Properties

Label 522.2.q.b
Level $522$
Weight $2$
Character orbit 522.q
Analytic conductor $4.168$
Analytic rank $0$
Dimension $180$
Inner twists $4$

Related objects

Downloads

Learn more

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(7,522)] 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("522.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([28, 18])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.q (of order \(21\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [180,15] 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: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(180\)
Relative dimension: \(15\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 180 q + 15 q^{2} + 2 q^{3} + 15 q^{4} + 2 q^{5} + 10 q^{6} + 5 q^{7} - 30 q^{8} - 6 q^{9} - 4 q^{10} - 4 q^{11} + 2 q^{12} - 2 q^{14} - q^{15} + 15 q^{16} + 12 q^{17} + 6 q^{18} - 6 q^{19} + 2 q^{20} - 13 q^{21}+ \cdots - 172 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 0.955573 + 0.294755i −1.73109 + 0.0578158i 0.826239 + 0.563320i −3.06329 + 2.84232i −1.67122 − 0.454999i 1.22536 − 0.835436i 0.623490 + 0.781831i 2.99331 − 0.200168i −3.76499 + 1.81312i
7.2 0.955573 + 0.294755i −1.71861 − 0.215323i 0.826239 + 0.563320i 0.552060 − 0.512236i −1.57879 − 0.712327i −0.169574 + 0.115613i 0.623490 + 0.781831i 2.90727 + 0.740115i 0.678517 − 0.326757i
7.3 0.955573 + 0.294755i −1.49455 − 0.875400i 0.826239 + 0.563320i −0.311988 + 0.289483i −1.17012 − 1.27703i 0.0671852 − 0.0458061i 0.623490 + 0.781831i 1.46735 + 2.61665i −0.383454 + 0.184662i
7.4 0.955573 + 0.294755i −1.24872 + 1.20029i 0.826239 + 0.563320i −0.0107276 + 0.00995374i −1.54704 + 0.778895i −3.71100 + 2.53011i 0.623490 + 0.781831i 0.118617 − 2.99765i −0.0131849 + 0.00634951i
7.5 0.955573 + 0.294755i −1.17480 + 1.27273i 0.826239 + 0.563320i 1.78975 − 1.66065i −1.49775 + 0.869907i 3.46103 − 2.35969i 0.623490 + 0.781831i −0.239680 − 2.99041i 2.19972 − 1.05933i
7.6 0.955573 + 0.294755i −0.658569 − 1.60196i 0.826239 + 0.563320i 3.10032 − 2.87668i −0.157123 − 1.72491i 0.333540 − 0.227404i 0.623490 + 0.781831i −2.13257 + 2.11001i 3.81050 − 1.83504i
7.7 0.955573 + 0.294755i −0.171896 − 1.72350i 0.826239 + 0.563320i −2.17317 + 2.01641i 0.343752 − 1.69760i −2.39093 + 1.63011i 0.623490 + 0.781831i −2.94090 + 0.592524i −2.67097 + 1.28627i
7.8 0.955573 + 0.294755i 0.0736744 + 1.73048i 0.826239 + 0.563320i −2.26108 + 2.09798i −0.439668 + 1.67532i −0.100199 + 0.0683148i 0.623490 + 0.781831i −2.98914 + 0.254984i −2.77902 + 1.33831i
7.9 0.955573 + 0.294755i 0.273396 + 1.71034i 0.826239 + 0.563320i 1.75313 − 1.62667i −0.242881 + 1.71494i −0.473122 + 0.322569i 0.623490 + 0.781831i −2.85051 + 0.935200i 2.15472 − 1.03766i
7.10 0.955573 + 0.294755i 0.348160 − 1.69670i 0.826239 + 0.563320i 0.399218 − 0.370420i 0.832802 − 1.51870i 3.43493 − 2.34189i 0.623490 + 0.781831i −2.75757 − 1.18144i 0.490666 − 0.236292i
