# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$672 = 2^{5} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 672.bb (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 $$\Q(\zeta_{6})$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 16q^{9} - 8q^{11} - 16q^{25} + 24q^{35} + 16q^{43} + 8q^{49} + 16q^{57} + 96q^{59} + 32q^{67} - 24q^{73} - 16q^{81} - 56q^{91} - 16q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
271.1 0 −0.866025 + 0.500000i 0 −2.08776 + 3.61611i 0 −2.39694 + 1.12013i 0 0.500000 0.866025i 0
271.2 0 −0.866025 + 0.500000i 0 −1.25150 + 2.16767i 0 1.36321 2.26752i 0 0.500000 0.866025i 0
271.3 0 −0.866025 + 0.500000i 0 −0.225540 + 0.390646i 0 −0.458196 2.60577i 0 0.500000 0.866025i 0
271.4 0 −0.866025 + 0.500000i 0 −0.155280 + 0.268953i 0 2.58581 + 0.560001i 0 0.500000 0.866025i 0
271.5 0 −0.866025 + 0.500000i 0 0.155280 0.268953i 0 −2.58581 0.560001i 0 0.500000 0.866025i 0
271.6 0 −0.866025 + 0.500000i 0 0.225540 0.390646i 0 0.458196 + 2.60577i 0 0.500000 0.866025i 0
271.7 0 −0.866025 + 0.500000i 0 1.25150 2.16767i 0 −1.36321 + 2.26752i 0 0.500000 0.866025i 0
271.8 0 −0.866025 + 0.500000i 0 2.08776 3.61611i 0 2.39694 1.12013i 0 0.500000 0.866025i 0
271.9 0 0.866025 0.500000i 0 −1.61398 + 2.79550i 0 1.82725 + 1.91341i 0 0.500000 0.866025i 0
271.10 0 0.866025 0.500000i 0 −1.44142 + 2.49662i 0 −2.63862 + 0.194181i 0 0.500000 0.866025i 0
271.11 0 0.866025 0.500000i 0 −1.14053 + 1.97545i 0 −1.95181 1.78618i 0 0.500000 0.866025i 0
271.12 0 0.866025 0.500000i 0 −0.128707 + 0.222928i 0 0.623918 2.57113i 0 0.500000 0.866025i 0
271.13 0 0.866025 0.500000i 0 0.128707 0.222928i 0 −0.623918 + 2.57113i 0 0.500000 0.866025i 0
271.14 0 0.866025 0.500000i 0 1.14053 1.97545i 0 1.95181 + 1.78618i 0 0.500000 0.866025i 0
271.15 0 0.866025 0.500000i 0 1.44142 2.49662i 0 2.63862 0.194181i 0 0.500000 0.866025i 0
271.16 0 0.866025 0.500000i 0 1.61398 2.79550i 0 −1.82725 1.91341i 0 0.500000 0.866025i 0
367.1 0 −0.866025 0.500000i 0 −2.08776 3.61611i 0 −2.39694 1.12013i 0 0.500000 + 0.866025i 0
367.2 0 −0.866025 0.500000i 0 −1.25150 2.16767i 0 1.36321 + 2.26752i 0 0.500000 + 0.866025i 0
367.3 0 −0.866025 0.500000i 0 −0.225540 0.390646i 0 −0.458196 + 2.60577i 0 0.500000 + 0.866025i 0
367.4 0 −0.866025 0.500000i 0 −0.155280 0.268953i 0 2.58581 0.560001i 0 0.500000 + 0.866025i 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 367.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
8.d odd 2 1 inner
56.m 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.bb.a 32
3.b odd 2 1 2016.2.bs.c 32
4.b odd 2 1 168.2.t.a 32
7.c even 3 1 4704.2.p.a 32
7.d odd 6 1 inner 672.2.bb.a 32
7.d odd 6 1 4704.2.p.a 32
8.b even 2 1 168.2.t.a 32
8.d odd 2 1 inner 672.2.bb.a 32
12.b even 2 1 504.2.bk.c 32
21.g even 6 1 2016.2.bs.c 32
24.f even 2 1 2016.2.bs.c 32
24.h odd 2 1 504.2.bk.c 32
28.f even 6 1 168.2.t.a 32
28.f even 6 1 1176.2.p.a 32
28.g odd 6 1 1176.2.p.a 32
56.j odd 6 1 168.2.t.a 32
56.j odd 6 1 1176.2.p.a 32
56.k odd 6 1 4704.2.p.a 32
56.m even 6 1 inner 672.2.bb.a 32
56.m even 6 1 4704.2.p.a 32
56.p even 6 1 1176.2.p.a 32
84.j odd 6 1 504.2.bk.c 32
168.ba even 6 1 504.2.bk.c 32
168.be odd 6 1 2016.2.bs.c 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.t.a 32 4.b odd 2 1
168.2.t.a 32 8.b even 2 1
168.2.t.a 32 28.f even 6 1
168.2.t.a 32 56.j odd 6 1
504.2.bk.c 32 12.b even 2 1
504.2.bk.c 32 24.h odd 2 1
504.2.bk.c 32 84.j odd 6 1
504.2.bk.c 32 168.ba even 6 1
672.2.bb.a 32 1.a even 1 1 trivial
672.2.bb.a 32 7.d odd 6 1 inner
672.2.bb.a 32 8.d odd 2 1 inner
672.2.bb.a 32 56.m even 6 1 inner
1176.2.p.a 32 28.f even 6 1
1176.2.p.a 32 28.g odd 6 1
1176.2.p.a 32 56.j odd 6 1
1176.2.p.a 32 56.p even 6 1
2016.2.bs.c 32 3.b odd 2 1
2016.2.bs.c 32 21.g even 6 1
2016.2.bs.c 32 24.f even 2 1
2016.2.bs.c 32 168.be odd 6 1
4704.2.p.a 32 7.c even 3 1
4704.2.p.a 32 7.d odd 6 1
4704.2.p.a 32 56.k odd 6 1
4704.2.p.a 32 56.m even 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