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Properties

Label 1380.3.l.a
Level $1380$
Weight $3$
Character orbit 1380.l
Analytic conductor $37.602$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1380,3,Mod(461,1380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1380.461");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1380.l (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: \(37.6022764817\)
Analytic rank: \(0\)
Dimension: \(60\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 8 q^{3} - 8 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 8 q^{3} - 8 q^{7} - 12 q^{9} - 32 q^{13} + 20 q^{15} + 136 q^{19} - 28 q^{21} - 300 q^{25} + 40 q^{27} - 72 q^{31} + 32 q^{33} - 160 q^{37} + 76 q^{39} - 64 q^{43} + 588 q^{49} - 180 q^{51} - 16 q^{57} - 64 q^{63} + 24 q^{67} + 336 q^{73} + 40 q^{75} + 8 q^{79} + 124 q^{81} - 120 q^{85} + 380 q^{87} - 456 q^{91} - 268 q^{93} - 176 q^{97} - 232 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
461.1 0 −2.93226 − 0.633895i 0 2.23607i 0 11.1081 0 8.19635 + 3.71750i 0
461.2 0 −2.93226 + 0.633895i 0 − 2.23607i 0 11.1081 0 8.19635 − 3.71750i 0
461.3 0 −2.88863 − 0.809829i 0 2.23607i 0 −1.46478 0 7.68836 + 4.67859i 0
461.4 0 −2.88863 + 0.809829i 0 − 2.23607i 0 −1.46478 0 7.68836 − 4.67859i 0
461.5 0 −2.84013 − 0.966249i 0 − 2.23607i 0 −3.83316 0 7.13272 + 5.48856i 0
461.6 0 −2.84013 + 0.966249i 0 2.23607i 0 −3.83316 0 7.13272 − 5.48856i 0
461.7 0 −2.82300 − 1.01522i 0 2.23607i 0 11.2985 0 6.93866 + 5.73193i 0
461.8 0 −2.82300 + 1.01522i 0 − 2.23607i 0 11.2985 0 6.93866 − 5.73193i 0
461.9 0 −2.78467 − 1.11607i 0 2.23607i 0 −9.40323 0 6.50877 + 6.21578i 0
461.10 0 −2.78467 + 1.11607i 0 − 2.23607i 0 −9.40323 0 6.50877 − 6.21578i 0
461.11 0 −2.73632 − 1.22987i 0 − 2.23607i 0 2.51654 0 5.97484 + 6.73062i 0
461.12 0 −2.73632 + 1.22987i 0 2.23607i 0 2.51654 0 5.97484 − 6.73062i 0
461.13 0 −2.37261 − 1.83595i 0 − 2.23607i 0 −12.1286 0 2.25858 + 8.71199i 0
461.14 0 −2.37261 + 1.83595i 0 2.23607i 0 −12.1286 0 2.25858 − 8.71199i 0
461.15 0 −2.30010 − 1.92602i 0 − 2.23607i 0 4.00501 0 1.58093 + 8.86006i 0
461.16 0 −2.30010 + 1.92602i 0 2.23607i 0 4.00501 0 1.58093 − 8.86006i 0
461.17 0 −1.72628 − 2.45356i 0 2.23607i 0 −2.88803 0 −3.03995 + 8.47105i 0
461.18 0 −1.72628 + 2.45356i 0 − 2.23607i 0 −2.88803 0 −3.03995 − 8.47105i 0
461.19 0 −1.67084 − 2.49165i 0 2.23607i 0 −0.285209 0 −3.41660 + 8.32627i 0
461.20 0 −1.67084 + 2.49165i 0 − 2.23607i 0 −0.285209 0 −3.41660 − 8.32627i 0
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 461.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1380.3.l.a ✓ 60
3.b odd 2 1 inner 1380.3.l.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.3.l.a ✓ 60 1.a even 1 1 trivial
1380.3.l.a ✓ 60 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(1380, [\chi])\).