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Properties

Label 225.6.a.c
Level $225$
Weight $6$
Character orbit 225.a
Self dual yes
Analytic conductor $36.086$
Analytic rank $0$
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] = [225,6,Mod(1,225)] 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("225.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-2,0,-28,0,0,132] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(36.0863594579\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} - 28 q^{4} + 132 q^{7} + 120 q^{8} - 472 q^{11} + 686 q^{13} - 264 q^{14} + 656 q^{16} - 1562 q^{17} - 2180 q^{19} + 944 q^{22} + 264 q^{23} - 1372 q^{26} - 3696 q^{28} - 170 q^{29} + 7272 q^{31}+ \cdots - 1234 q^{98}+O(q^{100}) \) 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
−2.00000 0 −28.0000 0 0 132.000 120.000 0 0
\(n\): e.g. 2-40 or 990-1000
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 225.6.a.c 1
3.b odd 2 1 75.6.a.c 1
5.b even 2 1 45.6.a.c 1
5.c odd 4 2 225.6.b.d 2
15.d odd 2 1 15.6.a.a ✓ 1
15.e even 4 2 75.6.b.d 2
20.d odd 2 1 720.6.a.w 1
60.h even 2 1 240.6.a.k 1
105.g even 2 1 735.6.a.a 1
120.i odd 2 1 960.6.a.v 1
120.m even 2 1 960.6.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.6.a.a ✓ 1 15.d odd 2 1
45.6.a.c 1 5.b even 2 1
75.6.a.c 1 3.b odd 2 1
75.6.b.d 2 15.e even 4 2
225.6.a.c 1 1.a even 1 1 trivial
225.6.b.d 2 5.c odd 4 2
240.6.a.k 1 60.h even 2 1
720.6.a.w 1 20.d odd 2 1
735.6.a.a 1 105.g even 2 1
960.6.a.m 1 120.m even 2 1
960.6.a.v 1 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2} + 2 \) Copy content Toggle raw display
\( T_{7} - 132 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 132 \) Copy content Toggle raw display
$11$ \( T + 472 \) Copy content Toggle raw display
$13$ \( T - 686 \) Copy content Toggle raw display
$17$ \( T + 1562 \) Copy content Toggle raw display
$19$ \( T + 2180 \) Copy content Toggle raw display
$23$ \( T - 264 \) Copy content Toggle raw display
$29$ \( T + 170 \) Copy content Toggle raw display
$31$ \( T - 7272 \) Copy content Toggle raw display
$37$ \( T - 142 \) Copy content Toggle raw display
$41$ \( T - 16198 \) Copy content Toggle raw display
$43$ \( T - 10316 \) Copy content Toggle raw display
$47$ \( T - 18568 \) Copy content Toggle raw display
$53$ \( T - 21514 \) Copy content Toggle raw display
$59$ \( T + 34600 \) Copy content Toggle raw display
$61$ \( T + 35738 \) Copy content Toggle raw display
$67$ \( T - 5772 \) Copy content Toggle raw display
$71$ \( T - 69088 \) Copy content Toggle raw display
$73$ \( T - 70526 \) Copy content Toggle raw display
$79$ \( T - 47640 \) Copy content Toggle raw display
$83$ \( T - 74004 \) Copy content Toggle raw display
$89$ \( T - 90030 \) Copy content Toggle raw display
$97$ \( T - 33502 \) Copy content Toggle raw display
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