# Properties

 Label 572.2.bb.b Level $572$ Weight $2$ Character orbit 572.bb Analytic conductor $4.567$ Analytic rank $0$ Dimension $304$ 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.bb (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$304$$ Relative dimension: $$76$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$304q - 14q^{4} + 72q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$304q - 14q^{4} + 72q^{9} - 20q^{12} - 10q^{13} + 2q^{14} + 14q^{16} + 20q^{17} - 66q^{22} + 72q^{25} - 34q^{26} - 100q^{29} - 60q^{30} + 62q^{36} - 14q^{38} - 10q^{40} - 68q^{42} + 4q^{48} + 40q^{49} - 30q^{52} - 12q^{53} - 132q^{56} - 20q^{61} + 130q^{62} - 38q^{64} + 86q^{66} - 60q^{68} + 56q^{69} + 40q^{74} + 68q^{77} - 48q^{78} - 48q^{81} - 70q^{82} - 102q^{88} + 180q^{90} - 2q^{92} - 190q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
51.1 −1.41328 0.0512829i −2.76420 0.898144i 1.99474 + 0.144954i −0.381188 0.524661i 3.86054 + 1.41109i 3.56329 1.15778i −2.81170 0.307158i 4.40710 + 3.20195i 0.511821 + 0.761043i
51.2 −1.41003 0.108636i 1.56442 + 0.508311i 1.97640 + 0.306361i −1.28003 1.76181i −2.15067 0.886688i 4.04252 1.31349i −2.75351 0.646686i −0.238023 0.172933i 1.61349 + 2.62328i
51.3 −1.40925 0.118332i 2.17390 + 0.706342i 1.97200 + 0.333520i 1.70523 + 2.34704i −2.97999 1.25266i −1.85344 + 0.602218i −2.73958 0.703365i 1.79986 + 1.30767i −2.12537 3.50936i
51.4 −1.39723 0.218517i −1.60576 0.521744i 1.90450 + 0.610638i −2.10647 2.89931i 2.12961 + 1.07988i −0.788330 + 0.256144i −2.52759 1.26937i −0.120789 0.0877584i 2.30967 + 4.51130i
51.5 −1.39058 + 0.257452i −0.978103 0.317805i 1.86744 0.716017i 0.745085 + 1.02552i 1.44195 + 0.190119i −4.15892 + 1.35131i −2.41248 + 1.47646i −1.57136 1.14166i −1.30012 1.23425i
51.6 −1.38768 + 0.272646i −0.00197950 0.000643177i 1.85133 0.756692i 0.569619 + 0.784014i 0.00292227 0.000352825i 0.226728 0.0736686i −2.36275 + 1.55481i −2.42705 1.76335i −1.00421 0.932658i
51.7 −1.38759 + 0.273140i 2.01726 + 0.655446i 1.85079 0.758009i −0.965607 1.32904i −2.97814 0.358495i 1.73497 0.563726i −2.36109 + 1.55733i 1.21266 + 0.881049i 1.70288 + 1.58042i
51.8 −1.33274 0.473076i −0.481761 0.156533i 1.55240 + 1.26098i 2.42980 + 3.34433i 0.568010 + 0.436528i 4.04744 1.31509i −1.47241 2.41496i −2.21946 1.61253i −1.65617 5.60661i
51.9 −1.32714 0.488563i 0.293133 + 0.0952447i 1.52261 + 1.29678i −1.67278 2.30238i −0.342496 0.269617i −2.15347 + 0.699705i −1.38716 2.46491i −2.35020 1.70752i 1.09516 + 3.87285i
51.10 −1.31807 0.512542i −2.25934 0.734104i 1.47460 + 1.35113i 0.349277 + 0.480738i 2.60170 + 2.12560i −1.15141 + 0.374115i −1.25111 2.53667i 2.13865 + 1.55382i −0.213972 0.812664i
