Properties

Label 242.2.g.a
Level $242$
Weight $2$
Character orbit 242.g
Analytic conductor $1.932$
Analytic rank $0$
Dimension $200$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 242 = 2 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 242.g (of order \(55\), degree \(40\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.93237972891\)
Analytic rank: \(0\)
Dimension: \(200\)
Relative dimension: \(5\) over \(\Q(\zeta_{55})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{55}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 200q - 5q^{2} + 10q^{3} + 5q^{4} - q^{5} + q^{6} - 5q^{8} - 40q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 200q - 5q^{2} + 10q^{3} + 5q^{4} - q^{5} + q^{6} - 5q^{8} - 40q^{9} - 4q^{10} - 11q^{11} + 4q^{12} - 32q^{13} - 11q^{14} - 24q^{15} + 5q^{16} + 2q^{17} - 4q^{18} + q^{19} - q^{20} - 8q^{21} - 22q^{22} + 41q^{23} + q^{24} + 2q^{25} - q^{26} - 59q^{27} + 3q^{29} + 13q^{30} + 18q^{31} + 20q^{32} + 8q^{34} + 4q^{35} + 4q^{36} + 17q^{37} - 34q^{38} - 172q^{39} - 10q^{40} + 6q^{41} + 42q^{42} - 20q^{43} + 11q^{44} + 64q^{45} - 9q^{46} + 7q^{47} - q^{48} - 43q^{49} - 2q^{50} - 135q^{51} + q^{52} - 80q^{53} - 16q^{54} - 22q^{55} + 11q^{56} + 41q^{57} - 157q^{58} + 30q^{59} + 31q^{60} + 40q^{61} + 4q^{62} - 45q^{63} + 5q^{64} - 7q^{65} + 55q^{66} - 19q^{67} + 2q^{68} - 8q^{69} - 70q^{70} + 5q^{71} - 4q^{72} + 34q^{73} + 5q^{74} - 11q^{75} - 15q^{76} - 198q^{77} + 16q^{78} - 47q^{79} - q^{80} - 18q^{81} - 28q^{82} + 9q^{83} - 174q^{84} - 100q^{85} - 5q^{86} + 37q^{87} - 11q^{88} - 67q^{89} - 28q^{90} - 166q^{91} - 46q^{92} - 47q^{93} - 18q^{94} + 129q^{95} + q^{96} - 3q^{97} - 40q^{98} + 22q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.516397 + 0.856349i −2.13866 1.55383i −0.466667 0.884433i −1.07424 + 0.878398i 2.43502 1.02905i −1.06439 + 1.38033i 0.998369 + 0.0570888i 1.23243 + 3.79304i −0.197483 1.37352i
5.2 −0.516397 + 0.856349i −0.433449 0.314919i −0.466667 0.884433i −1.12246 + 0.917831i 0.493513 0.208560i −1.54587 + 2.00472i 0.998369 + 0.0570888i −0.838347 2.58017i −0.206348 1.43518i
5.3 −0.516397 + 0.856349i −0.324490 0.235756i −0.466667 0.884433i 1.42498 1.16520i 0.369456 0.156133i 0.331985 0.430526i 0.998369 + 0.0570888i −0.877338 2.70017i 0.261962 + 1.82199i
5.4 −0.516397 + 0.856349i 1.49802 + 1.08837i −0.466667 0.884433i 1.49869 1.22548i −1.70560 + 0.720792i 1.72885 2.24201i 0.998369 + 0.0570888i 0.132446 + 0.407627i 0.275513 + 1.91624i
