Properties

Label 950.2.bb.e
Level $950$
Weight $2$
Character orbit 950.bb
Analytic conductor $7.586$
Analytic rank $0$
Dimension $120$
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.bb (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{36})\)
Twist minimal: no (minimal twist has level 190)
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120q + 12q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 120q + 12q^{7} - 36q^{17} - 96q^{21} - 24q^{22} - 12q^{26} + 96q^{33} - 12q^{41} + 72q^{43} + 24q^{47} + 24q^{51} - 36q^{53} - 84q^{57} + 48q^{61} + 24q^{62} - 36q^{63} - 24q^{66} + 96q^{67} + 12q^{68} + 36q^{73} + 12q^{76} - 96q^{78} + 144q^{81} - 48q^{82} - 24q^{83} + 48q^{86} - 72q^{87} + 72q^{91} - 72q^{92} - 156q^{93} - 120q^{97} - 24q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1 −0.573576 + 0.819152i −0.243772 + 2.78632i −0.342020 0.939693i 0 −2.14260 1.79785i 0.966713 + 3.60782i 0.965926 + 0.258819i −4.74974 0.837508i 0
143.2 −0.573576 + 0.819152i −0.197676 + 2.25945i −0.342020 0.939693i 0 −1.73745 1.45789i −0.925273 3.45316i 0.965926 + 0.258819i −2.11161 0.372333i 0
143.3 −0.573576 + 0.819152i 0.0801625 0.916262i −0.342020 0.939693i 0 0.704578 + 0.591211i −0.322803 1.20472i 0.965926 + 0.258819i 2.12131 + 0.374045i 0
143.4 −0.573576 + 0.819152i 0.0833140 0.952283i −0.342020 0.939693i 0 0.732278 + 0.614454i 0.790865 + 2.95155i 0.965926 + 0.258819i 2.05452 + 0.362268i 0
143.5 −0.573576 + 0.819152i 0.277971 3.17723i −0.342020 0.939693i 0 2.44319 + 2.05008i −1.05257 3.92825i 0.965926 + 0.258819i −7.06307 1.24541i 0
143.6 0.573576 0.819152i −0.255151 + 2.91639i −0.342020 0.939693i 0 2.24262 + 1.88178i −0.0607618 0.226766i −0.965926 0.258819i −5.48583 0.967300i 0
143.7 0.573576 0.819152i −0.0418747 + 0.478630i −0.342020 0.939693i 0 0.368052 + 0.308832i 0.971585 + 3.62600i −0.965926 0.258819i 2.72709 + 0.480860i 0
143.8 0.573576 0.819152i 0.0201525 0.230344i −0.342020 0.939693i 0 −0.177128 0.148628i −1.23887 4.62353i −0.965926 0.258819i 2.90177 + 0.511661i 0
143.9 0.573576 0.819152i 0.0709170 0.810585i −0.342020 0.939693i 0 −0.623316 0.523024i 0.211848 + 0.790626i −0.965926 0.258819i 2.30240 + 0.405976i 0
143.10 0.573576 0.819152i 0.205957 2.35409i −0.342020 0.939693i 0 −1.81023 1.51896i −0.0727825 0.271628i −0.965926 0.258819i −2.54492 0.448738i 0
193.1 −0.0871557 0.996195i −1.35298 + 2.90149i −0.984808 + 0.173648i 0 3.00836 + 1.09496i 1.43361 + 0.384135i 0.258819 + 0.965926i −4.65969 5.55320i 0
193.2 −0.0871557 0.996195i −0.507694 + 1.08875i −0.984808 + 0.173648i 0 1.12886 + 0.410871i −2.08332 0.558224i 0.258819 + 0.965926i 1.00073 + 1.19263i 0
