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Properties

Label 30.18.a.c
Level $30$
Weight $18$
Character orbit 30.a
Self dual yes
Analytic conductor $54.967$
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] = [30,18,Mod(1,30)] 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("30.1"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(30, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 30.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-256,6561,65536,390625] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(54.9666262034\)
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)
 
\(f(q)\) \(=\) \( q - 256 q^{2} + 6561 q^{3} + 65536 q^{4} + 390625 q^{5} - 1679616 q^{6} + 1929536 q^{7} - 16777216 q^{8} + 43046721 q^{9} - 100000000 q^{10} + 352985820 q^{11} + 429981696 q^{12} - 3500000218 q^{13} - 493961216 q^{14}+ \cdots + 15\!\cdots\!20 q^{99}+O(q^{100}) \) 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
−256.000 6561.00 65536.0 390625. −1.67962e6 1.92954e6 −1.67772e7 4.30467e7 −1.00000e8
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(5\) \( -1 \)

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 30.18.a.c ✓ 1
3.b odd 2 1 90.18.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.18.a.c ✓ 1 1.a even 1 1 trivial
90.18.a.f 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 1929536 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(30))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 256 \) Copy content Toggle raw display
$3$ \( T - 6561 \) Copy content Toggle raw display
$5$ \( T - 390625 \) Copy content Toggle raw display
$7$ \( T - 1929536 \) Copy content Toggle raw display
$11$ \( T - 352985820 \) Copy content Toggle raw display
$13$ \( T + 3500000218 \) Copy content Toggle raw display
$17$ \( T + 54970977894 \) Copy content Toggle raw display
$19$ \( T - 31092978236 \) Copy content Toggle raw display
$23$ \( T + 316297396800 \) Copy content Toggle raw display
$29$ \( T - 3237919791294 \) Copy content Toggle raw display
$31$ \( T + 6515774619112 \) Copy content Toggle raw display
$37$ \( T - 26233622989982 \) Copy content Toggle raw display
$41$ \( T - 44280220470714 \) Copy content Toggle raw display
$43$ \( T + 6124629403444 \) Copy content Toggle raw display
$47$ \( T + 44171951954040 \) Copy content Toggle raw display
$53$ \( T + 489680642744826 \) Copy content Toggle raw display
$59$ \( T + 820004055924180 \) Copy content Toggle raw display
$61$ \( T + 2460927715419250 \) Copy content Toggle raw display
$67$ \( T + 4871518631786044 \) Copy content Toggle raw display
$71$ \( T - 4555031256305160 \) Copy content Toggle raw display
$73$ \( T - 1598431505471162 \) Copy content Toggle raw display
$79$ \( T + 7625872763936872 \) Copy content Toggle raw display
$83$ \( T + 19\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T + 15\!\cdots\!02 \) Copy content Toggle raw display
$97$ \( T + 17\!\cdots\!78 \) Copy content Toggle raw display
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