# Properties

 Label 572.2.t.a Level $572$ Weight $2$ Character orbit 572.t Analytic conductor $4.567$ Analytic rank $0$ Dimension $160$ 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.t (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$160$$ Relative dimension: $$80$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$160q - 2q^{4} - 16q^{5} + 68q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$160q - 2q^{4} - 16q^{5} + 68q^{9} - 4q^{12} + 8q^{14} - 14q^{16} - 26q^{20} + 112q^{25} - 14q^{33} + 12q^{34} - 6q^{36} - 4q^{37} + 40q^{38} + 54q^{42} - 16q^{44} - 48q^{45} - 40q^{48} - 52q^{49} - 32q^{53} - 18q^{56} - 6q^{58} - 52q^{60} + 16q^{64} - 84q^{66} - 12q^{69} - 68q^{70} - 4q^{77} + 28q^{78} - 44q^{80} - 40q^{81} - 38q^{82} + 120q^{86} - 14q^{88} - 64q^{92} - 8q^{93} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
87.1 −1.41020 + 0.106478i 0.151443 + 0.0874355i 1.97732 0.300310i −4.25810 −0.222875 0.107176i 1.66734 + 2.88791i −2.75645 + 0.634039i −1.48471 2.57159i 6.00478 0.453394i
87.2 −1.40800 0.132444i 1.78274 + 1.02926i 1.96492 + 0.372961i 1.58498 −2.37377 1.68531i 1.72331 + 2.98486i −2.71720 0.785368i 0.618768 + 1.07174i −2.23164 0.209920i
87.3 −1.39957 0.203000i −2.41412 1.39379i 1.91758 + 0.568224i 2.71349 3.09579 + 2.44077i −1.13343 1.96316i −2.56844 1.18454i 2.38532 + 4.13149i −3.79771 0.550837i
87.4 −1.39945 0.203823i −0.783384 0.452287i 1.91691 + 0.570480i 2.92431 1.00412 + 0.792624i −0.351276 0.608428i −2.56634 1.18907i −1.09087 1.88945i −4.09242 0.596043i
87.5 −1.39552 + 0.229204i 2.08234 + 1.20224i 1.89493 0.639716i −2.36335 −3.18150 1.20047i −1.42248 2.46380i −2.49778 + 1.32706i 1.39077 + 2.40888i 3.29809 0.541689i
87.6 −1.36138 + 0.382925i −1.69448 0.978307i 1.70674 1.04262i −1.87383 2.68146 + 0.682994i 0.639155 + 1.10705i −1.92428 + 2.07296i 0.414171 + 0.717364i 2.55100 0.717536i
87.7 −1.35748 + 0.396562i −0.410459 0.236978i 1.68548 1.07664i −1.00051 0.651164 + 0.158920i −1.25701 2.17721i −1.86104 + 2.12991i −1.38768 2.40354i 1.35817 0.396764i
87.8 −1.35030 0.420348i −2.86468 1.65392i 1.64661 + 1.13519i −3.67952 3.17295 + 3.43745i −0.939669 1.62756i −1.74625 2.22500i 3.97092 + 6.87783i 4.96846 + 1.54668i
87.9 −1.34933 0.423444i 1.94116 + 1.12073i 1.64139 + 1.14273i −0.245835 −2.14470 2.33421i −0.228705 0.396129i −1.73089 2.23696i 1.01207 + 1.75295i 0.331712 + 0.104097i
87.10 −1.33912 + 0.454709i 0.665321 + 0.384123i 1.58648 1.21782i 2.62826 −1.06561 0.211859i −0.983531 1.70353i −1.57073 + 2.35219i −1.20490 2.08695i −3.51955 + 1.19509i
87.11 −1.33705 0.460748i −0.847243 0.489156i 1.57542 + 1.23209i 1.08222 0.907431 + 1.04439i 2.37503 + 4.11368i −1.53874 2.37324i −1.02145 1.76921i −1.44699 0.498633i
87.12 −1.30161 0.552996i 0.965312 + 0.557323i 1.38839 + 1.43957i −1.84918 −0.948265 1.25923i −0.798405 1.38288i −1.01107 2.64154i −0.878782 1.52209i 2.40691 + 1.02259i
87.13 −1.20939 + 0.733056i 2.20753 + 1.27452i 0.925257 1.77310i −0.492373 −3.60406 + 0.0768518i 2.32336 + 4.02417i 0.180786 + 2.82264i 1.74880 + 3.02900i 0.595472 0.360937i
87.14 −1.19068 + 0.763078i −2.37620 1.37190i 0.835423 1.81716i 1.57679 3.87615 0.179738i 0.286729 + 0.496630i 0.391917 + 2.80114i 2.26422 + 3.92174i −1.87745 + 1.20322i
87.15 −1.12971 0.850732i −0.965312 0.557323i 0.552511 + 1.92217i −1.84918 0.616395 + 1.45084i 0.798405 + 1.38288i 1.01107 2.64154i −0.878782 1.52209i 2.08904 + 1.57315i
87.16 −1.06755 0.927548i 0.847243 + 0.489156i 0.279310 + 1.98040i 1.08222 −0.450755 1.30805i −2.37503 4.11368i 1.53874 2.37324i −1.02145 1.76921i −1.15532 1.00382i
87.17 −1.05336 + 0.943626i −1.81974 1.05063i 0.219139 1.98796i −2.74224 2.90824 0.610464i −2.27804 3.94569i 1.64506 + 2.30082i 0.707629 + 1.22565i 2.88857 2.58765i
87.18 −1.04138 0.956833i −1.94116 1.12073i 0.168941 + 1.99285i −0.245835 0.949133 + 3.02447i 0.228705 + 0.396129i 1.73089 2.23696i 1.01207 + 1.75295i 0.256007 + 0.235223i
87.19 −1.03918 0.959219i 2.86468 + 1.65392i 0.159798 + 1.99361i −3.67952 −1.39045 4.46658i 0.939669 + 1.62756i 1.74625 2.22500i 3.97092 + 6.87783i 3.82369 + 3.52947i
87.20 −1.00532 + 0.994647i 2.84151 + 1.64055i 0.0213531 1.99989i 0.805960 −4.48841 + 1.17702i −0.565511 0.979494i 1.96771 + 2.03177i 3.88279 + 6.72520i −0.810251 + 0.801646i
See next 80 embeddings (of 160 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 263.80 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.b odd 2 1 inner
13.c even 3 1 inner
44.c even 2 1 inner
52.j odd 6 1 inner
143.k odd 6 1 inner
572.t even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.t.a 160
4.b odd 2 1 inner 572.2.t.a 160
11.b odd 2 1 inner 572.2.t.a 160
13.c even 3 1 inner 572.2.t.a 160
44.c even 2 1 inner 572.2.t.a 160
52.j odd 6 1 inner 572.2.t.a 160
143.k odd 6 1 inner 572.2.t.a 160
572.t even 6 1 inner 572.2.t.a 160

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.t.a 160 1.a even 1 1 trivial
572.2.t.a 160 4.b odd 2 1 inner
572.2.t.a 160 11.b odd 2 1 inner
572.2.t.a 160 13.c even 3 1 inner
572.2.t.a 160 44.c even 2 1 inner
572.2.t.a 160 52.j odd 6 1 inner
572.2.t.a 160 143.k odd 6 1 inner
572.2.t.a 160 572.t even 6 1 inner

## Hecke kernels

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