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Properties

Label 693.2.by.b
Level $693$
Weight $2$
Character orbit 693.by
Analytic conductor $5.534$
Analytic rank $0$
Dimension $40$
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] = [693,2,Mod(37,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 10, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.by (of order \(15\), degree \(8\), 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: \(40\)
Relative dimension: \(5\) over \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 3 q^{2} - 3 q^{4} - 4 q^{5} - 2 q^{7} + 38 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 3 q^{2} - 3 q^{4} - 4 q^{5} - 2 q^{7} + 38 q^{8} + 14 q^{10} + 9 q^{11} + 6 q^{13} + 3 q^{14} - 5 q^{16} + 7 q^{17} - 4 q^{19} + 30 q^{20} + 44 q^{22} + 14 q^{23} + 21 q^{25} + 16 q^{28} - 17 q^{31} + 30 q^{32} + 48 q^{34} + 14 q^{35} + 24 q^{37} - 12 q^{38} + 10 q^{40} - 60 q^{41} - 72 q^{43} - 18 q^{44} + 8 q^{46} - 13 q^{47} - 10 q^{49} - 6 q^{50} + 2 q^{52} - 33 q^{53} - 6 q^{55} - 24 q^{56} - 17 q^{58} - 21 q^{59} + 52 q^{62} + 94 q^{64} + 40 q^{65} - 38 q^{67} + 23 q^{68} - 3 q^{70} - 20 q^{71} + 11 q^{73} + 41 q^{74} - 96 q^{76} - 36 q^{77} + 21 q^{79} - 12 q^{80} + 6 q^{82} + 46 q^{83} - 78 q^{85} - 7 q^{86} + 32 q^{88} + 10 q^{89} - 14 q^{91} + 110 q^{92} + 37 q^{94} - 7 q^{95} - 54 q^{97} - 116 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −2.04369 + 0.909911i 0 2.01049 − 2.23287i −0.0905565 − 0.861587i 0 −2.63985 − 0.176636i −0.694498 + 2.13745i 0 0.969038 + 1.67842i
37.2 −1.16484 + 0.518618i 0 −0.250384 + 0.278080i 0.123874 + 1.17858i 0 2.27911 − 1.34375i 0.935477 − 2.87910i 0 −0.755525 − 1.30861i
37.3 0.399419 − 0.177833i 0 −1.21035 + 1.34423i −0.127084 − 1.20913i 0 −1.48810 − 2.18759i −0.514604 + 1.58379i 0 −0.265782 − 0.460348i
37.4 2.05127 − 0.913284i 0 2.03535 − 2.26049i 0.257667 + 2.45154i 0 2.29081 + 1.32370i 0.722861 − 2.22474i 0 2.76749 + 4.79344i
37.5 2.23599 − 0.995527i 0 2.67031 − 2.96568i −0.0346954 − 0.330104i 0 0.540741 − 2.58990i 1.50568 − 4.63401i 0 −0.406206 − 0.703570i
163.1 −1.76087 − 1.95564i 0 −0.514820 + 4.89819i −1.15065 − 0.244578i 0 0.510555 − 2.59602i 6.22764 − 4.52465i 0 1.54783 + 2.68093i
163.2 −0.391628 − 0.434946i 0 0.173251 − 1.64837i −3.87755 − 0.824199i 0 −1.49228 + 2.18474i −1.73180 + 1.25823i 0 1.16007 + 2.00931i
163.3 0.0508685 + 0.0564952i 0 0.208453 − 1.98330i 2.52173 + 0.536011i 0 2.37978 − 1.15613i 0.245656 − 0.178480i 0 0.0979948 + 0.169732i
163.4 0.450970 + 0.500853i 0 0.161577 − 1.53730i 1.65218 + 0.351182i 0 1.55484 + 2.14067i 1.93333 − 1.40464i 0 0.569194 + 0.985873i
163.5 1.23711 + 1.37395i 0 −0.148239 + 1.41040i −2.31107 − 0.491232i 0 −0.122985 − 2.64289i 0.870261 − 0.632281i 0 −2.18411 − 3.78299i
235.1 −0.255843 + 2.43419i 0 −3.90352 − 0.829718i 0.303226 + 0.135005i 0 −1.95978 − 1.77744i 1.50568 − 4.63401i 0 −0.406206 + 0.703570i
235.2 −0.234707 + 2.23309i 0 −2.97532 − 0.632424i −2.25193 − 1.00262i 0 −1.07525 + 2.41740i 0.722861 − 2.22474i 0 2.76749 − 4.79344i
235.3 −0.0457018 + 0.434823i 0 1.76931 + 0.376079i 1.11068 + 0.494505i 0 −0.0819410 − 2.64448i −0.514604 + 1.58379i 0 −0.265782 + 0.460348i
