Properties

 Label 656.2.bs.d Level 656 Weight 2 Character orbit 656.bs Analytic conductor 5.238 Analytic rank 0 Dimension 24 CM no Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ = $$656 = 2^{4} \cdot 41$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 656.bs (of order $$20$$, degree $$8$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$5.23818637260$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$3$$ over $$\Q(\zeta_{20})$$ Coefficient ring index: multiple of None Twist minimal: no (minimal twist has level 41) Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 6q^{3} - 10q^{5} + 8q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 6q^{3} - 10q^{5} + 8q^{7} + 16q^{11} - 8q^{15} + 8q^{17} - 16q^{19} - 10q^{21} - 12q^{23} - 8q^{25} + 6q^{27} + 40q^{29} + 12q^{31} + 10q^{33} + 36q^{35} + 50q^{39} - 4q^{41} + 16q^{45} + 12q^{47} - 30q^{49} + 24q^{51} - 26q^{53} - 20q^{55} + 10q^{57} - 6q^{59} + 30q^{61} - 92q^{63} + 68q^{65} + 22q^{67} - 38q^{69} - 4q^{71} - 4q^{75} - 20q^{77} + 2q^{79} + 28q^{81} - 80q^{83} - 56q^{85} + 10q^{87} - 72q^{89} - 6q^{93} + 40q^{95} - 22q^{97} - 14q^{99} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1 0 −2.22848 + 2.22848i 0 −2.57687 + 0.837277i 0 0.252316 1.59306i 0 6.93221i 0
33.2 0 0.983583 0.983583i 0 1.02071 0.331648i 0 −0.625712 + 3.95059i 0 1.06513i 0
33.3 0 2.02366 2.02366i 0 0.110420 0.0358775i 0 0.422339 2.66654i 0 5.19040i 0
49.1 0 −1.24382 1.24382i 0 −1.15434 + 0.375067i 0 −1.19485 0.189245i 0 0.0941838i 0
49.2 0 0.242613 + 0.242613i 0 2.26179 0.734900i 0 4.85225 + 0.768522i 0 2.88228i 0
49.3 0 0.604406 + 0.604406i 0 −3.27974 + 1.06565i 0 −1.70635 0.270260i 0 2.26939i 0
225.1 0 −0.964912 + 0.964912i 0 −2.03721 + 2.80398i 0 −1.29765 0.661185i 0 1.13789i 0
225.2 0 1.55151 1.55151i 0 1.49595 2.05900i 0 1.01547 + 0.517405i 0 1.81438i 0
225.3 0 1.67347 1.67347i 0 −1.68856 + 2.32411i 0 1.86997 + 0.952797i 0 2.60102i 0
241.1 0 −1.24382 + 1.24382i 0 −1.15434 0.375067i 0 −1.19485 + 0.189245i 0 0.0941838i 0
241.2 0 0.242613 0.242613i 0 2.26179 + 0.734900i 0 4.85225 0.768522i 0 2.88228i 0
241.3 0 0.604406 0.604406i 0 −3.27974 1.06565i 0 −1.70635 + 0.270260i 0 2.26939i 0
289.1 0 −1.49921 + 1.49921i 0 1.72514 + 2.37446i 0 1.11329 + 2.18496i 0 1.49525i 0
289.2 0 −0.0432913 + 0.0432913i 0 −0.422124 0.581004i 0 1.42228 + 2.79137i 0 2.99625i 0
289.3 0 1.90046 1.90046i 0 −0.455161 0.626475i 0 −2.12336 4.16732i 0 4.22349i 0
449.1 0 −0.964912 0.964912i 0 −2.03721 2.80398i 0 −1.29765 + 0.661185i 0 1.13789i 0
449.2 0 1.55151 + 1.55151i 0 1.49595 + 2.05900i 0 1.01547 0.517405i 0 1.81438i 0
449.3 0 1.67347 + 1.67347i 0 −1.68856 2.32411i 0 1.86997 0.952797i 0 2.60102i 0
497.1 0 −2.22848 2.22848i 0 −2.57687 0.837277i 0 0.252316 + 1.59306i 0 6.93221i 0
497.2 0 0.983583 + 0.983583i 0 1.02071 + 0.331648i 0 −0.625712 3.95059i 0 1.06513i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 513.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.g even 20 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 656.2.bs.d 24
4.b odd 2 1 41.2.g.a 24
12.b even 2 1 369.2.u.a 24
41.g even 20 1 inner 656.2.bs.d 24
164.n odd 20 1 41.2.g.a 24
164.o even 40 2 1681.2.a.m 24
492.y even 20 1 369.2.u.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.2.g.a 24 4.b odd 2 1
41.2.g.a 24 164.n odd 20 1
369.2.u.a 24 12.b even 2 1
369.2.u.a 24 492.y even 20 1
656.2.bs.d 24 1.a even 1 1 trivial
656.2.bs.d 24 41.g even 20 1 inner
1681.2.a.m 24 164.o even 40 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{24} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(656, [\chi])$$.

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database