193.3 −0.0871557 0.996195i −0.186197 + 0.399301i −0.984808 + 0.173648i 0 0.414010 + 0.150687i −1.97694 0.529719i 0.258819 + 0.965926i 1.80359 + 2.14944i 0
193.4 −0.0871557 0.996195i 0.684861 1.46869i −0.984808 + 0.173648i 0 −1.52279 0.554250i 3.13506 + 0.840038i 0.258819 + 0.965926i 0.240351 + 0.286439i 0
193.5 −0.0871557 0.996195i 1.36202 2.92085i −0.984808 + 0.173648i 0 −3.02844 1.10226i −1.04490 0.279979i 0.258819 + 0.965926i −4.74793 5.65836i 0
193.6 0.0871557 + 0.996195i −1.08832 + 2.33390i −0.984808 + 0.173648i 0 −2.41987 0.880761i 3.72976 + 0.999386i −0.258819 0.965926i −2.33430 2.78191i 0
193.7 0.0871557 + 0.996195i −0.602383 + 1.29181i −0.984808 + 0.173648i 0 −1.33940 0.487501i 1.39748 + 0.374454i −0.258819 0.965926i 0.622444 + 0.741800i 0
193.8 0.0871557 + 0.996195i 0.202610 0.434499i −0.984808 + 0.173648i 0 0.450504 + 0.163970i −3.80940 1.02072i −0.258819 0.965926i 1.78062 + 2.12207i 0
193.9 0.0871557 + 0.996195i 0.715886 1.53522i −0.984808 + 0.173648i 0 1.59177 + 0.579359i 4.29599 + 1.15111i −0.258819 0.965926i 0.0839459 + 0.100043i 0
193.10 0.0871557 + 0.996195i 0.772202 1.65599i −0.984808 + 0.173648i 0 1.71699 + 0.624934i −2.34531 0.628423i −0.258819 0.965926i −0.217652 0.259388i 0
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 907.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.f odd 18 1 inner
95.r even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.bb.e 120
5.b even 2 1 190.2.r.a 120
5.c odd 4 1 190.2.r.a 120
5.c odd 4 1 inner 950.2.bb.e 120
19.f odd 18 1 inner 950.2.bb.e 120
95.o odd 18 1 190.2.r.a 120
95.r even 36 1 190.2.r.a 120
95.r even 36 1 inner 950.2.bb.e 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.r.a 120 5.b even 2 1
190.2.r.a 120 5.c odd 4 1
190.2.r.a 120 95.o odd 18 1
190.2.r.a 120 95.r even 36 1
950.2.bb.e 120 1.a even 1 1 trivial
950.2.bb.e 120 5.c odd 4 1 inner
950.2.bb.e 120 19.f odd 18 1 inner
950.2.bb.e 120 95.r even 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(11\!\cdots\!84\)\( T_{3}^{95} - \)\(10\!\cdots\!80\)\( T_{3}^{94} - \)\(13\!\cdots\!32\)\( T_{3}^{93} - \)\(25\!\cdots\!00\)\( T_{3}^{92} + \)\(15\!\cdots\!16\)\( T_{3}^{91} + \)\(13\!\cdots\!32\)\( T_{3}^{90} + \)\(95\!\cdots\!00\)\( T_{3}^{89} - \)\(89\!\cdots\!34\)\( T_{3}^{88} - \)\(13\!\cdots\!28\)\( T_{3}^{87} + \)\(13\!\cdots\!12\)\( T_{3}^{86} + \)\(53\!\cdots\!80\)\( T_{3}^{85} - \)\(27\!\cdots\!52\)\( T_{3}^{84} - \)\(96\!\cdots\!32\)\( T_{3}^{83} - \)\(85\!\cdots\!56\)\( T_{3}^{82} + \)\(41\!\cdots\!28\)\( T_{3}^{81} + \)\(18\!\cdots\!40\)\( T_{3}^{80} + \)\(26\!\cdots\!92\)\( T_{3}^{79} - \)\(13\!\cdots\!36\)\( T_{3}^{78} - \)\(89\!\cdots\!32\)\( T_{3}^{77} + \)\(18\!\cdots\!44\)\( T_{3}^{76} + \)\(11\!\cdots\!16\)\( T_{3}^{75} + \)\(27\!\cdots\!00\)\( T_{3}^{74} + \)\(47\!