7.11 0.955573 + 0.294755i 1.26354 − 1.18468i 0.826239 + 0.563320i 1.22901 − 1.14035i 1.55659 − 0.759614i −2.20600 + 1.50403i 0.623490 + 0.781831i 0.193061 − 2.99378i 1.51053 − 0.727432i
7.12 0.955573 + 0.294755i 1.29635 + 1.14869i 0.826239 + 0.563320i −0.445334 + 0.413209i 0.900175 + 1.47976i 2.84393 − 1.93896i 0.623490 + 0.781831i 0.361043 + 2.97820i −0.547344 + 0.263587i
7.13 0.955573 + 0.294755i 1.56958 − 0.732412i 0.826239 + 0.563320i −2.35043 + 2.18088i 1.71573 − 0.237232i 2.58681 − 1.76365i 0.623490 + 0.781831i 1.92715 − 2.29915i −2.88884 + 1.39119i
7.14 0.955573 + 0.294755i 1.61552 + 0.624571i 0.826239 + 0.563320i −1.48102 + 1.37419i 1.35965 + 1.07301i −3.34141 + 2.27813i 0.623490 + 0.781831i 2.21982 + 2.01802i −1.82027 + 0.876597i
7.15 0.955573 + 0.294755i 1.72476 − 0.158731i 0.826239 + 0.563320i 1.80745 − 1.67707i 1.69492 + 0.356704i −0.0717182 + 0.0488967i 0.623490 + 0.781831i 2.94961 − 0.547545i 2.22148 − 1.06981i
25.1 −0.733052 − 0.680173i −1.66308 − 0.483918i 0.0747301 + 0.997204i −4.04144 + 1.24662i 0.889973 + 1.48592i 0.0301674 − 0.402556i 0.623490 − 0.781831i 2.53165 + 1.60959i 3.81050 + 1.83504i
25.2 −0.733052 − 0.680173i −1.61625 + 0.622682i 0.0747301 + 0.997204i 0.406694 − 0.125448i 1.60833 + 0.642871i 0.00607664 − 0.0810871i 0.623490 − 0.781831i 2.22453 − 2.01282i −0.383454 − 0.184662i
25.3 −0.733052 − 0.680173i −1.45466 − 0.940191i 0.0747301 + 0.997204i 2.83284 − 0.873816i 0.426850 + 1.67863i −0.216250 + 2.88566i 0.623490 − 0.781831i 1.23208 + 2.73532i −2.67097 − 1.28627i
25.4 −0.733052 − 0.680173i −1.23989 + 1.20942i 0.0747301 + 0.997204i −0.719639 + 0.221979i 1.73151 − 0.0432281i −0.0153373 + 0.204662i 0.623490 − 0.781831i 0.0746296 − 2.99907i 0.678517 + 0.326757i
25.5 −0.733052 − 0.680173i −1.10946 − 1.33008i 0.0747301 + 0.997204i −0.520403 + 0.160523i −0.0913911 + 1.72964i 0.310676 − 4.14568i 0.623490 − 0.781831i −0.538205 + 2.95133i 0.490666 + 0.236292i
See next 80 embeddings (of 180 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 7.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
29.d even 7 1 inner
261.q even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 522.2.q.b ✓ 180
9.c even 3 1 inner 522.2.q.b ✓ 180
29.d even 7 1 inner 522.2.q.b ✓ 180
261.q even 21 1 inner 522.2.q.b ✓ 180
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
522.2.q.b ✓ 180 1.a even 1 1 trivial
522.2.q.b ✓ 180 9.c even 3 1 inner
522.2.q.b ✓ 180 29.d even 7 1 inner
522.2.q.b ✓ 180 261.q even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{180} - 2 T_{5}^{179} - 43 T_{5}^{178} + 42 T_{5}^{177} + 840 T_{5}^{176} + 416 T_{5}^{175} + \cdots + 20\!\cdots\!61 \) acting on \(S_{2}^{\mathrm{new}}(522, [\chi])\). Copy content Toggle raw display