# Properties

 Label 572.2.bm.a Level $572$ Weight $2$ Character orbit 572.bm 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.bm (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$640$$ Relative dimension: $$80$$ over $$\Q(\zeta_{30})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$640q - 5q^{2} - 3q^{4} - 24q^{5} - 5q^{6} - 20q^{8} - 78q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$640q - 5q^{2} - 3q^{4} - 24q^{5} - 5q^{6} - 20q^{8} - 78q^{9} - 36q^{12} - 20q^{13} - 28q^{14} - 11q^{16} - 10q^{17} - 20q^{18} - 29q^{20} + 25q^{22} - 65q^{24} - 152q^{25} + 40q^{26} - 5q^{28} - 10q^{29} + 5q^{30} + 4q^{33} - 132q^{34} - 24q^{36} - 6q^{37} - 10q^{38} - 20q^{40} - 50q^{41} - 49q^{42} - 4q^{44} + 28q^{45} - 25q^{46} + 35q^{48} + 62q^{49} + 20q^{50} + 15q^{52} - 8q^{53} - 2q^{56} - 40q^{57} + 41q^{58} - 68q^{60} - 10q^{61} - 5q^{62} - 36q^{64} - 66q^{66} + 60q^{68} - 58q^{69} + 118q^{70} - 5q^{72} - 40q^{73} + 45q^{74} - 76q^{77} + 12q^{78} - 41q^{80} + 30q^{81} - 37q^{82} + 50q^{84} - 10q^{85} - 120q^{86} + 89q^{88} - 40q^{89} - 250q^{90} - 46q^{92} - 2q^{93} - 5q^{94} + 110q^{96} - 10q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
35.1 −1.41408 0.0197050i −2.48287 + 0.260960i 1.99922 + 0.0557288i −0.329087 1.01283i 3.51611 0.320092i −0.0624841 + 0.594497i −2.82596 0.118200i 3.16210 0.672124i 0.445397 + 1.43870i
35.2 −1.41200 + 0.0790274i −0.146087 + 0.0153543i 1.98751 0.223174i 1.02366 + 3.15050i 0.205061 0.0332252i 0.530846 5.05066i −2.78873 + 0.472190i −2.91334 + 0.619249i −1.69439 4.36763i
35.3 −1.40161 0.188417i 2.98914 0.314171i 1.92900 + 0.528172i 1.02633 + 3.15871i −4.24879 0.122859i −0.494362 + 4.70354i −2.60418 1.10374i 5.90180 1.25447i −0.843352 4.62064i
35.4 −1.40108 0.192317i 1.35212 0.142114i 1.92603 + 0.538900i −0.229666 0.706840i −1.92176 0.0609233i −0.0980876 + 0.933242i −2.59487 1.12545i −1.12640 + 0.239425i 0.185843 + 1.03451i
35.5 −1.39027 + 0.259143i 0.781568 0.0821461i 1.86569 0.720556i 0.119255 + 0.367030i −1.06530 + 0.316743i 0.0805893 0.766756i −2.40708 + 1.48525i −2.33034 + 0.495330i −0.260910 0.479366i
35.6 −1.36069 0.385373i −3.20597 + 0.336961i 1.70298 + 1.04875i 0.704069 + 2.16690i 4.49220 + 0.776993i −0.0455161 + 0.433057i −1.91307 2.08331i 7.23025 1.53684i −0.122957 3.21982i
35.7 −1.35772 0.395706i −0.705374 + 0.0741378i 1.68683 + 1.07452i −1.19366 3.67372i 0.987040 + 0.178462i 0.389527 3.70610i −1.86506 2.12639i −2.44239 + 0.519145i 0.166956 + 5.46025i
35.8 −1.35728 + 0.397241i −1.69383 + 0.178028i 1.68440 1.07833i −0.635019 1.95439i 2.22827 0.914490i −0.118464 + 1.12711i −1.85784 + 2.13271i −0.0970887 + 0.0206368i 1.63826 + 2.40039i
35.9 −1.33593 0.464004i 2.67160 0.280796i 1.56940 + 1.23975i −0.259301 0.798046i −3.69935 0.864509i 0.286196 2.72297i −1.52136 2.38442i 4.12414 0.876614i −0.0238895 + 1.18645i
