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Properties

Label 693.2.cx.a
Level $693$
Weight $2$
Character orbit 693.cx
Analytic conductor $5.534$
Analytic rank $0$
Dimension $736$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [693,2,Mod(2,693)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("693.2"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(693, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([5, 10, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.cx (of order \(30\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.53363286007\)
Analytic rank: \(0\)
Dimension: \(736\)
Relative dimension: \(92\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 736 q - 15 q^{2} - 3 q^{3} + 91 q^{4} - 20 q^{6} - 5 q^{7} - 9 q^{9} - 28 q^{12} - 10 q^{13} - 9 q^{14} - 24 q^{15} + 79 q^{16} - 5 q^{18} - 10 q^{19} - 36 q^{20} - 4 q^{22} - 25 q^{24} + 146 q^{25} + 24 q^{26}+ \cdots - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −2.65461 + 0.564255i −1.41923 + 0.992861i 4.90149 − 2.18228i −3.64002 + 1.18271i 3.20729 − 3.43647i 0.155448 + 2.64118i −7.38897 + 5.36840i 1.02845 − 2.81821i 8.99548 − 5.19354i
2.2 −2.63042 + 0.559114i −0.0288739 − 1.73181i 4.77944 − 2.12794i −1.05839 + 0.343891i 1.04423 + 4.53925i 0.683395 − 2.55597i −7.03098 + 5.10831i −2.99833 + 0.100008i 2.59173 − 1.49634i
2.3 −2.60676 + 0.554083i −1.37604 + 1.05192i 4.66108 − 2.07525i 3.87634 − 1.25950i 3.00414 − 3.50453i 2.62450 − 0.334677i −6.68839 + 4.85940i 0.786945 − 2.89495i −9.40681 + 5.43103i
2.4 −2.59523 + 0.551633i 1.24676 − 1.20233i 4.60382 − 2.04975i 1.53679 − 0.499335i −2.57238 + 3.80807i 1.53410 + 2.15558i −6.52426 + 4.74015i 0.108814 − 2.99803i −3.71288 + 2.14363i
2.5 −2.59388 + 0.551347i 1.63989 + 0.557469i 4.59715 − 2.04678i −1.07822 + 0.350336i −4.56103 − 0.541863i −2.63582 − 0.229008i −6.50522 + 4.72632i 2.37846 + 1.82837i 2.60363 − 1.50320i
2.6 −2.57563 + 0.547468i −1.32491 − 1.11562i 4.50708 − 2.00668i 3.29153 − 1.06948i 4.02325 + 2.14808i −2.64387 − 0.0997827i −6.24942 + 4.54047i 0.510784 + 2.95620i −7.89228 + 4.55661i
2.7 −2.53170 + 0.538130i 1.12243 + 1.31914i 4.29284 − 1.91130i −1.59371 + 0.517827i −3.55154 − 2.73567i 2.64560 − 0.0281597i −5.65177 + 4.10625i −0.480284 + 2.96130i 3.75613 − 2.16860i
2.8 −2.44909 + 0.520570i −0.340448 + 1.69826i 3.89996 − 1.73638i −0.462934 + 0.150416i −0.0502762 − 4.33643i −0.906398 − 2.48565i −4.59622 + 3.33935i −2.76819 − 1.15634i 1.05547 − 0.609373i
2.9 −2.29805 + 0.488466i 1.73145 − 0.0456392i 3.21535 − 1.43156i 3.27894 − 1.06539i −3.95666 + 0.950635i −1.30310 − 2.30259i −2.88836 + 2.09852i 2.99583 − 0.158044i −7.01477 + 4.04998i
2.10 −2.28104 + 0.484851i −1.35352 − 1.08073i 3.14098 − 1.39846i −0.152939 + 0.0496928i 3.61144 + 1.80893i 2.39886 + 1.11600i −2.71342 + 1.97141i 0.664060 + 2.92558i 0.324766 − 0.187504i
