Properties

Label 950.2.r.b
Level $950$
Weight $2$
Character orbit 950.r
Analytic conductor $7.586$
Analytic rank $0$
Dimension $200$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.r (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(200\)
Relative dimension: \(25\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 200q + 25q^{2} + 25q^{4} + q^{5} + 28q^{7} - 50q^{8} + 13q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 200q + 25q^{2} + 25q^{4} + q^{5} + 28q^{7} - 50q^{8} + 13q^{9} - 4q^{10} + 14q^{11} + 8q^{13} + q^{14} - 4q^{15} + 25q^{16} - 5q^{17} + 184q^{18} + 24q^{19} - 2q^{20} + 32q^{21} + 8q^{22} + 11q^{23} - 23q^{25} + 64q^{26} - 36q^{27} + 6q^{28} + 12q^{29} + 8q^{30} - 12q^{31} - 100q^{32} + 12q^{33} + 2q^{35} + 13q^{36} + 8q^{37} + 2q^{38} - 52q^{39} + q^{40} + 14q^{41} + 32q^{42} - 10q^{43} + 8q^{44} - 52q^{45} + 8q^{46} - 22q^{47} + 204q^{49} - 34q^{50} - 10q^{51} + 8q^{52} - 18q^{53} - 12q^{54} + 10q^{55} - 12q^{56} + 34q^{57} - 24q^{58} - 2q^{59} - 14q^{60} + 16q^{61} - 4q^{62} - 21q^{63} - 50q^{64} + 96q^{65} - 18q^{66} - 54q^{67} - 20q^{68} - 20q^{69} - 3q^{70} - 5q^{71} + 33q^{72} - 16q^{73} - 4q^{74} + 56q^{75} + 14q^{76} - 116q^{77} - 34q^{78} + q^{80} + 57q^{81} - 36q^{82} - 72q^{83} + 76q^{84} - 5q^{85} - 112q^{87} + 14q^{88} - 4q^{89} - 49q^{90} - 64q^{91} - 4q^{92} - 124q^{93} + 44q^{94} + 41q^{95} + 13q^{97} + 43q^{98} - 28q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −0.104528 0.994522i −1.98778 2.20765i −0.978148 + 0.207912i −1.48151 + 1.67485i −1.98778 + 2.20765i 5.05644 0.309017 + 0.951057i −0.608880 + 5.79310i 1.82053 + 1.29833i
11.2 −0.104528 0.994522i −1.96943 2.18727i −0.978148 + 0.207912i 2.20352 + 0.380149i −1.96943 + 2.18727i 1.45173 0.309017 + 0.951057i −0.591926 + 5.63180i 0.147736 2.23118i
11.3 −0.104528 0.994522i −1.82439 2.02619i −0.978148 + 0.207912i −2.20812 0.352454i −1.82439 + 2.02619i −1.52448 0.309017 + 0.951057i −0.463466 + 4.40959i −0.119712 + 2.23286i
11.4 −0.104528 0.994522i −1.56704 1.74038i −0.978148 + 0.207912i −1.01249 + 1.99370i −1.56704 + 1.74038i −2.70259 0.309017 + 0.951057i −0.259705 + 2.47093i 2.08862 + 0.798548i
11.5 −0.104528 0.994522i −1.54095 1.71140i −0.978148 + 0.207912i 1.52065 1.63939i −1.54095 + 1.71140i 0.0891276 0.309017 + 0.951057i −0.240774 + 2.29081i −1.78936 1.34096i
11.6 −0.104528 0.994522i −1.20234 1.33534i −0.978148 + 0.207912i −0.901641 2.04623i −1.20234 + 1.33534i 1.04913 0.309017 + 0.951057i −0.0239117 + 0.227505i −1.94077 + 1.11059i
