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Properties

Label 900.2.bj.e
Level $900$
Weight $2$
Character orbit 900.bj
Analytic conductor $7.187$
Analytic rank $0$
Dimension $224$
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] = [900,2,Mod(127,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 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("900.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.bj (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: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 224 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q - 4 q^{10} + 16 q^{13} - 16 q^{16} + 28 q^{22} - 32 q^{25} + 28 q^{28} - 100 q^{34} - 104 q^{37} + 60 q^{40} + 156 q^{52} + 144 q^{58} - 48 q^{61} + 60 q^{64} + 28 q^{70} + 40 q^{73} + 64 q^{82} + 136 q^{85} + 148 q^{88} + 40 q^{94} - 160 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1 −1.41097 + 0.0956789i 0 1.98169 − 0.270001i 2.15024 + 0.613575i 0 −2.54150 + 2.54150i −2.77028 + 0.570570i 0 −3.09264 − 0.660006i
127.2 −1.35252 + 0.413133i 0 1.65864 − 1.11754i −0.00168427 + 2.23607i 0 3.38133 − 3.38133i −1.78166 + 2.19675i 0 −0.921516 − 3.02503i
127.3 −1.31235 − 0.527011i 0 1.44452 + 1.38324i 2.15024 + 0.613575i 0 2.54150 − 2.54150i −1.16673 − 2.57658i 0 −2.49850 − 1.93842i
127.4 −1.25424 + 0.653372i 0 1.14621 − 1.63896i −0.326937 − 2.21204i 0 1.80142 − 1.80142i −0.366766 + 2.80455i 0 1.85534 + 2.56080i
127.5 −1.15866 − 0.810866i 0 0.684993 + 1.87904i −0.00168427 + 2.23607i 0 −3.38133 + 3.38133i 0.729973 − 2.73261i 0 1.81510 − 2.58948i
127.6 −1.07844 + 0.914857i 0 0.326072 − 1.97324i −1.52496 + 1.63539i 0 −1.57451 + 1.57451i 1.45358 + 2.42633i 0 0.148429 − 3.15879i
127.7 −0.999417 + 1.00058i 0 −0.00233243 − 2.00000i 2.16813 + 0.546990i 0 −0.919512 + 0.919512i 2.00350 + 1.99650i 0 −2.71418 + 1.62273i
127.8 −0.990945 − 1.00897i 0 −0.0360543 + 1.99967i −0.326937 − 2.21204i 0 −1.80142 + 1.80142i 2.05335 − 1.94519i 0 −1.90791 + 2.52188i
127.9 −0.742952 − 1.20334i 0 −0.896044 + 1.78804i −1.52496 + 1.63539i 0 1.57451 − 1.57451i 2.81734 − 0.250188i 0 3.10090 + 0.620023i
127.10 −0.641305 − 1.26045i 0 −1.17746 + 1.61666i 2.16813 + 0.546990i 0 0.919512 − 0.919512i 2.79283 + 0.447350i 0 −0.700982 − 3.08361i
127.11 −0.491133 + 1.32619i 0 −1.51758 − 1.30267i −1.22373 − 1.87149i 0 −1.80675 + 1.80675i 2.47293 − 1.37281i 0 3.08297 − 0.703756i
127.12 −0.486415 + 1.32793i 0 −1.52680 − 1.29185i 1.58602 − 1.57624i 0 1.29830 − 1.29830i 2.45815 − 1.39911i 0 1.32167 + 2.87284i
127.13 −0.0572791 − 1.41305i 0 −1.99344 + 0.161877i −1.22373 − 1.87149i 0 1.80675 − 1.80675i 0.342923 + 2.80756i 0 −2.57442 + 1.83640i
127.14 −0.0522554 − 1.41325i 0 −1.99454 + 0.147700i 1.58602 − 1.57624i 0 −1.29830 + 1.29830i 0.312962 + 2.81106i 0 −2.31050 − 2.15908i
127.15 0.0522554 + 1.41325i 0 −1.99454 + 0.147700i −1.58602 + 1.57624i 0 −1.29830 + 1.29830i −0.312962 − 2.81106i 0 −2.31050 − 2.15908i
127.16 0.0572791 + 1.41305i 0 −1.99344 + 0.161877i 1.22373 + 1.87149i 0 1.80675 − 1.80675i −0.342923 − 2.80756i 0 −2.57442 + 1.83640i
127.17 0.486415 − 1.32793i 0 −1.52680 − 1.29185i −1.58602 + 1.57624i 0 1.29830 − 1.29830i −2.45815 + 1.39911i 0 1.32167 + 2.87284i
127.18 0.491133 − 1.32619i 0 −1.51758 − 1.30267i 1.22373 + 1.87149i 0 −1.80675 + 1.80675i −2.47293 + 1.37281i 0 3.08297 − 0.703756i
127.19 0.641305 + 1.26045i 0 −1.17746 + 1.61666i −2.16813 − 0.546990i 0 0.919512 − 0.919512i −2.79283 − 0.447350i 0 −0.700982 − 3.08361i
127.20 0.742952 + 1.20334i 0 −0.896044 + 1.78804i 1.52496 − 1.63539i 0 1.57451 − 1.57451i −2.81734 + 0.250188i 0 3.10090 + 0.620023i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.28
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.bj.e ✓ 224
3.b odd 2 1 inner 900.2.bj.e ✓ 224
4.b odd 2 1 inner 900.2.bj.e ✓ 224
12.b even 2 1 inner 900.2.bj.e ✓ 224
25.f odd 20 1 inner 900.2.bj.e ✓ 224
75.l even 20 1 inner 900.2.bj.e ✓ 224
100.l even 20 1 inner 900.2.bj.e ✓ 224
300.u odd 20 1 inner 900.2.bj.e ✓ 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.2.bj.e ✓ 224 1.a even 1 1 trivial
900.2.bj.e ✓ 224 3.b odd 2 1 inner
900.2.bj.e ✓ 224 4.b odd 2 1 inner
900.2.bj.e ✓ 224 12.b even 2 1 inner
900.2.bj.e ✓ 224 25.f odd 20 1 inner
900.2.bj.e ✓ 224 75.l even 20 1 inner
900.2.bj.e ✓ 224 100.l even 20 1 inner
900.2.bj.e ✓ 224 300.u odd 20 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{112} + 3650 T_{7}^{108} + 5942355 T_{7}^{104} + 5712795030 T_{7}^{100} + 3622786310365 T_{7}^{96} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
\( T_{13}^{56} - 4 T_{13}^{55} + 8 T_{13}^{54} - 150 T_{13}^{53} - 922 T_{13}^{52} + 5372 T_{13}^{51} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
\( T_{17}^{112} + 40 T_{17}^{110} - 2658 T_{17}^{108} - 261640 T_{17}^{106} + 1393643 T_{17}^{104} + \cdots + 88\!\cdots\!16 \) Copy content Toggle raw display