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Properties

Label 150.6.e.c
Level $150$
Weight $6$
Character orbit 150.e
Analytic conductor $24.058$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [150,6,Mod(107,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.107");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 150.e (of order \(4\), degree \(2\), minimal)

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: no
Analytic conductor: \(24.0575729719\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 144 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 144 q^{6} - 6144 q^{16} - 12552 q^{21} - 8688 q^{31} + 25536 q^{36} - 120384 q^{46} + 77292 q^{51} + 11136 q^{61} + 54288 q^{66} - 57984 q^{76} - 189612 q^{81} + 290880 q^{91} + 36864 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −2.82843 + 2.82843i −12.6263 − 9.14207i − 16.0000i 0 61.5702 − 9.85477i −57.0619 − 57.0619i 45.2548 + 45.2548i 75.8449 + 230.860i 0
107.2 −2.82843 + 2.82843i −9.14207 − 12.6263i − 16.0000i 0 61.5702 + 9.85477i 57.0619 + 57.0619i 45.2548 + 45.2548i −75.8449 + 230.860i 0
107.3 −2.82843 + 2.82843i 4.59980 + 14.8944i − 16.0000i 0 −55.1378 − 29.1174i 83.3319 + 83.3319i 45.2548 + 45.2548i −200.684 + 137.022i 0
107.4 −2.82843 + 2.82843i 14.8944 + 4.59980i − 16.0000i 0 −55.1378 + 29.1174i −83.3319 − 83.3319i 45.2548 + 45.2548i 200.684 + 137.022i 0
107.5 −2.82843 + 2.82843i −5.82221 + 14.4604i − 16.0000i 0 −24.4324 − 57.3678i 44.8636 + 44.8636i 45.2548 + 45.2548i −175.204 − 168.382i 0
107.6 −2.82843 + 2.82843i 14.4604 − 5.82221i − 16.0000i 0 −24.4324 + 57.3678i −44.8636 − 44.8636i 45.2548 + 45.2548i 175.204 − 168.382i 0
107.7 2.82843 − 2.82843i 9.14207 + 12.6263i − 16.0000i 0 61.5702 + 9.85477i −57.0619 − 57.0619i −45.2548 − 45.2548i −75.8449 + 230.860i 0
107.8 2.82843 − 2.82843i 12.6263 + 9.14207i − 16.0000i 0 61.5702 − 9.85477i 57.0619 + 57.0619i −45.2548 − 45.2548i 75.8449 + 230.860i 0
107.9 2.82843 − 2.82843i −14.4604 + 5.82221i − 16.0000i 0 −24.4324 + 57.3678i 44.8636 + 44.8636i −45.2548 − 45.2548i 175.204 − 168.382i 0
107.10 2.82843 − 2.82843i 5.82221 − 14.4604i − 16.0000i 0 −24.4324 − 57.3678i −44.8636 − 44.8636i −45.2548 − 45.2548i −175.204 − 168.382i 0
107.11 2.82843 − 2.82843i −14.8944 − 4.59980i − 16.0000i 0 −55.1378 + 29.1174i 83.3319 + 83.3319i −45.2548 − 45.2548i 200.684 + 137.022i 0
107.12 2.82843 − 2.82843i −4.59980 − 14.8944i − 16.0000i 0 −55.1378 − 29.1174i −83.3319 − 83.3319i −45.2548 − 45.2548i −200.684 + 137.022i 0
143.1 −2.82843 − 2.82843i −12.6263 + 9.14207i 16.0000i 0 61.5702 + 9.85477i −57.0619 + 57.0619i 45.2548 − 45.2548i 75.8449 − 230.860i 0
143.2 −2.82843 − 2.82843i −9.14207 + 12.6263i 16.0000i 0 61.5702 − 9.85477i 57.0619 − 57.0619i 45.2548 − 45.2548i −75.8449 − 230.860i 0
143.3 −2.82843 − 2.82843i 4.59980 − 14.8944i 16.0000i 0 −55.1378 + 29.1174i 83.3319 − 83.3319i 45.2548 − 45.2548i −200.684 − 137.022i 0
143.4 −2.82843 − 2.82843i 14.8944 − 4.59980i 16.0000i 0 −55.1378 − 29.1174i −83.3319 + 83.3319i 45.2548 − 45.2548i 200.684 − 137.022i 0
143.5 −2.82843 − 2.82843i −5.82221 − 14.4604i 16.0000i 0 −24.4324 + 57.3678i 44.8636 − 44.8636i 45.2548 − 45.2548i −175.204 + 168.382i 0
143.6 −2.82843 − 2.82843i 14.4604 + 5.82221i 16.0000i 0 −24.4324 − 57.3678i −44.8636 + 44.8636i 45.2548 − 45.2548i 175.204 + 168.382i 0
143.7 2.82843 + 2.82843i 9.14207 − 12.6263i 16.0000i 0 61.5702 − 9.85477i −57.0619 + 57.0619i −45.2548 + 45.2548i −75.8449 − 230.860i 0
143.8 2.82843 + 2.82843i 12.6263 − 9.14207i 16.0000i 0 61.5702 + 9.85477i 57.0619 − 57.0619i −45.2548 + 45.2548i 75.8449 − 230.860i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.12
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 150.6.e.c ✓ 24
3.b odd 2 1 inner 150.6.e.c ✓ 24
5.b even 2 1 inner 150.6.e.c ✓ 24
5.c odd 4 2 inner 150.6.e.c ✓ 24
15.d odd 2 1 inner 150.6.e.c ✓ 24
15.e even 4 2 inner 150.6.e.c ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.6.e.c ✓ 24 1.a even 1 1 trivial
150.6.e.c ✓ 24 3.b odd 2 1 inner
150.6.e.c ✓ 24 5.b even 2 1 inner
150.6.e.c ✓ 24 5.c odd 4 2 inner
150.6.e.c ✓ 24 15.d odd 2 1 inner
150.6.e.c ✓ 24 15.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 251499762T_{7}^{8} + 11992759326158913T_{7}^{4} + 132551777852975874977424 \) acting on \(S_{6}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display