11.7 −0.104528 0.994522i −1.08024 1.19973i −0.978148 + 0.207912i 0.757710 + 2.10378i −1.08024 + 1.19973i 2.77680 0.309017 + 0.951057i 0.0411561 0.391575i 2.01305 0.973464i
11.8 −0.104528 0.994522i −1.00310 1.11405i −0.978148 + 0.207912i 2.10151 + 0.763969i −1.00310 + 1.11405i −3.65812 0.309017 + 0.951057i 0.0786767 0.748558i 0.540117 2.16986i
11.9 −0.104528 0.994522i −0.809690 0.899251i −0.978148 + 0.207912i −1.75482 1.38587i −0.809690 + 0.899251i 2.11907 0.309017 + 0.951057i 0.160530 1.52734i −1.19485 + 1.89006i
11.10 −0.104528 0.994522i −0.475190 0.527752i −0.978148 + 0.207912i 1.37456 1.76369i −0.475190 + 0.527752i −4.63378 0.309017 + 0.951057i 0.260869 2.48200i −1.89771 1.18267i
11.11 −0.104528 0.994522i −0.358798 0.398486i −0.978148 + 0.207912i −0.0794850 + 2.23465i −0.358798 + 0.398486i −0.651916 0.309017 + 0.951057i 0.283531 2.69761i 2.23072 0.154535i
11.12 −0.104528 0.994522i −0.302553 0.336019i −0.978148 + 0.207912i 0.565124 2.16348i −0.302553 + 0.336019i 3.20863 0.309017 + 0.951057i 0.292215 2.78024i −2.21070 0.335883i
11.13 −0.104528 0.994522i 0.220840 + 0.245267i −0.978148 + 0.207912i 2.16400 0.563111i 0.220840 0.245267i 4.03900 0.309017 + 0.951057i 0.302200 2.87524i −0.786226 2.09329i
11.14 −0.104528 0.994522i 0.343520 + 0.381518i −0.978148 + 0.207912i 1.76429 + 1.37378i 0.343520 0.381518i −3.26090 0.309017 + 0.951057i 0.286036 2.72145i 1.18184 1.89822i
11.15 −0.104528 0.994522i 0.370227 + 0.411179i −0.978148 + 0.207912i −1.78679 + 1.34439i 0.370227 0.411179i −1.46528 0.309017 + 0.951057i 0.281585 2.67911i 1.52379 + 1.63648i
11.16 −0.104528 0.994522i 0.459328 + 0.510136i −0.978148 + 0.207912i −2.23563 0.0442828i 0.459328 0.510136i 4.99570 0.309017 + 0.951057i 0.264329 2.51493i 0.189647 + 2.22801i
11.17 −0.104528 0.994522i 0.525231 + 0.583328i −0.978148 + 0.207912i −1.47087 1.68420i 0.525231 0.583328i −3.06829 0.309017 + 0.951057i 0.249181 2.37080i −1.52123 + 1.63886i
11.18 −0.104528 0.994522i 0.918911 + 1.02055i −0.978148 + 0.207912i −1.88261 + 1.20655i 0.918911 1.02055i −0.500112 0.309017 + 0.951057i 0.116452 1.10797i 1.39673 + 1.74618i
11.19 −0.104528 0.994522i 0.942721 + 1.04700i −0.978148 + 0.207912i −0.192905 2.22773i 0.942721 1.04700i −0.469500 0.309017 + 0.951057i 0.106104 1.00951i −2.19536 + 0.424709i
