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Properties

Label 357.2.bj.a
Level $357$
Weight $2$
Character orbit 357.bj
Analytic conductor $2.851$
Analytic rank $0$
Dimension $352$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [357,2,Mod(26,357)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(357, base_ring=CyclotomicField(24)) chi = DirichletCharacter(H, H._module([12, 20, 3])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("357.26"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 357 = 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 357.bj (of order \(24\), degree \(8\), 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: \(2.85065935216\)
Analytic rank: \(0\)
Dimension: \(352\)
Relative dimension: \(44\) over \(\Q(\zeta_{24})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 352 q - 12 q^{3} - 16 q^{7} - 4 q^{9} - 24 q^{10} - 12 q^{12} - 24 q^{15} + 112 q^{16} - 24 q^{18} - 24 q^{19} - 80 q^{22} - 60 q^{24} - 8 q^{25} - 72 q^{28} - 24 q^{31} + 24 q^{33} - 64 q^{36} - 40 q^{37}+ \cdots + 256 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −2.59415 + 0.695100i −0.872841 − 1.49604i 4.51440 − 2.60639i −1.23237 − 1.60605i 3.30418 + 3.27425i −2.49253 + 0.887284i −6.10122 + 6.10122i −1.47630 + 2.61162i 4.31331 + 3.30972i
26.2 −2.50767 + 0.671927i −1.60705 − 0.646066i 4.10485 − 2.36994i 2.62512 + 3.42112i 4.46404 + 0.540298i 2.59707 − 0.505196i −5.02969 + 5.02969i 2.16520 + 2.07652i −8.88165 − 6.81513i
26.3 −2.48935 + 0.667019i −0.496682 + 1.65931i 4.01990 − 2.32089i −0.903833 − 1.17790i 0.129623 − 4.46190i 2.54176 + 0.734486i −4.81420 + 4.81420i −2.50661 − 1.64830i 3.03564 + 2.32933i
26.4 −2.30329 + 0.617166i 1.12053 + 1.32077i 3.19222 − 1.84303i −0.799891 − 1.04244i −3.39603 − 2.35056i −2.48892 − 0.897372i −2.84291 + 2.84291i −0.488846 + 2.95990i 2.48574 + 1.90738i
26.5 −2.24242 + 0.600854i −0.427208 + 1.67854i 2.93536 − 1.69473i 1.99584 + 2.60102i −0.0505770 − 4.02067i −2.05255 + 1.66944i −2.28089 + 2.28089i −2.63499 − 1.43417i −6.03833 − 4.63338i
26.6 −2.18027 + 0.584202i −1.66007 + 0.494138i 2.68024 − 1.54744i −1.33356 − 1.73794i 3.33072 − 2.04717i −0.287919 − 2.63004i −1.74749 + 1.74749i 2.51165 − 1.64061i 3.92284 + 3.01010i
26.7 −2.15355 + 0.577042i 0.879586 − 1.49209i 2.57275 − 1.48538i 0.469021 + 0.611240i −1.03323 + 3.72084i 0.376382 − 2.61884i −1.53040 + 1.53040i −1.45266 − 2.62484i −1.36277 − 1.04569i
26.8 −1.97869 + 0.530188i 1.67147 + 0.454087i 1.90206 − 1.09815i 0.977708 + 1.27417i −3.54807 − 0.0123035i 1.39179 + 2.25009i −0.284351 + 0.284351i 2.58761 + 1.51798i −2.61013 − 2.00282i
26.9 −1.79777 + 0.481712i −0.0502743 − 1.73132i 1.26789 − 0.732016i 0.344456 + 0.448903i 0.924379 + 3.08830i 1.62504 + 2.08788i 0.705366 − 0.705366i −2.99494 + 0.174082i −0.835495 − 0.641098i
26.10 −1.61484 + 0.432695i 1.70456 − 0.307393i 0.688428 − 0.397464i −1.64076 − 2.13829i −2.61957 + 1.23394i −2.63988 − 0.176109i 1.42457 − 1.42457i 2.81102 − 1.04794i 3.57479 + 2.74304i
