# Properties

 Label 572.2.bk.a Level $572$ Weight $2$ Character orbit 572.bk Analytic conductor $4.567$ Analytic rank $0$ Dimension $640$ 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.bk (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$640$$ Relative dimension: $$80$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$640q - 6q^{2} - 12q^{5} - 14q^{6} - 6q^{8} + 120q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$640q - 6q^{2} - 12q^{5} - 14q^{6} - 6q^{8} + 120q^{9} - 12q^{13} + 4q^{14} - 36q^{16} - 10q^{18} - 60q^{20} - 56q^{21} - 44q^{22} - 14q^{24} + 26q^{26} - 14q^{28} - 24q^{29} - 16q^{32} - 28q^{33} + 12q^{34} - 12q^{37} + 4q^{40} - 36q^{41} - 8q^{42} + 26q^{44} - 16q^{45} - 50q^{46} + 48q^{48} + 36q^{50} - 52q^{52} - 40q^{53} + 116q^{54} - 36q^{57} - 70q^{58} + 52q^{60} - 8q^{61} - 48q^{65} - 228q^{66} - 44q^{68} - 68q^{70} - 38q^{72} - 12q^{73} - 12q^{74} + 108q^{76} + 108q^{78} + 62q^{80} - 168q^{81} - 36q^{84} - 52q^{85} - 62q^{86} + 16q^{89} - 80q^{92} - 4q^{93} - 124q^{94} - 132q^{96} - 4q^{97} - 108q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1 −1.41408 + 0.0192463i 0.0675205 + 0.0929340i 1.99926 0.0544317i 3.09570 + 1.57734i −0.0972682 0.130117i 0.303959 0.0481424i −2.82607 + 0.115449i 0.922973 2.84062i −4.40794 2.17091i
31.2 −1.40755 0.137128i −0.993999 1.36812i 1.96239 + 0.386029i −1.79157 0.912850i 1.21150 + 2.06201i 3.11515 0.493392i −2.70923 0.812454i 0.0433265 0.133345i 2.39655 + 1.53056i
31.3 −1.39732 + 0.217955i 0.585304 + 0.805601i 1.90499 0.609104i −0.214144 0.109112i −0.993440 0.998111i −4.72480 + 0.748335i −2.52912 + 1.26631i 0.620638 1.91013i 0.323009 + 0.105790i
31.4 −1.39464 + 0.234483i 0.111085 + 0.152895i 1.89004 0.654039i −2.01367 1.02602i −0.190775 0.187186i 3.86875 0.612750i −2.48256 + 1.35533i 0.916014 2.81920i 3.04893 + 0.958750i
31.5 −1.38260 + 0.297351i −1.11684 1.53720i 1.82316 0.822235i 1.61342 + 0.822078i 2.00123 + 1.79324i −1.26651 + 0.200595i −2.27621 + 1.67894i −0.188596 + 0.580439i −2.47516 0.656853i
31.6 −1.37530 0.329478i 1.80222 + 2.48055i 1.78289 + 0.906260i 1.79602 + 0.915119i −1.66131 4.00528i −0.175107 + 0.0277342i −2.15341 1.83380i −1.97806 + 6.08783i −2.16855 1.85031i
31.7 −1.37479 + 0.331572i 1.89320 + 2.60576i 1.78012 0.911688i −3.02502 1.54132i −3.46676 2.95466i −0.922158 + 0.146055i −2.14501 + 1.84362i −2.27875 + 7.01328i 4.66983 + 1.11599i
31.8 −1.35684 0.398721i −1.82403 2.51057i 1.68204 + 1.08200i −2.67101 1.36095i 1.47391 + 4.13373i −2.81984 + 0.446619i −1.85085 2.13877i −2.04880 + 6.30555i 3.08150 + 2.91158i
31.9 −1.33748 0.459517i 0.531376 + 0.731377i 1.57769 + 1.22919i 0.313818 + 0.159898i −0.374624 1.22238i −2.84457 + 0.450536i −1.54529 2.36898i 0.674500 2.07590i −0.346248 0.358064i
31.10 −1.33474 + 0.467394i −1.68703 2.32199i 1.56309 1.24770i 1.23417 + 0.628841i 3.33703 + 2.31076i −0.224881 + 0.0356177i −1.50315 + 2.39594i −1.61854 + 4.98136i −1.94122 0.262499i
