Properties

Label 950.2.r.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 200q - 25q^{2} + 25q^{4} + q^{5} - 36q^{7} + 50q^{8} + 37q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 200q - 25q^{2} + 25q^{4} + q^{5} - 36q^{7} + 50q^{8} + 37q^{9} + 4q^{10} - 2q^{11} + 8q^{13} + 7q^{14} + 10q^{15} + 25q^{16} - 13q^{17} - 216q^{18} - 36q^{19} - 2q^{20} + 4q^{22} - 13q^{23} + 33q^{25} - 64q^{26} - 2q^{28} + 12q^{29} - 20q^{30} + 12q^{31} + 100q^{32} - 12q^{33} + 8q^{34} - 22q^{35} + 37q^{36} + 8q^{37} + 18q^{38} + 28q^{39} - q^{40} + 2q^{41} + 70q^{43} - 4q^{44} + 60q^{45} + 24q^{46} - 22q^{47} + 156q^{49} + 26q^{50} - 42q^{51} + 8q^{52} - 18q^{53} - 10q^{55} - 4q^{56} + 74q^{57} + 24q^{58} + 50q^{59} + 12q^{61} - 4q^{62} - 45q^{63} - 50q^{64} + 120q^{65} + 2q^{66} - 18q^{67} - 84q^{68} + 4q^{69} - 13q^{70} + 19q^{71} - 17q^{72} - 24q^{73} + 4q^{74} - 136q^{75} - 6q^{76} - 12q^{77} - 26q^{78} - 8q^{79} + q^{80} - 63q^{81} + 28q^{82} + 16q^{83} - 20q^{84} - 5q^{85} + 20q^{86} - 64q^{87} + 2q^{88} + 16q^{89} - 65q^{90} + 72q^{91} + 12q^{92} - 8q^{93} - 44q^{94} - 73q^{95} - 11q^{97} - 27q^{98} + 16q^{99} + O(q^{100}) \)

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 −2.23670 2.48410i −0.978148 + 0.207912i 2.20697 0.359579i 2.23670 2.48410i −4.60253 −0.309017 0.951057i −0.854373 + 8.12881i 0.588300 + 2.15729i
11.2 0.104528 + 0.994522i −2.02768 2.25196i −0.978148 + 0.207912i −0.209287 2.22625i 2.02768 2.25196i 1.71696 −0.309017 0.951057i −0.646281 + 6.14895i 2.19218 0.440848i
11.3 0.104528 + 0.994522i −1.82987 2.03228i −0.978148 + 0.207912i −1.78813 + 1.34260i 1.82987 2.03228i 1.23976 −0.309017 0.951057i −0.468141 + 4.45407i −1.52216 1.63800i
11.4 0.104528 + 0.994522i −1.76134 1.95617i −0.978148 + 0.207912i −1.69383 + 1.45977i 1.76134 1.95617i −4.49296 −0.309017 0.951057i −0.410684 + 3.90739i −1.62883 1.53197i
11.5 0.104528 + 0.994522i −1.74362 1.93649i −0.978148 + 0.207912i −1.31558 1.80811i 1.74362 1.93649i −0.103099 −0.309017 0.951057i −0.396182 + 3.76942i 1.66069 1.49737i
11.6 0.104528 + 0.994522i −1.36974 1.52125i −0.978148 + 0.207912i 2.23379 0.100934i 1.36974 1.52125i 5.15541 −0.309017 0.951057i −0.124432 + 1.18389i 0.333876 + 2.21100i
11.7 0.104528 + 0.994522i −1.32901 1.47601i −0.978148 + 0.207912i 1.09596 + 1.94907i 1.32901 1.47601i −0.0211674 −0.309017 0.951057i −0.0987671 + 0.939706i −1.82383 + 1.29369i
