# Properties

 Label 572.2.bv.a Level $572$ Weight $2$ Character orbit 572.bv Analytic conductor $4.567$ Analytic rank $0$ Dimension $224$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$572 = 2^{2} \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 572.bv (of order $$60$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$224$$ Relative dimension: $$14$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$224q + 28q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$224q + 28q^{9} + 16q^{11} + 10q^{13} - 28q^{15} - 48q^{23} + 24q^{27} + 20q^{29} + 4q^{31} + 60q^{33} + 50q^{35} + 12q^{37} - 40q^{39} + 20q^{41} + 64q^{45} - 62q^{47} + 100q^{53} - 22q^{55} + 12q^{59} - 40q^{61} - 80q^{63} - 44q^{67} - 152q^{71} + 30q^{73} - 120q^{75} + 80q^{79} + 72q^{81} + 90q^{83} - 40q^{85} - 8q^{89} - 36q^{91} - 90q^{93} - 42q^{97} + 144q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1 0 −2.12932 + 2.36485i 0 1.33109 0.210824i 0 −1.93450 0.101383i 0 −0.744921 7.08745i 0
41.2 0 −1.60983 + 1.78790i 0 −1.34020 + 0.212267i 0 3.84208 + 0.201355i 0 −0.291440 2.77287i 0
41.3 0 −1.57791 + 1.75245i 0 −0.871666 + 0.138058i 0 1.64722 + 0.0863272i 0 −0.267683 2.54684i 0
41.4 0 −1.00688 + 1.11826i 0 3.41629 0.541087i 0 −1.08459 0.0568410i 0 0.0769002 + 0.731656i 0
41.5 0 −0.765371 + 0.850031i 0 −3.62842 + 0.574686i 0 −0.942248 0.0493811i 0 0.176826 + 1.68239i 0
41.6 0 −0.389674 + 0.432777i 0 3.67270 0.581698i 0 4.87664 + 0.255574i 0 0.278135 + 2.64628i 0
41.7 0 −0.170530 + 0.189393i 0 −0.481251 + 0.0762227i 0 −0.670222 0.0351249i 0 0.306796 + 2.91897i 0
41.8 0 0.183828 0.204162i 0 2.17308 0.344183i 0 −2.74555 0.143888i 0 0.305696 + 2.90850i 0
41.9 0 0.555859 0.617343i 0 −3.44415 + 0.545500i 0 1.95070 + 0.102232i 0 0.241451 + 2.29725i 0
41.10 0 0.632506 0.702469i 0 −1.21053 + 0.191729i 0 −4.38981 0.230060i 0 0.220187 + 2.09493i 0
41.11 0 1.13877 1.26473i 0 −0.297595 + 0.0471344i 0 3.35932 + 0.176054i 0 0.0108356 + 0.103094i 0
41.12 0 1.43269 1.59116i 0 −0.186319 + 0.0295100i 0 0.938903 + 0.0492058i 0 −0.165612 1.57570i 0
41.13 0 1.52983 1.69905i 0 3.46848 0.549353i 0 −0.174489 0.00914460i 0 −0.232798 2.21493i 0
41.14 0 2.17604 2.41674i 0 −2.60150 + 0.412038i 0 −4.67346 0.244925i 0 −0.791886 7.53429i 0
85.1 0 −3.18098 0.676138i 0 −0.412038 2.60150i 0 −3.92487 2.54884i 0 6.92083 + 3.08135i 0
85.2 0 −2.23633 0.475347i 0 0.549353 + 3.46848i 0 −0.146540 0.0951642i 0 2.03458 + 0.905855i 0
85.3 0 −2.09433 0.445163i 0 −0.0295100 0.186319i 0 0.788511 + 0.512065i 0 1.44740 + 0.644424i 0
85.4 0 −1.66467 0.353837i 0 −0.0471344 0.297595i 0 2.82123 + 1.83213i 0 −0.0946995 0.0421630i 0
85.5 0 −0.924609 0.196532i 0 −0.191729 1.21053i 0 −3.68665 2.39414i 0 −1.92436 0.856780i 0
85.6 0 −0.812564 0.172716i 0 −0.545500 3.44415i 0 1.63824 + 1.06389i 0 −2.11021 0.939524i 0
See next 80 embeddings (of 224 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 557.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
13.f odd 12 1 inner
143.w 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.bv.a 224
11.d odd 10 1 inner 572.2.bv.a 224
13.f odd 12 1 inner 572.2.bv.a 224
143.w even 60 1 inner 572.2.bv.a 224

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.bv.a 224 1.a even 1 1 trivial
572.2.bv.a 224 11.d odd 10 1 inner
572.2.bv.a 224 13.f odd 12 1 inner
572.2.bv.a 224 143.w even 60 1 inner

## Hecke kernels

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