5.5 −0.516397 + 0.856349i 2.20760 + 1.60392i −0.466667 0.884433i −3.30271 + 2.70061i −2.51351 + 1.06222i 1.14932 1.49047i 0.998369 + 0.0570888i 1.37390 + 4.22844i −0.607156 4.22286i
15.1 −0.0855750 0.996332i −0.871571 + 2.68242i −0.985354 + 0.170522i 1.17984 0.0674655i 2.74716 + 0.638826i 0.234620 + 0.343121i 0.254218 + 0.967147i −4.00869 2.91248i −0.168183 1.16974i
15.2 −0.0855750 0.996332i −0.287239 + 0.884029i −0.985354 + 0.170522i 1.80964 0.103479i 0.905367 + 0.210534i −2.83890 4.15177i 0.254218 + 0.967147i 1.72805 + 1.25550i −0.257959 1.79415i
15.3 −0.0855750 0.996332i −0.194167 + 0.597585i −0.985354 + 0.170522i −0.859975 + 0.0491752i 0.612009 + 0.142317i 1.98344 + 2.90069i 0.254218 + 0.967147i 2.10764 + 1.53129i 0.122587 + 0.852612i
15.4 −0.0855750 0.996332i 0.292119 0.899048i −0.985354 + 0.170522i 4.07280 0.232891i −0.920748 0.214111i 1.25754 + 1.83909i 0.254218 + 0.967147i 1.70410 + 1.23810i −0.580567 4.03793i
15.5 −0.0855750 0.996332i 0.751841 2.31393i −0.985354 + 0.170522i −1.44457 + 0.0826036i −2.36978 0.551069i −0.642307 0.939344i 0.254218 + 0.967147i −2.36195 1.71606i 0.205920 + 1.43220i
25.1 0.466667 + 0.884433i −0.816622 2.51330i −0.564443 + 0.825472i −0.759633 + 3.74894i 1.84176 1.89512i 0.840551 + 3.19779i −0.993482 0.113991i −3.22277 + 2.34148i −3.67018 + 1.07766i
25.2 0.466667 + 0.884433i −0.632586 1.94690i −0.564443 + 0.825472i 0.184594 0.911006i 1.42669 1.46803i −0.597482 2.27306i −0.993482 0.113991i −0.963202 + 0.699807i 0.891867 0.261876i
25.3 0.466667 + 0.884433i −0.0758337 0.233392i −0.564443 + 0.825472i 0.588996 2.90681i 0.171031 0.175986i 1.08715 + 4.13596i −0.993482 0.113991i 2.37833 1.72796i 2.84574 0.835586i
25.4 0.466667 + 0.884433i 0.313512 + 0.964891i −0.564443 + 0.825472i −0.337582 + 1.66603i −0.707075 + 0.727563i 0.220471 + 0.838759i −0.993482 0.113991i 1.59433 1.15835i −1.63103 + 0.478915i
25.5 0.466667 + 0.884433i 0.902513 + 2.77765i −0.564443 + 0.825472i −0.0878091 + 0.433355i −2.03547 + 2.09445i −0.274842 1.04561i −0.993482 0.113991i −4.47375 + 3.25037i −0.424251 + 0.124571i
31.1 −0.774142 + 0.633012i −1.86596 + 1.35570i 0.198590 0.980083i −0.339686 + 0.167012i 0.586344 2.23068i −0.378964 4.41220i 0.466667 + 0.884433i 0.716842 2.20621i 0.157244 0.344317i
31.2 −0.774142 + 0.633012i −1.59601 + 1.15957i 0.198590 0.980083i 1.05809 0.520226i 0.501516 1.90796i 0.173538 + 2.02047i 0.466667 + 0.884433i 0.275594 0.848191i −0.489799 + 1.07251i
31.3 −0.774142 + 0.633012i 0.108872 0.0791004i 0.198590 0.980083i 0.362789 0.178371i −0.0342111 + 0.130152i 0.152815 + 1.77919i 0.466667 + 0.884433i −0.921455 + 2.83595i −0.167939 + 0.367735i
