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Properties

Label 420.2.bf.a
Level $420$
Weight $2$
Character orbit 420.bf
Analytic conductor $3.354$
Analytic rank $0$
Dimension $128$
Inner twists $8$

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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] = [420,2,Mod(11,420)] 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("420.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(420, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.bf (of order \(6\), degree \(2\), 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: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(64\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q + 10 q^{12} + 16 q^{13} + 16 q^{16} + 10 q^{18} - 4 q^{21} - 56 q^{22} + 20 q^{24} + 64 q^{25} - 12 q^{28} - 48 q^{34} - 40 q^{36} + 8 q^{37} - 54 q^{42} - 4 q^{45} - 16 q^{46} - 36 q^{48} - 24 q^{49}+ \cdots - 96 q^{97}+O(q^{100}) \) 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 −1.41421 0.000293530i 0.807523 + 1.53229i 2.00000 0.000830229i 0.866025 − 0.500000i −1.14246 − 2.16675i −0.723852 + 2.54481i −2.82843 + 0.00176118i −1.69581 + 2.47472i −1.22460 + 0.707361i
11.2 −1.41297 − 0.0593533i 1.63498 − 0.571702i 1.99295 + 0.167729i −0.866025 + 0.500000i −2.34410 + 0.710755i −2.24626 + 1.39797i −2.80602 − 0.355284i 2.34631 − 1.86944i 1.25334 − 0.655082i
11.3 −1.40915 + 0.119537i 1.60246 − 0.657371i 1.97142 − 0.336893i 0.866025 − 0.500000i −2.17952 + 1.11789i 2.64036 − 0.168758i −2.73776 + 0.710393i 2.13573 − 2.10681i −1.16059 + 0.808099i
11.4 −1.40715 − 0.141211i −1.50915 + 0.849977i 1.96012 + 0.397408i 0.866025 − 0.500000i 2.24362 − 0.982933i −0.462734 − 2.60497i −2.70206 − 0.836001i 1.55508 − 2.56549i −1.28923 + 0.581281i
11.5 −1.39104 − 0.254992i −0.0941332 − 1.72949i 1.86996 + 0.709406i −0.866025 + 0.500000i −0.310064 + 2.42979i 1.21179 + 2.35193i −2.42028 − 1.46363i −2.98228 + 0.325605i 1.33217 − 0.474688i
11.6 −1.35951 + 0.389511i −1.33585 − 1.10249i 1.69656 − 1.05909i −0.866025 + 0.500000i 2.24555 + 0.978527i −1.80926 − 1.93044i −1.89397 + 2.10068i 0.569013 + 2.94554i 0.982619 − 1.01708i
11.7 −1.34679 − 0.431446i −1.40184 − 1.01727i 1.62771 + 1.16214i 0.866025 − 0.500000i 1.44909 + 1.97488i −2.00973 + 1.72075i −1.69079 − 2.26743i 0.930303 + 2.85211i −1.38208 + 0.299754i
11.8 −1.34659 − 0.432097i −0.839129 + 1.51521i 1.62658 + 1.16371i −0.866025 + 0.500000i 1.78468 − 1.67778i 2.38545 − 1.14438i −1.68750 − 2.26988i −1.59173 − 2.54291i 1.38223 − 0.299086i
11.9 −1.31394 + 0.523038i −0.988657 + 1.42217i 1.45286 − 1.37448i −0.866025 + 0.500000i 0.555187 − 2.38574i −1.77239 + 1.96434i −1.19007 + 2.56588i −1.04512 − 2.81207i 0.876385 − 1.10993i
11.10 −1.25626 + 0.649463i 0.457845 + 1.67044i 1.15640 − 1.63179i 0.866025 − 0.500000i −1.66006 − 1.80116i −0.775771 − 2.52946i −0.392949 + 2.80100i −2.58076 + 1.52961i −0.763225 + 1.19058i
