Properties

Label 750.2.l.a
Level $750$
Weight $2$
Character orbit 750.l
Analytic conductor $5.989$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [750,2,Mod(107,750)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(750, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("750.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 750.l (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: \(5.98878015160\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(10\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 150)
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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 4 q^{3} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 4 q^{3} - 4 q^{7} - 16 q^{12} + 20 q^{16} + 8 q^{18} + 40 q^{19} - 4 q^{22} + 56 q^{27} - 4 q^{28} + 96 q^{33} + 40 q^{34} + 64 q^{37} + 40 q^{39} + 4 q^{42} + 24 q^{43} - 16 q^{48} + 64 q^{57} - 20 q^{58} - 4 q^{63} + 104 q^{67} - 140 q^{69} - 8 q^{72} + 60 q^{73} + 60 q^{78} - 80 q^{79} - 40 q^{81} - 96 q^{82} - 60 q^{84} - 80 q^{87} - 24 q^{88} - 12 q^{93} + 12 q^{97}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −0.891007 0.453990i −1.65131 + 0.522656i 0.587785 + 0.809017i 0 1.70861 + 0.283990i −0.712495 0.712495i −0.156434 0.987688i 2.45366 1.72614i 0
107.2 −0.891007 0.453990i −0.974581 1.43185i 0.587785 + 0.809017i 0 0.218312 + 1.71824i −3.13589 3.13589i −0.156434 0.987688i −1.10038 + 2.79091i 0
107.3 −0.891007 0.453990i 0.102145 + 1.72904i 0.587785 + 0.809017i 0 0.693954 1.58696i 2.97677 + 2.97677i −0.156434 0.987688i −2.97913 + 0.353226i 0
107.4 −0.891007 0.453990i 0.646378 + 1.60692i 0.587785 + 0.809017i 0 0.153600 1.72523i −2.03922 2.03922i −0.156434 0.987688i −2.16439 + 2.07736i 0
107.5 −0.891007 0.453990i 1.59860 0.666690i 0.587785 + 0.809017i 0 −1.72703 0.131724i 1.51403 + 1.51403i −0.156434 0.987688i 2.11105 2.13154i 0
107.6 0.891007 + 0.453990i −1.31434 1.12805i 0.587785 + 0.809017i 0 −0.658960 1.60180i 1.51403 + 1.51403i 0.156434 + 0.987688i 0.454984 + 2.96530i 0
107.7 0.891007 + 0.453990i −1.11131 + 1.32853i 0.587785 + 0.809017i 0 −1.59332 + 0.679206i −2.03922 2.03922i 0.156434 + 0.987688i −0.529988 2.95281i 0
107.8 0.891007 + 0.453990i −0.631448 + 1.61285i 0.587785 + 0.809017i 0 −1.29484 + 1.15039i 2.97677 + 2.97677i 0.156434 + 0.987688i −2.20255 2.03686i 0
107.9 0.891007 + 0.453990i 1.36935 1.06061i 0.587785 + 0.809017i 0 1.70160 0.323338i −3.13589 3.13589i 0.156434 + 0.987688i 0.750223 2.90468i 0
107.10 0.891007 + 0.453990i 1.40898 + 1.00736i 0.587785 + 0.809017i 0 0.798080 + 1.53723i −0.712495 0.712495i 0.156434 + 0.987688i 0.970457 + 2.83870i 0
143.1 −0.453990 + 0.891007i −1.51894 0.832356i −0.587785 0.809017i 0 1.43122 0.975505i −0.0556476 + 0.0556476i 0.987688 0.156434i 1.61437 + 2.52860i 0
143.2 −0.453990 + 0.891007i −0.333634 + 1.69961i −0.587785 0.809017i 0 −1.36290 1.06888i −2.58285 + 2.58285i 0.987688 0.156434i −2.77738 1.13410i 0
