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Properties

Label 693.2.j.i
Level $693$
Weight $2$
Character orbit 693.j
Analytic conductor $5.534$
Analytic rank $0$
Dimension $30$
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(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: \(30\)
Relative dimension: \(15\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 2 q^{2} - 14 q^{4} + q^{5} - 2 q^{6} + 15 q^{7} + 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 2 q^{2} - 14 q^{4} + q^{5} - 2 q^{6} + 15 q^{7} + 18 q^{8} - 16 q^{10} + 15 q^{11} + 10 q^{13} + 2 q^{14} - 13 q^{15} - 12 q^{16} + 20 q^{17} - 11 q^{18} - 32 q^{19} + 2 q^{20} + 2 q^{22} - 2 q^{23} + 21 q^{24} - 6 q^{25} - 44 q^{26} + 24 q^{27} - 28 q^{28} - 8 q^{29} - 43 q^{30} + 13 q^{31} - 15 q^{32} + 24 q^{34} + 2 q^{35} + 48 q^{36} + 2 q^{37} + 14 q^{38} - 14 q^{39} + 45 q^{40} - 15 q^{41} - 13 q^{42} + 30 q^{43} - 28 q^{44} - 16 q^{45} - 48 q^{46} + 8 q^{47} - 59 q^{48} - 15 q^{49} - 24 q^{50} + 13 q^{51} + 40 q^{52} - 30 q^{53} + 16 q^{54} + 2 q^{55} + 9 q^{56} - 52 q^{57} + 5 q^{58} + 16 q^{59} + 27 q^{60} + 8 q^{61} + 90 q^{62} + 6 q^{63} + 22 q^{64} - 6 q^{65} - 13 q^{66} + 46 q^{67} + 14 q^{68} + 14 q^{69} - 8 q^{70} - 46 q^{71} + 18 q^{72} - 50 q^{73} + 8 q^{74} - 81 q^{75} + 49 q^{76} - 15 q^{77} + 28 q^{78} + 29 q^{79} - 138 q^{80} + 24 q^{81} - 68 q^{82} - 18 q^{83} - 12 q^{84} + 39 q^{86} + 4 q^{87} + 9 q^{88} + 20 q^{89} + 95 q^{90} + 20 q^{91} + 18 q^{92} - 34 q^{93} + 49 q^{94} - 8 q^{95} + 91 q^{96} + 48 q^{97} + 4 q^{98} + 6 q^{99}+O(q^{100}) \) 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 −1.38123 + 2.39236i 0.746114 + 1.56311i −2.81560 − 4.87675i 1.73908 + 3.01217i −4.77008 − 0.374041i 0.500000 − 0.866025i 10.0310 −1.88663 + 2.33252i −9.60826
232.2 −1.17242 + 2.03070i −1.23866 − 1.21066i −1.74916 − 3.02963i −0.281697 − 0.487913i 3.91074 − 1.09594i 0.500000 − 0.866025i 3.51333 0.0685806 + 2.99922i 1.32107
232.3 −1.12094 + 1.94152i 1.61490 − 0.626180i −1.51301 − 2.62061i 0.320499 + 0.555121i −0.594459 + 3.83727i 0.500000 − 0.866025i 2.30021 2.21580 − 2.02244i −1.43704
232.4 −0.944170 + 1.63535i −1.08062 + 1.35361i −0.782913 − 1.35605i −1.81301 − 3.14023i −1.19333 − 3.04523i 0.500000 − 0.866025i −0.819866 −0.664515 − 2.92548i 6.84716
232.5 −0.693796 + 1.20169i −1.71312 + 0.255376i 0.0372935 + 0.0645942i 0.0234767 + 0.0406628i 0.881673 − 2.23582i 0.500000 − 0.866025i −2.87868 2.86957 − 0.874982i −0.0651522
232.6 −0.506354 + 0.877030i −0.914975 − 1.47065i 0.487212 + 0.843876i 0.0799377 + 0.138456i 1.75311 − 0.0577902i 0.500000 − 0.866025i −3.01222 −1.32564 + 2.69122i −0.161907
232.7 −0.261485 + 0.452906i −0.544502 + 1.64424i 0.863251 + 1.49519i 1.91314 + 3.31366i −0.602305 − 0.676552i 0.500000 − 0.866025i −1.94885 −2.40703 − 1.79058i −2.00103
