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Properties

Label 693.2.l.c
Level $693$
Weight $2$
Character orbit 693.l
Analytic conductor $5.534$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [693,2,Mod(529,693)]
 
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, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.529");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.l (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: \(80\)
Relative dimension: \(40\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 80 q^{4} - 4 q^{5} + 6 q^{6} - q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 80 q^{4} - 4 q^{5} + 6 q^{6} - q^{7} - 4 q^{9} + 12 q^{10} + 40 q^{11} + 5 q^{12} + 21 q^{13} + 7 q^{14} - 5 q^{15} + 80 q^{16} - 3 q^{17} - 10 q^{18} + 18 q^{19} - 12 q^{20} - 33 q^{21} + 4 q^{23} - 2 q^{24} - 40 q^{25} - 12 q^{26} - 6 q^{27} + 6 q^{28} - 6 q^{29} + 14 q^{30} - 64 q^{31} - 70 q^{32} + 28 q^{34} - 18 q^{35} - 16 q^{36} - 7 q^{37} - 23 q^{38} + 42 q^{39} + 51 q^{40} - 2 q^{41} + 25 q^{42} + 9 q^{43} + 40 q^{44} - 14 q^{45} - 6 q^{46} + 52 q^{47} - 28 q^{48} - q^{49} + 2 q^{50} - 20 q^{51} + 44 q^{52} + 4 q^{53} - 2 q^{54} - 8 q^{55} + 6 q^{56} + 16 q^{57} - 16 q^{58} + 48 q^{59} - 70 q^{60} - 138 q^{61} + 12 q^{62} + 95 q^{63} + 44 q^{64} + 14 q^{65} + 3 q^{66} - 14 q^{67} - 17 q^{68} + 15 q^{69} - 40 q^{70} + 34 q^{71} - 19 q^{72} + 26 q^{73} - 4 q^{74} + 14 q^{75} + 52 q^{76} - 2 q^{77} - 60 q^{78} + 10 q^{79} - 47 q^{80} - 44 q^{81} + 60 q^{82} + 4 q^{83} - 205 q^{84} + 13 q^{85} + 82 q^{87} - 38 q^{89} - 43 q^{90} + 13 q^{91} - 21 q^{92} + 44 q^{93} - 112 q^{94} + 40 q^{95} - 78 q^{96} + 49 q^{97} - 84 q^{98} - 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1 −2.75841 −0.384232 − 1.68889i 5.60883 0.763681 − 1.32273i 1.05987 + 4.65866i −1.78814 − 1.95002i −9.95462 −2.70473 + 1.29785i −2.10654 + 3.64864i
529.2 −2.67810 1.52606 + 0.819228i 5.17224 −1.97264 + 3.41671i −4.08695 − 2.19398i −2.35489 + 1.20602i −8.49557 1.65773 + 2.50039i 5.28292 − 9.15029i
529.3 −2.56999 −0.594881 + 1.62669i 4.60484 0.0518560 − 0.0898173i 1.52884 − 4.18057i 2.32912 + 1.25508i −6.69440 −2.29223 − 1.93537i −0.133269 + 0.230829i
529.4 −2.54625 −1.71339 − 0.253555i 4.48339 −1.71038 + 2.96247i 4.36272 + 0.645616i 2.51806 − 0.812012i −6.32335 2.87142 + 0.868879i 4.35506 − 7.54319i
529.5 −2.25915 1.58280 − 0.703380i 3.10377 0.429263 − 0.743506i −3.57579 + 1.58904i 0.204124 + 2.63787i −2.49358 2.01051 − 2.22662i −0.969771 + 1.67969i
529.6 −2.11652 1.07526 − 1.35787i 2.47967 −1.28330 + 2.22274i −2.27581 + 2.87397i 0.502180 − 2.59766i −1.01522 −0.687640 − 2.92013i 2.71613 − 4.70448i
