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Properties

Label 1380.4.i.b
Level $1380$
Weight $4$
Character orbit 1380.i
Analytic conductor $81.423$
Analytic rank $0$
Dimension $48$
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] = [1380,4,Mod(1241,1380)]
 
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, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1380.1241");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1380.i (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: \(81.4226358079\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 240 q^{5} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 240 q^{5} + 14 q^{9} + 60 q^{11} + 116 q^{21} + 60 q^{23} + 1200 q^{25} + 168 q^{27} - 108 q^{31} + 656 q^{33} + 208 q^{39} + 70 q^{45} - 2820 q^{49} - 542 q^{51} + 672 q^{53} + 300 q^{55} - 224 q^{57} + 240 q^{63} - 414 q^{69} - 120 q^{73} + 1450 q^{81} - 1992 q^{83} - 2500 q^{87} - 4872 q^{89} + 512 q^{93} - 704 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1241.1 0 −5.19517 − 0.101265i 0 5.00000 0 8.12529i 0 26.9795 + 1.05218i 0
1241.2 0 −5.19517 + 0.101265i 0 5.00000 0 − 8.12529i 0 26.9795 − 1.05218i 0
1241.3 0 −5.16048 − 0.607818i 0 5.00000 0 − 33.1908i 0 26.2611 + 6.27327i 0
1241.4 0 −5.16048 + 0.607818i 0 5.00000 0 33.1908i 0 26.2611 − 6.27327i 0
1241.5 0 −4.98192 − 1.47665i 0 5.00000 0 − 13.9227i 0 22.6390 + 14.7131i 0
1241.6 0 −4.98192 + 1.47665i 0 5.00000 0 13.9227i 0 22.6390 − 14.7131i 0
1241.7 0 −4.62647 − 2.36553i 0 5.00000 0 − 1.34259i 0 15.8085 + 21.8881i 0
1241.8 0 −4.62647 + 2.36553i 0 5.00000 0 1.34259i 0 15.8085 − 21.8881i 0
1241.9 0 −4.21140 − 3.04370i 0 5.00000 0 20.8262i 0 8.47176 + 25.6365i 0
1241.10 0 −4.21140 + 3.04370i 0 5.00000 0 − 20.8262i 0 8.47176 − 25.6365i 0
1241.11 0 −3.86092 − 3.47754i 0 5.00000 0 5.97529i 0 2.81343 + 26.8530i 0
1241.12 0 −3.86092 + 3.47754i 0 5.00000 0 − 5.97529i 0 2.81343 − 26.8530i 0
1241.13 0 −3.29884 − 4.01468i 0 5.00000 0 24.8632i 0 −5.23524 + 26.4876i 0
1241.14 0 −3.29884 + 4.01468i 0 5.00000 0 − 24.8632i 0 −5.23524 − 26.4876i 0
1241.15 0 −3.23820 − 4.06375i 0 5.00000 0 − 22.9904i 0 −6.02813 + 26.3185i 0
1241.16 0 −3.23820 + 4.06375i 0 5.00000 0 22.9904i 0 −6.02813 − 26.3185i 0
1241.17 0 −2.09248 − 4.75621i 0 5.00000 0 − 32.2235i 0 −18.2431 + 19.9045i 0
1241.18 0 −2.09248 + 4.75621i 0 5.00000 0 32.2235i 0 −18.2431 − 19.9045i 0
1241.19 0 −1.55517 − 4.95797i 0 5.00000 0 − 0.870272i 0 −22.1629 + 15.4210i 0
1241.20 0 −1.55517 + 4.95797i 0 5.00000 0 0.870272i 0 −22.1629 − 15.4210i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1241.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
69.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1380.4.i.b yes 48
3.b odd 2 1 1380.4.i.a ✓ 48
23.b odd 2 1 1380.4.i.a ✓ 48
69.c even 2 1 inner 1380.4.i.b yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.4.i.a ✓ 48 3.b odd 2 1
1380.4.i.a ✓ 48 23.b odd 2 1
1380.4.i.b yes 48 1.a even 1 1 trivial
1380.4.i.b yes 48 69.c even 2 1 inner