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Properties

Label 168.2.bc.a
Level $168$
Weight $2$
Character orbit 168.bc
Analytic conductor $1.341$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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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] = [168,2,Mod(37,168)] 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("168.37"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(168, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 168.bc (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: \(1.34148675396\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{2} - 2 q^{4} - 16 q^{8} + 16 q^{9} + 6 q^{10} + 22 q^{14} - 10 q^{16} - 2 q^{18} - 40 q^{20} - 12 q^{22} - 8 q^{23} - 6 q^{24} + 16 q^{25} + 6 q^{26} - 26 q^{28} - 8 q^{30} - 24 q^{31} - 8 q^{32}+ \cdots - 64 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −1.40107 − 0.192386i 0.866025 − 0.500000i 1.92598 + 0.539091i −2.93503 − 1.69454i −1.30955 + 0.533922i −1.85242 − 1.88906i −2.59471 − 1.12583i 0.500000 − 0.866025i 3.78617 + 2.93882i
37.2 −1.33630 + 0.462932i −0.866025 + 0.500000i 1.57139 − 1.23723i −1.56250 − 0.902108i 0.925802 − 1.06906i 2.63683 − 0.217074i −1.52709 + 2.38076i 0.500000 − 0.866025i 2.50558 + 0.482155i
37.3 −1.31787 − 0.513056i −0.866025 + 0.500000i 1.47355 + 1.35228i −0.0402223 − 0.0232224i 1.39783 − 0.214614i −1.97032 + 1.76574i −1.24814 − 2.53814i 0.500000 − 0.866025i 0.0410933 + 0.0512403i
37.4 −1.00926 − 0.990649i 0.866025 − 0.500000i 0.0372299 + 1.99965i 0.586448 + 0.338586i −1.36937 − 0.353295i 2.23683 + 1.41301i 1.94338 − 2.05506i 0.500000 − 0.866025i −0.256462 − 0.922687i
37.5 −0.938973 + 1.05751i 0.866025 − 0.500000i −0.236659 − 1.98595i 1.98722 + 1.14732i −0.284419 + 1.38532i −1.05630 − 2.42574i 2.32238 + 1.61448i 0.500000 − 0.866025i −3.07926 + 1.02420i
37.6 −0.446345 + 1.34193i −0.866025 + 0.500000i −1.60155 − 1.19793i −1.98722 − 1.14732i −0.284419 − 1.38532i −1.05630 − 2.42574i 2.32238 − 1.61448i 0.500000 − 0.866025i 2.42662 − 2.15461i
37.7 −0.189716 − 1.40143i −0.866025 + 0.500000i −1.92802 + 0.531748i 3.09843 + 1.78888i 0.865014 + 1.11882i 0.993295 + 2.45222i 1.11098 + 2.60110i 0.500000 − 0.866025i 1.91917 − 4.68162i
37.8 0.267238 + 1.38873i 0.866025 − 0.500000i −1.85717 + 0.742246i 1.56250 + 0.902108i 0.925802 + 1.06906i 2.63683 − 0.217074i −1.52709 − 2.38076i 0.500000 − 0.866025i −0.835229 + 2.41097i
37.9 0.268038 − 1.38858i 0.866025 − 0.500000i −1.85631 − 0.744384i −1.23074 − 0.710569i −0.462163 − 1.33656i 1.39545 − 2.24783i −1.53120 + 2.37811i 0.500000 − 0.866025i −1.31657 + 1.51852i
37.10 0.491996 − 1.32587i −0.866025 + 0.500000i −1.51588 − 1.30465i −3.08781 − 1.78275i 0.236856 + 1.39424i −2.38336 + 1.14873i −2.47560 + 1.36799i 0.500000 − 0.866025i −3.88289 + 3.21694i
37.11 0.867144 + 1.11717i −0.866025 + 0.500000i −0.496121 + 1.93749i 2.93503 + 1.69454i −1.30955 − 0.533922i −1.85242 − 1.88906i −2.59471 + 1.12583i 0.500000 − 0.866025i 0.652012 + 4.74833i
37.12 0.902242 − 1.08902i 0.866025 − 0.500000i −0.371918 − 1.96512i 3.08781 + 1.78275i 0.236856 − 1.39424i −2.38336 + 1.14873i −2.47560 − 1.36799i 0.500000 − 0.866025i 4.72740 − 1.75421i
