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Properties

Label 75.6.a.d
Level $75$
Weight $6$
Character orbit 75.a
Self dual yes
Analytic conductor $12.029$
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] = [75,6,Mod(1,75)] 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("75.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,4] 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: \(12.0287864860\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} + 9 q^{3} - 16 q^{4} + 36 q^{6} - 225 q^{7} - 192 q^{8} + 81 q^{9} - 434 q^{11} - 144 q^{12} + 613 q^{13} - 900 q^{14} - 256 q^{16} - 878 q^{17} + 324 q^{18} - 731 q^{19} - 2025 q^{21} - 1736 q^{22}+ \cdots - 35154 q^{99}+O(q^{100}) \) 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
4.00000 9.00000 −16.0000 0 36.0000 −225.000 −192.000 81.0000 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( -1 \)

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 75.6.a.d yes 1
3.b odd 2 1 225.6.a.b 1
5.b even 2 1 75.6.a.b ✓ 1
5.c odd 4 2 75.6.b.c 2
15.d odd 2 1 225.6.a.g 1
15.e even 4 2 225.6.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.6.a.b ✓ 1 5.b even 2 1
75.6.a.d yes 1 1.a even 1 1 trivial
75.6.b.c 2 5.c odd 4 2
225.6.a.b 1 3.b odd 2 1
225.6.a.g 1 15.d odd 2 1
225.6.b.c 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 4 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(75))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T - 9 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 225 \) Copy content Toggle raw display
$11$ \( T + 434 \) Copy content Toggle raw display
$13$ \( T - 613 \) Copy content Toggle raw display
$17$ \( T + 878 \) Copy content Toggle raw display
$19$ \( T + 731 \) Copy content Toggle raw display
$23$ \( T - 2850 \) Copy content Toggle raw display
$29$ \( T + 7582 \) Copy content Toggle raw display
$31$ \( T - 2175 \) Copy content Toggle raw display
$37$ \( T + 9310 \) Copy content Toggle raw display
$41$ \( T + 12040 \) Copy content Toggle raw display
$43$ \( T + 1121 \) Copy content Toggle raw display
$47$ \( T - 29878 \) Copy content Toggle raw display
$53$ \( T - 5740 \) Copy content Toggle raw display
$59$ \( T + 5174 \) Copy content Toggle raw display
$61$ \( T + 38717 \) Copy content Toggle raw display
$67$ \( T - 31707 \) Copy content Toggle raw display
$71$ \( T - 64472 \) Copy content Toggle raw display
$73$ \( T + 19790 \) Copy content Toggle raw display
$79$ \( T + 105000 \) Copy content Toggle raw display
$83$ \( T + 3318 \) Copy content Toggle raw display
$89$ \( T + 65376 \) Copy content Toggle raw display
$97$ \( T + 89143 \) Copy content Toggle raw display
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