Properties

 Label 572.2.x.a Level $572$ Weight $2$ Character orbit 572.x Analytic conductor $4.567$ Analytic rank $0$ Dimension $56$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$572 = 2^{2} \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 572.x (of order $$10$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$14$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$56q - 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q - 2q^{9} + q^{13} - 10q^{17} + 12q^{23} + 2q^{25} + 12q^{27} + 44q^{29} - 42q^{35} + 15q^{39} + 48q^{43} - 2q^{49} - 12q^{51} - 22q^{53} - 40q^{55} - 4q^{61} - 6q^{65} + 8q^{69} + 20q^{75} - 2q^{77} + 48q^{79} - 130q^{81} - 20q^{87} + 47q^{91} + 12q^{95} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1 0 −0.798565 2.45773i 0 −1.49474 + 2.05733i 0 2.65396 + 0.862324i 0 −2.97568 + 2.16196i 0
25.2 0 −0.798565 2.45773i 0 1.49474 2.05733i 0 −2.65396 0.862324i 0 −2.97568 + 2.16196i 0
25.3 0 −0.741447 2.28194i 0 −0.419790 + 0.577792i 0 −2.31756 0.753022i 0 −2.23045 + 1.62052i 0
25.4 0 −0.741447 2.28194i 0 0.419790 0.577792i 0 2.31756 + 0.753022i 0 −2.23045 + 1.62052i 0
25.5 0 −0.129602 0.398873i 0 −2.17400 + 2.99226i 0 −1.13565 0.368994i 0 2.28475 1.65997i 0
25.6 0 −0.129602 0.398873i 0 2.17400 2.99226i 0 1.13565 + 0.368994i 0 2.28475 1.65997i 0
25.7 0 −0.0444965 0.136946i 0 −1.21904 + 1.67786i 0 −3.54683 1.15243i 0 2.41028 1.75117i 0
25.8 0 −0.0444965 0.136946i 0 1.21904 1.67786i 0 3.54683 + 1.15243i 0 2.41028 1.75117i 0
25.9 0 0.268029 + 0.824907i 0 −0.0443935 + 0.0611023i 0 2.23222 + 0.725291i 0 1.81842 1.32116i 0
25.10 0 0.268029 + 0.824907i 0 0.0443935 0.0611023i 0 −2.23222 0.725291i 0 1.81842 1.32116i 0
25.11 0 0.579681 + 1.78407i 0 −2.40717 + 3.31319i 0 3.67038 + 1.19258i 0 −0.419842 + 0.305033i 0
25.12 0 0.579681 + 1.78407i 0 2.40717 3.31319i 0 −3.67038 1.19258i 0 −0.419842 + 0.305033i 0
25.13 0 0.866400 + 2.66651i 0 −0.720035 + 0.991043i 0 −1.55809 0.506254i 0 −3.93255 + 2.85717i 0
25.14 0 0.866400 + 2.66651i 0 0.720035 0.991043i 0 1.55809 + 0.506254i 0 −3.93255 + 2.85717i 0
181.1 0 −2.67869 1.94618i 0 −2.67966 0.870675i 0 1.22618 + 1.68769i 0 2.46069 + 7.57323i 0
181.2 0 −2.67869 1.94618i 0 2.67966 + 0.870675i 0 −1.22618 1.68769i 0 2.46069 + 7.57323i 0
181.3 0 −1.46518 1.06452i 0 −0.739289 0.240210i 0 0.0268071 + 0.0368969i 0 0.0865070 + 0.266241i 0
181.4 0 −1.46518 1.06452i 0 0.739289 + 0.240210i 0 −0.0268071 0.0368969i 0 0.0865070 + 0.266241i 0
181.5 0 −0.480503 0.349106i 0 −1.45044 0.471277i 0 −0.743441 1.02326i 0 −0.818043 2.51768i 0
181.6 0 −0.480503 0.349106i 0 1.45044 + 0.471277i 0 0.743441 + 1.02326i 0 −0.818043 2.51768i 0
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 493.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
13.b even 2 1 inner
143.n even 10 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.x.a 56
11.c even 5 1 inner 572.2.x.a 56
13.b even 2 1 inner 572.2.x.a 56
143.n even 10 1 inner 572.2.x.a 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.x.a 56 1.a even 1 1 trivial
572.2.x.a 56 11.c even 5 1 inner
572.2.x.a 56 13.b even 2 1 inner
572.2.x.a 56 143.n even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(572, [\chi])$$.