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Properties

Label 25.22.a.a
Level $25$
Weight $22$
Character orbit 25.a
Self dual yes
Analytic conductor $69.869$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [25,22,Mod(1,25)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("25.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(25, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,288] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(69.8693360718\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 288 q^{2} + 128844 q^{3} - 2014208 q^{4} + 37107072 q^{6} + 768078808 q^{7} - 1184071680 q^{8} + 6140423133 q^{9} - 94724929188 q^{11} - 259518615552 q^{12} + 80621789794 q^{13} + 221206696704 q^{14}+ \cdots - 58\!\cdots\!04 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
0
288.000 128844. −2.01421e6 0 3.71071e7 7.68079e8 −1.18407e9 6.14042e9 0
\(n\): e.g. 2-40 or 80-90
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 25.22.a.a 1
5.b even 2 1 1.22.a.a ✓ 1
5.c odd 4 2 25.22.b.a 2
15.d odd 2 1 9.22.a.c 1
20.d odd 2 1 16.22.a.c 1
35.c odd 2 1 49.22.a.a 1
40.e odd 2 1 64.22.a.a 1
40.f even 2 1 64.22.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.22.a.a ✓ 1 5.b even 2 1
9.22.a.c 1 15.d odd 2 1
16.22.a.c 1 20.d odd 2 1
25.22.a.a 1 1.a even 1 1 trivial
25.22.b.a 2 5.c odd 4 2
49.22.a.a 1 35.c odd 2 1
64.22.a.a 1 40.e odd 2 1
64.22.a.g 1 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 288 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(25))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 288 \) Copy content Toggle raw display
$3$ \( T - 128844 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 768078808 \) Copy content Toggle raw display
$11$ \( T + 94724929188 \) Copy content Toggle raw display
$13$ \( T - 80621789794 \) Copy content Toggle raw display
$17$ \( T + 3052282930002 \) Copy content Toggle raw display
$19$ \( T + 7920788351740 \) Copy content Toggle raw display
$23$ \( T - 73845437470344 \) Copy content Toggle raw display
$29$ \( T + 4253031736469010 \) Copy content Toggle raw display
$31$ \( T - 1900541176310432 \) Copy content Toggle raw display
$37$ \( T + 22\!\cdots\!22 \) Copy content Toggle raw display
$41$ \( T + 20\!\cdots\!58 \) Copy content Toggle raw display
$43$ \( T - 19\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T + 14\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T + 20\!\cdots\!06 \) Copy content Toggle raw display
$59$ \( T + 59\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T - 61\!\cdots\!62 \) Copy content Toggle raw display
$67$ \( T + 16\!\cdots\!52 \) Copy content Toggle raw display
$71$ \( T + 56\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T - 43\!\cdots\!94 \) Copy content Toggle raw display
$79$ \( T + 51\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T + 48\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T + 50\!\cdots\!30 \) Copy content Toggle raw display
$97$ \( T + 80\!\cdots\!82 \) Copy content Toggle raw display
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