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Properties

Label 135.3.n.a
Level $135$
Weight $3$
Character orbit 135.n
Analytic conductor $3.678$
Analytic rank $0$
Dimension $204$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [135,3,Mod(14,135)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(135, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([17, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("135.14");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 135.n (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: \(3.67848356886\)
Analytic rank: \(0\)
Dimension: \(204\)
Relative dimension: \(34\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 204 q - 12 q^{4} + 3 q^{5} - 24 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 204 q - 12 q^{4} + 3 q^{5} - 24 q^{6} - 18 q^{9} - 3 q^{10} + 6 q^{11} - 48 q^{14} - 3 q^{15} + 12 q^{16} - 6 q^{19} + 63 q^{20} - 192 q^{21} + 42 q^{24} - 15 q^{25} + 96 q^{29} - 177 q^{30} - 102 q^{31} + 12 q^{34} - 252 q^{35} + 324 q^{36} - 258 q^{39} + 117 q^{40} + 96 q^{41} - 666 q^{44} - 279 q^{45} - 6 q^{46} + 60 q^{49} + 48 q^{50} + 270 q^{51} + 432 q^{54} - 12 q^{55} + 294 q^{56} + 510 q^{59} + 390 q^{60} + 132 q^{61} - 486 q^{64} + 147 q^{65} - 186 q^{66} - 84 q^{69} - 141 q^{70} - 18 q^{71} - 954 q^{74} - 285 q^{75} + 84 q^{76} - 48 q^{79} - 1026 q^{81} + 198 q^{84} + 69 q^{85} - 1506 q^{86} + 792 q^{89} - 180 q^{90} - 6 q^{91} + 492 q^{94} - 543 q^{95} + 654 q^{96} + 792 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −0.679314 − 3.85258i 1.39274 − 2.65712i −10.6221 + 3.86614i 0.627081 − 4.96052i −11.1829 − 3.56063i 1.72996 − 4.75304i 14.2863 + 24.7447i −5.12055 − 7.40135i −19.5368 + 0.953869i
14.2 −0.639775 − 3.62834i 1.24111 + 2.73123i −8.99679 + 3.27456i 4.64119 + 1.85994i 9.11582 − 6.25055i −3.70892 + 10.1902i 10.2685 + 17.7856i −5.91929 + 6.77953i 3.77917 − 18.0298i
14.3 −0.608257 − 3.44960i −2.64168 + 1.42181i −7.77098 + 2.82841i −2.88963 + 4.08044i 6.51148 + 8.24791i 2.09618 − 5.75921i 7.47800 + 12.9523i 4.95693 − 7.51191i 15.8335 + 7.48612i
14.4 −0.551287 − 3.12650i 1.55448 + 2.56585i −5.71233 + 2.07912i −4.23469 − 2.65846i 7.16517 − 6.27462i 3.19764 − 8.78543i 3.30004 + 5.71584i −4.16716 + 7.97714i −5.97715 + 14.7053i
14.5 −0.546138 − 3.09730i −0.936343 − 2.85013i −5.53626 + 2.01503i −1.29656 + 4.82897i −8.31636 + 4.45671i −2.34454 + 6.44158i 2.97457 + 5.15210i −7.24652 + 5.33741i 15.6649 + 1.37857i
14.6 −0.516357 − 2.92841i −2.72971 − 1.24446i −4.55017 + 1.65613i −2.42693 − 4.37150i −2.23477 + 8.63629i −2.19904 + 6.04183i 1.25216 + 2.16881i 5.90266 + 6.79402i −11.5484 + 9.36429i
14.7 −0.492910 − 2.79543i 2.73613 − 1.23029i −3.81270 + 1.38771i 2.98736 + 4.00945i −4.78784 − 7.04223i 1.44881 − 3.98057i 0.0814486 + 0.141073i 5.97279 − 6.73244i 9.73565 − 10.3272i
14.8 −0.460882 − 2.61379i −2.18216 + 2.05869i −2.86072 + 1.04122i 3.88116 − 3.15223i 6.38670 + 4.75489i 0.805322 − 2.21260i −1.26824 − 2.19666i 0.523606 − 8.98476i −10.0280 − 8.69174i
