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Properties

Label 693.2.k.b
Level $693$
Weight $2$
Character orbit 693.k
Analytic conductor $5.534$
Analytic rank $0$
Dimension $74$
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: \(74\)
Relative dimension: \(37\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 74 q - 40 q^{4} + 14 q^{5} - 23 q^{6} - q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 74 q - 40 q^{4} + 14 q^{5} - 23 q^{6} - q^{7} - 6 q^{8} + 2 q^{9} - 9 q^{10} + 74 q^{11} - 10 q^{12} - 25 q^{13} + 4 q^{14} + 18 q^{15} - 46 q^{16} - 6 q^{17} - 27 q^{18} - 13 q^{19} - 15 q^{20} - 13 q^{21} - 8 q^{23} + 6 q^{24} + 92 q^{25} - 12 q^{26} - 21 q^{27} - 26 q^{28} - 9 q^{29} + 49 q^{30} - 33 q^{31} - 4 q^{32} - 16 q^{34} - 30 q^{35} - 57 q^{36} + 18 q^{37} + 58 q^{38} + 16 q^{39} + 114 q^{40} - 2 q^{41} + 6 q^{42} - 10 q^{43} - 40 q^{44} + 14 q^{45} - 12 q^{46} + 7 q^{47} + 46 q^{48} - 7 q^{49} - 46 q^{50} - 67 q^{51} + 72 q^{52} + 13 q^{53} - 5 q^{54} + 14 q^{55} + 33 q^{56} - 63 q^{57} - 44 q^{58} - 33 q^{59} - 10 q^{60} - 55 q^{61} + 36 q^{62} + 12 q^{63} + 122 q^{64} + 23 q^{65} - 23 q^{66} + 34 q^{67} + 40 q^{68} - 15 q^{69} + 25 q^{70} - 8 q^{71} + 9 q^{72} - 66 q^{73} + 32 q^{74} - 18 q^{75} - 77 q^{76} - q^{77} + 94 q^{78} + 4 q^{79} - 29 q^{80} - 34 q^{81} - 54 q^{82} + 7 q^{83} - 39 q^{84} + 5 q^{85} + 2 q^{87} - 6 q^{88} - 26 q^{89} + 12 q^{90} - 35 q^{91} - 15 q^{92} + 2 q^{93} - 32 q^{94} - 38 q^{95} + 28 q^{96} - 29 q^{97} + 27 q^{98} + 2 q^{99}+O(q^{100}) \) 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 −1.34000 + 2.32095i 1.69699 + 0.346746i −2.59122 − 4.48813i 4.04212 −3.07875 + 3.47399i −2.36989 − 1.17628i 8.52897 2.75953 + 1.17685i −5.41646 + 9.38158i
67.2 −1.32915 + 2.30216i 1.33800 + 1.09989i −2.53329 − 4.38778i −1.89006 −4.31053 + 1.61836i 2.63550 + 0.232699i 8.15188 0.580477 + 2.94331i 2.51218 − 4.35122i
67.3 −1.31100 + 2.27072i −1.46886 + 0.917854i −2.43745 − 4.22179i 1.99006 −0.158516 − 4.53868i −0.509695 + 2.59619i 7.53801 1.31509 − 2.69640i −2.60898 + 4.51888i
67.4 −1.27862 + 2.21464i −1.32801 − 1.11193i −2.26975 − 3.93132i 0.518003 4.16055 − 1.51932i 0.707282 − 2.54946i 6.49410 0.527214 + 2.95331i −0.662329 + 1.14719i
67.5 −1.08807 + 1.88459i 0.750953 − 1.56079i −1.36778 − 2.36906i −0.0725817 2.12436 + 3.11348i −1.99910 + 1.73308i 1.60067 −1.87214 − 2.34416i 0.0789737 − 0.136787i
67.6 −1.05084 + 1.82010i −0.879629 + 1.49206i −1.20852 − 2.09322i 2.71814 −1.79136 − 3.16893i 1.68432 − 2.04036i 0.876481 −1.45251 − 2.62492i −2.85632 + 4.94729i
67.7 −1.04854 + 1.81613i −1.72882 − 0.105776i −1.19888 − 2.07653i −2.27657 2.00484 − 3.02885i −2.55370 − 0.691835i 0.834153 2.97762 + 0.365734i 2.38708 − 4.13455i
