# Properties

 Label 80.6.s.a Level 80 Weight 6 Character orbit 80.s Analytic conductor 12.831 Analytic rank 0 Dimension 116 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 80.s (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.8307055850$$ Analytic rank: $$0$$ Dimension: $$116$$ Relative dimension: $$58$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$116q - 2q^{2} - 4q^{3} + 20q^{4} - 2q^{5} - 4q^{6} - 4q^{7} - 248q^{8} + 8748q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$116q - 2q^{2} - 4q^{3} + 20q^{4} - 2q^{5} - 4q^{6} - 4q^{7} - 248q^{8} + 8748q^{9} + 62q^{10} - 4q^{11} - 1280q^{12} - 972q^{15} - 1224q^{16} - 4q^{17} - 4346q^{18} + 2360q^{19} + 832q^{20} - 4q^{21} - 3148q^{22} - 4q^{23} - 972q^{24} - 884q^{26} - 976q^{27} + 16508q^{28} + 19836q^{30} - 7972q^{32} - 4q^{33} + 12520q^{34} + 3860q^{35} + 2380q^{36} - 12256q^{38} + 16192q^{40} + 38424q^{42} - 8200q^{44} - 6738q^{45} - 35924q^{46} - 65256q^{47} + 17620q^{48} + 24194q^{50} + 10436q^{51} - 34384q^{52} - 4q^{53} - 33820q^{54} - 4q^{55} - 64684q^{56} - 972q^{57} - 30632q^{58} + 14480q^{59} + 1200q^{60} + 48076q^{61} - 6116q^{62} - 972q^{63} - 71920q^{64} - 4q^{65} + 72436q^{66} + 119464q^{68} + 21348q^{69} - 141252q^{70} - 143848q^{71} - 130864q^{72} + 10072q^{73} + 82508q^{74} + 160004q^{75} - 128004q^{76} - 67232q^{77} - 253108q^{78} - 171436q^{80} + 551116q^{81} - 152200q^{82} + 126436q^{83} + 80928q^{84} + 6248q^{85} - 85324q^{86} - 282188q^{87} - 247536q^{88} + 275094q^{90} - 164724q^{91} + 336476q^{92} + 106060q^{94} - 204760q^{95} - 62264q^{96} - 4q^{97} + 110122q^{98} + 168788q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −5.61166 + 0.713657i 5.58602 30.9814 8.00959i 24.7855 50.1067i −31.3468 + 3.98650i 145.876 145.876i −168.141 + 67.0572i −211.796 −103.329 + 298.870i
3.2 −5.60238 + 0.783133i 14.3156 30.7734 8.77482i 23.2238 + 50.8494i −80.2014 + 11.2110i −103.868 + 103.868i −165.533 + 73.2596i −38.0641 −169.930 266.690i
3.3 −5.58900 + 0.873525i −13.7617 30.4739 9.76427i −55.0423 9.76467i 76.9143 12.0212i −58.2575 + 58.2575i −161.789 + 81.1923i −53.6151 316.161 + 6.49396i
3.4 −5.54990 1.09479i −11.0755 29.6029 + 12.1520i 24.0182 + 50.4790i 61.4682 + 12.1254i 119.074 119.074i −150.989 99.8515i −120.332 −78.0344 306.448i
3.5 −5.53415 1.17184i 28.9277 29.2536 + 12.9702i −47.1500 + 30.0314i −160.090 33.8985i 139.505 139.505i −146.695 106.060i 593.809 296.127 110.946i
3.6 −5.46635 + 1.45570i −23.1807 27.7619 15.9147i 47.1776 29.9879i 126.714 33.7441i −125.489 + 125.489i −128.589 + 127.408i 294.343 −214.236 + 232.601i
3.7 −5.31218 1.94441i 16.7325 24.4385 + 20.6581i −7.11375 55.4472i −88.8859 32.5348i −104.316 + 104.316i −89.6538 157.258i 36.9755 −70.0228 + 308.378i
3.8 −5.11622 2.41336i −5.06764 20.3514 + 24.6946i −51.4203 + 21.9305i 25.9272 + 12.2301i −35.3013 + 35.3013i −44.5250 175.458i −217.319 316.004 + 11.8945i
3.9 −4.99964 + 2.64643i −25.1055 17.9928 26.4624i −17.6043 + 53.0574i 125.519 66.4399i 92.6452 92.6452i −19.9271 + 179.919i 387.286 −52.3971 311.857i
3.10 −4.95241 2.73379i −9.68797 17.0528 + 27.0777i 53.9138 14.7751i 47.9788 + 26.4849i −40.5624 + 40.5624i −10.4273 180.719i −149.143 −307.395 74.2169i
3.11 −4.94622 + 2.74497i 14.4255 16.9303 27.1545i −46.0646 31.6710i −71.3518 + 39.5976i −46.8577 + 46.8577i −9.20264 + 180.785i −34.9048 314.782 + 30.2059i
3.12 −4.88220 2.85730i −30.2943 15.6717 + 27.8998i −21.4296 51.6311i 147.903 + 86.5599i 106.513 106.513i 3.20578 180.991i 674.746 −42.9018 + 313.304i
3.13 −4.25790 + 3.72428i −2.96317 4.25947 31.7152i 54.1456 + 13.9016i 12.6169 11.0357i 6.13739 6.13739i 99.9801 + 150.904i −234.220 −282.320 + 142.462i
3.14 −4.25189 + 3.73115i 30.7007 4.15710 31.7288i 47.2290 29.9068i −130.536 + 114.549i −7.67296 + 7.67296i 100.709 + 150.418i 699.530 −89.2258 + 303.379i
3.15 −4.11912 3.87722i 21.0242 1.93432 + 31.9415i 54.6422 + 11.7996i −86.6012 81.5154i 19.5216 19.5216i 115.876 139.071i 199.016 −179.328 260.464i
3.16 −4.09704 + 3.90055i 12.0492 1.57141 31.9614i −35.9351 + 42.8214i −49.3660 + 46.9985i 63.2042 63.2042i 118.229 + 137.076i −97.8169 −19.7998 315.607i
3.17 −3.48504 4.45584i −24.9367 −7.70899 + 31.0576i −3.26981 + 55.8060i 86.9053 + 111.114i −112.850 + 112.850i 165.254 73.8868i 378.838 260.058 179.916i
3.18 −3.31543 + 4.58344i −14.5939 −10.0158 30.3922i −17.9275 52.9491i 48.3851 66.8903i 47.3215 47.3215i 172.507 + 54.8563i −30.0178 302.126 + 93.3792i
3.19 −3.26583 4.61891i 4.38609 −10.6687 + 30.1692i −47.6895 29.1670i −14.3242 20.2590i 121.663 121.663i 174.191 49.2495i −223.762 21.0260 + 315.528i
3.20 −2.93140 4.83807i 8.72573 −14.8138 + 28.3646i −5.22519 + 55.6570i −25.5786 42.2157i −6.29366 + 6.29366i 180.655 11.4775i −166.862 284.589 137.873i
See next 80 embeddings (of 116 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 27.58 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.s even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.6.s.a yes 116
5.c odd 4 1 80.6.j.a 116
16.f odd 4 1 80.6.j.a 116
80.s even 4 1 inner 80.6.s.a yes 116

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.6.j.a 116 5.c odd 4 1
80.6.j.a 116 16.f odd 4 1
80.6.s.a yes 116 1.a even 1 1 trivial
80.6.s.a yes 116 80.s even 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(80, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database