31.11 −1.29838 0.560535i −0.722183 0.993999i 1.37160 + 1.45558i 2.78387 + 1.41845i 0.380499 + 1.69540i 1.85332 0.293537i −0.964963 2.65873i 0.460564 1.41747i −2.81944 3.40216i
31.12 −1.26887 + 0.624477i −0.732346 1.00799i 1.22006 1.58476i −3.57348 1.82078i 1.55872 + 0.821671i −3.95380 + 0.626220i −0.558450 + 2.77275i 0.447343 1.37678i 5.67132 + 0.0787760i
31.13 −1.26761 + 0.627035i 0.675560 + 0.929829i 1.21365 1.58967i 1.46176 + 0.744806i −1.43938 0.755058i 1.82475 0.289012i −0.541659 + 2.77608i 0.518851 1.59686i −2.31996 0.0275439i
31.14 −1.25415 0.653531i 0.738802 + 1.01687i 1.14579 + 1.63925i −3.61425 1.84155i −0.262011 1.75814i −0.636645 + 0.100835i −0.365696 2.80469i 0.438847 1.35063i 3.32931 + 4.67161i
31.15 −1.19654 + 0.753844i 1.53446 + 2.11200i 0.863437 1.80402i 3.64099 + 1.85518i −3.42817 1.37036i −0.0406960 + 0.00644561i 0.326806 + 2.80948i −1.17893 + 3.62839i −5.75512 + 0.524938i
31.16 −1.19169 0.761498i 1.13752 + 1.56566i 0.840242 + 1.81494i −1.23921 0.631408i −0.163321 2.73200i 3.50929 0.555817i 0.380764 2.80268i −0.230299 + 0.708787i 0.995933 + 1.69609i
31.17 −1.13998 0.836921i −1.62851 2.24145i 0.599125 + 1.90815i 2.91084 + 1.48315i −0.0194442 + 3.91816i −4.55254 + 0.721051i 0.913981 2.67668i −1.44502 + 4.44730i −2.07704 4.12691i
31.18 −1.09010 + 0.900938i 1.25644 + 1.72935i 0.376620 1.96422i −0.781284 0.398084i −2.92768 0.753176i 2.81330 0.445583i 1.35909 + 2.48050i −0.484940 + 1.49249i 1.21032 0.269939i
31.19 −1.00757 + 0.992369i −0.167099 0.229991i 0.0304084 1.99977i −2.43805 1.24225i 0.396600 + 0.0659099i 0.467841 0.0740988i 1.95387 + 2.04509i 0.902077 2.77631i 3.68928 1.16779i
31.20 −0.986038 1.01377i −0.445909 0.613742i −0.0554586 + 1.99923i −0.483456 0.246333i −0.182509 + 1.05722i −2.49202 + 0.394697i 2.08144 1.91510i 0.749207 2.30582i 0.226981 + 0.733007i
See next 80 embeddings (of 640 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 515.80 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.c even 5 1 inner
13.d odd 4 1 inner
44.h odd 10 1 inner
52.f even 4 1 inner
143.r odd 20 1 inner
572.bk 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.bk.a 640
4.b odd 2 1 inner 572.2.bk.a 640
11.c even 5 1 inner 572.2.bk.a 640
13.d odd 4 1 inner 572.2.bk.a 640
44.h odd 10 1 inner 572.2.bk.a 640
52.f even 4 1 inner 572.2.bk.a 640
143.r odd 20 1 inner 572.2.bk.a 640
572.bk even 20 1 inner 572.2.bk.a 640

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.bk.a 640 1.a even 1 1 trivial
572.2.bk.a 640 4.b odd 2 1 inner
572.2.bk.a 640 11.c even 5 1 inner
572.2.bk.a 640 13.d odd 4 1 inner
572.2.bk.a 640 44.h odd 10 1 inner
572.2.bk.a 640 52.f even 4 1 inner
572.2.bk.a 640 143.r odd 20 1 inner
572.2.bk.a 640 572.bk even 20 1 inner

## Hecke kernels

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