# Properties

 Label 572.2.b.c Level $572$ Weight $2$ Character orbit 572.b Analytic conductor $4.567$ Analytic rank $0$ Dimension $56$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$572 = 2^{2} \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 572.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

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

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$56q + 4q^{4} - 32q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q + 4q^{4} - 32q^{9} - 12q^{14} - 4q^{16} - 4q^{22} - 192q^{25} + 4q^{26} + 28q^{36} + 24q^{38} + 88q^{42} + 56q^{48} + 40q^{49} - 8q^{53} - 68q^{56} + 28q^{64} - 76q^{66} - 16q^{69} + 32q^{77} + 108q^{78} - 152q^{81} - 60q^{82} + 52q^{88} + 132q^{92} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
571.1 −1.41153 0.0870114i 2.50968i 1.98486 + 0.245639i 2.92941i 0.218370 3.54249i 1.23786i −2.78032 0.519434i −3.29847 −0.254892 + 4.13496i
571.2 −1.41153 0.0870114i 2.50968i 1.98486 + 0.245639i 2.92941i 0.218370 3.54249i 1.23786i −2.78032 0.519434i −3.29847 0.254892 4.13496i
571.3 −1.41153 + 0.0870114i 2.50968i 1.98486 0.245639i 2.92941i 0.218370 + 3.54249i 1.23786i −2.78032 + 0.519434i −3.29847 0.254892 + 4.13496i
571.4 −1.41153 + 0.0870114i 2.50968i 1.98486 0.245639i 2.92941i 0.218370 + 3.54249i 1.23786i −2.78032 + 0.519434i −3.29847 −0.254892 4.13496i
571.5 −1.30250 0.550896i 0.566258i 1.39303 + 1.43509i 3.29519i 0.311949 0.737553i 2.02255i −1.02384 2.63662i 2.67935 −1.81531 + 4.29199i
571.6 −1.30250 0.550896i 0.566258i 1.39303 + 1.43509i 3.29519i 0.311949 0.737553i 2.02255i −1.02384 2.63662i 2.67935 1.81531 4.29199i
571.7 −1.30250 + 0.550896i 0.566258i 1.39303 1.43509i 3.29519i 0.311949 + 0.737553i 2.02255i −1.02384 + 2.63662i 2.67935 1.81531 + 4.29199i
571.8 −1.30250 + 0.550896i 0.566258i 1.39303 1.43509i 3.29519i 0.311949 + 0.737553i 2.02255i −1.02384 + 2.63662i 2.67935 −1.81531 4.29199i
571.9 −1.28473 0.591168i 1.41709i 1.30104 + 1.51898i 1.72500i −0.837739 + 1.82057i 3.58187i −0.773512 2.72060i 0.991852 −1.01976 + 2.21615i
571.10 −1.28473 0.591168i 1.41709i 1.30104 + 1.51898i 1.72500i −0.837739 + 1.82057i 3.58187i −0.773512 2.72060i 0.991852 1.01976 2.21615i
571.11 −1.28473 + 0.591168i 1.41709i 1.30104 1.51898i 1.72500i −0.837739 1.82057i 3.58187i −0.773512 + 2.72060i 0.991852 1.01976 + 2.21615i
571.12 −1.28473 + 0.591168i 1.41709i 1.30104 1.51898i 1.72500i −0.837739 1.82057i 3.58187i −0.773512 + 2.72060i 0.991852 −1.01976 2.21615i
571.13 −0.913620 1.07949i 1.67751i −0.330598 + 1.97249i 1.10479i 1.81085 1.53260i 1.27047i 2.43132 1.44523i 0.185967 −1.19260 + 1.00935i
571.14 −0.913620 1.07949i 1.67751i −0.330598 + 1.97249i 1.10479i 1.81085 1.53260i 1.27047i 2.43132 1.44523i 0.185967 1.19260 1.00935i
571.15 −0.913620 + 1.07949i 1.67751i −0.330598 1.97249i 1.10479i 1.81085 + 1.53260i 1.27047i 2.43132 + 1.44523i 0.185967 1.19260 + 1.00935i
571.16 −0.913620 + 1.07949i 1.67751i −0.330598 1.97249i 1.10479i 1.81085 + 1.53260i 1.27047i 2.43132 + 1.44523i 0.185967 −1.19260 1.00935i
571.17 −0.883550 1.10424i 2.47596i −0.438678 + 1.95130i 2.76928i −2.73405 + 2.18764i 0.534510i 2.54229 1.23966i −3.13038 −3.05794 + 2.44680i
571.18 −0.883550 1.10424i 2.47596i −0.438678 + 1.95130i 2.76928i −2.73405 + 2.18764i 0.534510i 2.54229 1.23966i −3.13038 3.05794 2.44680i
571.19 −0.883550 + 1.10424i 2.47596i −0.438678 1.95130i 2.76928i −2.73405 2.18764i 0.534510i 2.54229 + 1.23966i −3.13038 3.05794 + 2.44680i
571.20 −0.883550 + 1.10424i 2.47596i −0.438678 1.95130i 2.76928i −2.73405 2.18764i 0.534510i 2.54229 + 1.23966i −3.13038 −3.05794 2.44680i
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 571.56 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.b odd 2 1 inner
13.b even 2 1 inner
44.c even 2 1 inner
52.b odd 2 1 inner
143.d odd 2 1 inner
572.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.b.c 56
4.b odd 2 1 inner 572.2.b.c 56
11.b odd 2 1 inner 572.2.b.c 56
13.b even 2 1 inner 572.2.b.c 56
44.c even 2 1 inner 572.2.b.c 56
52.b odd 2 1 inner 572.2.b.c 56
143.d odd 2 1 inner 572.2.b.c 56
572.b even 2 1 inner 572.2.b.c 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.b.c 56 1.a even 1 1 trivial
572.2.b.c 56 4.b odd 2 1 inner
572.2.b.c 56 11.b odd 2 1 inner
572.2.b.c 56 13.b even 2 1 inner
572.2.b.c 56 44.c even 2 1 inner
572.2.b.c 56 52.b odd 2 1 inner
572.2.b.c 56 143.d odd 2 1 inner
572.2.b.c 56 572.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{14} + 25 T_{3}^{12} + 241 T_{3}^{10} + 1118 T_{3}^{8} + 2535 T_{3}^{6} + 2517 T_{3}^{4} + 743 T_{3}^{2} + 52$$ acting on $$S_{2}^{\mathrm{new}}(572, [\chi])$$.