31.4 −0.774142 + 0.633012i 1.89410 1.37614i 0.198590 0.980083i 2.54151 1.24958i −0.595185 + 2.26432i 0.000169160 0.00196950i 0.466667 + 0.884433i 0.766788 2.35993i −1.17649 + 2.57616i
31.5 −0.774142 + 0.633012i 2.26802 1.64781i 0.198590 0.980083i −2.60086 + 1.27876i −0.712682 + 2.71132i −0.380327 4.42807i 0.466667 + 0.884433i 1.50157 4.62137i 1.20396 2.63631i
See next 80 embeddings (of 200 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.g even 55 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 242.2.g.a 200
121.g even 55 1 inner 242.2.g.a 200
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
242.2.g.a 200 1.a even 1 1 trivial
242.2.g.a 200 121.g even 55 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(34\!\cdots\!17\)\( T_{3}^{184} - \)\(13\!\cdots\!43\)\( T_{3}^{183} + \)\(55\!\cdots\!09\)\( T_{3}^{182} - \)\(20\!\cdots\!68\)\( T_{3}^{181} + \)\(82\!\cdots\!08\)\( T_{3}^{180} - \)\(29\!\cdots\!64\)\( T_{3}^{179} + \)\(11\!\cdots\!51\)\( T_{3}^{178} - \)\(38\!\cdots\!70\)\( T_{3}^{177} + \)\(14\!\cdots\!19\)\( T_{3}^{176} - \)\(47\!\cdots\!23\)\( T_{3}^{175} + \)\(16\!\cdots\!95\)\( T_{3}^{174} - \)\(54\!\cdots\!35\)\( T_{3}^{173} + \)\(18\!\cdots\!93\)\( T_{3}^{172} - \)\(59\!\cdots\!59\)\( T_{3}^{171} + \)\(19\!\cdots\!07\)\( T_{3}^{170} - \)\(60\!\cdots\!27\)\( T_{3}^{169} + \)\(19\!\cdots\!58\)\( T_{3}^{168} - \)\(58\!\cdots\!70\)\( T_{3}^{167} + \)\(18\!\cdots\!52\)\( T_{3}^{166} - \)\(53\!\cdots\!16\)\( T_{3}^{165} + \)\(16\!\cdots\!13\)\( T_{3}^{164} - \)\(46\!\cdots\!81\)\( T_{3}^{163} + \)\(14\!\cdots\!80\)\( T_{3}^{162} - \)\(38\!\cdots\!01\)\( T_{3}^{161} + \)\(11\!\cdots\!00\)\( T_{3}^{160} - \)\(30\!\cdots\!09\)\( T_{3}^{159} + \)\(92\!\cdots\!40\)\( T_{3}^{158} - \)\(23\!\cdots\!50\)\( T_{3}^{157} + \)\(69\!\cdots\!70\)\( T_{3}^{156} - \)\(16\!\cdots\!49\)\( T_{3}^{155} + \)\(49\!\cdots\!00\)\( T_{3}^{154} - \)\(11\!\cdots\!96\)\( T_{3}^{153} + \)\(34\!\cdots\!47\)\( T_{3}^{152} - \)\(78\!\cdots\!67\)\( T_{3}^{151} + \)\(22\!\cdots\!15\)\( T_{3}^{150} - \)\(50\!\cdots\!43\)\( T_{3}^{149} + \)\(14\!\cdots\!00\)\( T_{3}^{148} - \)\(30\!\cdots\!03\)\( T_{3}^{147} + \)\(88\!\cdots\!01\)\( T_{3}^{146} - \)\(18\!\cdots\!46\)\( T_{3}^{145} + \)\(51\!\cdots\!19\)\( T_{3}^{144} - \)\(10\!\cdots\!30\)\( T_{3}^{143} + \)\(29\!\cdots\!35\)\( T_{3}^{142} - \)\(56\!\cdots\!38\)\( T_{3}^{141} + \)\(16\!\cdots\!50\)\( T_{3}^{140} - \)\(29\!\cdots\!11\)\( T_{3}^{139} + \)\(84\!\cdots\!78\)\( T_{3}^{138} - \)\(14\!\cdots\!91\)\( T_{3}^{137} + \)\(43\!