↑

Properties

Label 693.2.j.h
Level $693$
Weight $2$
Character orbit 693.j
Analytic conductor $5.534$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [693,2,Mod(232,693)]
 
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([4, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.232");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.j (of order \(3\), 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: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - q^{2} - 2 q^{3} - 17 q^{4} - 4 q^{5} + q^{6} + 14 q^{7} + 24 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - q^{2} - 2 q^{3} - 17 q^{4} - 4 q^{5} + q^{6} + 14 q^{7} + 24 q^{8} + 2 q^{9} + 20 q^{10} - 14 q^{11} + 20 q^{12} - 9 q^{13} + q^{14} - 5 q^{15} - 19 q^{16} + 24 q^{17} - 13 q^{18} + 32 q^{19} - 8 q^{20} - q^{21} - q^{22} + 12 q^{23} - 42 q^{24} - 16 q^{25} - 10 q^{26} - 26 q^{27} - 34 q^{28} - 8 q^{29} + 73 q^{30} - 15 q^{31} - 10 q^{32} + q^{33} - 14 q^{34} - 8 q^{35} - 17 q^{36} - 2 q^{37} + 14 q^{38} + 22 q^{39} + 3 q^{40} - 20 q^{41} - 13 q^{42} - 39 q^{43} + 34 q^{44} - 13 q^{45} + 8 q^{46} - 29 q^{47} + 38 q^{48} - 14 q^{49} - 7 q^{50} - 29 q^{51} - 59 q^{52} + 24 q^{53} + q^{54} + 8 q^{55} + 12 q^{56} + 6 q^{57} - 23 q^{58} - 3 q^{59} - q^{60} - 26 q^{61} - 8 q^{62} - 5 q^{63} + 36 q^{64} - 8 q^{65} + 13 q^{66} - 35 q^{67} + 16 q^{68} + 41 q^{69} + 10 q^{70} - 28 q^{71} + 21 q^{72} + 48 q^{73} - q^{74} + 23 q^{75} - 55 q^{76} + 14 q^{77} - 71 q^{78} + 15 q^{79} + 170 q^{80} - 10 q^{81} + 30 q^{82} - 6 q^{83} + 13 q^{84} + 28 q^{85} - 6 q^{86} - 12 q^{88} - 76 q^{89} - 19 q^{90} - 18 q^{91} + 94 q^{92} + 7 q^{93} - 32 q^{94} - 8 q^{95} + 65 q^{96} - 25 q^{97} + 2 q^{98} + 5 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
232.1 −1.39746 + 2.42048i −1.69348 + 0.363469i −2.90581 − 5.03302i −1.13954 − 1.97373i 1.48682 − 4.60698i 0.500000 − 0.866025i 10.6532 2.73578 − 1.23106i 6.36984
232.2 −1.20403 + 2.08543i 1.37488 + 1.05343i −1.89936 − 3.28978i −1.25365 − 2.17138i −3.85224 + 1.59886i 0.500000 − 0.866025i 4.33139 0.780573 + 2.89667i 6.03770
232.3 −1.11056 + 1.92354i −1.68729 − 0.391211i −1.46668 − 2.54036i 1.97074 + 3.41342i 2.62635 − 2.81112i 0.500000 − 0.866025i 2.07310 2.69391 + 1.32017i −8.75449
232.4 −0.992884 + 1.71973i −0.579798 + 1.63213i −0.971639 − 1.68293i 0.324058 + 0.561285i −2.23114 − 2.61761i 0.500000 − 0.866025i −0.112638 −2.32767 − 1.89261i −1.28701
232.5 −0.732515 + 1.26875i 1.37381 − 1.05482i −0.0731551 − 0.126708i −1.41252 − 2.44655i 0.331967 + 2.51570i 0.500000 − 0.866025i −2.71571 0.774716 − 2.89824i 4.13876
232.6 −0.313556 + 0.543094i −0.395002 − 1.68641i 0.803366 + 1.39147i −0.166433 − 0.288270i 1.03973 + 0.314260i 0.500000 − 0.866025i −2.26182 −2.68795 + 1.33227i 0.208744
232.7 −0.0403476 + 0.0698840i −1.72588 − 0.146116i 0.996744 + 1.72641i −1.85018 − 3.20460i 0.0798461 − 0.114716i 0.500000 − 0.866025i −0.322255 2.95730 + 0.504355i 0.298601
