↑

Properties

Label 600.2.bq.a
Level $600$
Weight $2$
Character orbit 600.bq
Analytic conductor $4.791$
Analytic rank $0$
Dimension $480$
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(67,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 10, 0, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.bq (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: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(60\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 480 q + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q + 12 q^{8} - 8 q^{10} + 8 q^{12} - 8 q^{17} + 20 q^{20} + 28 q^{22} + 8 q^{25} - 4 q^{28} - 8 q^{30} - 20 q^{32} + 48 q^{35} + 20 q^{38} - 108 q^{40} - 20 q^{42} + 32 q^{43} - 140 q^{44} - 16 q^{48} + 124 q^{50} - 132 q^{52} + 152 q^{58} - 20 q^{60} + 116 q^{62} - 60 q^{64} + 8 q^{65} - 48 q^{67} - 72 q^{68} - 4 q^{70} - 12 q^{72} + 40 q^{73} + 24 q^{78} - 76 q^{80} + 120 q^{81} - 64 q^{82} - 80 q^{83} - 80 q^{84} - 132 q^{88} + 200 q^{89} - 48 q^{90} - 216 q^{92} - 40 q^{94} + 20 q^{96} - 56 q^{97} - 192 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1 −1.41400 − 0.0245098i 0.156434 − 0.987688i 1.99880 + 0.0693137i 1.36582 − 1.77046i −0.245407 + 1.39276i −2.04585 + 2.04585i −2.82460 − 0.147000i −0.951057 − 0.309017i −1.97467 + 2.46996i
67.2 −1.41177 + 0.0830388i 0.156434 − 0.987688i 1.98621 − 0.234464i 1.96811 + 1.06140i −0.138834 + 1.40738i −0.856582 + 0.856582i −2.78461 + 0.495942i −0.951057 − 0.309017i −2.86666 − 1.33502i
67.3 −1.40718 − 0.140882i −0.156434 + 0.987688i 1.96030 + 0.396491i −2.05535 − 0.880639i 0.359278 − 1.36782i 0.433708 − 0.433708i −2.70264 − 0.834106i −0.951057 − 0.309017i 2.76818 + 1.52878i
67.4 −1.37413 + 0.334299i 0.156434 − 0.987688i 1.77649 − 0.918743i −0.826522 − 2.07771i 0.115221 + 1.40951i 2.09317 − 2.09317i −2.13400 + 1.85635i −0.951057 − 0.309017i 1.83033 + 2.57874i
67.5 −1.36825 − 0.357612i −0.156434 + 0.987688i 1.74423 + 0.978605i 2.04174 + 0.911759i 0.567251 − 1.29546i −0.598416 + 0.598416i −2.03658 − 1.96274i −0.951057 − 0.309017i −2.46756 − 1.97767i
67.6 −1.36370 − 0.374578i 0.156434 − 0.987688i 1.71938 + 1.02163i −2.23606 + 0.00551913i −0.583297 + 1.28832i −1.28065 + 1.28065i −1.96205 − 2.03724i −0.951057 − 0.309017i 3.05139 + 0.830053i
67.7 −1.32587 + 0.492011i −0.156434 + 0.987688i 1.51585 − 1.30468i −0.735417 + 2.11167i −0.278542 − 1.38651i 2.48226 − 2.48226i −1.36790 + 2.47565i −0.951057 − 0.309017i −0.0638993 − 3.16163i
67.8 −1.30912 + 0.534989i −0.156434 + 0.987688i 1.42757 − 1.40073i 0.932862 − 2.03218i −0.323611 − 1.37669i 1.82779 − 1.82779i −1.11949 + 2.59745i −0.951057 − 0.309017i −0.134030 + 3.15944i
