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Properties

Label 693.2.k.c
Level $693$
Weight $2$
Character orbit 693.k
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(67,693)]
 
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([2, 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("693.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.k (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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 40 q^{4} + 8 q^{5} + 6 q^{6} - q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 40 q^{4} + 8 q^{5} + 6 q^{6} - q^{7} + 2 q^{9} + 12 q^{10} - 80 q^{11} - 10 q^{12} + 21 q^{13} - 8 q^{14} - 5 q^{15} - 40 q^{16} - 3 q^{17} + 32 q^{18} + 18 q^{19} - 12 q^{20} + 18 q^{21} - 8 q^{23} - 14 q^{24} + 80 q^{25} - 12 q^{26} - 6 q^{27} + 6 q^{28} - 6 q^{29} + 20 q^{30} + 32 q^{31} + 35 q^{32} + 28 q^{34} - 18 q^{35} - 16 q^{36} - 7 q^{37} + 46 q^{38} - 48 q^{39} - 102 q^{40} - 2 q^{41} - 14 q^{42} + 9 q^{43} + 40 q^{44} - 5 q^{45} - 6 q^{46} - 26 q^{47} - 28 q^{48} - 19 q^{49} + 2 q^{50} + 22 q^{51} - 88 q^{52} + 4 q^{53} + 19 q^{54} - 8 q^{55} + 18 q^{56} + 16 q^{57} + 32 q^{58} - 24 q^{59} - 10 q^{60} + 69 q^{61} + 12 q^{62} + 20 q^{63} + 44 q^{64} - 7 q^{65} - 6 q^{66} + 7 q^{67} + 34 q^{68} + 15 q^{69} - 22 q^{70} + 34 q^{71} - 70 q^{72} + 26 q^{73} + 8 q^{74} - 64 q^{75} + 52 q^{76} + q^{77} - 60 q^{78} - 5 q^{79} - 47 q^{80} + 46 q^{81} + 60 q^{82} + 4 q^{83} + 29 q^{84} + 13 q^{85} - 38 q^{87} - 38 q^{89} - 43 q^{90} + 13 q^{91} - 21 q^{92} + 35 q^{93} + 56 q^{94} - 20 q^{95} + 15 q^{96} + 49 q^{97} - 84 q^{98} - 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1 −1.33880 + 2.31887i −0.248830 + 1.71408i −2.58477 − 4.47695i −1.33978 −3.64160 − 2.87182i −2.26239 − 1.37171i 8.48676 −2.87617 − 0.853031i 1.79369 − 3.10677i
67.2 −1.29986 + 2.25143i 0.715903 − 1.57718i −2.37928 − 4.12103i 0.311533 2.62032 + 3.66191i 1.85375 − 1.88776i 7.17148 −1.97497 − 2.25821i −0.404950 + 0.701394i
67.3 −1.24651 + 2.15901i 1.60939 − 0.640217i −2.10756 − 3.65040i 0.861122 −0.623873 + 4.27272i 1.22127 + 2.34702i 5.52231 2.18024 − 2.06071i −1.07339 + 1.85917i
67.4 −1.22200 + 2.11657i −1.64916 + 0.529420i −1.98659 − 3.44088i −3.27178 0.894720 − 4.13751i 2.64019 + 0.171508i 4.82247 2.43943 − 1.74619i 3.99814 − 6.92497i
67.5 −1.10481 + 1.91359i −1.58815 − 0.691220i −1.44122 − 2.49626i 3.44892 3.07732 − 2.27540i 2.28606 + 1.33188i 1.94985 2.04443 + 2.19552i −3.81040 + 6.59981i
67.6 −1.10420 + 1.91253i 1.69259 + 0.367597i −1.43852 − 2.49159i −3.67196 −2.57200 + 2.83124i −1.97051 + 1.76553i 1.93685 2.72974 + 1.24438i 4.05458 − 7.02274i
