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Properties

Label 1512.2.q.d
Level $1512$
Weight $2$
Character orbit 1512.q
Analytic conductor $12.073$
Analytic rank $0$
Dimension $22$
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] = [1512,2,Mod(793,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1512.793");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.q (of order \(3\), degree \(2\), not 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: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q - q^{5} + 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - q^{5} + 5 q^{7} - 3 q^{11} + 7 q^{13} + q^{17} + 13 q^{19} - 22 q^{25} + 7 q^{29} - 12 q^{31} - 2 q^{35} + 6 q^{37} - 4 q^{41} + 2 q^{43} + 34 q^{47} - 25 q^{49} - q^{53} + 2 q^{55} - 42 q^{59} - 62 q^{61} - 6 q^{65} + 52 q^{67} + 32 q^{71} + 17 q^{73} + q^{77} + 32 q^{79} + 36 q^{83} + 28 q^{85} + 2 q^{89} + 15 q^{91} - 48 q^{95} + 19 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
793.1 0 0 0 −2.11148 − 3.65719i 0 2.19338 − 1.47956i 0 0 0
793.2 0 0 0 −1.89970 − 3.29038i 0 −0.841809 + 2.50826i 0 0 0
793.3 0 0 0 −1.33425 − 2.31099i 0 2.54743 + 0.714566i 0 0 0
793.4 0 0 0 −0.891774 − 1.54460i 0 −2.54386 + 0.727153i 0 0 0
793.5 0 0 0 −0.234085 − 0.405446i 0 0.212345 − 2.63722i 0 0 0
793.6 0 0 0 −0.0309846 − 0.0536670i 0 −0.981674 − 2.45689i 0 0 0
793.7 0 0 0 0.263002 + 0.455533i 0 −0.333150 + 2.62469i 0 0 0
793.8 0 0 0 1.05220 + 1.82246i 0 2.58382 − 0.569079i 0 0 0
793.9 0 0 0 1.38590 + 2.40045i 0 1.74026 + 1.99286i 0 0 0
793.10 0 0 0 1.59750 + 2.76695i 0 −1.66645 + 2.05498i 0 0 0
793.11 0 0 0 1.70368 + 2.95086i 0 −0.410295 − 2.61374i 0 0 0
1369.1 0 0 0 −2.11148 + 3.65719i 0 2.19338 + 1.47956i 0 0 0
1369.2 0 0 0 −1.89970 + 3.29038i 0 −0.841809 − 2.50826i 0 0 0
1369.3 0 0 0 −1.33425 + 2.31099i 0 2.54743 − 0.714566i 0 0 0
1369.4 0 0 0 −0.891774 + 1.54460i 0 −2.54386 − 0.727153i 0 0 0
1369.5 0 0 0 −0.234085 + 0.405446i 0 0.212345 + 2.63722i 0 0 0
1369.6 0 0 0 −0.0309846 + 0.0536670i 0 −0.981674 + 2.45689i 0 0 0
1369.7 0 0 0 0.263002 − 0.455533i 0 −0.333150 − 2.62469i 0 0 0
1369.8 0 0 0 1.05220 − 1.82246i 0 2.58382 + 0.569079i 0 0 0
1369.9 0 0 0 1.38590 − 2.40045i 0 1.74026 − 1.99286i 0 0 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 793.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.q.d 22
3.b odd 2 1 504.2.q.c ✓ 22
4.b odd 2 1 3024.2.q.l 22
7.c even 3 1 1512.2.t.c 22
9.c even 3 1 1512.2.t.c 22
9.d odd 6 1 504.2.t.c yes 22
12.b even 2 1 1008.2.q.l 22
21.h odd 6 1 504.2.t.c yes 22
28.g odd 6 1 3024.2.t.k 22
36.f odd 6 1 3024.2.t.k 22
36.h even 6 1 1008.2.t.l 22
63.h even 3 1 inner 1512.2.q.d 22
63.j odd 6 1 504.2.q.c ✓ 22
84.n even 6 1 1008.2.t.l 22
252.u odd 6 1 3024.2.q.l 22
252.bb even 6 1 1008.2.q.l 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.q.c ✓ 22 3.b odd 2 1
504.2.q.c ✓ 22 63.j odd 6 1
504.2.t.c yes 22 9.d odd 6 1
504.2.t.c yes 22 21.h odd 6 1
1008.2.q.l 22 12.b even 2 1
1008.2.q.l 22 252.bb even 6 1
1008.2.t.l 22 36.h even 6 1
1008.2.t.l 22 84.n even 6 1
1512.2.q.d 22 1.a even 1 1 trivial
1512.2.q.d 22 63.h even 3 1 inner
1512.2.t.c 22 7.c even 3 1
1512.2.t.c 22 9.c even 3 1
3024.2.q.l 22 4.b odd 2 1
3024.2.q.l 22 252.u odd 6 1
3024.2.t.k 22 28.g odd 6 1
3024.2.t.k 22 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{22} + T_{5}^{21} + 39 T_{5}^{20} + 4 T_{5}^{19} + 952 T_{5}^{18} - 120 T_{5}^{17} + 14145 T_{5}^{16} + \cdots + 5476 \) acting on \(S_{2}^{\mathrm{new}}(1512, [\chi])\). Copy content Toggle raw display