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Properties

Label 605.2.m.g
Level $605$
Weight $2$
Character orbit 605.m
Analytic conductor $4.831$
Analytic rank $0$
Dimension $160$
CM no
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [605,2,Mod(112,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([5, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("605.112");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.m (of order \(20\), degree \(8\), 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: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 8 q^{3} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 8 q^{3} + 4 q^{5} - 128 q^{12} - 32 q^{15} + 24 q^{16} - 32 q^{20} - 96 q^{23} + 52 q^{25} + 8 q^{26} - 32 q^{27} - 16 q^{31} + 40 q^{36} - 36 q^{37} - 32 q^{38} - 32 q^{42} + 256 q^{45} - 32 q^{47} + 16 q^{48} + 68 q^{53} + 320 q^{56} - 132 q^{58} + 64 q^{60} - 352 q^{67} + 8 q^{70} + 16 q^{75} - 992 q^{78} + 164 q^{80} + 40 q^{81} - 100 q^{82} + 80 q^{86} + 96 q^{91} + 56 q^{92} - 24 q^{93} + 68 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
112.1 −1.23116 − 2.41628i −0.183968 + 1.16153i −3.14709 + 4.33160i −0.956130 + 2.02134i 3.03306 − 0.985502i −0.535369 + 0.0847942i 8.98397 + 1.42292i 1.53787 + 0.499685i 6.06127 − 0.178306i
112.2 −1.05904 − 2.07848i −0.318551 + 2.01125i −2.02296 + 2.78437i 2.18295 + 0.484484i 4.51772 − 1.46790i 4.54674 − 0.720133i 3.32162 + 0.526094i −1.09050 − 0.354324i −1.30484 − 5.05032i
112.3 −1.05076 − 2.06223i 0.159862 − 1.00933i −1.97313 + 2.71579i −1.98733 + 1.02495i −2.24944 + 0.730888i 2.99845 − 0.474908i 3.10187 + 0.491288i 1.85999 + 0.604346i 4.20189 + 3.02136i
112.4 −0.890440 − 1.74759i 0.459138 − 2.89888i −1.08561 + 1.49421i 0.260122 − 2.22089i −5.47488 + 1.77890i −2.07637 + 0.328864i −0.296505 − 0.0469618i −5.33954 − 1.73492i −4.11281 + 1.52298i
112.5 −0.832753 − 1.63437i −0.488992 + 3.08737i −0.802115 + 1.10402i −1.82276 + 1.29520i 5.45312 − 1.77183i 0.491081 − 0.0777795i −1.15109 − 0.182315i −6.43959 − 2.09235i 3.63475 + 1.90048i
112.6 −0.679174 − 1.33295i −0.340390 + 2.14914i −0.139917 + 0.192579i 0.485485 − 2.18273i 3.09589 − 1.00591i −2.32674 + 0.368519i −2.60345 − 0.412346i −1.64977 − 0.536043i −3.23920 + 0.835322i
112.7 −0.544755 − 1.06914i 0.0572260 − 0.361310i 0.329265 − 0.453195i 2.07465 − 0.834158i −0.417466 + 0.135643i 1.48586 − 0.235337i −3.03420 − 0.480570i 2.72590 + 0.885698i −2.02201 − 1.76368i
112.8 −0.323023 − 0.633967i 0.440005 − 2.77808i 0.877999 − 1.20846i −2.10640 − 0.750378i −1.90335 + 0.618434i 3.39829 − 0.538236i −2.45526 − 0.388874i −4.67097 − 1.51769i 0.204700 + 1.57778i
112.9 −0.280420 − 0.550355i −0.235077 + 1.48421i 0.951315 − 1.30937i −2.19624 − 0.420141i 0.882765 − 0.286828i −1.51988 + 0.240726i −2.20753 − 0.349639i 0.705537 + 0.229243i 0.384643 + 1.32653i
112.10 −0.120168 − 0.235842i 0.00828396 − 0.0523028i 1.13439 − 1.56135i 1.85452 + 1.24930i −0.0133307 + 0.00433140i −3.21766 + 0.509627i −1.02742 − 0.162727i 2.85050 + 0.926184i 0.0717827 − 0.587500i