2.11 −2.26655 + 0.481770i −0.491566 − 1.66083i 3.07806 − 1.37044i −3.34649 + 1.08734i 1.91430 + 3.52754i −2.17298 + 1.50936i −2.56705 + 1.86507i −2.51673 + 1.63282i 7.06115 − 4.07676i
2.12 −2.23339 + 0.474722i 0.700662 − 1.58401i 2.93558 − 1.30700i 1.27988 − 0.415858i −0.812889 + 3.87032i −1.35009 + 2.27536i −2.24139 + 1.62846i −2.01815 − 2.21970i −2.66105 + 1.53636i
2.13 −2.19012 + 0.465525i 1.52315 − 0.824638i 2.75284 − 1.22564i −3.48709 + 1.13302i −2.95199 + 2.51512i 1.92548 − 1.81453i −1.83563 + 1.33366i 1.63995 − 2.51209i 7.10971 − 4.10479i
2.14 −2.09851 + 0.446051i −1.72538 + 0.151919i 2.37768 − 1.05861i 0.837300 − 0.272055i 3.55295 − 1.08841i −1.57385 + 2.12673i −1.04607 + 0.760013i 2.95384 − 0.524233i −1.63573 + 0.944389i
2.15 −2.05012 + 0.435766i −1.61759 − 0.619190i 2.18601 − 0.973273i −1.55645 + 0.505721i 3.58608 + 0.564521i 2.44834 − 1.00282i −0.666184 + 0.484011i 2.23321 + 2.00319i 2.97053 − 1.71504i
2.16 −2.03507 + 0.432568i 0.717736 + 1.57634i 2.12732 − 0.947144i 2.43513 − 0.791221i −2.14252 − 2.89750i 1.17581 − 2.37012i −0.553165 + 0.401898i −1.96971 + 2.26279i −4.61341 + 2.66355i
2.17 −1.94654 + 0.413750i −1.58088 + 0.707691i 1.79074 − 0.797290i 0.404910 − 0.131563i 2.78444 − 2.03164i −1.86715 − 1.87450i 0.0640565 − 0.0465398i 1.99835 − 2.23755i −0.733740 + 0.423625i
2.18 −1.81463 + 0.385712i 1.28496 + 1.16141i 1.31703 − 0.586381i 3.55974 − 1.15663i −2.77970 − 1.61192i −1.01644 + 2.44271i 0.837976 − 0.608825i 0.302237 + 2.98474i −6.01351 + 3.47190i
2.19 −1.80409 + 0.383472i 0.327129 + 1.70088i 1.28061 − 0.570163i −1.83301 + 0.595581i −1.24241 − 2.94310i 1.60778 + 2.10120i 0.892605 − 0.648516i −2.78597 + 1.11281i 3.07853 − 1.77739i
2.20 −1.79760 + 0.382092i 1.73015 + 0.0811017i 1.25828 − 0.560223i −0.879960 + 0.285916i −3.14111 + 0.515288i −0.726845 + 2.54395i 0.925726 − 0.672580i 2.98685 + 0.280637i 1.47257 − 0.850188i
See next 80 embeddings (of 736 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.92
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
63.n odd 6 1 inner
693.cx even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.cx.a yes 736
7.c even 3 1 693.2.cb.a ✓ 736
9.d odd 6 1 693.2.cb.a ✓ 736
11.d odd 10 1 inner 693.2.cx.a yes 736
63.n odd 6 1 inner 693.2.cx.a yes 736
77.o odd 30 1 693.2.cb.a ✓ 736
99.p even 30 1 693.2.cb.a ✓ 736
693.cx even 30 1 inner 693.2.cx.a yes 736
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.cb.a ✓ 736 7.c even 3 1
693.2.cb.a ✓ 736 9.d odd 6 1
693.2.cb.a ✓ 736 77.o odd 30 1
693.2.cb.a ✓ 736 99.p even 30 1
693.2.cx.a yes 736 1.a even 1 1 trivial
693.2.cx.a yes 736 11.d odd 10 1 inner
693.2.cx.a yes 736 63.n odd 6 1 inner
693.2.cx.a yes 736 693.cx even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(693, [\chi])\).