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Properties

Label 693.2.by.c
Level $693$
Weight $2$
Character orbit 693.by
Analytic conductor $5.534$
Analytic rank $0$
Dimension $64$
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: \(64\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 4 q^{2} + 10 q^{4} + 4 q^{5} - q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 4 q^{2} + 10 q^{4} + 4 q^{5} - q^{7} - 8 q^{8} - 14 q^{10} - 11 q^{11} - 8 q^{13} - 6 q^{14} + 4 q^{17} - 2 q^{19} - 24 q^{20} - 14 q^{22} - 10 q^{25} - 4 q^{26} + 29 q^{28} + 58 q^{29} - 19 q^{31} + 64 q^{32} - 88 q^{34} - 17 q^{35} - 20 q^{37} - 29 q^{38} + 51 q^{40} + 68 q^{41} + 92 q^{43} + 21 q^{44} - 5 q^{46} + 26 q^{47} + 37 q^{49} + 10 q^{50} - 14 q^{52} + 3 q^{53} - 32 q^{55} - 24 q^{56} + 52 q^{58} - 7 q^{59} - 21 q^{61} - 92 q^{62} - 72 q^{64} + 66 q^{65} - 4 q^{67} + 17 q^{68} - q^{70} - 58 q^{71} - 3 q^{73} + 28 q^{74} + 168 q^{76} + 34 q^{77} + 9 q^{79} + 5 q^{80} - 42 q^{82} - 60 q^{83} + 110 q^{85} - 13 q^{86} + 92 q^{88} + 10 q^{89} + 10 q^{91} - 110 q^{92} - 46 q^{94} - 43 q^{95} + 64 q^{97} + 88 q^{98}+O(q^{100}) \) 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 −2.32951 + 1.03717i 0 3.01265 − 3.34589i −0.323917 − 3.08186i 0 1.89829 − 1.84296i −1.97180 + 6.06857i 0 3.95097 + 6.84328i
37.2 −2.24871 + 1.00119i 0 2.71605 − 3.01647i 0.346612 + 3.29780i 0 −0.636334 + 2.56809i −1.56623 + 4.82036i 0 −4.08115 − 7.06876i
37.3 −1.02024 + 0.454240i 0 −0.503705 + 0.559421i −0.0732968 − 0.697372i 0 2.46176 + 0.969406i 0.950004 − 2.92381i 0 0.391555 + 0.678193i
37.4 −0.874462 + 0.389335i 0 −0.725160 + 0.805372i −0.192025 − 1.82700i 0 −2.42288 + 1.06286i 0.912158 − 2.80733i 0 0.879234 + 1.52288i
37.5 −0.194910 + 0.0867797i 0 −1.30780 + 1.45246i 0.252674 + 2.40403i 0 −1.37600 − 2.25978i 0.260721 − 0.802417i 0 −0.257870 − 0.446643i
37.6 1.05983 − 0.471865i 0 −0.437687 + 0.486101i 0.113401 + 1.07894i 0 1.82236 − 1.91807i −0.951494 + 2.92840i 0 0.629300 + 1.08998i
37.7 1.43967 − 0.640984i 0 0.323537 − 0.359324i −0.433772 − 4.12706i 0 −2.64557 − 0.0310171i −0.738505 + 2.27288i 0 −3.26987 − 5.66358i
37.8 1.77664 − 0.791011i 0 1.19249 − 1.32439i 0.181119 + 1.72323i 0 −0.788829 + 2.52542i −0.130923 + 0.402938i 0 1.68488 + 2.91829i
163.1 −1.65532 − 1.83842i 0 −0.430646 + 4.09732i 3.98025 + 0.846028i 0 2.09656 + 1.61383i 4.24270 − 3.08250i 0 −5.03324 − 8.71783i
163.2 −1.48721 − 1.65172i 0 −0.307311 + 2.92387i −0.936836 − 0.199131i 0 −2.63004 − 0.287923i 1.69019 − 1.22800i 0 1.06437 + 1.84354i
163.3 −0.828639 − 0.920297i 0 0.0487532 − 0.463856i 0.125705 + 0.0267195i 0 1.98634 − 1.74770i −2.47103 + 1.79531i 0 −0.0795744 − 0.137827i
163.4 −0.319281 − 0.354597i 0 0.185258 − 1.76261i −0.186763 − 0.0396976i 0 −0.145117 + 2.64177i −1.45622 + 1.05801i 0 0.0455530 + 0.0789002i
163.5 0.205704 + 0.228458i 0 0.199178 − 1.89505i −2.20866 − 0.469464i 0 −1.70967 − 2.01917i 0.971327 − 0.705711i 0 −0.347077 − 0.601155i
163.6 0.792716 + 0.880401i 0 0.0623507 − 0.593228i 3.24422 + 0.689580i 0 −2.60045 − 0.487504i 2.48858 − 1.80806i 0 1.96464 + 3.40285i
163.7 1.49404 + 1.65930i 0 −0.312065 + 2.96910i 1.00537 + 0.213698i 0 2.49708 − 0.874397i −1.78011 + 1.29333i 0 1.14748 + 1.98749i
163.8 1.54241 + 1.71302i 0 −0.346348 + 3.29528i −1.85794 − 0.394917i 0 −1.03747 + 2.43386i −2.44937 + 1.77957i 0 −2.18920 − 3.79180i
235.1 −0.203284 + 1.93412i 0 −1.74320 − 0.370529i −1.58292 − 0.704762i 0 2.12258 + 1.57945i −0.130923 + 0.402938i 0 1.68488 − 2.91829i
235.2 −0.164728 + 1.56729i 0 −0.472952 − 0.100529i 3.79103 + 1.68787i 0 2.12208 − 1.58012i −0.738505 + 2.27288i 0 −3.26987 + 5.66358i
235.3 −0.121266 + 1.15377i 0 0.639819 + 0.135998i −0.991091 − 0.441262i 0 −2.60174 − 0.480599i −0.951494 + 2.92840i 0 0.629300 − 1.08998i
235.4 0.0223018 − 0.212187i 0 1.91177 + 0.406359i −2.20829 − 0.983193i 0 −0.215060 − 2.63700i 0.260721 − 0.802417i 0 −0.257870 + 0.446643i
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.c 64
3.b odd 2 1 231.2.y.b ✓ 64
7.c even 3 1 inner 693.2.by.c 64
11.c even 5 1 inner 693.2.by.c 64
21.h odd 6 1 231.2.y.b ✓ 64
33.h odd 10 1 231.2.y.b ✓ 64
77.m even 15 1 inner 693.2.by.c 64
231.z odd 30 1 231.2.y.b ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.y.b ✓ 64 3.b odd 2 1
231.2.y.b ✓ 64 21.h odd 6 1
231.2.y.b ✓ 64 33.h odd 10 1
231.2.y.b ✓ 64 231.z odd 30 1
693.2.by.c 64 1.a even 1 1 trivial
693.2.by.c 64 7.c even 3 1 inner
693.2.by.c 64 11.c even 5 1 inner
693.2.by.c 64 77.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{64} + 4 T_{2}^{63} - 5 T_{2}^{62} - 36 T_{2}^{61} - 11 T_{2}^{60} + 12 T_{2}^{59} - 34 T_{2}^{58} + \cdots + 923521 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display