Properties

 Label 143.4.w.a Level $143$ Weight $4$ Character orbit 143.w Analytic conductor $8.437$ Analytic rank $0$ Dimension $640$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$143 = 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 143.w (of order $$60$$, degree $$16$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$8.43727313082$$ Analytic rank: $$0$$ Dimension: $$640$$ Relative dimension: $$40$$ over $$\Q(\zeta_{60})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$640 q - 20 q^{2} - 6 q^{3} - 18 q^{4} - 12 q^{5} - 20 q^{6} - 20 q^{7} - 20 q^{8} + 642 q^{9}+O(q^{10})$$ 640 * q - 20 * q^2 - 6 * q^3 - 18 * q^4 - 12 * q^5 - 20 * q^6 - 20 * q^7 - 20 * q^8 + 642 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$640 q - 20 q^{2} - 6 q^{3} - 18 q^{4} - 12 q^{5} - 20 q^{6} - 20 q^{7} - 20 q^{8} + 642 q^{9} - 140 q^{11} - 60 q^{13} + 24 q^{14} - 144 q^{15} - 1310 q^{16} - 30 q^{17} - 20 q^{18} - 20 q^{19} + 942 q^{20} - 38 q^{22} - 960 q^{23} - 760 q^{24} - 516 q^{26} - 864 q^{27} - 20 q^{28} - 710 q^{29} - 30 q^{30} + 260 q^{31} - 1354 q^{33} + 1508 q^{34} - 1310 q^{35} + 3540 q^{36} - 348 q^{37} - 1860 q^{39} + 1240 q^{40} + 2200 q^{41} - 1174 q^{42} - 124 q^{44} - 936 q^{45} + 1340 q^{46} - 2564 q^{47} - 490 q^{48} - 18 q^{49} + 1230 q^{50} + 1430 q^{52} - 3912 q^{53} - 692 q^{55} + 1296 q^{56} - 20 q^{57} - 2556 q^{58} - 668 q^{59} + 6556 q^{60} + 470 q^{61} - 30 q^{62} - 5010 q^{63} + 24980 q^{66} - 5384 q^{67} - 250 q^{68} - 18 q^{69} + 1262 q^{70} + 1296 q^{71} + 11020 q^{72} - 1940 q^{73} - 10 q^{74} - 8862 q^{75} + 10476 q^{78} - 9400 q^{79} + 12112 q^{80} + 4294 q^{81} - 9414 q^{82} - 4500 q^{83} + 920 q^{84} + 140 q^{85} - 4708 q^{86} - 10182 q^{88} - 2208 q^{89} + 9988 q^{91} - 9440 q^{92} + 13400 q^{93} - 3210 q^{94} - 9330 q^{95} - 6270 q^{96} + 4872 q^{97} + 6876 q^{99}+O(q^{100})$$ 640 * q - 20 * q^2 - 6 * q^3 - 18 * q^4 - 12 * q^5 - 20 * q^6 - 20 * q^7 - 20 * q^8 + 642 * q^9 - 140 * q^11 - 60 * q^13 + 24 * q^14 - 144 * q^15 - 1310 * q^16 - 30 * q^17 - 20 * q^18 - 20 * q^19 + 942 * q^20 - 38 * q^22 - 960 * q^23 - 760 * q^24 - 516 * q^26 - 864 * q^27 - 20 * q^28 - 710 * q^29 - 30 * q^30 + 260 * q^31 - 1354 * q^33 + 1508 * q^34 - 1310 * q^35 + 3540 * q^36 - 348 * q^37 - 1860 * q^39 + 1240 * q^40 + 2200 * q^41 - 1174 * q^42 - 124 * q^44 - 936 * q^45 + 1340 * q^46 - 2564 * q^47 - 490 * q^48 - 18 * q^49 + 1230 * q^50 + 1430 * q^52 - 3912 * q^53 - 692 * q^55 + 1296 * q^56 - 20 * q^57 - 2556 * q^58 - 668 * q^59 + 6556 * q^60 + 470 * q^61 - 30 * q^62 - 5010 * q^63 + 24980 * q^66 - 5384 * q^67 - 250 * q^68 - 18 * q^69 + 1262 * q^70 + 1296 * q^71 + 11020 * q^72 - 1940 * q^73 - 10 * q^74 - 8862 * q^75 + 10476 * q^78 - 9400 * q^79 + 12112 * q^80 + 4294 * q^81 - 9414 * q^82 - 4500 * q^83 + 920 * q^84 + 140 * q^85 - 4708 * q^86 - 10182 * q^88 - 2208 * q^89 + 9988 * q^91 - 9440 * q^92 + 13400 * q^93 - 3210 * q^94 - 9330 * q^95 - 6270 * q^96 + 4872 * q^97 + 6876 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −4.61575 2.99751i −0.866584 0.385828i 9.06625 + 20.3631i −16.7403 8.52960i 2.84342 + 4.37848i 9.35497 24.3705i 12.3033 77.6801i −17.4644 19.3962i 51.7015 + 89.5496i
2.2 −4.50283 2.92417i −5.02636 2.23788i 8.47080 + 19.0257i 4.12727 + 2.10295i 16.0889 + 24.7747i −10.6643 + 27.7815i 10.7727 68.0164i 2.18964 + 2.43184i −12.4350 21.5381i
2.3 −4.35952 2.83111i 5.31423 + 2.36605i 7.73636 + 17.3762i 13.2395 + 6.74586i −16.4690 25.3600i 4.68860 12.2142i 8.96156 56.5811i 4.57635 + 5.08255i −38.6196 66.8911i
