Properties

Label 5.24.a.b
Level $5$
Weight $24$
Character orbit 5.a
Self dual yes
Analytic conductor $16.760$
Analytic rank $0$
Dimension $4$
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] = [5,24,Mod(1,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 24, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.1");
 
S:= CuspForms(chi, 24);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 5.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: \(16.7602018673\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 5761014x^{2} - 3205061410x + 2143006857425 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 5^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 195) q^{2} + ( - \beta_{2} - 39 \beta_1 - 51670) q^{3} + (23 \beta_{3} + \beta_{2} + 2062 \beta_1 + 3171448) q^{4} - 48828125 q^{5} + (1512 \beta_{3} - 2216 \beta_{2} + 43158 \beta_1 + 453750802) q^{6} + ( - 960 \beta_{3} - 9561 \beta_{2} - 1410383 \beta_1 - 252677650) q^{7} + ( - 70420 \beta_{3} - 39068 \beta_{2} + \cdots - 22741180440) q^{8}+ \cdots + (203136 \beta_{3} + 171052 \beta_{2} + \cdots + 58990837657) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 195) q^{2} + ( - \beta_{2} - 39 \beta_1 - 51670) q^{3} + (23 \beta_{3} + \beta_{2} + 2062 \beta_1 + 3171448) q^{4} - 48828125 q^{5} + (1512 \beta_{3} - 2216 \beta_{2} + 43158 \beta_1 + 453750802) q^{6} + ( - 960 \beta_{3} - 9561 \beta_{2} - 1410383 \beta_1 - 252677650) q^{7} + ( - 70420 \beta_{3} - 39068 \beta_{2} + \cdots - 22741180440) q^{8}+ \cdots + (51\!\cdots\!92 \beta_{3} + \cdots + 12\!\cdots\!24) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 780 q^{2} - 206680 q^{3} + 12685792 q^{4} - 195312500 q^{5} + 1815003208 q^{6} - 1010710600 q^{7} - 90964721760 q^{8} + 235963350628 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 780 q^{2} - 206680 q^{3} + 12685792 q^{4} - 195312500 q^{5} + 1815003208 q^{6} - 1010710600 q^{7} - 90964721760 q^{8} + 235963350628 q^{9} + 38085937500 q^{10} + 510770963328 q^{11} - 661143479360 q^{12} - 14153856943960 q^{13} + 64982579554584 q^{14} + 10091796875000 q^{15} + 113436824881024 q^{16} + 15332090016360 q^{17} + 13\!\cdots\!20 q^{18}+ \cdots + 48\!\cdots\!96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 5761014x^{2} - 3205061410x + 2143006857425 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 23\nu^{3} - 28517\nu^{2} - 91667033\nu + 26703215801 ) / 103518 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 19243\nu^{2} - 11083097\nu - 53011636315 ) / 103518 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 23\beta_{3} + \beta_{2} + 1674\beta _1 + 11522032 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 28517\beta_{3} + 19243\beta_{2} + 10046588\beta _1 + 9649750322 ) / 4 \) 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
2585.58
395.315
−1140.42
−1838.47
−5365.16 −220531. 2.03963e7 −4.88281e7 1.18318e9 −7.50656e9 −6.44232e10 −4.55095e10 2.61971e11
1.2 −984.630 38959.8 −7.41911e6 −4.88281e7 −3.83610e7 2.99523e8 1.55648e10 −9.26253e10 4.80776e10
1.3 2086.84 −542682. −4.03369e6 −4.88281e7 −1.13249e9 −2.45126e9 −2.59234e10 2.00360e11 −1.01897e11
1.4 3482.94 517572. 3.74229e6 −4.88281e7 1.80268e9 8.64758e9 −1.61829e10 1.73738e11 −1.70066e11
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)

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 5.24.a.b 4
3.b odd 2 1 45.24.a.d 4
5.b even 2 1 25.24.a.c 4
5.c odd 4 2 25.24.b.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.24.a.b 4 1.a even 1 1 trivial
25.24.a.c 4 5.b even 2 1
25.24.b.c 8 5.c odd 4 2
45.24.a.d 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 780T_{2}^{3} - 22815912T_{2}^{2} + 16729054720T_{2} + 38396524609536 \) acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(5))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 780 T^{3} + \cdots + 38396524609536 \) Copy content Toggle raw display
$3$ \( T^{4} + 206680 T^{3} + \cdots + 24\!\cdots\!56 \) Copy content Toggle raw display
$5$ \( (T + 48828125)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 1010710600 T^{3} + \cdots + 47\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{4} - 510770963328 T^{3} + \cdots + 37\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{4} + 14153856943960 T^{3} + \cdots - 12\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{4} - 15332090016360 T^{3} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots - 25\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots - 13\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 13\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 61\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 10\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 94\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 10\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 35\!\cdots\!24 \) Copy content Toggle raw display
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