# Properties

 Label 40.6.k.a Level 40 Weight 6 Character orbit 40.k Analytic conductor 6.415 Analytic rank 0 Dimension 56 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$40 = 2^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 40.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.41535279252$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$28$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$56q - 2q^{2} - 4q^{3} + 176q^{6} + 244q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q - 2q^{2} - 4q^{3} + 176q^{6} + 244q^{8} - 570q^{10} - 8q^{11} - 308q^{12} + 456q^{16} - 408q^{17} + 3370q^{18} + 1360q^{20} - 3148q^{22} - 3120q^{25} - 5084q^{26} + 968q^{27} - 10020q^{28} + 21100q^{30} + 7968q^{32} - 976q^{33} - 4780q^{35} + 23132q^{36} - 49524q^{38} - 58980q^{40} - 8q^{41} + 42060q^{42} + 1308q^{43} + 42416q^{46} + 93776q^{48} - 94010q^{50} + 20872q^{51} - 67180q^{52} - 43384q^{56} + 968q^{57} + 86100q^{58} + 118180q^{60} + 6240q^{62} + 17680q^{65} - 41128q^{66} + 89252q^{67} - 158164q^{68} + 116980q^{70} + 1180q^{72} - 25184q^{73} - 127740q^{75} + 41592q^{76} - 161440q^{78} - 167360q^{80} - 67792q^{81} + 23412q^{82} - 126444q^{83} - 192704q^{86} + 100176q^{88} - 82330q^{90} + 329432q^{91} + 226020q^{92} + 358176q^{96} + 212576q^{97} + 65834q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −5.65636 0.0746695i −5.85746 5.85746i 31.9888 + 0.844715i −13.4330 54.2637i 32.6945 + 33.5693i 100.487 + 100.487i −180.877 7.16660i 174.380i 71.9302 + 307.938i
3.2 −5.63573 0.488434i 17.2238 + 17.2238i 31.5229 + 5.50537i 54.8737 10.6715i −88.6558 105.481i −14.7411 14.7411i −174.965 46.4236i 350.316i −314.465 + 33.3393i
3.3 −5.57667 + 0.949067i −3.12794 3.12794i 30.1985 10.5853i −34.4559 + 44.0204i 20.4121 + 14.4749i −19.1929 19.1929i −158.361 + 87.6910i 223.432i 150.371 278.188i
3.4 −4.95505 + 2.72901i −17.2715 17.2715i 17.1050 27.0447i 55.8976 0.673734i 132.715 + 38.4469i −135.845 135.845i −10.9508 + 180.688i 353.606i −275.137 + 155.884i
3.5 −4.60750 3.28191i −7.52871 7.52871i 10.4581 + 30.2428i 50.5551 + 23.8576i 9.97997 + 59.3971i 51.0903 + 51.0903i 51.0685 173.666i 129.637i −154.634 275.841i
3.6 −4.60651 3.28330i 11.5932 + 11.5932i 10.4398 + 30.2491i −48.9682 + 26.9650i −15.3402 91.4681i −75.8554 75.8554i 51.2259 173.620i 25.8041i 314.107 + 36.5629i
3.7 −4.18096 + 3.81045i 10.0657 + 10.0657i 2.96087 31.8627i −23.9125 50.5291i −80.4392 3.72940i −96.6855 96.6855i 109.032 + 144.499i 40.3635i 292.516 + 120.143i
3.8 −3.87156 4.12444i −18.3638 18.3638i −2.02198 + 31.9361i −43.5466 35.0528i −4.64371 + 146.837i −112.979 112.979i 139.546 115.303i 431.456i 24.0203 + 315.314i
3.9 −3.81045 + 4.18096i 10.0657 + 10.0657i −2.96087 31.8627i 23.9125 + 50.5291i −80.4392 + 3.72940i 96.6855 + 96.6855i 144.499 + 109.032i 40.3635i −302.378 92.5617i
3.10 −2.72901 + 4.95505i −17.2715 17.2715i −17.1050 27.0447i −55.8976 + 0.673734i 132.715 38.4469i 135.845 + 135.845i 180.688 10.9508i 353.606i 149.207 278.814i
3.11 −2.49882 5.07503i 10.6033 + 10.6033i −19.5118 + 25.3631i −15.3629 53.7493i 27.3164 80.3079i 159.941 + 159.941i 177.475 + 35.6453i 18.1393i −234.390 + 212.276i
3.12 −0.949067 + 5.57667i −3.12794 3.12794i −30.1985 10.5853i 34.4559 44.0204i 20.4121 14.4749i 19.1929 + 19.1929i 87.6910 158.361i 223.432i 212.786 + 233.927i
3.13 −0.946365 5.57713i 4.45554 + 4.45554i −30.2088 + 10.5560i 49.9881 25.0237i 20.6326 29.0657i −169.279 169.279i 87.4607 + 158.489i 203.296i −186.868 255.109i
3.14 −0.540478 5.63098i −8.83424 8.83424i −31.4158 + 6.08684i −16.1587 + 53.5154i −44.9707 + 54.5201i 48.2986 + 48.2986i 51.2543 + 173.612i 86.9124i 310.077 + 62.0653i
3.15 0.0746695 + 5.65636i −5.85746 5.85746i −31.9888 + 0.844715i 13.4330 + 54.2637i 32.6945 33.5693i −100.487 100.487i −7.16660 180.877i 174.380i −305.932 + 80.0339i
3.16 0.488434 + 5.63573i 17.2238 + 17.2238i −31.5229 + 5.50537i −54.8737 + 10.6715i −88.6558 + 105.481i 14.7411 + 14.7411i −46.4236 174.965i 350.316i −86.9437 304.041i
3.17 1.30166 5.50506i 20.2357 + 20.2357i −28.6114 14.3314i 6.76469 + 55.4909i 137.739 85.0587i 37.6042 + 37.6042i −116.138 + 138.853i 575.965i 314.286 + 34.9902i
3.18 2.14651 5.23378i −17.0411 17.0411i −22.7850 22.4688i 28.8684 47.8708i −125.768 + 52.6104i 87.4493 + 87.4493i −166.505 + 71.0220i 337.797i −188.579 253.846i
3.19 3.12884 4.71279i 2.84747 + 2.84747i −12.4207 29.4911i −51.8149 20.9812i 22.3288 4.51023i −44.8183 44.8183i −177.848 33.7370i 226.784i −261.001 + 178.546i
3.20 3.28191 + 4.60750i −7.52871 7.52871i −10.4581 + 30.2428i −50.5551 23.8576i 9.97997 59.3971i −51.0903 51.0903i −173.666 + 51.0685i 129.637i −55.9936 311.231i
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 27.28 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.c odd 4 1 inner
8.d odd 2 1 inner
40.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.6.k.a 56
4.b odd 2 1 160.6.o.a 56
5.c odd 4 1 inner 40.6.k.a 56
8.b even 2 1 160.6.o.a 56
8.d odd 2 1 inner 40.6.k.a 56
20.e even 4 1 160.6.o.a 56
40.i odd 4 1 160.6.o.a 56
40.k even 4 1 inner 40.6.k.a 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.k.a 56 1.a even 1 1 trivial
40.6.k.a 56 5.c odd 4 1 inner
40.6.k.a 56 8.d odd 2 1 inner
40.6.k.a 56 40.k even 4 1 inner
160.6.o.a 56 4.b odd 2 1
160.6.o.a 56 8.b even 2 1
160.6.o.a 56 20.e even 4 1
160.6.o.a 56 40.i odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(40, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database