Properties

Label 588.2.x.a
Level $588$
Weight $2$
Character orbit 588.x
Analytic conductor $4.695$
Analytic rank $0$
Dimension $168$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 588.x (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.69520363885\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(28\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 168 q - 28 q^{3} + 2 q^{7} + 6 q^{8} - 28 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 168 q - 28 q^{3} + 2 q^{7} + 6 q^{8} - 28 q^{9} + 20 q^{10} - 12 q^{14} + 36 q^{16} + 12 q^{19} - 25 q^{20} + 2 q^{21} - 6 q^{22} - 15 q^{24} + 32 q^{25} + 6 q^{26} - 28 q^{27} - 66 q^{28} - 8 q^{30} - 4 q^{31} + 25 q^{32} - 68 q^{34} - 12 q^{35} - 10 q^{37} + 35 q^{38} + 14 q^{39} + 16 q^{40} + 9 q^{42} + 20 q^{44} - 28 q^{46} - 8 q^{47} + 8 q^{48} - 8 q^{49} + 114 q^{50} + 20 q^{52} - 8 q^{53} - q^{56} + 12 q^{57} - 6 q^{58} + 20 q^{59} + 10 q^{60} - 14 q^{61} - 16 q^{62} - 12 q^{63} + 42 q^{64} - 8 q^{65} - 6 q^{66} - 16 q^{68} + 59 q^{70} + 28 q^{71} - 15 q^{72} + 22 q^{74} + 18 q^{75} + 7 q^{76} + 8 q^{77} + 6 q^{78} + 26 q^{80} - 28 q^{81} + 12 q^{82} + 10 q^{83} + 11 q^{84} - 24 q^{85} - 6 q^{86} - 242 q^{88} + 20 q^{90} - 16 q^{91} + 7 q^{92} - 4 q^{93} - 53 q^{94} - 10 q^{96} - 118 q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1 −1.40988 0.110588i −0.900969 0.433884i 1.97554 + 0.311833i 1.52760 3.17209i 1.22228 + 0.711362i −1.99561 + 1.73711i −2.75080 0.658120i 0.623490 + 0.781831i −2.50453 + 4.30333i
55.2 −1.39924 0.205230i −0.900969 0.433884i 1.91576 + 0.574332i −1.72652 + 3.58517i 1.17163 + 0.792014i 1.74761 + 1.98642i −2.56275 1.19680i 0.623490 + 0.781831i 3.15161 4.66218i
55.3 −1.35491 0.405223i −0.900969 0.433884i 1.67159 + 1.09809i −0.997385 + 2.07109i 1.04492 + 0.952969i −0.739943 2.54017i −1.81989 2.16518i 0.623490 + 0.781831i 2.19063 2.40199i
55.4 −1.27947 + 0.602464i −0.900969 0.433884i 1.27407 1.54167i 0.0432081 0.0897226i 1.41416 + 0.0123389i 0.103018 + 2.64374i −0.701339 + 2.74010i 0.623490 + 0.781831i −0.00122877 + 0.140828i
55.5 −1.20909 + 0.733551i −0.900969 0.433884i 0.923805 1.77386i −0.0202251 + 0.0419978i 1.40763 0.136302i 2.50508 0.851222i 0.184253 + 2.82242i 0.623490 + 0.781831i −0.00635357 0.0656153i
55.6 −1.17933 0.780496i −0.900969 0.433884i 0.781653 + 1.84093i −0.352634 + 0.732252i 0.723898 + 1.21490i −1.67777 + 2.04575i 0.515009 2.78114i 0.623490 + 0.781831i 0.987392 0.588339i
55.7 −1.11034 0.875865i −0.900969 0.433884i 0.465720 + 1.94502i 0.698680 1.45082i 0.620360 + 1.27089i 1.90070 1.84047i 1.18647 2.56755i 0.623490 + 0.781831i −2.04650 + 0.998961i
55.8 −0.977782 + 1.02174i −0.900969 0.433884i −0.0878851 1.99807i −1.73990 + 3.61294i 1.32427 0.496308i −2.50782 0.843102i 2.12743 + 1.86388i 0.623490 + 0.781831i −1.99022 5.31038i
