↑

Properties

Label 600.2.bm.a
Level $600$
Weight $2$
Character orbit 600.bm
Analytic conductor $4.791$
Analytic rank $0$
Dimension $240$
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] = [600,2,Mod(61,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.bm (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: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(60\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 18 q^{8} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 18 q^{8} + 60 q^{9} - 2 q^{10} - 8 q^{12} + 12 q^{14} - 12 q^{16} + 8 q^{17} + 16 q^{20} - 38 q^{22} - 4 q^{25} - 20 q^{26} + 28 q^{28} - 36 q^{30} - 24 q^{31} - 20 q^{32} - 16 q^{34} - 46 q^{38} + 16 q^{39} + 40 q^{40} - 8 q^{41} - 20 q^{42} + 42 q^{44} + 20 q^{46} + 48 q^{47} - 16 q^{48} + 240 q^{49} - 144 q^{50} + 16 q^{52} - 16 q^{55} + 36 q^{56} - 94 q^{58} - 2 q^{60} - 66 q^{62} + 42 q^{64} + 84 q^{65} + 8 q^{66} + 16 q^{68} + 4 q^{70} - 18 q^{72} + 40 q^{73} - 44 q^{74} + 8 q^{76} - 24 q^{78} - 40 q^{79} - 16 q^{80} - 60 q^{81} - 80 q^{82} - 24 q^{84} - 60 q^{86} + 22 q^{88} - 60 q^{89} - 28 q^{90} + 14 q^{92} - 50 q^{94} - 184 q^{95} + 40 q^{96} + 32 q^{97} - 140 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 −1.41274 − 0.0644444i −0.951057 − 0.309017i 1.99169 + 0.182087i −2.08111 − 0.817909i 1.32369 + 0.497852i 3.46491 −2.80202 − 0.385596i 0.809017 + 0.587785i 2.88737 + 1.28961i
61.2 −1.41021 − 0.106287i 0.951057 + 0.309017i 1.97741 + 0.299775i 0.718382 + 2.11753i −1.30835 − 0.536865i −1.37109 −2.75670 − 0.632920i 0.809017 + 0.587785i −0.788007 − 3.06252i
61.3 −1.40935 − 0.117131i 0.951057 + 0.309017i 1.97256 + 0.330158i 1.93965 − 1.11254i −1.30418 − 0.546913i −3.27587 −2.74137 − 0.696357i 0.809017 + 0.587785i −2.86397 + 1.34077i
61.4 −1.37700 − 0.322293i −0.951057 − 0.309017i 1.79225 + 0.887595i −2.23466 + 0.0794374i 1.21001 + 0.732035i −4.30924 −2.18187 − 1.79985i 0.809017 + 0.587785i 3.10272 + 0.610829i
61.5 −1.36761 + 0.360054i −0.951057 − 0.309017i 1.74072 − 0.984828i 0.207132 − 2.22645i 1.41194 + 0.0801834i −2.14729 −2.02604 + 1.97362i 0.809017 + 0.587785i 0.518368 + 3.11950i
61.6 −1.36652 − 0.364183i −0.951057 − 0.309017i 1.73474 + 0.995324i 2.15049 + 0.612698i 1.18710 + 0.768635i 4.57226 −2.00808 − 1.99189i 0.809017 + 0.587785i −2.71555 − 1.62043i
61.7 −1.32863 + 0.484503i −0.951057 − 0.309017i 1.53051 − 1.28745i 2.21490 + 0.306977i 1.41332 − 0.0502205i −1.70149 −1.40971 + 2.45208i 0.809017 + 0.587785i −3.09151 + 0.665266i
61.8 −1.32488 + 0.494658i 0.951057 + 0.309017i 1.51063 − 1.31073i −1.21629 − 1.87633i −1.41290 + 0.0610369i −1.57996 −1.35304 + 2.48380i 0.809017 + 0.587785i 2.53959 + 1.88427i
