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Properties

Label 693.2.l.b
Level $693$
Weight $2$
Character orbit 693.l
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(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: \(74\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 74 q + 80 q^{4} - 7 q^{5} - 23 q^{6} - q^{7} - 6 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 74 q + 80 q^{4} - 7 q^{5} - 23 q^{6} - q^{7} - 6 q^{8} - 4 q^{9} - 9 q^{10} - 37 q^{11} - 4 q^{12} - 25 q^{13} + 10 q^{14} + 18 q^{15} + 92 q^{16} - 6 q^{17} - 6 q^{18} - 13 q^{19} - 15 q^{20} + 11 q^{21} + 4 q^{23} - 33 q^{24} - 46 q^{25} - 12 q^{26} - 21 q^{27} - 26 q^{28} - 9 q^{29} - 17 q^{30} + 66 q^{31} + 8 q^{32} - 16 q^{34} - 30 q^{35} - 57 q^{36} + 18 q^{37} - 29 q^{38} - 14 q^{39} - 57 q^{40} - 2 q^{41} - 48 q^{42} - 10 q^{43} - 40 q^{44} + 32 q^{45} - 12 q^{46} - 14 q^{47} + 46 q^{48} + 11 q^{49} - 46 q^{50} + 14 q^{51} - 36 q^{52} + 13 q^{53} + 10 q^{54} + 14 q^{55} + 3 q^{56} - 63 q^{57} + 22 q^{58} + 66 q^{59} + 122 q^{60} + 110 q^{61} + 36 q^{62} + 18 q^{63} + 122 q^{64} - 46 q^{65} + 16 q^{66} - 68 q^{67} - 20 q^{68} - 15 q^{69} + 43 q^{70} - 8 q^{71} + 45 q^{72} - 66 q^{73} - 16 q^{74} + 15 q^{75} - 77 q^{76} + 2 q^{77} + 94 q^{78} - 8 q^{79} - 29 q^{80} - 28 q^{81} - 54 q^{82} + 7 q^{83} + 75 q^{84} + 5 q^{85} - 10 q^{87} + 3 q^{88} - 26 q^{89} + 12 q^{90} - 35 q^{91} - 15 q^{92} - 37 q^{93} + 64 q^{94} + 76 q^{95} - 140 q^{96} - 29 q^{97} + 27 q^{98} + 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1 −2.74307 1.69212 − 0.369786i 5.52442 0.600382 − 1.03989i −4.64159 + 1.01435i 2.10573 − 1.60184i −9.66773 2.72652 − 1.25144i −1.64689 + 2.85249i
529.2 −2.66364 0.718874 + 1.57582i 5.09496 1.71010 − 2.96197i −1.91482 − 4.19742i −2.26384 + 1.36932i −8.24386 −1.96644 + 2.26564i −4.55508 + 7.88963i
529.3 −2.62848 0.0151618 − 1.73198i 4.90890 −1.18295 + 2.04893i −0.0398523 + 4.55248i 1.21627 + 2.34961i −7.64599 −2.99954 − 0.0525198i 3.10936 − 5.38556i
529.4 −2.35824 −1.40944 + 1.00671i 3.56128 −0.331494 + 0.574164i 3.32380 − 2.37406i −1.50390 + 2.17676i −3.68186 0.973069 − 2.83780i 0.781741 − 1.35401i
529.5 −2.33211 −1.59509 − 0.675045i 3.43872 −1.00469 + 1.74018i 3.71992 + 1.57428i −2.54010 − 0.740210i −3.35524 2.08863 + 2.15352i 2.34305 − 4.05828i
529.6 −2.27594 0.708304 + 1.58060i 3.17991 −0.924037 + 1.60048i −1.61206 − 3.59736i 1.22912 − 2.34292i −2.68540 −1.99661 + 2.23910i 2.10305 − 3.64259i
529.7 −2.03369 0.953190 − 1.44618i 2.13588 1.72479 − 2.98743i −1.93849 + 2.94107i −2.57136 + 0.622982i −0.276340 −1.18286 − 2.75696i −3.50769 + 6.07550i