11.20 −0.104528 0.994522i 1.04186 + 1.15710i −0.978148 + 0.207912i 0.550620 + 2.16721i 1.04186 1.15710i 3.43419 0.309017 + 0.951057i 0.0601717 0.572496i 2.09779 0.774140i
See next 80 embeddings (of 200 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.25
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner
25.d even 5 1 inner
475.r even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.r.b 200
19.c even 3 1 inner 950.2.r.b 200
25.d even 5 1 inner 950.2.r.b 200
475.r even 15 1 inner 950.2.r.b 200
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.r.b 200 1.a even 1 1 trivial
950.2.r.b 200 19.c even 3 1 inner
950.2.r.b 200 25.d even 5 1 inner
950.2.r.b 200 475.r even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(20\!\cdots\!60\)\( T_{3}^{178} - \)\(10\!\cdots\!35\)\( T_{3}^{177} - \)\(59\!\cdots\!97\)\( T_{3}^{176} + \)\(42\!\cdots\!19\)\( T_{3}^{175} + \)\(58\!\cdots\!97\)\( T_{3}^{174} - \)\(41\!\cdots\!52\)\( T_{3}^{173} - \)\(18\!\cdots\!97\)\( T_{3}^{172} + \)\(11\!\cdots\!64\)\( T_{3}^{171} - \)\(26\!\cdots\!65\)\( T_{3}^{170} + \)\(17\!\cdots\!07\)\( T_{3}^{169} + \)\(47\!\cdots\!26\)\( T_{3}^{168} - \)\(25\!\cdots\!02\)\( T_{3}^{167} - \)\(34\!\cdots\!16\)\( T_{3}^{166} + \)\(13\!\cdots\!02\)\( T_{3}^{165} + \)\(43\!\cdots\!67\)\( T_{3}^{164} + \)\(22\!\cdots\!53\)\( T_{3}^{163} + \)\(18\!\cdots\!67\)\( T_{3}^{162} - \)\(89\!\cdots\!77\)\( T_{3}^{161} - \)\(22\!\cdots\!56\)\( T_{3}^{160} + \)\(56\!\cdots\!71\)\( T_{3}^{159} + \)\(11\!\cdots\!54\)\( T_{3}^{158} + \)\(75\!\cdots\!85\)\( T_{3}^{157} + \)\(12\!\cdots\!43\)\( T_{3}^{156} - \)\(34\!\cdots\!72\)\( T_{3}^{155} - \)\(75\!\cdots\!63\)\( T_{3}^{154} + \)\(16\!\cdots\!42\)\( T_{3}^{153} + \)\(63\!\cdots\!56\)\( T_{3}^{152} + \)\(91\!\cdots\!17\)\( T_{3}^{151} - \)\(21\!\cdots\!95\)\( T_{3}^{150} - \)\(17\!\cdots\!40\)\( T_{3}^{149} - \)\(10\!\cdots\!01\)\( T_{3}^{148} + \)\(88\!\cdots\!96\)\( T_{3}^{147} + \)\(18\!\cdots\!41\)\( T_{3}^{146} + \)\(22\!\cdots\!33\)\( T_{3}^{145} - \)\(10\!\cdots\!79\)\( T_{3}^{144} - \)\(67\!\cdots\!56\)\( T_{3}^{143} + \)\(14\!\cdots\!57\)\( T_{3}^{142} + \)\(46\!\cdots\!30\)\( T_{3}^{141} + \)\(27\!\cdots\!55\)\( T_{3}^{140} - \)\(77\!\cdots\!16\)\( T_{3}^{139} - \)\(25\!\cdots\!29\)\( T_{3}^{138} - \)\(13\!\cdots\!98\)\( T_{3}^{137} + \)\(96\!\cdots\!04\)\( T_{3}^{136} + \)\(13\!\cdots\!15\)\( T_{3}^{135} + \)\(11\!\cdots\!40\)\( T_{3}^{134} - \)\(51\!\cdots\!62\)\( T_{3}^{133} - \)\(35\!\cdots\!88\)\( T_{3}^{132} - \)\(64\!