67.8 −0.861771 + 1.49263i −0.422966 + 1.67961i −0.485299 − 0.840562i −3.83204 −2.14254 − 2.07877i −1.40726 + 2.24045i −1.77422 −2.64220 − 1.42084i 3.30234 − 5.71982i
67.9 −0.843962 + 1.46179i 1.63697 − 0.565968i −0.424545 − 0.735334i −2.75591 −0.554220 + 2.87056i 2.23244 − 1.41993i −1.94265 2.35936 − 1.85295i 2.32589 − 4.02855i
67.10 −0.812514 + 1.40732i 1.34387 − 1.09271i −0.320357 − 0.554875i 4.16639 0.445878 + 2.77909i 2.45557 − 0.984981i −2.20888 0.611963 − 2.93692i −3.38525 + 5.86343i
67.11 −0.791597 + 1.37109i 1.32409 + 1.11659i −0.253251 − 0.438644i 2.10948 −2.57909 + 0.931550i 0.954987 + 2.46739i −2.36450 0.506434 + 2.95695i −1.66986 + 2.89228i
67.12 −0.561808 + 0.973081i 0.557414 + 1.63991i 0.368742 + 0.638681i −2.55173 −1.90892 − 0.378904i −1.42340 − 2.23023i −3.07588 −2.37858 + 1.82821i 1.43358 − 2.48304i
67.13 −0.559218 + 0.968593i 1.72270 + 0.179737i 0.374551 + 0.648742i 0.701145 −1.13746 + 1.56808i −2.56077 + 0.665164i −3.07469 2.93539 + 0.619267i −0.392092 + 0.679124i
67.14 −0.414766 + 0.718396i −1.60968 − 0.639465i 0.655939 + 1.13612i 3.51740 1.12703 − 0.891162i −1.81415 + 1.92584i −2.74731 2.18217 + 2.05867i −1.45890 + 2.52688i
67.15 −0.409923 + 0.710007i 0.742803 − 1.56469i 0.663926 + 1.14995i −1.37744 0.806447 + 1.16880i −2.00201 − 1.72973i −2.72833 −1.89649 − 2.32451i 0.564642 − 0.977989i
67.16 −0.262143 + 0.454045i −1.49504 − 0.874554i 0.862562 + 1.49400i −4.06754 0.789001 − 0.449559i 2.64569 − 0.0179644i −1.95303 1.47031 + 2.61499i 1.06628 − 1.84684i
67.17 −0.154582 + 0.267744i 0.0866237 + 1.72988i 0.952209 + 1.64927i 1.38834 −0.476557 − 0.244216i 2.56044 + 0.666459i −1.20711 −2.98499 + 0.299698i −0.214614 + 0.371722i
67.18 −0.0572140 + 0.0990976i −1.15820 + 1.28785i 0.993453 + 1.72071i 0.785057 −0.0613579 − 0.188458i −1.94857 − 1.78972i −0.456214 −0.317136 − 2.98319i −0.0449163 + 0.0777973i
67.19 −0.0242054 + 0.0419250i 0.395432 − 1.68631i 0.998828 + 1.73002i 2.84199 0.0611268 + 0.0573962i −0.329708 + 2.62513i −0.193530 −2.68727 − 1.33364i −0.0687915 + 0.119150i
67.20 −0.0190526 + 0.0330002i −1.70817 + 0.286645i 0.999274 + 1.73079i 2.92461 0.0230858 − 0.0618311i 1.96751 − 1.76887i −0.152366 2.83567 − 0.979275i −0.0557216 + 0.0965127i
See all 74 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.37
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.b ✓ 74
7.c even 3 1 693.2.l.b yes 74
9.c even 3 1 693.2.l.b yes 74
63.g even 3 1 inner 693.2.k.b ✓ 74
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.k.b ✓ 74 1.a even 1 1 trivial
693.2.k.b ✓ 74 63.g even 3 1 inner
693.2.l.b yes 74 7.c even 3 1
693.2.l.b yes 74 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{74} + 57 T_{2}^{72} + 2 T_{2}^{71} + 1770 T_{2}^{70} + 110 T_{2}^{69} + 38014 T_{2}^{68} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display