Properties

Label 405.2.p.a
Level $405$
Weight $2$
Character orbit 405.p
Analytic conductor $3.234$
Analytic rank $0$
Dimension $96$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [405,2,Mod(19,405)] 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("405.19"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([16, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.p (of order \(18\), degree \(6\), 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: \(3.23394128186\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 12 q^{4} + 9 q^{5} - 3 q^{10} + 6 q^{11} + 18 q^{14} - 24 q^{16} - 6 q^{19} + 57 q^{20} + 3 q^{25} - 48 q^{26} + 30 q^{29} - 30 q^{31} - 24 q^{34} + 12 q^{35} - 9 q^{40} + 12 q^{41} - 78 q^{44} - 6 q^{46}+ \cdots - 87 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.911584 − 2.50456i 0 −3.90973 + 3.28066i 1.21121 − 1.87962i 0 1.98995 − 2.37153i 7.16422 + 4.13626i 0 −5.81174 − 1.32011i
19.2 −0.846914 − 2.32688i 0 −3.16501 + 2.65576i −2.17927 + 0.500797i 0 1.41496 − 1.68629i 4.57119 + 2.63918i 0 3.01095 + 4.64676i
19.3 −0.691324 − 1.89940i 0 −1.59769 + 1.34062i −0.648657 − 2.13992i 0 −2.82126 + 3.36225i 0.149906 + 0.0865484i 0 −3.61612 + 2.71143i
19.4 −0.613751 − 1.68627i 0 −0.934715 + 0.784319i 1.93472 + 1.12109i 0 −0.337711 + 0.402468i −1.21189 − 0.699685i 0 0.703028 − 3.95053i
19.5 −0.557159 − 1.53078i 0 −0.500777 + 0.420202i −0.326837 + 2.21205i 0 −1.77805 + 2.11900i −1.89930 − 1.09656i 0 3.56827 − 0.732149i
19.6 −0.289634 − 0.795763i 0 0.982738 − 0.824615i 2.23533 + 0.0572668i 0 3.00798 − 3.58478i −2.40759 − 1.39002i 0 −0.601858 − 1.79538i
19.7 −0.228876 − 0.628833i 0 1.18904 − 0.997725i −1.63133 + 1.52931i 0 0.158590 − 0.189000i −2.05862 − 1.18854i 0 1.33505 + 0.675809i
19.8 −0.141924 − 0.389932i 0 1.40018 − 1.17489i 0.459147 − 2.18842i 0 0.408725 − 0.487099i −1.37557 − 0.794189i 0 −0.918499 + 0.131552i
19.9 0.141924 + 0.389932i 0 1.40018 − 1.17489i −1.17994 − 1.89940i 0 −0.408725 + 0.487099i 1.37557 + 0.794189i 0 0.573177 − 0.729668i
19.10 0.228876 + 0.628833i 0 1.18904 − 0.997725i 2.05600 + 0.879131i 0 −0.158590 + 0.189000i 2.05862 + 1.18854i 0 −0.0822570 + 1.49409i
19.11 0.289634 + 0.795763i 0 0.982738 − 0.824615i −2.08094 + 0.818343i 0 −3.00798 + 3.58478i 2.40759 + 1.39002i 0 −1.25392 − 1.41892i
19.12 0.557159 + 1.53078i 0 −0.500777 + 0.420202i 1.06369 + 1.96687i 0 1.77805 − 2.11900i 1.89930 + 1.09656i 0 −2.41819 + 2.72414i
19.13 0.613751 + 1.68627i 0 −0.934715 + 0.784319i −1.43461 + 1.71520i 0 0.337711 − 0.402468i 1.21189 + 0.699685i 0 −3.77277 − 1.36642i
19.14 0.691324 + 1.89940i 0 −1.59769 + 1.34062i −0.122356 − 2.23272i 0 2.82126 − 3.36225i −0.149906 − 0.0865484i 0 4.15623 − 1.77593i
19.15 0.846914 + 2.32688i 0 −3.16501 + 2.65576i 2.21912 − 0.274758i 0 −1.41496 + 1.68629i −4.57119 − 2.63918i 0 2.51874 + 4.93093i
19.16 0.911584 + 2.50456i 0 −3.90973 + 3.28066i −1.78103 − 1.35201i 0 −1.98995 + 2.37153i −7.16422 − 4.13626i 0 1.76262 − 5.69317i
64.1 −0.911584 + 2.50456i 0 −3.90973 − 3.28066i 1.21121 + 1.87962i 0 1.98995 + 2.37153i 7.16422 − 4.13626i 0 −5.81174 + 1.32011i
64.2 −0.846914 + 2.32688i 0 −3.16501 − 2.65576i −2.17927 − 0.500797i 0 1.41496 + 1.68629i 4.57119 − 2.63918i 0 3.01095 − 4.64676i
64.3 −0.691324 + 1.89940i 0 −1.59769 − 1.34062i −0.648657 + 2.13992i 0 −2.82126 − 3.36225i 0.149906 − 0.0865484i 0 −3.61612 − 2.71143i
64.4 −0.613751 + 1.68627i 0 −0.934715 − 0.784319i 1.93472 − 1.12109i 0 −0.337711 − 0.402468i −1.21189 + 0.699685i 0 0.703028 + 3.95053i
See all 96 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 19.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
27.e even 9 1 inner
135.p even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.2.p.a 96
3.b odd 2 1 135.2.p.a ✓ 96
5.b even 2 1 inner 405.2.p.a 96
15.d odd 2 1 135.2.p.a ✓ 96
15.e even 4 2 675.2.l.h 96
27.e even 9 1 inner 405.2.p.a 96
27.f odd 18 1 135.2.p.a ✓ 96
135.n odd 18 1 135.2.p.a ✓ 96
135.p even 18 1 inner 405.2.p.a 96
135.q even 36 2 675.2.l.h 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.p.a ✓ 96 3.b odd 2 1
135.2.p.a ✓ 96 15.d odd 2 1
135.2.p.a ✓ 96 27.f odd 18 1
135.2.p.a ✓ 96 135.n odd 18 1
405.2.p.a 96 1.a even 1 1 trivial
405.2.p.a 96 5.b even 2 1 inner
405.2.p.a 96 27.e even 9 1 inner
405.2.p.a 96 135.p even 18 1 inner
675.2.l.h 96 15.e even 4 2
675.2.l.h 96 135.q even 36 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(405, [\chi])\).