2.4 −4.06520 2.63997i 8.74527 + 3.89365i 6.30249 + 14.1556i −10.6455 5.42415i −25.2721 38.9157i −7.80705 + 20.3381i 5.68342 35.8837i 43.2528 + 48.0371i 28.9564 + 50.1540i
2.5 −3.75529 2.43872i −3.69699 1.64600i 4.90100 + 11.0078i 8.76581 + 4.46640i 9.86913 + 15.1971i 6.22368 16.2132i 2.83656 17.9093i −7.10815 7.89440i −22.0259 38.1500i
2.6 −3.73512 2.42562i 2.56251 + 1.14090i 4.81363 + 10.8116i −2.56515 1.30701i −6.80388 10.4771i −4.48558 + 11.6853i 2.67168 16.8683i −12.8017 14.2178i 6.41083 + 11.1039i
2.7 −3.61764 2.34932i −8.83031 3.93151i 4.31410 + 9.68962i −6.26308 3.19120i 22.7085 + 34.9680i 2.63177 6.85599i 1.75890 11.1053i 44.4510 + 49.3679i 15.1604 + 26.2586i
2.8 −3.12805 2.03138i 5.19148 + 2.31140i 2.40430 + 5.40015i 0.717477 + 0.365573i −11.5439 17.7760i 4.70905 12.2675i −1.21874 + 7.69482i 3.54240 + 3.93424i −1.50169 2.60100i
2.9 −2.96020 1.92238i −4.52412 2.01427i 1.81337 + 4.07289i −3.12928 1.59445i 9.52013 + 14.6597i −1.90673 + 4.96720i −1.95556 + 12.3469i −1.65614 1.83933i 6.19818 + 10.7356i
2.10 −2.94463 1.91226i 1.58522 + 0.705785i 1.76019 + 3.95345i 14.9771 + 7.63121i −3.31823 5.10963i −8.30045 + 21.6234i −2.01708 + 12.7353i −16.0517 17.8273i −29.5091 51.1113i
2.11 −2.62944 1.70758i 1.89477 + 0.843607i 0.744240 + 1.67159i −16.7269 8.52278i −3.54166 5.45368i 1.54432 4.02309i −3.02625 + 19.1070i −15.1880 16.8680i 29.4291 + 50.9727i
2.12 −2.05191 1.33253i −4.55695 2.02888i −0.819179 1.83991i −15.5428 7.91944i 6.64692 + 10.2354i −9.19210 + 23.9462i −3.83273 + 24.1989i −1.41711 1.57386i 21.3395 + 36.9612i
2.13 −2.03945 1.32443i −3.45137 1.53665i −0.848666 1.90614i 7.68362 + 3.91500i 5.00371 + 7.70503i 11.4289 29.7734i −3.83703 + 24.2261i −8.51585 9.45781i −10.4852 18.1609i
2.14 −1.60685 1.04350i 8.46099 + 3.76707i −1.76082 3.95486i 14.7384 + 7.50959i −9.66460 14.8822i 4.46294 11.6264i −3.69530 + 23.3312i 39.3309 + 43.6814i −15.8462 27.4464i
2.15 −1.46478 0.951240i −9.44006 4.20299i −2.01317 4.52165i 13.4542 + 6.85525i 9.82957 + 15.1362i −4.19617 + 10.9314i −3.53809 + 22.3386i 53.3831 + 59.2880i −13.1864 22.8396i
2.16 −1.41854 0.921209i 4.78676 + 2.13120i −2.09027 4.69482i −6.68912 3.40828i −4.82692 7.43280i 7.57564 19.7352i −3.47655 + 21.9501i 0.304533 + 0.338218i 6.34904 + 10.9969i
2.17 −1.28263 0.832949i 7.41783 + 3.30263i −2.30256 5.17164i −1.35664 0.691245i −6.76340 10.4147i −10.4457 + 27.2119i −3.26833 + 20.6354i 26.0503 + 28.9318i 1.16430 + 2.01662i
2.18 −1.26284 0.820099i −2.99685 1.33428i −2.33169 5.23705i 14.0544 + 7.16106i 2.69030 + 4.14269i −4.84976 + 12.6340i −3.23478 + 20.4236i −10.8658 12.0676i −11.8757 20.5692i
2.19 −0.293417 0.190547i 0.878279 + 0.391035i −3.20411 7.19654i 0.705520 + 0.359480i −0.183191 0.282090i −6.01679 + 15.6743i −0.868983 + 5.48654i −17.4481 19.3780i −0.138514 0.239913i
2.20 −0.257450 0.167190i −4.17549 1.85905i −3.21557 7.22228i −8.58006 4.37176i 0.764164 + 1.17671i 1.42206 3.70460i −0.763815 + 4.82254i −4.08788 4.54005i 1.47802 + 2.56000i
See next 80 embeddings (of 640 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 128.40 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.d odd 10 1 inner
13.f odd 12 1 inner
143.w even 60 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 143.4.w.a 640
11.d odd 10 1 inner 143.4.w.a 640
13.f odd 12 1 inner 143.4.w.a 640
143.w even 60 1 inner 143.4.w.a 640

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.w.a 640 1.a even 1 1 trivial
143.4.w.a 640 11.d odd 10 1 inner
143.4.w.a 640 13.f odd 12 1 inner
143.4.w.a 640 143.w even 60 1 inner

Hecke kernels

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