↑

Properties

Label 600.2.bp.c
Level $600$
Weight $2$
Character orbit 600.bp
Analytic conductor $4.791$
Analytic rank $0$
Dimension $896$
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [600,2,Mod(53,600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("600.53"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([0, 10, 10, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.bp (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [896,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(896\)
Relative dimension: \(112\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 896 q - 20 q^{4} - 6 q^{6} - 40 q^{7} - 20 q^{9} - 24 q^{10} - 2 q^{12} + 8 q^{15} - 44 q^{16} + 10 q^{18} + 32 q^{22} - 32 q^{25} - 112 q^{28} + 22 q^{30} - 24 q^{31} - 28 q^{33} - 20 q^{34} + 26 q^{36}+ \cdots + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −1.41039 − 0.103925i 1.13385 − 1.30934i 1.97840 + 0.293150i 2.19952 − 0.402631i −1.73525 + 1.72884i −3.63618 + 3.63618i −2.75985 − 0.619061i −0.428746 − 2.96920i −3.14402 + 0.339281i
53.2 −1.40850 + 0.127001i 1.10205 + 1.33622i 1.96774 − 0.357761i 1.18027 − 1.89920i −1.72194 − 1.74211i 1.93901 − 1.93901i −2.72613 + 0.753812i −0.570981 + 2.94516i −1.42122 + 2.82491i
53.3 −1.40658 − 0.146750i 1.69508 + 0.355958i 1.95693 + 0.412830i −1.37046 − 1.76687i −2.33203 − 0.749435i −2.54321 + 2.54321i −2.69199 − 0.867856i 2.74659 + 1.20675i 1.66837 + 2.68636i
53.4 −1.39853 − 0.210015i −0.152390 + 1.72533i 1.91179 + 0.587427i −1.39171 + 1.75019i 0.575469 − 2.38093i 3.12071 − 3.12071i −2.55033 − 1.22304i −2.95355 − 0.525846i 2.31391 − 2.15541i
53.5 −1.39679 + 0.221320i −1.47971 − 0.900260i 1.90204 − 0.618274i 2.12572 + 0.693764i 2.26608 + 0.929984i 1.12641 − 1.12641i −2.51990 + 1.28456i 1.37907 + 2.66424i −3.12273 − 0.498578i
53.6 −1.39560 − 0.228703i −0.0393175 − 1.73160i 1.89539 + 0.638355i −2.08185 + 0.816027i −0.341151 + 2.42562i 0.792870 − 0.792870i −2.49921 − 1.32437i −2.99691 + 0.136165i 3.09205 − 0.662720i
53.7 −1.39191 − 0.250145i −0.817494 − 1.52699i 1.87485 + 0.696362i 0.572751 − 2.16147i 0.755912 + 2.32994i 0.0999903 − 0.0999903i −2.43545 − 1.43826i −1.66341 + 2.49661i −1.33790 + 2.86531i
53.8 −1.38540 + 0.284038i 1.70512 + 0.304266i 1.83864 − 0.787011i −0.805971 + 2.08576i −2.44868 + 0.0627894i −1.45684 + 1.45684i −2.32371 + 1.61257i 2.81484 + 1.03762i 0.524152 − 3.11854i
53.9 −1.37519 + 0.329945i −1.73079 − 0.0661578i 1.78227 − 0.907473i −2.23595 − 0.0225130i 2.40198 − 0.480086i 0.386054 − 0.386054i −2.15154 + 1.83600i 2.99125 + 0.229010i 3.08228 − 0.706783i
53.10 −1.37000 − 0.350871i −1.34261 + 1.09426i 1.75378 + 0.961384i −1.02476 + 1.98743i 2.22331 − 1.02805i −2.21859 + 2.21859i −2.06535 − 1.93244i 0.605191 − 2.93832i 2.10124 − 2.36321i
53.11 −1.36883 + 0.355379i 0.505169 − 1.65674i 1.74741 − 0.972909i 1.28180 + 1.83221i −0.102721 + 2.44733i 2.18260 − 2.18260i −2.04617 + 1.95274i −2.48961 − 1.67387i −2.40570 − 2.05246i
