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Properties

Label 539.2.f.i
Level $539$
Weight $2$
Character orbit 539.f
Analytic conductor $4.304$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [539,2,Mod(148,539)] 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("539.148"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(539, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.f (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 16 q^{4} + 20 q^{8} - 8 q^{11} + 16 q^{15} - 20 q^{16} + 42 q^{18} - 4 q^{22} - 12 q^{25} - 80 q^{32} - 160 q^{36} + 20 q^{37} + 36 q^{39} - 56 q^{43} + 50 q^{44} + 22 q^{46} - 84 q^{50} + 60 q^{51}+ \cdots + 76 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
148.1 −0.805554 − 2.47924i −2.32273 − 1.68756i −3.87968 + 2.81875i −0.591639 + 1.82088i −2.31279 + 7.11804i 0 5.89572 + 4.28349i 1.62016 + 4.98635i 4.99099
148.2 −0.805554 − 2.47924i 2.32273 + 1.68756i −3.87968 + 2.81875i 0.591639 − 1.82088i 2.31279 − 7.11804i 0 5.89572 + 4.28349i 1.62016 + 4.98635i −4.99099
148.3 −0.626013 − 1.92667i −2.28466 − 1.65990i −1.70213 + 1.23667i 0.835006 − 2.56988i −1.76786 + 5.44091i 0 0.170359 + 0.123773i 1.53735 + 4.73148i −5.47404
148.4 −0.626013 − 1.92667i 2.28466 + 1.65990i −1.70213 + 1.23667i −0.835006 + 2.56988i 1.76786 − 5.44091i 0 0.170359 + 0.123773i 1.53735 + 4.73148i 5.47404
148.5 −0.535336 − 1.64759i −0.227905 − 0.165583i −0.809949 + 0.588462i −1.31046 + 4.03318i −0.150807 + 0.464137i 0 −1.39991 − 1.01710i −0.902528 − 2.77770i 7.34658
148.6 −0.535336 − 1.64759i 0.227905 + 0.165583i −0.809949 + 0.588462i 1.31046 − 4.03318i 0.150807 − 0.464137i 0 −1.39991 − 1.01710i −0.902528 − 2.77770i −7.34658
148.7 −0.0509017 − 0.156659i −0.472322 − 0.343162i 1.59608 − 1.15962i −0.585481 + 1.80193i −0.0297175 + 0.0914612i 0 −0.529434 − 0.384656i −0.821723 − 2.52900i 0.312091
148.8 −0.0509017 − 0.156659i 0.472322 + 0.343162i 1.59608 − 1.15962i 0.585481 − 1.80193i 0.0297175 − 0.0914612i 0 −0.529434 − 0.384656i −0.821723 − 2.52900i −0.312091
148.9 0.303110 + 0.932876i −1.59837 − 1.16128i 0.839652 − 0.610043i −0.516586 + 1.58989i 0.598851 − 1.84307i 0 2.41070 + 1.75148i 0.279150 + 0.859136i −1.63975
148.10 0.303110 + 0.932876i 1.59837 + 1.16128i 0.839652 − 0.610043i 0.516586 − 1.58989i −0.598851 + 1.84307i 0 2.41070 + 1.75148i 0.279150 + 0.859136i 1.63975
148.11 0.596661 + 1.83633i −1.75290 − 1.27355i −1.39808 + 1.01576i −0.251935 + 0.775375i 1.29278 − 3.97878i 0 0.424696 + 0.308559i 0.523654 + 1.61164i −1.57417
148.12 0.596661 + 1.83633i 1.75290 + 1.27355i −1.39808 + 1.01576i 0.251935 − 0.775375i −1.29278 + 3.97878i 0 0.424696 + 0.308559i 0.523654 + 1.61164i 1.57417
