# Properties

 Label 576.2.bl.a Level $576$ Weight $2$ Character orbit 576.bl Analytic conductor $4.599$ Analytic rank $0$ Dimension $1504$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 576.bl (of order $$48$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.59938315643$$ Analytic rank: $$0$$ Dimension: $$1504$$ Relative dimension: $$94$$ over $$\Q(\zeta_{48})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{48}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$1504 q - 24 q^{2} - 16 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{9}+O(q^{10})$$ 1504 * q - 24 * q^2 - 16 * q^3 - 8 * q^4 - 24 * q^5 - 16 * q^6 - 8 * q^7 - 16 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$1504 q - 24 q^{2} - 16 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{9} - 32 q^{10} - 24 q^{11} - 16 q^{12} - 8 q^{13} - 24 q^{14} - 16 q^{15} - 8 q^{16} - 16 q^{18} - 32 q^{19} - 24 q^{20} - 16 q^{21} - 8 q^{22} - 24 q^{23} - 96 q^{24} - 8 q^{25} - 16 q^{27} - 32 q^{28} - 24 q^{29} + 64 q^{30} - 16 q^{31} - 24 q^{32} - 8 q^{34} + 64 q^{36} - 32 q^{37} - 24 q^{38} - 16 q^{39} - 8 q^{40} - 24 q^{41} - 96 q^{42} - 8 q^{43} - 16 q^{45} - 32 q^{46} - 24 q^{47} - 16 q^{48} - 8 q^{49} - 24 q^{50} - 16 q^{51} - 8 q^{52} - 16 q^{54} - 32 q^{55} - 24 q^{56} - 16 q^{57} + 64 q^{58} - 24 q^{59} - 16 q^{60} - 8 q^{61} - 32 q^{64} - 48 q^{65} - 8 q^{67} - 24 q^{68} - 16 q^{69} - 8 q^{70} - 16 q^{72} - 32 q^{73} - 24 q^{74} - 16 q^{75} - 56 q^{76} - 24 q^{77} - 40 q^{78} - 8 q^{79} - 16 q^{81} - 32 q^{82} - 24 q^{83} - 128 q^{84} - 8 q^{85} - 24 q^{86} - 16 q^{87} - 8 q^{88} - 160 q^{90} - 32 q^{91} - 480 q^{92} + 8 q^{93} - 8 q^{94} - 48 q^{95} - 152 q^{96} - 16 q^{99}+O(q^{100})$$ 1504 * q - 24 * q^2 - 16 * q^3 - 8 * q^4 - 24 * q^5 - 16 * q^6 - 8 * q^7 - 16 * q^9 - 32 * q^10 - 24 * q^11 - 16 * q^12 - 8 * q^13 - 24 * q^14 - 16 * q^15 - 8 * q^16 - 16 * q^18 - 32 * q^19 - 24 * q^20 - 16 * q^21 - 8 * q^22 - 24 * q^23 - 96 * q^24 - 8 * q^25 - 16 * q^27 - 32 * q^28 - 24 * q^29 + 64 * q^30 - 16 * q^31 - 24 * q^32 - 8 * q^34 + 64 * q^36 - 32 * q^37 - 24 * q^38 - 16 * q^39 - 8 * q^40 - 24 * q^41 - 96 * q^42 - 8 * q^43 - 16 * q^45 - 32 * q^46 - 24 * q^47 - 16 * q^48 - 8 * q^49 - 24 * q^50 - 16 * q^51 - 8 * q^52 - 16 * q^54 - 32 * q^55 - 24 * q^56 - 16 * q^57 + 64 * q^58 - 24 * q^59 - 16 * q^60 - 8 * q^61 - 32 * q^64 - 48 * q^65 - 8 * q^67 - 24 * q^68 - 16 * q^69 - 8 * q^70 - 16 * q^72 - 32 * q^73 - 24 * q^74 - 16 * q^75 - 56 * q^76 - 24 * q^77 - 40 * q^78 - 8 * q^79 - 16 * q^81 - 32 * q^82 - 24 * q^83 - 128 * q^84 - 8 * q^85 - 24 * q^86 - 16 * q^87 - 8 * q^88 - 160 * q^90 - 32 * q^91 - 480 * q^92 + 8 * q^93 - 8 * q^94 - 48 * q^95 - 152 * q^96 - 16 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −1.41417 + 0.0110611i 1.36345 1.06817i 1.99976 0.0312846i 2.78390 + 1.37287i −1.91634 + 1.52566i 0.492179 + 0.641420i −2.82765 + 0.0663612i 0.718010 2.91281i −3.95209 1.91067i
11.2 −1.41165 + 0.0851593i −1.66601 0.473724i 1.98550 0.240430i −1.72799 0.852148i 2.39216 + 0.526855i 0.749706 + 0.977036i −2.78234 + 0.508485i 2.55117 + 1.57846i 2.51187 + 1.05578i
11.3 −1.40351 0.173668i −0.728794 + 1.57126i 1.93968 + 0.487489i −1.79173 0.883583i 1.29575 2.07871i −0.0778075 0.101401i −2.63770 1.02106i −1.93772 2.29025i 2.36126 + 1.55128i
11.4 −1.40127 0.190903i −0.555798 1.64045i 1.92711 + 0.535013i −0.879249 0.433598i 0.465655 + 2.40482i −3.05138 3.97664i −2.59827 1.11759i −2.38218 + 1.82352i 1.14929 + 0.775439i
