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Properties

Label 669.2.i.b
Level $669$
Weight $2$
Character orbit 669.i
Analytic conductor $5.342$
Analytic rank $0$
Dimension $720$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [669,2,Mod(4,669)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(669, base_ring=CyclotomicField(74)) chi = DirichletCharacter(H, H._module([0, 46])) 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("669.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 669 = 3 \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 669.i (of order \(37\), degree \(36\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [720] 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: \(5.34199189522\)
Analytic rank: \(0\)
Dimension: \(720\)
Relative dimension: \(20\) over \(\Q(\zeta_{37})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{37}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 720 q - q^{2} - 20 q^{3} - 23 q^{4} - 4 q^{5} - q^{6} - 12 q^{7} - 9 q^{8} - 20 q^{9} - 16 q^{10} - 14 q^{11} - 23 q^{12} - 14 q^{13} - 28 q^{14} - 4 q^{15} - 37 q^{16} - 20 q^{17} - q^{18} + 42 q^{19}+ \cdots - 14 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.48607 + 0.878383i 0.372856 − 0.927889i 3.85289 − 3.11099i −1.70596 − 2.77066i −0.111904 + 2.63430i −0.329622 + 2.57407i −4.08104 + 6.62803i −0.721956 − 0.691939i 6.67483 + 5.38955i
4.2 −2.48440 + 0.877794i 0.372856 − 0.927889i 3.84565 − 3.10514i 1.33305 + 2.16501i −0.111829 + 2.63254i 0.112969 − 0.882194i −4.06544 + 6.60270i −0.721956 − 0.691939i −5.21227 − 4.20861i
4.3 −2.24774 + 0.794178i 0.372856 − 0.927889i 2.86556 − 2.31378i −0.143408 − 0.232909i −0.101176 + 2.38177i 0.373427 − 2.91614i −2.10367 + 3.41658i −0.721956 − 0.691939i 0.507314 + 0.409628i
4.4 −1.77809 + 0.628240i 0.372856 − 0.927889i 1.21085 − 0.977695i 0.784626 + 1.27431i −0.0800362 + 1.88411i −0.541066 + 4.22526i 0.438712 − 0.712514i −0.721956 − 0.691939i −2.19571 − 1.77291i
4.5 −1.72125 + 0.608157i 0.372856 − 0.927889i 1.03678 − 0.837141i 1.37137 + 2.22725i −0.0774777 + 1.82389i −0.171615 + 1.34017i 0.638833 − 1.03753i −0.721956 − 0.691939i −3.71499 − 2.99965i
4.6 −1.27818 + 0.451610i 0.372856 − 0.927889i −0.126276 + 0.101961i −2.03001 − 3.29695i −0.0575340 + 1.35440i 0.0912925 − 0.712916i 1.53688 − 2.49605i −0.721956 − 0.691939i 4.08366 + 3.29733i
4.7 −1.17140 + 0.413884i 0.372856 − 0.927889i −0.355182 + 0.286790i 0.695338 + 1.12930i −0.0527278 + 1.24125i −0.102959 + 0.804025i 1.60013 − 2.59878i −0.721956 − 0.691939i −1.28192 − 1.03508i
4.8 −0.647316 + 0.228711i 0.372856 − 0.927889i −1.18936 + 0.960343i −0.504580 − 0.819490i −0.0291373 + 0.685914i −0.0340902 + 0.266216i 1.27016 − 2.06287i −0.721956 − 0.691939i 0.514050 + 0.415066i
4.9 −0.398222 + 0.140701i 0.372856 − 0.927889i −1.41729 + 1.14438i 0.296585 + 0.481685i −0.0179249 + 0.421967i 0.443730 − 3.46515i 0.846260 − 1.37441i −0.721956 − 0.691939i −0.185880 − 0.150088i
