# Properties

 Label 552.2.t.b Level $552$ Weight $2$ Character orbit 552.t Analytic conductor $4.408$ Analytic rank $0$ Dimension $240$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$552 = 2^{3} \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 552.t (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.40774219157$$ Analytic rank: $$0$$ Dimension: $$240$$ Relative dimension: $$24$$ over $$\Q(\zeta_{22})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$240q + 4q^{2} - 24q^{3} - 4q^{4} - 7q^{6} + 4q^{8} - 24q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$240q + 4q^{2} - 24q^{3} - 4q^{4} - 7q^{6} + 4q^{8} - 24q^{9} - 4q^{12} - 4q^{16} - 44q^{17} + 4q^{18} + 55q^{20} + 4q^{24} - 24q^{25} - 24q^{27} + 4q^{32} - 55q^{34} + 7q^{36} + 22q^{38} + 11q^{40} - 44q^{42} + 77q^{44} - 48q^{46} + 51q^{48} + 8q^{49} - 33q^{50} + 165q^{52} - 7q^{54} - 22q^{56} - 59q^{58} + 11q^{60} + 29q^{62} - 4q^{64} + 55q^{66} + 132q^{70} + 4q^{72} - 144q^{73} - 66q^{74} - 24q^{75} - 55q^{76} - 24q^{81} - 26q^{82} - 44q^{84} - 110q^{86} - 198q^{88} - 47q^{92} - 73q^{94} + 4q^{96} + 53q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −1.40456 0.164932i −0.959493 + 0.281733i 1.94559 + 0.463315i −1.02510 0.658795i 1.39414 0.237460i −0.612113 4.25734i −2.65630 0.971645i 0.841254 0.540641i 1.33117 + 1.09439i
19.2 −1.40243 + 0.182193i −0.959493 + 0.281733i 1.93361 0.511026i 2.78195 + 1.78785i 1.29429 0.569923i −0.0254555 0.177047i −2.61865 + 1.06897i 0.841254 0.540641i −4.22722 2.00048i
19.3 −1.34789 0.428007i −0.959493 + 0.281733i 1.63362 + 1.15381i −0.781202 0.502048i 1.41388 + 0.0309246i 0.233208 + 1.62200i −1.70810 2.25441i 0.841254 0.540641i 0.838095 + 1.01107i
19.4 −1.19352 + 0.758628i −0.959493 + 0.281733i 0.848969 1.81087i 0.579145 + 0.372194i 0.931442 1.06415i −0.117047 0.814077i 0.360517 + 2.80536i 0.841254 0.540641i −0.973576 0.00486464i
19.5 −1.07273 0.921553i −0.959493 + 0.281733i 0.301482 + 1.97715i 3.13971 + 2.01777i 1.28890 + 0.582001i −0.427596 2.97400i 1.49864 2.39877i 0.841254 0.540641i −1.50857 5.05792i
19.6 −1.05559 + 0.941132i −0.959493 + 0.281733i 0.228541 1.98690i −3.23381 2.07824i 0.747684 1.20040i −0.275643 1.91714i 1.62869 + 2.31244i 0.841254 0.540641i 5.36947 0.849668i
19.7 −0.753178 1.19696i −0.959493 + 0.281733i −0.865447 + 1.80305i 0.282372 + 0.181470i 1.05989 + 0.936284i 0.179788 + 1.25045i 2.81003 0.322111i 0.841254 0.540641i 0.00453619 0.474669i
