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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [420,2,Mod(31,420)] 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("420.31"); 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, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.bi (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [28,-4] 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: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 4 q^{2} - 14 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{7} - 4 q^{8} - 14 q^{9} - 4 q^{10} + 2 q^{12} + 28 q^{14} - 2 q^{16} - 24 q^{17} + 2 q^{18} + 14 q^{19} - 4 q^{20} + 16 q^{22} - 16 q^{24} + 14 q^{25}+ \cdots + 2 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −1.41270 − 0.0654937i −0.500000 + 0.866025i 1.99142 + 0.185045i −0.866025 + 0.500000i 0.763067 − 1.19068i 2.61608 − 0.395104i −2.80115 − 0.391838i −0.500000 − 0.866025i 1.25618 − 0.649629i
31.2 −1.41178 + 0.0829396i −0.500000 + 0.866025i 1.98624 − 0.234185i 0.866025 − 0.500000i 0.634062 − 1.26411i −0.693280 + 2.55330i −2.78471 + 0.495356i −0.500000 − 0.866025i −1.18117 + 0.777718i
31.3 −1.21645 + 0.721286i −0.500000 + 0.866025i 0.959493 − 1.75481i −0.866025 + 0.500000i −0.0164277 − 1.41412i −2.64546 + 0.0390617i 0.0985483 + 2.82671i −0.500000 − 0.866025i 0.692832 − 1.23288i
31.4 −1.10407 + 0.883758i −0.500000 + 0.866025i 0.437945 − 1.95146i 0.866025 − 0.500000i −0.213321 − 1.39803i 2.27168 − 1.35627i 1.24110 + 2.54159i −0.500000 − 0.866025i −0.514275 + 1.31739i
31.5 −1.08061 − 0.912296i −0.500000 + 0.866025i 0.335431 + 1.97167i 0.866025 − 0.500000i 1.33038 − 0.479687i −2.63725 − 0.211889i 1.43628 − 2.43662i −0.500000 − 0.866025i −1.39198 − 0.249767i
31.6 −0.668090 + 1.24646i −0.500000 + 0.866025i −1.10731 − 1.66549i 0.866025 − 0.500000i −0.745419 − 1.20181i −1.71433 − 2.01521i 2.81575 − 0.267520i −0.500000 − 0.866025i 0.0446459 + 1.41351i
31.7 −0.535729 − 1.30881i −0.500000 + 0.866025i −1.42599 + 1.40234i −0.866025 + 0.500000i 1.40133 + 0.190453i 0.482420 + 2.60140i 2.59934 + 1.11508i −0.500000 − 0.866025i 1.11836 + 0.865602i
31.8 0.00862847 + 1.41419i −0.500000 + 0.866025i −1.99985 + 0.0244045i −0.866025 + 0.500000i −1.22904 − 0.699621i 0.189415 − 2.63896i −0.0517682 − 2.82795i −0.500000 − 0.866025i −0.714566 − 1.22041i
31.9 0.252003 − 1.39158i −0.500000 + 0.866025i −1.87299 − 0.701364i 0.866025 − 0.500000i 1.07914 + 0.914031i −2.29181 − 1.32197i −1.44800 + 2.42967i −0.500000 − 0.866025i −0.477549 − 1.33114i
31.10 0.483616 − 1.32895i −0.500000 + 0.866025i −1.53223 − 1.28541i −0.866025 + 0.500000i 0.909099 + 1.08330i −1.79146 + 1.94696i −2.44925 + 1.41462i −0.500000 − 0.866025i 0.245653 + 1.39271i
31.11 0.905086 − 1.08666i −0.500000 + 0.866025i −0.361639 − 1.96703i 0.866025 − 0.500000i 0.488528 + 1.32715i 2.52676 + 0.784530i −2.46480 − 1.38736i −0.500000 − 0.866025i 0.240500 − 1.39361i
31.12 1.14443 + 0.830825i −0.500000 + 0.866025i 0.619461 + 1.90165i −0.866025 + 0.500000i −1.29173 + 0.575697i −2.50660 + 0.846737i −0.871005 + 2.69098i −0.500000 − 0.866025i −1.40652 − 0.147298i
31.13 1.24143 + 0.677377i −0.500000 + 0.866025i 1.08232 + 1.68184i 0.866025 − 0.500000i −1.20734 + 0.736426i 1.03822 + 2.43353i 0.204394 + 2.82103i −0.500000 − 0.866025i 1.41380 − 0.0340920i
31.14 1.39422 + 0.236963i −0.500000 + 0.866025i 1.88770 + 0.660757i −0.866025 + 0.500000i −0.902326 + 1.08895i 2.15561 − 1.53406i 2.47529 + 1.36855i −0.500000 − 0.866025i −1.32591 + 0.491894i
271.1 −1.41270 + 0.0654937i −0.500000 − 0.866025i 1.99142 − 0.185045i −0.866025 − 0.500000i 0.763067 + 1.19068i 2.61608 + 0.395104i −2.80115 + 0.391838i −0.500000 + 0.866025i 1.25618 + 0.649629i
271.2 −1.41178 − 0.0829396i −0.500000 − 0.866025i 1.98624 + 0.234185i 0.866025 + 0.500000i 0.634062 + 1.26411i −0.693280 − 2.55330i −2.78471 − 0.495356i −0.500000 + 0.866025i −1.18117 − 0.777718i
271.3 −1.21645 − 0.721286i −0.500000 − 0.866025i 0.959493 + 1.75481i −0.866025 − 0.500000i −0.0164277 + 1.41412i −2.64546 − 0.0390617i 0.0985483 − 2.82671i −0.500000 + 0.866025i 0.692832 + 1.23288i
271.4 −1.10407 − 0.883758i −0.500000 − 0.866025i 0.437945 + 1.95146i 0.866025 + 0.500000i −0.213321 + 1.39803i 2.27168 + 1.35627i 1.24110 − 2.54159i −0.500000 + 0.866025i −0.514275 − 1.31739i
271.5 −1.08061 + 0.912296i −0.500000 − 0.866025i 0.335431 − 1.97167i 0.866025 + 0.500000i 1.33038 + 0.479687i −2.63725 + 0.211889i 1.43628 + 2.43662i −0.500000 + 0.866025i −1.39198 + 0.249767i
271.6 −0.668090 − 1.24646i −0.500000 − 0.866025i −1.10731 + 1.66549i 0.866025 + 0.500000i −0.745419 + 1.20181i −1.71433 + 2.01521i 2.81575 + 0.267520i −0.500000 + 0.866025i 0.0446459 − 1.41351i
See all 28 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 31.14
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f 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.bi.c ✓ 28
4.b odd 2 1 420.2.bi.d yes 28
7.d odd 6 1 420.2.bi.d yes 28
28.f even 6 1 inner 420.2.bi.c ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bi.c ✓ 28 1.a even 1 1 trivial
420.2.bi.c ✓ 28 28.f even 6 1 inner
420.2.bi.d yes 28 4.b odd 2 1
420.2.bi.d yes 28 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(420, [\chi])\):

\( T_{11}^{28} - 110 T_{11}^{26} + 7351 T_{11}^{24} - 720 T_{11}^{23} - 320470 T_{11}^{22} + \cdots + 12068340736 \) Copy content Toggle raw display
\( T_{19}^{28} - 14 T_{19}^{27} + 253 T_{19}^{26} - 2506 T_{19}^{25} + 30520 T_{19}^{24} + \cdots + 96477596934601 \) Copy content Toggle raw display