# Properties

 Label 847.2.f.z Level 847 Weight 2 Character orbit 847.f Analytic conductor 6.763 Analytic rank 0 Dimension 24 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$847 = 7 \cdot 11^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 847.f (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.76332905120$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$6$$ over $$\Q(\zeta_{5})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 4q^{2} + 2q^{3} - 4q^{4} + 4q^{5} - 6q^{6} + 6q^{7} + 12q^{8} - 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 4q^{2} + 2q^{3} - 4q^{4} + 4q^{5} - 6q^{6} + 6q^{7} + 12q^{8} - 8q^{9} + 32q^{10} - 56q^{12} + 4q^{13} - 4q^{14} - 2q^{15} - 8q^{16} + 22q^{17} + 24q^{18} + 6q^{19} - 2q^{20} + 8q^{21} + 8q^{23} - 20q^{24} - 4q^{25} - 6q^{26} + 2q^{27} + 4q^{28} + 12q^{29} + 20q^{30} + 2q^{31} - 32q^{32} + 96q^{34} - 4q^{35} - 18q^{36} - 14q^{37} + 22q^{38} + 20q^{39} + 18q^{40} + 26q^{41} + 6q^{42} + 16q^{43} - 144q^{45} + 12q^{46} + 16q^{47} + 24q^{48} - 6q^{49} - 4q^{50} - 4q^{51} + 12q^{52} - 4q^{53} + 128q^{54} + 48q^{56} + 20q^{57} + 2q^{58} + 4q^{59} - 24q^{60} - 8q^{61} + 20q^{62} + 8q^{63} - 26q^{64} - 96q^{65} + 24q^{67} + 12q^{68} + 14q^{69} + 8q^{70} - 22q^{71} + 16q^{72} + 14q^{73} + 44q^{74} + 20q^{75} + 120q^{76} + 128q^{78} - 28q^{79} + 4q^{80} + 6q^{81} + 4q^{82} + 22q^{83} - 14q^{84} - 24q^{85} + 30q^{86} - 88q^{87} - 22q^{90} - 4q^{91} - 10q^{92} + 50q^{93} - 38q^{94} - 24q^{95} - 62q^{96} + 4q^{97} - 16q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
148.1 −0.835334 2.57089i 2.27106 + 1.65003i −4.29367 + 3.11953i −0.137535 + 0.423289i 2.34494 7.21698i 0.809017 0.587785i 7.23277 + 5.25492i 1.50810 + 4.64146i 1.20312
148.2 −0.651837 2.00615i −1.37413 0.998361i −1.98170 + 1.43979i 0.152157 0.468292i −1.10715 + 3.40747i 0.809017 0.587785i 0.767119 + 0.557345i −0.0355535 0.109422i −1.03864
148.3 −0.394480 1.21408i 2.08406 + 1.51415i 0.299647 0.217706i −1.26432 + 3.89119i 1.01619 3.12752i 0.809017 0.587785i −2.44804 1.77861i 1.12357 + 3.45799i 5.22298
148.4 −0.0371933 0.114469i −2.23923 1.62690i 1.60631 1.16706i −0.867882 + 2.67107i −0.102945 + 0.316832i 0.809017 0.587785i −0.388082 0.281958i 1.44031 + 4.43281i 0.338034
148.5 0.254493 + 0.783248i 0.777181 + 0.564655i 1.06932 0.776908i 0.922616 2.83952i −0.244478 + 0.752427i 0.809017 0.587785i 2.21319 + 1.60798i −0.641876 1.97549i 2.45885
148.6 0.428283 + 1.31812i 0.0990877 + 0.0719914i 0.0640220 0.0465147i −0.0411006 + 0.126495i −0.0524557 + 0.161442i 0.809017 0.587785i 2.33125 + 1.69375i −0.922415 2.83890i −0.184338
323.1 −1.12126 0.814642i −0.0378481 + 0.116485i −0.0244542 0.0752624i 0.107603 0.0781781i 0.137331 0.0997767i −0.309017 0.951057i −0.890458 + 2.74055i 2.41491 + 1.75454i −0.184338
323.2 −0.666271 0.484074i −0.296857 + 0.913631i −0.408445 1.25706i −2.41544 + 1.75492i 0.640052 0.465025i −0.309017 0.951057i −0.845363 + 2.60176i 1.68045 + 1.22092i 2.45885