11.8 0.104528 + 0.994522i −0.955473 1.06116i −0.978148 + 0.207912i 0.663784 2.13527i 0.955473 1.06116i −1.22437 −0.309017 0.951057i 0.100453 0.955746i 2.19296 + 0.436951i
11.9 0.104528 + 0.994522i −0.588897 0.654036i −0.978148 + 0.207912i −2.19070 0.448160i 0.588897 0.654036i −3.02472 −0.309017 0.951057i 0.232622 2.21325i 0.216715 2.22554i
11.10 0.104528 + 0.994522i −0.286801 0.318525i −0.978148 + 0.207912i −1.03010 + 1.98466i 0.286801 0.318525i 3.29125 −0.309017 0.951057i 0.294382 2.80086i −2.08147 0.817006i
11.11 0.104528 + 0.994522i −0.278343 0.309131i −0.978148 + 0.207912i −1.78815 1.34258i 0.278343 0.309131i 2.38873 −0.309017 0.951057i 0.295498 2.81148i 1.14831 1.91869i
11.12 0.104528 + 0.994522i −0.261407 0.290322i −0.978148 + 0.207912i 1.28550 + 1.82962i 0.261407 0.290322i −0.691099 −0.309017 0.951057i 0.297632 2.83178i −1.68522 + 1.46970i
11.13 0.104528 + 0.994522i −0.175744 0.195183i −0.978148 + 0.207912i 1.83434 1.27875i 0.175744 0.195183i −2.84995 −0.309017 0.951057i 0.306375 2.91496i 1.46349 + 1.69062i
11.14 0.104528 + 0.994522i −0.0896612 0.0995788i −0.978148 + 0.207912i −1.17911 + 1.89992i 0.0896612 0.0995788i 2.27794 −0.309017 0.951057i 0.311709 2.96571i −2.01277 0.974050i
11.15 0.104528 + 0.994522i 0.388315 + 0.431268i −0.978148 + 0.207912i 1.69879 + 1.45400i −0.388315 + 0.431268i −3.61865 −0.309017 0.951057i 0.278382 2.64863i −1.26846 + 1.84147i
11.16 0.104528 + 0.994522i 0.831705 + 0.923702i −0.978148 + 0.207912i 2.14903 + 0.617800i −0.831705 + 0.923702i 0.754763 −0.309017 0.951057i 0.152093 1.44707i −0.389781 + 2.20183i
11.17 0.104528 + 0.994522i 0.856524 + 0.951266i −0.978148 + 0.207912i −1.58933 1.57290i −0.856524 + 0.951266i 1.37782 −0.309017 0.951057i 0.142311 1.35400i 1.39816 1.74504i
11.18 0.104528 + 0.994522i 0.895592 + 0.994656i −0.978148 + 0.207912i −2.22420 + 0.230050i −0.895592 + 0.994656i −2.57007 −0.309017 0.951057i 0.126330 1.20195i −0.461283 2.18797i
11.19 0.104528 + 0.994522i 1.14496 + 1.27160i −0.978148 + 0.207912i 1.79074 1.33913i −1.14496 + 1.27160i 1.05033 −0.309017 0.951057i 0.00753716 0.0717113i 1.51897 + 1.64095i
11.20 0.104528 + 0.994522i 1.32430 + 1.47079i −0.978148 + 0.207912i 0.202278 2.22690i −1.32430 + 1.47079i −4.37854 −0.309017 0.951057i −0.0958531 + 0.911982i 2.23584 0.0316046i