19.8 −0.617779 + 1.27214i −0.959493 + 0.281733i −1.23670 1.57181i 2.92140 + 1.87747i 0.234350 1.39466i 0.376896 + 2.62137i 2.76357 0.602229i 0.841254 0.540641i −4.19319 + 2.55658i
19.9 −0.562315 + 1.29761i −0.959493 + 0.281733i −1.36760 1.45934i −1.62830 1.04644i 0.173957 1.40347i 0.707273 + 4.91919i 2.66268 0.954016i 0.841254 0.540641i 2.27350 1.52447i
19.10 −0.411379 1.35306i −0.959493 + 0.281733i −1.66154 + 1.11324i −0.282372 0.181470i 0.775916 + 1.18235i −0.179788 1.25045i 2.18980 + 1.79019i 0.841254 0.540641i −0.129377 + 0.456719i
19.11 −0.218084 + 1.39730i −0.959493 + 0.281733i −1.90488 0.609457i −0.659581 0.423887i −0.184414 1.40214i −0.167612 1.16577i 1.26702 2.52877i 0.841254 0.540641i 0.736140 0.829187i
19.12 0.00602308 1.41420i −0.959493 + 0.281733i −1.99993 0.0170357i −3.13971 2.01777i 0.392647 + 1.35861i 0.427596 + 2.97400i −0.0361376 + 2.82820i 0.841254 0.540641i −2.87244 + 4.42803i
19.13 0.175979 + 1.40322i −0.959493 + 0.281733i −1.93806 + 0.493874i 1.87195 + 1.20303i −0.564184 1.29680i −0.493720 3.43390i −1.03407 2.63262i 0.841254 0.540641i −1.35870 + 2.83848i
19.14 0.492284 + 1.32577i −0.959493 + 0.281733i −1.51531 + 1.30531i 2.38725 + 1.53419i −0.845855 1.13337i 0.401276 + 2.79093i −2.47650 1.36637i 0.841254 0.540641i −0.858775 + 3.92019i
19.15 0.559215 1.29895i −0.959493 + 0.281733i −1.37456 1.45279i 0.781202 + 0.502048i −0.170606 + 1.40389i −0.233208 1.62200i −2.65578 + 0.973064i 0.841254 0.540641i 1.08900 0.733992i
19.16 0.679570 + 1.24024i −0.959493 + 0.281733i −1.07637 + 1.68565i −2.38725 1.53419i −1.00146 0.998540i −0.401276 2.79093i −2.82208 0.189432i 0.841254 0.540641i 0.280457 4.00334i
19.17 0.795146 1.16951i −0.959493 + 0.281733i −0.735486 1.85986i 1.02510 + 0.658795i −0.433449 + 1.34615i 0.612113 + 4.25734i −2.75993 0.618702i 0.841254 0.540641i 1.58557 0.675027i
19.18 0.945243 + 1.05191i −0.959493 + 0.281733i −0.213032 + 1.98862i −1.87195 1.20303i −1.20331 0.742995i 0.493720 + 3.43390i −2.29322 + 1.65564i 0.841254 0.540641i −0.503969 3.10629i
19.19 1.05609 0.940573i −0.959493 + 0.281733i 0.230643 1.98666i −2.78195 1.78785i −0.748319 + 1.20001i 0.0254555 + 0.177047i −1.62502 2.31502i 0.841254 0.540641i −4.61959 + 0.728501i