235.4 0.133281 − 1.26809i 0 0.366016 + 0.0777992i −1.08262 − 0.482012i 0 −2.63367 + 0.252511i 0.935477 − 2.87910i 0 −0.755525 + 1.30861i
235.5 0.233841 − 2.22485i 0 −2.93896 − 0.624696i 0.791435 + 0.352369i 0 2.03186 − 1.69457i −0.694498 + 2.13745i 0 0.969038 − 1.67842i
289.1 −0.255843 − 2.43419i 0 −3.90352 + 0.829718i 0.303226 − 0.135005i 0 −1.95978 + 1.77744i 1.50568 + 4.63401i 0 −0.406206 − 0.703570i
289.2 −0.234707 − 2.23309i 0 −2.97532 + 0.632424i −2.25193 + 1.00262i 0 −1.07525 − 2.41740i 0.722861 + 2.22474i 0 2.76749 + 4.79344i
289.3 −0.0457018 − 0.434823i 0 1.76931 − 0.376079i 1.11068 − 0.494505i 0 −0.0819410 + 2.64448i −0.514604 − 1.58379i 0 −0.265782 − 0.460348i
289.4 0.133281 + 1.26809i 0 0.366016 − 0.0777992i −1.08262 + 0.482012i 0 −2.63367 − 0.252511i 0.935477 + 2.87910i 0 −0.755525 − 1.30861i
289.5 0.233841 + 2.22485i 0 −2.93896 + 0.624696i 0.791435 − 0.352369i 0 2.03186 + 1.69457i −0.694498 − 2.13745i 0 0.969038 + 1.67842i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.by.b 40
3.b odd 2 1 77.2.m.b ✓ 40
7.c even 3 1 inner 693.2.by.b 40
11.c even 5 1 inner 693.2.by.b 40
21.c even 2 1 539.2.q.h 40
21.g even 6 1 539.2.f.g 20
21.g even 6 1 539.2.q.h 40
21.h odd 6 1 77.2.m.b ✓ 40
21.h odd 6 1 539.2.f.h 20
33.d even 2 1 847.2.n.j 40
33.f even 10 1 847.2.e.h 20
33.f even 10 2 847.2.n.h 40
33.f even 10 1 847.2.n.j 40
33.h odd 10 1 77.2.m.b ✓ 40
33.h odd 10 1 847.2.e.i 20
33.h odd 10 2 847.2.n.i 40
77.m even 15 1 inner 693.2.by.b 40
231.l even 6 1 847.2.n.j 40
231.u even 10 1 539.2.q.h 40
231.z odd 30 1 77.2.m.b ✓ 40
231.z odd 30 1 539.2.f.h 20
231.z odd 30 1 847.2.e.i 20
231.z odd 30 2 847.2.n.i 40
231.z odd 30 1 5929.2.a.bw 10
231.bc even 30 1 539.2.f.g 20
231.bc even 30 1 539.2.q.h 40
231.bc even 30 1 5929.2.a.bx 10
231.be even 30 1 847.2.e.h 20
231.be even 30 2 847.2.n.h 40
231.be even 30 1 847.2.n.j 40
231.be even 30 1 5929.2.a.by 10
231.bf odd 30 1 5929.2.a.bz 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.m.b ✓ 40 3.b odd 2 1
77.2.m.b ✓ 40 21.h odd 6 1
77.2.m.b ✓ 40 33.h odd 10 1
77.2.m.b ✓ 40 231.z odd 30 1
539.2.f.g 20 21.g even 6 1
539.2.f.g 20 231.bc even 30 1
539.2.f.h 20 21.h odd 6 1
539.2.f.h 20 231.z odd 30 1
539.2.q.h 40 21.c even 2 1
539.2.q.h 40 21.g even 6 1
539.2.q.h 40 231.u even 10 1
539.2.q.h 40 231.bc even 30 1
693.2.by.b 40 1.a even 1 1 trivial
693.2.by.b 40 7.c even 3 1 inner
693.2.by.b 40 11.c even 5 1 inner
693.2.by.b 40 77.m even 15 1 inner
847.2.e.h 20 33.f even 10 1
847.2.e.h 20 231.be even 30 1
847.2.e.i 20 33.h odd 10 1
847.2.e.i 20 231.z odd 30 1
847.2.n.h 40 33.f even 10 2
847.2.n.h 40 231.be even 30 2
847.2.n.i 40 33.h odd 10 2
847.2.n.i 40 231.z odd 30 2
847.2.n.j 40 33.d even 2 1
847.2.n.j 40 33.f even 10 1
847.2.n.j 40 231.l even 6 1
847.2.n.j 40 231.be even 30 1
5929.2.a.bw 10 231.z odd 30 1
5929.2.a.bx 10 231.bc even 30 1
5929.2.a.by 10 231.be even 30 1
5929.2.a.bz 10 231.bf odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 3 T_{2}^{39} + T_{2}^{38} - 12 T_{2}^{37} + 21 T_{2}^{36} + 50 T_{2}^{35} + 238 T_{2}^{34} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display