232.8 −0.133914 + 0.231946i 1.72714 + 0.130301i 0.964134 + 1.66993i −1.47834 − 2.56057i −0.261512 + 0.383155i 0.500000 − 0.866025i −1.05210 2.96604 + 0.450096i 0.791885
232.9 0.327238 − 0.566793i −1.70482 + 0.305905i 0.785830 + 1.36110i 0.729270 + 1.26313i −0.384499 + 1.06639i 0.500000 − 0.866025i 2.33757 2.81284 − 1.04303i 0.954581
232.10 0.331654 − 0.574442i 1.73190 + 0.0225526i 0.780011 + 1.35102i 1.43426 + 2.48420i 0.587349 − 0.987399i 0.500000 − 0.866025i 2.36139 2.99898 + 0.0781178i 1.90271
232.11 0.473628 − 0.820348i 0.122376 − 1.72772i 0.551353 + 0.954971i −1.73523 − 3.00550i −1.35937 − 0.918689i 0.500000 − 0.866025i 2.93906 −2.97005 − 0.422864i −3.28741
232.12 0.808256 − 1.39994i −1.04767 − 1.37927i −0.306556 − 0.530971i 1.10536 + 1.91454i −2.77768 + 0.351877i 0.500000 − 0.866025i 2.24192 −0.804765 + 2.89004i 3.57365
232.13 0.871222 − 1.50900i 1.42104 + 0.990268i −0.518054 − 0.897296i −0.303514 − 0.525702i 2.73236 − 1.28161i 0.500000 − 0.866025i 1.67953 1.03874 + 2.81443i −1.05771
232.14 1.14486 − 1.98295i 0.161121 + 1.72454i −1.62140 − 2.80834i −0.955494 − 1.65496i 3.60414 + 1.65486i 0.500000 − 0.866025i −2.84564 −2.94808 + 0.555719i −4.37562
232.15 1.25746 − 2.17798i 0.719779 − 1.57541i −2.16240 − 3.74539i −0.277728 − 0.481039i −2.52612 − 3.54868i 0.500000 − 0.866025i −5.84667 −1.96384 − 2.26789i −1.39692
463.1 −1.38123 − 2.39236i 0.746114 − 1.56311i −2.81560 + 4.87675i 1.73908 − 3.01217i −4.77008 + 0.374041i 0.500000 + 0.866025i 10.0310 −1.88663 − 2.33252i −9.60826
463.2 −1.17242 − 2.03070i −1.23866 + 1.21066i −1.74916 + 3.02963i −0.281697 + 0.487913i 3.91074 + 1.09594i 0.500000 + 0.866025i 3.51333 0.0685806 − 2.99922i 1.32107
463.3 −1.12094 − 1.94152i 1.61490 + 0.626180i −1.51301 + 2.62061i 0.320499 − 0.555121i −0.594459 − 3.83727i 0.500000 + 0.866025i 2.30021 2.21580 + 2.02244i −1.43704
463.4 −0.944170 − 1.63535i −1.08062 − 1.35361i −0.782913 + 1.35605i −1.81301 + 3.14023i −1.19333 + 3.04523i 0.500000 + 0.866025i −0.819866 −0.664515 + 2.92548i 6.84716
463.5 −0.693796 − 1.20169i −1.71312 − 0.255376i 0.0372935 − 0.0645942i 0.0234767 − 0.0406628i 0.881673 + 2.23582i 0.500000 + 0.866025i −2.87868 2.86957 + 0.874982i −0.0651522
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 232.15
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.i ✓ 30
9.c even 3 1 inner 693.2.j.i ✓ 30
9.c even 3 1 6237.2.a.bh 15
9.d odd 6 1 6237.2.a.bg 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.j.i ✓ 30 1.a even 1 1 trivial
693.2.j.i ✓ 30 9.c even 3 1 inner
6237.2.a.bg 15 9.d odd 6 1
6237.2.a.bh 15 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} + 2 T_{2}^{29} + 24 T_{2}^{28} + 34 T_{2}^{27} + 318 T_{2}^{26} + 375 T_{2}^{25} + \cdots + 1296 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display