Properties

Label 207.3.p.a
Level $207$
Weight $3$
Character orbit 207.p
Analytic conductor $5.640$
Analytic rank $0$
Dimension $920$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [207,3,Mod(7,207)] 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("207.7"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(207, base_ring=CyclotomicField(66)) chi = DirichletCharacter(H, H._module([44, 57])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 207.p (of order \(66\), degree \(20\), 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: \(5.64034147226\)
Analytic rank: \(0\)
Dimension: \(920\)
Relative dimension: \(46\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 920 q - 9 q^{2} - 16 q^{3} + 79 q^{4} - 11 q^{5} - 13 q^{6} - 11 q^{7} - 70 q^{8} + 8 q^{9} - 44 q^{10} - 11 q^{11} - 58 q^{12} - 9 q^{13} - 11 q^{14} + 132 q^{15} + 159 q^{16} - 44 q^{17} - 165 q^{18}+ \cdots - 2134 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.27268 − 3.67716i 2.99759 − 0.120256i −8.75762 + 6.88707i 2.12854 + 1.51572i −4.25717 − 10.8696i 8.14249 + 1.97535i 23.3766 + 15.0233i 8.97108 − 0.720957i 2.86462 − 9.75601i
7.2 −1.22574 − 3.54155i −1.10465 + 2.78922i −7.89594 + 6.20944i 4.52852 + 3.22475i 11.2322 + 0.493293i −4.86503 − 1.18024i 19.0585 + 12.2481i −6.55951 − 6.16221i 5.86981 − 19.9907i
7.3 −1.15544 − 3.33843i −2.83228 − 0.989034i −6.66588 + 5.24211i 1.71939 + 1.22437i −0.0292864 + 10.5982i 9.73668 + 2.36209i 13.3148 + 8.55689i 7.04362 + 5.60244i 2.10083 − 7.15478i
7.4 −1.10845 − 3.20264i −0.0772054 + 2.99901i −5.88406 + 4.62728i −7.11239 − 5.06471i 9.69033 − 3.07697i 6.69124 + 1.62328i 9.93753 + 6.38646i −8.98808 − 0.463079i −8.33676 + 28.3924i
7.5 −1.09022 − 3.15000i −2.99991 − 0.0231242i −5.58968 + 4.39578i −5.03024 − 3.58202i 3.19774 + 9.47492i −9.32741 − 2.26280i 8.72400 + 5.60658i 8.99893 + 0.138741i −5.79926 + 19.7505i
7.6 −1.07231 − 3.09825i −1.03595 − 2.81546i −5.30508 + 4.17196i 1.66303 + 1.18424i −7.61212 + 6.22870i −5.47715 − 1.32874i 7.58207 + 4.87270i −6.85360 + 5.83336i 1.88578 − 6.42238i
7.7 −1.05439 − 3.04647i 2.58572 + 1.52121i −5.02501 + 3.95171i −2.46457 − 1.75501i 1.90794 − 9.48125i −8.89932 − 2.15895i 6.48905 + 4.17026i 4.37186 + 7.86682i −2.74796 + 9.35869i
7.8 −0.972781 − 2.81067i 2.06860 − 2.17276i −3.80933 + 2.99569i −4.42049 − 3.14782i −8.11920 − 3.70053i 0.779816 + 0.189181i 2.11714 + 1.36060i −0.441770 − 8.98915i −4.54730 + 15.4867i
7.9 −0.903763 − 2.61125i 0.786458 − 2.89508i −2.85764 + 2.24727i 6.70533 + 4.77485i −8.27056 + 0.562825i 5.78622 + 1.40372i −0.847469 − 0.544635i −7.76297 − 4.55372i 6.40830 − 21.8246i
7.10 −0.829996 − 2.39812i 2.06495 + 2.17623i −1.91786 + 1.50822i 4.91069 + 3.49688i 3.50495 − 6.75826i 1.67111 + 0.405408i −3.33065 − 2.14048i −0.471956 + 8.98762i 4.31009 − 14.6788i
7.11 −0.744077 − 2.14987i −1.89905 + 2.32242i −0.924073 + 0.726699i 3.30593 + 2.35415i 6.40593 + 2.35464i −3.91580 − 0.949962i −5.40549 − 3.47390i −1.78725 − 8.82076i 2.60123 − 8.85899i
7.12 −0.715440 − 2.06713i −2.62803 + 1.44689i −0.616954 + 0.485178i −0.803977 − 0.572510i 4.87109 + 4.39731i 8.28005 + 2.00872i −5.91643 − 3.80226i 4.81304 − 7.60491i −0.608253 + 2.07152i
7.13 −0.609575 − 1.76125i 2.92012 − 0.687660i 0.413788 − 0.325406i 2.10535 + 1.49921i −2.99118 − 4.72389i −5.01490 − 1.21660i −7.09692 − 4.56092i 8.05425 − 4.01610i 1.35712 − 4.62193i
7.14 −0.586173 − 1.69364i −1.31237 − 2.69772i 0.619406 − 0.487106i −4.74071 − 3.37584i −3.79968 + 3.80401i 3.38933 + 0.822242i −7.21886 − 4.63928i −5.55536 + 7.08081i −2.93857 + 10.0079i
7.15 −0.532010 − 1.53714i −2.86155 − 0.900848i 1.06445 − 0.837089i 6.19695 + 4.41283i 0.137642 + 4.87787i −11.4735 − 2.78343i −7.32656 − 4.70849i 7.37695 + 5.15565i 3.48630 − 11.8732i
7.16 −0.473892 − 1.36922i 0.970202 + 2.83879i 1.49402 − 1.17491i −3.88296 − 2.76504i 3.42715 − 2.67370i −9.94573 − 2.41281i −7.19232 − 4.62222i −7.11742 + 5.50839i −1.94585 + 6.62695i
7.17 −0.404726 − 1.16938i 2.70540 + 1.29646i 1.94057 − 1.52608i −3.91559 − 2.78828i 0.421104 − 3.68835i 12.1228 + 2.94097i −6.73396 − 4.32765i 5.63840 + 7.01488i −1.67581 + 5.70729i
7.18 −0.288402 − 0.833282i 2.50659 − 1.64833i 2.53303 − 1.99200i 3.73171 + 2.65734i −2.09643 − 1.61332i 4.57304 + 1.10941i −5.35762 − 3.44313i 3.56602 − 8.26338i 1.13808 − 3.87595i
7.19 −0.273874 − 0.791306i −0.519571 + 2.95467i 2.59305 − 2.03920i −2.94145 − 2.09460i 2.48034 − 0.398065i −2.46692 − 0.598468i −5.14153 − 3.30426i −8.46009 − 3.07032i −0.851882 + 2.90124i
7.20 −0.245059 − 0.708051i −2.38873 − 1.81493i 2.70293 − 2.12561i 4.94001 + 3.51777i −0.699684 + 2.13611i 8.42154 + 2.04304i −4.68868 − 3.01323i 2.41205 + 8.67076i 1.28016 − 4.35984i
See next 80 embeddings (of 920 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 7.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
23.d odd 22 1 inner
207.p odd 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.3.p.a ✓ 920
9.c even 3 1 inner 207.3.p.a ✓ 920
23.d odd 22 1 inner 207.3.p.a ✓ 920
207.p odd 66 1 inner 207.3.p.a ✓ 920
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.3.p.a ✓ 920 1.a even 1 1 trivial
207.3.p.a ✓ 920 9.c even 3 1 inner
207.3.p.a ✓ 920 23.d odd 22 1 inner
207.3.p.a ✓ 920 207.p odd 66 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(207, [\chi])\).