Properties

Label 522.2.r.b
Level $522$
Weight $2$
Character orbit 522.r
Analytic conductor $4.168$
Analytic rank $0$
Dimension $72$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [522,2,Mod(89,522)] 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("522.89"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(28)) chi = DirichletCharacter(H, H._module([14, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.r (of order \(28\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(6\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 8 q^{7} + 4 q^{10} + 12 q^{16} - 40 q^{19} - 20 q^{25} - 28 q^{31} - 12 q^{37} - 4 q^{40} - 72 q^{43} + 16 q^{46} + 52 q^{49} + 4 q^{52} + 140 q^{55} + 44 q^{58} + 24 q^{61} + 140 q^{67} - 20 q^{70}+ \cdots + 28 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1 −0.111964 + 0.993712i 0 −0.974928 − 0.222521i −2.49224 − 3.12517i 0 0.207179 + 0.907709i 0.330279 − 0.943883i 0 3.38457 − 2.12666i
89.2 −0.111964 + 0.993712i 0 −0.974928 − 0.222521i −0.139983 − 0.175534i 0 −0.379369 − 1.66212i 0.330279 − 0.943883i 0 0.190103 − 0.119450i
89.3 −0.111964 + 0.993712i 0 −0.974928 − 0.222521i 1.39309 + 1.74688i 0 1.11420 + 4.88164i 0.330279 − 0.943883i 0 −1.89187 + 1.18874i
89.4 0.111964 − 0.993712i 0 −0.974928 − 0.222521i −1.39309 − 1.74688i 0 1.11420 + 4.88164i −0.330279 + 0.943883i 0 −1.89187 + 1.18874i
89.5 0.111964 − 0.993712i 0 −0.974928 − 0.222521i 0.139983 + 0.175534i 0 −0.379369 − 1.66212i −0.330279 + 0.943883i 0 0.190103 − 0.119450i
89.6 0.111964 − 0.993712i 0 −0.974928 − 0.222521i 2.49224 + 3.12517i 0 0.207179 + 0.907709i −0.330279 + 0.943883i 0 3.38457 − 2.12666i
143.1 −0.993712 − 0.111964i 0 0.974928 + 0.222521i −2.74885 − 3.44694i 0 −0.313020 − 1.37143i −0.943883 − 0.330279i 0 2.34563 + 3.73304i
143.2 −0.993712 − 0.111964i 0 0.974928 + 0.222521i 0.543443 + 0.681456i 0 0.457765 + 2.00560i −0.943883 − 0.330279i 0 −0.463727 − 0.738017i
143.3 −0.993712 − 0.111964i 0 0.974928 + 0.222521i 2.06579 + 2.59041i 0 −0.284820 − 1.24788i −0.943883 − 0.330279i 0 −1.76276 − 2.80542i
143.4 0.993712 + 0.111964i 0 0.974928 + 0.222521i −2.06579 − 2.59041i 0 −0.284820 − 1.24788i 0.943883 + 0.330279i 0 −1.76276 − 2.80542i
143.5 0.993712 + 0.111964i 0 0.974928 + 0.222521i −0.543443 − 0.681456i 0 0.457765 + 2.00560i 0.943883 + 0.330279i 0 −0.463727 − 0.738017i
143.6 0.993712 + 0.111964i 0 0.974928 + 0.222521i 2.74885 + 3.44694i 0 −0.313020 − 1.37143i 0.943883 + 0.330279i 0 2.34563 + 3.73304i
251.1 −0.846724 + 0.532032i 0 0.433884 − 0.900969i −0.501312 + 2.19639i 0 2.25090 − 1.08398i 0.111964 + 0.993712i 0 −0.744077 − 2.12645i
251.2 −0.846724 + 0.532032i 0 0.433884 − 0.900969i 0.0961712 − 0.421354i 0 0.802475 − 0.386452i 0.111964 + 0.993712i 0 0.142743 + 0.407937i
251.3 −0.846724 + 0.532032i 0 0.433884 − 0.900969i 0.641917 − 2.81242i 0 −3.82892 + 1.84391i 0.111964 + 0.993712i 0 0.952772 + 2.72287i
251.4 0.846724 − 0.532032i 0 0.433884 − 0.900969i −0.641917 + 2.81242i 0 −3.82892 + 1.84391i −0.111964 − 0.993712i 0 0.952772 + 2.72287i
251.5 0.846724 − 0.532032i 0 0.433884 − 0.900969i −0.0961712 + 0.421354i 0 0.802475 − 0.386452i −0.111964 − 0.993712i 0 0.142743 + 0.407937i
251.6 0.846724 − 0.532032i 0 0.433884 − 0.900969i 0.501312 − 2.19639i 0 2.25090 − 1.08398i −0.111964 − 0.993712i 0 −0.744077 − 2.12645i
269.1 −0.330279 + 0.943883i 0 −0.781831 − 0.623490i −2.61274 + 1.25823i 0 2.60118 + 3.26177i 0.846724 − 0.532032i 0 −0.324689 − 2.88169i
269.2 −0.330279 + 0.943883i 0 −0.781831 − 0.623490i −1.40539 + 0.676801i 0 −0.643463 − 0.806877i 0.846724 − 0.532032i 0 −0.174650 − 1.55006i
See all 72 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 89.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
29.f odd 28 1 inner
87.k even 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 522.2.r.b ✓ 72
3.b odd 2 1 inner 522.2.r.b ✓ 72
29.f odd 28 1 inner 522.2.r.b ✓ 72
87.k even 28 1 inner 522.2.r.b ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
522.2.r.b ✓ 72 1.a even 1 1 trivial
522.2.r.b ✓ 72 3.b odd 2 1 inner
522.2.r.b ✓ 72 29.f odd 28 1 inner
522.2.r.b ✓ 72 87.k even 28 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{72} + 40 T_{5}^{70} + 1367 T_{5}^{68} + 29644 T_{5}^{66} + 514785 T_{5}^{64} + \cdots + 46\!\cdots\!61 \) acting on \(S_{2}^{\mathrm{new}}(522, [\chi])\). Copy content Toggle raw display