↑

Properties

Label 693.2.by.d
Level $693$
Weight $2$
Character orbit 693.by
Analytic conductor $5.534$
Analytic rank $0$
Dimension $64$
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(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: \(64\)
Relative dimension: \(8\) over \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 231)
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 10 q^{4} - q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 10 q^{4} - q^{7} - 8 q^{8} - 22 q^{10} + 13 q^{11} - 8 q^{13} + 26 q^{14} - 4 q^{17} + 10 q^{19} - 24 q^{20} - 38 q^{22} + 8 q^{23} - 2 q^{25} - 4 q^{26} - 67 q^{28} - 2 q^{29} + 25 q^{31} - 72 q^{32} - 56 q^{34} - 19 q^{35} - 12 q^{37} + 37 q^{38} - 9 q^{40} - 20 q^{41} - 100 q^{43} + 5 q^{44} - 33 q^{46} + 18 q^{47} + 29 q^{49} + 46 q^{50} + 26 q^{52} + 49 q^{53} - 24 q^{55} + 48 q^{56} - 40 q^{58} - q^{59} + 3 q^{61} - 4 q^{62} - 24 q^{64} - 82 q^{65} + 76 q^{67} + 39 q^{68} + 59 q^{70} - 70 q^{71} - 3 q^{73} - 32 q^{74} + 104 q^{76} - 38 q^{77} - 15 q^{79} + 83 q^{80} + 42 q^{82} - 68 q^{83} + 62 q^{85} + 47 q^{86} - 64 q^{88} - 82 q^{89} - 10 q^{91} - 190 q^{92} - 6 q^{94} + 53 q^{95} - 32 q^{97} + 152 q^{98}+O(q^{100}) \) 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 −1.83841 + 0.818513i 0 1.37153 − 1.52323i 0.0520867 + 0.495572i 0 0.483130 + 2.60127i −0.0309144 + 0.0951446i 0 −0.501389 − 0.868431i
37.2 −1.54620 + 0.688411i 0 0.578555 − 0.642550i 0.317169 + 3.01766i 0 −1.40565 − 2.24146i 0.593816 − 1.82758i 0 −2.56780 − 4.44756i
37.3 −1.16334 + 0.517953i 0 −0.253175 + 0.281180i −0.397932 − 3.78607i 0 0.522748 − 2.59359i 0.935917 − 2.88046i 0 2.42393 + 4.19837i
37.4 −0.0592521 + 0.0263807i 0 −1.33545 + 1.48316i −0.117657 − 1.11943i 0 −1.35463 + 2.27266i 0.0800864 − 0.246481i 0 0.0365029 + 0.0632249i
37.5 0.122947 − 0.0547397i 0 −1.32614 + 1.47283i 0.379677 + 3.61238i 0 2.62556 + 0.326267i −0.165600 + 0.509665i 0 0.244421 + 0.423350i
37.6 1.54210 − 0.686586i 0 0.568401 − 0.631273i 0.279932 + 2.66338i 0 −2.55686 + 0.680067i −0.600157 + 1.84709i 0 2.26032 + 3.91499i
37.7 1.70284 − 0.758153i 0 0.986607 − 1.09574i −0.363031 − 3.45401i 0 2.64131 + 0.153270i −0.302713 + 0.931655i 0 −3.23685 − 5.60640i
37.8 2.50180 − 1.11387i 0 3.68004 − 4.08710i −0.150245 − 1.42949i 0 −2.64282 − 0.124535i 2.96170 − 9.11518i 0 −1.96815 − 3.40894i
163.1 −1.55128 − 1.72288i 0 −0.352761 + 3.35629i −0.651377 − 0.138454i 0 −0.438868 + 2.60910i 2.57853 − 1.87341i 0 0.771931 + 1.33702i
163.2 −0.872627 − 0.969151i 0 0.0312821 − 0.297629i −0.151225 − 0.0321438i 0 2.55326 + 0.693447i −2.42586 + 1.76249i 0 0.100810 + 0.174609i
163.3 −0.654921 − 0.727363i 0 0.108921 − 1.03631i 3.19676 + 0.679492i 0 −2.61318 + 0.413884i −2.40878 + 1.75008i 0 −1.59938 − 2.77022i
163.4 0.285304 + 0.316862i 0 0.190054 − 1.80824i −0.861829 − 0.183187i 0 −0.532095 − 2.59169i 1.31708 − 0.956916i 0 −0.187838 − 0.325345i
163.5 0.805605 + 0.894715i 0 0.0575414 − 0.547470i −3.19313 − 0.678721i 0 2.33800 + 1.23845i 2.48423 − 1.80490i 0 −1.96514 − 3.40372i
163.6 1.18104 + 1.31168i 0 −0.116586 + 1.10924i 0.872919 + 0.185545i 0 −0.544420 + 2.58913i 1.26323 − 0.917790i 0 0.787577 + 1.36412i
163.7 1.40081 + 1.55575i 0 −0.249052 + 2.36957i 3.68128 + 0.782479i 0 0.128636 − 2.64262i −0.648035 + 0.470825i 0 3.93941 + 6.82326i
163.8 1.82701 + 2.02911i 0 −0.570229 + 5.42537i −2.89339 − 0.615009i 0 −2.43409 − 1.03693i −7.63253 + 5.54536i 0 −4.03835 − 6.99462i
235.1 −0.286258 + 2.72356i 0 −5.37956 − 1.14346i 1.31309 + 0.584627i 0 2.06489 − 1.65416i 2.96170 − 9.11518i 0 −1.96815 + 3.40894i
235.2 −0.194840 + 1.85378i 0 −1.44224 − 0.306558i 3.17278 + 1.41261i 0 −2.04677 + 1.67652i −0.302713 + 0.931655i 0 −3.23685 + 5.60640i
235.3 −0.176448 + 1.67879i 0 −0.830899 − 0.176613i −2.44652 − 1.08926i 0 2.46827 − 0.952696i −0.600157 + 1.84709i 0 2.26032 − 3.91499i
235.4 −0.0140677 + 0.133845i 0 1.93858 + 0.412058i −3.31825 − 1.47738i 0 −1.93235 + 1.80722i −0.165600 + 0.509665i 0 0.244421 − 0.423350i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.8
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.d 64
3.b odd 2 1 231.2.y.a ✓ 64
7.c even 3 1 inner 693.2.by.d 64
11.c even 5 1 inner 693.2.by.d 64
21.h odd 6 1 231.2.y.a ✓ 64
33.h odd 10 1 231.2.y.a ✓ 64
77.m even 15 1 inner 693.2.by.d 64
231.z odd 30 1 231.2.y.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.y.a ✓ 64 3.b odd 2 1
231.2.y.a ✓ 64 21.h odd 6 1
231.2.y.a ✓ 64 33.h odd 10 1
231.2.y.a ✓ 64 231.z odd 30 1
693.2.by.d 64 1.a even 1 1 trivial
693.2.by.d 64 7.c even 3 1 inner
693.2.by.d 64 11.c even 5 1 inner
693.2.by.d 64 77.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{64} - 13 T_{2}^{62} + 16 T_{2}^{61} + 61 T_{2}^{60} - 96 T_{2}^{59} + 238 T_{2}^{58} - 614 T_{2}^{57} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display