232.8 0.195781 − 0.339102i 1.04757 − 1.37935i 0.923340 + 1.59927i 0.0326924 + 0.0566249i −0.262648 − 0.625282i 0.500000 − 0.866025i 1.50621 −0.805215 − 2.88992i 0.0256022
232.9 0.322465 − 0.558527i 0.688479 + 1.58934i 0.792032 + 1.37184i −0.163512 − 0.283211i 1.10970 + 0.127973i 0.500000 − 0.866025i 2.31147 −2.05199 + 2.18845i −0.210907
232.10 0.653993 − 1.13275i 1.05240 − 1.37567i 0.144587 + 0.250433i 1.25128 + 2.16728i −0.870024 − 2.09178i 0.500000 − 0.866025i 2.99421 −0.784917 − 2.89550i 3.27331
232.11 0.806067 − 1.39615i −1.27956 + 1.16736i −0.299487 − 0.518726i −0.635904 − 1.10142i 0.598400 + 2.72742i 0.500000 − 0.866025i 2.25864 0.274536 − 2.98741i −2.05032
232.12 0.834555 − 1.44549i 0.196526 + 1.72087i −0.392963 − 0.680632i 2.19459 + 3.80115i 2.65151 + 1.15208i 0.500000 − 0.866025i 2.02642 −2.92276 + 0.676389i 7.32603
232.13 1.19436 − 2.06869i −1.09789 − 1.33964i −1.85297 − 3.20944i −1.50255 − 2.60250i −4.08257 + 0.671186i 0.500000 − 0.866025i −4.07502 −0.589267 + 2.94156i −7.17834
232.14 1.28413 − 2.22419i 1.72525 − 0.153379i −2.29800 − 3.98026i 0.350915 + 0.607802i 1.87430 − 4.03423i 0.500000 − 0.866025i −6.66723 2.95295 − 0.529233i 1.80249
463.1 −1.39746 − 2.42048i −1.69348 − 0.363469i −2.90581 + 5.03302i −1.13954 + 1.97373i 1.48682 + 4.60698i 0.500000 + 0.866025i 10.6532 2.73578 + 1.23106i 6.36984
463.2 −1.20403 − 2.08543i 1.37488 − 1.05343i −1.89936 + 3.28978i −1.25365 + 2.17138i −3.85224 − 1.59886i 0.500000 + 0.866025i 4.33139 0.780573 − 2.89667i 6.03770
463.3 −1.11056 − 1.92354i −1.68729 + 0.391211i −1.46668 + 2.54036i 1.97074 − 3.41342i 2.62635 + 2.81112i 0.500000 + 0.866025i 2.07310 2.69391 − 1.32017i −8.75449
463.4 −0.992884 − 1.71973i −0.579798 − 1.63213i −0.971639 + 1.68293i 0.324058 − 0.561285i −2.23114 + 2.61761i 0.500000 + 0.866025i −0.112638 −2.32767 + 1.89261i −1.28701
463.5 −0.732515 − 1.26875i 1.37381 + 1.05482i −0.0731551 + 0.126708i −1.41252 + 2.44655i 0.331967 − 2.51570i 0.500000 + 0.866025i −2.71571 0.774716 + 2.89824i 4.13876
463.6 −0.313556 − 0.543094i −0.395002 + 1.68641i 0.803366 − 1.39147i −0.166433 + 0.288270i 1.03973 − 0.314260i 0.500000 + 0.866025i −2.26182 −2.68795 − 1.33227i 0.208744
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 232.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.j.h ✓ 28
9.c even 3 1 inner 693.2.j.h ✓ 28
9.c even 3 1 6237.2.a.bf 14
9.d odd 6 1 6237.2.a.be 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.j.h ✓ 28 1.a even 1 1 trivial
693.2.j.h ✓ 28 9.c even 3 1 inner
6237.2.a.be 14 9.d odd 6 1
6237.2.a.bf 14 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} + T_{2}^{27} + 23 T_{2}^{26} + 12 T_{2}^{25} + 312 T_{2}^{24} + 95 T_{2}^{23} + 2746 T_{2}^{22} + \cdots + 144 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display