246.1 −1.73786 − 1.26263i −0.394686 + 1.21472i 0.807891 + 2.48643i −2.99750 + 2.17781i 2.21965 − 1.61267i 0 0.407834 − 1.25518i 1.10729 + 0.804493i 7.95900
246.2 −1.73786 − 1.26263i 0.394686 − 1.21472i 0.807891 + 2.48643i 2.99750 − 2.17781i −2.21965 + 1.61267i 0 0.407834 − 1.25518i 1.10729 + 0.804493i −7.95900
246.3 −0.922272 − 0.670070i −0.305751 + 0.941006i −0.216442 − 0.666140i 2.21755 − 1.61115i 0.912526 − 0.662989i 0 −0.951295 + 2.92778i 1.63504 + 1.18793i −3.12477
246.4 −0.922272 − 0.670070i 0.305751 − 0.941006i −0.216442 − 0.666140i −2.21755 + 1.61115i −0.912526 + 0.662989i 0 −0.951295 + 2.92778i 1.63504 + 1.18793i 3.12477
246.5 −0.346956 − 0.252078i −0.733140 + 2.25637i −0.561199 − 1.72719i 0.263947 − 0.191769i 0.823148 − 0.598052i 0 −0.505727 + 1.55647i −2.12667 − 1.54512i −0.139919
246.6 −0.346956 − 0.252078i 0.733140 − 2.25637i −0.561199 − 1.72719i −0.263947 + 0.191769i −0.823148 + 0.598052i 0 −0.505727 + 1.55647i −2.12667 − 1.54512i 0.139919
246.7 0.727552 + 0.528598i −1.00861 + 3.10418i −0.368117 − 1.13295i −1.09386 + 0.794734i −2.37468 + 1.72531i 0 0.886850 − 2.72944i −6.19160 − 4.49846i −1.21593
246.8 0.727552 + 0.528598i 1.00861 − 3.10418i −0.368117 − 1.13295i 1.09386 − 0.794734i 2.37468 − 1.72531i 0 0.886850 − 2.72944i −6.19160 − 4.49846i 1.21593
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 148.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.c even 5 1 inner
77.j odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.f.i ✓ 48
7.b odd 2 1 inner 539.2.f.i ✓ 48
7.c even 3 2 539.2.q.i 96
7.d odd 6 2 539.2.q.i 96
11.c even 5 1 inner 539.2.f.i ✓ 48
11.c even 5 1 5929.2.a.cg 24
11.d odd 10 1 5929.2.a.ch 24
77.j odd 10 1 inner 539.2.f.i ✓ 48
77.j odd 10 1 5929.2.a.cg 24
77.l even 10 1 5929.2.a.ch 24
77.m even 15 2 539.2.q.i 96
77.p odd 30 2 539.2.q.i 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
539.2.f.i ✓ 48 1.a even 1 1 trivial
539.2.f.i ✓ 48 7.b odd 2 1 inner
539.2.f.i ✓ 48 11.c even 5 1 inner
539.2.f.i ✓ 48 77.j odd 10 1 inner
539.2.q.i 96 7.c even 3 2
539.2.q.i 96 7.d odd 6 2
539.2.q.i 96 77.m even 15 2
539.2.q.i 96 77.p odd 30 2
5929.2.a.cg 24 11.c even 5 1
5929.2.a.cg 24 77.j odd 10 1
5929.2.a.ch 24 11.d odd 10 1
5929.2.a.ch 24 77.l even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(539, [\chi])\):

\( T_{2}^{24} + 10 T_{2}^{22} - 10 T_{2}^{21} + 69 T_{2}^{20} - 4 T_{2}^{19} + 488 T_{2}^{18} + 130 T_{2}^{17} + \cdots + 121 \) Copy content Toggle raw display
\( T_{3}^{48} + 18 T_{3}^{46} + 252 T_{3}^{44} + 2968 T_{3}^{42} + 33944 T_{3}^{40} + 259420 T_{3}^{38} + \cdots + 3748096 \) Copy content Toggle raw display