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Properties

Label 150.2.e.b
Level $150$
Weight $2$
Character orbit 150.e
Analytic conductor $1.198$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [150,2,Mod(107,150)] 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("150.107"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(150, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 150.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.19775603032\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{24}^{5} - \zeta_{24}) q^{2} + (\zeta_{24}^{7} - 2 \zeta_{24}^{3}) q^{3} - \zeta_{24}^{6} q^{4} + ( - \zeta_{24}^{4} + 2) q^{6} + (2 \zeta_{24}^{5} + 2 \zeta_{24}) q^{7} + \zeta_{24}^{3} q^{8} + \cdots + (18 \zeta_{24}^{6} - 9 \zeta_{24}^{2}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{6} - 8 q^{16} - 24 q^{21} - 16 q^{31} + 12 q^{36} - 48 q^{46} - 36 q^{51} + 112 q^{61} + 36 q^{66} - 8 q^{76} + 36 q^{81} - 12 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(\zeta_{24}^{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
0.965926 + 0.258819i
−0.258819 − 0.965926i
0.258819 + 0.965926i
−0.965926 − 0.258819i
0.965926 − 0.258819i
−0.258819 + 0.965926i
0.258819 − 0.965926i
−0.965926 + 0.258819i
−0.707107 + 0.707107i −1.67303 − 0.448288i − 1.00000i 0 1.50000 − 0.866025i 2.44949 + 2.44949i 0.707107 + 0.707107i 2.59808 + 1.50000i 0
107.2 −0.707107 + 0.707107i −0.448288 − 1.67303i − 1.00000i 0 1.50000 + 0.866025i −2.44949 − 2.44949i 0.707107 + 0.707107i −2.59808 + 1.50000i 0
107.3 0.707107 − 0.707107i 0.448288 + 1.67303i − 1.00000i 0 1.50000 + 0.866025i 2.44949 + 2.44949i −0.707107 − 0.707107i −2.59808 + 1.50000i 0
107.4 0.707107 − 0.707107i 1.67303 + 0.448288i − 1.00000i 0 1.50000 − 0.866025i −2.44949 − 2.44949i −0.707107 − 0.707107i 2.59808 + 1.50000i 0
143.1 −0.707107 − 0.707107i −1.67303 + 0.448288i 1.00000i 0 1.50000 + 0.866025i 2.44949 − 2.44949i 0.707107 − 0.707107i 2.59808 − 1.50000i 0
143.2 −0.707107 − 0.707107i −0.448288 + 1.67303i 1.00000i 0 1.50000 − 0.866025i −2.44949 + 2.44949i 0.707107 − 0.707107i −2.59808 − 1.50000i 0
143.3 0.707107 + 0.707107i 0.448288 − 1.67303i 1.00000i 0 1.50000 − 0.866025i 2.44949 − 2.44949i −0.707107 + 0.707107i −2.59808 − 1.50000i 0
143.4 0.707107 + 0.707107i 1.67303 − 0.448288i 1.00000i 0 1.50000 + 0.866025i −2.44949 + 2.44949i −0.707107 + 0.707107i 2.59808 − 1.50000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.2.e.b ✓ 8
3.b odd 2 1 inner 150.2.e.b ✓ 8
4.b odd 2 1 1200.2.v.l 8
5.b even 2 1 inner 150.2.e.b ✓ 8
5.c odd 4 2 inner 150.2.e.b ✓ 8
12.b even 2 1 1200.2.v.l 8
15.d odd 2 1 inner 150.2.e.b ✓ 8
15.e even 4 2 inner 150.2.e.b ✓ 8
20.d odd 2 1 1200.2.v.l 8
20.e even 4 2 1200.2.v.l 8
60.h even 2 1 1200.2.v.l 8
60.l odd 4 2 1200.2.v.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.e.b ✓ 8 1.a even 1 1 trivial
150.2.e.b ✓ 8 3.b odd 2 1 inner
150.2.e.b ✓ 8 5.b even 2 1 inner
150.2.e.b ✓ 8 5.c odd 4 2 inner
150.2.e.b ✓ 8 15.d odd 2 1 inner
150.2.e.b ✓ 8 15.e even 4 2 inner
1200.2.v.l 8 4.b odd 2 1
1200.2.v.l 8 12.b even 2 1
1200.2.v.l 8 20.d odd 2 1
1200.2.v.l 8 20.e even 4 2
1200.2.v.l 8 60.h even 2 1
1200.2.v.l 8 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 144 \) acting on \(S_{2}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 9T^{4} + 81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 27)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 81)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T + 2)^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 27)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$61$ \( (T - 14)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 729)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 5625)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 196)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 81)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 243)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 2304)^{2} \) Copy content Toggle raw display
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