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Properties

Label 825.2.bs.i
Level $825$
Weight $2$
Character orbit 825.bs
Analytic conductor $6.588$
Analytic rank $0$
Dimension $112$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,2,Mod(74,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.74");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.bs (of order \(10\), degree \(4\), 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: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 40 q^{4} + 30 q^{6} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q + 40 q^{4} + 30 q^{6} + 20 q^{9} - 88 q^{16} + 10 q^{24} - 16 q^{31} + 176 q^{34} - 78 q^{36} + 110 q^{39} - 140 q^{46} - 84 q^{49} - 10 q^{51} - 60 q^{61} + 52 q^{64} - 130 q^{66} - 14 q^{69} - 180 q^{79} + 36 q^{81} - 200 q^{84} + 112 q^{91} - 380 q^{94} + 120 q^{96} - 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1 −1.59709 − 2.19821i −1.71189 − 0.263503i −1.66338 + 5.11937i 0 2.15481 + 4.18393i 0.444924 − 1.36934i 8.74172 − 2.84036i 2.86113 + 0.902176i 0
74.2 −1.59709 − 2.19821i 1.23006 − 1.21940i −1.66338 + 5.11937i 0 −4.64503 − 0.756442i −0.444924 + 1.36934i 8.74172 − 2.84036i 0.0261182 − 2.99989i 0
74.3 −1.34332 − 1.84892i 0.421391 + 1.68001i −0.995969 + 3.06528i 0 2.54014 − 3.03591i −0.525615 + 1.61768i 2.65829 − 0.863730i −2.64486 + 1.41588i 0
74.4 −1.34332 − 1.84892i 0.646572 + 1.60684i −0.995969 + 3.06528i 0 2.10237 − 3.35397i 0.525615 − 1.61768i 2.65829 − 0.863730i −2.16389 + 2.07788i 0
74.5 −1.29688 − 1.78501i −1.56962 + 0.732325i −0.886306 + 2.72777i 0 3.34281 + 1.85204i 1.09208 − 3.36108i 1.82172 − 0.591913i 1.92740 − 2.29894i 0
74.6 −1.29688 − 1.78501i 1.70030 − 0.330135i −0.886306 + 2.72777i 0 −2.79438 − 2.60689i −1.09208 + 3.36108i 1.82172 − 0.591913i 2.78202 − 1.12265i 0
74.7 −0.796890 − 1.09683i −1.70212 + 0.320599i 0.0500420 − 0.154013i 0 1.70804 + 1.61145i −0.843620 + 2.59640i −2.78760 + 0.905745i 2.79443 − 1.09140i 0
74.8 −0.796890 − 1.09683i 1.56549 − 0.741112i 0.0500420 − 0.154013i 0 −2.06039 − 1.12648i 0.843620 − 2.59640i −2.78760 + 0.905745i 1.90151 − 2.32040i 0
74.9 −0.447894 − 0.616474i −0.946216 + 1.45075i 0.438604 − 1.34988i 0 1.31815 − 0.0664655i 1.38365 − 4.25843i −2.47803 + 0.805161i −1.20935 − 2.74545i 0
74.10 −0.447894 − 0.616474i 1.61823 + 0.617510i 0.438604 − 1.34988i 0 −0.344119 − 1.27418i −1.38365 + 4.25843i −2.47803 + 0.805161i 2.23736 + 1.99855i 0
74.11 −0.381067 − 0.524493i 0.529800 + 1.64903i 0.488153 − 1.50238i 0 0.663018 − 0.906269i −1.25954 + 3.87647i −2.20716 + 0.717151i −2.43862 + 1.74732i 0
74.12 −0.381067 − 0.524493i 0.540660 + 1.64550i 0.488153 − 1.50238i 0 0.657029 − 0.910620i 1.25954 − 3.87647i −2.20716 + 0.717151i −2.41537 + 1.77932i 0
