Properties

Label 552.2.t.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 240q + 24q^{3} - 4q^{4} + 11q^{6} - 24q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 240q + 24q^{3} - 4q^{4} + 11q^{6} - 24q^{9} + 4q^{12} - 4q^{16} + 44q^{17} - 55q^{20} - 24q^{25} + 24q^{27} + 33q^{34} + 7q^{36} - 132q^{38} + 11q^{40} - 66q^{42} + 77q^{44} + 37q^{48} - 56q^{49} - 37q^{50} - 55q^{52} - 11q^{54} + 132q^{56} + 37q^{58} + 11q^{60} + 51q^{62} - 4q^{64} - 55q^{66} - 132q^{70} + 144q^{73} + 44q^{74} + 24q^{75} - 55q^{76} - 198q^{80} - 24q^{81} - 26q^{82} + 44q^{84} - 110q^{86} + 22q^{88} - 139q^{92} - 117q^{94} - 209q^{98} + O(q^{100}) \)

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 −1.40793 0.133210i 0.959493 0.281733i 1.96451 + 0.375100i 2.14099 + 1.37593i −1.38842 + 0.268844i 0.0311079 + 0.216360i −2.71592 0.789806i 0.841254 0.540641i −2.83107 2.22241i
19.2 −1.40783 0.134195i 0.959493 0.281733i 1.96398 + 0.377848i −2.06189 1.32509i −1.38861 + 0.267873i 0.593065 + 4.12486i −2.71425 0.795503i 0.841254 0.540641i 2.72497 + 2.14220i
19.3 −1.36604 + 0.365971i 0.959493 0.281733i 1.73213 0.999862i −0.824449 0.529841i −1.20760 + 0.736005i −0.536086 3.72856i −2.00024 + 1.99976i 0.841254 0.540641i 1.32014 + 0.422060i
19.4 −1.23359 + 0.691558i 0.959493 0.281733i 1.04350 1.70620i −1.23207 0.791805i −0.988788 + 1.01109i 0.0805522 + 0.560253i −0.107310 + 2.82639i 0.841254 0.540641i 2.06745 + 0.124714i
19.5 −1.05818 0.938216i 0.959493 0.281733i 0.239503 + 1.98561i 0.0143700 + 0.00923506i −1.27965 0.602087i −0.416387 2.89604i 1.60949 2.32584i 0.841254 0.540641i −0.00654165 0.0232546i
19.6 −0.969131 1.02994i 0.959493 0.281733i −0.121568 + 1.99630i −1.74343 1.12044i −1.22004 0.715188i 0.425534 + 2.95966i 2.17389 1.80947i 0.841254 0.540641i 0.535630 + 2.88149i
19.7 −0.933371 + 1.06246i 0.959493 0.281733i −0.257636 1.98334i 2.60551 + 1.67446i −0.596234 + 1.28238i 0.418423 + 2.91019i 2.34768 + 1.57746i 0.841254 0.540641i −4.21095 + 1.20535i
19.8 −0.865054 1.11879i 0.959493 0.281733i −0.503365 + 1.93562i 3.59098 + 2.30779i −1.14521 0.829754i 0.294270 + 2.04669i 2.60098 1.11126i 0.841254 0.540641i −0.524475 6.01390i
19.9 −0.675020 + 1.24272i 0.959493 0.281733i −1.08870 1.67772i −0.991910 0.637462i −0.297563 + 1.38255i −0.126423 0.879292i 2.81982 0.220446i 0.841254 0.540641i 1.46175 0.802365i
19.10 −0.279033 1.38641i 0.959493 0.281733i −1.84428 + 0.773709i −3.59098 2.30779i −0.658327 1.25164i −0.294270 2.04669i 1.58729 + 2.34105i 0.841254 0.540641i −2.19754 + 5.62253i
19.11 −0.196694 + 1.40047i 0.959493 0.281733i −1.92262 0.550929i −3.35394 2.15545i 0.205831 + 1.39915i 0.412102 + 2.86624i 1.14973 2.58421i 0.841254 0.540641i 3.67834 4.27312i
19.12 −0.180888 + 1.40260i 0.959493 0.281733i −1.93456 0.507426i 1.54978 + 0.995986i 0.221597 + 1.39674i −0.318386 2.21442i 1.06165 2.62162i 0.841254 0.540641i −1.67730 + 1.99356i
19.13 −0.143733 1.40689i 0.959493 0.281733i −1.95868 + 0.404434i 1.74343 + 1.12044i −0.534278 1.30941i −0.425534 2.95966i 0.850523 + 2.69752i 0.841254 0.540641i 1.32574 2.61386i