55.9 −0.782021 + 1.17832i −0.900969 0.433884i −0.776887 1.84295i 1.10901 2.30289i 1.21583 0.722326i −0.953739 2.46787i 2.77913 + 0.525799i 0.623490 + 0.781831i 1.84628 + 3.10769i
55.10 −0.712285 1.22174i −0.900969 0.433884i −0.985301 + 1.74045i 1.07827 2.23906i 0.111653 + 1.40980i −2.56980 0.629382i 2.82820 0.0359170i 0.623490 + 0.781831i −3.50359 + 0.277476i
55.11 −0.537130 1.30824i −0.900969 0.433884i −1.42298 + 1.40539i −1.35460 + 2.81286i −0.0836867 + 1.41174i 2.57260 0.617858i 2.60291 + 1.10673i 0.623490 + 0.781831i 4.40750 + 0.261274i
55.12 −0.365015 + 1.36630i −0.900969 0.433884i −1.73353 0.997438i −0.617993 + 1.28328i 0.921681 1.07262i 2.62009 0.367586i 1.99556 2.00443i 0.623490 + 0.781831i −1.52776 1.31278i
55.13 −0.0562843 1.41309i −0.900969 0.433884i −1.99366 + 0.159070i 0.302206 0.627538i −0.562408 + 1.29757i 1.01653 + 2.44268i 0.336993 + 2.80828i 0.623490 + 0.781831i −0.903779 0.391725i
55.14 −0.0312592 + 1.41387i −0.900969 0.433884i −1.99805 0.0883928i 1.63442 3.39390i 0.641618 1.26029i 0.364664 + 2.62050i 0.187433 2.82221i 0.623490 + 0.781831i 4.74743 + 2.41694i
55.15 0.0809236 1.41190i −0.900969 0.433884i −1.98690 0.228511i −0.881995 + 1.83148i −0.685509 + 1.23696i −1.70818 2.02043i −0.483422 + 2.78681i 0.623490 + 0.781831i 2.51449 + 1.39349i
55.16 0.131027 + 1.40813i −0.900969 0.433884i −1.96566 + 0.369007i −1.40445 + 2.91638i 0.492913 1.32553i −0.602515 + 2.57623i −0.777166 2.71956i 0.623490 + 0.781831i −4.29066 1.59553i
55.17 0.311114 + 1.37957i −0.900969 0.433884i −1.80642 + 0.858405i 0.477768 0.992096i 0.318268 1.37794i −2.27188 1.35593i −1.74623 2.22501i 0.623490 + 0.781831i 1.51730 + 0.350459i
55.18 0.505797 1.32067i −0.900969 0.433884i −1.48834 1.33598i 1.55098 3.22064i −1.02872 + 0.970425i 1.05984 2.42420i −2.51719 + 1.28987i 0.623490 + 0.781831i −3.46892 3.67732i
55.19 0.652439 + 1.25472i −0.900969 0.433884i −1.14865 + 1.63726i −0.0842420 + 0.174930i −0.0434247 1.41355i 2.61856 + 0.378316i −2.80372 0.373018i 0.623490 + 0.781831i −0.274451 + 0.00843125i
55.20 0.745941 1.20149i −0.900969 0.433884i −0.887143 1.79248i 0.0475607 0.0987608i −1.19338 + 0.758851i −1.69059 + 2.03517i −2.81540 0.271193i 0.623490 + 0.781831i −0.0831823 0.130813i
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
196.j even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.x.a 168
4.b odd 2 1 588.2.x.b yes 168
49.f odd 14 1 588.2.x.b yes 168
196.j even 14 1 inner 588.2.x.a 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.x.a 168 1.a even 1 1 trivial
588.2.x.a 168 196.j even 14 1 inner
588.2.x.b yes 168 4.b odd 2 1
