Properties

Label 54.3.f.a
Level $54$
Weight $3$
Character orbit 54.f
Analytic conductor $1.471$
Analytic rank $0$
Dimension $36$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [54,3,Mod(5,54)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(54, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("54.5");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 54.f (of order \(18\), degree \(6\), 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: \(1.47139342755\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 18 q^{5} + 12 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 18 q^{5} + 12 q^{6} - 12 q^{9} - 18 q^{11} - 12 q^{12} - 36 q^{14} - 18 q^{15} - 48 q^{18} - 72 q^{20} - 228 q^{21} + 36 q^{22} - 180 q^{23} + 18 q^{25} + 54 q^{27} + 144 q^{29} + 144 q^{30} - 90 q^{31} + 324 q^{33} - 72 q^{34} + 486 q^{35} + 168 q^{36} + 180 q^{38} + 102 q^{39} - 90 q^{41} + 48 q^{42} + 90 q^{43} - 378 q^{45} - 378 q^{47} - 24 q^{48} + 72 q^{49} - 54 q^{51} - 36 q^{54} - 72 q^{56} + 72 q^{57} + 252 q^{59} + 36 q^{60} - 144 q^{61} + 318 q^{63} + 144 q^{64} + 18 q^{65} - 432 q^{66} - 594 q^{67} - 180 q^{68} - 522 q^{69} - 360 q^{70} - 648 q^{71} - 192 q^{72} + 126 q^{73} - 504 q^{74} - 438 q^{75} - 72 q^{76} - 342 q^{77} - 288 q^{78} - 72 q^{79} + 324 q^{81} + 594 q^{83} + 216 q^{84} + 360 q^{85} + 540 q^{86} + 1062 q^{87} + 144 q^{88} + 648 q^{89} + 720 q^{90} - 198 q^{91} + 396 q^{92} + 462 q^{93} + 504 q^{94} + 252 q^{95} + 96 q^{96} + 702 q^{97} + 648 q^{98} + 126 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.909039 1.08335i −1.44503 + 2.62905i −0.347296 + 1.96962i 0.696976 + 1.91493i 4.16177 0.824439i 2.23222 + 12.6596i 2.44949 1.41421i −4.82380 7.59809i 1.44096 2.49581i
5.2 −0.909039 1.08335i 1.30968 2.69902i −0.347296 + 1.96962i −1.54033 4.23203i −4.11454 + 1.03467i 0.223815 + 1.26932i 2.44949 1.41421i −5.56946 7.06974i −3.18455 + 5.51580i
5.3 −0.909039 1.08335i 2.80846 + 1.05478i −0.347296 + 1.96962i 2.86430 + 7.86960i −1.41030 4.00138i −1.95208 11.0708i 2.44949 1.41421i 6.77487 + 5.92462i 5.92177 10.2568i
5.4 0.909039 + 1.08335i −2.95590 + 0.512490i −0.347296 + 1.96962i 2.71293 + 7.45370i −3.24224 2.73640i 0.0787775 + 0.446769i −2.44949 + 1.41421i 8.47471 3.02974i −5.60882 + 9.71475i
5.5 0.909039 + 1.08335i 1.54884 + 2.56926i −0.347296 + 1.96962i −1.07911 2.96482i −1.37545 + 4.01349i −0.250410 1.42015i −2.44949 + 1.41421i −4.20220 + 7.95874i 2.23099 3.86419i
5.6 0.909039 + 1.08335i 2.14542 2.09694i −0.347296 + 1.96962i 0.387127 + 1.06362i 4.22200 + 0.418040i −0.332318 1.88467i −2.44949 + 1.41421i 0.205664 8.99765i −0.800362 + 1.38627i
11.1 −0.909039 + 1.08335i −1.44503 2.62905i −0.347296 1.96962i 0.696976 1.91493i 4.16177 + 0.824439i 2.23222 12.6596i 2.44949 + 1.41421i −4.82380 + 7.59809i 1.44096 + 2.49581i