\cdots\!64\)\( T_{3}^{73} + \)\(98\!\cdots\!48\)\( T_{3}^{72} + \)\(11\!\cdots\!04\)\( T_{3}^{71} - \)\(93\!\cdots\!80\)\( T_{3}^{70} - \)\(12\!\cdots\!20\)\( T_{3}^{69} - \)\(44\!\cdots\!96\)\( T_{3}^{68} - \)\(10\!\cdots\!28\)\( T_{3}^{67} - \)\(15\!\cdots\!96\)\( T_{3}^{66} - \)\(28\!\cdots\!48\)\( T_{3}^{65} - \)\(60\!\cdots\!06\)\( T_{3}^{64} - \)\(77\!\cdots\!76\)\( T_{3}^{63} + \)\(39\!\cdots\!64\)\( T_{3}^{62} + \)\(25\!\cdots\!52\)\( T_{3}^{61} + \)\(54\!\cdots\!12\)\( T_{3}^{60} + \)\(18\!\cdots\!56\)\( T_{3}^{59} + \)\(61\!\cdots\!24\)\( T_{3}^{58} + \)\(13\!\cdots\!08\)\( T_{3}^{57} + \)\(24\!\cdots\!85\)\( T_{3}^{56} + \)\(47\!\cdots\!84\)\( T_{3}^{55} + \)\(94\!\cdots\!24\)\( T_{3}^{54} + \)\(15\!\cdots\!24\)\( T_{3}^{53} + \)\(23\!\cdots\!20\)\( T_{3}^{52} + \)\(37\!\cdots\!20\)\( T_{3}^{51} + \)\(61\!\cdots\!84\)\( T_{3}^{50} + \)\(94\!\cdots\!16\)\( T_{3}^{49} + \)\(13\!\cdots\!50\)\( T_{3}^{48} + \)\(20\!\cdots\!00\)\( T_{3}^{47} + \)\(30\!\cdots\!12\)\( T_{3}^{46} + \)\(44\!\cdots\!68\)\( T_{3}^{45} + \)\(57\!\cdots\!44\)\( T_{3}^{44} + \)\(75\!\cdots\!56\)\( T_{3}^{43} + \)\(10\!\cdots\!32\)\( T_{3}^{42} + \)\(13\!\cdots\!64\)\( T_{3}^{41} + \)\(16\!\cdots\!19\)\( T_{3}^{40} + \)\(19\!\cdots\!12\)\( T_{3}^{39} + \)\(23\!\cdots\!72\)\( T_{3}^{38} + \)\(28\!\cdots\!68\)\( T_{3}^{37} + \)\(32\!\cdots\!20\)\( T_{3}^{36} + \)\(35\!\cdots\!80\)\( T_{3}^{35} + \)\(38\!\cdots\!64\)\( T_{3}^{34} + \)\(39\!\cdots\!92\)\( T_{3}^{33} + \)\(41\!\cdots\!78\)\( T_{3}^{32} + \)\(44\!\cdots\!08\)\( T_{3}^{31} + \)\(46\!\cdots\!12\)\( T_{3}^{30} + \)\(44\!\cdots\!44\)\( T_{3}^{29} + \)\(37\!\cdots\!92\)\( T_{3}^{28} + \)\(26\!\cdots\!44\)\( T_{3}^{27} + \)\(15\!\cdots\!68\)\( T_{3}^{26} + \)\(58\!\cdots\!32\)\( T_{3}^{25} + \)\(45\!\cdots\!41\)\( T_{3}^{24} - \)\(14\!\cdots\!84\)\( T_{3}^{23} - \)\(13\!\cdots\!28\)\( T_{3}^{22} - \)\(71\!\cdots\!52\)\( T_{3}^{21} - \)\(22\!\cdots\!72\)\( T_{3}^{20} - \)\(10\!\cdots\!36\)\( T_{3}^{19} + \)\(34\!\cdots\!32\)\( T_{3}^{18} + \)\(23\!\cdots\!60\)\( T_{3}^{17} + \)\(83\!\cdots\!52\)\( T_{3}^{16} + \)\(57\!\cdots\!72\)\( T_{3}^{15} - \)\(13\!\cdots\!80\)\( T_{3}^{14} - \)\(96\!\cdots\!76\)\( T_{3}^{13} - \)\(29\!\cdots\!60\)\( T_{3}^{12} + \)\(21\!\cdots\!64\)\( T_{3}^{11} + \)\(73\!\cdots\!76\)\( T_{3}^{10} + \)\(41\!\cdots\!04\)\( T_{3}^{9} + \)\(14\!\cdots\!40\)\( T_{3}^{8} + \)\(36\!\cdots\!72\)\( T_{3}^{7} + \)\(67\!\cdots\!08\)\( T_{3}^{6} + \)\(94\!\cdots\!52\)\( T_{3}^{5} + \)\(71\!\cdots\!12\)\( T_{3}^{4} + \)\(11\!\cdots\!76\)\( T_{3}^{3} + \)\(10\!\cdots\!12\)\( T_{3}^{2} + \)\(47\!\cdots\!32\)\( T_{3} + 110075314176 \)">\(T_{3}^{120} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\).