↑

Properties

Label 693.2.m.k
Level $693$
Weight $2$
Character orbit 693.m
Analytic conductor $5.534$
Analytic rank $0$
Dimension $32$
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] = [693,2,Mod(64,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.m (of order \(5\), degree \(4\), 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: \(8\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 10 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 10 q^{4} - 8 q^{7} - 12 q^{10} + 10 q^{13} - 26 q^{16} - 10 q^{19} - 6 q^{22} - 52 q^{25} - 10 q^{28} - 20 q^{31} + 24 q^{34} - 4 q^{37} + 124 q^{40} + 32 q^{43} + 30 q^{46} - 8 q^{49} + 24 q^{52} - 12 q^{55} - 104 q^{58} + 34 q^{61} - 52 q^{64} + 104 q^{67} + 18 q^{70} + 2 q^{73} - 28 q^{76} - 42 q^{79} - 172 q^{82} - 30 q^{85} - 16 q^{88} - 10 q^{91} + 150 q^{94} - 74 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −0.804651 + 2.47646i 0 −3.86736 − 2.80980i 1.33452 + 4.10723i 0 −0.809017 − 0.587785i 5.85703 − 4.25538i 0 −11.2452
64.2 −0.643412 + 1.98022i 0 −1.88925 − 1.37262i −0.411680 − 1.26702i 0 −0.809017 − 0.587785i 0.564713 − 0.410288i 0 2.77386
64.3 −0.405703 + 1.24862i 0 0.223566 + 0.162430i −1.06900 − 3.29005i 0 −0.809017 − 0.587785i −2.41780 + 1.75664i 0 4.54174
64.4 −0.123907 + 0.381345i 0 1.48796 + 1.08107i 0.712476 + 2.19277i 0 −0.809017 − 0.587785i −1.24541 + 0.904845i 0 −0.924485
64.5 0.123907 − 0.381345i 0 1.48796 + 1.08107i −0.712476 − 2.19277i 0 −0.809017 − 0.587785i 1.24541 − 0.904845i 0 −0.924485
64.6 0.405703 − 1.24862i 0 0.223566 + 0.162430i 1.06900 + 3.29005i 0 −0.809017 − 0.587785i 2.41780 − 1.75664i 0 4.54174
64.7 0.643412 − 1.98022i 0 −1.88925 − 1.37262i 0.411680 + 1.26702i 0 −0.809017 − 0.587785i −0.564713 + 0.410288i 0 2.77386
64.8 0.804651 − 2.47646i 0 −3.86736 − 2.80980i −1.33452 − 4.10723i 0 −0.809017 − 0.587785i −5.85703 + 4.25538i 0 −11.2452
190.1 −2.16885 + 1.57576i 0 1.60284 − 4.93305i 0.946867 + 0.687939i 0 0.309017 − 0.951057i 2.64012 + 8.12545i 0 −3.13763
190.2 −1.70216 + 1.23669i 0 0.749912 − 2.30799i −3.03419 − 2.20447i 0 0.309017 − 0.951057i 0.277470 + 0.853964i 0 7.89093
190.3 −0.727975 + 0.528905i 0 −0.367826 + 1.13205i 0.650418 + 0.472557i 0 0.309017 − 0.951057i −0.887104 − 2.73023i 0 −0.723426
190.4 −0.614338 + 0.446343i 0 −0.439845 + 1.35370i 2.31804 + 1.68415i 0 0.309017 − 0.951057i −0.803314 − 2.47235i 0 −2.17577
190.5 0.614338 − 0.446343i 0 −0.439845 + 1.35370i −2.31804 − 1.68415i 0 0.309017 − 0.951057i 0.803314 + 2.47235i 0 −2.17577
190.6 0.727975 − 0.528905i 0 −0.367826 + 1.13205i −0.650418 − 0.472557i 0 0.309017 − 0.951057i 0.887104 + 2.73023i 0 −0.723426
190.7 1.70216 − 1.23669i 0 0.749912 − 2.30799i 3.03419 + 2.20447i 0 0.309017 − 0.951057i −0.277470 − 0.853964i 0 7.89093
190.8 2.16885 − 1.57576i 0 1.60284 − 4.93305i −0.946867 − 0.687939i 0 0.309017 − 0.951057i −2.64012 − 8.12545i 0 −3.13763
379.1 −0.804651 − 2.47646i 0 −3.86736 + 2.80980i 1.33452 − 4.10723i 0 −0.809017 + 0.587785i 5.85703 + 4.25538i 0 −11.2452
379.2 −0.643412 − 1.98022i 0 −1.88925 + 1.37262i −0.411680 + 1.26702i 0 −0.809017 + 0.587785i 0.564713 + 0.410288i 0 2.77386
379.3 −0.405703 − 1.24862i 0 0.223566 − 0.162430i −1.06900 + 3.29005i 0 −0.809017 + 0.587785i −2.41780 − 1.75664i 0 4.54174
379.4 −0.123907 − 0.381345i 0 1.48796 − 1.08107i 0.712476 − 2.19277i 0 −0.809017 + 0.587785i −1.24541 − 0.904845i 0 −0.924485
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.m.k ✓ 32
3.b odd 2 1 inner 693.2.m.k ✓ 32
11.c even 5 1 inner 693.2.m.k ✓ 32
11.c even 5 1 7623.2.a.dc 16
11.d odd 10 1 7623.2.a.db 16
33.f even 10 1 7623.2.a.db 16
33.h odd 10 1 inner 693.2.m.k ✓ 32
33.h odd 10 1 7623.2.a.dc 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.m.k ✓ 32 1.a even 1 1 trivial
693.2.m.k ✓ 32 3.b odd 2 1 inner
693.2.m.k ✓ 32 11.c even 5 1 inner
693.2.m.k ✓ 32 33.h odd 10 1 inner
7623.2.a.db 16 11.d odd 10 1
7623.2.a.db 16 33.f even 10 1
7623.2.a.dc 16 11.c even 5 1
7623.2.a.dc 16 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 13 T_{2}^{30} + 122 T_{2}^{28} + 1019 T_{2}^{26} + 7741 T_{2}^{24} + 38323 T_{2}^{22} + \cdots + 14641 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display