# Properties

 Label 143.4.b.a Level $143$ Weight $4$ Character orbit 143.b Analytic conductor $8.437$ Analytic rank $0$ Dimension $36$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$143 = 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 143.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.43727313082$$ Analytic rank: $$0$$ Dimension: $$36$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$36 q - 152 q^{4} + 360 q^{9}+O(q^{10})$$ 36 * q - 152 * q^4 + 360 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$36 q - 152 q^{4} + 360 q^{9} - 112 q^{10} - 108 q^{12} - 50 q^{13} + 8 q^{14} + 728 q^{16} + 276 q^{17} + 44 q^{22} - 472 q^{23} - 1172 q^{25} + 152 q^{26} - 12 q^{27} - 572 q^{29} + 712 q^{30} + 68 q^{35} - 430 q^{36} - 50 q^{38} + 640 q^{39} - 216 q^{40} + 1126 q^{42} + 920 q^{43} + 1674 q^{48} - 2164 q^{49} - 340 q^{51} - 800 q^{52} + 2432 q^{53} + 440 q^{55} - 2274 q^{56} - 1844 q^{61} + 2796 q^{62} - 2592 q^{64} + 2264 q^{65} + 1078 q^{66} - 4548 q^{68} - 3288 q^{69} - 4036 q^{74} + 820 q^{75} - 616 q^{77} + 2222 q^{78} + 360 q^{79} + 852 q^{81} + 1948 q^{82} - 2480 q^{87} + 264 q^{88} - 496 q^{90} + 4600 q^{91} + 454 q^{92} - 488 q^{94} + 952 q^{95}+O(q^{100})$$ 36 * q - 152 * q^4 + 360 * q^9 - 112 * q^10 - 108 * q^12 - 50 * q^13 + 8 * q^14 + 728 * q^16 + 276 * q^17 + 44 * q^22 - 472 * q^23 - 1172 * q^25 + 152 * q^26 - 12 * q^27 - 572 * q^29 + 712 * q^30 + 68 * q^35 - 430 * q^36 - 50 * q^38 + 640 * q^39 - 216 * q^40 + 1126 * q^42 + 920 * q^43 + 1674 * q^48 - 2164 * q^49 - 340 * q^51 - 800 * q^52 + 2432 * q^53 + 440 * q^55 - 2274 * q^56 - 1844 * q^61 + 2796 * q^62 - 2592 * q^64 + 2264 * q^65 + 1078 * q^66 - 4548 * q^68 - 3288 * q^69 - 4036 * q^74 + 820 * q^75 - 616 * q^77 + 2222 * q^78 + 360 * q^79 + 852 * q^81 + 1948 * q^82 - 2480 * q^87 + 264 * q^88 - 496 * q^90 + 4600 * q^91 + 454 * q^92 - 488 * q^94 + 952 * q^95

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
12.1 5.51522i 3.00574 −22.4176 18.8802i 16.5773i 0.324979i 79.5165i −17.9655 104.128
12.2 5.22956i 1.03843 −19.3483 12.1782i 5.43054i 36.1879i 59.3469i −25.9217 −63.6866
12.3 4.98751i −4.47034 −16.8752 20.1605i 22.2959i 23.8481i 44.2654i −7.01607 −100.551
12.4 4.83238i 9.74664 −15.3519 8.44122i 47.0995i 12.5767i 35.5272i 67.9971 −40.7912
12.5 4.83029i −7.08819 −15.3317 9.07266i 34.2380i 2.15508i 35.4141i 23.2424 43.8235
12.6 4.27424i 4.33951 −10.2691 0.295728i 18.5481i 11.5737i 9.69883i −8.16862 1.26401
12.7 3.98165i −8.48796 −7.85352 2.25841i 33.7961i 28.5605i 0.583227i 45.0454 −8.99219
12.8 3.97618i −1.00306 −7.81003 0.345299i 3.98837i 17.7538i 0.755338i −25.9939 1.37297
12.9 3.37885i 7.92479 −3.41661 15.9579i 26.7767i 29.7230i 15.4866i 35.8023 53.9193
12.10 2.40428i 7.21858 2.21943 9.84845i 17.3555i 8.26104i 24.5704i 25.1078 −23.6784
12.11 2.37174i −6.00967 2.37485 22.0017i 14.2534i 21.9243i 24.6064i 9.11608 52.1824
12.12 2.32803i −4.42417 2.58028 12.1881i 10.2996i 11.2388i 24.6312i −7.42671 −28.3743
12.13 2.22320i 2.89827 3.05739 19.1319i 6.44342i 4.29594i 24.5828i −18.6000 −42.5339
12.14 2.18512i −9.62593 3.22525 10.6961i 21.0338i 26.7047i 24.5285i 65.6585 −23.3723
12.15 1.26989i 3.96847 6.38737 4.99906i 5.03953i 32.3730i 18.2704i −11.2513 6.34827
12.16 0.901974i 8.17246 7.18644 11.3696i 7.37135i 8.81530i 13.6978i 39.7891 10.2551
12.17 0.468651i −6.65460 7.78037 3.45502i 3.11869i 11.5900i 7.39549i 17.2837 −1.61920
12.18 0.370518i −0.548979 7.86272 11.6192i 0.203406i 19.9902i 5.87742i −26.6986 4.30511
12.19 0.370518i −0.548979 7.86272 11.6192i 0.203406i 19.9902i 5.87742i −26.6986 4.30511
12.20 0.468651i −6.65460 7.78037 3.45502i 3.11869i 11.5900i 7.39549i 17.2837 −1.61920
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 12.36 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 143.4.b.a 36
13.b even 2 1 inner 143.4.b.a 36
13.d odd 4 1 1859.4.a.j 18
13.d odd 4 1 1859.4.a.k 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.b.a 36 1.a even 1 1 trivial
143.4.b.a 36 13.b even 2 1 inner
1859.4.a.j 18 13.d odd 4 1
1859.4.a.k 18 13.d odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(143, [\chi])$$.