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Properties

Label 1380.2.r.c
Level $1380$
Weight $2$
Character orbit 1380.r
Analytic conductor $11.019$
Analytic rank $0$
Dimension $80$
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] = [1380,2,Mod(737,1380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1380, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1380.737");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1380.r (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: \(11.0193554789\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 8 q^{3}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 8 q^{3} + 32 q^{13} + 24 q^{21} - 32 q^{25} - 28 q^{27} - 32 q^{31} - 44 q^{33} + 24 q^{37} - 32 q^{43} + 88 q^{45} + 16 q^{51} + 8 q^{55} + 16 q^{57} - 32 q^{61} - 12 q^{63} - 16 q^{67} - 32 q^{73} + 4 q^{75} - 64 q^{81} - 32 q^{85} + 64 q^{91} + 8 q^{93} - 96 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
737.1 0 −1.72892 − 0.104128i 0 1.51193 + 1.64744i 0 3.27368 + 3.27368i 0 2.97831 + 0.360057i 0
737.2 0 −1.71368 − 0.251606i 0 0.257544 − 2.22119i 0 −2.77975 − 2.77975i 0 2.87339 + 0.862345i 0
737.3 0 −1.67967 − 0.422758i 0 2.05360 − 0.884710i 0 0.924264 + 0.924264i 0 2.64255 + 1.42018i 0
737.4 0 −1.57061 − 0.730184i 0 −0.115647 + 2.23308i 0 −0.487335 − 0.487335i 0 1.93366 + 2.29368i 0
737.5 0 −1.53876 + 0.795128i 0 1.56652 − 1.59562i 0 0.769646 + 0.769646i 0 1.73554 − 2.44702i 0
737.6 0 −1.50371 − 0.859570i 0 −0.917834 − 2.03901i 0 0.160200 + 0.160200i 0 1.52228 + 2.58509i 0
737.7 0 −1.43700 + 0.966965i 0 1.81165 + 1.31070i 0 −2.64771 − 2.64771i 0 1.12996 − 2.77906i 0
737.8 0 −1.43142 + 0.975211i 0 0.832554 + 2.07530i 0 0.0455472 + 0.0455472i 0 1.09793 − 2.79187i 0
737.9 0 −1.31796 − 1.12382i 0 2.16179 + 0.571544i 0 −3.03605 − 3.03605i 0 0.474038 + 2.96231i 0
737.10 0 −1.29281 − 1.15267i 0 −0.532114 + 2.17183i 0 −0.268444 − 0.268444i 0 0.342726 + 2.98036i 0
737.11 0 −1.21045 − 1.23887i 0 −2.23574 + 0.0384947i 0 1.22712 + 1.22712i 0 −0.0696199 + 2.99919i 0
737.12 0 −0.975211 + 1.43142i 0 −0.832554 − 2.07530i 0 0.0455472 + 0.0455472i 0 −1.09793 − 2.79187i 0
737.13 0 −0.966965 + 1.43700i 0 −1.81165 − 1.31070i 0 −2.64771 − 2.64771i 0 −1.12996 − 2.77906i 0
737.14 0 −0.795128 + 1.53876i 0 −1.56652 + 1.59562i 0 0.769646 + 0.769646i 0 −1.73554 − 2.44702i 0
737.15 0 −0.727046 − 1.57207i 0 1.08694 − 1.95412i 0 2.39152 + 2.39152i 0 −1.94281 + 2.28593i 0
737.16 0 −0.136960 − 1.72663i 0 −2.01176 − 0.976134i 0 −0.818985 − 0.818985i 0 −2.96248 + 0.472958i 0
737.17 0 −0.118851 − 1.72797i 0 0.238514 + 2.22331i 0 2.86460 + 2.86460i 0 −2.97175 + 0.410741i 0
737.18 0 −0.0819537 − 1.73011i 0 −1.50202 + 1.65649i 0 −2.00070 − 2.00070i 0 −2.98657 + 0.283578i 0
737.19 0 0.104128 + 1.72892i 0 −1.51193 − 1.64744i 0 3.27368 + 3.27368i 0 −2.97831 + 0.360057i 0
737.20 0 0.251606 + 1.71368i 0 −0.257544 + 2.22119i 0 −2.77975 − 2.77975i 0 −2.87339 + 0.862345i 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 737.40
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1380.2.r.c ✓ 80
3.b odd 2 1 inner 1380.2.r.c ✓ 80
5.c odd 4 1 inner 1380.2.r.c ✓ 80
15.e even 4 1 inner 1380.2.r.c ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.r.c ✓ 80 1.a even 1 1 trivial
1380.2.r.c ✓ 80 3.b odd 2 1 inner
1380.2.r.c ✓ 80 5.c odd 4 1 inner
1380.2.r.c ✓ 80 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{40} - 20 T_{7}^{37} + 1084 T_{7}^{36} - 172 T_{7}^{35} + 200 T_{7}^{34} - 20844 T_{7}^{33} + \cdots + 136048896 \) acting on \(S_{2}^{\mathrm{new}}(1380, [\chi])\). Copy content Toggle raw display