588.2.x.b yes 168 49.f odd 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(60\!\cdots\!46\)\( T_{11}^{154} - \)\(46\!\cdots\!52\)\( T_{11}^{153} + \)\(22\!\cdots\!34\)\( T_{11}^{152} + \)\(20\!\cdots\!88\)\( T_{11}^{151} - \)\(75\!\cdots\!26\)\( T_{11}^{150} - \)\(82\!\cdots\!36\)\( T_{11}^{149} + \)\(23\!\cdots\!22\)\( T_{11}^{148} + \)\(30\!\cdots\!12\)\( T_{11}^{147} - \)\(63\!\cdots\!74\)\( T_{11}^{146} - \)\(10\!\cdots\!92\)\( T_{11}^{145} + \)\(15\!\cdots\!46\)\( T_{11}^{144} + \)\(34\!\cdots\!36\)\( T_{11}^{143} - \)\(30\!\cdots\!80\)\( T_{11}^{142} - \)\(10\!\cdots\!24\)\( T_{11}^{141} + \)\(41\!\cdots\!44\)\( T_{11}^{140} + \)\(30\!\cdots\!80\)\( T_{11}^{139} + \)\(48\!\cdots\!68\)\( T_{11}^{138} - \)\(81\!\cdots\!00\)\( T_{11}^{137} - \)\(31\!\cdots\!23\)\( T_{11}^{136} + \)\(20\!\cdots\!56\)\( T_{11}^{135} + \)\(14\!\cdots\!82\)\( T_{11}^{134} - \)\(50\!\cdots\!60\)\( T_{11}^{133} - \)\(51\!\cdots\!33\)\( T_{11}^{132} + \)\(11\!\cdots\!92\)\( T_{11}^{131} + \)\(14\!\cdots\!86\)\( T_{11}^{130} - \)\(24\!\cdots\!92\)\( T_{11}^{129} - \)\(38\!\cdots\!23\)\( T_{11}^{128} + \)\(49\!\cdots\!68\)\( T_{11}^{127} + \)\(90\!\cdots\!54\)\( T_{11}^{126} - \)\(97\!\cdots\!20\)\( T_{11}^{125} - \)\(19\!\cdots\!61\)\( T_{11}^{124} + \)\(18\!\cdots\!20\)\( T_{11}^{123} + \)\(39\!\cdots\!10\)\( T_{11}^{122} - \)\(32\!\cdots\!32\)\( T_{11}^{121} - \)\(74\!\cdots\!19\)\( T_{11}^{120} + \)\(55\!\cdots\!84\)\( T_{11}^{119} + \)\(13\!\cdots\!42\)\( T_{11}^{118} - \)\(90\!\cdots\!32\)\( T_{11}^{117} - \)\(22\!\cdots\!62\)\( T_{11}^{116} + \)\(14\!\cdots\!16\)\( T_{11}^{115} + \)\(36\!\cdots\!18\)\( T_{11}^{114} - \)\(21\!\cdots\!80\)\( T_{11}^{113} - \)\(55\!\cdots\!99\)\( T_{11}^{112} + \)\(32\!\cdots\!60\)\( T_{11}^{111} + \)\(80\!\cdots\!44\)\( T_{11}^{110} - \)\(45\!\cdots\!24\)\( T_{11}^{109} - \)\(11\!\cdots\!24\)\( T_{11}^{108} + \)\(62\!\cdots\!52\)\( T_{11}^{107} + \)\(14\!\cdots\!84\)\( T_{11}^{106} - \)\(83\!\cdots\!76\)\( T_{11}^{105} - \)\(17\!\cdots\!31\)\( T_{11}^{104} + \)\(10\!\cdots\!64\)\( T_{11}^{103} + \)\(20\!\cdots\!86\)\( T_{11}^{102} - \)\(13\!\cdots\!52\)\( T_{11}^{101} - \)\(22\!\cdots\!15\)\( T_{11}^{100} + \)\(15\!\cdots\!40\)\( T_{11}^{99} + \)\(22\!\cdots\!32\)\( T_{11}^{98} - \)\(18\!\cdots\!80\)\( T_{11}^{97} - \)\(20\!\cdots\!00\)\( T_{11}^{96} + \)\(19\!\cdots\!08\)\( T_{11}^{95} + \)\(15\!\cdots\!90\)\( T_{11}^{94} - \)\(20\!\cdots\!44\)\( T_{11}^{93} - \)\(92\!\cdots\!11\)\( T_{11}^{92} + \)\(20\!\cdots\!00\)\( T_{11}^{91} + \)\(15\!\cdots\!66\)\( T_{11}^{90} - \)\(19\!\cdots\!60\)\( T_{11}^{89} + \)\(62\!\cdots\!38\)\( T_{11}^{88} + \)\(17\!\cdots\!28\)\( T_{11}^{87} - \)\(13\!\cdots\!64\)\( T_{11}^{86} - \)\(14\!\cdots\!68\)\( T_{11}^{85} + \)\(17\!\cdots\!60\)\( T_{11}^{84} + \)\(10\!\cdots\!24\)\( T_{11}^{83} - \)\(19\!\cdots\!84\)\( T_{11}^{82} - \)\(68\!\cdots\!84\)\( T_{11}^{81} + \)\(18\!\cdots\!41\)\( T_{11}^{80} + \)\(37\!\cdots\!48\)\( T_{11}^{79} - \)\(15\!\cdots\!24\)\( T_{11}^{78} - \)\(16\!\cdots\!56\)\( T_{11}^{77} + \)\(10\!\cdots\!69\)\( T_{11}^{76} + \)\(35\!\cdots\!64\)\( T_{11}^{75} - \)\(68\!