51.11 −1.26197 + 0.638312i −2.95888 0.961399i 1.18512 1.61106i 2.44539 + 3.36579i 4.34768 0.675437i 0.969947 0.315155i −0.467220 + 2.78957i 5.40364 + 3.92598i −5.23443 2.68659i
51.12 −1.24129 0.677637i 3.02416 + 0.982609i 1.08162 + 1.68229i 0.633792 + 0.872340i −3.08802 3.26899i 2.38490 0.774902i −0.202620 2.82116i 5.75297 + 4.17978i −0.195592 1.51231i
51.13 −1.23705 0.685344i 0.814117 + 0.264523i 1.06061 + 1.69562i 0.470337 + 0.647364i −0.825818 0.885179i −2.23927 + 0.727584i −0.149950 2.82445i −1.83424 1.33265i −0.138166 1.12317i
51.14 −1.21901 + 0.716951i 0.925845 + 0.300825i 0.971962 1.74794i 2.00580 + 2.76075i −1.34429 + 0.297077i 2.03870 0.662414i 0.0683572 + 2.82760i −1.66036 1.20632i −4.42442 1.92732i
51.15 −1.21395 + 0.725477i 1.24296 + 0.403861i 0.947367 1.76139i −1.74166 2.39719i −1.80188 + 0.411468i −2.61403 + 0.849351i 0.127789 + 2.82554i −1.04521 0.759392i 3.85340 + 1.64654i
51.16 −1.18926 + 0.765279i −1.85594 0.603030i 0.828695 1.82024i −1.55303 2.13756i 2.66868 0.703148i 3.68649 1.19781i 0.407454 + 2.79893i 0.653799 + 0.475013i 3.48278 + 1.35362i
51.17 −1.16754 + 0.798029i −2.45868 0.798874i 0.726298 1.86346i −0.870035 1.19750i 3.50813 1.02938i −3.66344 + 1.19032i 0.639116 + 2.75527i 2.97986 + 2.16500i 1.97144 + 0.703816i
51.18 −1.10151 + 0.886943i 2.81972 + 0.916182i 0.426665 1.95396i 1.33159 + 1.83277i −3.91856 + 1.49174i 0.109055 0.0354341i 1.26307 + 2.53074i 4.68437 + 3.40339i −3.09232 0.837781i
51.19 −1.03407 0.964726i −0.814117 0.264523i 0.138608 + 1.99519i 0.470337 + 0.647364i 0.586663 + 1.05893i 2.23927 0.727584i 1.78148 2.19689i −1.83424 1.33265i 0.138166 1.12317i
51.20 −1.02805 0.971139i −3.02416 0.982609i 0.113780 + 1.99676i 0.633792 + 0.872340i 2.15474 + 3.94705i −2.38490 + 0.774902i 1.82216 2.16327i 5.75297 + 4.17978i 0.195592 1.51231i
See next 80 embeddings (of 304 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 519.76 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
13.b even 2 1 inner
44.g even 10 1 inner
52.b odd 2 1 inner
143.l odd 10 1 inner
572.bb 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.bb.b 304
4.b odd 2 1 inner 572.2.bb.b 304
11.d odd 10 1 inner 572.2.bb.b 304
13.b even 2 1 inner 572.2.bb.b 304
44.g even 10 1 inner 572.2.bb.b 304
52.b odd 2 1 inner 572.2.bb.b 304
143.l odd 10 1 inner 572.2.bb.b 304
572.bb even 10 1 inner 572.2.bb.b 304

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.bb.b 304 1.a even 1 1 trivial
572.2.bb.b 304 4.b odd 2 1 inner
572.2.bb.b 304 11.d odd 10 1 inner
572.2.bb.b 304 13.b even 2 1 inner
