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Properties

Label 675.2.w.a
Level $675$
Weight $2$
Character orbit 675.w
Analytic conductor $5.390$
Analytic rank $0$
Dimension $160$
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] = [675,2,Mod(53,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.w (of order \(20\), degree \(8\), 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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(20\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 8 q^{7} + 16 q^{10} + 16 q^{13} + 40 q^{16} + 40 q^{19} - 32 q^{22} - 8 q^{25} + 80 q^{28} - 40 q^{34} + 16 q^{37} - 72 q^{40} - 8 q^{43} - 8 q^{52} + 32 q^{55} + 72 q^{58} + 80 q^{64} + 80 q^{67} + 152 q^{70} + 32 q^{73} + 40 q^{79} - 48 q^{82} - 88 q^{85} + 272 q^{88} - 160 q^{94} + 248 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −2.45463 + 1.25069i 0 3.28538 − 4.52194i −1.52896 + 1.63165i 0 −1.84987 + 1.84987i −1.54690 + 9.76672i 0 1.71233 − 5.91735i
53.2 −2.09866 + 1.06932i 0 2.08536 − 2.87025i 0.939638 − 2.02906i 0 1.97617 − 1.97617i −0.570314 + 3.60082i 0 0.197735 + 5.26308i
53.3 −2.04267 + 1.04079i 0 1.91368 − 2.63395i −2.03246 − 0.932267i 0 2.61557 − 2.61557i −0.450349 + 2.84339i 0 5.12193 − 0.211050i
53.4 −1.55336 + 0.791478i 0 0.610927 − 0.840868i 1.74829 − 1.39408i 0 −0.868157 + 0.868157i 0.261987 − 1.65412i 0 −1.61235 + 3.54925i
53.5 −1.53433 + 0.781780i 0 0.567415 − 0.780980i 2.10202 + 0.762559i 0 −2.40762 + 2.40762i 0.278719 − 1.75976i 0 −3.82135 + 0.473303i
53.6 −1.53318 + 0.781192i 0 0.564797 − 0.777376i −2.22336 + 0.238059i 0 −3.26299 + 3.26299i 0.279708 − 1.76601i 0 3.22283 − 2.10186i
53.7 −1.16573 + 0.593969i 0 −0.169443 + 0.233218i −0.783107 + 2.09446i 0 1.73390 − 1.73390i 0.468336 − 2.95696i 0 −0.331151 − 2.90671i
53.8 −0.694266 + 0.353746i 0 −0.818702 + 1.12685i −1.59289 − 1.56930i 0 0.326191 − 0.326191i 0.413564 − 2.61114i 0 1.66102 + 0.526032i
53.9 −0.499496 + 0.254506i 0 −0.990847 + 1.36378i 2.23539 + 0.0551483i 0 2.65202 − 2.65202i 0.323227 − 2.04077i 0 −1.13060 + 0.541373i
53.10 −0.0810707 + 0.0413076i 0 −1.17070 + 1.61134i −0.325627 − 2.21223i 0 −1.35768 + 1.35768i 0.0568166 − 0.358726i 0 0.117781 + 0.165896i
53.11 0.0810707 − 0.0413076i 0 −1.17070 + 1.61134i 0.325627 + 2.21223i 0 −1.35768 + 1.35768i −0.0568166 + 0.358726i 0 0.117781 + 0.165896i
53.12 0.499496 − 0.254506i 0 −0.990847 + 1.36378i −2.23539 − 0.0551483i 0 2.65202 − 2.65202i −0.323227 + 2.04077i 0 −1.13060 + 0.541373i
53.13 0.694266 − 0.353746i 0 −0.818702 + 1.12685i 1.59289 + 1.56930i 0 0.326191 − 0.326191i −0.413564 + 2.61114i 0 1.66102 + 0.526032i
53.14 1.16573 − 0.593969i 0 −0.169443 + 0.233218i 0.783107 − 2.09446i 0 1.73390 − 1.73390i −0.468336 + 2.95696i 0 −0.331151 − 2.90671i
53.15 1.53318 − 0.781192i 0 0.564797 − 0.777376i 2.22336 − 0.238059i 0 −3.26299 + 3.26299i −0.279708 + 1.76601i 0 3.22283 − 2.10186i
53.16 1.53433 − 0.781780i 0 0.567415 − 0.780980i −2.10202 − 0.762559i 0 −2.40762 + 2.40762i −0.278719 + 1.75976i 0 −3.82135 + 0.473303i
53.17 1.55336 − 0.791478i 0 0.610927 − 0.840868i −1.74829 + 1.39408i 0 −0.868157 + 0.868157i −0.261987 + 1.65412i 0 −1.61235 + 3.54925i
53.18 2.04267 − 1.04079i 0 1.91368 − 2.63395i 2.03246 + 0.932267i 0 2.61557 − 2.61557i 0.450349 − 2.84339i 0 5.12193 − 0.211050i
53.19 2.09866 − 1.06932i 0 2.08536 − 2.87025i −0.939638 + 2.02906i 0 1.97617 − 1.97617i 0.570314 − 3.60082i 0 0.197735 + 5.26308i
53.20 2.45463 − 1.25069i 0 3.28538 − 4.52194i 1.52896 − 1.63165i 0 −1.84987 + 1.84987i 1.54690 − 9.76672i 0 1.71233 − 5.91735i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.f odd 20 1 inner
75.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.w.a ✓ 160
3.b odd 2 1 inner 675.2.w.a ✓ 160
25.f odd 20 1 inner 675.2.w.a ✓ 160
75.l even 20 1 inner 675.2.w.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
675.2.w.a ✓ 160 1.a even 1 1 trivial
675.2.w.a ✓ 160 3.b odd 2 1 inner
675.2.w.a ✓ 160 25.f odd 20 1 inner
675.2.w.a ✓ 160 75.l even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{160} - 150 T_{2}^{156} - 80 T_{2}^{154} + 13980 T_{2}^{152} + 12000 T_{2}^{150} + \cdots + 31816650390625 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display