Properties

Label 670.2.h.c
Level $670$
Weight $2$
Character orbit 670.h
Analytic conductor $5.350$
Analytic rank $0$
Dimension $52$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [670,2,Mod(29,670)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(670, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("670.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 670 = 2 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 670.h (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.34997693543\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(26\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 52 q + 26 q^{4} - 16 q^{5} - 6 q^{6} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q + 26 q^{4} - 16 q^{5} - 6 q^{6} - 48 q^{9} - 10 q^{10} + 24 q^{11} + 24 q^{14} + 28 q^{15} - 26 q^{16} - 24 q^{19} - 8 q^{20} + 4 q^{21} - 12 q^{24} - 28 q^{25} + 4 q^{26} - 20 q^{29} - 10 q^{30} - 30 q^{31} - 16 q^{34} - 14 q^{35} - 24 q^{36} - 8 q^{39} - 20 q^{40} - 10 q^{41} - 24 q^{44} - 16 q^{45} - 8 q^{46} + 14 q^{49} - 8 q^{50} - 44 q^{51} + 42 q^{54} + 16 q^{55} + 12 q^{56} + 80 q^{59} + 14 q^{60} - 20 q^{61} - 52 q^{64} + 10 q^{65} - 40 q^{66} - 4 q^{69} + 48 q^{70} - 74 q^{71} + 34 q^{74} - 16 q^{75} - 48 q^{76} + 32 q^{79} + 8 q^{80} + 4 q^{81} - 4 q^{84} + 20 q^{85} + 12 q^{86} + 84 q^{89} - 30 q^{90} - 16 q^{91} + 136 q^{94} + 2 q^{95} - 6 q^{96} - 96 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −0.866025 0.500000i 3.40294i 0.500000 + 0.866025i 1.60001 + 1.56204i −1.70147 + 2.94703i −3.81692 + 2.20370i 1.00000i −8.57997 −0.604635 2.15277i
29.2 −0.866025 0.500000i 2.72042i 0.500000 + 0.866025i −1.66450 1.49313i −1.36021 + 2.35596i 0.550668 0.317928i 1.00000i −4.40071 0.694933 + 2.12534i
29.3 −0.866025 0.500000i 2.35660i 0.500000 + 0.866025i −1.42939 + 1.71955i −1.17830 + 2.04088i 3.13521 1.81012i 1.00000i −2.55358 2.09766 0.774481i
29.4 −0.866025 0.500000i 2.00096i 0.500000 + 0.866025i 1.72962 1.41719i −1.00048 + 1.73288i −0.879107 + 0.507552i 1.00000i −1.00384 −2.20649 + 0.362514i
29.5 −0.866025 0.500000i 1.19283i 0.500000 + 0.866025i −1.06930 + 1.96382i −0.596414 + 1.03302i −0.996756 + 0.575477i 1.00000i 1.57716 1.90795 1.16607i
29.6 −0.866025 0.500000i 1.04987i 0.500000 + 0.866025i −0.0350949 2.23579i −0.524935 + 0.909214i −0.779249 + 0.449900i 1.00000i 1.89777 −1.08750 + 1.95380i
29.7 −0.866025 0.500000i 0.0712992i 0.500000 + 0.866025i 0.722984 2.11596i −0.0356496 + 0.0617469i 3.87744 2.23864i 1.00000i 2.99492 −1.68410 + 1.47098i
29.8 −0.866025 0.500000i 0.433437i 0.500000 + 0.866025i −2.23506 0.0672889i 0.216718 0.375367i 0.301725 0.174201i 1.00000i 2.81213 1.90197 + 1.17580i
29.9 −0.866025 0.500000i 0.719082i 0.500000 + 0.866025i 0.192333 2.22778i 0.359541 0.622744i −3.77588 + 2.18000i 1.00000i 2.48292 −1.28046 + 1.83315i
29.10 −0.866025 0.500000i 1.76407i 0.500000 + 0.866025i 0.428350 + 2.19466i 0.882037 1.52773i −2.08791 + 1.20545i 1.00000i −0.111955 0.726366 2.11480i
29.11 −0.866025 0.500000i 1.90212i 0.500000 + 0.866025i −2.20047 0.397387i 0.951061 1.64729i −2.70090 + 1.55936i 1.00000i −0.618066 1.70697 + 1.44438i
29.12 −0.866025 0.500000i 2.24783i 0.500000 + 0.866025i 1.83882 1.27230i 1.12391 1.94667i −0.323048 + 0.186512i 1.00000i −2.05272 −2.22861 + 0.182431i
29.13 −0.866025 0.500000i 2.72838i 0.500000 + 0.866025i −1.87831 1.21324i 1.36419 2.36285i 2.29856 1.32708i 1.00000i −4.44406 1.02005 + 1.98985i
29.14 0.866025 + 0.500000i 2.72838i 0.500000 + 0.866025i −1.87831 + 1.21324i 1.36419 2.36285i −2.29856 + 1.32708i 1.00000i −4.44406 −2.23328 + 0.111537i
29.15 0.866025 + 0.500000i 2.24783i 0.500000 + 0.866025i 1.83882 + 1.27230i 1.12391 1.94667i 0.323048 0.186512i 1.00000i −2.05272 0.956317 + 2.02125i
29.16 0.866025 + 0.500000i 1.90212i 0.500000 + 0.866025i −2.20047 + 0.397387i 0.951061 1.64729i 2.70090 1.55936i 1.00000i −0.618066 −2.10436 0.756089i
29.17 0.866025 + 0.500000i 1.76407i 0.500000 + 0.866025i 0.428350 2.19466i 0.882037 1.52773i 2.08791 1.20545i 1.00000i −0.111955 1.46829 1.68645i
29.18 0.866025 + 0.500000i 0.719082i 0.500000 + 0.866025i 0.192333 + 2.22778i 0.359541 0.622744i 3.77588 2.18000i 1.00000i 2.48292 −0.947326 + 2.02548i
29.19 0.866025 + 0.500000i 0.433437i 0.500000 + 0.866025i −2.23506 + 0.0672889i 0.216718 0.375367i −0.301725 + 0.174201i 1.00000i 2.81213 −1.96926 1.05925i
29.20 0.866025 + 0.500000i 0.0712992i 0.500000 + 0.866025i 0.722984 + 2.11596i −0.0356496 + 0.0617469i −3.87744 + 2.23864i 1.00000i 2.99492 −0.431858 + 2.19397i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
67.c even 3 1 inner
335.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 670.2.h.c 52
5.b even 2 1 inner 670.2.h.c 52
67.c even 3 1 inner 670.2.h.c 52
335.i even 6 1 inner 670.2.h.c 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
670.2.h.c 52 1.a even 1 1 trivial
670.2.h.c 52 5.b even 2 1 inner
670.2.h.c 52 67.c even 3 1 inner
670.2.h.c 52 335.i even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{26} + 51 T_{3}^{24} + 1129 T_{3}^{22} + 14301 T_{3}^{20} + 114897 T_{3}^{18} + 612675 T_{3}^{16} + 2202906 T_{3}^{14} + 5313734 T_{3}^{12} + 8389455 T_{3}^{10} + 8263965 T_{3}^{8} + 4677865 T_{3}^{6} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(670, [\chi])\). Copy content Toggle raw display