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Properties

Label 531.2.i.b
Level $531$
Weight $2$
Character orbit 531.i
Analytic conductor $4.240$
Analytic rank $0$
Dimension $140$
Inner twists $2$

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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] = [531,2,Mod(19,531)] 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("531.19"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(531, base_ring=CyclotomicField(58)) chi = DirichletCharacter(H, H._module([0, 38])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 531.i (of order \(29\), degree \(28\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [140,-1] 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: \(4.24005634733\)
Analytic rank: \(0\)
Dimension: \(140\)
Relative dimension: \(5\) over \(\Q(\zeta_{29})\)
Twist minimal: no (minimal twist has level 177)
Sato-Tate group: $\mathrm{SU}(2)[C_{29}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 140 q - q^{2} - q^{4} - 2 q^{5} - 2 q^{7} + 3 q^{8} - 116 q^{10} - 2 q^{11} + 4 q^{13} + 43 q^{14} + 7 q^{16} - 2 q^{19} - 4 q^{20} + 6 q^{22} - 6 q^{23} - 57 q^{25} - 12 q^{26} - 10 q^{28} + 4 q^{29}+ \cdots + 143 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.09892 + 2.07278i 0 −1.96641 − 2.90024i 2.10012 − 0.462272i 0 0.717585 − 0.545494i 3.50786 − 0.381503i 0 −1.34967 + 4.86109i
19.2 −0.390110 + 0.735825i 0 0.733121 + 1.08127i 0.577946 − 0.127216i 0 0.804132 − 0.611285i −2.73754 + 0.297726i 0 −0.131854 + 0.474895i
19.3 0.166156 − 0.313404i 0 1.05176 + 1.55123i −2.73415 + 0.601833i 0 −2.80376 + 2.13136i 1.36621 − 0.148584i 0 −0.265680 + 0.956892i
19.4 0.715981 − 1.35048i 0 −0.188804 − 0.278465i 3.13782 − 0.690687i 0 −2.11659 + 1.60899i 2.52792 − 0.274928i 0 1.31386 − 4.73210i
19.5 1.07530 − 2.02823i 0 −1.83506 − 2.70652i −1.52678 + 0.336069i 0 3.14104 − 2.38776i −2.89830 + 0.315209i 0 −0.960116 + 3.45803i
28.1 −1.09892 − 2.07278i 0 −1.96641 + 2.90024i 2.10012 + 0.462272i 0 0.717585 + 0.545494i 3.50786 + 0.381503i 0 −1.34967 − 4.86109i
28.2 −0.390110 − 0.735825i 0 0.733121 − 1.08127i 0.577946 + 0.127216i 0 0.804132 + 0.611285i −2.73754 − 0.297726i 0 −0.131854 − 0.474895i
28.3 0.166156 + 0.313404i 0 1.05176 − 1.55123i −2.73415 − 0.601833i 0 −2.80376 − 2.13136i 1.36621 + 0.148584i 0 −0.265680 − 0.956892i
28.4 0.715981 + 1.35048i 0 −0.188804 + 0.278465i 3.13782 + 0.690687i 0 −2.11659 − 1.60899i 2.52792 + 0.274928i 0 1.31386 + 4.73210i
28.5 1.07530 + 2.02823i 0 −1.83506 + 2.70652i −1.52678 − 0.336069i 0 3.14104 + 2.38776i −2.89830 − 0.315209i 0 −0.960116 − 3.45803i
46.1 −1.08904 − 1.28211i 0 −0.134248 + 0.818877i 1.86649 − 3.52057i 0 −3.71509 + 0.404041i −1.68672 + 1.01487i 0 −6.54644 + 1.44098i
46.2 −0.595995 − 0.701659i 0 0.186448 − 1.13728i −0.899692 + 1.69700i 0 2.37631 − 0.258439i −2.48678 + 1.49625i 0 1.72693 − 0.380125i
46.3 0.263983 + 0.310785i 0 0.296664 − 1.80957i 0.967651 − 1.82518i 0 4.66773 − 0.507647i 1.33950 − 0.805950i 0 0.822684 − 0.181087i
46.4 0.363660 + 0.428134i 0 0.272514 − 1.66226i −1.18543 + 2.23597i 0 −4.45607 + 0.484627i 1.77343 − 1.06704i 0 −1.38839 + 0.305607i
46.5 1.70477 + 2.00701i 0 −0.798289 + 4.86935i −1.68034 + 3.16946i 0 2.57060 − 0.279570i −6.62100 + 3.98372i 0 −9.22574 + 2.03074i
64.1 −1.91605 − 0.645591i 0 1.66226 + 1.26362i −0.0203200 + 0.0509993i 0 −3.48406 + 1.61190i −0.0998732 − 0.147302i 0 0.0718588 − 0.0845987i
64.2 −1.32319 − 0.445835i 0 −0.0401232 − 0.0305009i 1.32107 − 3.31564i 0 3.77207 − 1.74515i 1.60664 + 2.36962i 0 −3.22625 + 3.79824i
64.3 −0.815839 − 0.274888i 0 −1.00216 − 0.761819i −1.45570 + 3.65354i 0 0.847760 − 0.392216i 1.57444 + 2.32213i 0 2.19193 − 2.58054i
64.4 1.21498 + 0.409376i 0 −0.283590 − 0.215579i 0.879726 − 2.20794i 0 −0.895277 + 0.414199i −1.69530 − 2.50038i 0 1.97273 − 2.32248i
64.5 1.89244 + 0.637637i 0 1.58256 + 1.20303i −1.39663 + 3.50528i 0 −2.04500 + 0.946119i −0.0135559 − 0.0199934i 0 −4.87813 + 5.74298i
See next 80 embeddings (of 140 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.5
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.c even 29 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 531.2.i.b 140
3.b odd 2 1 177.2.e.b ✓ 140
59.c even 29 1 inner 531.2.i.b 140
177.h odd 58 1 177.2.e.b ✓ 140
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.e.b ✓ 140 3.b odd 2 1
177.2.e.b ✓ 140 177.h odd 58 1
531.2.i.b 140 1.a even 1 1 trivial
531.2.i.b 140 59.c even 29 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{140} + T_{2}^{139} + 6 T_{2}^{138} + 6 T_{2}^{137} + 28 T_{2}^{136} - 3 T_{2}^{135} + \cdots + 335952241 \) acting on \(S_{2}^{\mathrm{new}}(531, [\chi])\). Copy content Toggle raw display