Newform orbit 950.2.r.b

• 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: $4$

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.

\(\operatorname{Tr}(f)(q) = \) \( 200q + 25q^{2} + 25q^{4} + q^{5} + 28q^{7} - 50q^{8} + 13q^{9} + O(q^{10}) \)

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

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 \(S_{2}^{\mathrm{new}}(950, [\chi])\).