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Properties

Label 162.4.e.b
Level $162$
Weight $4$
Character orbit 162.e
Analytic conductor $9.558$
Analytic rank $0$
Dimension $30$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [162,4,Mod(19,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([16])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.19"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 162.e (of order \(9\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.55830942093\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{9})\)
Twist minimal: no (minimal twist has level 54)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 12 q^{5} + 33 q^{7} - 120 q^{8} - 30 q^{10} + 39 q^{11} - 60 q^{13} + 66 q^{14} - 102 q^{17} - 171 q^{19} - 96 q^{20} + 78 q^{22} - 48 q^{23} - 432 q^{25} + 468 q^{26} + 336 q^{28} + 381 q^{29} - 801 q^{31}+ \cdots - 4002 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.87939 + 0.684040i 0 3.06418 − 2.57115i −3.05422 − 17.3214i 0 −19.9833 − 16.7680i −4.00000 + 6.92820i 0 17.5886 + 30.4643i
19.2 −1.87939 + 0.684040i 0 3.06418 − 2.57115i −2.39299 − 13.5713i 0 20.6081 + 17.2922i −4.00000 + 6.92820i 0 13.7807 + 23.8688i
19.3 −1.87939 + 0.684040i 0 3.06418 − 2.57115i −0.127307 − 0.721992i 0 1.18538 + 0.994655i −4.00000 + 6.92820i 0 0.733130 + 1.26982i
19.4 −1.87939 + 0.684040i 0 3.06418 − 2.57115i 2.36712 + 13.4246i 0 −10.7136 − 8.98975i −4.00000 + 6.92820i 0 −13.6317 − 23.6108i
19.5 −1.87939 + 0.684040i 0 3.06418 − 2.57115i 3.01146 + 17.0788i 0 16.7015 + 14.0142i −4.00000 + 6.92820i 0 −17.3423 − 30.0378i
37.1 1.53209 − 1.28558i 0 0.694593 − 3.93923i −13.2521 + 4.82337i 0 3.00195 + 17.0249i −4.00000 − 6.92820i 0 −14.1026 + 24.4264i
37.2 1.53209 − 1.28558i 0 0.694593 − 3.93923i −11.4993 + 4.18541i 0 3.15689 + 17.9036i −4.00000 − 6.92820i 0 −12.2373 + 21.1956i
37.3 1.53209 − 1.28558i 0 0.694593 − 3.93923i −3.40278 + 1.23851i 0 −3.51598 − 19.9401i −4.00000 − 6.92820i 0 −3.62116 + 6.27204i
37.4 1.53209 − 1.28558i 0 0.694593 − 3.93923i 7.46345 − 2.71647i 0 −2.66046 − 15.0882i −4.00000 − 6.92820i 0 7.94244 − 13.7567i
37.5 1.53209 − 1.28558i 0 0.694593 − 3.93923i 17.2977 − 6.29584i 0 6.03855 + 34.2463i −4.00000 − 6.92820i 0 18.4078 − 31.8832i
73.1 0.347296 − 1.96962i 0 −3.75877 − 1.36808i −7.73816 + 6.49309i 0 7.54967 − 2.74785i −4.00000 + 6.92820i 0 10.1015 + 17.4962i
73.2 0.347296 − 1.96962i 0 −3.75877 − 1.36808i −6.77242 + 5.68273i 0 30.7336 − 11.1861i −4.00000 + 6.92820i 0 8.84076 + 15.3127i
73.3 0.347296 − 1.96962i 0 −3.75877 − 1.36808i −0.657259 + 0.551505i 0 −20.5288 + 7.47187i −4.00000 + 6.92820i 0 0.857990 + 1.48608i
73.4 0.347296 − 1.96962i 0 −3.75877 − 1.36808i 9.84993 − 8.26508i 0 −26.8516 + 9.77320i −4.00000 + 6.92820i 0 −12.8582 − 22.2710i
73.5 0.347296 − 1.96962i 0 −3.75877 − 1.36808i 14.9069 − 12.5084i 0 11.7781 − 4.28689i −4.00000 + 6.92820i 0 −19.4596 − 33.7050i
91.1 0.347296 + 1.96962i 0 −3.75877 + 1.36808i −7.73816 − 6.49309i 0 7.54967 + 2.74785i −4.00000 − 6.92820i 0 10.1015 − 17.4962i
91.2 0.347296 + 1.96962i 0 −3.75877 + 1.36808i −6.77242 − 5.68273i 0 30.7336 + 11.1861i −4.00000 − 6.92820i 0 8.84076 − 15.3127i
91.3 0.347296 + 1.96962i 0 −3.75877 + 1.36808i −0.657259 − 0.551505i 0 −20.5288 − 7.47187i −4.00000 − 6.92820i 0 0.857990 − 1.48608i
91.4 0.347296 + 1.96962i 0 −3.75877 + 1.36808i 9.84993 + 8.26508i 0 −26.8516 − 9.77320i −4.00000 − 6.92820i 0 −12.8582 + 22.2710i
91.5 0.347296 + 1.96962i 0 −3.75877 + 1.36808i 14.9069 + 12.5084i 0 11.7781 + 4.28689i −4.00000 − 6.92820i 0 −19.4596 + 33.7050i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.5
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.4.e.b 30
3.b odd 2 1 54.4.e.b ✓ 30
27.e even 9 1 inner 162.4.e.b 30
27.e even 9 1 1458.4.a.j 15
27.f odd 18 1 54.4.e.b ✓ 30
27.f odd 18 1 1458.4.a.i 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.4.e.b ✓ 30 3.b odd 2 1
54.4.e.b ✓ 30 27.f odd 18 1
162.4.e.b 30 1.a even 1 1 trivial
162.4.e.b 30 27.e even 9 1 inner
1458.4.a.i 15 27.f odd 18 1
1458.4.a.j 15 27.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{30} - 12 T_{5}^{29} + 288 T_{5}^{28} - 1077 T_{5}^{27} - 7965 T_{5}^{26} + 1743363 T_{5}^{25} + \cdots + 54\!\cdots\!96 \) acting on \(S_{4}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display