\cdots\!73\)\( T_{3}^{131} + \)\(20\!\cdots\!90\)\( T_{3}^{130} + \)\(20\!\cdots\!42\)\( T_{3}^{129} - \)\(39\!\cdots\!63\)\( T_{3}^{128} - \)\(11\!\cdots\!98\)\( T_{3}^{127} - \)\(24\!\cdots\!17\)\( T_{3}^{126} + \)\(24\!\cdots\!98\)\( T_{3}^{125} + \)\(24\!\cdots\!86\)\( T_{3}^{124} + \)\(12\!\cdots\!18\)\( T_{3}^{123} - \)\(89\!\cdots\!47\)\( T_{3}^{122} - \)\(13\!\cdots\!54\)\( T_{3}^{121} + \)\(25\!\cdots\!89\)\( T_{3}^{120} + \)\(52\!\cdots\!45\)\( T_{3}^{119} + \)\(15\!\cdots\!16\)\( T_{3}^{118} - \)\(42\!\cdots\!34\)\( T_{3}^{117} - \)\(84\!\cdots\!19\)\( T_{3}^{116} - \)\(66\!\cdots\!92\)\( T_{3}^{115} + \)\(18\!\cdots\!26\)\( T_{3}^{114} + \)\(41\!\cdots\!14\)\( T_{3}^{113} + \)\(26\!\cdots\!37\)\( T_{3}^{112} - \)\(10\!\cdots\!40\)\( T_{3}^{111} - \)\(36\!\cdots\!36\)\( T_{3}^{110} + \)\(29\!\cdots\!49\)\( T_{3}^{109} + \)\(13\!\cdots\!75\)\( T_{3}^{108} + \)\(11\!\cdots\!85\)\( T_{3}^{107} - \)\(20\!\cdots\!98\)\( T_{3}^{106} - \)\(48\!\cdots\!70\)\( T_{3}^{105} - \)\(40\!\cdots\!67\)\( T_{3}^{104} + \)\(99\!\cdots\!86\)\( T_{3}^{103} + \)\(33\!\cdots\!70\)\( T_{3}^{102} - \)\(30\!\cdots\!26\)\( T_{3}^{101} - \)\(99\!\cdots\!57\)\( T_{3}^{100} - \)\(55\!\cdots\!09\)\( T_{3}^{99} + \)\(13\!\cdots\!30\)\( T_{3}^{98} + \)\(21\!\cdots\!96\)\( T_{3}^{97} + \)\(18\!\cdots\!14\)\( T_{3}^{96} - \)\(40\!\cdots\!74\)\( T_{3}^{95} - \)\(14\!\cdots\!52\)\( T_{3}^{94} + \)\(24\!\cdots\!26\)\( T_{3}^{93} + \)\(36\!\cdots\!13\)\( T_{3}^{92} + \)\(10\!\cdots\!82\)\( T_{3}^{91} - \)\(41\!\cdots\!91\)\( T_{3}^{90} - \)\(37\!\cdots\!31\)\( T_{3}^{89} - \)\(53\!\cdots\!11\)\( T_{3}^{88} + \)\(67\!\cdots\!26\)\( T_{3}^{87} + \)\(35\!\cdots\!93\)\( T_{3}^{86} - \)\(56\!\cdots\!42\)\( T_{3}^{85} - \)\(81\!\cdots\!71\)\( T_{3}^{84} - \)\(38\!\cdots\!30\)\( T_{3}^{83} + \)\(93\!\cdots\!07\)\( T_{3}^{82} + \)\(18\!\cdots\!39\)\( T_{3}^{81} + \)\(34\!\cdots\!38\)\( T_{3}^{80} - \)\(26\!\cdots\!80\)\( T_{3}^{79} - \)\(37\!\cdots\!69\)\( T_{3}^{78} + \)\(19\!\cdots\!28\)\( T_{3}^{77} + \)\(77\!\cdots\!78\)\( T_{3}^{76} - \)\(16\!\cdots\!44\)\( T_{3}^{75} - \)\(75\!\cdots\!02\)\( T_{3}^{74} + \)\(48\!\cdots\!84\)\( T_{3}^{73} - \)\(24\!\cdots\!90\)\( T_{3}^{72} - \)\(14\!\cdots\!11\)\( T_{3}^{71} + \)\(21\!\cdots\!64\)\( T_{3}^{70} + \)\(26\!\cdots\!86\)\( T_{3}^{69} - \)\(38\!\cdots\!34\)\( T_{3}^{68} - \)\(16\!\cdots\!12\)\( T_{3}^{67} + \)\(33\!\cdots\!91\)\( T_{3}^{66} - \)\(68\!