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Properties

Label 297.3.l.a
Level $297$
Weight $3$
Character orbit 297.l
Analytic conductor $8.093$
Analytic rank $0$
Dimension $64$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [297,3,Mod(28,297)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(297, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 9])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("297.28"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 297.l (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [64,0,0,32,0,0,0,0,0,0,0,0,0,0,0,-140] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(16)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.09266385150\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 32 q^{4} - 140 q^{16} - 84 q^{22} + 100 q^{25} + 520 q^{28} - 92 q^{31} + 136 q^{34} - 156 q^{37} - 480 q^{40} - 160 q^{46} - 168 q^{49} - 100 q^{52} + 100 q^{55} - 240 q^{58} - 360 q^{61} - 36 q^{64}+ \cdots - 120 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1 −3.54471 + 1.15175i 0 8.00236 − 5.81406i −1.37260 + 4.22442i 0 7.83056 + 10.7778i −12.9067 + 17.7646i 0 − 16.5552i
28.2 −3.36249 + 1.09254i 0 6.87662 − 4.99616i −0.167460 + 0.515388i 0 −2.87959 − 3.96341i −9.35153 + 12.8713i 0 − 1.91594i
28.3 −2.46532 + 0.801032i 0 2.20009 − 1.59846i 2.02943 − 6.24594i 0 0.446735 + 0.614878i 1.95110 − 2.68545i 0 17.0239i
28.4 −2.36517 + 0.768489i 0 1.76737 − 1.28407i −1.62812 + 5.01083i 0 −7.38827 − 10.1691i 2.65368 − 3.65248i 0 − 13.1026i
28.5 −2.17481 + 0.706638i 0 0.994385 − 0.722463i 1.84605 − 5.68157i 0 1.94543 + 2.67766i 3.72434 − 5.12612i 0 13.6608i
28.6 −1.52193 + 0.494505i 0 −1.16433 + 0.845936i −1.94332 + 5.98091i 0 7.34667 + 10.1118i 5.11613 − 7.04175i 0 − 10.0635i
28.7 −0.906042 + 0.294391i 0 −2.50182 + 1.81768i −0.862169 + 2.65348i 0 −4.17645 − 5.74839i 3.97151 − 5.46631i 0 − 2.65798i
28.8 −0.0795886 + 0.0258599i 0 −3.23040 + 2.34702i 1.48372 − 4.56642i 0 1.34705 + 1.85405i 0.393163 − 0.541143i 0 0.401804i
28.9 0.0795886 − 0.0258599i 0 −3.23040 + 2.34702i −1.48372 + 4.56642i 0 1.34705 + 1.85405i −0.393163 + 0.541143i 0 0.401804i
28.10 0.906042 − 0.294391i 0 −2.50182 + 1.81768i 0.862169 − 2.65348i 0 −4.17645 − 5.74839i −3.97151 + 5.46631i 0 − 2.65798i
28.11 1.52193 − 0.494505i 0 −1.16433 + 0.845936i 1.94332 − 5.98091i 0 7.34667 + 10.1118i −5.11613 + 7.04175i 0 − 10.0635i
28.12 2.17481 − 0.706638i 0 0.994385 − 0.722463i −1.84605 + 5.68157i 0 1.94543 + 2.67766i −3.72434 + 5.12612i 0 13.6608i
28.13 2.36517 − 0.768489i 0 1.76737 − 1.28407i 1.62812 − 5.01083i 0 −7.38827 − 10.1691i −2.65368 + 3.65248i 0 − 13.1026i
28.14 2.46532 − 0.801032i 0 2.20009 − 1.59846i −2.02943 + 6.24594i 0 0.446735 + 0.614878i −1.95110 + 2.68545i 0 17.0239i
28.15 3.36249 − 1.09254i 0 6.87662 − 4.99616i 0.167460 − 0.515388i 0 −2.87959 − 3.96341i 9.35153 − 12.8713i 0 − 1.91594i
28.16 3.54471 − 1.15175i 0 8.00236 − 5.81406i 1.37260 − 4.22442i 0 7.83056 + 10.7778i 12.9067 − 17.7646i 0 − 16.5552i
217.1 −2.18358 − 3.00544i 0 −3.02859 + 9.32103i −7.33251 − 5.32738i 0 −7.07395 − 2.29846i 20.4945 − 6.65908i 0 33.6702i
217.2 −2.12418 − 2.92369i 0 −2.79972 + 8.61665i 5.78682 + 4.20437i 0 −8.46490 − 2.75041i 17.3915 − 5.65085i 0 − 25.8497i
217.3 −1.76521 − 2.42960i 0 −1.55092 + 4.77325i −3.34009 − 2.42672i 0 7.42789 + 2.41347i 2.91011 − 0.945551i 0 12.3987i
217.4 −1.49206 − 2.05364i 0 −0.755134 + 2.32406i 1.98794 + 1.44433i 0 3.94202 + 1.28084i −3.75730 + 1.22082i 0 − 6.23753i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.3.l.a ✓ 64
3.b odd 2 1 inner 297.3.l.a ✓ 64
11.d odd 10 1 inner 297.3.l.a ✓ 64
33.f even 10 1 inner 297.3.l.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.3.l.a ✓ 64 1.a even 1 1 trivial
297.3.l.a ✓ 64 3.b odd 2 1 inner
297.3.l.a ✓ 64 11.d odd 10 1 inner
297.3.l.a ✓ 64 33.f even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{64} - 48 T_{2}^{62} + 1411 T_{2}^{60} - 32946 T_{2}^{58} + 668298 T_{2}^{56} - 11457474 T_{2}^{54} + \cdots + 214358881 \) acting on \(S_{3}^{\mathrm{new}}(297, [\chi])\). Copy content Toggle raw display