11.11 −1.20965 − 0.732624i 0.751444 + 1.56055i 0.926524 + 1.77244i −0.866025 + 0.500000i 0.234313 − 2.43826i −2.51577 − 0.819076i 0.177761 − 2.82284i −1.87066 + 2.34534i 1.41390 + 0.0296442i
11.12 −1.19058 + 0.763225i −0.457845 − 1.67044i 0.834976 − 1.81736i 0.866025 − 0.500000i 1.82002 + 1.63936i 0.775771 + 2.52946i 0.392949 + 2.80100i −2.58076 + 1.52961i −0.649463 + 1.25626i
11.13 −1.17511 − 0.786840i 1.65504 + 0.510711i 0.761766 + 1.84925i 0.866025 − 0.500000i −1.54301 − 1.90240i −0.255054 − 2.63343i 0.559902 − 2.77246i 2.47835 + 1.69050i −1.41109 − 0.0938684i
11.14 −1.10993 + 0.876385i 0.988657 − 1.42217i 0.463900 − 1.94546i −0.866025 + 0.500000i 0.149022 + 2.44495i 1.77239 − 1.96434i 1.19007 + 2.56588i −1.04512 − 2.81207i 0.523038 − 1.31394i
11.15 −1.10230 − 0.885971i −0.727404 + 1.57190i 0.430110 + 1.95320i 0.866025 − 0.500000i 2.19448 − 1.08824i 1.67318 + 2.04950i 1.25637 − 2.53407i −1.94177 − 2.28682i −1.39760 − 0.216126i
11.16 −1.01708 + 0.982619i 1.33585 + 1.10249i 0.0689203 − 1.99881i −0.866025 + 0.500000i −2.44201 + 0.191306i 1.80926 + 1.93044i 1.89397 + 2.10068i 0.569013 + 2.94554i 0.389511 − 1.35951i
11.17 −0.974322 − 1.02504i −1.46311 − 0.926981i −0.101394 + 1.99743i −0.866025 + 0.500000i 0.475356 + 2.40292i 1.76764 − 1.96862i 2.14622 − 1.84220i 1.28141 + 2.71256i 1.35630 + 0.400546i
11.18 −0.928294 − 1.06690i 1.73135 − 0.0492062i −0.276542 + 1.98079i −0.866025 + 0.500000i −1.65970 − 1.80150i 2.44888 + 1.00148i 2.37001 − 1.54371i 2.99516 − 0.170386i 1.33737 + 0.459814i
11.19 −0.811460 − 1.15825i 0.383950 − 1.68896i −0.683064 + 1.87974i 0.866025 − 0.500000i −2.26779 + 0.925816i 2.28363 + 1.33605i 2.73148 − 0.734180i −2.70517 − 1.29695i −1.28187 − 0.597340i
11.20 −0.808099 + 1.16059i −1.60246 + 0.657371i −0.693953 − 1.87575i 0.866025 − 0.500000i 0.532003 − 2.39102i −2.64036 + 0.168758i 2.73776 + 0.710393i 2.13573 − 2.10681i −0.119537 + 1.40915i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 11.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
7.c even 3 1 inner
12.b even 2 1 inner
21.h odd 6 1 inner
28.g odd 6 1 inner
84.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.bf.a ✓ 128
3.b odd 2 1 inner 420.2.bf.a ✓ 128
4.b odd 2 1 inner 420.2.bf.a ✓ 128
7.c even 3 1 inner 420.2.bf.a ✓ 128
12.b even 2 1 inner 420.2.bf.a ✓ 128
21.h odd 6 1 inner 420.2.bf.a ✓ 128
28.g odd 6 1 inner 420.2.bf.a ✓ 128
84.n even 6 1 inner 420.2.bf.a ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bf.a ✓ 128 1.a even 1 1 trivial
420.2.bf.a ✓ 128 3.b odd 2 1 inner
420.2.bf.a ✓ 128 4.b odd 2 1 inner
420.2.bf.a ✓ 128 7.c even 3 1 inner
420.2.bf.a ✓ 128 12.b even 2 1 inner
420.2.bf.a ✓ 128 21.h odd 6 1 inner
420.2.bf.a ✓ 128 28.g odd 6 1 inner
420.2.bf.a ✓ 128 84.n even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(420, [\chi])\).