67.9 −1.27604 − 0.609682i −0.156434 + 0.987688i 1.25658 + 1.55596i −0.786800 + 2.09307i 0.801793 − 1.16496i −2.82227 + 2.82227i −0.654805 − 2.75159i −0.951057 − 0.309017i 2.28010 − 2.19115i
67.10 −1.25246 + 0.656773i 0.156434 − 0.987688i 1.13730 − 1.64516i −1.61862 + 1.54274i 0.452760 + 1.33978i 0.875917 − 0.875917i −0.343918 + 2.80744i −0.951057 − 0.309017i 1.01402 − 2.99529i
67.11 −1.21959 + 0.715968i −0.156434 + 0.987688i 0.974780 − 1.74637i 1.86666 + 1.23110i −0.516368 − 1.31657i −2.13082 + 2.13082i 0.0615147 + 2.82776i −0.951057 − 0.309017i −3.15797 − 0.164961i
67.12 −1.20047 − 0.747580i 0.156434 − 0.987688i 0.882248 + 1.79489i 2.16585 − 0.555978i −0.926171 + 1.06874i 3.40458 − 3.40458i 0.282714 − 2.81426i −0.951057 − 0.309017i −3.01567 − 0.951709i
67.13 −1.13103 − 0.848976i −0.156434 + 0.987688i 0.558480 + 1.92044i −0.164444 − 2.23001i 1.01546 − 0.984301i 1.00526 − 1.00526i 0.998749 − 2.64622i −0.951057 − 0.309017i −1.70723 + 2.66183i
67.14 −1.12794 − 0.853088i 0.156434 − 0.987688i 0.544481 + 1.92446i 0.709006 + 2.12069i −1.01903 + 0.980597i −1.11085 + 1.11085i 1.02759 − 2.63516i −0.951057 − 0.309017i 1.00942 − 2.99684i
67.15 −1.05549 − 0.941247i 0.156434 − 0.987688i 0.228107 + 1.98695i −1.03358 − 1.98285i −1.09477 + 0.895249i −3.05339 + 3.05339i 1.62945 − 2.31190i −0.951057 − 0.309017i −0.775421 + 3.06573i
67.16 −0.975187 + 1.02421i 0.156434 − 0.987688i −0.0980218 − 1.99760i 1.77995 + 1.35343i 0.859050 + 1.12340i 0.707964 − 0.707964i 2.14155 + 1.84763i −0.951057 − 0.309017i −3.12199 + 0.503199i
67.17 −0.880155 − 1.10695i −0.156434 + 0.987688i −0.450655 + 1.94857i −1.76274 + 1.37578i 1.23100 − 0.696154i 2.32720 − 2.32720i 2.55360 − 1.21619i −0.951057 − 0.309017i 3.07439 + 0.740356i
67.18 −0.811220 + 1.15841i −0.156434 + 0.987688i −0.683845 − 1.87946i −2.20028 + 0.398472i −1.01725 − 0.982448i −1.21844 + 1.21844i 2.73194 + 0.732477i −0.951057 − 0.309017i 1.32331 − 2.87208i
67.19 −0.765205 + 1.18931i −0.156434 + 0.987688i −0.828921 − 1.82013i −0.162207 − 2.23018i −1.05496 − 0.941834i −1.78241 + 1.78241i 2.79900 + 0.406932i −0.951057 − 0.309017i 2.77650 + 1.51363i
67.20 −0.669159 − 1.24588i 0.156434 − 0.987688i −1.10445 + 1.66739i −1.28288 + 1.83145i −1.33522 + 0.466021i 0.620319 − 0.620319i 2.81643 + 0.260274i −0.951057 − 0.309017i 3.14023 + 0.372786i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
25.f odd 20 1 inner
200.v even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.bq.a ✓ 480
8.d odd 2 1 inner 600.2.bq.a ✓ 480
25.f odd 20 1 inner 600.2.bq.a ✓ 480
200.v even 20 1 inner 600.2.bq.a ✓ 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.bq.a ✓ 480 1.a even 1 1 trivial
600.2.bq.a ✓ 480 8.d odd 2 1 inner
600.2.bq.a ✓ 480 25.f odd 20 1 inner
600.2.bq.a ✓ 480 200.v even 20 1 inner

Hecke kernels

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