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Properties

Label 150.6.a.d
Level $150$
Weight $6$
Character orbit 150.a
Self dual yes
Analytic conductor $24.058$
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] = [150,6,Mod(1,150)] 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("150.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
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, 6, names="a")
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 150.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-4,9,16,0,-36,-176] 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: \(24.0575729719\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
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)
 
\(f(q)\) \(=\) \( q - 4 q^{2} + 9 q^{3} + 16 q^{4} - 36 q^{6} - 176 q^{7} - 64 q^{8} + 81 q^{9} - 60 q^{11} + 144 q^{12} + 658 q^{13} + 704 q^{14} + 256 q^{16} + 414 q^{17} - 324 q^{18} + 956 q^{19} - 1584 q^{21} + 240 q^{22}+ \cdots - 4860 q^{99}+O(q^{100}) \) 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 −176.000 −64.0000 81.0000 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 150.6.a.d 1
3.b odd 2 1 450.6.a.m 1
5.b even 2 1 6.6.a.a ✓ 1
5.c odd 4 2 150.6.c.b 2
15.d odd 2 1 18.6.a.b 1
15.e even 4 2 450.6.c.j 2
20.d odd 2 1 48.6.a.c 1
35.c odd 2 1 294.6.a.m 1
35.i odd 6 2 294.6.e.a 2
35.j even 6 2 294.6.e.g 2
40.e odd 2 1 192.6.a.g 1
40.f even 2 1 192.6.a.o 1
45.h odd 6 2 162.6.c.h 2
45.j even 6 2 162.6.c.e 2
55.d odd 2 1 726.6.a.a 1
60.h even 2 1 144.6.a.j 1
65.d even 2 1 1014.6.a.c 1
80.k odd 4 2 768.6.d.p 2
80.q even 4 2 768.6.d.c 2
105.g even 2 1 882.6.a.a 1
120.i odd 2 1 576.6.a.j 1
120.m even 2 1 576.6.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.6.a.a ✓ 1 5.b even 2 1
18.6.a.b 1 15.d odd 2 1
48.6.a.c 1 20.d odd 2 1
144.6.a.j 1 60.h even 2 1
150.6.a.d 1 1.a even 1 1 trivial
150.6.c.b 2 5.c odd 4 2
162.6.c.e 2 45.j even 6 2
162.6.c.h 2 45.h odd 6 2
192.6.a.g 1 40.e odd 2 1
192.6.a.o 1 40.f even 2 1
294.6.a.m 1 35.c odd 2 1
294.6.e.a 2 35.i odd 6 2
294.6.e.g 2 35.j even 6 2
450.6.a.m 1 3.b odd 2 1
450.6.c.j 2 15.e even 4 2
576.6.a.i 1 120.m even 2 1
576.6.a.j 1 120.i odd 2 1
726.6.a.a 1 55.d odd 2 1
768.6.d.c 2 80.q even 4 2
768.6.d.p 2 80.k odd 4 2
882.6.a.a 1 105.g even 2 1
1014.6.a.c 1 65.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 176 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(150))\). 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
$7$ \( T + 176 \) Copy content Toggle raw display
$11$ \( T + 60 \) Copy content Toggle raw display
$13$ \( T - 658 \) Copy content Toggle raw display
$17$ \( T - 414 \) Copy content Toggle raw display
$19$ \( T - 956 \) Copy content Toggle raw display
$23$ \( T + 600 \) Copy content Toggle raw display
$29$ \( T - 5574 \) Copy content Toggle raw display
$31$ \( T + 3592 \) Copy content Toggle raw display
$37$ \( T - 8458 \) Copy content Toggle raw display
$41$ \( T - 19194 \) Copy content Toggle raw display
$43$ \( T + 13316 \) Copy content Toggle raw display
$47$ \( T - 19680 \) Copy content Toggle raw display
$53$ \( T - 31266 \) Copy content Toggle raw display
$59$ \( T - 26340 \) Copy content Toggle raw display
$61$ \( T + 31090 \) Copy content Toggle raw display
$67$ \( T - 16804 \) Copy content Toggle raw display
$71$ \( T - 6120 \) Copy content Toggle raw display
$73$ \( T - 25558 \) Copy content Toggle raw display
$79$ \( T - 74408 \) Copy content Toggle raw display
$83$ \( T - 6468 \) Copy content Toggle raw display
$89$ \( T + 32742 \) Copy content Toggle raw display
$97$ \( T + 166082 \) Copy content Toggle raw display
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