323.3 0.0973732 + 0.0707458i 0.855309 2.63237i −0.613557 1.88834i 2.27214 1.65081i 0.269513 0.195813i −0.309017 0.951057i 0.148234 0.456218i −3.77078 2.73963i 0.338034
323.4 1.03276 + 0.750346i −0.796038 + 2.44995i −0.114455 0.352256i 3.31004 2.40489i −2.66043 + 1.93292i −0.309017 0.951057i 0.935069 2.87785i −2.94155 2.13716i 5.22298
323.5 1.70653 + 1.23987i 0.524869 1.61538i 0.756943 + 2.32963i −0.398353 + 0.289420i 2.89857 2.10593i −0.309017 0.951057i −0.293014 + 0.901803i 0.0930802 + 0.0676268i −1.03864
323.6 2.18693 + 1.58890i −0.867470 + 2.66980i 1.64004 + 5.04751i 0.360071 0.261607i −6.13913 + 4.46034i −0.309017 0.951057i −2.76267 + 8.50263i −3.94826 2.86858i 1.20312
372.1 −0.835334 + 2.57089i 2.27106 1.65003i −4.29367 3.11953i −0.137535 0.423289i 2.34494 + 7.21698i 0.809017 + 0.587785i 7.23277 5.25492i 1.50810 4.64146i 1.20312
372.2 −0.651837 + 2.00615i −1.37413 + 0.998361i −1.98170 1.43979i 0.152157 + 0.468292i −1.10715 3.40747i 0.809017 + 0.587785i 0.767119 0.557345i −0.0355535 + 0.109422i −1.03864
372.3 −0.394480 + 1.21408i 2.08406 1.51415i 0.299647 + 0.217706i −1.26432 3.89119i 1.01619 + 3.12752i 0.809017 + 0.587785i −2.44804 + 1.77861i 1.12357 3.45799i 5.22298
372.4 −0.0371933 + 0.114469i −2.23923 + 1.62690i 1.60631 + 1.16706i −0.867882 2.67107i −0.102945 0.316832i 0.809017 + 0.587785i −0.388082 + 0.281958i 1.44031 4.43281i 0.338034
372.5 0.254493 0.783248i 0.777181 0.564655i 1.06932 + 0.776908i 0.922616 + 2.83952i −0.244478 0.752427i 0.809017 + 0.587785i 2.21319 1.60798i −0.641876 + 1.97549i 2.45885
372.6 0.428283 1.31812i 0.0990877 0.0719914i 0.0640220 + 0.0465147i −0.0411006 0.126495i −0.0524557 0.161442i 0.809017 + 0.587785i 2.33125 1.69375i −0.922415 + 2.83890i −0.184338
729.1 −1.12126 + 0.814642i −0.0378481 0.116485i −0.0244542 + 0.0752624i 0.107603 + 0.0781781i 0.137331 + 0.0997767i −0.309017 + 0.951057i −0.890458 2.74055i 2.41491 1.75454i −0.184338
729.2 −0.666271 + 0.484074i −0.296857 0.913631i −0.408445 + 1.25706i −2.41544 1.75492i 0.640052 + 0.465025i −0.309017 + 0.951057i −0.845363 2.60176i 1.68045 1.22092i 2.45885
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 729.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.f.z 24
11.b odd 2 1 847.2.f.y 24
11.c even 5 1 847.2.a.m 6
11.c even 5 3 inner 847.2.f.z 24
11.d odd 10 1 847.2.a.n yes 6
11.d odd 10 3 847.2.f.y 24
33.f even 10 1 7623.2.a.cp 6
33.h odd 10 1 7623.2.a.cs 6
77.j odd 10 1 5929.2.a.bj 6
77.l even 10 1 5929.2.a.bm 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.m 6 11.c even 5 1
847.2.a.n yes 6 11.d odd 10 1
847.2.f.y 24 11.b odd 2 1
847.2.f.y 24 11.d odd 10 3
847.2.f.z 24 1.a even 1 1 trivial
847.2.f.z 24 11.c even 5 3 inner
5929.2.a.bj 6 77.j odd 10 1
5929.2.a.bm 6 77.l even 10 1
7623.2.a.cp 6 33.f even 10 1
7623.2.a.cs 6 33.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(847, [\chi])$$:

 $$T_{2}^{24} - \cdots$$ $$T_{3}^{24} - \cdots$$ $$T_{13}^{24} - \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database