# Properties

 Label 572.2.bf.a Level $572$ Weight $2$ Character orbit 572.bf Analytic conductor $4.567$ Analytic rank $0$ Dimension $280$ 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.bf (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$280$$ Relative dimension: $$70$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$280q - 4q^{5} + 12q^{6} - 12q^{8} + 140q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$280q - 4q^{5} + 12q^{6} - 12q^{8} + 140q^{9} - 16q^{14} - 24q^{18} - 16q^{20} + 16q^{21} - 28q^{26} - 40q^{28} - 20q^{32} - 16q^{34} - 36q^{36} - 36q^{37} + 80q^{40} - 40q^{41} + 20q^{42} + 8q^{44} - 40q^{45} + 60q^{46} - 20q^{48} - 144q^{49} + 20q^{50} + 108q^{52} + 8q^{53} + 20q^{54} + 108q^{56} - 64q^{57} + 60q^{58} + 108q^{60} - 20q^{61} - 36q^{62} - 40q^{65} - 40q^{66} - 76q^{68} - 44q^{70} - 20q^{72} - 100q^{73} - 32q^{74} + 32q^{76} - 140q^{78} - 156q^{80} - 140q^{81} - 260q^{84} - 12q^{85} + 56q^{86} - 72q^{88} + 60q^{89} + 80q^{92} + 80q^{93} + 32q^{94} + 80q^{96} + 44q^{97} + 16q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1 −1.41338 + 0.0484642i −2.30576 + 1.33123i 1.99530 0.136997i 0.670105 0.670105i 3.19441 1.99329i 1.27385 + 4.75407i −2.81349 + 0.290330i 2.04436 3.54093i −0.914639 + 0.979591i
67.2 −1.41251 0.0694338i 1.39356 0.804570i 1.99036 + 0.196152i −1.40920 + 1.40920i −2.02427 + 1.03970i −0.135295 0.504929i −2.79778 0.415264i −0.205335 + 0.355651i 2.08835 1.89266i
67.3 −1.40492 0.161847i −0.747603 + 0.431629i 1.94761 + 0.454764i −1.77001 + 1.77001i 1.12018 0.485407i −0.913100 3.40773i −2.66264 0.954122i −1.12739 + 1.95270i 2.77319 2.20025i
67.4 −1.39566 + 0.228338i −2.91789 + 1.68465i 1.89572 0.637363i 1.61438 1.61438i 3.68771 3.01745i −1.10646 4.12937i −2.50025 + 1.32241i 4.17606 7.23315i −1.88450 + 2.62174i
67.5 −1.39359 + 0.240656i 2.08344 1.20287i 1.88417 0.670751i 1.79541 1.79541i −2.61397 + 2.17770i −0.472870 1.76477i −2.46433 + 1.38819i 1.39380 2.41414i −2.06998 + 2.93413i
67.6 −1.39307 + 0.243637i −1.43186 + 0.826687i 1.88128 0.678806i −1.59868 + 1.59868i 1.79327 1.50049i −0.175707 0.655746i −2.45537 + 1.40397i −0.133176 + 0.230667i 1.83757 2.61657i
67.7 −1.36347 + 0.375447i −0.276166 + 0.159445i 1.71808 1.02382i 0.625185 0.625185i 0.316680 0.321083i 0.928719 + 3.46603i −1.95815 + 2.04099i −1.44915 + 2.51001i −0.617695 + 1.08714i
67.8 −1.33591 + 0.464039i 2.67338 1.54348i 1.56934 1.23983i −1.23437 + 1.23437i −2.85518 + 3.30251i 0.844074 + 3.15013i −1.52117 + 2.38454i 3.26465 5.65455i 1.07622 2.22181i
67.9 −1.29762 0.562298i 0.747603 0.431629i 1.36764 + 1.45930i −1.77001 + 1.77001i −1.21281 + 0.139715i 0.913100 + 3.40773i −0.954122 2.66264i −1.12739 + 1.95270i 3.29207 1.30153i
67.10 −1.28912 + 0.581515i −0.101031 + 0.0583303i 1.32368 1.49929i 1.51781 1.51781i 0.0963216 0.133946i −0.585492 2.18508i −0.834531 + 2.70251i −1.49320 + 2.58629i −1.07402 + 2.83927i
67.11 −1.25798 0.646123i −1.39356 + 0.804570i 1.16505 + 1.62562i −1.40920 + 1.40920i 2.27292 0.111729i 0.135295 + 0.504929i −0.415264 2.79778i −0.205335 + 0.355651i 2.68326 0.862235i
67.12 −1.19979 0.748663i 2.30576 1.33123i 0.879008 + 1.79648i 0.670105 0.670105i −3.76308 0.129034i −1.27385 4.75407i 0.290330 2.81349i 2.04436 3.54093i −1.30567 + 0.302305i
67.13 −1.16821 + 0.797043i 1.53999 0.889113i 0.729444 1.86223i −2.51999 + 2.51999i −1.09037 + 2.26611i −1.06525 3.97557i 0.632133 + 2.75688i 0.0810425 0.140370i 0.935344 4.95242i
67.14 −1.09451 0.895575i 2.91789 1.68465i 0.395890 + 1.96043i 1.61438 1.61438i −4.70238 0.769336i 1.10646 + 4.12937i 1.32241 2.50025i 4.17606 7.23315i −3.21274 + 0.321150i
67.15 −1.08655 0.905208i −2.08344 + 1.20287i 0.361197 + 1.96711i 1.79541 1.79541i 3.35261 + 0.578958i 0.472870 + 1.76477i 1.38819 2.46433i 1.39380 2.41414i −3.57602 + 0.325589i
67.16 −1.08461 0.907530i 1.43186 0.826687i 0.352778 + 1.96864i −1.59868 + 1.59868i −2.30327 0.402823i 0.175707 + 0.655746i 1.40397 2.45537i −0.133176 + 0.230667i 3.18480 0.283101i
67.17 −1.07247 + 0.921853i 0.790362 0.456316i 0.300374 1.97732i 0.301911 0.301911i −0.426981 + 1.21798i 0.768929 + 2.86968i 1.50065 + 2.39751i −1.08355 + 1.87677i −0.0454722 + 0.602108i
67.18 −1.01057 + 0.989319i −1.57035 + 0.906642i 0.0424954 1.99955i 2.19271 2.19271i 0.689987 2.46980i −0.0595437 0.222220i 1.93525 + 2.06272i 0.143999 0.249414i −0.0465927 + 4.38518i
67.19 −0.993073 1.00688i 0.276166 0.159445i −0.0276135 + 1.99981i 0.625185 0.625185i −0.434795 0.119726i −0.928719 3.46603i 2.04099 1.95815i −1.44915 + 2.51001i −1.25034 0.00863198i
67.20 −0.924916 1.06983i −2.67338 + 1.54348i −0.289059 + 1.97900i −1.23437 + 1.23437i 4.12391 + 1.43247i −0.844074 3.15013i 2.38454 1.52117i 3.26465 5.65455i 2.46226 + 0.178873i
See next 80 embeddings (of 280 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 375.70 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
13.f odd 12 1 inner
52.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.bf.a 280
4.b odd 2 1 inner 572.2.bf.a 280
13.f odd 12 1 inner 572.2.bf.a 280
52.l even 12 1 inner 572.2.bf.a 280

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.bf.a 280 1.a even 1 1 trivial
572.2.bf.a 280 4.b odd 2 1 inner
572.2.bf.a 280 13.f odd 12 1 inner
572.2.bf.a 280 52.l even 12 1 inner

## Hecke kernels

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