↑

Properties

Label 135.5.h.a
Level $135$
Weight $5$
Character orbit 135.h
Analytic conductor $13.955$
Analytic rank $0$
Dimension $44$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [135,5,Mod(44,135)] 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("135.44"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(135, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 3])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 135.h (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: \(13.9549450163\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 45)
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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 162 q^{4} - 6 q^{5} + 28 q^{10} - 228 q^{11} - 282 q^{14} - 1058 q^{16} - 8 q^{19} + 2196 q^{20} - 148 q^{25} - 2370 q^{29} - 1112 q^{31} - 436 q^{34} - 850 q^{40} - 1830 q^{41} - 5668 q^{46} + 5396 q^{49}+ \cdots - 58746 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1 −3.75365 − 6.50151i 0 −20.1797 + 34.9523i −22.9075 + 10.0124i 0 60.4872 − 34.9223i 182.874 0 151.082 + 111.350i
44.2 −3.64116 − 6.30667i 0 −18.5161 + 32.0708i −0.540999 − 24.9941i 0 −38.4567 + 22.2030i 153.163 0 −155.660 + 94.4195i
44.3 −3.12350 − 5.41005i 0 −11.5125 + 19.9402i 24.9883 − 0.765363i 0 10.1314 − 5.84936i 43.8846 0 −82.1915 − 132.797i
44.4 −2.87361 − 4.97724i 0 −8.51525 + 14.7488i −24.3310 + 5.74467i 0 −41.4151 + 23.9110i 5.92246 0 98.5104 + 104.593i
44.5 −2.61690 − 4.53261i 0 −5.69638 + 9.86641i 23.1420 + 9.45767i 0 −50.0684 + 28.9070i −24.1135 0 −17.6925 − 129.644i
44.6 −2.30921 − 3.99967i 0 −2.66491 + 4.61577i −1.07184 + 24.9770i 0 45.6166 − 26.3368i −49.2794 0 102.375 − 53.3902i
44.7 −1.64811 − 2.85461i 0 2.56747 − 4.44699i 9.73566 − 23.0264i 0 64.7475 − 37.3820i −69.6654 0 −81.7769 + 10.1586i
44.8 −1.59606 − 2.76446i 0 2.90517 − 5.03190i −6.24274 − 24.2080i 0 −21.0893 + 12.1759i −69.6213 0 −56.9583 + 55.8953i
44.9 −1.02189 − 1.76997i 0 5.91148 − 10.2390i −24.7463 − 3.55264i 0 2.09732 − 1.21089i −56.8640 0 19.0000 + 47.4305i
44.10 −0.547257 − 0.947877i 0 7.40102 − 12.8189i −6.73050 + 24.0770i 0 16.9207 − 9.76914i −33.7133 0 26.5053 − 6.79660i
44.11 −0.316502 − 0.548197i 0 7.79965 − 13.5094i 18.0085 + 17.3406i 0 −73.2178 + 42.2723i −20.0025 0 3.80633 − 15.3605i
44.12 0.316502 + 0.548197i 0 7.79965 − 13.5094i 24.0216 + 6.92551i 0 73.2178 − 42.2723i 20.0025 0 3.80633 + 15.3605i
44.13 0.547257 + 0.947877i 0 7.40102 − 12.8189i 17.4860 − 17.8673i 0 −16.9207 + 9.76914i 33.7133 0 26.5053 + 6.79660i
44.14 1.02189 + 1.76997i 0 5.91148 − 10.2390i −15.4498 − 19.6546i 0 −2.09732 + 1.21089i 56.8640 0 19.0000 − 47.4305i
44.15 1.59606 + 2.76446i 0 2.90517 − 5.03190i −24.0861 + 6.69764i 0 21.0893 − 12.1759i 69.6213 0 −56.9583 − 55.8953i
44.16 1.64811 + 2.85461i 0 2.56747 − 4.44699i −15.0736 + 19.9446i 0 −64.7475 + 37.3820i 69.6654 0 −81.7769 − 10.1586i
44.17 2.30921 + 3.99967i 0 −2.66491 + 4.61577i 21.0948 − 13.4168i 0 −45.6166 + 26.3368i 49.2794 0 102.375 + 53.3902i
44.18 2.61690 + 4.53261i 0 −5.69638 + 9.86641i 19.7616 + 15.3127i 0 50.0684 − 28.9070i 24.1135 0 −17.6925 + 129.644i
44.19 2.87361 + 4.97724i 0 −8.51525 + 14.7488i −7.19048 − 23.9436i 0 41.4151 − 23.9110i −5.92246 0 98.5104 − 104.593i
44.20 3.12350 + 5.41005i 0 −11.5125 + 19.9402i 11.8313 + 22.0232i 0 −10.1314 + 5.84936i −43.8846 0 −82.1915 + 132.797i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 44.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.d odd 6 1 inner
45.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.5.h.a 44
3.b odd 2 1 45.5.h.a ✓ 44
5.b even 2 1 inner 135.5.h.a 44
9.c even 3 1 45.5.h.a ✓ 44
9.c even 3 1 405.5.d.a 44
9.d odd 6 1 inner 135.5.h.a 44
9.d odd 6 1 405.5.d.a 44
15.d odd 2 1 45.5.h.a ✓ 44
45.h odd 6 1 inner 135.5.h.a 44
45.h odd 6 1 405.5.d.a 44
45.j even 6 1 45.5.h.a ✓ 44
45.j even 6 1 405.5.d.a 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.5.h.a ✓ 44 3.b odd 2 1
45.5.h.a ✓ 44 9.c even 3 1
45.5.h.a ✓ 44 15.d odd 2 1
45.5.h.a ✓ 44 45.j even 6 1
135.5.h.a 44 1.a even 1 1 trivial
135.5.h.a 44 5.b even 2 1 inner
135.5.h.a 44 9.d odd 6 1 inner
135.5.h.a 44 45.h odd 6 1 inner
405.5.d.a 44 9.c even 3 1
405.5.d.a 44 9.d odd 6 1
405.5.d.a 44 45.h odd 6 1
405.5.d.a 44 45.j even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(135, [\chi])\).