\cdots\!45\)\( T_{3}^{136} - \)\(71\!\cdots\!12\)\( T_{3}^{135} + \)\(21\!\cdots\!52\)\( T_{3}^{134} - \)\(33\!\cdots\!66\)\( T_{3}^{133} + \)\(99\!\cdots\!82\)\( T_{3}^{132} - \)\(14\!\cdots\!88\)\( T_{3}^{131} + \)\(45\!\cdots\!67\)\( T_{3}^{130} - \)\(63\!\cdots\!13\)\( T_{3}^{129} + \)\(19\!\cdots\!79\)\( T_{3}^{128} - \)\(26\!\cdots\!97\)\( T_{3}^{127} + \)\(84\!\cdots\!47\)\( T_{3}^{126} - \)\(10\!\cdots\!93\)\( T_{3}^{125} + \)\(34\!\cdots\!38\)\( T_{3}^{124} - \)\(39\!\cdots\!38\)\( T_{3}^{123} + \)\(13\!\cdots\!04\)\( T_{3}^{122} - \)\(14\!\cdots\!09\)\( T_{3}^{121} + \)\(51\!\cdots\!01\)\( T_{3}^{120} - \)\(49\!\cdots\!15\)\( T_{3}^{119} + \)\(18\!\cdots\!79\)\( T_{3}^{118} - \)\(16\!\cdots\!17\)\( T_{3}^{117} + \)\(66\!\cdots\!88\)\( T_{3}^{116} - \)\(52\!\cdots\!02\)\( T_{3}^{115} + \)\(22\!\cdots\!84\)\( T_{3}^{114} - \)\(15\!\cdots\!98\)\( T_{3}^{113} + \)\(73\!\cdots\!01\)\( T_{3}^{112} - \)\(46\!\cdots\!41\)\( T_{3}^{111} + \)\(22\!\cdots\!37\)\( T_{3}^{110} - \)\(12\!\cdots\!06\)\( T_{3}^{109} + \)\(68\!\cdots\!05\)\( T_{3}^{108} - \)\(33\!\cdots\!13\)\( T_{3}^{107} + \)\(19\!\cdots\!31\)\( T_{3}^{106} - \)\(85\!\cdots\!75\)\( T_{3}^{105} + \)\(55\!\cdots\!40\)\( T_{3}^{104} - \)\(20\!\cdots\!23\)\( T_{3}^{103} + \)\(14\!\cdots\!53\)\( T_{3}^{102} - \)\(46\!\cdots\!59\)\( T_{3}^{101} + \)\(37\!\cdots\!37\)\( T_{3}^{100} - \)\(10\!\cdots\!86\)\( T_{3}^{99} + \)\(91\!\cdots\!85\)\( T_{3}^{98} - \)\(21\!\cdots\!34\)\( T_{3}^{97} + \)\(21\!\cdots\!51\)\( T_{3}^{96} - \)\(41\!\cdots\!75\)\( T_{3}^{95} + \)\(47\!\cdots\!37\)\( T_{3}^{94} - \)\(78\!\cdots\!61\)\( T_{3}^{93} + \)\(10\!\cdots\!63\)\( T_{3}^{92} - \)\(14\!\cdots\!64\)\( T_{3}^{91} + \)\(20\!\cdots\!69\)\( T_{3}^{90} - \)\(24\!\cdots\!92\)\( T_{3}^{89} + \)\(39\!\cdots\!46\)\( T_{3}^{88} - \)\(42\!\cdots\!01\)\( T_{3}^{87} + \)\(73\!\cdots\!47\)\( T_{3}^{86} - \)\(68\!\cdots\!15\)\( T_{3}^{85} + \)\(12\!\cdots\!96\)\( T_{3}^{84} - \)\(10\!\cdots\!66\)\( T_{3}^{83} + \)\(21\!\cdots\!29\)\( T_{3}^{82} - \)\(16\!\cdots\!98\)\( T_{3}^{81} + \)\(34\!\cdots\!23\)\( T_{3}^{80} - \)\(24\!\cdots\!16\)\( T_{3}^{79} + \)\(50\!\cdots\!31\)\( T_{3}^{78} - \)\(33\!\cdots\!45\)\( T_{3}^{77} + \)\(72\!\cdots\!83\)\( T_{3}^{76} - \)\(41\!\cdots\!38\)\( T_{3}^{75} + \)\(96\!\cdots\!11\)\( T_{3}^{74} - \)\(45\!\cdots\!55\)\( T_{3}^{73} + \)\(12\!\cdots\!35\)\( T_{3}^{72} - \)\(38\!\cdots\!65\)\( T_{3}^{71} + \)\(14\!\cdots\!96\)\( T_{3}^{70} - \)\(12\!\cdots\!42\)\( T_{3}^{69} + \)\(16\!\cdots\!24\)\( T_{3}^{68} + \)\(36\!