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Properties

Label 330.3.h.a
Level $330$
Weight $3$
Character orbit 330.h
Analytic conductor $8.992$
Analytic rank $0$
Dimension $24$
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] = [330,3,Mod(109,330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(330, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("330.109");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 330.h (of order \(2\), degree \(1\), 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: \(8.99184872389\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{4} - 8 q^{5} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{4} - 8 q^{5} - 72 q^{9} + 48 q^{11} + 32 q^{14} + 96 q^{16} - 16 q^{20} - 8 q^{25} - 64 q^{26} + 96 q^{31} - 32 q^{34} - 144 q^{36} + 96 q^{44} + 24 q^{45} + 392 q^{49} - 112 q^{55} + 64 q^{56} - 352 q^{59} + 192 q^{64} - 48 q^{66} - 384 q^{69} + 256 q^{70} - 640 q^{71} - 192 q^{75} - 32 q^{80} + 216 q^{81} + 512 q^{86} + 48 q^{89} + 800 q^{91} - 144 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 −1.41421 − 1.73205i 2.00000 −4.77529 − 1.48210i 2.44949i −8.82358 −2.82843 −3.00000 6.75328 + 2.09601i
109.2 −1.41421 − 1.73205i 2.00000 −4.40599 + 2.36374i 2.44949i 7.34126 −2.82843 −3.00000 6.23101 − 3.34283i
109.3 −1.41421 − 1.73205i 2.00000 −1.22977 + 4.84641i 2.44949i 0.577720 −2.82843 −3.00000 1.73916 − 6.85385i
109.4 −1.41421 − 1.73205i 2.00000 0.724623 − 4.94721i 2.44949i 11.3031 −2.82843 −3.00000 −1.02477 + 6.99642i
109.5 −1.41421 − 1.73205i 2.00000 3.52238 − 3.54863i 2.44949i −6.74939 −2.82843 −3.00000 −4.98140 + 5.01853i
109.6 −1.41421 − 1.73205i 2.00000 4.16404 + 2.76780i 2.44949i −9.30599 −2.82843 −3.00000 −5.88885 − 3.91426i
109.7 −1.41421 1.73205i 2.00000 −4.77529 + 1.48210i − 2.44949i −8.82358 −2.82843 −3.00000 6.75328 − 2.09601i
109.8 −1.41421 1.73205i 2.00000 −4.40599 − 2.36374i − 2.44949i 7.34126 −2.82843 −3.00000 6.23101 + 3.34283i
109.9 −1.41421 1.73205i 2.00000 −1.22977 − 4.84641i − 2.44949i 0.577720 −2.82843 −3.00000 1.73916 + 6.85385i
109.10 −1.41421 1.73205i 2.00000 0.724623 + 4.94721i − 2.44949i 11.3031 −2.82843 −3.00000 −1.02477 − 6.99642i
109.11 −1.41421 1.73205i 2.00000 3.52238 + 3.54863i − 2.44949i −6.74939 −2.82843 −3.00000 −4.98140 − 5.01853i
109.12 −1.41421 1.73205i 2.00000 4.16404 − 2.76780i − 2.44949i −9.30599 −2.82843 −3.00000 −5.88885 + 3.91426i
109.13 1.41421 − 1.73205i 2.00000 −4.77529 − 1.48210i − 2.44949i 8.82358 2.82843 −3.00000 −6.75328 − 2.09601i
109.14 1.41421 − 1.73205i 2.00000 −4.40599 + 2.36374i − 2.44949i −7.34126 2.82843 −3.00000 −6.23101 + 3.34283i
109.15 1.41421 − 1.73205i 2.00000 −1.22977 + 4.84641i − 2.44949i −0.577720 2.82843 −3.00000 −1.73916 + 6.85385i
109.16 1.41421 − 1.73205i 2.00000 0.724623 − 4.94721i − 2.44949i −11.3031 2.82843 −3.00000 1.02477 − 6.99642i
109.17 1.41421 − 1.73205i 2.00000 3.52238 − 3.54863i − 2.44949i 6.74939 2.82843 −3.00000 4.98140 − 5.01853i
109.18 1.41421 − 1.73205i 2.00000 4.16404 + 2.76780i − 2.44949i 9.30599 2.82843 −3.00000 5.88885 + 3.91426i
109.19 1.41421 1.73205i 2.00000 −4.77529 + 1.48210i 2.44949i 8.82358 2.82843 −3.00000 −6.75328 + 2.09601i
109.20 1.41421 1.73205i 2.00000 −4.40599 − 2.36374i 2.44949i −7.34126 2.82843 −3.00000 −6.23101 − 3.34283i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.b odd 2 1 inner
55.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 330.3.h.a ✓ 24
3.b odd 2 1 990.3.h.d 24
5.b even 2 1 inner 330.3.h.a ✓ 24
5.c odd 4 2 1650.3.b.e 24
11.b odd 2 1 inner 330.3.h.a ✓ 24
15.d odd 2 1 990.3.h.d 24
33.d even 2 1 990.3.h.d 24
55.d odd 2 1 inner 330.3.h.a ✓ 24
55.e even 4 2 1650.3.b.e 24
165.d even 2 1 990.3.h.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.3.h.a ✓ 24 1.a even 1 1 trivial
330.3.h.a ✓ 24 5.b even 2 1 inner
330.3.h.a ✓ 24 11.b odd 2 1 inner
330.3.h.a ✓ 24 55.d odd 2 1 inner
990.3.h.d 24 3.b odd 2 1
990.3.h.d 24 15.d odd 2 1
990.3.h.d 24 33.d even 2 1
990.3.h.d 24 165.d even 2 1
1650.3.b.e 24 5.c odd 4 2
1650.3.b.e 24 55.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(330, [\chi])\).