53.12 −1.36226 + 0.379784i −1.25714 + 1.19146i 1.71153 − 1.03473i 1.37760 − 1.76131i 1.26006 − 2.10053i −1.89543 + 1.89543i −1.93858 + 2.05959i 0.160825 − 2.99569i −1.20774 + 2.92256i
53.13 −1.32924 − 0.482820i −1.72662 + 0.137047i 1.53377 + 1.28357i 1.88813 + 1.19790i 2.36127 + 0.651477i −0.872055 + 0.872055i −1.41902 − 2.44671i 2.96244 − 0.473257i −1.93141 − 2.50393i
53.14 −1.32528 − 0.493584i −0.141468 + 1.72626i 1.51275 + 1.30828i −1.46479 − 1.68950i 1.03954 − 2.21796i −1.23990 + 1.23990i −1.35908 − 2.48051i −2.95997 − 0.488421i 1.10735 + 2.96206i
53.15 −1.31325 + 0.524774i 0.408903 + 1.68309i 1.44923 − 1.37831i 1.76041 + 1.37875i −1.42023 − 1.99573i −2.12021 + 2.12021i −1.17989 + 2.57058i −2.66560 + 1.37644i −3.03538 − 0.886823i
53.16 −1.30127 + 0.553810i 1.19755 − 1.25135i 1.38659 − 1.44131i −0.637143 − 2.14337i −0.865315 + 2.29156i 1.49090 − 1.49090i −1.00611 + 2.64343i −0.131768 − 2.99710i 2.01611 + 2.43624i
53.17 −1.29732 − 0.562982i 1.44129 + 0.960565i 1.36610 + 1.46074i 1.10392 + 1.94457i −1.32904 − 2.05758i 0.394035 − 0.394035i −0.949905 − 2.66415i 1.15463 + 2.76890i −0.337381 − 3.14423i
53.18 −1.25481 − 0.652262i −1.69783 + 0.342590i 1.14911 + 1.63693i −0.779635 − 2.09575i 2.35392 + 0.677544i 3.10884 − 3.10884i −0.374208 − 2.80356i 2.76526 − 1.16332i −0.388682 + 3.13830i
53.19 −1.23979 − 0.680381i 1.57634 − 0.717740i 1.07416 + 1.68706i 2.01194 − 0.975746i −2.44267 − 0.182665i 2.46251 − 2.46251i −0.183894 − 2.82244i 1.96970 − 2.26281i −3.15827 − 0.159167i
53.20 −1.21291 + 0.727226i −1.19755 + 1.25135i 0.942285 − 1.76411i 0.637143 + 2.14337i 0.542495 − 2.38866i 1.49090 − 1.49090i 0.140005 + 2.82496i −0.131768 − 2.99710i −2.33151 − 2.13637i
See next 80 embeddings (of 896 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.112
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner
25.f odd 20 1 inner
75.l even 20 1 inner
200.x odd 20 1 inner
600.bp 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.bp.c ✓ 896
3.b odd 2 1 inner 600.2.bp.c ✓ 896
8.b even 2 1 inner 600.2.bp.c ✓ 896
24.h odd 2 1 inner 600.2.bp.c ✓ 896
25.f odd 20 1 inner 600.2.bp.c ✓ 896
75.l even 20 1 inner 600.2.bp.c ✓ 896
200.x odd 20 1 inner 600.2.bp.c ✓ 896
600.bp even 20 1 inner 600.2.bp.c ✓ 896
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.bp.c ✓ 896 1.a even 1 1 trivial
600.2.bp.c ✓ 896 3.b odd 2 1 inner
600.2.bp.c ✓ 896 8.b even 2 1 inner
600.2.bp.c ✓ 896 24.h odd 2 1 inner
600.2.bp.c ✓ 896 25.f odd 20 1 inner
600.2.bp.c ✓ 896 75.l even 20 1 inner
600.2.bp.c ✓ 896 200.x odd 20 1 inner
600.2.bp.c ✓ 896 600.bp even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\):

\( T_{7}^{224} + 10 T_{7}^{223} + 50 T_{7}^{222} + 104 T_{7}^{221} + 5807 T_{7}^{220} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
\( T_{11}^{448} + 638 T_{11}^{446} + 210288 T_{11}^{444} + 47714846 T_{11}^{442} + 8380203960 T_{11}^{440} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display