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 $4$

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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:

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 \(S_{2}^{\mathrm{new}}(552, [\chi])\).