67.7 −1.09944 + 1.90428i −0.320518 − 1.70214i −1.41753 − 2.45523i 3.30405 3.59374 + 1.26104i −2.64520 + 0.0539971i 1.83618 −2.79454 + 1.09113i −3.63260 + 6.29185i
67.8 −1.05164 + 1.82149i 0.591977 + 1.62775i −1.21188 − 2.09904i 2.17153 −3.58747 − 0.633521i −1.06886 + 2.42024i 0.891289 −2.29913 + 1.92718i −2.28366 + 3.95542i
67.9 −0.952385 + 1.64958i −0.322344 − 1.70179i −0.814073 − 1.41002i −3.95283 3.11423 + 1.08903i −0.889024 − 2.49191i −0.708295 −2.79219 + 1.09712i 3.76462 − 6.52051i
67.10 −0.857961 + 1.48603i −1.44096 + 0.961064i −0.472196 − 0.817867i 1.78430 −0.191886 − 2.96586i −2.53403 − 0.760734i −1.81134 1.15271 − 2.76970i −1.53086 + 2.65153i
67.11 −0.793665 + 1.37467i −0.410682 + 1.68266i −0.259808 − 0.450001i −0.790103 −1.98715 − 1.90002i 2.64443 − 0.0834765i −2.34986 −2.66268 − 1.38208i 0.627077 − 1.08613i
67.12 −0.614445 + 1.06425i −1.72029 − 0.201530i 0.244914 + 0.424204i 0.507707 1.27150 − 1.70699i 0.592251 − 2.57861i −3.05973 2.91877 + 0.693380i −0.311958 + 0.540327i
67.13 −0.560588 + 0.970967i 1.22782 − 1.22166i 0.371482 + 0.643425i 2.04342 0.497894 + 1.87702i −1.67464 − 2.04831i −3.07535 0.0150779 − 2.99996i −1.14552 + 1.98410i
67.14 −0.545667 + 0.945123i −1.28728 − 1.15884i 0.404495 + 0.700606i −1.30912 1.79767 − 0.584302i −1.49335 + 2.18401i −3.06555 0.314195 + 2.98350i 0.714343 − 1.23728i
67.15 −0.488901 + 0.846801i 1.13815 − 1.30561i 0.521953 + 0.904048i −2.33520 0.549154 + 1.60210i 0.709762 + 2.54877i −2.97633 −0.409251 − 2.97195i 1.14168 − 1.97745i
67.16 −0.402449 + 0.697061i 1.70254 + 0.318365i 0.676070 + 1.17099i 1.72034 −0.907105 + 1.05865i 2.28270 + 1.33763i −2.69813 2.79729 + 1.08406i −0.692349 + 1.19918i
67.17 −0.270594 + 0.468683i 0.806282 + 1.53294i 0.853557 + 1.47840i 4.26443 −0.936639 − 0.0369142i −0.399242 − 2.61546i −2.00625 −1.69982 + 2.47197i −1.15393 + 1.99867i
67.18 −0.170037 + 0.294513i 1.59444 + 0.676575i 0.942175 + 1.63189i −2.75363 −0.470375 + 0.354542i 2.24055 − 1.40710i −1.32097 2.08449 + 2.15752i 0.468221 − 0.810982i
67.19 −0.152741 + 0.264555i −0.898620 + 1.48070i 0.953340 + 1.65123i −3.17515 −0.254471 − 0.463899i 0.356113 − 2.62168i −1.19342 −1.38496 − 2.66118i 0.484975 − 0.840002i
67.20 0.0316257 − 0.0547774i −0.506515 − 1.65633i 0.998000 + 1.72859i −0.643747 −0.106749 − 0.0246372i −2.63953 + 0.181392i 0.252753 −2.48689 + 1.67792i −0.0203590 + 0.0352628i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.40
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{80} + 60 T_{2}^{78} + 1950 T_{2}^{76} - 7 T_{2}^{75} + 43846 T_{2}^{74} - 492 T_{2}^{73} + \cdots + 793881 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display