# Properties

 Label 143.4.j.a Level $143$ Weight $4$ Character orbit 143.j Analytic conductor $8.437$ Analytic rank $0$ Dimension $72$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$143 = 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 143.j (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.43727313082$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$36$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$72 q - 12 q^{3} + 152 q^{4} + 90 q^{6} - 36 q^{7} - 360 q^{9}+O(q^{10})$$ 72 * q - 12 * q^3 + 152 * q^4 + 90 * q^6 - 36 * q^7 - 360 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$72 q - 12 q^{3} + 152 q^{4} + 90 q^{6} - 36 q^{7} - 360 q^{9} - 56 q^{10} - 132 q^{12} - 46 q^{13} + 328 q^{14} - 644 q^{16} + 138 q^{17} + 492 q^{19} + 540 q^{20} - 44 q^{22} + 46 q^{23} + 720 q^{24} - 1636 q^{25} - 902 q^{26} + 48 q^{27} - 714 q^{28} - 262 q^{29} + 104 q^{30} + 144 q^{32} - 68 q^{35} + 1960 q^{36} + 630 q^{37} - 448 q^{38} - 1612 q^{39} + 216 q^{40} + 126 q^{41} - 82 q^{42} + 436 q^{43} - 570 q^{45} - 1590 q^{46} + 1944 q^{48} + 1192 q^{49} + 1290 q^{50} - 1424 q^{51} + 590 q^{52} + 292 q^{53} - 528 q^{54} - 440 q^{55} - 102 q^{56} - 1128 q^{58} + 504 q^{59} + 590 q^{61} + 1776 q^{62} + 1884 q^{63} - 5028 q^{64} + 994 q^{65} + 2156 q^{66} + 3396 q^{67} + 1530 q^{68} + 12 q^{69} - 1014 q^{71} - 1062 q^{72} - 1568 q^{74} + 4688 q^{75} + 7872 q^{76} - 1232 q^{77} - 4964 q^{78} - 2928 q^{79} - 558 q^{80} - 3780 q^{81} - 4072 q^{82} + 696 q^{84} + 4662 q^{85} + 1832 q^{87} + 924 q^{88} + 1014 q^{89} + 6844 q^{90} + 2900 q^{91} - 1336 q^{92} - 4092 q^{93} + 4166 q^{94} - 256 q^{95} - 5070 q^{97} - 8550 q^{98}+O(q^{100})$$ 72 * q - 12 * q^3 + 152 * q^4 + 90 * q^6 - 36 * q^7 - 360 * q^9 - 56 * q^10 - 132 * q^12 - 46 * q^13 + 328 * q^14 - 644 * q^16 + 138 * q^17 + 492 * q^19 + 540 * q^20 - 44 * q^22 + 46 * q^23 + 720 * q^24 - 1636 * q^25 - 902 * q^26 + 48 * q^27 - 714 * q^28 - 262 * q^29 + 104 * q^30 + 144 * q^32 - 68 * q^35 + 1960 * q^36 + 630 * q^37 - 448 * q^38 - 1612 * q^39 + 216 * q^40 + 126 * q^41 - 82 * q^42 + 436 * q^43 - 570 * q^45 - 1590 * q^46 + 1944 * q^48 + 1192 * q^49 + 1290 * q^50 - 1424 * q^51 + 590 * q^52 + 292 * q^53 - 528 * q^54 - 440 * q^55 - 102 * q^56 - 1128 * q^58 + 504 * q^59 + 590 * q^61 + 1776 * q^62 + 1884 * q^63 - 5028 * q^64 + 994 * q^65 + 2156 * q^66 + 3396 * q^67 + 1530 * q^68 + 12 * q^69 - 1014 * q^71 - 1062 * q^72 - 1568 * q^74 + 4688 * q^75 + 7872 * q^76 - 1232 * q^77 - 4964 * q^78 - 2928 * q^79 - 558 * q^80 - 3780 * q^81 - 4072 * q^82 + 696 * q^84 + 4662 * q^85 + 1832 * q^87 + 924 * q^88 + 1014 * q^89 + 6844 * q^90 + 2900 * q^91 - 1336 * q^92 - 4092 * q^93 + 4166 * q^94 - 256 * q^95 - 5070 * q^97 - 8550 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1 −4.68493 + 2.70485i −4.52492 7.83739i 10.6324 18.4159i 14.9789i 42.3979 + 24.4784i −18.3113 10.5720i 71.7586i −27.4498 + 47.5444i −40.5158 70.1754i
23.2 −4.60341 + 2.65778i 3.48806 + 6.04150i 10.1276 17.5415i 11.0780i −32.1139 18.5410i −13.4565 7.76914i 65.1431i −10.8331 + 18.7635i 29.4429 + 50.9966i
23.3 −4.40125 + 2.54106i 0.865871 + 1.49973i 8.91398 15.4395i 8.37251i −7.62183 4.40046i −3.75563 2.16832i 49.9469i 12.0005 20.7855i −21.2751 36.8495i