35.10 −1.32463 + 0.495335i 0.111975 0.0117691i 1.50929 1.31227i 0.137946 + 0.424554i −0.142496 + 0.0710549i −0.311356 + 2.96235i −1.34923 + 2.48587i −2.92204 + 0.621099i −0.393024 0.494048i
35.11 −1.30497 + 0.545018i 3.13196 0.329183i 1.40591 1.42247i −0.288212 0.887025i −3.90772 + 2.13655i 0.347803 3.30913i −1.05941 + 2.62253i 6.76639 1.43824i 0.859553 + 1.00046i
35.12 −1.29084 0.577698i −0.134266 + 0.0141119i 1.33253 + 1.49143i 1.07700 + 3.31468i 0.181468 + 0.0593488i −0.0713495 + 0.678845i −0.858484 2.69500i −2.91661 + 0.619946i 0.524645 4.90090i
35.13 −1.27612 0.609521i −1.34702 + 0.141577i 1.25697 + 1.55564i −0.804721 2.47668i 1.80525 + 0.640366i −0.453255 + 4.31244i −0.655846 2.75134i −1.14003 + 0.242321i −0.482665 + 3.65103i
35.14 −1.26262 + 0.637013i −2.11369 + 0.222158i 1.18843 1.60861i 1.25520 + 3.86310i 2.52728 1.62695i −0.342039 + 3.25429i −0.475830 + 2.78812i 1.48390 0.315412i −4.04568 4.07806i
35.15 −1.14774 + 0.826246i 1.80992 0.190230i 0.634634 1.89664i 0.746857 + 2.29859i −1.92014 + 1.71377i −0.161700 + 1.53847i 0.838694 + 2.70122i 0.305164 0.0648647i −2.75640 2.02111i
35.16 −1.14220 0.833894i 0.971799 0.102140i 0.609240 + 1.90495i −0.384851 1.18445i −1.19516 0.693713i −0.230566 + 2.19369i 0.892651 2.68387i −2.00048 + 0.425215i −0.548129 + 1.67380i
35.17 −1.12305 + 0.859505i 0.297649 0.0312842i 0.522503 1.93054i −1.00335 3.08799i −0.307388 + 0.290965i 0.422442 4.01927i 1.07251 + 2.61720i −2.84683 + 0.605112i 3.78095 + 2.60559i
35.18 −1.10128 + 0.887234i −2.91603 + 0.306487i 0.425631 1.95418i −0.642340 1.97692i 2.93943 2.92472i 0.368936 3.51019i 1.26508 + 2.52974i 5.47483 1.16371i 2.46138 + 1.60723i
35.19 −1.06158 0.934374i −2.14948 + 0.225919i 0.253889 + 1.98382i −0.00262404 0.00807598i 2.49293 + 1.76858i 0.302942 2.88230i 1.58411 2.34320i 1.63476 0.347479i −0.00476036 + 0.0110251i
35.20 −1.05823 0.938168i −0.852202 + 0.0895701i 0.239682 + 1.98559i 0.526525 + 1.62048i 0.985854 + 0.704723i 0.121128 1.15246i 1.60917 2.32606i −2.21622 + 0.471071i 0.963098 2.20880i
See next 80 embeddings (of 640 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 523.80 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.d odd 10 1 inner
13.c even 3 1 inner
44.g even 10 1 inner
52.j odd 6 1 inner
143.t odd 30 1 inner
572.bm even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.bm.a 640
4.b odd 2 1 inner 572.2.bm.a 640
11.d odd 10 1 inner 572.2.bm.a 640
13.c even 3 1 inner 572.2.bm.a 640
44.g even 10 1 inner 572.2.bm.a 640
52.j odd 6 1 inner 572.2.bm.a 640
143.t odd 30 1 inner 572.2.bm.a 640
572.bm even 30 1 inner 572.2.bm.a 640

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.bm.a 640 1.a even 1 1 trivial
572.2.bm.a 640 4.b odd 2 1 inner
572.2.bm.a 640 11.d odd 10 1 inner
572.2.bm.a 640 13.c even 3 1 inner
572.2.bm.a 640 44.g even 10 1 inner
572.2.bm.a 640 52.j odd 6 1 inner
572.2.bm.a 640 143.t odd 30 1 inner
572.2.bm.a 640 572.bm even 30 1 inner

## Hecke kernels

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