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Properties

Label 27.15.d.a
Level $27$
Weight $15$
Character orbit 27.d
Analytic conductor $33.569$
Analytic rank $0$
Dimension $26$
Inner twists $2$

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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] = [27,15,Mod(8,27)] 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("27.8"); S:= CuspForms(chi, 15); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 15, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 27.d (of order \(6\), degree \(2\), not 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: \(33.5688214010\)
Analytic rank: \(0\)
Dimension: \(26\)
Relative dimension: \(13\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 26 q + 3 q^{2} + 98303 q^{4} + 107994 q^{5} - 146330 q^{7} - 32772 q^{10} + 15978711 q^{11} - 23281436 q^{13} + 349442850 q^{14} - 671072257 q^{16} + 195706222 q^{19} - 2822935854 q^{20} - 204235521 q^{22}+ \cdots + 7621755375583 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −209.734 − 121.090i 0 21133.6 + 36604.4i 69749.9 − 40270.1i 0 −376400. + 651944.i − 6.26837e6i 0 −1.95052e7
8.2 −195.306 − 112.760i 0 17237.5 + 29856.3i −103308. + 59645.1i 0 163246. − 282751.i − 4.07989e6i 0 2.69023e7
8.3 −129.548 − 74.7948i 0 2996.54 + 5190.15i −31950.1 + 18446.4i 0 153772. − 266342.i 1.55438e6i 0 5.51879e6
8.4 −104.562 − 60.3690i 0 −903.169 − 1564.33i 116056. − 67005.0i 0 −391422. + 677963.i 2.19626e6i 0 −1.61801e7
8.5 −104.538 − 60.3552i 0 −906.505 − 1570.11i 10010.1 − 5779.31i 0 308433. − 534222.i 2.19657e6i 0 −1.39525e6
8.6 −25.4810 − 14.7114i 0 −7759.15 − 13439.2i −68095.1 + 39314.7i 0 −576494. + 998517.i 938657.i 0 2.31351e6
8.7 36.6279 + 21.1471i 0 −7297.60 − 12639.8i −53737.7 + 31025.5i 0 −26704.6 + 46253.7i − 1.31024e6i 0 −2.62440e6
8.8 40.5755 + 23.4263i 0 −7094.42 − 12287.9i 113204. − 65358.5i 0 742609. − 1.28624e6i − 1.43242e6i 0 6.12443e6
8.9 47.7787 + 27.5850i 0 −6670.13 − 11553.0i −20922.0 + 12079.3i 0 105744. − 183153.i − 1.63989e6i 0 −1.33284e6
8.10 122.622 + 70.7960i 0 1832.14 + 3173.36i 101737. − 58738.1i 0 −525106. + 909510.i − 1.80101e6i 0 1.66337e7
8.11 164.198 + 94.7998i 0 9782.02 + 16943.0i −93735.9 + 54118.4i 0 584546. − 1.01246e6i 602934.i 0 −2.05217e7
8.12 165.573 + 95.5937i 0 10084.3 + 17466.5i −5072.07 + 2928.36i 0 −596946. + 1.03394e6i 723570.i 0 −1.11973e6
8.13 193.294 + 111.598i 0 16716.4 + 28953.6i 20060.8 − 11582.1i 0 361557. − 626235.i 3.80522e6i 0 5.17016e6
17.1 −209.734 + 121.090i 0 21133.6 − 36604.4i 69749.9 + 40270.1i 0 −376400. − 651944.i 6.26837e6i 0 −1.95052e7
17.2 −195.306 + 112.760i 0 17237.5 − 29856.3i −103308. − 59645.1i 0 163246. + 282751.i 4.07989e6i 0 2.69023e7
17.3 −129.548 + 74.7948i 0 2996.54 − 5190.15i −31950.1 − 18446.4i 0 153772. + 266342.i − 1.55438e6i 0 5.51879e6
17.4 −104.562 + 60.3690i 0 −903.169 + 1564.33i 116056. + 67005.0i 0 −391422. − 677963.i − 2.19626e6i 0 −1.61801e7
17.5 −104.538 + 60.3552i 0 −906.505 + 1570.11i 10010.1 + 5779.31i 0 308433. + 534222.i − 2.19657e6i 0 −1.39525e6
17.6 −25.4810 + 14.7114i 0 −7759.15 + 13439.2i −68095.1 − 39314.7i 0 −576494. − 998517.i − 938657.i 0 2.31351e6
17.7 36.6279 − 21.1471i 0 −7297.60 + 12639.8i −53737.7 − 31025.5i 0 −26704.6 − 46253.7i 1.31024e6i 0 −2.62440e6
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.13
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 27.15.d.a 26
3.b odd 2 1 9.15.d.a ✓ 26
9.c even 3 1 9.15.d.a ✓ 26
9.c even 3 1 81.15.b.a 26
9.d odd 6 1 inner 27.15.d.a 26
9.d odd 6 1 81.15.b.a 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.15.d.a ✓ 26 3.b odd 2 1
9.15.d.a ✓ 26 9.c even 3 1
27.15.d.a 26 1.a even 1 1 trivial
27.15.d.a 26 9.d odd 6 1 inner
81.15.b.a 26 9.c even 3 1
81.15.b.a 26 9.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{15}^{\mathrm{new}}(27, [\chi])\).