# Properties

 Label 572.2.bs.a Level $572$ Weight $2$ Character orbit 572.bs Analytic conductor $4.567$ Analytic rank $0$ Dimension $1280$ 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.bs (of order $$60$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$1280$$ Relative dimension: $$80$$ over $$\Q(\zeta_{60})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$1280q - 12q^{2} - 18q^{4} - 24q^{5} - 4q^{6} - 12q^{8} - 156q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$1280q - 12q^{2} - 18q^{4} - 24q^{5} - 4q^{6} - 12q^{8} - 156q^{9} - 48q^{10} - 24q^{13} - 40q^{14} + 18q^{16} - 36q^{17} - 8q^{18} + 18q^{20} - 40q^{21} - 10q^{22} - 100q^{24} - 44q^{26} - 4q^{28} - 12q^{29} - 114q^{30} - 92q^{32} - 20q^{33} + 36q^{34} - 72q^{36} - 24q^{37} + 8q^{40} + 26q^{42} - 20q^{44} - 80q^{45} + 32q^{46} + 30q^{48} - 36q^{49} - 78q^{50} + 34q^{52} - 80q^{53} - 188q^{54} - 48q^{56} - 92q^{58} + 20q^{60} - 28q^{61} + 42q^{62} - 48q^{65} + 204q^{66} + 26q^{68} - 36q^{69} + 38q^{70} - 136q^{72} - 24q^{73} - 6q^{74} - 276q^{76} - 156q^{78} - 80q^{80} + 60q^{81} + 162q^{82} - 108q^{84} + 16q^{85} - 88q^{86} + 6q^{88} - 64q^{89} - 16q^{92} + 64q^{93} + 34q^{94} + 114q^{96} - 8q^{97} - 24q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1 −1.40935 + 0.117195i −0.693741 3.26379i 1.97253 0.330337i 0.187333 1.18277i 1.36022 + 4.51852i 2.19944 1.42833i −2.74127 + 0.696730i −7.43044 + 3.30824i −0.125403 + 1.68889i
15.2 −1.40745 + 0.138159i −0.392951 1.84869i 1.96182 0.388905i 0.0502636 0.317352i 0.808473 + 2.54765i −3.23254 + 2.09923i −2.70744 + 0.818408i −0.522613 + 0.232682i −0.0268983 + 0.453601i
15.3 −1.40470 0.163726i −0.0924543 0.434963i 1.94639 + 0.459972i −0.177808 + 1.12264i 0.0586564 + 0.626132i −1.44036 + 0.935381i −2.65879 0.964798i 2.55999 1.13978i 0.433572 1.54786i
15.4 −1.40184 + 0.186682i 0.154145 + 0.725193i 1.93030 0.523395i 0.469822 2.96634i −0.351466 0.987827i −2.19466 + 1.42523i −2.60826 + 1.09407i 2.23849 0.996641i −0.104853 + 4.24603i
15.5 −1.40168 + 0.187875i 0.525158 + 2.47067i 1.92941 0.526680i −0.0408544 + 0.257944i −1.20028 3.36443i 2.76404 1.79499i −2.60546 + 1.10072i −3.08781 + 1.37478i 0.00880344 0.369231i
15.6 −1.39320 0.242899i −0.355188 1.67103i 1.88200 + 0.676813i −0.634502 + 4.00609i 0.0889558 + 2.41435i 0.869682 0.564778i −2.45760 1.40007i 0.0744566 0.0331502i 1.85706 5.42715i
15.7 −1.39159 + 0.251945i 0.265282 + 1.24805i 1.87305 0.701210i −0.578915 + 3.65512i −0.683606 1.66994i 2.36802 1.53781i −2.42985 + 1.44770i 1.25337 0.558037i −0.115280 5.23229i
15.8 −1.36269 + 0.378247i 0.565515 + 2.66054i 1.71386 1.03087i −0.216128 + 1.36458i −1.77696 3.41159i −3.01627 + 1.95879i −1.94554 + 2.05302i −4.01801 + 1.78893i −0.221632 1.94125i
15.9 −1.36092 0.384566i −0.238841 1.12366i 1.70422 + 1.04673i 0.255413 1.61261i −0.107076 + 1.62106i 3.96409 2.57431i −1.91677 2.07990i 1.53508 0.683460i −0.967752 + 2.09642i
