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Properties

Label 828.2.k.a
Level $828$
Weight $2$
Character orbit 828.k
Analytic conductor $6.612$
Analytic rank $0$
Dimension $48$
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] = [828,2,Mod(137,828)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(828, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("828.137");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 828.k (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: \(6.61161328736\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{3} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{3} + 6 q^{9} + 21 q^{23} - 30 q^{25} - 2 q^{27} - 6 q^{29} - 6 q^{31} - 18 q^{39} - 12 q^{41} - 48 q^{47} + 12 q^{49} - 12 q^{55} - 36 q^{59} + 11 q^{69} + 64 q^{75} + 30 q^{77} - 26 q^{81} - 4 q^{87} + 38 q^{93} - 84 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1 0 −1.72869 + 0.107863i 0 −0.921249 − 1.59565i 0 −4.26677 − 2.46342i 0 2.97673 − 0.372924i 0
137.2 0 −1.72869 + 0.107863i 0 0.921249 + 1.59565i 0 4.26677 + 2.46342i 0 2.97673 − 0.372924i 0
137.3 0 −1.57122 − 0.728872i 0 −1.93423 − 3.35018i 0 2.86158 + 1.65213i 0 1.93749 + 2.29044i 0
137.4 0 −1.57122 − 0.728872i 0 1.93423 + 3.35018i 0 −2.86158 − 1.65213i 0 1.93749 + 2.29044i 0
137.5 0 −1.36517 + 1.06598i 0 −0.653156 − 1.13130i 0 0.100014 + 0.0577433i 0 0.727379 − 2.91048i 0
137.6 0 −1.36517 + 1.06598i 0 0.653156 + 1.13130i 0 −0.100014 − 0.0577433i 0 0.727379 − 2.91048i 0
137.7 0 −1.20469 + 1.24447i 0 −1.88303 − 3.26150i 0 1.71151 + 0.988141i 0 −0.0974246 − 2.99842i 0
137.8 0 −1.20469 + 1.24447i 0 1.88303 + 3.26150i 0 −1.71151 − 0.988141i 0 −0.0974246 − 2.99842i 0
137.9 0 −1.03886 − 1.38592i 0 −0.441594 − 0.764863i 0 0.785017 + 0.453230i 0 −0.841529 + 2.87955i 0
137.10 0 −1.03886 − 1.38592i 0 0.441594 + 0.764863i 0 −0.785017 − 0.453230i 0 −0.841529 + 2.87955i 0
137.11 0 0.0317653 + 1.73176i 0 −0.973353 − 1.68590i 0 −1.18310 − 0.683064i 0 −2.99798 + 0.110020i 0
137.12 0 0.0317653 + 1.73176i 0 0.973353 + 1.68590i 0 1.18310 + 0.683064i 0 −2.99798 + 0.110020i 0
137.13 0 0.0548135 − 1.73118i 0 −2.11067 − 3.65578i 0 −2.25960 − 1.30458i 0 −2.99399 − 0.189784i 0
137.14 0 0.0548135 − 1.73118i 0 2.11067 + 3.65578i 0 2.25960 + 1.30458i 0 −2.99399 − 0.189784i 0
137.15 0 0.519232 − 1.65239i 0 −0.222766 − 0.385842i 0 2.71024 + 1.56476i 0 −2.46080 − 1.71595i 0
137.16 0 0.519232 − 1.65239i 0 0.222766 + 0.385842i 0 −2.71024 − 1.56476i 0 −2.46080 − 1.71595i 0
137.17 0 1.07060 + 1.36155i 0 −1.25406 − 2.17209i 0 −2.66517 − 1.53873i 0 −0.707611 + 2.91535i 0
137.18 0 1.07060 + 1.36155i 0 1.25406 + 2.17209i 0 2.66517 + 1.53873i 0 −0.707611 + 2.91535i 0
137.19 0 1.48015 − 0.899534i 0 −0.499827 − 0.865725i 0 3.53115 + 2.03871i 0 1.38168 − 2.66289i 0
137.20 0 1.48015 − 0.899534i 0 0.499827 + 0.865725i 0 −3.53115 − 2.03871i 0 1.38168 − 2.66289i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
23.b odd 2 1 inner
207.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 828.2.k.a ✓ 48
3.b odd 2 1 2484.2.k.a 48
9.c even 3 1 2484.2.k.a 48
9.c even 3 1 7452.2.g.b 48
9.d odd 6 1 inner 828.2.k.a ✓ 48
9.d odd 6 1 7452.2.g.b 48
23.b odd 2 1 inner 828.2.k.a ✓ 48
69.c even 2 1 2484.2.k.a 48
207.f odd 6 1 2484.2.k.a 48
207.f odd 6 1 7452.2.g.b 48
207.g even 6 1 inner 828.2.k.a ✓ 48
207.g even 6 1 7452.2.g.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
828.2.k.a ✓ 48 1.a even 1 1 trivial
828.2.k.a ✓ 48 9.d odd 6 1 inner
828.2.k.a ✓ 48 23.b odd 2 1 inner
828.2.k.a ✓ 48 207.g even 6 1 inner
2484.2.k.a 48 3.b odd 2 1
2484.2.k.a 48 9.c even 3 1
2484.2.k.a 48 69.c even 2 1
2484.2.k.a 48 207.f odd 6 1
7452.2.g.b 48 9.c even 3 1
7452.2.g.b 48 9.d odd 6 1
7452.2.g.b 48 207.f odd 6 1
7452.2.g.b 48 207.g even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(828, [\chi])\).