19.20 1.19882 + 0.750218i −0.959493 + 0.281733i 0.874346 + 1.79875i 0.659581 + 0.423887i −1.36162 0.382082i 0.167612 + 1.16577i −0.301274 + 2.81234i 0.841254 0.540641i 0.472712 + 1.00299i
See next 80 embeddings (of 240 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 523.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
23.d odd 22 1 inner
184.j even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 552.2.t.b 240
8.d odd 2 1 inner 552.2.t.b 240
23.d odd 22 1 inner 552.2.t.b 240
184.j even 22 1 inner 552.2.t.b 240

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.t.b 240 1.a even 1 1 trivial
552.2.t.b 240 8.d odd 2 1 inner
552.2.t.b 240 23.d odd 22 1 inner
552.2.t.b 240 184.j even 22 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$12\!\cdots\!84$$$$T_{5}^{222} +$$$$26\!\cdots\!52$$$$T_{5}^{220} +$$$$51\!\cdots\!80$$$$T_{5}^{218} +$$$$95\!\cdots\!65$$$$T_{5}^{216} +$$$$17\!\cdots\!20$$$$T_{5}^{214} +$$$$29\!\cdots\!98$$$$T_{5}^{212} +$$$$49\!\cdots\!28$$$$T_{5}^{210} +$$$$81\!\cdots\!22$$$$T_{5}^{208} +$$$$12\!\cdots\!96$$$$T_{5}^{206} +$$$$19\!\cdots\!00$$$$T_{5}^{204} +$$$$29\!\cdots\!80$$$$T_{5}^{202} +$$$$43\!\cdots\!77$$$$T_{5}^{200} +$$$$62\!\cdots\!80$$$$T_{5}^{198} +$$$$85\!\cdots\!36$$$$T_{5}^{196} +$$$$11\!\cdots\!72$$$$T_{5}^{194} +$$$$14\!\cdots\!74$$$$T_{5}^{192} +$$$$18\!\cdots\!32$$$$T_{5}^{190} +$$$$22\!\cdots\!06$$$$T_{5}^{188} +$$$$27\!\cdots\!52$$$$T_{5}^{186} +$$$$31\!\cdots\!76$$$$T_{5}^{184} +$$$$35\!\cdots\!52$$$$T_{5}^{182} +$$$$38\!\cdots\!80$$$$T_{5}^{180} +$$$$39\!\cdots\!40$$$$T_{5}^{178} +$$$$39\!\cdots\!32$$$$T_{5}^{176} +$$$$38\!\cdots\!12$$$$T_{5}^{174} +$$$$34\!\cdots\!00$$$$T_{5}^{172} +$$$$30\!\cdots\!40$$$$T_{5}^{170} +$$$$25\!\cdots\!86$$$$T_{5}^{168} +$$$$19\!\cdots\!64$$$$T_{5}^{166} +$$$$15\!\cdots\!18$$$$T_{5}^{164} +$$$$11\!\cdots\!52$$$$T_{5}^{162} +$$$$80\!\cdots\!62$$$$T_{5}^{160} +$$$$55\!\cdots\!32$$$$T_{5}^{158} +$$$$37\!\cdots\!90$$$$T_{5}^{156} +$$$$24\!\cdots\!56$$$$T_{5}^{154} +$$$$16\!\cdots\!52$$$$T_{5}^{152} +$$$$10\!\cdots\!84$$$$T_{5}^{150} +$$$$61\!\cdots\!36$$$$T_{5}^{148} +$$$$36\!\cdots\!28$$$$T_{5}^{146} +$$$$20\!\cdots\!95$$$$T_{5}^{144} +$$$$11\!\cdots\!60$$$$T_{5}^{142} +$$$$65\!\cdots\!00$$$$T_{5}^{140} +$$$$35\!\cdots\!08$$$$T_{5}^{138} +$$$$18\!\cdots\!61$$$$T_{5}^{136} +$$$$95\!\cdots\!48$$$$T_{5}^{134} +$$$$47\!\cdots\!62$$$$T_{5}^{132} +$$$$23\!