19.14 −0.0160934 1.41412i 0.959493 0.281733i −1.99948 + 0.0455160i −0.0143700 0.00923506i −0.413846 1.35231i 0.416387 + 2.89604i 0.0965435 + 2.82678i 0.841254 0.540641i −0.0128282 + 0.0204696i
19.15 0.578628 + 1.29042i 0.959493 0.281733i −1.33038 + 1.49335i −1.60117 1.02901i 0.918744 + 1.07513i −0.727375 5.05900i −2.69685 0.852656i 0.841254 0.540641i 0.401374 2.66159i
19.16 0.596315 + 1.28234i 0.959493 0.281733i −1.28882 + 1.52936i 1.60117 + 1.02901i 0.933438 + 1.06240i 0.727375 + 5.05900i −2.72971 0.740725i 0.841254 0.540641i −0.364742 + 2.66686i
19.17 0.820516 1.15185i 0.959493 0.281733i −0.653506 1.89022i 2.06189 + 1.32509i 0.462767 1.33636i −0.593065 4.12486i −2.71346 0.798217i 0.841254 0.540641i 3.21812 1.28772i
19.18 0.821322 1.15127i 0.959493 0.281733i −0.650861 1.89113i −2.14099 1.37593i 0.463701 1.33603i −0.0311079 0.216360i −2.71178 0.803909i 0.841254 0.540641i −3.34252 + 1.33478i
19.19 1.17115 0.792724i 0.959493 0.281733i 0.743177 1.85679i 0.824449 + 0.529841i 0.900373 1.09056i 0.536086 + 3.72856i −0.601555 2.76372i 0.841254 0.540641i 1.38557 0.0330378i
19.20 1.17847 + 0.781800i 0.959493 0.281733i 0.777577 + 1.84265i −1.54978 0.995986i 1.35099 + 0.418119i 0.318386 + 2.21442i −0.524237 + 2.77942i 0.841254 0.540641i −1.04771 2.38536i
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.a 240
8.d odd 2 1 inner 552.2.t.a 240
23.d odd 22 1 inner 552.2.t.a 240
184.j even 22 1 inner 552.2.t.a 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.t.a 240 1.a even 1 1 trivial
552.2.t.a 240 8.d odd 2 1 inner
552.2.t.a 240 23.d odd 22 1 inner
552.2.t.a 240 184.j even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(10\!\cdots\!48\)\( T_{5}^{222} + \)\(17\!\cdots\!88\)\( T_{5}^{220} + \)\(29\!\cdots\!92\)\( T_{5}^{218} + \)\(47\!\cdots\!05\)\( T_{5}^{216} + \)\(74\!\cdots\!36\)\( T_{5}^{214} + \)\(11\!\cdots\!94\)\( T_{5}^{212} + \)\(16\!\cdots\!08\)\( T_{5}^{210} + \)\(21\!\cdots\!14\)\( T_{5}^{208} + \)\(28\!\cdots\!04\)\( T_{5}^{206} + \)\(34\!\cdots\!76\)\( T_{5}^{204} + \)\(42\!\cdots\!76\)\( T_{5}^{202} + \)\(50\!\cdots\!53\)\( T_{5}^{200} + \)\(60\!\cdots\!16\)\( T_{5}^{198} + \)\(69\!\cdots\!00\)\( T_{5}^{196} + \)\(75\!\cdots\!88\)\( T_{5}^{194} + \)\(80\!\cdots\!54\)\( T_{5}^{192} + \)\(83\!\cdots\!40\)\( T_{5}^{190} + \)\(86\!\cdots\!38\)\( T_{5}^{188} + \)\(87\!\cdots\!04\)\( T_{5}^{186} + \)\(86\!\cdots\!56\)\( T_{5}^{184} + \)\(83\!\cdots\!84\)\( T_{5}^{182} + \)\(79\!\cdots\!96\)\( T_{5}^{180} + \)\(73\!\cdots\!96\)\( T_{5}^{178} + \)\(66\!\cdots\!20\)\( T_{5}^{176} + \)\(58\!\cdots\!24\)\( T_{5}^{174} + \)\(51\!\cdots\!72\)\( T_{5}^{172} + \)\(44\!\cdots\!16\)\( T_{5}^{170} + \)\(37\!\cdots\!42\)\( T_{5}^{168} + \)\(30\!\cdots\!80\)\( T_{5}^{166} + \)\(24\!\cdots\!14\)\( T_{5}^{164} + \)\(19\!\cdots\!28\)\( T_{5}^{162} + \)\(15\!\cdots\!02\)\( T_{5}^{160} + \)\(11\!\cdots\!08\)\( T_{5}^{158} + \)\(85\!