# Properties

 Label 572.2.bh.a Level $572$ Weight $2$ Character orbit 572.bh Analytic conductor $4.567$ Analytic rank $0$ Dimension $112$ 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.bh (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$112$$ Relative dimension: $$14$$ over $$\Q(\zeta_{20})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$112q - 28q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$112q - 28q^{9} + 8q^{11} - 10q^{13} + 4q^{15} - 24q^{27} - 20q^{29} - 16q^{31} - 54q^{33} + 100q^{35} - 12q^{37} + 40q^{39} - 20q^{41} - 4q^{45} - 10q^{47} - 76q^{53} - 20q^{55} + 18q^{59} + 40q^{61} + 80q^{63} + 92q^{67} + 8q^{71} - 30q^{73} - 80q^{79} + 12q^{81} + 40q^{85} + 32q^{89} - 12q^{91} - 114q^{93} + 54q^{97} - 90q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
57.1 0 −2.33596 + 1.69718i 0 0.452121 + 0.230367i 0 1.95312 0.309344i 0 1.64927 5.07592i 0
57.2 0 −2.07031 + 1.50417i 0 −3.28934 1.67600i 0 −2.19381 + 0.347466i 0 1.09660 3.37499i 0
57.3 0 −1.91160 + 1.38886i 0 −0.585089 0.298118i 0 1.57586 0.249591i 0 0.798235 2.45671i 0
57.4 0 −1.79433 + 1.30366i 0 2.19779 + 1.11983i 0 −3.30914 + 0.524116i 0 0.593048 1.82521i 0
57.5 0 −0.703928 + 0.511434i 0 0.428276 + 0.218218i 0 3.57177 0.565712i 0 −0.693101 + 2.13314i 0
57.6 0 −0.673845 + 0.489577i 0 −1.19803 0.610425i 0 −1.96669 + 0.311493i 0 −0.712670 + 2.19337i 0
57.7 0 −0.400850 + 0.291234i 0 3.42455 + 1.74489i 0 3.82155 0.605273i 0 −0.851188 + 2.61969i 0
57.8 0 0.308779 0.224341i 0 −3.55885 1.81333i 0 −0.0145346 + 0.00230205i 0 −0.882035 + 2.71463i 0
57.9 0 0.403100 0.292869i 0 −0.455199 0.231936i 0 −2.63712 + 0.417678i 0 −0.850334 + 2.61706i 0
57.10 0 0.762321 0.553859i 0 1.51772 + 0.773316i 0 −0.624041 + 0.0988384i 0 −0.652677 + 2.00873i 0
57.11 0 1.76806 1.28457i 0 3.38384 + 1.72415i 0 −2.59890 + 0.411625i 0 0.548857 1.68921i 0
57.12 0 1.91590 1.39198i 0 −1.08106 0.550829i 0 1.99865 0.316555i 0 0.806006 2.48063i 0
57.13 0 2.15330 1.56447i 0 1.45337 + 0.740531i 0 3.81051 0.603525i 0 1.26211 3.88437i 0
57.14 0 2.57936 1.87402i 0 −2.69008 1.37067i 0 −3.38722 + 0.536483i 0 2.21412 6.81437i 0
73.1 0 −0.986426 + 3.03591i 0 −2.94028 0.465695i 0 −1.57983 + 3.10060i 0 −5.81665 4.22604i 0
73.2 0 −0.776972 + 2.39127i 0 3.56342 + 0.564390i 0 −0.372910 + 0.731878i 0 −2.68746 1.95255i 0
73.3 0 −0.746064 + 2.29615i 0 −0.201105 0.0318519i 0 1.79289 3.51875i 0 −2.28863 1.66279i 0
73.4 0 −0.501869 + 1.54459i 0 1.48121 + 0.234601i 0 −1.00750 + 1.97734i 0 0.293155 + 0.212990i 0
73.5 0 −0.372507 + 1.14646i 0 −3.01780 0.477973i 0 1.67505 3.28747i 0 1.25145 + 0.909229i 0
73.6 0 −0.269308 + 0.828844i 0 −2.25685 0.357450i 0 −0.534177 + 1.04838i 0 1.81260 + 1.31693i 0
See next 80 embeddings (of 112 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 541.14 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.d odd 10 1 inner
13.d odd 4 1 inner
143.s even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.bh.a 112
11.d odd 10 1 inner 572.2.bh.a 112
13.d odd 4 1 inner 572.2.bh.a 112
143.s even 20 1 inner 572.2.bh.a 112

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.bh.a 112 1.a even 1 1 trivial
572.2.bh.a 112 11.d odd 10 1 inner
572.2.bh.a 112 13.d odd 4 1 inner
572.2.bh.a 112 143.s even 20 1 inner

## Hecke kernels

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