572.2.bb.b 304 44.g even 10 1 inner
572.2.bb.b 304 52.b odd 2 1 inner
572.2.bb.b 304 143.l odd 10 1 inner
572.2.bb.b 304 572.bb even 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$30\!\cdots\!27$$$$T_{3}^{134} +$$$$39\!\cdots\!20$$$$T_{3}^{132} -$$$$47\!\cdots\!54$$$$T_{3}^{130} +$$$$53\!\cdots\!56$$$$T_{3}^{128} -$$$$55\!\cdots\!45$$$$T_{3}^{126} +$$$$55\!\cdots\!63$$$$T_{3}^{124} -$$$$51\!\cdots\!21$$$$T_{3}^{122} +$$$$45\!\cdots\!41$$$$T_{3}^{120} -$$$$38\!\cdots\!92$$$$T_{3}^{118} +$$$$30\!\cdots\!52$$$$T_{3}^{116} -$$$$23\!\cdots\!25$$$$T_{3}^{114} +$$$$16\!\cdots\!28$$$$T_{3}^{112} -$$$$11\!\cdots\!68$$$$T_{3}^{110} +$$$$74\!\cdots\!90$$$$T_{3}^{108} -$$$$45\!\cdots\!89$$$$T_{3}^{106} +$$$$27\!\cdots\!48$$$$T_{3}^{104} -$$$$15\!\cdots\!16$$$$T_{3}^{102} +$$$$81\!\cdots\!52$$$$T_{3}^{100} -$$$$41\!\cdots\!99$$$$T_{3}^{98} +$$$$20\!\cdots\!59$$$$T_{3}^{96} -$$$$92\!\cdots\!93$$$$T_{3}^{94} +$$$$40\!\cdots\!88$$$$T_{3}^{92} -$$$$16\!\cdots\!35$$$$T_{3}^{90} +$$$$66\!\cdots\!92$$$$T_{3}^{88} -$$$$24\!\cdots\!55$$$$T_{3}^{86} +$$$$88\!\cdots\!70$$$$T_{3}^{84} -$$$$29\!\cdots\!55$$$$T_{3}^{82} +$$$$95\!\cdots\!95$$$$T_{3}^{80} -$$$$28\!\cdots\!26$$$$T_{3}^{78} +$$$$81\!\cdots\!12$$$$T_{3}^{76} -$$$$22\!\cdots\!77$$$$T_{3}^{74} +$$$$56\!\cdots\!34$$$$T_{3}^{72} -$$$$13\!\cdots\!28$$$$T_{3}^{70} +$$$$31\!\cdots\!59$$$$T_{3}^{68} -$$$$68\!\cdots\!34$$$$T_{3}^{66} +$$$$13\!\cdots\!98$$$$T_{3}^{64} -$$$$27\!\cdots\!12$$$$T_{3}^{62} +$$$$49\!\cdots\!43$$$$T_{3}^{60} -$$$$86\!\cdots\!29$$$$T_{3}^{58} +$$$$14\!\cdots\!79$$$$T_{3}^{56} -$$$$22\!\cdots\!59$$$$T_{3}^{54} +$$$$32\!\cdots\!72$$$$T_{3}^{52} -$$$$44\!\cdots\!83$$$$T_{3}^{50} +$$$$57\!\cdots\!26$$$$T_{3}^{48} -$$$$69\!\cdots\!49$$$$T_{3}^{46} +$$$$79\!\cdots\!73$$$$T_{3}^{44} -$$$$83\!\cdots\!26$$$$T_{3}^{42} +$$$$81\!\cdots\!72$$$$T_{3}^{40} -$$$$73\!\cdots\!87$$$$T_{3}^{38} +$$$$60\!\cdots\!00$$$$T_{3}^{36} -$$$$45\!\cdots\!24$$$$T_{3}^{34} +$$$$30\!\cdots\!33$$$$T_{3}^{32} -$$$$17\!\cdots\!34$$$$T_{3}^{30} +$$$$91\!\cdots\!88$$$$T_{3}^{28} -$$$$39\!\cdots\!20$$$$T_{3}^{26} +$$$$14\!\cdots\!08$$$$T_{3}^{24} -$$$$48\!\cdots\!44$$$$T_{3}^{22} +$$$$14\!\cdots\!80$$$$T_{3}^{20} -$$$$35\!\cdots\!32$$$$T_{3}^{18} +$$$$78\!\cdots\!12$$$$T_{3}^{16} -$$$$15\!\cdots\!12$$$$T_{3}^{14} +$$$$26\!\cdots\!76$$$$T_{3}^{12} -$$$$34\!\cdots\!92$$$$T_{3}^{10} +$$$$46\!\cdots\!88$$$$T_{3}^{8} -$$$$40\!\cdots\!92$$$$T_{3}^{6} +$$$$17\!\cdots\!76$$$$T_{3}^{4} -$$$$12\!\cdots\!48$$$$T_{3}^{2} +$$$$33\!\cdots\!36$$">$$T_{3}^{152} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(572, [\chi])$$.