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Properties

Label 150.4.a.c
Level $150$
Weight $4$
Character orbit 150.a
Self dual yes
Analytic conductor $8.850$
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] = [150,4,Mod(1,150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("150.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 150.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-2,3,4,0,-6,-23] 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: \(8.85028650086\)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + 3 q^{3} + 4 q^{4} - 6 q^{6} - 23 q^{7} - 8 q^{8} + 9 q^{9} - 30 q^{11} + 12 q^{12} - 29 q^{13} + 46 q^{14} + 16 q^{16} - 78 q^{17} - 18 q^{18} + 149 q^{19} - 69 q^{21} + 60 q^{22} - 150 q^{23}+ \cdots - 270 q^{99}+O(q^{100}) \) 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
−2.00000 3.00000 4.00000 0 −6.00000 −23.0000 −8.00000 9.00000 0
\(n\): 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 150.4.a.c ✓ 1
3.b odd 2 1 450.4.a.l 1
4.b odd 2 1 1200.4.a.r 1
5.b even 2 1 150.4.a.g yes 1
5.c odd 4 2 150.4.c.b 2
15.d odd 2 1 450.4.a.i 1
15.e even 4 2 450.4.c.h 2
20.d odd 2 1 1200.4.a.v 1
20.e even 4 2 1200.4.f.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.4.a.c ✓ 1 1.a even 1 1 trivial
150.4.a.g yes 1 5.b even 2 1
150.4.c.b 2 5.c odd 4 2
450.4.a.i 1 15.d odd 2 1
450.4.a.l 1 3.b odd 2 1
450.4.c.h 2 15.e even 4 2
1200.4.a.r 1 4.b odd 2 1
1200.4.a.v 1 20.d odd 2 1
1200.4.f.q 2 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 23 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(150))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 23 \) Copy content Toggle raw display
$11$ \( T + 30 \) Copy content Toggle raw display
$13$ \( T + 29 \) Copy content Toggle raw display
$17$ \( T + 78 \) Copy content Toggle raw display
$19$ \( T - 149 \) Copy content Toggle raw display
$23$ \( T + 150 \) Copy content Toggle raw display
$29$ \( T + 234 \) Copy content Toggle raw display
$31$ \( T + 217 \) Copy content Toggle raw display
$37$ \( T + 146 \) Copy content Toggle raw display
$41$ \( T + 156 \) Copy content Toggle raw display
$43$ \( T - 433 \) Copy content Toggle raw display
$47$ \( T + 30 \) Copy content Toggle raw display
$53$ \( T - 552 \) Copy content Toggle raw display
$59$ \( T + 270 \) Copy content Toggle raw display
$61$ \( T - 275 \) Copy content Toggle raw display
$67$ \( T + 803 \) Copy content Toggle raw display
$71$ \( T - 660 \) Copy content Toggle raw display
$73$ \( T - 646 \) Copy content Toggle raw display
$79$ \( T - 992 \) Copy content Toggle raw display
$83$ \( T - 846 \) Copy content Toggle raw display
$89$ \( T + 1488 \) Copy content Toggle raw display
$97$ \( T - 319 \) Copy content Toggle raw display
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