11.5 −1.38867 0.267570i −1.55208 + 0.768797i 1.85681 + 0.743133i 2.74800 + 1.35516i 2.36103 0.652317i −1.12886 1.47116i −2.37966 1.52879i 1.81790 2.38647i −3.45347 2.61716i
11.6 −1.38521 0.284957i 0.799756 + 1.53636i 1.83760 + 0.789449i 1.54938 + 0.764070i −0.670032 2.35607i 2.14122 + 2.79049i −2.32050 1.61719i −1.72078 + 2.45742i −1.92849 1.49990i
11.7 −1.37786 + 0.318576i −0.139389 1.72643i 1.79702 0.877909i −0.245576 0.121104i 0.742059 + 2.33438i 0.961138 + 1.25258i −2.19637 + 1.78213i −2.96114 + 0.481292i 0.376951 + 0.0886311i
11.8 −1.36414 0.372980i 1.49848 + 0.868647i 1.72177 + 1.01760i −3.68730 1.81838i −1.72016 1.74386i 0.0571406 + 0.0744670i −1.96920 2.03033i 1.49090 + 2.60331i 4.35179 + 3.85582i
11.9 −1.35939 + 0.389931i 1.72496 + 0.156592i 1.69591 1.06014i 1.33127 + 0.656508i −2.40596 + 0.459744i −1.06088 1.38257i −1.89203 + 2.10244i 2.95096 + 0.540229i −2.06571 0.373352i
11.10 −1.35284 + 0.412086i −0.492104 + 1.66067i 1.66037 1.11497i 1.67682 + 0.826917i −0.0186000 2.44942i −0.0289848 0.0377738i −1.78676 + 2.19260i −2.51567 1.63445i −2.60924 0.427695i
11.11 −1.34003 + 0.452023i 0.961303 + 1.44080i 1.59135 1.21145i −1.07061 0.527968i −1.93945 1.49618i −2.72460 3.55077i −1.58485 + 2.34270i −1.15179 + 2.77009i 1.67331 + 0.223550i
11.12 −1.30367 0.548138i 1.29483 1.15039i 1.39909 + 1.42918i −1.03041 0.508143i −2.31860 + 0.789982i 1.75332 + 2.28498i −1.04056 2.63006i 0.353186 2.97914i 1.06478 + 1.22726i
11.13 −1.28421 + 0.592280i −1.57042 0.730613i 1.29841 1.52123i 3.02548 + 1.49200i 2.44948 + 0.00813819i −2.08079 2.71174i −0.766443 + 2.72260i 1.93241 + 2.29473i −4.76905 0.124119i
11.14 −1.26628 + 0.629706i 1.72410 + 0.165818i 1.20694 1.59477i −1.13197 0.558226i −2.28761 + 0.875702i 3.03857 + 3.95994i −0.524089 + 2.77945i 2.94501 + 0.571771i 1.78491 0.00593726i
11.15 −1.24238 0.675648i 1.24724 1.20182i 1.08700 + 1.67882i −0.558818 0.275579i −2.36156 + 0.650421i −2.33627 3.04469i −0.216171 2.82015i 0.111234 2.99794i 0.508068 + 0.719937i
11.16 −1.24017 0.679688i −0.405173 1.68399i 1.07605 + 1.68586i 3.53047 + 1.74104i −0.642106 + 2.36383i 0.212417 + 0.276827i −0.188628 2.82213i −2.67167 + 1.36462i −3.19503 4.55880i
11.17 −1.21036 0.731448i −1.70840 0.285283i 0.929967 + 1.77064i 1.45581 + 0.717926i 1.85911 + 1.59490i 2.68057 + 3.49338i 0.169531 2.82334i 2.83723 + 0.974751i −1.23693 1.93380i
11.18 −1.18516 0.771613i −0.450297 1.67249i 0.809226 + 1.82898i −3.24873 1.60210i −0.756842 + 2.32963i 1.96855 + 2.56546i 0.452197 2.79205i −2.59447 + 1.50624i 2.61408 + 4.40551i
11.19 −1.14192 + 0.834271i −1.59577 + 0.673430i 0.607985 1.90535i −1.78193 0.878751i 1.26043 2.10031i −0.225755 0.294210i 0.895304 + 2.68299i 2.09299 2.14928i 2.76795 0.483145i
11.20 −1.11718 0.867123i 1.58760 + 0.692470i 0.496195 + 1.93747i 0.901826 + 0.444731i −1.17319 2.15026i −0.914237 1.19146i 1.12569 2.59477i 2.04097 + 2.19873i −0.621867 1.27884i
See next 80 embeddings (of 1504 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 563.94 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
64.j odd 16 1 inner
576.bl even 48 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.bl.a 1504
9.d odd 6 1 inner 576.2.bl.a 1504
64.j odd 16 1 inner 576.2.bl.a 1504
576.bl even 48 1 inner 576.2.bl.a 1504

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.bl.a 1504 1.a even 1 1 trivial
576.2.bl.a 1504 9.d odd 6 1 inner
576.2.bl.a 1504 64.j odd 16 1 inner
576.2.bl.a 1504 576.bl even 48 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(576, [\chi])$$.