23.4 −4.25281 + 2.45536i −2.48451 4.30330i 8.05759 13.9562i 14.5097i 21.1323 + 12.2007i 21.5060 + 12.4165i 39.8514i 1.15443 1.99953i 35.6265 + 61.7070i
23.5 −4.05448 + 2.34086i −0.806074 1.39616i 6.95923 12.0537i 5.38645i 6.53643 + 3.77381i −1.17956 0.681020i 27.7085i 12.2005 21.1319i 12.6089 + 21.8393i
23.6 −3.65232 + 2.10867i 4.62634 + 8.01306i 4.89294 8.47483i 13.5656i −33.7937 19.5108i 13.5120 + 7.80115i 7.53168i −29.3061 + 50.7597i −28.6052 49.5457i
23.7 −3.43971 + 1.98592i −0.953208 1.65100i 3.88775 6.73378i 18.9601i 6.55752 + 3.78599i 20.6923 + 11.9467i 0.891668i 11.6828 20.2352i −37.6533 65.2174i
23.8 −3.05274 + 1.76250i −3.30496 5.72436i 2.21282 3.83272i 15.7848i 20.1784 + 11.6500i −30.7733 17.7670i 12.5996i −8.34556 + 14.4549i 27.8208 + 48.1870i
23.9 −2.99291 + 1.72796i 2.70873 + 4.69166i 1.97166 3.41501i 9.57590i −16.2140 9.36113i −24.1322 13.9328i 14.0195i −1.17444 + 2.03419i −16.5467 28.6598i
23.10 −2.76240 + 1.59487i −4.96121 8.59307i 1.08722 1.88313i 1.24067i 27.4097 + 15.8250i 23.6555 + 13.6575i 18.5820i −35.7273 + 61.8814i −1.97871 3.42722i
23.11 −2.65952 + 1.53547i −2.90866 5.03795i 0.715353 1.23903i 4.09794i 15.4713 + 8.93234i −10.5127 6.06949i 20.1739i −3.42061 + 5.92468i −6.29227 10.8985i
23.12 −2.21820 + 1.28068i 0.591773 + 1.02498i −0.719733 + 1.24661i 4.85042i −2.62534 1.51574i 5.49965 + 3.17522i 24.1778i 12.7996 22.1696i 6.21183 + 10.7592i
23.13 −1.40127 + 0.809025i 1.16324 + 2.01478i −2.69096 + 4.66087i 8.11969i −3.26002 1.88217i −7.84996 4.53218i 21.6526i 10.7938 18.6954i 6.56903 + 11.3779i
23.14 −0.943417 + 0.544682i −1.41111 2.44412i −3.40664 + 5.90048i 21.6301i 2.66254 + 1.53722i −24.7951 14.3155i 16.1371i 9.51752 16.4848i −11.7815 20.4062i
23.15 −0.753305 + 0.434921i 4.96974 + 8.60785i −3.62169 + 6.27295i 11.3907i −7.48747 4.32289i −24.1895 13.9658i 13.2593i −35.8967 + 62.1749i 4.95404 + 8.58064i
23.16 −0.737315 + 0.425689i −2.76813 4.79455i −3.63758 + 6.30047i 7.50257i 4.08197 + 2.35673i 7.20298 + 4.15864i 13.0049i −1.82513 + 3.16122i −3.19376 5.53176i
23.17 −0.586414 + 0.338566i −2.81095 4.86870i −3.77075 + 6.53112i 21.4343i 3.29676 + 1.90338i 9.06253 + 5.23226i 10.5236i −2.30284 + 3.98864i 7.25695 + 12.5694i
23.18 −0.575952 + 0.332526i 1.60738 + 2.78407i −3.77885 + 6.54517i 10.5229i −1.85155 1.06899i 25.1101 + 14.4973i 10.3467i 8.33263 14.4325i −3.49915 6.06070i
23.19 −0.204845 + 0.118267i 3.87126 + 6.70523i −3.97203 + 6.87975i 11.7890i −1.58602 0.915689i 4.94948 + 2.85759i 3.77132i −16.4734 + 28.5327i −1.39425 2.41491i
23.20 0.200845 0.115958i −4.34402 7.52406i −3.97311 + 6.88162i 0.297636i −1.74495 1.00745i −6.82685 3.94148i 3.69817i −24.2410 + 41.9867i −0.0345132 0.0597786i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 56.36 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.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 143.4.j.a 72
13.e even 6 1 inner 143.4.j.a 72
13.f odd 12 1 1859.4.a.l 36
13.f odd 12 1 1859.4.a.m 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.j.a 72 1.a even 1 1 trivial
143.4.j.a 72 13.e even 6 1 inner
1859.4.a.l 36 13.f odd 12 1
1859.4.a.m 36 13.f odd 12 1

## Hecke kernels

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