# 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 $$\Q(\zeta_{5})$$ 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.

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

## 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 $$S_{2}^{\mathrm{new}}(847, [\chi])$$:

 $$T_{2}^{24} - 4 T_{2}^{23} + 16 T_{2}^{22} - 52 T_{2}^{21} + 164 T_{2}^{20} - 304 T_{2}^{19} + 687 T_{2}^{18} - 1112 T_{2}^{17} + 1856 T_{2}^{16} - 1656 T_{2}^{15} + 3916 T_{2}^{14} - 3032 T_{2}^{13} + 7889 T_{2}^{12} - 4924 T_{2}^{11} + \cdots + 1$$ T2^24 - 4*T2^23 + 16*T2^22 - 52*T2^21 + 164*T2^20 - 304*T2^19 + 687*T2^18 - 1112*T2^17 + 1856*T2^16 - 1656*T2^15 + 3916*T2^14 - 3032*T2^13 + 7889*T2^12 - 4924*T2^11 + 10368*T2^10 - 4376*T2^9 + 10312*T2^8 + 3392*T2^7 + 6895*T2^6 + 640*T2^5 + 4688*T2^4 - 564*T2^3 + 68*T2^2 - 8*T2 + 1 $$T_{3}^{24} - 2 T_{3}^{23} + 15 T_{3}^{22} - 32 T_{3}^{21} + 164 T_{3}^{20} - 254 T_{3}^{19} + 1365 T_{3}^{18} - 1976 T_{3}^{17} + 10879 T_{3}^{16} - 15620 T_{3}^{15} + 57101 T_{3}^{14} - 67166 T_{3}^{13} + 243744 T_{3}^{12} + \cdots + 256$$ T3^24 - 2*T3^23 + 15*T3^22 - 32*T3^21 + 164*T3^20 - 254*T3^19 + 1365*T3^18 - 1976*T3^17 + 10879*T3^16 - 15620*T3^15 + 57101*T3^14 - 67166*T3^13 + 243744*T3^12 - 77534*T3^11 + 584211*T3^10 - 315012*T3^9 + 1601637*T3^8 - 1551956*T3^7 + 1632652*T3^6 - 1150240*T3^5 + 1278480*T3^4 - 156544*T3^3 + 19136*T3^2 - 2304*T3 + 256 $$T_{13}^{24} - 4 T_{13}^{23} + 35 T_{13}^{22} - 144 T_{13}^{21} + 989 T_{13}^{20} - 1744 T_{13}^{19} + 16771 T_{13}^{18} - 35252 T_{13}^{17} + 373741 T_{13}^{16} - 1049368 T_{13}^{15} + 2711988 T_{13}^{14} - 5654432 T_{13}^{13} + \cdots + 65536$$ T13^24 - 4*T13^23 + 35*T13^22 - 144*T13^21 + 989*T13^20 - 1744*T13^19 + 16771*T13^18 - 35252*T13^17 + 373741*T13^16 - 1049368*T13^15 + 2711988*T13^14 - 5654432*T13^13 + 12905408*T13^12 - 12965632*T13^11 + 15177664*T13^10 - 13670400*T13^9 + 11676416*T13^8 - 2916352*T13^7 + 4058112*T13^6 - 1310720*T13^5 + 2048000*T13^4 - 819200*T13^3 + 409600*T13^2 - 131072*T13 + 65536