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Properties

Label 1716.3.j.a
Level $1716$
Weight $3$
Character orbit 1716.j
Analytic conductor $46.758$
Analytic rank $0$
Dimension $92$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1716,3,Mod(1013,1716)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1716, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1716.1013");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1716.j (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: \(46.7576133642\)
Analytic rank: \(0\)
Dimension: \(92\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 92 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 92 q + 16 q^{13} + 548 q^{25} + 24 q^{27} - 96 q^{39} - 88 q^{43} - 612 q^{49} + 332 q^{51} + 352 q^{61} + 228 q^{69} - 584 q^{75} - 128 q^{79} - 112 q^{81} - 596 q^{87} + 40 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1013.1 0 −2.99496 − 0.173870i 0 −3.21311 0 10.3613i 0 8.93954 + 1.04147i 0
1013.2 0 −2.99496 − 0.173870i 0 3.21311 0 − 10.3613i 0 8.93954 + 1.04147i 0
1013.3 0 −2.99496 + 0.173870i 0 −3.21311 0 − 10.3613i 0 8.93954 − 1.04147i 0
1013.4 0 −2.99496 + 0.173870i 0 3.21311 0 10.3613i 0 8.93954 − 1.04147i 0
1013.5 0 −2.97736 − 0.367864i 0 −8.84066 0 7.29382i 0 8.72935 + 2.19053i 0
1013.6 0 −2.97736 − 0.367864i 0 8.84066 0 − 7.29382i 0 8.72935 + 2.19053i 0
1013.7 0 −2.97736 + 0.367864i 0 −8.84066 0 − 7.29382i 0 8.72935 − 2.19053i 0
1013.8 0 −2.97736 + 0.367864i 0 8.84066 0 7.29382i 0 8.72935 − 2.19053i 0
1013.9 0 −2.93023 − 0.643244i 0 −4.20494 0 4.24859i 0 8.17248 + 3.76970i 0
1013.10 0 −2.93023 − 0.643244i 0 4.20494 0 − 4.24859i 0 8.17248 + 3.76970i 0
1013.11 0 −2.93023 + 0.643244i 0 −4.20494 0 − 4.24859i 0 8.17248 − 3.76970i 0
1013.12 0 −2.93023 + 0.643244i 0 4.20494 0 4.24859i 0 8.17248 − 3.76970i 0
1013.13 0 −2.44845 − 1.73352i 0 −2.26980 0 − 5.76719i 0 2.98983 + 8.48887i 0
1013.14 0 −2.44845 − 1.73352i 0 2.26980 0 5.76719i 0 2.98983 + 8.48887i 0
1013.15 0 −2.44845 + 1.73352i 0 −2.26980 0 5.76719i 0 2.98983 − 8.48887i 0
1013.16 0 −2.44845 + 1.73352i 0 2.26980 0 − 5.76719i 0 2.98983 − 8.48887i 0
1013.17 0 −2.41056 − 1.78584i 0 −9.85885 0 − 1.65146i 0 2.62156 + 8.60973i 0
1013.18 0 −2.41056 − 1.78584i 0 9.85885 0 1.65146i 0 2.62156 + 8.60973i 0
1013.19 0 −2.41056 + 1.78584i 0 −9.85885 0 1.65146i 0 2.62156 − 8.60973i 0
1013.20 0 −2.41056 + 1.78584i 0 9.85885 0 − 1.65146i 0 2.62156 − 8.60973i 0
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1013.92
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.b even 2 1 inner
39.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1716.3.j.a ✓ 92
3.b odd 2 1 inner 1716.3.j.a ✓ 92
13.b even 2 1 inner 1716.3.j.a ✓ 92
39.d odd 2 1 inner 1716.3.j.a ✓ 92
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1716.3.j.a ✓ 92 1.a even 1 1 trivial
1716.3.j.a ✓ 92 3.b odd 2 1 inner
1716.3.j.a ✓ 92 13.b even 2 1 inner
1716.3.j.a ✓ 92 39.d odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(1716, [\chi])\).