# Properties

 Label 80.6.q.a Level 80 Weight 6 Character orbit 80.q Analytic conductor 12.831 Analytic rank 0 Dimension 116 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 80.q (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.

 $$\operatorname{Tr}(f)(q) =$$ $$116q - 4q^{4} - 2q^{5} - 4q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$116q - 4q^{4} - 2q^{5} - 4q^{6} + 532q^{10} - 4q^{11} + 1108q^{14} - 4q^{15} + 1216q^{16} + 2356q^{19} + 824q^{20} - 976q^{21} - 2864q^{24} - 12104q^{26} - 4q^{29} + 736q^{30} + 23056q^{31} - 38304q^{34} - 3864q^{35} - 20588q^{36} + 15144q^{40} - 25536q^{44} + 6734q^{45} - 36052q^{46} + 220884q^{49} + 56116q^{50} - 9472q^{51} - 104360q^{54} - 2552q^{56} + 14476q^{59} + 1008q^{60} - 48084q^{61} + 72944q^{64} - 27692q^{65} - 56736q^{66} - 23296q^{69} + 135260q^{70} - 186128q^{74} - 33244q^{75} - 35824q^{76} + 427312q^{79} - 129128q^{80} - 551132q^{81} - 418448q^{84} + 6248q^{85} + 177516q^{86} + 449576q^{90} - 231952q^{91} + 607356q^{94} + 38420q^{95} - 440264q^{96} + 406924q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 −5.65319 0.203680i −12.8690 + 12.8690i 31.9170 + 2.30288i −50.2546 + 24.4842i 75.3721 70.1298i −156.045 −179.964 19.5195i 88.2233i 289.086 128.178i
29.2 −5.64889 + 0.300072i −16.1243 + 16.1243i 31.8199 3.39015i 33.0407 + 45.0922i 86.2458 95.9226i 106.654 −178.730 + 28.6988i 276.984i −200.174 244.806i
29.3 −5.61624 0.676673i 18.5525 18.5525i 31.0842 + 7.60071i 8.85788 55.1955i −116.749 + 91.6413i −206.395 −169.433 63.7213i 445.391i −87.0972 + 303.997i
29.4 −5.57469 0.960646i −2.26113 + 2.26113i 30.1543 + 10.7106i −30.6126 46.7747i 14.7772 10.4329i 66.4138 −157.812 88.6759i 232.775i 125.722 + 290.162i
29.5 −5.48884 1.36846i 6.61300 6.61300i 28.2546 + 15.0225i 49.7455 + 25.5026i −45.3473 + 27.2480i 180.633 −134.527 121.121i 155.536i −238.146 208.054i
29.6 −5.44124 + 1.54691i 9.88218 9.88218i 27.2142 16.8342i 27.0375 + 48.9282i −38.4845 + 69.0581i −163.839 −122.038 + 133.697i 47.6849i −222.805 224.406i
29.7 −5.29707 + 1.98521i 7.43164 7.43164i 24.1179 21.0316i −55.3405 + 7.90104i −24.6125 + 54.1193i 59.4492 −86.0017 + 159.285i 132.541i 277.457 151.715i
29.8 −5.17693 2.28021i 18.1244 18.1244i 21.6013 + 23.6090i −35.0084 + 43.5822i −135.156 + 52.5013i 75.8099 −57.9951 171.478i 413.986i 280.613 145.796i
29.9 −5.15884 + 2.32087i −9.83803 + 9.83803i 21.2272 23.9459i 46.4875 31.0470i 27.9200 73.5856i −45.7194 −53.9321 + 172.799i 49.4262i −167.765 + 268.057i
29.10 −4.87261 + 2.87361i 14.7569 14.7569i 15.4847 28.0040i 24.3169 50.3357i −29.4990 + 114.310i 197.707 5.02143 + 180.950i 192.530i 26.1584 + 315.144i
29.11 −4.74064 3.08648i −1.81560 + 1.81560i 12.9473 + 29.2637i 48.1357 28.4245i 14.2109 3.00330i −66.7879 28.9433 178.690i 236.407i −315.926 13.8197i
29.12 −4.71172 3.13045i −21.0051 + 21.0051i 12.4006 + 29.4996i 9.85062 55.0270i 164.725 33.2148i −32.3244 33.9186 177.813i 639.428i −218.672 + 228.435i
29.13 −4.30125 3.67413i −2.38770 + 2.38770i 5.00149 + 31.6067i 4.21193 + 55.7428i 19.0429 1.49737i −204.311 94.6147 154.325i 231.598i 186.690 255.239i
29.14 −4.04395 3.95557i −11.3559 + 11.3559i 0.706997 + 31.9922i −38.8598 + 40.1860i 90.8416 1.00363i 230.224 123.688 132.171i 14.9127i 316.105 8.79746i
29.15 −4.03469 + 3.96501i −19.4902 + 19.4902i 0.557416 31.9951i −49.1605 26.6128i 1.35807 155.915i 138.003 124.612 + 131.301i 516.733i 303.867 87.5474i
29.16 −3.76715 + 4.22003i −3.83997 + 3.83997i −3.61723 31.7949i −6.34052 + 55.5410i −1.73905 30.6705i 92.8374 147.802 + 104.511i 213.509i −210.499 235.988i
29.17 −3.62558 + 4.34225i −3.49000 + 3.49000i −5.71035 31.4864i −25.3489 49.8240i −2.50120 27.8078i −221.972 157.425 + 89.3605i 218.640i 308.253 + 70.5698i
29.18 −3.40088 4.52040i 8.86425 8.86425i −8.86807 + 30.7467i −48.7561 27.3468i −70.2162 9.92375i −41.3923 169.146 64.4784i 85.8500i 42.1948 + 313.400i
29.19 −2.80183 + 4.91424i 12.4641 12.4641i −16.2995 27.5377i 55.1914 + 8.88296i 26.3294 + 96.1740i −61.0801 180.995 2.94408i 67.7088i −198.290 + 246.336i
29.20 −2.78634 4.92304i 16.5974 16.5974i −16.4726 + 27.4345i 55.5811 5.97797i −127.956 35.4637i 50.7548 180.960 + 4.65334i 307.949i −184.298 256.971i
See next 80 embeddings (of 116 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 69.58 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
16.e even 4 1 inner
80.q 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.q.a 116
5.b even 2 1 inner 80.6.q.a 116
16.e even 4 1 inner 80.6.q.a 116
80.q even 4 1 inner 80.6.q.a 116

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.6.q.a 116 1.a even 1 1 trivial
80.6.q.a 116 5.b even 2 1 inner
80.6.q.a 116 16.e even 4 1 inner
80.6.q.a 116 80.q 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