61.9 −1.28979 + 0.580027i 0.951057 + 0.309017i 1.32714 − 1.49623i −1.11784 + 1.93660i −1.40591 + 0.153070i 4.51209 −0.843882 + 2.69960i 0.809017 + 0.587785i 0.318507 − 3.14620i
61.10 −1.19482 + 0.756580i 0.951057 + 0.309017i 0.855174 − 1.80795i 2.02065 − 0.957578i −1.37013 + 0.350332i 1.19999 0.346081 + 2.80717i 0.809017 + 0.587785i −1.68983 + 2.67292i
61.11 −1.15902 − 0.810359i 0.951057 + 0.309017i 0.686637 + 1.87844i 0.663628 − 2.13532i −0.851875 − 1.12885i 3.38107 0.726386 − 2.73356i 0.809017 + 0.587785i −2.49953 + 1.93710i
61.12 −1.15246 − 0.819659i 0.951057 + 0.309017i 0.656319 + 1.88924i −1.25453 − 1.85099i −0.842764 − 1.13567i −3.81300 0.792155 − 2.71523i 0.809017 + 0.587785i −0.0713897 + 3.16147i
61.13 −1.08877 + 0.902542i −0.951057 − 0.309017i 0.370835 − 1.96532i −2.02065 + 0.957578i 1.31438 − 0.521921i 1.19999 1.37003 + 2.47447i 0.809017 + 0.587785i 1.33577 − 2.86631i
61.14 −1.02235 − 0.977139i −0.951057 − 0.309017i 0.0903997 + 1.99796i 2.23571 + 0.0401037i 0.670360 + 1.24524i −3.15183 1.85986 − 2.13094i 0.809017 + 0.587785i −2.24649 − 2.22560i
61.15 −0.967261 − 1.03170i 0.951057 + 0.309017i −0.128814 + 1.99585i 1.68421 + 1.47086i −0.601106 − 1.28011i 2.08501 2.18371 − 1.79761i 0.809017 + 0.587785i −0.111588 − 3.16031i
61.16 −0.950206 + 1.04743i −0.951057 − 0.309017i −0.194215 − 1.99055i 1.11784 − 1.93660i 1.22737 − 0.702534i 4.51209 2.26950 + 1.68800i 0.809017 + 0.587785i 0.966272 + 3.01103i
61.17 −0.884319 − 1.10362i −0.951057 − 0.309017i −0.435960 + 1.95191i −0.126928 + 2.23246i 0.499999 + 1.32288i 2.60298 2.53969 − 1.24497i 0.809017 + 0.587785i 2.57604 − 1.83413i
61.18 −0.883166 − 1.10454i −0.951057 − 0.309017i −0.440035 + 1.95099i −1.08015 − 1.95787i 0.498618 + 1.32340i 0.659576 2.54358 − 1.23701i 0.809017 + 0.587785i −1.20861 + 2.92220i
61.19 −0.879859 + 1.10718i −0.951057 − 0.309017i −0.451696 − 1.94833i 1.21629 + 1.87633i 1.17893 − 0.781099i −1.57996 2.55458 + 1.21414i 0.809017 + 0.587785i −3.14761 − 0.304253i
61.20 −0.871359 + 1.11388i 0.951057 + 0.309017i −0.481467 − 1.94118i −2.21490 − 0.306977i −1.17292 + 0.790100i −1.70149 2.58178 + 1.15517i 0.809017 + 0.587785i 2.27191 − 2.19965i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
25.d even 5 1 inner
200.t even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.bm.a ✓ 240
8.b even 2 1 inner 600.2.bm.a ✓ 240
25.d even 5 1 inner 600.2.bm.a ✓ 240
200.t even 10 1 inner 600.2.bm.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.bm.a ✓ 240 1.a even 1 1 trivial
600.2.bm.a ✓ 240 8.b even 2 1 inner
600.2.bm.a ✓ 240 25.d even 5 1 inner
600.2.bm.a ✓ 240 200.t even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(600, [\chi])\).