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Properties

Label 150.11.b.c
Level $150$
Weight $11$
Character orbit 150.b
Analytic conductor $95.304$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [150,11,Mod(149,150)]
 
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([1, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.149");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 150.b (of order \(2\), degree \(1\), not 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: \(95.3035879011\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 14336 q^{4} - 13568 q^{6} + 237916 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 14336 q^{4} - 13568 q^{6} + 237916 q^{9} + 7340032 q^{16} - 4090604 q^{19} - 9429644 q^{21} - 6946816 q^{24} + 5866364 q^{31} - 162179328 q^{34} + 121812992 q^{36} + 634324820 q^{39} - 778760448 q^{46} - 2654075760 q^{49} + 1211082024 q^{51} + 347951360 q^{54} - 1491726508 q^{61} + 3758096384 q^{64} + 6985556736 q^{66} + 1970895984 q^{69} - 2094389248 q^{76} - 7727532776 q^{79} - 17531586364 q^{81} - 4827977728 q^{84} - 22922302940 q^{91} + 7710044160 q^{94} - 3556769792 q^{96} - 8154948216 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1 −22.6274 −239.008 − 43.8665i 512.000 0 5408.13 + 992.586i 17330.1i −11585.2 55200.5 + 20968.9i 0
149.2 −22.6274 −239.008 + 43.8665i 512.000 0 5408.13 − 992.586i − 17330.1i −11585.2 55200.5 − 20968.9i 0
149.3 −22.6274 −191.244 − 149.915i 512.000 0 4327.37 + 3392.19i 18361.5i −11585.2 14099.9 + 57340.9i 0
149.4 −22.6274 −191.244 + 149.915i 512.000 0 4327.37 − 3392.19i − 18361.5i −11585.2 14099.9 − 57340.9i 0
149.5 −22.6274 −98.4300 − 222.172i 512.000 0 2227.22 + 5027.19i − 23249.5i −11585.2 −39672.1 + 43736.8i 0
149.6 −22.6274 −98.4300 + 222.172i 512.000 0 2227.22 − 5027.19i 23249.5i −11585.2 −39672.1 − 43736.8i 0
149.7 −22.6274 54.0703 − 236.908i 512.000 0 −1223.47 + 5360.62i − 5641.65i −11585.2 −53201.8 − 25619.4i 0
149.8 −22.6274 54.0703 + 236.908i 512.000 0 −1223.47 − 5360.62i 5641.65i −11585.2 −53201.8 + 25619.4i 0
149.9 −22.6274 200.542 − 137.229i 512.000 0 −4537.76 + 3105.13i − 22976.9i −11585.2 21385.5 − 55040.4i 0
149.10 −22.6274 200.542 + 137.229i 512.000 0 −4537.76 − 3105.13i 22976.9i −11585.2 21385.5 + 55040.4i 0
149.11 −22.6274 210.463 − 121.467i 512.000 0 −4762.24 + 2748.49i 2558.79i −11585.2 29540.5 − 51128.7i 0
149.12 −22.6274 210.463 + 121.467i 512.000 0 −4762.24 − 2748.49i − 2558.79i −11585.2 29540.5 + 51128.7i 0
149.13 −22.6274 213.513 − 116.023i 512.000 0 −4831.24 + 2625.29i 29941.9i −11585.2 32126.5 − 49544.7i 0
149.14 −22.6274 213.513 + 116.023i 512.000 0 −4831.24 − 2625.29i − 29941.9i −11585.2 32126.5 + 49544.7i 0
149.15 22.6274 −213.513 − 116.023i 512.000 0 −4831.24 − 2625.29i 29941.9i 11585.2 32126.5 + 49544.7i 0
149.16 22.6274 −213.513 + 116.023i 512.000 0 −4831.24 + 2625.29i − 29941.9i 11585.2 32126.5 − 49544.7i 0
149.17 22.6274 −210.463 − 121.467i 512.000 0 −4762.24 − 2748.49i 2558.79i 11585.2 29540.5 + 51128.7i 0
149.18 22.6274 −210.463 + 121.467i 512.000 0 −4762.24 + 2748.49i − 2558.79i 11585.2 29540.5 − 51128.7i 0
149.19 22.6274 −200.542 − 137.229i 512.000 0 −4537.76 − 3105.13i − 22976.9i 11585.2 21385.5 + 55040.4i 0
149.20 22.6274 −200.542 + 137.229i 512.000 0 −4537.76 + 3105.13i 22976.9i 11585.2 21385.5 − 55040.4i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.28
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
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.11.b.c 28
3.b odd 2 1 inner 150.11.b.c 28
5.b even 2 1 inner 150.11.b.c 28
5.c odd 4 1 150.11.d.c ✓ 14
5.c odd 4 1 150.11.d.d yes 14
15.d odd 2 1 inner 150.11.b.c 28
15.e even 4 1 150.11.d.c ✓ 14
15.e even 4 1 150.11.d.d yes 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.11.b.c 28 1.a even 1 1 trivial
150.11.b.c 28 3.b odd 2 1 inner
150.11.b.c 28 5.b even 2 1 inner
150.11.b.c 28 15.d odd 2 1 inner
150.11.d.c ✓ 14 5.c odd 4 1
150.11.d.c ✓ 14 15.e even 4 1
150.11.d.d yes 14 5.c odd 4 1
150.11.d.d yes 14 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{14} + 2640845683 T_{7}^{12} + \cdots + 53\!\cdots\!24 \) acting on \(S_{11}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display