\cdots\!40\)\( T_{11}^{74} + \)\(19\!\cdots\!64\)\( T_{11}^{73} + \)\(38\!\cdots\!90\)\( T_{11}^{72} - \)\(32\!\cdots\!60\)\( T_{11}^{71} - \)\(19\!\cdots\!62\)\( T_{11}^{70} + \)\(27\!\cdots\!36\)\( T_{11}^{69} + \)\(85\!\cdots\!52\)\( T_{11}^{68} - \)\(19\!\cdots\!36\)\( T_{11}^{67} - \)\(31\!\cdots\!88\)\( T_{11}^{66} + \)\(11\!\cdots\!40\)\( T_{11}^{65} + \)\(79\!\cdots\!23\)\( T_{11}^{64} - \)\(58\!\cdots\!84\)\( T_{11}^{63} + \)\(40\!\cdots\!26\)\( T_{11}^{62} + \)\(25\!\cdots\!24\)\( T_{11}^{61} - \)\(18\!\cdots\!78\)\( T_{11}^{60} - \)\(86\!\cdots\!48\)\( T_{11}^{59} + \)\(12\!\cdots\!56\)\( T_{11}^{58} + \)\(23\!\cdots\!24\)\( T_{11}^{57} - \)\(59\!\cdots\!70\)\( T_{11}^{56} - \)\(49\!\cdots\!12\)\( T_{11}^{55} + \)\(22\!\cdots\!92\)\( T_{11}^{54} + \)\(61\!\cdots\!48\)\( T_{11}^{53} - \)\(81\!\cdots\!41\)\( T_{11}^{52} + \)\(23\!\cdots\!92\)\( T_{11}^{51} + \)\(27\!\cdots\!56\)\( T_{11}^{50} - \)\(28\!\cdots\!64\)\( T_{11}^{49} - \)\(67\!\cdots\!60\)\( T_{11}^{48} + \)\(15\!\cdots\!12\)\( T_{11}^{47} + \)\(54\!\cdots\!92\)\( T_{11}^{46} - \)\(49\!\cdots\!88\)\( T_{11}^{45} + \)\(42\!\cdots\!73\)\( T_{11}^{44} + \)\(66\!\cdots\!72\)\( T_{11}^{43} - \)\(16\!\cdots\!42\)\( T_{11}^{42} + \)\(72\!\cdots\!36\)\( T_{11}^{41} + \)\(15\!\cdots\!81\)\( T_{11}^{40} - \)\(24\!\cdots\!00\)\( T_{11}^{39} + \)\(10\!\cdots\!38\)\( T_{11}^{38} - \)\(55\!\cdots\!32\)\( T_{11}^{37} + \)\(22\!\cdots\!85\)\( T_{11}^{36} - \)\(21\!\cdots\!88\)\( T_{11}^{35} - \)\(16\!\cdots\!56\)\( T_{11}^{34} + \)\(35\!\cdots\!88\)\( T_{11}^{33} - \)\(27\!\cdots\!72\)\( T_{11}^{32} - \)\(30\!\cdots\!80\)\( T_{11}^{31} + \)\(43\!\cdots\!04\)\( T_{11}^{30} + \)\(26\!\cdots\!24\)\( T_{11}^{29} - \)\(30\!\cdots\!80\)\( T_{11}^{28} - \)\(24\!\cdots\!16\)\( T_{11}^{27} + \)\(44\!\cdots\!72\)\( T_{11}^{26} + \)\(21\!\cdots\!92\)\( T_{11}^{25} - \)\(43\!\cdots\!00\)\( T_{11}^{24} - \)\(18\!\cdots\!40\)\( T_{11}^{23} + \)\(63\!\cdots\!28\)\( T_{11}^{22} + \)\(10\!\cdots\!52\)\( T_{11}^{21} - \)\(42\!\cdots\!76\)\( T_{11}^{20} - \)\(47\!\cdots\!00\)\( T_{11}^{19} + \)\(19\!\cdots\!48\)\( T_{11}^{18} + \)\(13\!\cdots\!20\)\( T_{11}^{17} - \)\(70\!\cdots\!08\)\( T_{11}^{16} - \)\(22\!\cdots\!92\)\( T_{11}^{15} + \)\(21\!\cdots\!76\)\( T_{11}^{14} + \)\(61\!\cdots\!36\)\( T_{11}^{13} - \)\(33\!\cdots\!20\)\( T_{11}^{12} - \)\(24\!\cdots\!64\)\( T_{11}^{11} + \)\(10\!\cdots\!72\)\( T_{11}^{10} + \)\(44\!\cdots\!28\)\( T_{11}^{9} - \)\(18\!\cdots\!40\)\( T_{11}^{8} - \)\(49\!\cdots\!76\)\( T_{11}^{7} + \)\(18\!\cdots\!44\)\( T_{11}^{6} + \)\(41\!\cdots\!16\)\( T_{11}^{5} - \)\(11\!\cdots\!76\)\( T_{11}^{4} - \)\(27\!\cdots\!84\)\( T_{11}^{3} + \)\(38\!\cdots\!36\)\( T_{11}^{2} + \)\(11\!\cdots\!24\)\( T_{11} + \)\(89\!\cdots\!56\)\( \)">\(T_{11}^{168} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(588, [\chi])\).