Properties

Label 522.2.v.a
Level $522$
Weight $2$
Character orbit 522.v
Analytic conductor $4.168$
Analytic rank $0$
Dimension $360$
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(13,522)] 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("522.13"); 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([14, 27])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.v (of order \(42\), degree \(12\), 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: \(360\)
Relative dimension: \(30\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 360 q - 30 q^{4} + 4 q^{5} - 20 q^{6} - 20 q^{9} + 30 q^{16} - 4 q^{20} - 14 q^{21} - 30 q^{23} - 10 q^{24} + 30 q^{25} + 126 q^{27} + 12 q^{29} - 40 q^{30} - 96 q^{33} - 64 q^{35} + 12 q^{36} + 18 q^{38}+ \cdots - 112 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −0.997204 + 0.0747301i −1.73154 − 0.0420571i 0.988831 − 0.149042i 1.36050 − 0.927574i 1.72984 − 0.0874587i −2.17835 − 0.328333i −0.974928 + 0.222521i 2.99646 + 0.145647i −1.28738 + 1.02665i
13.2 −0.997204 + 0.0747301i −1.59969 + 0.664071i 0.988831 − 0.149042i −1.55000 + 1.05677i 1.54559 − 0.781759i 2.51948 + 0.379751i −0.974928 + 0.222521i 2.11802 − 2.12462i 1.46669 − 1.16965i
13.3 −0.997204 + 0.0747301i −1.50842 − 0.851272i 0.988831 − 0.149042i 1.76113 − 1.20072i 1.56782 + 0.736167i 3.51819 + 0.530282i −0.974928 + 0.222521i 1.55067 + 2.56815i −1.66648 + 1.32897i
13.4 −0.997204 + 0.0747301i −1.44683 − 0.952196i 0.988831 − 0.149042i −2.37584 + 1.61982i 1.51394 + 0.841411i −4.31009 − 0.649642i −0.974928 + 0.222521i 1.18665 + 2.75533i 2.24814 − 1.79284i
13.5 −0.997204 + 0.0747301i −0.764678 − 1.55411i 0.988831 − 0.149042i −1.97155 + 1.34418i 0.878679 + 1.49262i 0.980512 + 0.147788i −0.974928 + 0.222521i −1.83053 + 2.37679i 1.86559 − 1.48776i
13.6 −0.997204 + 0.0747301i −0.699550 + 1.58450i 0.988831 − 0.149042i 2.07112 − 1.41207i 0.579184 − 1.63234i −3.18907 − 0.480675i −0.974928 + 0.222521i −2.02126 − 2.21687i −1.95981 + 1.56289i
13.7 −0.997204 + 0.0747301i 0.337107 + 1.69893i 0.988831 − 0.149042i −1.88482 + 1.28505i −0.463125 − 1.66899i 3.00761 + 0.453324i −0.974928 + 0.222521i −2.77272 + 1.14544i 1.78352 − 1.42231i
13.8 −0.997204 + 0.0747301i 0.492451 + 1.66057i 0.988831 − 0.149042i −0.330575 + 0.225382i −0.615169 − 1.61913i −2.66165 − 0.401179i −0.974928 + 0.222521i −2.51498 + 1.63550i 0.312808 − 0.249456i
13.9 −0.997204 + 0.0747301i 0.508316 − 1.65578i 0.988831 − 0.149042i 2.97182 − 2.02615i −0.383157 + 1.68914i −2.01914 − 0.304337i −0.974928 + 0.222521i −2.48323 − 1.68332i −2.81210 + 2.24257i
13.10 −0.997204 + 0.0747301i 0.677320 − 1.59413i 0.988831 − 0.149042i 0.0493567 − 0.0336508i −0.556296 + 1.64028i 4.13469 + 0.623204i −0.974928 + 0.222521i −2.08248 − 2.15947i −0.0467040 + 0.0372452i
13.11 −0.997204 + 0.0747301i 1.07633 + 1.35702i 0.988831 − 0.149042i 3.44465 − 2.34852i −1.17473 − 1.27279i 1.37560 + 0.207339i −0.974928 + 0.222521i −0.683024 + 2.92121i −3.25951 + 2.59938i
13.12 −0.997204 + 0.0747301i 1.34263 − 1.09423i 0.988831 − 0.149042i −3.30404 + 2.25265i −1.25710 + 1.19151i 0.909310 + 0.137056i −0.974928 + 0.222521i 0.605314 − 2.93830i 3.12646 − 2.49327i
13.13 −0.997204 + 0.0747301i 1.58230 + 0.704512i 0.988831 − 0.149042i −1.22253 + 0.833504i −1.63052 − 0.584297i −2.95193 − 0.444931i −0.974928 + 0.222521i 2.00733 + 2.22949i 1.15682 − 0.922533i
13.14 −0.997204 + 0.0747301i 1.69403 − 0.360898i 0.988831 − 0.149042i 1.42688 − 0.972832i −1.66233 + 0.486484i −1.86317 − 0.280827i −0.974928 + 0.222521i 2.73951 − 1.22275i −1.35019 + 1.07674i
13.15 −0.997204 + 0.0747301i 1.71760 − 0.223248i 0.988831 − 0.149042i 1.20636 − 0.822482i −1.69612 + 0.350980i 4.27421 + 0.644233i −0.974928 + 0.222521i 2.90032 − 0.766902i −1.14152 + 0.910334i
13.16 0.997204 − 0.0747301i −1.69199 − 0.370375i 0.988831 − 0.149042i 1.79611 − 1.22457i −1.71493 − 0.242897i 0.210006 + 0.0316533i 0.974928 − 0.222521i 2.72564 + 1.25334i 1.69958 − 1.35537i
13.17 0.997204 − 0.0747301i −1.63051 − 0.584326i 0.988831 − 0.149042i −1.23179 + 0.839819i −1.66962 − 0.460844i −3.40320 − 0.512949i 0.974928 − 0.222521i 2.31713 + 1.90550i −1.16558 + 0.929522i
13.18 0.997204 − 0.0747301i −1.43998 + 0.962531i 0.988831 − 0.149042i −1.07868 + 0.735435i −1.36402 + 1.06745i 3.42178 + 0.515750i 0.974928 − 0.222521i 1.14707 − 2.77205i −1.02071 + 0.813988i
13.19 0.997204 − 0.0747301i −1.35011 + 1.08499i 0.988831 − 0.149042i 3.22004 − 2.19539i −1.26525 + 1.18285i −0.459445 − 0.0692501i 0.974928 − 0.222521i 0.645597 − 2.92971i 3.04698 − 2.42988i
13.20 0.997204 − 0.0747301i −1.15516 + 1.29058i 0.988831 − 0.149042i −0.909662 + 0.620197i −1.05548 + 1.37330i −4.75480 − 0.716670i 0.974928 − 0.222521i −0.331218 − 2.98166i −0.860771 + 0.686442i
See next 80 embeddings (of 360 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 13.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
29.e even 14 1 inner
261.u even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 522.2.v.a ✓ 360
9.c even 3 1 inner 522.2.v.a ✓ 360
29.e even 14 1 inner 522.2.v.a ✓ 360
261.u even 42 1 inner 522.2.v.a ✓ 360
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
522.2.v.a ✓ 360 1.a even 1 1 trivial
522.2.v.a ✓ 360 9.c even 3 1 inner
522.2.v.a ✓ 360 29.e even 14 1 inner
522.2.v.a ✓ 360 261.u even 42 1 inner

Hecke kernels

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