14.9 −0.350376 − 1.98708i −1.37848 − 2.66454i −0.0669610 + 0.0243718i 4.99959 − 0.0637194i −4.81168 + 3.67275i 3.13621 − 8.61666i −3.96358 − 6.86512i −5.19958 + 7.34604i −1.87835 − 9.91228i
14.10 −0.345749 − 1.96084i 2.98719 + 0.276957i 0.0334196 − 0.0121637i 1.63766 − 4.72420i −0.489748 − 5.95316i −1.87604 + 5.15436i −4.01758 − 6.95866i 8.84659 + 1.65465i −9.82962 − 1.57780i
14.11 −0.288230 − 1.63463i −0.482171 + 2.96100i 1.16982 − 0.425780i −4.92583 + 0.857995i 4.97912 − 0.0652755i −4.46162 + 12.2582i −4.35287 − 7.53940i −8.53502 − 2.85542i 2.82228 + 7.80464i
14.12 −0.268479 − 1.52262i 1.45114 − 2.62568i 1.51247 − 0.550495i −4.97898 − 0.457943i −4.38752 − 1.50460i 0.141471 − 0.388687i −4.33649 − 7.51102i −4.78837 − 7.62047i 0.639481 + 7.70406i
14.13 −0.251769 − 1.42785i 2.23580 + 2.00030i 1.78340 − 0.649105i −0.659178 + 4.95636i 2.29323 − 3.69600i 2.31785 − 6.36824i −4.27558 − 7.40553i 0.997584 + 8.94454i 7.24290 − 0.306648i
14.14 −0.182248 − 1.03358i −2.99855 + 0.0933385i 2.72369 − 0.991344i 3.73342 + 3.32589i 0.642953 + 3.08223i −2.98175 + 8.19229i −3.62008 − 6.27015i 8.98258 − 0.559760i 2.75717 − 4.46493i
14.15 −0.106122 − 0.601847i −2.94655 − 0.563801i 3.40781 − 1.24034i −3.49169 + 3.57884i −0.0266292 + 1.83320i 2.94775 − 8.09887i −2.33040 − 4.03637i 8.36426 + 3.32253i 2.52446 + 1.72167i
14.16 −0.0311484 − 0.176651i −1.88685 + 2.33234i 3.72854 − 1.35708i −3.02537 − 3.98085i 0.470783 + 0.260666i 2.38238 − 6.54554i −0.714620 − 1.23776i −1.87958 − 8.80154i −0.608987 + 0.658432i
14.17 −0.0283722 − 0.160907i 0.950057 + 2.84559i 3.73368 − 1.35895i 4.77107 − 1.49564i 0.430920 − 0.233606i 0.871180 − 2.39355i −0.651376 − 1.12822i −7.19478 + 5.40695i −0.376024 − 0.725263i
14.18 0.0283722 + 0.160907i −0.950057 − 2.84559i 3.73368 − 1.35895i 0.644429 − 4.95830i 0.430920 − 0.233606i −0.871180 + 2.39355i 0.651376 + 1.12822i −7.19478 + 5.40695i 0.816108 − 0.0369849i
14.19 0.0311484 + 0.176651i 1.88685 − 2.33234i 3.72854 − 1.35708i 4.44572 + 2.28814i 0.470783 + 0.260666i −2.38238 + 6.54554i 0.714620 + 1.23776i −1.87958 − 8.80154i −0.265726 + 0.856614i
14.20 0.106122 + 0.601847i 2.94655 + 0.563801i 3.40781 − 1.24034i −2.91814 + 4.06010i −0.0266292 + 1.83320i −2.94775 + 8.09887i 2.33040 + 4.03637i 8.36426 + 3.32253i −2.75324 − 1.32541i
See next 80 embeddings (of 204 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
27.f odd 18 1 inner
135.n odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.3.n.a ✓ 204
3.b odd 2 1 405.3.n.a 204
5.b even 2 1 inner 135.3.n.a ✓ 204
15.d odd 2 1 405.3.n.a 204
27.e even 9 1 405.3.n.a 204
27.f odd 18 1 inner 135.3.n.a ✓ 204
135.n odd 18 1 inner 135.3.n.a ✓ 204
135.p even 18 1 405.3.n.a 204
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.3.n.a ✓ 204 1.a even 1 1 trivial
135.3.n.a ✓ 204 5.b even 2 1 inner
135.3.n.a ✓ 204 27.f odd 18 1 inner
135.3.n.a ✓ 204 135.n odd 18 1 inner
405.3.n.a 204 3.b odd 2 1
405.3.n.a 204 15.d odd 2 1
405.3.n.a 204 27.e even 9 1
405.3.n.a 204 135.p even 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(135, [\chi])\).