\cdots\!30\)\( T_{5}^{156} + \)\(62\!\cdots\!32\)\( T_{5}^{154} + \)\(43\!\cdots\!48\)\( T_{5}^{152} + \)\(31\!\cdots\!36\)\( T_{5}^{150} + \)\(21\!\cdots\!12\)\( T_{5}^{148} + \)\(13\!\cdots\!88\)\( T_{5}^{146} + \)\(94\!\cdots\!51\)\( T_{5}^{144} + \)\(61\!\cdots\!40\)\( T_{5}^{142} + \)\(37\!\cdots\!76\)\( T_{5}^{140} + \)\(24\!\cdots\!76\)\( T_{5}^{138} + \)\(14\!\cdots\!77\)\( T_{5}^{136} + \)\(83\!\cdots\!40\)\( T_{5}^{134} + \)\(49\!\cdots\!54\)\( T_{5}^{132} + \)\(27\!\cdots\!40\)\( T_{5}^{130} + \)\(14\!\cdots\!95\)\( T_{5}^{128} + \)\(81\!\cdots\!00\)\( T_{5}^{126} + \)\(43\!\cdots\!56\)\( T_{5}^{124} + \)\(21\!\cdots\!60\)\( T_{5}^{122} + \)\(10\!\cdots\!72\)\( T_{5}^{120} + \)\(51\!\cdots\!08\)\( T_{5}^{118} + \)\(23\!\cdots\!38\)\( T_{5}^{116} + \)\(10\!\cdots\!72\)\( T_{5}^{114} + \)\(45\!\cdots\!36\)\( T_{5}^{112} + \)\(19\!\cdots\!44\)\( T_{5}^{110} + \)\(78\!\cdots\!28\)\( T_{5}^{108} + \)\(30\!\cdots\!16\)\( T_{5}^{106} + \)\(11\!\cdots\!24\)\( T_{5}^{104} + \)\(40\!\cdots\!36\)\( T_{5}^{102} + \)\(13\!\cdots\!22\)\( T_{5}^{100} + \)\(40\!\cdots\!88\)\( T_{5}^{98} + \)\(11\!\cdots\!56\)\( T_{5}^{96} + \)\(31\!\cdots\!96\)\( T_{5}^{94} + \)\(77\!\cdots\!56\)\( T_{5}^{92} + \)\(17\!\cdots\!72\)\( T_{5}^{90} + \)\(36\!\cdots\!70\)\( T_{5}^{88} + \)\(68\!\cdots\!24\)\( T_{5}^{86} + \)\(11\!\cdots\!50\)\( T_{5}^{84} + \)\(18\!\cdots\!04\)\( T_{5}^{82} + \)\(26\!\cdots\!03\)\( T_{5}^{80} + \)\(34\!\cdots\!60\)\( T_{5}^{78} + \)\(41\!\cdots\!28\)\( T_{5}^{76} + \)\(44\!\cdots\!80\)\( T_{5}^{74} + \)\(44\!\cdots\!73\)\( T_{5}^{72} + \)\(41\!\cdots\!68\)\( T_{5}^{70} + \)\(37\!\cdots\!78\)\( T_{5}^{68} + \)\(33\!\cdots\!72\)\( T_{5}^{66} + \)\(29\!\cdots\!73\)\( T_{5}^{64} + \)\(24\!\cdots\!72\)\( T_{5}^{62} + \)\(20\!\cdots\!28\)\( T_{5}^{60} + \)\(15\!\cdots\!12\)\( T_{5}^{58} + \)\(99\!\cdots\!70\)\( T_{5}^{56} + \)\(52\!\cdots\!56\)\( T_{5}^{54} + \)\(22\!\cdots\!02\)\( T_{5}^{52} + \)\(10\!\cdots\!92\)\( T_{5}^{50} + \)\(58\!\cdots\!15\)\( T_{5}^{48} + \)\(38\!\cdots\!72\)\( T_{5}^{46} + \)\(21\!\cdots\!64\)\( T_{5}^{44} + \)\(87\!\cdots\!88\)\( T_{5}^{42} + \)\(24\!\cdots\!65\)\( T_{5}^{40} + \)\(49\!\cdots\!40\)\( T_{5}^{38} + \)\(13\!\cdots\!88\)\( T_{5}^{36} + \)\(71\!\cdots\!44\)\( T_{5}^{34} + \)\(32\!\cdots\!39\)\( T_{5}^{32} + \)\(97\!\cdots\!36\)\( T_{5}^{30} + \)\(21\!\cdots\!10\)\( T_{5}^{28} + \)\(31\!\cdots\!20\)\( T_{5}^{26} + \)\(29\!\cdots\!75\)\( T_{5}^{24} + \)\(29\!\cdots\!64\)\( T_{5}^{22} + \)\(61\!\cdots\!58\)\( T_{5}^{20} + \)\(17\!\cdots\!04\)\( T_{5}^{18} + \)\(33\!\cdots\!05\)\( T_{5}^{16} + \)\(34\!\cdots\!76\)\( T_{5}^{14} + \)\(17\!\cdots\!04\)\( T_{5}^{12} + \)\(50\!\cdots\!24\)\( T_{5}^{10} + \)\(14\!\cdots\!19\)\( T_{5}^{8} + \)\(97\!\cdots\!56\)\( T_{5}^{6} + \)\(16\!\cdots\!66\)\( T_{5}^{4} - \)\(39\!\cdots\!36\)\( T_{5}^{2} + \)\(14\!\cdots\!01\)\( \)">\(T_{5}^{240} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(552, [\chi])\).