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.a 200
19.c even 3 1 inner 950.2.r.a 200
25.d even 5 1 inner 950.2.r.a 200
475.r even 15 1 inner 950.2.r.a 200
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.r.a 200 1.a even 1 1 trivial
950.2.r.a 200 19.c even 3 1 inner
950.2.r.a 200 25.d even 5 1 inner
950.2.r.a 200 475.r even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(11\!\cdots\!72\)\( T_{3}^{178} - \)\(12\!\cdots\!85\)\( T_{3}^{177} + \)\(58\!\cdots\!71\)\( T_{3}^{176} + \)\(26\!\cdots\!45\)\( T_{3}^{175} + \)\(33\!\cdots\!23\)\( T_{3}^{174} - \)\(16\!\cdots\!00\)\( T_{3}^{173} - \)\(10\!\cdots\!09\)\( T_{3}^{172} - \)\(94\!\cdots\!30\)\( T_{3}^{171} + \)\(10\!\cdots\!99\)\( T_{3}^{170} + \)\(26\!\cdots\!35\)\( T_{3}^{169} - \)\(40\!\cdots\!22\)\( T_{3}^{168} - \)\(16\!\cdots\!80\)\( T_{3}^{167} - \)\(29\!\cdots\!62\)\( T_{3}^{166} - \)\(12\!\cdots\!20\)\( T_{3}^{165} + \)\(63\!\cdots\!95\)\( T_{3}^{164} + \)\(31\!\cdots\!45\)\( T_{3}^{163} - \)\(51\!\cdots\!43\)\( T_{3}^{162} - \)\(25\!\cdots\!25\)\( T_{3}^{161} + \)\(15\!\cdots\!18\)\( T_{3}^{160} + \)\(37\!\cdots\!95\)\( T_{3}^{159} + \)\(14\!\cdots\!04\)\( T_{3}^{158} + \)\(15\!\cdots\!95\)\( T_{3}^{157} - \)\(23\!\cdots\!99\)\( T_{3}^{156} - \)\(20\!\cdots\!70\)\( T_{3}^{155} + \)\(14\!\cdots\!55\)\( T_{3}^{154} + \)\(11\!\cdots\!20\)\( T_{3}^{153} - \)\(24\!\cdots\!46\)\( T_{3}^{152} + \)\(60\!\cdots\!25\)\( T_{3}^{151} - \)\(45\!\cdots\!47\)\( T_{3}^{150} - \)\(63\!\cdots\!30\)\( T_{3}^{149} + \)\(50\!\cdots\!63\)\( T_{3}^{148} + \)\(54\!\cdots\!60\)\( T_{3}^{147} - \)\(24\!\cdots\!45\)\( T_{3}^{146} - \)\(18\!\cdots\!55\)\( T_{3}^{145} + \)\(57\!\cdots\!31\)\( T_{3}^{144} - \)\(69\!\cdots\!80\)\( T_{3}^{143} + \)\(86\!\cdots\!69\)\( T_{3}^{142} + \)\(12\!\cdots\!80\)\( T_{3}^{141} - \)\(69\!\cdots\!81\)\( T_{3}^{140} - \)\(73\!\cdots\!60\)\( T_{3}^{139} + \)\(23\!\cdots\!43\)\( T_{3}^{138} + \)\(11\!\cdots\!10\)\( T_{3}^{137} + \)\(43\!\cdots\!02\)\( T_{3}^{136} + \)\(14\!\cdots\!65\)\( T_{3}^{135} - \)\(10\!\cdots\!42\)\( T_{3}^{134} - \)\(13\!\cdots\!30\)\( T_{3}^{133} + \)\(58\!\cdots\!40\)\( T_{3}^{132} + \)\(48\!\cdots\!15\)\( T_{3}^{131} - \)\(12\!\cdots\!52\)\( T_{3}^{130} + \)\(34\!\cdots\!60\)\( T_{3}^{129} - \)\(70\!\cdots\!91\)\( T_{3}^{128} - \)\(13\!\cdots\!20\)\( T_{3}^{127} + \)\(73\!