↑

Properties

Label 1340.2.d.c
Level $1340$
Weight $2$
Character orbit 1340.d
Analytic conductor $10.700$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1340,2,Mod(269,1340)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1340, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1340.269");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1340 = 2^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1340.d (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: \(10.6999538709\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 10 q^{5} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 10 q^{5} - 40 q^{9} - 12 q^{11} + 10 q^{15} + 6 q^{19} - 34 q^{21} - 32 q^{25} - 4 q^{29} + 16 q^{31} + 2 q^{35} - 12 q^{39} + 16 q^{41} + 18 q^{45} + 8 q^{49} + 16 q^{51} + 32 q^{55} + 8 q^{59} + 16 q^{61} + 12 q^{65} - 24 q^{69} - 38 q^{71} - 34 q^{75} - 12 q^{79} + 24 q^{81} + 6 q^{85} - 4 q^{89} - 8 q^{91} - 20 q^{95} - 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1 0 − 3.35830i 0 −1.05839 + 1.96972i 0 0.243981i 0 −8.27816 0
269.2 0 − 3.03520i 0 −2.22635 + 0.208198i 0 − 2.33405i 0 −6.21241 0
269.3 0 − 2.81686i 0 1.18993 − 1.89316i 0 − 3.49884i 0 −4.93469 0
269.4 0 − 2.70814i 0 0.126352 − 2.23250i 0 1.82447i 0 −4.33401 0
269.5 0 − 2.32962i 0 1.79784 + 1.32957i 0 − 4.45400i 0 −2.42711 0
269.6 0 − 2.24630i 0 0.328614 + 2.21179i 0 − 1.08146i 0 −2.04585 0
269.7 0 − 1.96434i 0 −2.20896 + 0.347144i 0 3.89601i 0 −0.858615 0
269.8 0 − 1.45417i 0 −0.526262 + 2.17326i 0 0.104425i 0 0.885383 0
269.9 0 − 1.33786i 0 −1.18468 − 1.89645i 0 2.67141i 0 1.21012 0
269.10 0 − 1.03387i 0 −0.459633 − 2.18832i 0 − 3.69864i 0 1.93111 0
269.11 0 − 0.849469i 0 −1.81697 + 1.30332i 0 − 1.11505i 0 2.27840 0
269.12 0 − 0.462771i 0 1.03851 + 1.98028i 0 0.815435i 0 2.78584 0
269.13 0 0.462771i 0 1.03851 − 1.98028i 0 − 0.815435i 0 2.78584 0
269.14 0 0.849469i 0 −1.81697 − 1.30332i 0 1.11505i 0 2.27840 0
269.15 0 1.03387i 0 −0.459633 + 2.18832i 0 3.69864i 0 1.93111 0
269.16 0 1.33786i 0 −1.18468 + 1.89645i 0 − 2.67141i 0 1.21012 0
269.17 0 1.45417i 0 −0.526262 − 2.17326i 0 − 0.104425i 0 0.885383 0
269.18 0 1.96434i 0 −2.20896 − 0.347144i 0 − 3.89601i 0 −0.858615 0
269.19 0 2.24630i 0 0.328614 − 2.21179i 0 1.08146i 0 −2.04585 0
269.20 0 2.32962i 0 1.79784 − 1.32957i 0 4.45400i 0 −2.42711 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 269.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1340.2.d.c ✓ 24
5.b even 2 1 inner 1340.2.d.c ✓ 24
5.c odd 4 1 6700.2.a.ba 12
5.c odd 4 1 6700.2.a.bb 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1340.2.d.c ✓ 24 1.a even 1 1 trivial
1340.2.d.c ✓ 24 5.b even 2 1 inner
6700.2.a.ba 12 5.c odd 4 1
6700.2.a.bb 12 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 56 T_{3}^{22} + 1364 T_{3}^{20} + 19003 T_{3}^{18} + 167534 T_{3}^{16} + 976759 T_{3}^{14} + \cdots + 399424 \) acting on \(S_{2}^{\mathrm{new}}(1340, [\chi])\). Copy content Toggle raw display