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Properties

Label 405.3.n.a
Level $405$
Weight $3$
Character orbit 405.n
Analytic conductor $11.035$
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] = [405,3,Mod(44,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([5, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.44");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.n (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: \(11.0354507066\)
Analytic rank: \(0\)
Dimension: \(204\)
Relative dimension: \(34\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 135)
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.

\(\operatorname{Tr}(f)(q) = \) \( 204 q - 12 q^{4} - 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 204 q - 12 q^{4} - 3 q^{5} - 3 q^{10} - 6 q^{11} + 48 q^{14} + 12 q^{16} - 6 q^{19} - 63 q^{20} - 15 q^{25} - 96 q^{29} - 102 q^{31} + 12 q^{34} + 252 q^{35} + 117 q^{40} - 96 q^{41} + 666 q^{44} - 6 q^{46} + 60 q^{49} - 48 q^{50} - 12 q^{55} - 294 q^{56} - 510 q^{59} + 132 q^{61} - 486 q^{64} - 147 q^{65} - 141 q^{70} + 18 q^{71} + 954 q^{74} + 84 q^{76} - 48 q^{79} + 69 q^{85} + 1506 q^{86} - 792 q^{89} - 6 q^{91} + 492 q^{94} + 543 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1 −2.88363 + 2.41965i 0 1.76601 − 10.0155i −4.89166 − 1.03521i 0 2.97739 − 0.524994i 11.6130 + 20.1142i 0 16.6106 − 8.85095i
44.2 −2.86556 + 2.40449i 0 1.73527 − 9.84122i 4.22671 − 2.67113i 0 9.13223 − 1.61026i 11.2092 + 19.4148i 0 −5.68919 + 17.8174i
44.3 −2.74136 + 2.30028i 0 1.52921 − 8.67255i −0.434430 − 4.98109i 0 −10.1296 + 1.78612i 8.59997 + 14.8956i 0 12.6488 + 12.6557i
44.4 −2.58191 + 2.16648i 0 1.27804 − 7.24812i 0.870514 + 4.92364i 0 0.216881 − 0.0382419i 5.66225 + 9.80731i 0 −12.9146 − 10.8265i
44.5 −2.30394 + 1.93324i 0 0.876151 − 4.96890i 0.897007 + 4.91888i 0 −11.3470 + 2.00078i 1.57229 + 2.72329i 0 −11.5760 − 9.59868i
44.6 −2.16032 + 1.81272i 0 0.686422 − 3.89289i −4.89925 + 0.998663i 0 −0.700520 + 0.123521i −0.0663500 − 0.114922i 0 8.77365 − 11.0384i
44.7 −1.98092 + 1.66219i 0 0.466579 − 2.64610i 2.60786 − 4.26604i 0 6.66174 − 1.17465i −1.69775 − 2.94059i 0 1.92501 + 12.7854i
44.8 −1.97854 + 1.66019i 0 0.463792 − 2.63030i 4.48038 + 2.21950i 0 2.79477 − 0.492794i −1.71644 − 2.97296i 0 −12.5494 + 3.04693i
44.9 −1.67712 + 1.40727i 0 0.137724 − 0.781069i 3.56412 − 3.50672i 0 −3.12327 + 0.550716i −3.51044 − 6.08027i 0 −1.04254 + 10.8968i
44.10 −1.47511 + 1.23777i 0 −0.0507010 + 0.287540i −4.66663 − 1.79514i 0 −8.65815 + 1.52666i −4.13236 − 7.15746i 0 9.10578 − 3.12817i
44.11 −1.47196 + 1.23512i 0 −0.0534492 + 0.303126i −0.707202 + 4.94973i 0 12.3105 − 2.17068i −4.13874 − 7.16851i 0 −5.07255 − 8.15930i
44.12 −1.03717 + 0.870287i 0 −0.376276 + 2.13397i −2.99716 − 4.00213i 0 6.59562 − 1.16299i −4.17475 − 7.23088i 0 6.59155 + 1.54249i
44.13 −0.860081 + 0.721694i 0 −0.475695 + 2.69780i −3.46701 + 3.60276i 0 −4.07589 + 0.718690i −3.78336 − 6.55297i 0 0.381820 − 5.60078i
44.14 −0.701263 + 0.588430i 0 −0.549072 + 3.11394i −1.10288 − 4.87685i 0 4.03271 − 0.711076i −3.27816 − 5.67794i 0 3.64309 + 2.77099i
44.15 −0.586390 + 0.492039i 0 −0.592843 + 3.36218i 4.09662 − 2.86665i 0 −12.4299 + 2.19173i −2.83764 − 4.91494i 0 −0.991714 + 3.69667i
44.16 −0.331013 + 0.277753i 0 −0.662170 + 3.75535i 3.69651 + 3.36687i 0 −5.28964 + 0.932706i −1.68809 − 2.92385i 0 −2.15875 − 0.0877598i
44.17 −0.201716 + 0.169260i 0 −0.682552 + 3.87095i 4.99982 − 0.0424447i 0 5.20453 − 0.917698i −1.04416 − 1.80853i 0 −1.00136 + 0.854829i
44.18 0.201716 − 0.169260i 0 −0.682552 + 3.87095i −3.85737 + 3.18131i 0 −5.20453 + 0.917698i 1.04416 + 1.80853i 0 −0.239625 + 1.29462i
44.19 0.331013 − 0.277753i 0 −0.662170 + 3.75535i −0.667512 + 4.95524i 0 5.28964 − 0.932706i 1.68809 + 2.92385i 0 1.15538 + 1.82565i
44.20 0.586390 − 0.492039i 0 −0.592843 + 3.36218i −4.98084 + 0.437279i 0 12.4299 − 2.19173i 2.83764 + 4.91494i 0 −2.70556 + 2.70719i
See next 80 embeddings (of 204 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 44.34
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 405.3.n.a 204
3.b odd 2 1 135.3.n.a ✓ 204
5.b even 2 1 inner 405.3.n.a 204
15.d odd 2 1 135.3.n.a ✓ 204
27.e even 9 1 135.3.n.a ✓ 204
27.f odd 18 1 inner 405.3.n.a 204
135.n odd 18 1 inner 405.3.n.a 204
135.p even 18 1 135.3.n.a ✓ 204
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.3.n.a ✓ 204 3.b odd 2 1
135.3.n.a ✓ 204 15.d odd 2 1
135.3.n.a ✓ 204 27.e even 9 1
135.3.n.a ✓ 204 135.p even 18 1
405.3.n.a 204 1.a even 1 1 trivial
405.3.n.a 204 5.b even 2 1 inner
405.3.n.a 204 27.f odd 18 1 inner
405.3.n.a 204 135.n odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(405, [\chi])\).