\cdots\!55\)\( T_{3}^{126} + \)\(71\!\cdots\!50\)\( T_{3}^{125} - \)\(28\!\cdots\!12\)\( T_{3}^{124} - \)\(13\!\cdots\!50\)\( T_{3}^{123} + \)\(12\!\cdots\!17\)\( T_{3}^{122} - \)\(65\!\cdots\!80\)\( T_{3}^{121} + \)\(49\!\cdots\!13\)\( T_{3}^{120} + \)\(57\!\cdots\!95\)\( T_{3}^{119} - \)\(29\!\cdots\!72\)\( T_{3}^{118} - \)\(17\!\cdots\!60\)\( T_{3}^{117} + \)\(73\!\cdots\!19\)\( T_{3}^{116} - \)\(56\!\cdots\!00\)\( T_{3}^{115} + \)\(11\!\cdots\!64\)\( T_{3}^{114} + \)\(27\!\cdots\!30\)\( T_{3}^{113} - \)\(17\!\cdots\!63\)\( T_{3}^{112} - \)\(10\!\cdots\!70\)\( T_{3}^{111} + \)\(69\!\cdots\!34\)\( T_{3}^{110} + \)\(67\!\cdots\!25\)\( T_{3}^{109} - \)\(80\!\cdots\!27\)\( T_{3}^{108} + \)\(95\!\cdots\!35\)\( T_{3}^{107} - \)\(53\!\cdots\!54\)\( T_{3}^{106} - \)\(38\!\cdots\!20\)\( T_{3}^{105} + \)\(33\!\cdots\!01\)\( T_{3}^{104} + \)\(70\!\cdots\!10\)\( T_{3}^{103} - \)\(80\!\cdots\!42\)\( T_{3}^{102} + \)\(46\!\cdots\!20\)\( T_{3}^{101} - \)\(24\!\cdots\!77\)\( T_{3}^{100} - \)\(16\!\cdots\!95\)\( T_{3}^{99} + \)\(89\!\cdots\!44\)\( T_{3}^{98} - \)\(50\!\cdots\!00\)\( T_{3}^{97} - \)\(30\!\cdots\!56\)\( T_{3}^{96} + \)\(22\!\cdots\!50\)\( T_{3}^{95} + \)\(28\!\cdots\!06\)\( T_{3}^{94} - \)\(71\!\cdots\!80\)\( T_{3}^{93} + \)\(14\!\cdots\!05\)\( T_{3}^{92} + \)\(17\!\cdots\!30\)\( T_{3}^{91} - \)\(64\!\cdots\!17\)\( T_{3}^{90} + \)\(54\!\cdots\!75\)\( T_{3}^{89} + \)\(88\!\cdots\!93\)\( T_{3}^{88} - \)\(17\!\cdots\!20\)\( T_{3}^{87} + \)\(14\!\cdots\!13\)\( T_{3}^{86} + \)\(15\!\cdots\!60\)\( T_{3}^{85} - \)\(85\!\cdots\!63\)\( T_{3}^{84} + \)\(60\!\cdots\!30\)\( T_{3}^{83} + \)\(14\!\cdots\!05\)\( T_{3}^{82} - \)\(24\!\cdots\!05\)\( T_{3}^{81} + \)\(23\!\cdots\!74\)\( T_{3}^{80} + \)\(37\!\cdots\!90\)\( T_{3}^{79} - \)\(70\!\cdots\!57\)\( T_{3}^{78} + \)\(15\!\cdots\!20\)\( T_{3}^{77} + \)\(16\!\cdots\!10\)\( T_{3}^{76} - \)\(20\!\cdots\!00\)\( T_{3}^{75} - \)\(11\!\cdots\!06\)\( T_{3}^{74} + \)\(43\!\cdots\!90\)\( T_{3}^{73} - \)\(33\!\cdots\!74\)\( T_{3}^{72} - \)\(29\!\cdots\!85\)\( T_{3}^{71} + \)\(11\!\cdots\!78\)\( T_{3}^{70} - \)\(72\!\cdots\!70\)\( T_{3}^{69} - \)\(13\!\cdots\!26\)\( T_{3}^{68} + \)\(23\!\cdots\!80\)\( T_{3}^{67} - \)\(20\!\cdots\!17\)\( T_{3}^{66} - \)\(28\!\cdots\!95\)\( T_{3}^{65} + \)\(32\!\cdots\!19\)\( T_{3}^{64} + \)\(11\!\cdots\!55\)\( T_{3}^{63} - \)\(52\!