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Properties

Label 693.2.bg.c
Level $693$
Weight $2$
Character orbit 693.bg
Analytic conductor $5.534$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [693,2,Mod(10,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, 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("693.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.bg (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: \(5.53363286007\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 20 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 20 q^{4} - 20 q^{16} - 16 q^{22} - 8 q^{25} - 12 q^{31} + 24 q^{37} + 52 q^{49} + 20 q^{58} - 104 q^{64} - 20 q^{67} + 48 q^{70} - 144 q^{82} - 24 q^{88} - 84 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −2.29990 + 1.32785i 0 2.52635 − 4.37577i −1.83579 + 1.05989i 0 −2.61780 + 0.383592i 8.10705i 0 2.81475 − 4.87529i
10.2 −2.29990 + 1.32785i 0 2.52635 − 4.37577i 1.83579 − 1.05989i 0 2.61780 − 0.383592i 8.10705i 0 −2.81475 + 4.87529i
10.3 −1.58056 + 0.912538i 0 0.665451 − 1.15259i −2.74464 + 1.58462i 0 2.35572 + 1.20440i − 1.22115i 0 2.89205 − 5.00918i
10.4 −1.58056 + 0.912538i 0 0.665451 − 1.15259i 2.74464 − 1.58462i 0 −2.35572 − 1.20440i − 1.22115i 0 −2.89205 + 5.00918i
10.5 −1.36355 + 0.787245i 0 0.239510 − 0.414844i −0.347571 + 0.200670i 0 −0.805228 − 2.52024i − 2.39477i 0 0.315953 − 0.547247i
10.6 −1.36355 + 0.787245i 0 0.239510 − 0.414844i 0.347571 − 0.200670i 0 0.805228 + 2.52024i − 2.39477i 0 −0.315953 + 0.547247i
10.7 −0.320979 + 0.185317i 0 −0.931315 + 1.61308i −1.57353 + 0.908480i 0 −2.04923 + 1.67352i − 1.43162i 0 0.336714 − 0.583206i
10.8 −0.320979 + 0.185317i 0 −0.931315 + 1.61308i 1.57353 − 0.908480i 0 2.04923 − 1.67352i − 1.43162i 0 −0.336714 + 0.583206i
10.9 0.320979 − 0.185317i 0 −0.931315 + 1.61308i −1.57353 + 0.908480i 0 2.04923 − 1.67352i 1.43162i 0 −0.336714 + 0.583206i
10.10 0.320979 − 0.185317i 0 −0.931315 + 1.61308i 1.57353 − 0.908480i 0 −2.04923 + 1.67352i 1.43162i 0 0.336714 − 0.583206i
10.11 1.36355 − 0.787245i 0 0.239510 − 0.414844i −0.347571 + 0.200670i 0 0.805228 + 2.52024i 2.39477i 0 −0.315953 + 0.547247i
10.12 1.36355 − 0.787245i 0 0.239510 − 0.414844i 0.347571 − 0.200670i 0 −0.805228 − 2.52024i 2.39477i 0 0.315953 − 0.547247i
10.13 1.58056 − 0.912538i 0 0.665451 − 1.15259i −2.74464 + 1.58462i 0 −2.35572 − 1.20440i 1.22115i 0 −2.89205 + 5.00918i
10.14 1.58056 − 0.912538i 0 0.665451 − 1.15259i 2.74464 − 1.58462i 0 2.35572 + 1.20440i 1.22115i 0 2.89205 − 5.00918i
10.15 2.29990 − 1.32785i 0 2.52635 − 4.37577i −1.83579 + 1.05989i 0 2.61780 − 0.383592i − 8.10705i 0 −2.81475 + 4.87529i
10.16 2.29990 − 1.32785i 0 2.52635 − 4.37577i 1.83579 − 1.05989i 0 −2.61780 + 0.383592i − 8.10705i 0 2.81475 − 4.87529i
208.1 −2.29990 − 1.32785i 0 2.52635 + 4.37577i −1.83579 − 1.05989i 0 −2.61780 − 0.383592i − 8.10705i 0 2.81475 + 4.87529i
208.2 −2.29990 − 1.32785i 0 2.52635 + 4.37577i 1.83579 + 1.05989i 0 2.61780 + 0.383592i − 8.10705i 0 −2.81475 − 4.87529i
208.3 −1.58056 − 0.912538i 0 0.665451 + 1.15259i −2.74464 − 1.58462i 0 2.35572 − 1.20440i 1.22115i 0 2.89205 + 5.00918i
208.4 −1.58056 − 0.912538i 0 0.665451 + 1.15259i 2.74464 + 1.58462i 0 −2.35572 + 1.20440i 1.22115i 0 −2.89205 − 5.00918i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
11.b odd 2 1 inner
21.g even 6 1 inner
33.d even 2 1 inner
77.i even 6 1 inner
231.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.bg.c ✓ 32
3.b odd 2 1 inner 693.2.bg.c ✓ 32
7.d odd 6 1 inner 693.2.bg.c ✓ 32
11.b odd 2 1 inner 693.2.bg.c ✓ 32
21.g even 6 1 inner 693.2.bg.c ✓ 32
33.d even 2 1 inner 693.2.bg.c ✓ 32
77.i even 6 1 inner 693.2.bg.c ✓ 32
231.k odd 6 1 inner 693.2.bg.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.bg.c ✓ 32 1.a even 1 1 trivial
693.2.bg.c ✓ 32 3.b odd 2 1 inner
693.2.bg.c ✓ 32 7.d odd 6 1 inner
693.2.bg.c ✓ 32 11.b odd 2 1 inner
693.2.bg.c ✓ 32 21.g even 6 1 inner
693.2.bg.c ✓ 32 33.d even 2 1 inner
693.2.bg.c ✓ 32 77.i even 6 1 inner
693.2.bg.c ✓ 32 231.k odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 13T_{2}^{14} + 118T_{2}^{12} - 533T_{2}^{10} + 1748T_{2}^{8} - 3107T_{2}^{6} + 3817T_{2}^{4} - 520T_{2}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display