# Properties

 Label 143.4.o.a Level $143$ Weight $4$ Character orbit 143.o Analytic conductor $8.437$ Analytic rank $0$ Dimension $160$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$143 = 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 143.o (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.43727313082$$ Analytic rank: $$0$$ Dimension: $$160$$ Relative dimension: $$40$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$160 q - 4 q^{3} - 12 q^{4} - 8 q^{5} - 652 q^{9}+O(q^{10})$$ 160 * q - 4 * q^3 - 12 * q^4 - 8 * q^5 - 652 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$160 q - 4 q^{3} - 12 q^{4} - 8 q^{5} - 652 q^{9} - 80 q^{11} - 64 q^{14} - 76 q^{15} + 940 q^{16} + 68 q^{20} + 28 q^{22} - 240 q^{23} + 496 q^{26} + 824 q^{27} - 280 q^{31} - 266 q^{33} + 2212 q^{34} - 2760 q^{36} + 328 q^{37} + 1164 q^{42} + 104 q^{44} + 896 q^{45} + 4 q^{47} + 2080 q^{48} - 12 q^{49} - 6528 q^{53} + 682 q^{55} - 1356 q^{56} + 1096 q^{58} - 1392 q^{59} + 4 q^{60} - 1880 q^{66} + 304 q^{67} - 12 q^{69} - 1932 q^{70} - 5076 q^{71} + 8832 q^{75} - 10876 q^{78} + 4588 q^{80} - 4624 q^{81} - 7716 q^{82} + 5608 q^{86} + 10152 q^{88} + 7268 q^{89} - 1008 q^{91} + 8120 q^{92} + 2740 q^{93} + 2728 q^{97} - 6996 q^{99}+O(q^{100})$$ 160 * q - 4 * q^3 - 12 * q^4 - 8 * q^5 - 652 * q^9 - 80 * q^11 - 64 * q^14 - 76 * q^15 + 940 * q^16 + 68 * q^20 + 28 * q^22 - 240 * q^23 + 496 * q^26 + 824 * q^27 - 280 * q^31 - 266 * q^33 + 2212 * q^34 - 2760 * q^36 + 328 * q^37 + 1164 * q^42 + 104 * q^44 + 896 * q^45 + 4 * q^47 + 2080 * q^48 - 12 * q^49 - 6528 * q^53 + 682 * q^55 - 1356 * q^56 + 1096 * q^58 - 1392 * q^59 + 4 * q^60 - 1880 * q^66 + 304 * q^67 - 12 * q^69 - 1932 * q^70 - 5076 * q^71 + 8832 * q^75 - 10876 * q^78 + 4588 * q^80 - 4624 * q^81 - 7716 * q^82 + 5608 * q^86 + 10152 * q^88 + 7268 * q^89 - 1008 * q^91 + 8120 * q^92 + 2740 * q^93 + 2728 * q^97 - 6996 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1 −5.35695 + 1.43539i 1.66657 2.88658i 19.7084 11.3787i −3.25582 + 3.25582i −4.78435 + 17.8554i 5.43090 + 1.45520i −57.8717 + 57.8717i 7.94512 + 13.7614i 12.7679 22.1147i
32.2 −4.87783 + 1.30701i −4.77756 + 8.27498i 15.1567 8.75074i −11.5708 + 11.5708i 12.4886 46.6082i 20.5354 + 5.50244i −33.9281 + 33.9281i −32.1502 55.6857i 41.3170 71.5632i
32.3 −4.78621 + 1.28246i −3.24999 + 5.62914i 14.3349 8.27626i 13.3984 13.3984i 8.33597 31.1102i −2.07549 0.556126i −29.9659 + 29.9659i −7.62484 13.2066i −46.9448 + 81.3108i
32.4 −4.63702 + 1.24249i −1.54154 + 2.67002i 13.0300 7.52287i 1.08150 1.08150i 3.83067 14.2963i −13.7466 3.68339i −23.9170 + 23.9170i 8.74733 + 15.1508i −3.67118 + 6.35868i
32.5 −4.40813 + 1.18115i 2.35756 4.08341i 11.1083 6.41337i 2.88937 2.88937i −5.56928 + 20.7848i 13.7739 + 3.69069i −15.5758 + 15.5758i 2.38386 + 4.12898i −9.32392 + 16.1495i
32.6 −4.16335 + 1.11557i 5.11454 8.85864i 9.16082 5.28900i −7.67818 + 7.67818i −11.4112 + 42.5873i −0.565842 0.151617i −7.85719 + 7.85719i −38.8170 67.2330i 23.4015 40.5325i