\cdots\!82\)\( T_{3}^{62} + \)\(45\!\cdots\!80\)\( T_{3}^{61} + \)\(34\!\cdots\!18\)\( T_{3}^{60} - \)\(69\!\cdots\!65\)\( T_{3}^{59} + \)\(16\!\cdots\!63\)\( T_{3}^{58} + \)\(46\!\cdots\!55\)\( T_{3}^{57} - \)\(53\!\cdots\!83\)\( T_{3}^{56} + \)\(78\!\cdots\!75\)\( T_{3}^{55} + \)\(57\!\cdots\!95\)\( T_{3}^{54} - \)\(43\!\cdots\!05\)\( T_{3}^{53} - \)\(41\!\cdots\!02\)\( T_{3}^{52} + \)\(35\!\cdots\!25\)\( T_{3}^{51} + \)\(14\!\cdots\!23\)\( T_{3}^{50} - \)\(11\!\cdots\!15\)\( T_{3}^{49} + \)\(78\!\cdots\!44\)\( T_{3}^{48} - \)\(34\!\cdots\!10\)\( T_{3}^{47} - \)\(12\!\cdots\!79\)\( T_{3}^{46} + \)\(63\!\cdots\!95\)\( T_{3}^{45} + \)\(50\!\cdots\!32\)\( T_{3}^{44} - \)\(32\!\cdots\!10\)\( T_{3}^{43} + \)\(10\!\cdots\!87\)\( T_{3}^{42} - \)\(14\!\cdots\!45\)\( T_{3}^{41} - \)\(13\!\cdots\!44\)\( T_{3}^{40} + \)\(79\!\cdots\!30\)\( T_{3}^{39} + \)\(32\!\cdots\!05\)\( T_{3}^{38} - \)\(30\!\cdots\!35\)\( T_{3}^{37} + \)\(83\!\cdots\!65\)\( T_{3}^{36} - \)\(34\!\cdots\!45\)\( T_{3}^{35} - \)\(33\!\cdots\!85\)\( T_{3}^{34} + \)\(37\!\cdots\!00\)\( T_{3}^{33} + \)\(33\!\cdots\!90\)\( T_{3}^{32} - \)\(73\!\cdots\!75\)\( T_{3}^{31} - \)\(33\!\cdots\!50\)\( T_{3}^{30} - \)\(99\!\cdots\!00\)\( T_{3}^{29} + \)\(65\!\cdots\!25\)\( T_{3}^{28} + \)\(44\!\cdots\!25\)\( T_{3}^{27} + \)\(79\!\cdots\!25\)\( T_{3}^{26} - \)\(17\!\cdots\!00\)\( T_{3}^{25} - \)\(54\!\cdots\!00\)\( T_{3}^{24} + \)\(27\!\cdots\!00\)\( T_{3}^{23} + \)\(11\!\cdots\!75\)\( T_{3}^{22} + \)\(13\!\cdots\!25\)\( T_{3}^{21} - \)\(11\!\cdots\!50\)\( T_{3}^{20} - \)\(29\!\cdots\!00\)\( T_{3}^{19} + \)\(80\!\cdots\!75\)\( T_{3}^{18} - \)\(12\!\cdots\!00\)\( T_{3}^{17} + \)\(29\!\cdots\!00\)\( T_{3}^{16} - \)\(14\!\cdots\!75\)\( T_{3}^{15} + \)\(33\!\cdots\!25\)\( T_{3}^{14} - \)\(15\!\cdots\!25\)\( T_{3}^{13} + \)\(75\!\cdots\!25\)\( T_{3}^{12} - \)\(19\!\cdots\!50\)\( T_{3}^{11} + \)\(61\!\cdots\!00\)\( T_{3}^{10} - \)\(23\!\cdots\!00\)\( T_{3}^{9} + \)\(50\!\cdots\!25\)\( T_{3}^{8} - \)\(73\!\cdots\!25\)\( T_{3}^{7} + \)\(26\!\cdots\!75\)\( T_{3}^{6} - \)\(76\!\cdots\!75\)\( T_{3}^{5} + \)\(82\!\cdots\!75\)\( T_{3}^{4} - \)\(26\!\cdots\!75\)\( T_{3}^{3} + \)\(99\!\cdots\!00\)\( T_{3}^{2} - \)\(23\!\cdots\!75\)\( T_{3} + \)\(13\!\cdots\!25\)\( \)">\(T_{3}^{200} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\).