\cdots\!08$$$$T_{5}^{130} +$$$$10\!\cdots\!43$$$$T_{5}^{128} +$$$$50\!\cdots\!32$$$$T_{5}^{126} +$$$$22\!\cdots\!00$$$$T_{5}^{124} +$$$$10\!\cdots\!96$$$$T_{5}^{122} +$$$$42\!\cdots\!08$$$$T_{5}^{120} +$$$$17\!\cdots\!92$$$$T_{5}^{118} +$$$$69\!\cdots\!02$$$$T_{5}^{116} +$$$$26\!\cdots\!24$$$$T_{5}^{114} +$$$$96\!\cdots\!16$$$$T_{5}^{112} +$$$$34\!\cdots\!16$$$$T_{5}^{110} +$$$$12\!\cdots\!00$$$$T_{5}^{108} +$$$$41\!\cdots\!52$$$$T_{5}^{106} +$$$$13\!\cdots\!40$$$$T_{5}^{104} +$$$$40\!\cdots\!40$$$$T_{5}^{102} +$$$$11\!\cdots\!26$$$$T_{5}^{100} +$$$$29\!\cdots\!96$$$$T_{5}^{98} +$$$$67\!\cdots\!92$$$$T_{5}^{96} +$$$$14\!\cdots\!88$$$$T_{5}^{94} +$$$$28\!\cdots\!44$$$$T_{5}^{92} +$$$$57\!\cdots\!84$$$$T_{5}^{90} +$$$$11\!\cdots\!34$$$$T_{5}^{88} +$$$$22\!\cdots\!92$$$$T_{5}^{86} +$$$$42\!\cdots\!74$$$$T_{5}^{84} +$$$$73\!\cdots\!20$$$$T_{5}^{82} +$$$$11\!\cdots\!83$$$$T_{5}^{80} +$$$$17\!\cdots\!16$$$$T_{5}^{78} +$$$$25\!\cdots\!04$$$$T_{5}^{76} +$$$$34\!\cdots\!72$$$$T_{5}^{74} +$$$$48\!\cdots\!53$$$$T_{5}^{72} +$$$$59\!\cdots\!80$$$$T_{5}^{70} +$$$$86\!\cdots\!02$$$$T_{5}^{68} +$$$$93\!\cdots\!84$$$$T_{5}^{66} +$$$$12\!\cdots\!97$$$$T_{5}^{64} +$$$$10\!\cdots\!84$$$$T_{5}^{62} +$$$$11\!\cdots\!44$$$$T_{5}^{60} +$$$$71\!\cdots\!16$$$$T_{5}^{58} +$$$$72\!\cdots\!06$$$$T_{5}^{56} +$$$$31\!\cdots\!80$$$$T_{5}^{54} +$$$$30\!\cdots\!82$$$$T_{5}^{52} +$$$$63\!\cdots\!68$$$$T_{5}^{50} +$$$$96\!\cdots\!47$$$$T_{5}^{48} +$$$$10\!\cdots\!28$$$$T_{5}^{46} +$$$$19\!\cdots\!16$$$$T_{5}^{44} -$$$$12\!\cdots\!48$$$$T_{5}^{42} +$$$$52\!\cdots\!97$$$$T_{5}^{40} +$$$$46\!\cdots\!60$$$$T_{5}^{38} +$$$$19\!\cdots\!80$$$$T_{5}^{36} +$$$$24\!\cdots\!12$$$$T_{5}^{34} +$$$$39\!\cdots\!47$$$$T_{5}^{32} -$$$$44\!\cdots\!60$$$$T_{5}^{30} -$$$$14\!\cdots\!02$$$$T_{5}^{28} +$$$$31\!\cdots\!76$$$$T_{5}^{26} +$$$$14\!\cdots\!83$$$$T_{5}^{24} -$$$$38\!\cdots\!64$$$$T_{5}^{22} +$$$$37\!\cdots\!38$$$$T_{5}^{20} -$$$$36\!\cdots\!28$$$$T_{5}^{18} +$$$$59\!\cdots\!09$$$$T_{5}^{16} +$$$$56\!\cdots\!16$$$$T_{5}^{14} +$$$$57\!\cdots\!48$$$$T_{5}^{12} +$$$$73\!\cdots\!44$$$$T_{5}^{10} +$$$$11\!\cdots\!39$$$$T_{5}^{8} +$$$$90\!\cdots\!32$$$$T_{5}^{6} +$$$$25\!\cdots\!54$$$$T_{5}^{4} +$$$$11\!\cdots\!12$$$$T_{5}^{2} +$$$$14\!\cdots\!41$$">$$T_{5}^{240} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(552, [\chi])$$.