529.8 −1.80662 0.189816 − 1.72162i 1.26386 −0.196591 + 0.340506i −0.342924 + 3.11030i 2.59423 + 0.519577i 1.32992 −2.92794 − 0.653580i 0.355165 − 0.615163i
529.9 −1.68857 −0.576770 − 1.63320i 0.851271 0.727462 − 1.26000i 0.973918 + 2.75777i 0.118080 − 2.64312i 1.93971 −2.33467 + 1.88396i −1.22837 + 2.12760i
529.10 −1.65607 1.71487 + 0.243363i 0.742554 −0.177394 + 0.307255i −2.83994 − 0.403024i −2.24110 − 1.40623i 2.08241 2.88155 + 0.834670i 0.293776 − 0.508835i
529.11 −1.30005 −1.14730 + 1.29758i −0.309875 −1.85334 + 3.21008i 1.49154 − 1.68691i 1.56772 − 2.13126i 3.00295 −0.367411 − 2.97742i 2.40943 − 4.17325i
529.12 −1.21788 1.51319 − 0.842776i −0.516765 −1.77839 + 3.08026i −1.84288 + 1.02640i −1.06252 + 2.42302i 3.06512 1.57946 − 2.55055i 2.16587 − 3.75139i
529.13 −1.13760 −0.435388 + 1.67644i −0.705877 1.19588 − 2.07133i 0.495295 − 1.90711i −1.69266 − 2.03344i 3.07819 −2.62088 − 1.45980i −1.36043 + 2.35634i
529.14 −1.12117 −1.70386 − 0.311217i −0.742971 −0.944822 + 1.63648i 1.91032 + 0.348928i 1.03991 + 2.43282i 3.07535 2.80629 + 1.06054i 1.05931 − 1.83478i
529.15 −1.01227 1.64194 − 0.551389i −0.975309 1.11106 − 1.92441i −1.66209 + 0.558155i 2.57972 − 0.587423i 3.01182 2.39194 − 1.81070i −1.12469 + 1.94803i
529.16 −0.244158 −1.69817 − 0.340912i −1.94039 1.55186 − 2.68789i 0.414622 + 0.0832364i 2.04610 − 1.67734i 0.962077 2.76756 + 1.15785i −0.378898 + 0.656271i
529.17 −0.218432 1.54647 + 0.780028i −1.95229 0.711601 − 1.23253i −0.337798 − 0.170383i −0.578773 + 2.58167i 0.863307 1.78311 + 2.41257i −0.155437 + 0.269224i
529.18 0.0381053 0.605842 + 1.62264i −1.99855 −1.46231 + 2.53279i 0.0230858 + 0.0618311i −2.51564 − 0.819479i −0.152366 −2.26591 + 1.96612i −0.0557216 + 0.0965127i
529.19 0.0484108 1.26267 − 1.18561i −1.99766 −1.42100 + 2.46124i 0.0611268 − 0.0573962i 2.43828 − 1.02703i −0.193530 0.188669 − 2.99406i −0.0687915 + 0.119150i
529.20 0.114428 −0.536213 + 1.64696i −1.98691 −0.392529 + 0.679880i −0.0613579 + 0.188458i −0.575657 + 2.58237i −0.456214 −2.42495 − 1.76624i −0.0449163 + 0.0777973i
See all 74 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.37
Significant digits:
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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.b yes 74
7.c even 3 1 693.2.k.b ✓ 74
9.c even 3 1 693.2.k.b ✓ 74
63.h even 3 1 inner 693.2.l.b yes 74
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.k.b ✓ 74 7.c even 3 1
693.2.k.b ✓ 74 9.c even 3 1
693.2.l.b yes 74 1.a even 1 1 trivial
693.2.l.b yes 74 63.h even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{37} - 57 T_{2}^{35} + T_{2}^{34} + 1479 T_{2}^{33} - 53 T_{2}^{32} - 23145 T_{2}^{31} + 1272 T_{2}^{30} + \cdots + 27 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display