{4} - 1) q^{5} + (\beta_{6} - \beta_{3} - \beta_{2} + 1) q^{7} - q^{8} + (\beta_{4} + 1) q^{10} + ( - \beta_{5} + \beta_{4}) q^{11} + (\beta_{3} + \beta_1 - 1) q^{13} + ( - \beta_{6} + \beta_{3} + \beta_{2} - 1) q^{14}+ \cdots + ( - 3 \beta_{5} - \beta_{4} - \beta_{2} + \cdots - 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 7 q^{2} + 7 q^{4} - 7 q^{5} + 6 q^{7} - 7 q^{8} + 7 q^{10} + 3 q^{11} - 9 q^{13} - 6 q^{14} + 7 q^{16} - 8 q^{17} - 11 q^{19} - 7 q^{20} - 3 q^{22} - 12 q^{23} + 22 q^{25} + 9 q^{26} + 6 q^{28} + 5 q^{29}+ \cdots - 15 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 15x^{5} + 16x^{4} + 49x^{3} - 53x^{2} - 44x + 48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + \nu^{5} - 15\nu^{4} - 12\nu^{3} + 51\nu^{2} + 21\nu - 48 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{6} - \nu^{5} + 45\nu^{4} + 10\nu^{3} - 151\nu^{2} - 17\nu + 138 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{6} + \nu^{5} - 73\nu^{4} - 8\nu^{3} + 225\nu^{2} + 13\nu - 186 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17\nu^{6} + 3\nu^{5} - 251\nu^{4} - 24\nu^{3} + 801\nu^{2} + 47\nu - 696 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -21\nu^{6} - 3\nu^{5} + 311\nu^{4} + 20\nu^{3} - 997\nu^{2} - 39\nu + 856 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{5} + 2\beta_{4} - 3\beta_{3} - 2\beta_{2} + 6\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 13\beta_{6} + 13\beta_{5} + \beta_{4} - 11\beta_{3} - 12\beta_{2} + 44 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{6} - 27\beta_{5} + 26\beta_{4} - 37\beta_{3} - 22\beta_{2} + 55\beta _1 - 41 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 145\beta_{6} + 147\beta_{5} + 13\beta_{4} - 113\beta_{3} - 129\beta_{2} - 4\beta _1 + 458 \) 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
3.22542
1.19866
−1.26381
1.14366
−1.59223
−3.33118
1.61949
−1.00000 0 1.00000 −4.15424 0 4.87818 −1.00000 0 4.15424
1.2 −1.00000 0 1.00000 −3.84433 0 −4.95403 −1.00000 0 3.84433
1.3 −1.00000 0 1.00000 −3.00565 0 −0.324393 −1.00000 0 3.00565
1.4 −1.00000 0 1.00000 −0.725043 0 1.29978 −1.00000 0 0.725043
1.5 −1.00000 0 1.00000 −0.0316474 0 1.08414 −1.00000 0 0.0316474
1.6 −1.00000 0 1.00000 0.953965 0 3.52894 −1.00000 0 −0.953965
1.7 −1.00000 0 1.00000 3.80694 0 0.487379 −1.00000 0 −3.80694
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(269\) \( -1 \)

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 4842.2.a.n 7
3.b odd 2 1 538.2.a.e 7
12.b even 2 1 4304.2.a.h 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.2.a.e 7 3.b odd 2 1
4304.2.a.h 7 12.b even 2 1
4842.2.a.n 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4842))\):

\( T_{5}^{7} + 7T_{5}^{6} - 4T_{5}^{5} - 109T_{5}^{4} - 161T_{5}^{3} + 109T_{5}^{2} + 130T_{5} + 4 \) Copy content Toggle raw display
\( T_{7}^{7} - 6T_{7}^{6} - 14T_{7}^{5} + 142T_{7}^{4} - 258T_{7}^{3} + 137T_{7}^{2} + 18T_{7} - 19 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( T^{7} + 7 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{7} - 6 T^{6} + \cdots - 19 \) Copy content Toggle raw display
$11$ \( T^{7} - 3 T^{6} + \cdots - 288 \) Copy content Toggle raw display
$13$ \( T^{7} + 9 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( T^{7} + 8 T^{6} + \cdots + 22016 \) Copy content Toggle raw display
$19$ \( T^{7} + 11 T^{6} + \cdots - 20 \) Copy content Toggle raw display
$23$ \( T^{7} + 12 T^{6} + \cdots - 2432 \) Copy content Toggle raw display
$29$ \( T^{7} - 5 T^{6} + \cdots + 27040 \) Copy content Toggle raw display
$31$ \( T^{7} - 14 T^{6} + \cdots + 250907 \) Copy content Toggle raw display
$37$ \( T^{7} - 13 T^{6} + \cdots + 2236 \) Copy content Toggle raw display
$41$ \( T^{7} + 12 T^{6} + \cdots + 288917 \) Copy content Toggle raw display
$43$ \( T^{7} + 11 T^{6} + \cdots - 227616 \) Copy content Toggle raw display
$47$ \( T^{7} + 2 T^{6} + \cdots - 64896 \) Copy content Toggle raw display
$53$ \( T^{7} + 19 T^{6} + \cdots + 26988 \) Copy content Toggle raw display
$59$ \( T^{7} - 9 T^{6} + \cdots + 146660 \) Copy content Toggle raw display
$61$ \( T^{7} + 3 T^{6} + \cdots + 210068 \) Copy content Toggle raw display
$67$ \( T^{7} + 33 T^{6} + \cdots - 251936 \) Copy content Toggle raw display
$71$ \( T^{7} + 28 T^{6} + \cdots - 28293 \) Copy content Toggle raw display
$73$ \( T^{7} + 14 T^{6} + \cdots - 382603 \) Copy content Toggle raw display
$79$ \( T^{7} - 2 T^{6} + \cdots + 14720 \) Copy content Toggle raw display
$83$ \( T^{7} - 7 T^{6} + \cdots + 441152 \) Copy content Toggle raw display
$89$ \( T^{7} + 18 T^{6} + \cdots - 1143155 \) Copy content Toggle raw display
$97$ \( T^{7} + 4 T^{6} + \cdots + 15927 \) Copy content Toggle raw display
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