143.3 −0.453990 + 0.891007i 0.00452789 1.73204i −0.587785 0.809017i 0 1.54121 + 0.790366i 0.152718 0.152718i 0.987688 0.156434i −2.99996 0.0156850i 0
143.4 −0.453990 + 0.891007i 1.05251 + 1.37558i −0.587785 0.809017i 0 −1.70348 + 0.313291i −0.462249 + 0.462249i 0.987688 0.156434i −0.784450 + 2.89562i 0
143.5 −0.453990 + 0.891007i 1.69234 0.368756i −0.587785 0.809017i 0 −0.439743 + 1.67530i 2.72680 2.72680i 0.987688 0.156434i 2.72804 1.24812i 0
143.6 0.453990 0.891007i −1.70181 + 0.322239i −0.587785 0.809017i 0 −0.485490 + 1.66262i −0.0556476 + 0.0556476i −0.987688 + 0.156434i 2.79232 1.09678i 0
143.7 0.453990 0.891007i −0.530925 + 1.64867i −0.587785 0.809017i 0 1.22794 + 1.22154i 0.152718 0.152718i −0.987688 + 0.156434i −2.43624 1.75064i 0
143.8 0.453990 0.891007i 0.207905 1.71953i −0.587785 0.809017i 0 −1.43772 0.965894i −2.58285 + 2.58285i −0.987688 + 0.156434i −2.91355 0.714995i 0
143.9 0.453990 0.891007i 1.42607 0.983013i −0.587785 0.809017i 0 −0.228447 1.71692i −0.462249 + 0.462249i −0.987688 + 0.156434i 1.06737 2.80370i 0
143.10 0.453990 0.891007i 1.49556 + 0.873670i −0.587785 0.809017i 0 1.45742 0.935916i 2.72680 2.72680i −0.987688 + 0.156434i 1.47340 + 2.61325i 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 750.2.l.a 80
3.b odd 2 1 inner 750.2.l.a 80
5.b even 2 1 750.2.l.c 80
5.c odd 4 1 150.2.l.a 80
5.c odd 4 1 750.2.l.b 80
15.d odd 2 1 750.2.l.c 80
15.e even 4 1 150.2.l.a 80
15.e even 4 1 750.2.l.b 80
25.d even 5 1 750.2.l.b 80
25.e even 10 1 150.2.l.a 80
25.f odd 20 1 inner 750.2.l.a 80
25.f odd 20 1 750.2.l.c 80
75.h odd 10 1 150.2.l.a 80
75.j odd 10 1 750.2.l.b 80
75.l even 20 1 inner 750.2.l.a 80
75.l even 20 1 750.2.l.c 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.l.a 80 5.c odd 4 1
150.2.l.a 80 15.e even 4 1
150.2.l.a 80 25.e even 10 1
150.2.l.a 80 75.h odd 10 1
750.2.l.a 80 1.a even 1 1 trivial
750.2.l.a 80 3.b odd 2 1 inner
750.2.l.a 80 25.f odd 20 1 inner
750.2.l.a 80 75.l even 20 1 inner
750.2.l.b 80 5.c odd 4 1
750.2.l.b 80 15.e even 4 1
750.2.l.b 80 25.d even 5 1
750.2.l.b 80 75.j odd 10 1
750.2.l.c 80 5.b even 2 1
750.2.l.c 80 15.d odd 2 1
750.2.l.c 80 25.f odd 20 1
750.2.l.c 80 75.l even 20 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(750, [\chi])\):

\( T_{7}^{40} + 2 T_{7}^{39} + 2 T_{7}^{38} - 20 T_{7}^{37} + 1183 T_{7}^{36} + 2404 T_{7}^{35} + 2642 T_{7}^{34} - 27954 T_{7}^{33} + 487851 T_{7}^{32} + 908296 T_{7}^{31} + 1164092 T_{7}^{30} - 12300768 T_{7}^{29} + \cdots + 5776 \) Copy content Toggle raw display
\( T_{13}^{40} - 20 T_{13}^{38} - 240 T_{13}^{37} - 735 T_{13}^{36} + 600 T_{13}^{35} + 35250 T_{13}^{34} + 197100 T_{13}^{33} + 1117575 T_{13}^{32} - 3114500 T_{13}^{31} - 21023750 T_{13}^{30} - 45746000 T_{13}^{29} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display