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Properties

Label 675.2.n.a
Level $675$
Weight $2$
Character orbit 675.n
Analytic conductor $5.390$
Analytic rank $0$
Dimension $80$
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(109,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.n (of order \(10\), degree \(4\), 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: \(80\)
Relative dimension: \(20\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 20 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 20 q^{4} + 4 q^{10} - 32 q^{16} - 12 q^{19} + 20 q^{22} + 4 q^{25} - 60 q^{28} + 12 q^{31} - 4 q^{34} - 60 q^{37} - 32 q^{40} - 52 q^{46} - 72 q^{49} + 100 q^{52} + 36 q^{55} + 120 q^{58} + 12 q^{61} + 92 q^{64} + 20 q^{67} - 36 q^{70} + 100 q^{73} + 16 q^{76} - 4 q^{79} + 32 q^{85} - 20 q^{88} + 52 q^{91} + 60 q^{94} - 140 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 −1.64441 + 2.26334i 0 −1.80057 − 5.54160i −2.13024 − 0.679761i 0 − 3.07753i 10.1820 + 3.30832i 0 5.04152 − 3.70365i
109.2 −1.36272 + 1.87562i 0 −1.04292 − 3.20977i 1.80946 + 1.31372i 0 1.87708i 3.03166 + 0.985045i 0 −4.92982 + 1.60364i
109.3 −1.35525 + 1.86534i 0 −1.02476 − 3.15388i −0.672644 + 2.13250i 0 − 2.14429i 2.88619 + 0.937780i 0 −3.06623 − 4.14477i
109.4 −1.26199 + 1.73698i 0 −0.806449 − 2.48200i −1.61590 − 1.54559i 0 4.22695i 1.24502 + 0.404533i 0 4.72392 − 0.856266i
109.5 −1.01246 + 1.39353i 0 −0.298817 − 0.919665i 1.40432 − 1.74008i 0 − 1.28467i −1.69226 − 0.549850i 0 1.00303 + 3.71872i
109.6 −0.797835 + 1.09813i 0 0.0486945 + 0.149866i 0.855506 + 2.06594i 0 − 0.723732i −2.78527 − 0.904989i 0 −2.95122 − 0.708826i
109.7 −0.492711 + 0.678159i 0 0.400899 + 1.23384i 2.02539 − 0.947528i 0 1.51748i −2.62871 − 0.854121i 0 −0.355356 + 1.84039i
109.8 −0.491954 + 0.677116i 0 0.401566 + 1.23589i −1.96477 + 1.06755i 0 − 3.80357i −2.62639 − 0.853366i 0 0.243719 − 1.85557i
109.9 −0.465386 + 0.640549i 0 0.424315 + 1.30591i −2.06834 − 0.849690i 0 − 2.77851i −2.53999 − 0.825293i 0 1.50685 − 0.929439i
109.10 −0.106593 + 0.146713i 0 0.607871 + 1.87084i −1.05206 − 1.97311i 0 2.38657i −0.684214 − 0.222315i 0 0.401623 + 0.0559705i
109.11 0.106593 − 0.146713i 0 0.607871 + 1.87084i 1.05206 + 1.97311i 0 2.38657i 0.684214 + 0.222315i 0 0.401623 + 0.0559705i
109.12 0.465386 − 0.640549i 0 0.424315 + 1.30591i 2.06834 + 0.849690i 0 − 2.77851i 2.53999 + 0.825293i 0 1.50685 − 0.929439i
109.13 0.491954 − 0.677116i 0 0.401566 + 1.23589i 1.96477 − 1.06755i 0 − 3.80357i 2.62639 + 0.853366i 0 0.243719 − 1.85557i
109.14 0.492711 − 0.678159i 0 0.400899 + 1.23384i −2.02539 + 0.947528i 0 1.51748i 2.62871 + 0.854121i 0 −0.355356 + 1.84039i
109.15 0.797835 − 1.09813i 0 0.0486945 + 0.149866i −0.855506 − 2.06594i 0 − 0.723732i 2.78527 + 0.904989i 0 −2.95122 − 0.708826i
109.16 1.01246 − 1.39353i 0 −0.298817 − 0.919665i −1.40432 + 1.74008i 0 − 1.28467i 1.69226 + 0.549850i 0 1.00303 + 3.71872i
109.17 1.26199 − 1.73698i 0 −0.806449 − 2.48200i 1.61590 + 1.54559i 0 4.22695i −1.24502 − 0.404533i 0 4.72392 − 0.856266i
109.18 1.35525 − 1.86534i 0 −1.02476 − 3.15388i 0.672644 − 2.13250i 0 − 2.14429i −2.88619 − 0.937780i 0 −3.06623 − 4.14477i
109.19 1.36272 − 1.87562i 0 −1.04292 − 3.20977i −1.80946 − 1.31372i 0 1.87708i −3.03166 − 0.985045i 0 −4.92982 + 1.60364i
109.20 1.64441 − 2.26334i 0 −1.80057 − 5.54160i 2.13024 + 0.679761i 0 − 3.07753i −10.1820 − 3.30832i 0 5.04152 − 3.70365i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.e even 10 1 inner
75.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.n.a ✓ 80
3.b odd 2 1 inner 675.2.n.a ✓ 80
25.e even 10 1 inner 675.2.n.a ✓ 80
75.h odd 10 1 inner 675.2.n.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
675.2.n.a ✓ 80 1.a even 1 1 trivial
675.2.n.a ✓ 80 3.b odd 2 1 inner
675.2.n.a ✓ 80 25.e even 10 1 inner
675.2.n.a ✓ 80 75.h odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{80} - 30 T_{2}^{78} + 528 T_{2}^{76} - 7162 T_{2}^{74} + 82634 T_{2}^{72} - 819064 T_{2}^{70} + \cdots + 81450625 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display