4.10 −0.0522189 + 0.0184501i 0.372856 − 0.927889i −1.55369 + 1.25451i −0.648842 − 1.05379i −0.00235050 + 0.0553326i −0.513131 + 4.00711i 0.116061 − 0.188495i −0.721956 − 0.691939i 0.0533243 + 0.0430564i
4.11 0.148918 − 0.0526161i 0.372856 − 0.927889i −1.53666 + 1.24077i 2.16037 + 3.50867i 0.00670316 − 0.157798i 0.0434417 − 0.339242i −0.329171 + 0.534607i −0.721956 − 0.691939i 0.506331 + 0.408834i
4.12 0.789887 − 0.279085i 0.372856 − 0.927889i −1.01004 + 0.815549i 0.639958 + 1.03936i 0.0355547 − 0.836986i 0.502075 − 3.92077i −1.44868 + 2.35280i −0.721956 − 0.691939i 0.795563 + 0.642372i
4.13 0.917855 − 0.324299i 0.372856 − 0.927889i −0.818784 + 0.661122i −1.41847 − 2.30374i 0.0413149 − 0.972584i 0.193950 − 1.51458i −1.55791 + 2.53021i −0.721956 − 0.691939i −2.04905 − 1.65449i
4.14 0.923374 − 0.326249i 0.372856 − 0.927889i −0.809890 + 0.653940i −0.542942 − 0.881794i 0.0415633 − 0.978433i −0.395566 + 3.08903i −1.56141 + 2.53589i −0.721956 − 0.691939i −0.789024 − 0.637092i
4.15 1.49775 − 0.529190i 0.372856 − 0.927889i 0.407155 − 0.328755i 1.79333 + 2.91256i 0.0674176 − 1.58706i −0.0506160 + 0.395268i −1.22987 + 1.99744i −0.721956 − 0.691939i 4.22727 + 3.41328i
4.16 1.85393 − 0.655035i 0.372856 − 0.927889i 1.45192 − 1.17234i −1.03921 − 1.68778i 0.0834499 − 1.96448i 0.225320 − 1.75955i −0.138010 + 0.224142i −0.721956 − 0.691939i −3.03217 − 2.44831i
4.17 1.90813 − 0.674184i 0.372856 − 0.927889i 1.63035 − 1.31641i −2.07608 − 3.37177i 0.0858894 − 2.02190i −0.211836 + 1.65426i 0.101294 − 0.164512i −0.721956 − 0.691939i −6.23462 − 5.03410i
4.18 2.21986 − 0.784325i 0.372856 − 0.927889i 2.75653 − 2.22574i 0.591087 + 0.959987i 0.0999212 − 2.35222i 0.541183 − 4.22618i 1.90459 − 3.09326i −0.721956 − 0.691939i 2.06507 + 1.66743i
4.19 2.30006 − 0.812663i 0.372856 − 0.927889i 3.07379 − 2.48191i 1.32211 + 2.14724i 0.103531 − 2.43721i −0.299249 + 2.33688i 2.49495 − 4.05205i −0.721956 − 0.691939i 4.78591 + 3.86435i
4.20 2.64801 − 0.935601i 0.372856 − 0.927889i 4.58053 − 3.69852i −1.35613 − 2.20249i 0.119193 − 2.80590i −0.231384 + 1.80691i 5.72398 − 9.29634i −0.721956 − 0.691939i −5.65170 − 4.56343i
See next 80 embeddings (of 720 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.20
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
223.e even 37 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 669.2.i.b ✓ 720
223.e even 37 1 inner 669.2.i.b ✓ 720
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
669.2.i.b ✓ 720 1.a even 1 1 trivial
669.2.i.b ✓ 720 223.e even 37 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{720} + T_{2}^{719} + 32 T_{2}^{718} + 36 T_{2}^{717} + 600 T_{2}^{716} + 754 T_{2}^{715} + \cdots + 16\!\cdots\!69 \) acting on \(S_{2}^{\mathrm{new}}(669, [\chi])\). Copy content Toggle raw display