\cdots\!59\)\( T_{3}^{65} - \)\(14\!\cdots\!05\)\( T_{3}^{64} + \)\(21\!\cdots\!03\)\( T_{3}^{63} - \)\(34\!\cdots\!90\)\( T_{3}^{62} - \)\(28\!\cdots\!50\)\( T_{3}^{61} + \)\(46\!\cdots\!26\)\( T_{3}^{60} + \)\(16\!\cdots\!55\)\( T_{3}^{59} - \)\(30\!\cdots\!09\)\( T_{3}^{58} + \)\(40\!\cdots\!67\)\( T_{3}^{57} + \)\(56\!\cdots\!37\)\( T_{3}^{56} - \)\(14\!\cdots\!73\)\( T_{3}^{55} + \)\(10\!\cdots\!07\)\( T_{3}^{54} + \)\(12\!\cdots\!69\)\( T_{3}^{53} - \)\(12\!\cdots\!24\)\( T_{3}^{52} - \)\(64\!\cdots\!73\)\( T_{3}^{51} + \)\(61\!\cdots\!87\)\( T_{3}^{50} + \)\(17\!\cdots\!85\)\( T_{3}^{49} - \)\(55\!\cdots\!54\)\( T_{3}^{48} + \)\(41\!\cdots\!70\)\( T_{3}^{47} - \)\(16\!\cdots\!55\)\( T_{3}^{46} - \)\(74\!\cdots\!25\)\( T_{3}^{45} + \)\(16\!\cdots\!40\)\( T_{3}^{44} + \)\(42\!\cdots\!80\)\( T_{3}^{43} - \)\(89\!\cdots\!15\)\( T_{3}^{42} - \)\(15\!\cdots\!55\)\( T_{3}^{41} + \)\(32\!\cdots\!60\)\( T_{3}^{40} + \)\(36\!\cdots\!00\)\( T_{3}^{39} - \)\(65\!\cdots\!75\)\( T_{3}^{38} - \)\(42\!\cdots\!75\)\( T_{3}^{37} - \)\(19\!\cdots\!75\)\( T_{3}^{36} + \)\(49\!\cdots\!75\)\( T_{3}^{35} + \)\(71\!\cdots\!25\)\( T_{3}^{34} - \)\(17\!\cdots\!50\)\( T_{3}^{33} - \)\(30\!\cdots\!50\)\( T_{3}^{32} + \)\(20\!\cdots\!25\)\( T_{3}^{31} + \)\(71\!\cdots\!00\)\( T_{3}^{30} - \)\(97\!\cdots\!00\)\( T_{3}^{29} - \)\(57\!\cdots\!25\)\( T_{3}^{28} + \)\(34\!\cdots\!25\)\( T_{3}^{27} - \)\(25\!\cdots\!25\)\( T_{3}^{26} - \)\(84\!\cdots\!50\)\( T_{3}^{25} + \)\(12\!\cdots\!50\)\( T_{3}^{24} + \)\(12\!\cdots\!50\)\( T_{3}^{23} - \)\(28\!\cdots\!75\)\( T_{3}^{22} - \)\(60\!\cdots\!25\)\( T_{3}^{21} + \)\(39\!\cdots\!00\)\( T_{3}^{20} - \)\(19\!\cdots\!00\)\( T_{3}^{19} - \)\(78\!\cdots\!75\)\( T_{3}^{18} + \)\(13\!\cdots\!50\)\( T_{3}^{17} - \)\(86\!\cdots\!50\)\( T_{3}^{16} - \)\(36\!\cdots\!75\)\( T_{3}^{15} + \)\(16\!\cdots\!25\)\( T_{3}^{14} + \)\(44\!\cdots\!75\)\( T_{3}^{13} - \)\(14\!\cdots\!25\)\( T_{3}^{12} - \)\(14\!\cdots\!00\)\( T_{3}^{11} + \)\(28\!\cdots\!00\)\( T_{3}^{10} + \)\(28\!\cdots\!00\)\( T_{3}^{9} + \)\(19\!\cdots\!25\)\( T_{3}^{8} + \)\(59\!\cdots\!75\)\( T_{3}^{7} - \)\(23\!\cdots\!75\)\( T_{3}^{6} + \)\(33\!\cdots\!25\)\( T_{3}^{5} + \)\(14\!\cdots\!25\)\( T_{3}^{4} + \)\(10\!\cdots\!75\)\( T_{3}^{3} + \)\(47\!\cdots\!50\)\( T_{3}^{2} + \)\(21\!\cdots\!25\)\( T_{3} + \)\(57\!\cdots\!25\)\( \)">\(T_{3}^{200} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\).