Properties

Label 672.2.bk.a
Level 672
Weight 2
Character orbit 672.bk
Analytic conductor 5.366
Analytic rank 0
Dimension 32
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 672.bk (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 168)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32q + 16q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q + 16q^{9} + 8q^{23} + 16q^{25} + 24q^{31} + 24q^{47} + 8q^{49} + 64q^{55} - 16q^{57} + 80q^{71} + 8q^{73} - 8q^{79} - 16q^{81} - 24q^{87} - 24q^{95} - 48q^{97} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1 0 −0.866025 0.500000i 0 −3.09843 + 1.78888i 0 −0.993295 + 2.45222i 0 0.500000 + 0.866025i 0
529.2 0 −0.866025 0.500000i 0 −2.93503 + 1.69454i 0 1.85242 1.88906i 0 0.500000 + 0.866025i 0
529.3 0 −0.866025 0.500000i 0 −1.23074 + 0.710569i 0 −1.39545 2.24783i 0 0.500000 + 0.866025i 0
529.4 0 −0.866025 0.500000i 0 0.0402223 0.0232224i 0 1.97032 + 1.76574i 0 0.500000 + 0.866025i 0
529.5 0 −0.866025 0.500000i 0 0.586448 0.338586i 0 −2.23683 + 1.41301i 0 0.500000 + 0.866025i 0
529.6 0 −0.866025 0.500000i 0 1.56250 0.902108i 0 −2.63683 0.217074i 0 0.500000 + 0.866025i 0
529.7 0 −0.866025 0.500000i 0 1.98722 1.14732i 0 1.05630 2.42574i 0 0.500000 + 0.866025i 0
529.8 0 −0.866025 0.500000i 0 3.08781 1.78275i 0 2.38336 + 1.14873i 0 0.500000 + 0.866025i 0
529.9 0 0.866025 + 0.500000i 0 −3.08781 + 1.78275i 0 2.38336 + 1.14873i 0 0.500000 + 0.866025i 0
529.10 0 0.866025 + 0.500000i 0 −1.98722 + 1.14732i 0 1.05630 2.42574i 0 0.500000 + 0.866025i 0
529.11 0 0.866025 + 0.500000i 0 −1.56250 + 0.902108i 0 −2.63683 0.217074i 0 0.500000 + 0.866025i 0
529.12 0 0.866025 + 0.500000i 0 −0.586448 + 0.338586i 0 −2.23683 + 1.41301i 0 0.500000 + 0.866025i 0
529.13 0 0.866025 + 0.500000i 0 −0.0402223 + 0.0232224i 0 1.97032 + 1.76574i 0 0.500000 + 0.866025i 0
529.14 0 0.866025 + 0.500000i 0 1.23074 0.710569i 0 −1.39545 2.24783i 0 0.500000 + 0.866025i 0
529.15 0 0.866025 + 0.500000i 0 2.93503 1.69454i 0 1.85242 1.88906i 0 0.500000 + 0.866025i 0
529.16 0 0.866025 + 0.500000i 0 3.09843 1.78888i 0 −0.993295 + 2.45222i 0 0.500000 + 0.866025i 0
625.1 0 −0.866025 + 0.500000i 0 −3.09843 1.78888i 0 −0.993295 2.45222i 0 0.500000 0.866025i 0
625.2 0 −0.866025 + 0.500000i 0 −2.93503 1.69454i 0 1.85242 + 1.88906i 0 0.500000 0.866025i 0
625.3 0 −0.866025 + 0.500000i 0 −1.23074 0.710569i 0 −1.39545 + 2.24783i 0 0.500000 0.866025i 0
625.4 0 −0.866025 + 0.500000i 0 0.0402223 + 0.0232224i 0 1.97032 1.76574i 0 0.500000 0.866025i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 625.16
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 672.2.bk.a 32
3.b odd 2 1 2016.2.cr.e 32
4.b odd 2 1 168.2.bc.a 32
7.c even 3 1 inner 672.2.bk.a 32
7.c even 3 1 4704.2.c.e 16
7.d odd 6 1 4704.2.c.f 16
8.b even 2 1 inner 672.2.bk.a 32
8.d odd 2 1 168.2.bc.a 32
12.b even 2 1 504.2.cj.e 32
21.h odd 6 1 2016.2.cr.e 32
24.f even 2 1 504.2.cj.e 32
24.h odd 2 1 2016.2.cr.e 32
28.f even 6 1 1176.2.c.f 16
28.g odd 6 1 168.2.bc.a 32
28.g odd 6 1 1176.2.c.e 16
56.j odd 6 1 4704.2.c.f 16
56.k odd 6 1 168.2.bc.a 32
56.k odd 6 1 1176.2.c.e 16
56.m even 6 1 1176.2.c.f 16
56.p even 6 1 inner 672.2.bk.a 32
56.p even 6 1 4704.2.c.e 16
84.n even 6 1 504.2.cj.e 32
168.s odd 6 1 2016.2.cr.e 32
168.v even 6 1 504.2.cj.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.bc.a 32 4.b odd 2 1
168.2.bc.a 32 8.d odd 2 1
168.2.bc.a 32 28.g odd 6 1
168.2.bc.a 32 56.k odd 6 1
504.2.cj.e 32 12.b even 2 1
504.2.cj.e 32 24.f even 2 1
504.2.cj.e 32 84.n even 6 1
504.2.cj.e 32 168.v even 6 1
672.2.bk.a 32 1.a even 1 1 trivial
672.2.bk.a 32 7.c even 3 1 inner
672.2.bk.a 32 8.b even 2 1 inner
672.2.bk.a 32 56.p even 6 1 inner
1176.2.c.e 16 28.g odd 6 1
1176.2.c.e 16 56.k odd 6 1
1176.2.c.f 16 28.f even 6 1
1176.2.c.f 16 56.m even 6 1
2016.2.cr.e 32 3.b odd 2 1
2016.2.cr.e 32 21.h odd 6 1
2016.2.cr.e 32 24.h odd 2 1
2016.2.cr.e 32 168.s odd 6 1
4704.2.c.e 16 7.c even 3 1
4704.2.c.e 16 56.p even 6 1
4704.2.c.f 16 7.d odd 6 1
4704.2.c.f 16 56.j odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(672, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database