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Properties

Label 135.5.i.a
Level $135$
Weight $5$
Character orbit 135.i
Analytic conductor $13.955$
Analytic rank $0$
Dimension $32$
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] = [135,5,Mod(71,135)] 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("135.71"); 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, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 135.i (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: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 128 q^{4} - 26 q^{7} + 738 q^{11} + 10 q^{13} - 1998 q^{14} - 1024 q^{16} + 508 q^{19} + 672 q^{22} - 1998 q^{23} + 2000 q^{25} - 1664 q^{28} + 270 q^{29} - 1472 q^{31} + 6048 q^{32} - 594 q^{34}+ \cdots + 23266 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1 −6.87716 + 3.97053i 0 23.5302 − 40.7555i −9.68246 − 5.59017i 0 10.1236 + 17.5346i 246.652i 0 88.7837
71.2 −5.93733 + 3.42792i 0 15.5013 − 26.8490i 9.68246 + 5.59017i 0 12.5523 + 21.7412i 102.855i 0 −76.6506
71.3 −5.22878 + 3.01884i 0 10.2268 − 17.7133i 9.68246 + 5.59017i 0 −8.13139 − 14.0840i 26.8889i 0 −67.5033
71.4 −4.78725 + 2.76392i 0 7.27848 − 12.6067i −9.68246 − 5.59017i 0 −0.820328 − 1.42085i − 7.97692i 0 61.8031
71.5 −3.10572 + 1.79309i 0 −1.56968 + 2.71877i 9.68246 + 5.59017i 0 45.9054 + 79.5105i − 68.6371i 0 −40.0946
71.6 −2.59623 + 1.49894i 0 −3.50638 + 6.07323i −9.68246 − 5.59017i 0 −1.63719 − 2.83570i − 68.9893i 0 33.5172
71.7 −0.904609 + 0.522276i 0 −7.45446 + 12.9115i 9.68246 + 5.59017i 0 −36.4983 − 63.2168i − 32.2860i 0 −11.6784
71.8 −0.817210 + 0.471817i 0 −7.55478 + 13.0853i −9.68246 − 5.59017i 0 31.6912 + 54.8908i − 29.3560i 0 10.5501
71.9 1.06605 − 0.615486i 0 −7.24235 + 12.5441i 9.68246 + 5.59017i 0 −21.1314 − 36.6007i 37.5258i 0 13.7627
71.10 2.15054 − 1.24161i 0 −4.91680 + 8.51614i −9.68246 − 5.59017i 0 −15.7848 − 27.3401i 64.1506i 0 −27.7633
71.11 3.23643 − 1.86855i 0 −1.01703 + 1.76154i 9.68246 + 5.59017i 0 8.02923 + 13.9070i 67.3951i 0 41.7821
71.12 3.23824 − 1.86960i 0 −1.00919 + 1.74797i −9.68246 − 5.59017i 0 −19.7075 − 34.1344i 67.3743i 0 −41.8055
71.13 3.37336 − 1.94761i 0 −0.413632 + 0.716431i −9.68246 − 5.59017i 0 15.5174 + 26.8770i 65.5459i 0 −43.5499
71.14 5.11159 − 2.95118i 0 9.41890 − 16.3140i 9.68246 + 5.59017i 0 36.1606 + 62.6320i − 16.7497i 0 65.9903
71.15 5.76237 − 3.32690i 0 14.1366 − 24.4853i 9.68246 + 5.59017i 0 −43.3865 − 75.1475i − 81.6633i 0 74.3918
71.16 6.31571 − 3.64638i 0 18.5921 − 32.2025i −9.68246 − 5.59017i 0 −25.8824 − 44.8297i − 154.491i 0 −81.5354
116.1 −6.87716 − 3.97053i 0 23.5302 + 40.7555i −9.68246 + 5.59017i 0 10.1236 − 17.5346i − 246.652i 0 88.7837
116.2 −5.93733 − 3.42792i 0 15.5013 + 26.8490i 9.68246 − 5.59017i 0 12.5523 − 21.7412i − 102.855i 0 −76.6506
116.3 −5.22878 − 3.01884i 0 10.2268 + 17.7133i 9.68246 − 5.59017i 0 −8.13139 + 14.0840i − 26.8889i 0 −67.5033
116.4 −4.78725 − 2.76392i 0 7.27848 + 12.6067i −9.68246 + 5.59017i 0 −0.820328 + 1.42085i 7.97692i 0 61.8031
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.16
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 135.5.i.a 32
3.b odd 2 1 45.5.i.a ✓ 32
9.c even 3 1 45.5.i.a ✓ 32
9.c even 3 1 405.5.c.b 32
9.d odd 6 1 inner 135.5.i.a 32
9.d odd 6 1 405.5.c.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.5.i.a ✓ 32 3.b odd 2 1
45.5.i.a ✓ 32 9.c even 3 1
135.5.i.a 32 1.a even 1 1 trivial
135.5.i.a 32 9.d odd 6 1 inner
405.5.c.b 32 9.c even 3 1
405.5.c.b 32 9.d odd 6 1

Hecke kernels

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