112.11 0.120168 + 0.235842i 0.00828396 − 0.0523028i 1.13439 − 1.56135i 1.85452 + 1.24930i 0.0133307 − 0.00433140i 3.21766 − 0.509627i 1.02742 + 0.162727i 2.85050 + 0.926184i −0.0717827 + 0.587500i
112.12 0.280420 + 0.550355i −0.235077 + 1.48421i 0.951315 − 1.30937i −2.19624 − 0.420141i −0.882765 + 0.286828i 1.51988 − 0.240726i 2.20753 + 0.349639i 0.705537 + 0.229243i −0.384643 − 1.32653i
112.13 0.323023 + 0.633967i 0.440005 − 2.77808i 0.877999 − 1.20846i −2.10640 − 0.750378i 1.90335 − 0.618434i −3.39829 + 0.538236i 2.45526 + 0.388874i −4.67097 − 1.51769i −0.204700 − 1.57778i
112.14 0.544755 + 1.06914i 0.0572260 − 0.361310i 0.329265 − 0.453195i 2.07465 − 0.834158i 0.417466 − 0.135643i −1.48586 + 0.235337i 3.03420 + 0.480570i 2.72590 + 0.885698i 2.02201 + 1.76368i
112.15 0.679174 + 1.33295i −0.340390 + 2.14914i −0.139917 + 0.192579i 0.485485 − 2.18273i −3.09589 + 1.00591i 2.32674 − 0.368519i 2.60345 + 0.412346i −1.64977 − 0.536043i 3.23920 − 0.835322i
112.16 0.832753 + 1.63437i −0.488992 + 3.08737i −0.802115 + 1.10402i −1.82276 + 1.29520i −5.45312 + 1.77183i −0.491081 + 0.0777795i 1.15109 + 0.182315i −6.43959 − 2.09235i −3.63475 − 1.90048i
112.17 0.890440 + 1.74759i 0.459138 − 2.89888i −1.08561 + 1.49421i 0.260122 − 2.22089i 5.47488 − 1.77890i 2.07637 − 0.328864i 0.296505 + 0.0469618i −5.33954 − 1.73492i 4.11281 − 1.52298i
112.18 1.05076 + 2.06223i 0.159862 − 1.00933i −1.97313 + 2.71579i −1.98733 + 1.02495i 2.24944 − 0.730888i −2.99845 + 0.474908i −3.10187 − 0.491288i 1.85999 + 0.604346i −4.20189 − 3.02136i
112.19 1.05904 + 2.07848i −0.318551 + 2.01125i −2.02296 + 2.78437i 2.18295 + 0.484484i −4.51772 + 1.46790i −4.54674 + 0.720133i −3.32162 − 0.526094i −1.09050 − 0.354324i 1.30484 + 5.05032i
112.20 1.23116 + 2.41628i −0.183968 + 1.16153i −3.14709 + 4.33160i −0.956130 + 2.02134i −3.03306 + 0.985502i 0.535369 − 0.0847942i −8.98397 − 1.42292i 1.53787 + 0.499685i −6.06127 + 0.178306i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 112.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
55.e even 4 1 inner
55.k odd 20 3 inner
55.l even 20 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.m.g 160
5.c odd 4 1 inner 605.2.m.g 160
11.b odd 2 1 inner 605.2.m.g 160
11.c even 5 1 605.2.e.c ✓ 40
11.c even 5 3 inner 605.2.m.g 160
11.d odd 10 1 605.2.e.c ✓ 40
11.d odd 10 3 inner 605.2.m.g 160
55.e even 4 1 inner 605.2.m.g 160
55.k odd 20 1 605.2.e.c ✓ 40
55.k odd 20 3 inner 605.2.m.g 160
55.l even 20 1 605.2.e.c ✓ 40
55.l even 20 3 inner 605.2.m.g 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.e.c ✓ 40 11.c even 5 1
605.2.e.c ✓ 40 11.d odd 10 1
605.2.e.c ✓ 40 55.k odd 20 1
605.2.e.c ✓ 40 55.l even 20 1
605.2.m.g 160 1.a even 1 1 trivial
605.2.m.g 160 5.c odd 4 1 inner
605.2.m.g 160 11.b odd 2 1 inner
605.2.m.g 160 11.c even 5 3 inner
605.2.m.g 160 11.d odd 10 3 inner
605.2.m.g 160 55.e even 4 1 inner
605.2.m.g 160 55.k odd 20 3 inner
605.2.m.g 160 55.l even 20 3 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{160} - 146 T_{2}^{156} + 13159 T_{2}^{152} - 954708 T_{2}^{148} + 61579675 T_{2}^{144} + \cdots + 45\!\cdots\!61 \) acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\). Copy content Toggle raw display