↑

Properties

Label 825.2.bi.g
Level $825$
Weight $2$
Character orbit 825.bi
Analytic conductor $6.588$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,2,Mod(101,825)]
 
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, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.bi (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: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 8 q^{3} - 20 q^{4} + 15 q^{6} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 8 q^{3} - 20 q^{4} + 15 q^{6} - 10 q^{9} + 14 q^{12} - 44 q^{16} - 25 q^{18} + 38 q^{22} - 5 q^{24} - 16 q^{27} - 40 q^{28} - 8 q^{31} - 6 q^{33} - 88 q^{34} - 39 q^{36} - 50 q^{37} - 55 q^{39} - 17 q^{42} - 70 q^{46} + 54 q^{48} + 42 q^{49} - 5 q^{51} + 25 q^{57} + 4 q^{58} - 30 q^{61} - 15 q^{63} - 26 q^{64} - 65 q^{66} - 80 q^{67} + 7 q^{69} - 80 q^{72} + 10 q^{73} - 34 q^{78} + 90 q^{79} + 18 q^{81} + 122 q^{82} + 100 q^{84} - 54 q^{88} + 56 q^{91} - 59 q^{93} + 190 q^{94} + 60 q^{96} + 146 q^{97} + 30 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 −0.807319 + 2.48467i 1.33193 − 1.10724i −3.90380 − 2.83628i 0 1.67583 + 4.20329i 1.04772 − 1.44206i 5.97165 − 4.33866i 0.548058 − 2.94951i 0
101.2 −0.732552 + 2.25456i −1.65857 − 0.499153i −2.92839 − 2.12760i 0 2.34036 − 3.37369i −0.206810 + 0.284650i 3.10630 − 2.25686i 2.50169 + 1.65576i 0
101.3 −0.665909 + 2.04946i −0.632101 + 1.61259i −2.13881 − 1.55394i 0 −2.88402 − 2.36930i 2.75318 − 3.78943i 1.12224 − 0.815353i −2.20090 − 2.03864i 0
101.4 −0.511433 + 1.57403i 0.0466623 − 1.73142i −0.597967 − 0.434448i 0 2.70144 + 0.958954i −2.59991 + 3.57847i −1.68824 + 1.22658i −2.99565 − 0.161584i 0
101.5 −0.435880 + 1.34150i 1.31553 + 1.12667i 0.00840377 + 0.00610569i 0 −2.08483 + 1.27370i −0.261256 + 0.359588i −2.29415 + 1.66680i 0.461252 + 2.96433i 0
101.6 −0.245316 + 0.755005i 1.66445 − 0.479178i 1.10818 + 0.805141i 0 −0.0465342 + 1.37422i 0.695031 − 0.956629i −2.16423 + 1.57241i 2.54078 − 1.59513i 0
101.7 −0.127531 + 0.392501i −1.30029 − 1.14423i 1.48024 + 1.07546i 0 0.614938 − 0.364439i 0.808116 − 1.11228i −1.27866 + 0.929000i 0.381484 + 2.97565i 0
101.8 0.127531 − 0.392501i 0.686415 + 1.59023i 1.48024 + 1.07546i 0 0.711707 − 0.0666145i 0.808116 − 1.11228i 1.27866 − 0.929000i −2.05767 + 2.18312i 0
101.9 0.245316 − 0.755005i 0.970068 − 1.43491i 1.10818 + 0.805141i 0 −0.845392 − 1.08441i 0.695031 − 0.956629i 2.16423 − 1.57241i −1.11794 − 2.78392i 0
101.10 0.435880 − 1.34150i −0.665000 − 1.59930i 0.00840377 + 0.00610569i 0 −2.43533 + 0.194993i −0.261256 + 0.359588i 2.29415 − 1.66680i −2.11555 + 2.12708i 0
101.11 0.511433 − 1.57403i 1.66110 + 0.490660i −0.597967 − 0.434448i 0 1.62185 − 2.36368i −2.59991 + 3.57847i 1.68824 − 1.22658i 2.51850 + 1.63007i 0
101.12 0.665909 − 2.04946i −1.72899 + 0.102846i −2.13881 − 1.55394i 0 −0.940576 + 3.61199i 2.75318 − 3.78943i −1.12224 + 0.815353i 2.97885 − 0.355639i 0
101.13 0.732552 − 2.25456i −0.0378025 + 1.73164i −2.92839 − 2.12760i 0 3.87639 + 1.35374i −0.206810 + 0.284650i −3.10630 + 2.25686i −2.99714 − 0.130921i 0
101.14 0.807319 − 2.48467i 1.46463 − 0.924583i −3.90380 − 2.83628i 0 −1.11486 − 4.38556i 1.04772 − 1.44206i −5.97165 + 4.33866i 1.29029 − 2.70835i 0
326.1 −2.19821 − 1.59709i 1.21940 − 1.23006i 1.66338 + 5.11937i 0 −4.64503 + 0.756442i 1.36934 − 0.444924i 2.84036 − 8.74172i −0.0261182 − 2.99989i 0
326.2 −1.84892 − 1.34332i −1.68001 − 0.421391i 0.995969 + 3.06528i 0 2.54014 + 3.03591i 1.61768 − 0.525615i 0.863730 − 2.65829i 2.64486 + 1.41588i 0
326.3 −1.78501 − 1.29688i −0.732325 + 1.56962i 0.886306 + 2.72777i 0 3.34281 − 1.85204i −3.36108 + 1.09208i 0.591913 − 1.82172i −1.92740 − 2.29894i 0
326.4 −1.09683 − 0.796890i 0.741112 − 1.56549i −0.0500420 − 0.154013i 0 −2.06039 + 1.12648i −2.59640 + 0.843620i −0.905745 + 2.78760i −1.90151 − 2.32040i 0
326.5 −0.616474 − 0.447894i −0.617510 − 1.61823i −0.438604 − 1.34988i 0 −0.344119 + 1.27418i 4.25843 − 1.38365i −0.805161 + 2.47803i −2.23736 + 1.99855i 0
326.6 −0.524493 − 0.381067i −1.64550 − 0.540660i −0.488153 − 1.50238i 0 0.657029 + 0.910620i −3.87647 + 1.25954i −0.717151 + 2.20716i 2.41537 + 1.77932i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.14
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{56} + 24 T_{2}^{54} + 342 T_{2}^{52} + 3831 T_{2}^{50} + 37214 T_{2}^{48} + 304294 T_{2}^{46} + \cdots + 366025 \) Copy content Toggle raw display
\( T_{7}^{28} - 35 T_{7}^{26} + 30 T_{7}^{25} + 829 T_{7}^{24} - 1050 T_{7}^{23} - 17603 T_{7}^{22} + \cdots + 2251205 \) Copy content Toggle raw display