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Properties

Label 225.2.s.a
Level $225$
Weight $2$
Character orbit 225.s
Analytic conductor $1.797$
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] = [225,2,Mod(8,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 225.s (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: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 8 q^{7} + 8 q^{10} - 4 q^{13} + 20 q^{16} - 40 q^{19} - 88 q^{22} - 16 q^{25} - 128 q^{28} - 20 q^{34} - 4 q^{37} + 12 q^{40} + 32 q^{43} - 4 q^{52} - 8 q^{55} - 12 q^{58} + 160 q^{64} + 64 q^{67} + 136 q^{70} + 76 q^{73} + 80 q^{79} + 84 q^{82} + 52 q^{85} - 56 q^{88} - 80 q^{94} - 236 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −1.25673 + 2.46647i 0 −3.32854 − 4.58134i 2.13687 − 0.658625i 0 1.31781 − 1.31781i 10.0146 − 1.58616i 0 −1.06099 + 6.09824i
8.2 −0.938638 + 1.84218i 0 −1.33702 − 1.84025i −0.979600 − 2.01007i 0 3.46588 − 3.46588i 0.560907 − 0.0888390i 0 4.62241 + 0.0821281i
8.3 −0.688810 + 1.35187i 0 −0.177511 − 0.244323i −0.939716 + 2.02902i 0 −0.353567 + 0.353567i −2.54455 + 0.403017i 0 −2.09568 − 2.66798i
8.4 −0.320524 + 0.629064i 0 0.882585 + 1.21477i 1.75143 + 1.39014i 0 1.59720 − 1.59720i −2.44171 + 0.386728i 0 −1.43586 + 0.656192i
8.5 −0.232387 + 0.456085i 0 1.02156 + 1.40606i −2.22023 − 0.265626i 0 −3.23371 + 3.23371i −1.88983 + 0.299319i 0 0.637101 − 0.950888i
8.6 0.232387 − 0.456085i 0 1.02156 + 1.40606i 2.22023 + 0.265626i 0 −3.23371 + 3.23371i 1.88983 − 0.299319i 0 0.637101 − 0.950888i
8.7 0.320524 − 0.629064i 0 0.882585 + 1.21477i −1.75143 − 1.39014i 0 1.59720 − 1.59720i 2.44171 − 0.386728i 0 −1.43586 + 0.656192i
8.8 0.688810 − 1.35187i 0 −0.177511 − 0.244323i 0.939716 − 2.02902i 0 −0.353567 + 0.353567i 2.54455 − 0.403017i 0 −2.09568 − 2.66798i
8.9 0.938638 − 1.84218i 0 −1.33702 − 1.84025i 0.979600 + 2.01007i 0 3.46588 − 3.46588i −0.560907 + 0.0888390i 0 4.62241 + 0.0821281i
8.10 1.25673 − 2.46647i 0 −3.32854 − 4.58134i −2.13687 + 0.658625i 0 1.31781 − 1.31781i −10.0146 + 1.58616i 0 −1.06099 + 6.09824i
17.1 −2.13458 − 1.08762i 0 2.19794 + 3.02520i −1.74195 + 1.40200i 0 −1.66247 − 1.66247i −0.651858 − 4.11567i 0 5.24319 − 1.09808i
17.2 −1.92481 − 0.980737i 0 1.56746 + 2.15742i −0.522555 − 2.17415i 0 −0.767908 − 0.767908i −0.225311 − 1.42256i 0 −1.12646 + 4.69731i
17.3 −1.62573 − 0.828352i 0 0.781267 + 1.07532i 1.49058 − 1.66679i 0 3.11978 + 3.11978i 0.191475 + 1.20893i 0 −3.80396 + 1.47503i
17.4 −1.00047 − 0.509765i 0 −0.434489 − 0.598023i −1.16530 + 1.90842i 0 −0.146546 − 0.146546i 0.481149 + 3.03785i 0 2.13869 − 1.31529i
17.5 −0.0559700 − 0.0285181i 0 −1.17325 − 1.61484i −2.22329 − 0.238739i 0 −0.100394 − 0.100394i 0.0392679 + 0.247928i 0 0.117629 + 0.0767662i
17.6 0.0559700 + 0.0285181i 0 −1.17325 − 1.61484i 2.22329 + 0.238739i 0 −0.100394 − 0.100394i −0.0392679 − 0.247928i 0 0.117629 + 0.0767662i
17.7 1.00047 + 0.509765i 0 −0.434489 − 0.598023i 1.16530 − 1.90842i 0 −0.146546 − 0.146546i −0.481149 − 3.03785i 0 2.13869 − 1.31529i
17.8 1.62573 + 0.828352i 0 0.781267 + 1.07532i −1.49058 + 1.66679i 0 3.11978 + 3.11978i −0.191475 − 1.20893i 0 −3.80396 + 1.47503i
17.9 1.92481 + 0.980737i 0 1.56746 + 2.15742i 0.522555 + 2.17415i 0 −0.767908 − 0.767908i 0.225311 + 1.42256i 0 −1.12646 + 4.69731i
17.10 2.13458 + 1.08762i 0 2.19794 + 3.02520i 1.74195 − 1.40200i 0 −1.66247 − 1.66247i 0.651858 + 4.11567i 0 5.24319 − 1.09808i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.10
Significant digits:
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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 225.2.s.a ✓ 80
3.b odd 2 1 inner 225.2.s.a ✓ 80
25.f odd 20 1 inner 225.2.s.a ✓ 80
75.l even 20 1 inner 225.2.s.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.s.a ✓ 80 1.a even 1 1 trivial
225.2.s.a ✓ 80 3.b odd 2 1 inner
225.2.s.a ✓ 80 25.f odd 20 1 inner
225.2.s.a ✓ 80 75.l even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(225, [\chi])\).