26.11 −1.50134 + 0.402282i −1.59673 + 0.671159i 0.360133 − 0.207923i 1.05067 + 1.36926i 2.12724 − 1.64997i −2.10371 − 1.60449i 1.74107 − 1.74107i 2.09909 − 2.14332i −2.12823 − 1.63305i
26.12 −1.33178 + 0.356849i −1.46457 − 0.924684i −0.0857553 + 0.0495108i 0.477156 + 0.621842i 2.28045 + 0.708845i −1.52209 + 2.16408i 2.04640 − 2.04640i 1.28992 + 2.70852i −0.857371 − 0.657884i
26.13 −1.30480 + 0.349620i −1.14963 − 1.29551i −0.151786 + 0.0876334i −2.14653 − 2.79741i 1.95297 + 1.28845i 1.60809 − 2.10097i 2.07777 − 2.07777i −0.356717 + 2.97872i 3.77882 + 2.89959i
26.14 −1.16177 + 0.311296i −0.864243 + 1.50103i −0.479242 + 0.276691i −1.84223 − 2.40084i 0.536790 − 2.01289i −0.732066 + 2.54246i 2.17159 − 2.17159i −1.50617 − 2.59451i 2.88763 + 2.21575i
26.15 −1.14393 + 0.306515i 1.09234 + 1.34417i −0.517430 + 0.298738i −1.89391 − 2.46819i −1.66156 − 1.20282i 2.58352 − 0.570481i 2.17516 − 2.17516i −0.613608 + 2.93658i 2.92304 + 2.24293i
26.16 −1.09272 + 0.292792i 1.72364 + 0.170509i −0.623749 + 0.360122i 2.45108 + 3.19431i −1.93337 + 0.318350i −0.403386 − 2.61482i 2.17599 − 2.17599i 2.94185 + 0.587793i −3.61361 − 2.77282i
26.17 −0.619501 + 0.165995i 1.29832 − 1.14646i −1.37582 + 0.794332i 0.0203376 + 0.0265045i −0.614001 + 0.925748i −1.84272 + 1.89852i 1.62748 − 1.62748i 0.371250 − 2.97694i −0.0169988 − 0.0130436i
26.18 −0.585485 + 0.156880i −1.65492 + 0.511129i −1.41387 + 0.816298i 0.951646 + 1.24021i 0.888742 − 0.558881i 2.44542 + 1.00991i 1.55695 − 1.55695i 2.47750 − 1.69175i −0.751738 − 0.576829i
26.19 −0.585455 + 0.156872i 1.33711 − 1.10097i −1.41390 + 0.816317i −0.642183 − 0.836909i −0.610106 + 0.854324i 2.51007 − 0.836394i 1.55688 − 1.55688i 0.575727 − 2.94424i 0.507256 + 0.389232i
26.20 −0.420640 + 0.112710i 0.492071 + 1.66068i −1.56782 + 0.905179i 0.104105 + 0.135672i −0.394161 − 0.643088i −2.33635 − 1.24156i 1.17332 − 1.17332i −2.51573 + 1.63435i −0.0590824 − 0.0453355i
See next 80 embeddings (of 352 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
17.d even 8 1 inner
21.g even 6 1 inner
51.g odd 8 1 inner
119.r odd 24 1 inner
357.bj even 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 357.2.bj.a ✓ 352
3.b odd 2 1 inner 357.2.bj.a ✓ 352
7.d odd 6 1 inner 357.2.bj.a ✓ 352
17.d even 8 1 inner 357.2.bj.a ✓ 352
21.g even 6 1 inner 357.2.bj.a ✓ 352
51.g odd 8 1 inner 357.2.bj.a ✓ 352
119.r odd 24 1 inner 357.2.bj.a ✓ 352
357.bj even 24 1 inner 357.2.bj.a ✓ 352
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
357.2.bj.a ✓ 352 1.a even 1 1 trivial
357.2.bj.a ✓ 352 3.b odd 2 1 inner
357.2.bj.a ✓ 352 7.d odd 6 1 inner
357.2.bj.a ✓ 352 17.d even 8 1 inner
357.2.bj.a ✓ 352 21.g even 6 1 inner
357.2.bj.a ✓ 352 51.g odd 8 1 inner
357.2.bj.a ✓ 352 119.r odd 24 1 inner
357.2.bj.a ✓ 352 357.bj even 24 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(357, [\chi])\).