# Properties

 Label 572.2.bg.a Level $572$ Weight $2$ Character orbit 572.bg 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.bg (of order $$15$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$112$$ Relative dimension: $$14$$ over $$\Q(\zeta_{15})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$112q + 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$112q + 8q^{9} - 10q^{11} + 11q^{13} - 2q^{15} + 4q^{17} - 12q^{19} - 40q^{21} + 10q^{23} - 16q^{25} - 12q^{27} + q^{29} + 4q^{31} + 35q^{33} - 5q^{35} - 12q^{37} + 21q^{39} - 10q^{41} - 32q^{43} + 34q^{45} + 70q^{47} + 16q^{49} - 48q^{51} - 26q^{53} + 10q^{55} - 12q^{57} - 5q^{59} + 28q^{61} + 34q^{63} + 22q^{65} - 68q^{67} - 58q^{69} + 44q^{71} + 42q^{73} - 24q^{75} + 46q^{77} - 24q^{79} + 64q^{81} - 114q^{83} + 4q^{85} - 30q^{87} - 6q^{89} + 77q^{91} - 5q^{93} - 36q^{95} - 15q^{97} - 40q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1 0 −0.351299 3.34239i 0 −0.419191 1.29014i 0 −0.149591 + 1.42326i 0 −8.11369 + 1.72462i 0
9.2 0 −0.263130 2.50351i 0 1.21356 + 3.73495i 0 0.197160 1.87585i 0 −3.26389 + 0.693762i 0
9.3 0 −0.229534 2.18387i 0 −0.547837 1.68607i 0 −0.0708172 + 0.673781i 0 −1.78216 + 0.378809i 0
9.4 0 −0.165782 1.57731i 0 −1.14854 3.53484i 0 0.493072 4.69127i 0 0.474019 0.100756i 0
9.5 0 −0.153632 1.46171i 0 0.347814 + 1.07046i 0 0.456258 4.34101i 0 0.821453 0.174605i 0
9.6 0 −0.0694093 0.660385i 0 −0.596743 1.83659i 0 −0.373949 + 3.55789i 0 2.50315 0.532061i 0
9.7 0 −0.0692906 0.659256i 0 0.701430 + 2.15878i 0 −0.337098 + 3.20727i 0 2.50463 0.532375i 0
9.8 0 0.0211820 + 0.201533i 0 −0.370496 1.14027i 0 −0.131751 + 1.25353i 0 2.89428 0.615197i 0
9.9 0 0.0911637 + 0.867364i 0 0.671959 + 2.06808i 0 0.257245 2.44753i 0 2.19043 0.465591i 0
9.10 0 0.151007 + 1.43674i 0 −0.310916 0.956900i 0 0.0904890 0.860946i 0 0.893027 0.189819i 0
9.11 0 0.161126 + 1.53301i 0 1.30886 + 4.02825i 0 −0.295810 + 2.81444i 0 0.610280 0.129719i 0
9.12 0 0.251214 + 2.39014i 0 −0.900429 2.77123i 0 0.251542 2.39326i 0 −2.71524 + 0.577141i 0
9.13 0 0.277675 + 2.64191i 0 0.389171 + 1.19775i 0 0.0870458 0.828185i 0 −3.96812 + 0.843450i 0
9.14 0 0.348708 + 3.31773i 0 −0.338638 1.04222i 0 −0.473796 + 4.50786i 0 −7.95129 + 1.69010i 0
81.1 0 −2.59967 + 0.552578i 0 −1.82163 + 1.32349i 0 −4.40077 0.935412i 0 3.71233 1.65283i 0
81.2 0 −2.59692 + 0.551993i 0 1.71674 1.24728i 0 −1.20270 0.255642i 0 3.69867 1.64675i 0
81.3 0 −2.50220 + 0.531859i 0 −0.767747 + 0.557801i 0 3.31034 + 0.703635i 0 3.23748 1.44142i 0
81.4 0 −2.03406 + 0.432352i 0 3.08678 2.24268i 0 1.74512 + 0.370936i 0 1.20982 0.538647i 0
81.5 0 −0.978574 + 0.208002i 0 −3.20054 + 2.32533i 0 0.357037 + 0.0758905i 0 −1.82629 + 0.813119i 0
81.6 0 −0.807770 + 0.171697i 0 0.437086 0.317562i 0 −2.23753 0.475601i 0 −2.11762 + 0.942827i 0
See next 80 embeddings (of 112 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 477.14 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.c even 3 1 inner
143.q even 15 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.bg.a 112
11.c even 5 1 inner 572.2.bg.a 112
13.c even 3 1 inner 572.2.bg.a 112
143.q even 15 1 inner 572.2.bg.a 112

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.bg.a 112 1.a even 1 1 trivial
572.2.bg.a 112 11.c even 5 1 inner
572.2.bg.a 112 13.c even 3 1 inner
572.2.bg.a 112 143.q even 15 1 inner

## Hecke kernels

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