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Properties

Label 162.3.f.a
Level $162$
Weight $3$
Character orbit 162.f
Analytic conductor $4.414$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,3,Mod(17,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 162.f (of order \(18\), degree \(6\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.41418028264\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 54)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 18 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 18 q^{5} + 18 q^{11} + 36 q^{14} + 72 q^{20} + 36 q^{22} + 180 q^{23} + 18 q^{25} - 144 q^{29} - 90 q^{31} - 72 q^{34} - 486 q^{35} - 180 q^{38} + 90 q^{41} + 90 q^{43} + 378 q^{47} + 72 q^{49} + 72 q^{56} - 252 q^{59} - 144 q^{61} + 144 q^{64} - 18 q^{65} - 594 q^{67} + 180 q^{68} - 360 q^{70} + 648 q^{71} + 126 q^{73} + 504 q^{74} - 72 q^{76} + 342 q^{77} - 72 q^{79} - 594 q^{83} + 360 q^{85} - 540 q^{86} + 144 q^{88} - 648 q^{89} - 198 q^{91} - 396 q^{92} + 504 q^{94} - 252 q^{95} + 702 q^{97} - 648 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −0.483690 + 1.32893i 0 −1.53209 − 1.28558i −5.90332 − 1.04091i 0 5.59840 − 4.69762i 2.44949 − 1.41421i 0 4.23867 − 7.34160i
17.2 −0.483690 + 1.32893i 0 −1.53209 − 1.28558i −0.891042 − 0.157115i 0 −5.50064 + 4.61559i 2.44949 − 1.41421i 0 0.639781 − 1.10813i
17.3 −0.483690 + 1.32893i 0 −1.53209 − 1.28558i 2.99623 + 0.528316i 0 6.30359 − 5.28934i 2.44949 − 1.41421i 0 −2.15134 + 3.72623i
17.4 0.483690 − 1.32893i 0 −1.53209 − 1.28558i −7.52350 − 1.32660i 0 4.40811 − 3.69885i −2.44949 + 1.41421i 0 −5.40199 + 9.35652i
17.5 0.483690 − 1.32893i 0 −1.53209 − 1.28558i −3.98669 − 0.702961i 0 −10.1193 + 8.49107i −2.44949 + 1.41421i 0 −2.86250 + 4.95800i
17.6 0.483690 − 1.32893i 0 −1.53209 − 1.28558i 7.71206 + 1.35984i 0 −0.690206 + 0.579152i −2.44949 + 1.41421i 0 5.53738 − 9.59102i
35.1 −0.909039 + 1.08335i 0 −0.347296 − 1.96962i −2.71293 + 7.45370i 0 0.0787775 − 0.446769i 2.44949 + 1.41421i 0 −5.60882 − 9.71475i
35.2 −0.909039 + 1.08335i 0 −0.347296 − 1.96962i −0.387127 + 1.06362i 0 −0.332318 + 1.88467i 2.44949 + 1.41421i 0 −0.800362 − 1.38627i
35.3 −0.909039 + 1.08335i 0 −0.347296 − 1.96962i 1.07911 − 2.96482i 0 −0.250410 + 1.42015i 2.44949 + 1.41421i 0 2.23099 + 3.86419i
35.4 0.909039 − 1.08335i 0 −0.347296 − 1.96962i −2.86430 + 7.86960i 0 −1.95208 + 11.0708i −2.44949 − 1.41421i 0 5.92177 + 10.2568i
35.5 0.909039 − 1.08335i 0 −0.347296 − 1.96962i −0.696976 + 1.91493i 0 2.23222 − 12.6596i −2.44949 − 1.41421i 0 1.44096 + 2.49581i
35.6 0.909039 − 1.08335i 0 −0.347296 − 1.96962i 1.54033 − 4.23203i 0 0.223815 − 1.26932i −2.44949 − 1.41421i 0 −3.18455 − 5.51580i
71.1 −1.39273 + 0.245576i 0 1.87939 − 0.684040i −1.85971 + 2.21631i 0 2.17427 + 0.791369i −2.44949 + 1.41421i 0 2.04580 − 3.54342i
71.2 −1.39273 + 0.245576i 0 1.87939 − 0.684040i −0.379008 + 0.451684i 0 2.96297 + 1.07843i −2.44949 + 1.41421i 0 0.416932 − 0.722148i
71.3 −1.39273 + 0.245576i 0 1.87939 − 0.684040i 3.55779 − 4.24001i 0 −10.2625 − 3.73526i −2.44949 + 1.41421i 0 −3.91380 + 6.77890i
71.4 1.39273 − 0.245576i 0 1.87939 − 0.684040i −5.54906 + 6.61311i 0 7.83131 + 2.85036i 2.44949 − 1.41421i 0 −6.10431 + 10.5730i
71.5 1.39273 − 0.245576i 0 1.87939 − 0.684040i 1.49345 − 1.77982i 0 5.91054 + 2.15126i 2.44949 − 1.41421i 0 1.64289 − 2.84556i
71.6 1.39273 − 0.245576i 0 1.87939 − 0.684040i 5.37469 − 6.40531i 0 −8.61655 − 3.13617i 2.44949 − 1.41421i 0 5.91250 − 10.2407i
89.1 −1.39273 − 0.245576i 0 1.87939 + 0.684040i −1.85971 − 2.21631i 0 2.17427 − 0.791369i −2.44949 − 1.41421i 0 2.04580 + 3.54342i
89.2 −1.39273 − 0.245576i 0 1.87939 + 0.684040i −0.379008 − 0.451684i 0 2.96297 − 1.07843i −2.44949 − 1.41421i 0 0.416932 + 0.722148i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.3.f.a 36
3.b odd 2 1 54.3.f.a ✓ 36
12.b even 2 1 432.3.bc.c 36
27.e even 9 1 54.3.f.a ✓ 36
27.e even 9 1 1458.3.b.c 36
27.f odd 18 1 inner 162.3.f.a 36
27.f odd 18 1 1458.3.b.c 36
108.j odd 18 1 432.3.bc.c 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.3.f.a ✓ 36 3.b odd 2 1
54.3.f.a ✓ 36 27.e even 9 1
162.3.f.a 36 1.a even 1 1 trivial
162.3.f.a 36 27.f odd 18 1 inner
432.3.bc.c 36 12.b even 2 1
432.3.bc.c 36 108.j odd 18 1
1458.3.b.c 36 27.e even 9 1
1458.3.b.c 36 27.f odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(162, [\chi])\).