32.7 −4.08019 + 1.09328i −0.854728 + 1.48043i 8.52451 4.92163i −15.4234 + 15.4234i 1.86892 6.97492i −23.9363 6.41370i −5.50565 + 5.50565i 12.0389 + 20.8519i 46.0684 79.7929i
32.8 −3.61125 + 0.967632i 3.54274 6.13621i 5.17664 2.98873i 3.98133 3.98133i −6.85614 + 25.5875i −33.6422 9.01441i 5.34682 5.34682i −11.6020 20.0953i −10.5251 + 18.2300i
32.9 −3.51428 + 0.941650i 2.24550 3.88932i 4.53529 2.61845i 14.9878 14.9878i −4.22895 + 15.7827i 11.9754 + 3.20880i 7.10844 7.10844i 3.41546 + 5.91574i −38.5582 + 66.7847i
32.10 −3.10849 + 0.832916i −2.01807 + 3.49539i 2.04073 1.17821i −2.38665 + 2.38665i 3.36176 12.5463i 25.0107 + 6.70161i 12.8423 12.8423i 5.35482 + 9.27482i 5.43099 9.40675i
32.11 −2.93374 + 0.786092i −2.23560 + 3.87217i 1.06067 0.612376i 4.09175 4.09175i 3.51478 13.1173i −10.9623 2.93733i 14.5508 14.5508i 3.50418 + 6.06942i −8.78763 + 15.2206i
32.12 −2.78174 + 0.745366i 0.496440 0.859860i 0.254316 0.146829i −7.62492 + 7.62492i −0.740059 + 2.76194i 25.1138 + 6.72922i 15.6930 15.6930i 13.0071 + 22.5289i 15.5272 26.8939i
32.13 −2.64615 + 0.709034i −4.79462 + 8.30452i −0.428815 + 0.247577i 0.391471 0.391471i 6.79910 25.3746i −16.3415 4.37870i 16.4561 16.4561i −32.4767 56.2513i −0.758325 + 1.31346i
32.14 −1.81132 + 0.485341i 1.94678 3.37192i −3.88289 + 2.24179i −10.0128 + 10.0128i −1.88970 + 7.05246i −12.3598 3.31179i 16.5529 16.5529i 5.92011 + 10.2539i 13.2767 22.9960i
32.15 −1.41412 + 0.378911i 1.05641 1.82975i −5.07206 + 2.92835i 4.68157 4.68157i −0.800569 + 2.98776i −20.9341 5.60929i 14.3445 14.3445i 11.2680 + 19.5168i −4.84638 + 8.39418i
32.16 −1.32670 + 0.355489i 3.59215 6.22179i −5.29444 + 3.05674i −7.26866 + 7.26866i −2.55394 + 9.53143i 15.2517 + 4.08668i 13.7072 13.7072i −12.3071 21.3165i 7.05942 12.2273i
32.17 −1.15192 + 0.308657i −3.89437 + 6.74525i −5.69654 + 3.28890i 12.2876 12.2876i 2.40405 8.97204i 32.4067 + 8.68335i 12.2930 12.2930i −16.8323 29.1543i −10.3617 + 17.9470i
32.18 −0.924432 + 0.247701i 4.37390 7.57581i −6.13498 + 3.54203i 8.53113 8.53113i −2.16684 + 8.08675i 20.7059 + 5.54814i 10.2079 10.2079i −24.7620 42.8890i −5.77328 + 9.99962i
32.19 −0.740490 + 0.198414i −0.446262 + 0.772948i −6.41925 + 3.70615i 7.28686 7.28686i 0.177089 0.660905i −5.35131 1.43388i 8.35464 8.35464i 13.1017 + 22.6928i −3.95003 + 6.84166i
32.20 −0.300514 + 0.0805224i −3.07985 + 5.33445i −6.84438 + 3.95160i −9.38759 + 9.38759i 0.495993 1.85107i −9.89724 2.65196i 3.49857 3.49857i −5.47090 9.47587i 2.06519 3.57701i
See next 80 embeddings (of 160 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 98.40 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
13.f odd 12 1 inner
143.o even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 143.4.o.a 160
11.b odd 2 1 inner 143.4.o.a 160
13.f odd 12 1 inner 143.4.o.a 160
143.o even 12 1 inner 143.4.o.a 160

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.o.a 160 1.a even 1 1 trivial
143.4.o.a 160 11.b odd 2 1 inner
143.4.o.a 160 13.f odd 12 1 inner
143.4.o.a 160 143.o even 12 1 inner

## Hecke kernels

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