# Properties

 Label 572.2.y.a Level $572$ Weight $2$ Character orbit 572.y Analytic conductor $4.567$ Analytic rank $0$ Dimension $288$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$572 = 2^{2} \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 572.y (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$288$$ Relative dimension: $$72$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$288q - 4q^{4} + 80q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$288q - 4q^{4} + 80q^{9} + 8q^{12} - 6q^{14} - 32q^{16} - 50q^{18} + 16q^{22} - 88q^{25} - 50q^{28} - 20q^{30} - 4q^{33} + 30q^{36} - 48q^{37} + 38q^{38} + 70q^{40} + 56q^{42} + 66q^{44} - 96q^{45} + 70q^{46} + 76q^{48} - 56q^{49} + 70q^{50} - 20q^{52} - 80q^{53} + 44q^{56} - 20q^{57} - 2q^{58} - 144q^{60} - 150q^{62} - 100q^{64} + 30q^{66} + 24q^{69} - 28q^{70} - 100q^{72} + 40q^{73} - 140q^{74} - 24q^{77} - 40q^{78} + 118q^{80} - 76q^{81} + 56q^{82} + 120q^{84} + 80q^{85} + 44q^{88} - 8q^{89} + 80q^{90} - 14q^{92} + 72q^{93} + 50q^{94} - 140q^{96} + 116q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1 −1.41285 + 0.0620317i 1.52448 + 2.09827i 1.99230 0.175283i −0.870694 2.67972i −2.28403 2.86998i −2.76319 2.00757i −2.80396 + 0.371235i −1.15164 + 3.54438i 1.39639 + 3.73204i
79.2 −1.40628 0.149624i 0.772280 + 1.06295i 1.95523 + 0.420826i −1.08359 3.33494i −0.926995 1.61036i 3.42488 + 2.48832i −2.68662 0.884347i 0.393600 1.21138i 1.02484 + 4.85198i
79.3 −1.39375 + 0.239692i −0.399378 0.549696i 1.88510 0.668144i −0.0281832 0.0867389i 0.688392 + 0.670413i 0.612187 + 0.444780i −2.46721 + 1.38307i 0.784388 2.41410i 0.0600710 + 0.114137i
79.4 −1.38715 + 0.275341i −0.322447 0.443810i 1.84837 0.763879i 1.31077 + 4.03414i 0.569482 + 0.526849i −2.94501 2.13968i −2.35365 + 1.56855i 0.834056 2.56696i −2.92901 5.23506i
79.5 −1.38067 + 0.306183i 1.45995 + 2.00944i 1.81250 0.845476i 0.521225 + 1.60416i −2.63096 2.32737i 0.136261 + 0.0989992i −2.24360 + 1.72228i −0.979372 + 3.01420i −1.21081 2.05523i
79.6 −1.36071 0.385317i −0.235269 0.323820i 1.70306 + 1.04861i 0.0217417 + 0.0669141i 0.195360 + 0.531278i −2.49703 1.81420i −1.91333 2.08307i 0.877543 2.70080i −0.00380103 0.0994281i
79.7 −1.35672 0.399148i 1.61480 + 2.22258i 1.68136 + 1.08306i 0.659364 + 2.02932i −1.30369 3.65996i 0.326711 + 0.237370i −1.84883 2.14052i −1.40524 + 4.32488i −0.0845732 3.01639i
79.8 −1.33222 0.474540i −1.61480 2.22258i 1.54962 + 1.26438i 0.659364 + 2.02932i 1.09656 + 3.72726i −0.326711 0.237370i −1.46444 2.41980i −1.40524 + 4.32488i 0.0845732 3.01639i
79.9 −1.32732 0.488077i 0.235269 + 0.323820i 1.52356 + 1.29567i 0.0217417 + 0.0669141i −0.154228 0.544643i 2.49703 + 1.81420i −1.38987 2.46339i 0.877543 2.70080i 0.00380103 0.0994281i
79.10 −1.24676 + 0.667521i 0.741969 + 1.02123i 1.10883 1.66448i 0.885652 + 2.72576i −1.60675 0.777954i 2.30040 + 1.67134i −0.271377 + 2.81538i 0.434653 1.33773i −2.92369 2.80718i
79.11 −1.22565 0.705540i −0.772280 1.06295i 1.00443 + 1.72949i −1.08359 3.33494i 0.196588 + 1.84768i −3.42488 2.48832i −0.0108521 2.82841i 0.393600 1.21138i −1.02484 + 4.85198i
79.12 −1.21290 + 0.727232i −1.60147 2.20423i 0.942269 1.76412i 0.216071 + 0.664997i 3.54541 + 1.50888i −3.90509 2.83722i 0.140045 + 2.82496i −1.36689 + 4.20685i −0.745679 0.649443i
79.13 −1.18675 + 0.769166i −1.73629 2.38980i 0.816767 1.82562i 0.954824 + 2.93865i 3.89869 + 1.50060i 3.05312 + 2.21822i 0.434904 + 2.79479i −1.76937 + 5.44558i −3.39345 2.75303i
79.14 −1.18418 + 0.773121i −0.0644943 0.0887688i 0.804567 1.83103i −1.31803 4.05649i 0.145002 + 0.0552564i −0.673809 0.489551i 0.462856 + 2.79030i 0.923331 2.84172i 4.69695 + 3.78462i
79.15 −1.10656 0.880639i −1.52448 2.09827i 0.448952 + 1.94896i −0.870694 2.67972i −0.160886 + 3.66438i 2.76319 + 2.00757i 1.21954 2.55201i −1.15164 + 3.54438i −1.39639 + 3.73204i
79.16 −1.10068 + 0.887979i −0.684477 0.942102i 0.422987 1.95476i −0.0530453 0.163257i 1.58996 + 0.429150i 0.351116 + 0.255100i 1.27021 + 2.52717i 0.508004 1.56347i 0.203354 + 0.132590i
79.17 −1.08817 + 0.903263i 1.08813 + 1.49768i 0.368233 1.96581i −0.149634 0.460526i −2.53687 0.646867i −3.21984 2.33935i 1.37494 + 2.47175i −0.131974 + 0.406174i 0.578804 + 0.365972i
79.18 −0.986682 1.01314i 0.399378 + 0.549696i −0.0529163 + 1.99930i −0.0281832 0.0867389i 0.162862 0.947002i −0.612187 0.444780i 2.07779 1.91906i 0.784388 2.41410i −0.0600710 + 0.114137i
79.19 −0.960387 1.03810i 0.322447 + 0.443810i −0.155313 + 1.99396i 1.31077 + 4.03414i 0.151046 0.760963i 2.94501 + 2.13968i 2.21910 1.75374i 0.834056 2.56696i 2.92901 5.23506i
79.20 −0.937016 1.05925i −1.45995 2.00944i −0.244001 + 1.98506i 0.521225 + 1.60416i −0.760501 + 3.42932i −0.136261 0.0989992i 2.33130 1.60158i −0.979372 + 3.01420i 1.21081 2.05523i
See next 80 embeddings (of 288 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 547.72 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
44.g even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.y.a 288
4.b odd 2 1 inner 572.2.y.a 288
11.d odd 10 1 inner 572.2.y.a 288
44.g even 10 1 inner 572.2.y.a 288

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.y.a 288 1.a even 1 1 trivial
572.2.y.a 288 4.b odd 2 1 inner
572.2.y.a 288 11.d odd 10 1 inner
572.2.y.a 288 44.g even 10 1 inner

## Hecke kernels

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