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Properties

Label 1260.2.n.a
Level $1260$
Weight $2$
Character orbit 1260.n
Analytic conductor $10.061$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1260,2,Mod(71,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.71");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.n (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: \(10.0611506547\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{4} - 4 q^{10} - 4 q^{14} + 4 q^{20} - 4 q^{22} - 24 q^{25} - 40 q^{26} + 4 q^{28} + 40 q^{32} + 8 q^{34} - 24 q^{35} + 4 q^{40} + 48 q^{44} + 36 q^{46} + 16 q^{47} - 24 q^{49} + 32 q^{52} + 4 q^{56}+ \cdots + 12 q^{88}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1 −1.31265 − 0.526260i 0 1.44610 + 1.38159i 1.00000i 0 1.00000i −1.17115 − 2.57457i 0 0.526260 − 1.31265i
71.2 −1.31265 + 0.526260i 0 1.44610 − 1.38159i − 1.00000i 0 − 1.00000i −1.17115 + 2.57457i 0 0.526260 + 1.31265i
71.3 −1.26282 − 0.636627i 0 1.18941 + 1.60789i − 1.00000i 0 − 1.00000i −0.478383 − 2.78768i 0 −0.636627 + 1.26282i
71.4 −1.26282 + 0.636627i 0 1.18941 − 1.60789i 1.00000i 0 1.00000i −0.478383 + 2.78768i 0 −0.636627 − 1.26282i
71.5 −1.13949 − 0.837595i 0 0.596870 + 1.90886i − 1.00000i 0 − 1.00000i 0.918725 − 2.67506i 0 −0.837595 + 1.13949i
71.6 −1.13949 + 0.837595i 0 0.596870 − 1.90886i 1.00000i 0 1.00000i 0.918725 + 2.67506i 0 −0.837595 − 1.13949i
71.7 −0.862943 − 1.12041i 0 −0.510659 + 1.93371i 1.00000i 0 1.00000i 2.60722 − 1.09653i 0 1.12041 − 0.862943i
71.8 −0.862943 + 1.12041i 0 −0.510659 − 1.93371i − 1.00000i 0 − 1.00000i 2.60722 + 1.09653i 0 1.12041 + 0.862943i
71.9 −0.428129 − 1.34785i 0 −1.63341 + 1.15411i − 1.00000i 0 − 1.00000i 2.25488 + 1.70749i 0 −1.34785 + 0.428129i
71.10 −0.428129 + 1.34785i 0 −1.63341 − 1.15411i 1.00000i 0 1.00000i 2.25488 − 1.70749i 0 −1.34785 − 0.428129i
71.11 −0.132404 − 1.40800i 0 −1.96494 + 0.372850i − 1.00000i 0 − 1.00000i 0.785140 + 2.71727i 0 −1.40800 + 0.132404i
71.12 −0.132404 + 1.40800i 0 −1.96494 − 0.372850i 1.00000i 0 1.00000i 0.785140 − 2.71727i 0 −1.40800 − 0.132404i
71.13 0.298198 − 1.38242i 0 −1.82216 − 0.824468i 1.00000i 0 1.00000i −1.68312 + 2.27313i 0 1.38242 + 0.298198i
71.14 0.298198 + 1.38242i 0 −1.82216 + 0.824468i − 1.00000i 0 − 1.00000i −1.68312 − 2.27313i 0 1.38242 − 0.298198i
71.15 0.463902 − 1.33596i 0 −1.56959 − 1.23951i − 1.00000i 0 − 1.00000i −2.38408 + 1.52190i 0 −1.33596 − 0.463902i
71.16 0.463902 + 1.33596i 0 −1.56959 + 1.23951i 1.00000i 0 1.00000i −2.38408 − 1.52190i 0 −1.33596 + 0.463902i
71.17 0.692735 − 1.23293i 0 −1.04024 − 1.70819i 1.00000i 0 1.00000i −2.82669 + 0.0992196i 0 1.23293 + 0.692735i
71.18 0.692735 + 1.23293i 0 −1.04024 + 1.70819i − 1.00000i 0 − 1.00000i −2.82669 − 0.0992196i 0 1.23293 − 0.692735i
71.19 0.935018 − 1.06101i 0 −0.251481 − 1.98413i − 1.00000i 0 − 1.00000i −2.34032 − 1.58837i 0 −1.06101 − 0.935018i
71.20 0.935018 + 1.06101i 0 −0.251481 + 1.98413i 1.00000i 0 1.00000i −2.34032 + 1.58837i 0 −1.06101 + 0.935018i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.n.a ✓ 24
3.b odd 2 1 1260.2.n.b yes 24
4.b odd 2 1 1260.2.n.b yes 24
12.b even 2 1 inner 1260.2.n.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.n.a ✓ 24 1.a even 1 1 trivial
1260.2.n.a ✓ 24 12.b even 2 1 inner
1260.2.n.b yes 24 3.b odd 2 1
1260.2.n.b yes 24 4.b odd 2 1