15.10 −1.35262 0.412811i 0.680250 + 3.20032i 1.65917 + 1.11675i 0.568211 3.58755i 0.401007 4.60964i −0.245673 + 0.159542i −1.78323 2.19547i −7.03870 + 3.13383i −2.24955 + 4.61803i
15.11 −1.31882 + 0.510612i −0.0308046 0.144924i 1.47855 1.34681i 0.670289 4.23204i 0.114626 + 0.175399i 2.29438 1.48999i −1.26224 + 2.53115i 2.72058 1.21128i 1.27694 + 5.92354i
15.12 −1.31691 0.515500i 0.427935 + 2.01327i 1.46852 + 1.35774i −0.499033 + 3.15077i 0.474290 2.87191i −3.19451 + 2.07454i −1.23400 2.54504i −1.12951 + 0.502889i 2.28140 3.89204i
15.13 −1.25157 0.658464i 0.151963 + 0.714928i 1.13285 + 1.64823i 0.124235 0.784389i 0.280563 0.994844i 2.56674 1.66686i −0.332541 2.80881i 2.25261 1.00293i −0.671981 + 0.899913i
15.14 −1.21980 0.715597i 0.271650 + 1.27801i 0.975843 + 1.74578i 0.291800 1.84235i 0.583182 1.75332i −2.48336 + 1.61271i 0.0589342 2.82781i 1.18112 0.525867i −1.67432 + 2.03850i
15.15 −1.20807 + 0.735234i 0.000709742 0.00333907i 0.918860 1.77643i −0.193859 + 1.22398i −0.00331242 0.00351200i 0.441648 0.286810i 0.196044 + 2.82162i 2.74063 1.22021i −0.665717 1.62118i
15.16 −1.16781 + 0.797640i −0.533943 2.51200i 0.727542 1.86298i 0.273613 1.72753i 2.62722 + 2.50764i −2.14674 + 1.39411i 0.636355 + 2.75591i −3.28443 + 1.46232i 1.05842 + 2.23566i
15.17 −1.13651 0.841633i −0.364332 1.71405i 0.583308 + 1.91305i 0.0328233 0.207238i −1.02853 + 2.25466i 0.127656 0.0829010i 0.947149 2.66513i −0.0645808 + 0.0287532i −0.211723 + 0.207903i
15.18 −1.12541 0.856423i −0.683893 3.21746i 0.533078 + 1.92765i −0.317452 + 2.00431i −1.98585 + 4.20665i −0.619612 + 0.402380i 1.05095 2.62593i −7.14372 + 3.18059i 2.07380 1.98379i
15.19 −1.06237 + 0.933467i 0.467247 + 2.19822i 0.257278 1.98338i 0.319292 2.01593i −2.54836 1.89918i 1.25573 0.815481i 1.57810 + 2.34726i −1.87323 + 0.834017i 1.54260 + 2.43972i
15.20 −1.06204 0.933852i −0.403488 1.89826i 0.255840 + 1.98357i 0.592510 3.74096i −1.34418 + 2.39282i −2.79408 + 1.81450i 1.58065 2.34554i −0.699952 + 0.311639i −4.12277 + 3.41972i
See next 80 embeddings (of 1280 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 531.80 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.f odd 12 1 inner
44.h odd 10 1 inner
52.l even 12 1 inner
143.x odd 60 1 inner
572.bs even 60 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.bs.a 1280
4.b odd 2 1 inner 572.2.bs.a 1280
11.c even 5 1 inner 572.2.bs.a 1280
13.f odd 12 1 inner 572.2.bs.a 1280
44.h odd 10 1 inner 572.2.bs.a 1280
52.l even 12 1 inner 572.2.bs.a 1280
143.x odd 60 1 inner 572.2.bs.a 1280
572.bs even 60 1 inner 572.2.bs.a 1280

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.bs.a 1280 1.a even 1 1 trivial
572.2.bs.a 1280 4.b odd 2 1 inner
572.2.bs.a 1280 11.c even 5 1 inner
572.2.bs.a 1280 13.f odd 12 1 inner
572.2.bs.a 1280 44.h odd 10 1 inner
572.2.bs.a 1280 52.l even 12 1 inner
572.2.bs.a 1280 143.x odd 60 1 inner
572.2.bs.a 1280 572.bs even 60 1 inner

## Hecke kernels

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