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 $4$

Related objects

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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