\cdots\!18\)\( T_{3}^{67} + \)\(16\!\cdots\!55\)\( T_{3}^{66} + \)\(11\!\cdots\!35\)\( T_{3}^{65} + \)\(16\!\cdots\!85\)\( T_{3}^{64} + \)\(19\!\cdots\!74\)\( T_{3}^{63} + \)\(14\!\cdots\!53\)\( T_{3}^{62} + \)\(28\!\cdots\!75\)\( T_{3}^{61} + \)\(12\!\cdots\!50\)\( T_{3}^{60} + \)\(35\!\cdots\!81\)\( T_{3}^{59} + \)\(99\!\cdots\!95\)\( T_{3}^{58} + \)\(37\!\cdots\!32\)\( T_{3}^{57} + \)\(74\!\cdots\!60\)\( T_{3}^{56} + \)\(35\!\cdots\!20\)\( T_{3}^{55} + \)\(51\!\cdots\!92\)\( T_{3}^{54} + \)\(30\!\cdots\!61\)\( T_{3}^{53} + \)\(34\!\cdots\!11\)\( T_{3}^{52} + \)\(22\!\cdots\!86\)\( T_{3}^{51} + \)\(22\!\cdots\!51\)\( T_{3}^{50} + \)\(15\!\cdots\!00\)\( T_{3}^{49} + \)\(13\!\cdots\!13\)\( T_{3}^{48} + \)\(97\!\cdots\!40\)\( T_{3}^{47} + \)\(78\!\cdots\!01\)\( T_{3}^{46} + \)\(55\!\cdots\!53\)\( T_{3}^{45} + \)\(42\!\cdots\!18\)\( T_{3}^{44} + \)\(29\!\cdots\!43\)\( T_{3}^{43} + \)\(20\!\cdots\!89\)\( T_{3}^{42} + \)\(14\!\cdots\!56\)\( T_{3}^{41} + \)\(96\!\cdots\!19\)\( T_{3}^{40} + \)\(62\!\cdots\!75\)\( T_{3}^{39} + \)\(39\!\cdots\!12\)\( T_{3}^{38} + \)\(24\!\cdots\!71\)\( T_{3}^{37} + \)\(14\!\cdots\!51\)\( T_{3}^{36} + \)\(85\!\cdots\!82\)\( T_{3}^{35} + \)\(48\!\cdots\!49\)\( T_{3}^{34} + \)\(25\!\cdots\!53\)\( T_{3}^{33} + \)\(13\!\cdots\!99\)\( T_{3}^{32} + \)\(65\!\cdots\!93\)\( T_{3}^{31} + \)\(30\!\cdots\!85\)\( T_{3}^{30} + \)\(13\!\cdots\!84\)\( T_{3}^{29} + \)\(54\!\cdots\!79\)\( T_{3}^{28} + \)\(20\!\cdots\!39\)\( T_{3}^{27} + \)\(76\!\cdots\!02\)\( T_{3}^{26} + \)\(27\!\cdots\!39\)\( T_{3}^{25} + \)\(93\!\cdots\!75\)\( T_{3}^{24} + \)\(30\!\cdots\!30\)\( T_{3}^{23} + \)\(95\!\cdots\!61\)\( T_{3}^{22} + \)\(27\!\cdots\!73\)\( T_{3}^{21} + \)\(69\!\cdots\!27\)\( T_{3}^{20} + \)\(15\!\cdots\!74\)\( T_{3}^{19} + \)\(29\!\cdots\!37\)\( T_{3}^{18} + \)\(50\!\cdots\!51\)\( T_{3}^{17} + \)\(73\!\cdots\!05\)\( T_{3}^{16} + \)\(96\!\cdots\!15\)\( T_{3}^{15} + \)\(12\!\cdots\!73\)\( T_{3}^{14} + \)\(17\!\cdots\!06\)\( T_{3}^{13} + \)\(24\!\cdots\!05\)\( T_{3}^{12} + \)\(32\!\cdots\!29\)\( T_{3}^{11} + \)\(36\!\cdots\!26\)\( T_{3}^{10} + \)\(29\!\cdots\!77\)\( T_{3}^{9} + \)\(23\!\cdots\!47\)\( T_{3}^{8} + \)\(16\!\cdots\!29\)\( T_{3}^{7} + \)\(28\!\cdots\!45\)\( T_{3}^{6} + \)\(22\!\cdots\!24\)\( T_{3}^{5} + \)\(24\!\cdots\!26\)\( T_{3}^{4} - \)\(11\!\cdots\!48\)\( T_{3}^{3} + \)\(62\!\cdots\!27\)\( T_{3}^{2} + \)\(49\!\cdots\!74\)\( T_{3} + \)\(10\!\cdots\!41\)\( \)">\(T_{3}^{200} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(242, [\chi])\).