# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$12.8307055850$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$40$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$80q + 44q^{4} - 228q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q + 44q^{4} - 228q^{6} - 100q^{10} + 1208q^{11} + 1572q^{12} - 916q^{14} - 1800q^{15} + 2960q^{16} + 6400q^{18} + 2360q^{19} - 3800q^{20} - 3260q^{22} - 13880q^{24} - 880q^{26} - 7464q^{27} + 18764q^{28} - 8144q^{29} + 22400q^{32} - 7440q^{34} + 27948q^{36} + 21296q^{37} - 34820q^{38} - 7260q^{42} + 32072q^{43} + 64392q^{44} + 149268q^{46} - 88360q^{47} + 94040q^{48} - 192080q^{49} + 2500q^{50} - 5920q^{51} - 55016q^{52} - 49456q^{53} - 250784q^{54} - 250352q^{56} - 115988q^{58} + 44984q^{59} + 7700q^{60} + 48080q^{61} + 444328q^{62} + 158760q^{63} + 218096q^{64} + 253440q^{66} + 61160q^{67} - 119792q^{68} - 22320q^{69} - 66200q^{70} - 629480q^{72} - 156664q^{74} + 223232q^{76} - 14896q^{77} + 516176q^{78} + 177680q^{79} + 48400q^{80} - 524880q^{81} + 254476q^{82} - 329240q^{83} - 173248q^{84} + 132400q^{85} - 765740q^{86} - 390400q^{88} + 71100q^{90} + 364832q^{91} + 333532q^{92} - 362352q^{93} + 537844q^{94} + 288800q^{95} + 1151112q^{96} + 52420q^{98} + 659000q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
21.1 −5.65572 + 0.113075i −18.3942 + 18.3942i 31.9744 1.27904i 17.6777 + 17.6777i 101.953 106.113i 47.3499i −180.694 + 10.8494i 433.697i −101.979 97.9811i
21.2 −5.65514 + 0.139396i 19.6483 19.6483i 31.9611 1.57661i −17.6777 17.6777i −108.375 + 113.853i 152.043i −180.525 + 13.3712i 529.109i 102.434 + 97.5054i
21.3 −5.62667 + 0.583586i −13.2011 + 13.2011i 31.3189 6.56730i −17.6777 17.6777i 66.5744 81.9824i 190.109i −172.388 + 55.2293i 105.540i 109.783 + 89.1500i
21.4 −5.55875 + 1.04896i 9.77316 9.77316i 29.7994 11.6618i 17.6777 + 17.6777i −44.0749 + 64.5782i 103.649i −153.414 + 96.0835i 51.9707i −116.809 79.7225i
21.5 −5.42543 1.60146i 3.86003 3.86003i 26.8706 + 17.3773i −17.6777 17.6777i −27.1241 + 14.7606i 20.3706i −117.956 137.312i 213.200i 67.5988 + 124.219i
21.6 −5.27573 2.04124i 6.55397 6.55397i 23.6666 + 21.5381i 17.6777 + 17.6777i −47.9552 + 21.1987i 188.280i −80.8943 161.939i 157.091i −57.1782 129.347i
21.7 −4.61831 3.26668i −15.8638 + 15.8638i 10.6576 + 30.1731i −17.6777 17.6777i 125.086 21.4420i 207.627i 49.3459 174.164i 260.322i 23.8936 + 139.388i
21.8 −4.51701 + 3.40538i −7.38373 + 7.38373i 8.80677 30.7643i 17.6777 + 17.6777i 8.20797 58.4968i 74.1496i 64.9838 + 168.953i 133.961i −140.049 19.6510i
21.9 −4.43266 + 3.51447i 11.6877 11.6877i 7.29694 31.1569i −17.6777 17.6777i −10.7314 + 92.8834i 158.930i 77.1554 + 163.753i 30.2029i 140.487 + 16.2314i
21.10 −4.24419 3.73990i 21.1465 21.1465i 4.02635 + 31.7457i 17.6777 + 17.6777i −168.835 + 10.6641i 100.160i 101.637 149.793i 651.346i −8.91479 141.140i
21.11 −4.01718 3.98275i 10.2463 10.2463i 0.275419 + 31.9988i −17.6777 17.6777i −81.9696 + 0.352756i 107.195i 126.337 129.642i 33.0270i 0.608601 + 141.420i
21.12 −3.40887 4.51438i −13.5228 + 13.5228i −8.75923 + 30.7778i 17.6777 + 17.6777i 107.145 + 14.9496i 90.4254i 168.802 65.3752i 122.733i 19.5428 140.065i
21.13 −3.36114 + 4.55002i −18.2668 + 18.2668i −9.40545 30.5866i −17.6777 17.6777i −21.7171 144.512i 13.9157i 170.783 + 60.0107i 424.353i 139.851 21.0167i
21.14 −3.10357 + 4.72947i −7.07478 + 7.07478i −12.7357 29.3564i 17.6777 + 17.6777i −11.5029 55.4170i 216.974i 178.367 + 30.8765i 142.895i −138.470 + 28.7421i
21.15 −2.38570 5.12917i 2.82595 2.82595i −20.6168 + 24.4734i 17.6777 + 17.6777i −21.2367 7.75292i 197.416i 174.714 + 47.3612i 227.028i 48.4982 132.846i
21.16 −2.24441 + 5.19255i 1.65480 1.65480i −21.9252 23.3084i −17.6777 17.6777i 4.87860 + 12.3067i 111.270i 170.240 61.5343i 237.523i 131.468 52.1163i
21.17 −1.86060 5.34211i −6.56300 + 6.56300i −25.0763 + 19.8791i −17.6777 17.6777i 47.2714 + 22.8492i 42.0449i 152.853 + 96.9737i 156.854i −61.5451 + 127.327i
21.18 −0.605604 + 5.62434i 4.24643 4.24643i −31.2665 6.81225i 17.6777 + 17.6777i 21.3117 + 26.4550i 137.329i 57.2496 171.728i 206.936i −110.131 + 88.7196i
21.19 −0.350398 + 5.64599i −18.9032 + 18.9032i −31.7544 3.95669i 17.6777 + 17.6777i −100.104 113.351i 174.451i 33.4661 177.899i 471.664i −106.002 + 93.6138i
21.20 −0.0219870 + 5.65681i 9.88924 9.88924i −31.9990 0.248752i −17.6777 17.6777i 55.7241 + 56.1590i 122.335i 2.11071 181.007i 47.4058i 100.388 99.6106i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 61.40 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e 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.l.a 80
4.b odd 2 1 320.6.l.a 80
16.e even 4 1 inner 80.6.l.a 80
16.f odd 4 1 320.6.l.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.6.l.a 80 1.a even 1 1 trivial
80.6.l.a 80 16.e even 4 1 inner
320.6.l.a 80 4.b odd 2 1
320.6.l.a 80 16.f odd 4 1

## 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