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Properties

Label 380.2.y.b
Level $380$
Weight $2$
Character orbit 380.y
Analytic conductor $3.034$
Analytic rank $0$
Dimension $36$
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] = [380,2,Mod(217,380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(380, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("380.217");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.y (of order \(12\), degree \(4\), 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: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(9\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 6 q^{3} - 4 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 6 q^{3} - 4 q^{5} + 4 q^{7} - 12 q^{11} - 6 q^{13} - 12 q^{15} + 6 q^{17} + 48 q^{21} + 16 q^{23} + 4 q^{25} - 54 q^{33} + 16 q^{35} - 2 q^{43} + 100 q^{45} + 24 q^{47} - 108 q^{51} + 14 q^{55} - 30 q^{57} + 34 q^{61} - 26 q^{63} - 78 q^{67} - 42 q^{71} + 16 q^{73} - 20 q^{77} + 14 q^{81} - 28 q^{83} + 10 q^{85} - 124 q^{87} + 96 q^{91} - 26 q^{93} - 32 q^{95} + 54 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1 0 −2.89008 + 0.774393i 0 2.13589 + 0.661803i 0 −2.81505 − 2.81505i 0 5.15478 − 2.97611i 0
217.2 0 −2.08775 + 0.559410i 0 1.02821 − 1.98565i 0 1.41342 + 1.41342i 0 1.44767 − 0.835811i 0
217.3 0 −1.83970 + 0.492946i 0 −2.13067 + 0.678412i 0 −0.607167 − 0.607167i 0 0.543427 − 0.313748i 0
217.4 0 −1.24341 + 0.333170i 0 −1.30148 − 1.81828i 0 0.474734 + 0.474734i 0 −1.16302 + 0.671470i 0
217.5 0 −0.660873 + 0.177080i 0 1.44130 + 1.70958i 0 1.38808 + 1.38808i 0 −2.19268 + 1.26594i 0
217.6 0 0.500957 − 0.134231i 0 −1.65073 + 1.50834i 0 −3.08317 − 3.08317i 0 −2.36514 + 1.36551i 0
217.7 0 1.33262 − 0.357073i 0 2.06883 − 0.848495i 0 0.519503 + 0.519503i 0 −0.949712 + 0.548316i 0
217.8 0 1.42686 − 0.382326i 0 −2.03585 + 0.924834i 0 3.57243 + 3.57243i 0 −0.708320 + 0.408949i 0
217.9 0 3.09534 − 0.829395i 0 1.17655 + 1.90150i 0 0.137224 + 0.137224i 0 6.29517 − 3.63452i 0
293.1 0 −0.774393 − 2.89008i 0 −1.64108 − 1.51883i 0 −2.81505 + 2.81505i 0 −5.15478 + 2.97611i 0
293.2 0 −0.559410 − 2.08775i 0 1.20552 − 1.88328i 0 1.41342 − 1.41342i 0 −1.44767 + 0.835811i 0
293.3 0 −0.492946 − 1.83970i 0 0.477813 + 2.18442i 0 −0.607167 + 0.607167i 0 −0.543427 + 0.313748i 0
293.4 0 −0.333170 − 1.24341i 0 2.22542 + 0.217974i 0 0.474734 − 0.474734i 0 1.16302 − 0.671470i 0
293.5 0 −0.177080 − 0.660873i 0 −2.20119 − 0.393414i 0 1.38808 − 1.38808i 0 2.19268 − 1.26594i 0
293.6 0 0.134231 + 0.500957i 0 −0.480898 + 2.18374i 0 −3.08317 + 3.08317i 0 2.36514 − 1.36551i 0
293.7 0 0.357073 + 1.33262i 0 −0.299597 − 2.21591i 0 0.519503 − 0.519503i 0 0.949712 − 0.548316i 0
293.8 0 0.382326 + 1.42686i 0 0.216995 + 2.22551i 0 3.57243 − 3.57243i 0 0.708320 − 0.408949i 0
293.9 0 0.829395 + 3.09534i 0 −2.23503 − 0.0681732i 0 0.137224 − 0.137224i 0 −6.29517 + 3.63452i 0
297.1 0 −0.774393 + 2.89008i 0 −1.64108 + 1.51883i 0 −2.81505 − 2.81505i 0 −5.15478 − 2.97611i 0
297.2 0 −0.559410 + 2.08775i 0 1.20552 + 1.88328i 0 1.41342 + 1.41342i 0 −1.44767 − 0.835811i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 217.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.d odd 6 1 inner
95.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.y.b ✓ 36
5.c odd 4 1 inner 380.2.y.b ✓ 36
19.d odd 6 1 inner 380.2.y.b ✓ 36
95.l even 12 1 inner 380.2.y.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.y.b ✓ 36 1.a even 1 1 trivial
380.2.y.b ✓ 36 5.c odd 4 1 inner
380.2.y.b ✓ 36 19.d odd 6 1 inner
380.2.y.b ✓ 36 95.l even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} + 6 T_{3}^{35} + 18 T_{3}^{34} + 36 T_{3}^{33} - 62 T_{3}^{32} - 648 T_{3}^{31} + \cdots + 1822500 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\). Copy content Toggle raw display