37.13 1.06853 − 0.926418i −0.866025 + 0.500000i 0.283500 − 1.97980i 1.23074 + 0.710569i −0.462163 + 1.33656i 1.39545 − 2.24783i −1.53120 − 2.37811i 0.500000 − 0.866025i 1.97336 − 0.380919i
37.14 1.10325 + 0.884778i 0.866025 − 0.500000i 0.434335 + 1.95227i 0.0402223 + 0.0232224i 1.39783 + 0.214614i −1.97032 + 1.76574i −1.24814 + 2.53814i 0.500000 − 0.866025i 0.0238287 + 0.0612079i
37.15 1.30853 − 0.536416i 0.866025 − 0.500000i 1.42451 − 1.40384i −3.09843 − 1.78888i 0.865014 − 1.11882i 0.993295 + 2.45222i 1.11098 − 2.60110i 0.500000 − 0.866025i −5.01398 − 0.678758i
37.16 1.36256 + 0.378724i −0.866025 + 0.500000i 1.71314 + 1.03207i −0.586448 − 0.338586i −1.36937 + 0.353295i 2.23683 + 1.41301i 1.94338 + 2.05506i 0.500000 − 0.866025i −0.670840 − 0.683446i
109.1 −1.40107 + 0.192386i 0.866025 + 0.500000i 1.92598 − 0.539091i −2.93503 + 1.69454i −1.30955 − 0.533922i −1.85242 + 1.88906i −2.59471 + 1.12583i 0.500000 + 0.866025i 3.78617 − 2.93882i
109.2 −1.33630 − 0.462932i −0.866025 − 0.500000i 1.57139 + 1.23723i −1.56250 + 0.902108i 0.925802 + 1.06906i 2.63683 + 0.217074i −1.52709 − 2.38076i 0.500000 + 0.866025i 2.50558 − 0.482155i
109.3 −1.31787 + 0.513056i −0.866025 − 0.500000i 1.47355 − 1.35228i −0.0402223 + 0.0232224i 1.39783 + 0.214614i −1.97032 − 1.76574i −1.24814 + 2.53814i 0.500000 + 0.866025i 0.0410933 − 0.0512403i
109.4 −1.00926 + 0.990649i 0.866025 + 0.500000i 0.0372299 − 1.99965i 0.586448 − 0.338586i −1.36937 + 0.353295i 2.23683 − 1.41301i 1.94338 + 2.05506i 0.500000 + 0.866025i −0.256462 + 0.922687i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
8.b even 2 1 inner
56.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.2.bc.a ✓ 32
3.b odd 2 1 504.2.cj.e 32
4.b odd 2 1 672.2.bk.a 32
7.c even 3 1 inner 168.2.bc.a ✓ 32
7.c even 3 1 1176.2.c.e 16
7.d odd 6 1 1176.2.c.f 16
8.b even 2 1 inner 168.2.bc.a ✓ 32
8.d odd 2 1 672.2.bk.a 32
12.b even 2 1 2016.2.cr.e 32
21.h odd 6 1 504.2.cj.e 32
24.f even 2 1 2016.2.cr.e 32
24.h odd 2 1 504.2.cj.e 32
28.f even 6 1 4704.2.c.f 16
28.g odd 6 1 672.2.bk.a 32
28.g odd 6 1 4704.2.c.e 16
56.j odd 6 1 1176.2.c.f 16
56.k odd 6 1 672.2.bk.a 32
56.k odd 6 1 4704.2.c.e 16
56.m even 6 1 4704.2.c.f 16
56.p even 6 1 inner 168.2.bc.a ✓ 32
56.p even 6 1 1176.2.c.e 16
84.n even 6 1 2016.2.cr.e 32
168.s odd 6 1 504.2.cj.e 32
168.v even 6 1 2016.2.cr.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.bc.a ✓ 32 1.a even 1 1 trivial
168.2.bc.a ✓ 32 7.c even 3 1 inner
168.2.bc.a ✓ 32 8.b even 2 1 inner
168.2.bc.a ✓ 32 56.p even 6 1 inner
504.2.cj.e 32 3.b odd 2 1
504.2.cj.e 32 21.h odd 6 1
504.2.cj.e 32 24.h odd 2 1
504.2.cj.e 32 168.s odd 6 1
672.2.bk.a 32 4.b odd 2 1
672.2.bk.a 32 8.d odd 2 1
672.2.bk.a 32 28.g odd 6 1
672.2.bk.a 32 56.k odd 6 1
1176.2.c.e 16 7.c even 3 1
1176.2.c.e 16 56.p even 6 1
1176.2.c.f 16 7.d odd 6 1
1176.2.c.f 16 56.j odd 6 1
2016.2.cr.e 32 12.b even 2 1
2016.2.cr.e 32 24.f even 2 1
2016.2.cr.e 32 84.n even 6 1
2016.2.cr.e 32 168.v even 6 1
4704.2.c.e 16 28.g odd 6 1
4704.2.c.e 16 56.k odd 6 1
4704.2.c.f 16 28.f even 6 1
4704.2.c.f 16 56.m even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(168, [\chi])\).