74.13 −0.154349 − 0.212443i −1.62148 − 0.608933i 0.596726 − 1.83653i 0 0.120910 + 0.438460i −0.114516 + 0.352444i −0.981744 + 0.318988i 2.25840 + 1.97475i 0
74.14 −0.154349 − 0.212443i 0.953884 − 1.44572i 0.596726 − 1.83653i 0 −0.454363 + 0.0204993i 0.114516 − 0.352444i −0.981744 + 0.318988i −1.18021 − 2.75810i 0
74.15 0.154349 + 0.212443i −0.953884 + 1.44572i 0.596726 − 1.83653i 0 −0.454363 + 0.0204993i −0.114516 + 0.352444i 0.981744 − 0.318988i −1.18021 − 2.75810i 0
74.16 0.154349 + 0.212443i 1.62148 + 0.608933i 0.596726 − 1.83653i 0 0.120910 + 0.438460i 0.114516 − 0.352444i 0.981744 − 0.318988i 2.25840 + 1.97475i 0
74.17 0.381067 + 0.524493i −0.540660 − 1.64550i 0.488153 − 1.50238i 0 0.657029 − 0.910620i −1.25954 + 3.87647i 2.20716 − 0.717151i −2.41537 + 1.77932i 0
74.18 0.381067 + 0.524493i −0.529800 − 1.64903i 0.488153 − 1.50238i 0 0.663018 − 0.906269i 1.25954 − 3.87647i 2.20716 − 0.717151i −2.43862 + 1.74732i 0
74.19 0.447894 + 0.616474i −1.61823 − 0.617510i 0.438604 − 1.34988i 0 −0.344119 − 1.27418i 1.38365 − 4.25843i 2.47803 − 0.805161i 2.23736 + 1.99855i 0
74.20 0.447894 + 0.616474i 0.946216 − 1.45075i 0.438604 − 1.34988i 0 1.31815 − 0.0664655i −1.38365 + 4.25843i 2.47803 − 0.805161i −1.20935 − 2.74545i 0
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 74.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
11.d odd 10 1 inner
15.d odd 2 1 inner
33.f even 10 1 inner
55.h odd 10 1 inner
165.r even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.bs.i 112
3.b odd 2 1 inner 825.2.bs.i 112
5.b even 2 1 inner 825.2.bs.i 112
5.c odd 4 1 825.2.bi.f ✓ 56
5.c odd 4 1 825.2.bi.g yes 56
11.d odd 10 1 inner 825.2.bs.i 112
15.d odd 2 1 inner 825.2.bs.i 112
15.e even 4 1 825.2.bi.f ✓ 56
15.e even 4 1 825.2.bi.g yes 56
33.f even 10 1 inner 825.2.bs.i 112
55.h odd 10 1 inner 825.2.bs.i 112
55.l even 20 1 825.2.bi.f ✓ 56
55.l even 20 1 825.2.bi.g yes 56
165.r even 10 1 inner 825.2.bs.i 112
165.u odd 20 1 825.2.bi.f ✓ 56
165.u odd 20 1 825.2.bi.g yes 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.bi.f ✓ 56 5.c odd 4 1
825.2.bi.f ✓ 56 15.e even 4 1
825.2.bi.f ✓ 56 55.l even 20 1
825.2.bi.f ✓ 56 165.u odd 20 1
825.2.bi.g yes 56 5.c odd 4 1
825.2.bi.g yes 56 15.e even 4 1
825.2.bi.g yes 56 55.l even 20 1
825.2.bi.g yes 56 165.u odd 20 1
825.2.bs.i 112 1.a even 1 1 trivial
825.2.bs.i 112 3.b odd 2 1 inner
825.2.bs.i 112 5.b even 2 1 inner
825.2.bs.i 112 11.d odd 10 1 inner
825.2.bs.i 112 15.d odd 2 1 inner
825.2.bs.i 112 33.f even 10 1 inner
825.2.bs.i 112 55.h odd 10 1 inner
825.2.bs.i 112 165.r even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} - 24 T_{2}^{54} + 342 T_{2}^{52} - 3831 T_{2}^{50} + 37214 T_{2}^{48} - 304294 T_{2}^{46} + \cdots + 366025 \) acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\). Copy content Toggle raw display