11.2 −0.909039 + 1.08335i 1.30968 + 2.69902i −0.347296 1.96962i −1.54033 + 4.23203i −4.11454 1.03467i 0.223815 1.26932i 2.44949 + 1.41421i −5.56946 + 7.06974i −3.18455 5.51580i
11.3 −0.909039 + 1.08335i 2.80846 1.05478i −0.347296 1.96962i 2.86430 7.86960i −1.41030 + 4.00138i −1.95208 + 11.0708i 2.44949 + 1.41421i 6.77487 5.92462i 5.92177 + 10.2568i
11.4 0.909039 1.08335i −2.95590 0.512490i −0.347296 1.96962i 2.71293 7.45370i −3.24224 + 2.73640i 0.0787775 0.446769i −2.44949 1.41421i 8.47471 + 3.02974i −5.60882 9.71475i
11.5 0.909039 1.08335i 1.54884 2.56926i −0.347296 1.96962i −1.07911 + 2.96482i −1.37545 4.01349i −0.250410 + 1.42015i −2.44949 1.41421i −4.20220 7.95874i 2.23099 + 3.86419i
11.6 0.909039 1.08335i 2.14542 + 2.09694i −0.347296 1.96962i 0.387127 1.06362i 4.22200 0.418040i −0.332318 + 1.88467i −2.44949 1.41421i 0.205664 + 8.99765i −0.800362 1.38627i
23.1 −0.483690 + 1.32893i −2.93655 0.613727i −1.53209 1.28558i −7.71206 1.35984i 2.23598 3.60561i −0.690206 + 0.579152i 2.44949 1.41421i 8.24668 + 3.60448i 5.53738 9.59102i
23.2 −0.483690 + 1.32893i −0.570511 + 2.94525i −1.53209 1.28558i 3.98669 + 0.702961i −3.63807 2.18276i −10.1193 + 8.49107i 2.44949 1.41421i −8.34903 3.36060i −2.86250 + 4.95800i
23.3 −0.483690 + 1.32893i −0.391734 2.97431i −1.53209 1.28558i 7.52350 + 1.32660i 4.14212 + 0.918059i 4.40811 3.69885i 2.44949 1.41421i −8.69309 + 2.33028i −5.40199 + 9.35652i
23.4 0.483690 1.32893i −1.47299 2.61348i −1.53209 1.28558i −2.99623 0.528316i −4.18560 + 0.693383i 6.30359 5.28934i −2.44949 + 1.41421i −4.66059 + 7.69928i −2.15134 + 3.72623i
23.5 0.483690 1.32893i 0.320982 + 2.98278i −1.53209 1.28558i 5.90332 + 1.04091i 4.11915 + 1.01618i 5.59840 4.69762i −2.44949 + 1.41421i −8.79394 + 1.91484i 4.23867 7.34160i
23.6 0.483690 1.32893i 2.82413 1.01208i −1.53209 1.28558i 0.891042 + 0.157115i 0.0210163 4.24259i −5.50064 + 4.61559i −2.44949 + 1.41421i 6.95137 5.71650i 0.639781 1.10813i
29.1 −1.39273 0.245576i −2.97067 0.418457i 1.87939 + 0.684040i 5.54906 + 6.61311i 4.03458 + 1.31232i 7.83131 2.85036i −2.44949 1.41421i 8.64979 + 2.48620i −6.10431 10.5730i
29.2 −1.39273 0.245576i −2.10518 + 2.13733i 1.87939 + 0.684040i −5.37469 6.40531i 3.45683 2.45974i −8.61655 + 3.13617i −2.44949 1.41421i −0.136395 8.99897i 5.91250 + 10.2407i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.6
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 54.3.f.a 36
3.b odd 2 1 162.3.f.a 36
4.b odd 2 1 432.3.bc.c 36
27.e even 9 1 162.3.f.a 36
27.e even 9 1 1458.3.b.c 36
27.f odd 18 1 inner 54.3.f.a 36
27.f odd 18 1 1458.3.b.c 36
108.l even 18 1 432.3.bc.c 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.3.f.a 36 1.a even 1 1 trivial
54.3.f.a 36 27.f odd 18 1 inner
162.3.f.a 36 3.b odd 2 1
162.3.f.a 36 27.e even 9 1
432.3.bc.c 36 4.b odd 2 1
432.3.bc.c 36 108.l even 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 \(S_{3}^{\mathrm{new}}(54, [\chi])\).