↑

Properties

Label 1140.2.bk.a
Level $1140$
Weight $2$
Character orbit 1140.bk
Analytic conductor $9.103$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1140,2,Mod(221,1140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1140, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1140.221");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1140 = 2^{2} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1140.bk (of order \(6\), 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: \(9.10294583043\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 6 q^{3} + 8 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 6 q^{3} + 8 q^{7} - 2 q^{9} - 12 q^{13} - 20 q^{19} + 28 q^{25} + 24 q^{33} - 4 q^{39} - 28 q^{43} - 8 q^{45} + 48 q^{49} + 12 q^{51} + 28 q^{57} - 8 q^{61} + 96 q^{67} + 36 q^{73} - 48 q^{79} + 18 q^{81} + 60 q^{87} - 48 q^{91} - 8 q^{93} + 12 q^{97} - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1 0 −1.72947 + 0.0945827i 0 −0.866025 + 0.500000i 0 −0.767732 0 2.98211 − 0.327155i 0
221.2 0 −1.68444 − 0.403299i 0 0.866025 − 0.500000i 0 0.868725 0 2.67470 + 1.35867i 0
221.3 0 −1.61414 − 0.628140i 0 0.866025 − 0.500000i 0 −1.75465 0 2.21088 + 2.02781i 0
221.4 0 −1.60349 + 0.654843i 0 −0.866025 + 0.500000i 0 2.14830 0 2.14236 − 2.10007i 0
221.5 0 −1.36886 + 1.06124i 0 0.866025 − 0.500000i 0 2.14830 0 0.747534 − 2.90537i 0
221.6 0 −1.26163 − 1.18672i 0 −0.866025 + 0.500000i 0 −4.48838 0 0.183402 + 2.99439i 0
221.7 0 −1.21987 − 1.22960i 0 −0.866025 + 0.500000i 0 3.36850 0 −0.0238174 + 2.99991i 0
221.8 0 −1.10599 − 1.33296i 0 0.866025 − 0.500000i 0 1.05093 0 −0.553557 + 2.94849i 0
221.9 0 −0.946644 + 1.45047i 0 0.866025 − 0.500000i 0 −0.767732 0 −1.20773 − 2.74616i 0
221.10 0 −0.513736 − 1.65411i 0 −0.866025 + 0.500000i 0 −2.10594 0 −2.47215 + 1.69955i 0
221.11 0 −0.492954 + 1.66042i 0 −0.866025 + 0.500000i 0 0.868725 0 −2.51399 − 1.63702i 0
221.12 0 −0.263083 + 1.71195i 0 −0.866025 + 0.500000i 0 −1.75465 0 −2.86157 − 0.900773i 0
221.13 0 −0.133562 − 1.72689i 0 0.866025 − 0.500000i 0 3.23712 0 −2.96432 + 0.461296i 0
221.14 0 0.396915 + 1.68596i 0 0.866025 − 0.500000i 0 −4.48838 0 −2.68492 + 1.33836i 0
221.15 0 0.415198 − 1.68155i 0 −0.866025 + 0.500000i 0 2.66248 0 −2.65522 − 1.39635i 0
221.16 0 0.454925 + 1.67124i 0 0.866025 − 0.500000i 0 3.36850 0 −2.58609 + 1.52058i 0
221.17 0 0.601379 + 1.62430i 0 −0.866025 + 0.500000i 0 1.05093 0 −2.27669 + 1.95364i 0
221.18 0 0.841532 − 1.51388i 0 0.866025 − 0.500000i 0 −3.99962 0 −1.58365 − 2.54795i 0
221.19 0 0.917447 − 1.46911i 0 0.866025 − 0.500000i 0 4.81442 0 −1.31658 − 2.69566i 0
221.20 0 0.987173 − 1.42320i 0 −0.866025 + 0.500000i 0 −0.380464 0 −1.05098 − 2.80988i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 221.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.d odd 6 1 inner
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1140.2.bk.a ✓ 56
3.b odd 2 1 inner 1140.2.bk.a ✓ 56
19.d odd 6 1 inner 1140.2.bk.a ✓ 56
57.f even 6 1 inner 1140.2.bk.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1140.2.bk.a ✓ 56 1.a even 1 1 trivial
1140.2.bk.a ✓ 56 3.b odd 2 1 inner
1140.2.bk.a ✓ 56 19.d odd 6 1 inner
1140.2.bk.a ✓ 56 57.f even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1140, [\chi])\).