529.7 −2.10851 −1.07420 − 1.35871i 2.44582 0.719032 − 1.24540i 2.26497 + 2.86485i −1.35129 + 2.27465i −0.940022 −0.692175 + 2.91906i −1.51609 + 2.62594i
529.8 −1.90277 1.37046 + 1.05916i 1.62055 2.11250 − 3.65896i −2.60768 − 2.01535i 2.39774 − 1.11842i 0.722016 0.756345 + 2.90309i −4.01961 + 6.96217i
529.9 −1.84837 −0.762290 + 1.55529i 1.41647 −1.42703 + 2.47168i 1.40899 − 2.87474i −2.64112 + 0.156395i 1.07858 −1.83783 − 2.37116i 2.63768 − 4.56859i
529.10 −1.58034 −1.38125 − 1.04506i 0.497465 0.313392 − 0.542811i 2.18285 + 1.65154i 2.31222 − 1.28594i 2.37451 0.815718 + 2.88697i −0.495265 + 0.857824i
529.11 −1.46279 −1.68075 + 0.418424i 0.139743 1.53982 − 2.66705i 2.45858 − 0.612064i −0.365345 + 2.62041i 2.72116 2.64984 − 1.40653i −2.25243 + 3.90132i
529.12 −1.26725 0.979738 + 1.42833i −0.394081 0.775254 − 1.34278i −1.24157 − 1.81004i −2.18139 + 1.49717i 3.03390 −1.08023 + 2.79877i −0.982439 + 1.70163i
529.13 −1.16349 −1.71293 + 0.256683i −0.646282 −0.307505 + 0.532614i 1.99298 − 0.298649i −2.08142 − 1.63331i 3.07893 2.86823 − 0.879357i 0.357780 − 0.619694i
529.14 −0.986601 0.236503 + 1.71583i −1.02662 −1.67507 + 2.90130i −0.233334 − 1.69284i 1.60167 + 2.10586i 2.98606 −2.88813 + 0.811597i 1.65262 − 2.86242i
529.15 −0.919189 0.438094 − 1.67573i −1.15509 −0.535841 + 0.928104i −0.402691 + 1.54031i −2.58581 + 0.559984i 2.90013 −2.61615 − 1.46826i 0.492539 − 0.853103i
529.16 −0.791239 1.26501 − 1.18311i −1.37394 1.50100 − 2.59981i −1.00092 + 0.936125i −0.573359 − 2.58288i 2.66959 0.200492 − 2.99329i −1.18765 + 2.05707i
529.17 −0.596766 1.66865 + 0.464330i −1.64387 −1.29351 + 2.24042i −0.995794 − 0.277096i 0.0409165 − 2.64543i 2.17454 2.56880 + 1.54961i 0.771922 − 1.33701i
529.18 −0.506145 0.548571 + 1.64288i −1.74382 0.192922 − 0.334151i −0.277656 − 0.831538i 1.04940 − 2.42873i 1.89491 −2.39814 + 1.80248i −0.0976467 + 0.169129i
529.19 −0.324486 −0.480575 − 1.66405i −1.89471 −1.54056 + 2.66833i 0.155940 + 0.539960i 2.53680 + 0.751428i 1.26378 −2.53809 + 1.59940i 0.499890 − 0.865836i
529.20 −0.103762 −1.39980 + 1.02007i −1.98923 −0.0658653 + 0.114082i 0.145246 − 0.105845i 2.60824 + 0.443956i 0.413930 0.918906 − 2.85580i 0.00683431 − 0.0118374i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h 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.l.c yes 80
7.c even 3 1 693.2.k.c ✓ 80
9.c even 3 1 693.2.k.c ✓ 80
63.h even 3 1 inner 693.2.l.c yes 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.k.c ✓ 80 7.c even 3 1
693.2.k.c ✓ 80 9.c even 3 1
693.2.l.c yes 80 1.a even 1 1 trivial
693.2.l.c yes 80 63.h even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 60 T_{2}^{38} + 1650 T_{2}^{36} + 7 T_{2}^{35} - 27577 T_{2}^{34} - 348 T_{2}^{33} + \cdots - 891 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display