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Properties

Label 522.2.x.a
Level $522$
Weight $2$
Character orbit 522.x
Analytic conductor $4.168$
Analytic rank $0$
Dimension $720$
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] = [522,2,Mod(11,522)] 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("522.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(84)) chi = DirichletCharacter(H, H._module([14, 75])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.x (of order \(84\), degree \(24\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(720\)
Relative dimension: \(30\) over \(\Q(\zeta_{84})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{84}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 720 q + 24 q^{11} - 12 q^{14} - 20 q^{15} - 60 q^{16} + 84 q^{21} + 20 q^{24} + 60 q^{25} - 144 q^{27} - 12 q^{29} + 72 q^{30} + 84 q^{33} - 40 q^{36} + 40 q^{39} - 12 q^{41} - 104 q^{45} + 24 q^{46} - 24 q^{47}+ \cdots - 84 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −0.982566 − 0.185912i −1.71642 − 0.232147i 0.930874 + 0.365341i −0.184951 − 2.46800i 1.64334 + 0.547203i −1.56577 − 3.98951i −0.846724 − 0.532032i 2.89222 + 0.796925i −0.277103 + 2.45936i
11.2 −0.982566 − 0.185912i −1.50597 + 0.855594i 0.930874 + 0.365341i −0.0639284 − 0.853065i 1.63878 − 0.560699i 0.298111 + 0.759576i −0.846724 − 0.532032i 1.53592 − 2.57700i −0.0957808 + 0.850078i
11.3 −0.982566 − 0.185912i −1.44169 − 0.959969i 0.930874 + 0.365341i −0.275865 − 3.68116i 1.23808 + 1.21126i 1.68705 + 4.29853i −0.846724 − 0.532032i 1.15692 + 2.76795i −0.413315 + 3.66827i
11.4 −0.982566 − 0.185912i −1.29418 − 1.15113i 0.930874 + 0.365341i 0.0856576 + 1.14302i 1.05761 + 1.37167i −0.125872 − 0.320716i −0.846724 − 0.532032i 0.349797 + 2.97954i 0.128337 − 1.13902i
11.5 −0.982566 − 0.185912i −1.23504 − 1.21436i 0.930874 + 0.365341i 0.204402 + 2.72756i 0.987747 + 1.42280i 0.592755 + 1.51031i −0.846724 − 0.532032i 0.0506548 + 2.99957i 0.306246 − 2.71801i
11.6 −0.982566 − 0.185912i −0.820653 + 1.52530i 0.930874 + 0.365341i 0.213523 + 2.84927i 1.08992 − 1.34614i −1.60496 − 4.08938i −0.846724 − 0.532032i −1.65306 − 2.50348i 0.319911 − 2.83929i
11.7 −0.982566 − 0.185912i −0.518316 + 1.65268i 0.930874 + 0.365341i 0.0879327 + 1.17338i 0.816532 − 1.52751i −0.445691 − 1.13560i −0.846724 − 0.532032i −2.46270 − 1.71322i 0.131745 − 1.16927i
11.8 −0.982566 − 0.185912i 0.0343479 − 1.73171i 0.930874 + 0.365341i −0.136577 − 1.82250i −0.355694 + 1.69513i −1.64863 − 4.20064i −0.846724 − 0.532032i −2.99764 − 0.118961i −0.204627 + 1.81612i
11.9 −0.982566 − 0.185912i 0.157136 + 1.72491i 0.930874 + 0.365341i −0.0842626 − 1.12441i 0.166284 − 1.72405i 1.53953 + 3.92267i −0.846724 − 0.532032i −2.95062 + 0.542091i −0.126247 + 1.12047i
11.10 −0.982566 − 0.185912i 0.555505 − 1.64055i 0.930874 + 0.365341i −0.0412137 − 0.549958i −0.850818 + 1.50868i 0.841625 + 2.14443i −0.846724 − 0.532032i −2.38283 − 1.82267i −0.0617484 + 0.548033i
11.11 −0.982566 − 0.185912i 0.767664 − 1.55264i 0.930874 + 0.365341i 0.318141 + 4.24529i −1.04293 + 1.38285i −0.310906 − 0.792175i −0.846724 − 0.532032i −1.82138 − 2.38381i 0.476655 − 4.23043i
11.12 −0.982566 − 0.185912i 1.05165 + 1.37624i 0.930874 + 0.365341i −0.0759607 − 1.01362i −0.777456 − 1.54776i −0.646748 − 1.64789i −0.846724 − 0.532032i −0.788070 + 2.89464i −0.113808 + 1.01008i
11.13 −0.982566 − 0.185912i 1.56532 − 0.741469i 0.930874 + 0.365341i −0.303679 − 4.05232i −1.67588 + 0.437531i −0.0290033 − 0.0738992i −0.846724 − 0.532032i 1.90045 − 2.32127i −0.454988 + 4.03813i
11.14 −0.982566 − 0.185912i 1.59878 + 0.666256i 0.930874 + 0.365341i 0.242739 + 3.23913i −1.44704 − 0.951873i 1.06501 + 2.71360i −0.846724 − 0.532032i 2.11221 + 2.13040i 0.363684 − 3.22778i
11.15 −0.982566 − 0.185912i 1.73184 − 0.0268280i 0.930874 + 0.365341i 0.0140425 + 0.187384i −1.70664 − 0.295609i 0.495198 + 1.26174i −0.846724 − 0.532032i 2.99856 − 0.0929238i 0.0210392 − 0.186728i
11.16 0.982566 + 0.185912i −1.67126 − 0.454848i 0.930874 + 0.365341i 0.195862 + 2.61359i −1.55756 − 0.757625i −1.29177 − 3.29137i 0.846724 + 0.532032i 2.58623 + 1.52034i −0.293450 + 2.60444i
11.17 0.982566 + 0.185912i −1.65841 − 0.499689i 0.930874 + 0.365341i −0.0777480 − 1.03748i −1.53660 − 0.799295i 1.12244 + 2.85994i 0.846724 + 0.532032i 2.50062 + 1.65738i 0.116486 − 1.03384i
11.18 0.982566 + 0.185912i −1.40938 + 1.00680i 0.930874 + 0.365341i −0.0704594 − 0.940215i −1.57199 + 0.727222i −1.16593 − 2.97075i 0.846724 + 0.532032i 0.972727 − 2.83792i 0.105566 − 0.936923i
11.19 0.982566 + 0.185912i −1.07799 − 1.35570i 0.930874 + 0.365341i −0.186309 − 2.48612i −0.807159 − 1.53248i −0.574533 − 1.46389i 0.846724 + 0.532032i −0.675861 + 2.92288i 0.279138 − 2.47742i
11.20 0.982566 + 0.185912i −0.924014 + 1.46499i 0.930874 + 0.365341i 0.254779 + 3.39979i −1.18026 + 1.26767i 0.116202 + 0.296079i 0.846724 + 0.532032i −1.29239 − 2.70734i −0.381723 + 3.38789i
See next 80 embeddings (of 720 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
29.f odd 28 1 inner
261.x even 84 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 522.2.x.a ✓ 720
9.d odd 6 1 inner 522.2.x.a ✓ 720
29.f odd 28 1 inner 522.2.x.a ✓ 720
261.x even 84 1 inner 522.2.x.a ✓ 720
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
522.2.x.a ✓ 720 1.a even 1 1 trivial
522.2.x.a ✓ 720 9.d odd 6 1 inner
522.2.x.a ✓ 720 29.f odd 28 1 inner
522.2.x.a ✓ 720 261.x even 84 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(522, [\chi])\).