Properties

Label 150.8.a.r
Level $150$
Weight $8$
Character orbit 150.a
Self dual yes
Analytic conductor $46.858$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [150,8,Mod(1,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 150.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(46.8577538226\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2641}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 660 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: no (minimal twist has level 30)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 10\sqrt{2641}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 q^{2} + 27 q^{3} + 64 q^{4} - 216 q^{6} + ( - 2 \beta - 282) q^{7} - 512 q^{8} + 729 q^{9} + ( - 7 \beta - 1860) q^{11} + 1728 q^{12} + ( - 3 \beta - 3696) q^{13} + (16 \beta + 2256) q^{14} + 4096 q^{16}+ \cdots + ( - 5103 \beta - 1355940) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{2} + 54 q^{3} + 128 q^{4} - 432 q^{6} - 564 q^{7} - 1024 q^{8} + 1458 q^{9} - 3720 q^{11} + 3456 q^{12} - 7392 q^{13} + 4512 q^{14} + 8192 q^{16} + 24176 q^{17} - 11664 q^{18} + 34728 q^{19}+ \cdots - 2711880 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
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
26.1953
−25.1953
−8.00000 27.0000 64.0000 0 −216.000 −1309.81 −512.000 729.000 0
1.2 −8.00000 27.0000 64.0000 0 −216.000 745.813 −512.000 729.000 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 150.8.a.r 2
3.b odd 2 1 450.8.a.bh 2
5.b even 2 1 150.8.a.s 2
5.c odd 4 2 30.8.c.b 4
15.d odd 2 1 450.8.a.be 2
15.e even 4 2 90.8.c.b 4
20.e even 4 2 240.8.f.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.8.c.b 4 5.c odd 4 2
90.8.c.b 4 15.e even 4 2
150.8.a.r 2 1.a even 1 1 trivial
150.8.a.s 2 5.b even 2 1
240.8.f.d 4 20.e even 4 2
450.8.a.be 2 15.d odd 2 1
450.8.a.bh 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 564T_{7} - 976876 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(150))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 8)^{2} \) Copy content Toggle raw display
$3$ \( (T - 27)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 564T - 976876 \) Copy content Toggle raw display
$11$ \( T^{2} + 3720 T - 9481300 \) Copy content Toggle raw display
$13$ \( T^{2} + 7392 T + 11283516 \) Copy content Toggle raw display
$17$ \( T^{2} - 24176 T - 141485156 \) Copy content Toggle raw display
$19$ \( T^{2} - 34728 T + 63818496 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 5901290000 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 19057296384 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 6515247296 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 29580347516 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 36687881884 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 234400112144 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 360984396800 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 1668312344196 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 5193452974700 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 8831371838500 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 457291398976 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 20010069370800 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 17059237628944 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 58253255294976 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 23716859094736 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 25448685498556 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 30105038810816 \) Copy content Toggle raw display
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