Properties

 Label 143.4.q.a Level $143$ Weight $4$ Character orbit 143.q Analytic conductor $8.437$ Analytic rank $0$ Dimension $320$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$143 = 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 143.q (of order $$15$$, degree $$8$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$8.43727313082$$ Analytic rank: $$0$$ Dimension: $$320$$ Relative dimension: $$40$$ over $$\Q(\zeta_{15})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$320 q - 3 q^{2} - 3 q^{3} + 149 q^{4} - 12 q^{5} - 35 q^{6} - 3 q^{7} - 44 q^{8} + 369 q^{9}+O(q^{10})$$ 320 * q - 3 * q^2 - 3 * q^3 + 149 * q^4 - 12 * q^5 - 35 * q^6 - 3 * q^7 - 44 * q^8 + 369 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$320 q - 3 q^{2} - 3 q^{3} + 149 q^{4} - 12 q^{5} - 35 q^{6} - 3 q^{7} - 44 q^{8} + 369 q^{9} - 80 q^{10} + 30 q^{11} - 220 q^{12} - 114 q^{13} + 4 q^{14} - 33 q^{15} + 637 q^{16} + 143 q^{17} + 72 q^{18} + 17 q^{19} - 377 q^{20} - 524 q^{21} + 283 q^{22} + 196 q^{23} - 1137 q^{24} - 1324 q^{25} + 632 q^{26} + 192 q^{27} - 279 q^{28} + 943 q^{29} + 1617 q^{30} + 260 q^{31} - 52 q^{32} - 1244 q^{33} - 3508 q^{34} + 1253 q^{35} + 282 q^{36} + 333 q^{37} - 694 q^{38} - 1106 q^{39} - 296 q^{40} - 1751 q^{41} - 895 q^{42} - 2200 q^{43} - 252 q^{44} - 282 q^{45} + 485 q^{46} + 2116 q^{47} - 825 q^{48} + 1607 q^{49} + 348 q^{50} + 1768 q^{51} - 2431 q^{52} + 428 q^{53} + 8326 q^{54} - 1379 q^{55} + 7630 q^{56} - 888 q^{57} - 331 q^{58} - 1097 q^{59} - 1820 q^{60} + 885 q^{61} - 437 q^{62} + 540 q^{63} + 1148 q^{64} + 2324 q^{65} - 1686 q^{66} - 5184 q^{67} + 3834 q^{68} - 4905 q^{69} + 4078 q^{70} + 843 q^{71} + 1155 q^{72} - 972 q^{73} + 5631 q^{74} - 3431 q^{75} - 3090 q^{76} + 602 q^{77} + 1252 q^{78} + 1140 q^{79} + 8631 q^{80} - 3809 q^{81} + 4015 q^{82} - 5292 q^{83} - 5060 q^{84} + 391 q^{85} + 3388 q^{86} - 7600 q^{87} - 7229 q^{88} + 6276 q^{89} + 7374 q^{90} - 4602 q^{91} + 8158 q^{92} - 2599 q^{93} + 11081 q^{94} + 2305 q^{95} - 30114 q^{96} + 1377 q^{97} + 5102 q^{98} - 5812 q^{99}+O(q^{100})$$ 320 * q - 3 * q^2 - 3 * q^3 + 149 * q^4 - 12 * q^5 - 35 * q^6 - 3 * q^7 - 44 * q^8 + 369 * q^9 - 80 * q^10 + 30 * q^11 - 220 * q^12 - 114 * q^13 + 4 * q^14 - 33 * q^15 + 637 * q^16 + 143 * q^17 + 72 * q^18 + 17 * q^19 - 377 * q^20 - 524 * q^21 + 283 * q^22 + 196 * q^23 - 1137 * q^24 - 1324 * q^25 + 632 * q^26 + 192 * q^27 - 279 * q^28 + 943 * q^29 + 1617 * q^30 + 260 * q^31 - 52 * q^32 - 1244 * q^33 - 3508 * q^34 + 1253 * q^35 + 282 * q^36 + 333 * q^37 - 694 * q^38 - 1106 * q^39 - 296 * q^40 - 1751 * q^41 - 895 * q^42 - 2200 * q^43 - 252 * q^44 - 282 * q^45 + 485 * q^46 + 2116 * q^47 - 825 * q^48 + 1607 * q^49 + 348 * q^50 + 1768 * q^51 - 2431 * q^52 + 428 * q^53 + 8326 * q^54 - 1379 * q^55 + 7630 * q^56 - 888 * q^57 - 331 * q^58 - 1097 * q^59 - 1820 * q^60 + 885 * q^61 - 437 * q^62 + 540 * q^63 + 1148 * q^64 + 2324 * q^65 - 1686 * q^66 - 5184 * q^67 + 3834 * q^68 - 4905 * q^69 + 4078 * q^70 + 843 * q^71 + 1155 * q^72 - 972 * q^73 + 5631 * q^74 - 3431 * q^75 - 3090 * q^76 + 602 * q^77 + 1252 * q^78 + 1140 * q^79 + 8631 * q^80 - 3809 * q^81 + 4015 * q^82 - 5292 * q^83 - 5060 * q^84 + 391 * q^85 + 3388 * q^86 - 7600 * q^87 - 7229 * q^88 + 6276 * q^89 + 7374 * q^90 - 4602 * q^91 + 8158 * q^92 - 2599 * q^93 + 11081 * q^94 + 2305 * q^95 - 30114 * q^96 + 1377 * q^97 + 5102 * q^98 - 5812 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −5.10212 2.27161i 1.38169 1.53452i 15.5183 + 17.2349i −0.783047 0.568917i −10.5353 + 4.69064i −13.2389 14.7033i −26.2187 80.6929i 2.37658 + 22.6117i 2.70284 + 4.68145i
3.2 −4.67876 2.08312i 6.53114 7.25356i 12.1984 + 13.5477i −0.321594 0.233652i −45.6677 + 20.3325i 19.0762 + 21.1863i −16.1907 49.8299i −7.13615 67.8960i 1.01794 + 1.76312i
3.3 −4.47044 1.99037i −5.27677 + 5.86045i 10.6702 + 11.8505i 11.7028 + 8.50257i 35.2539 15.6961i 10.5278 + 11.6923i −12.0164 36.9826i −3.67827 34.9964i −35.3934 61.3031i
3.4 −4.29137 1.91064i −0.857939 + 0.952838i 9.41224 + 10.4533i −6.11754 4.44465i 5.50226 2.44976i 3.47282 + 3.85696i −8.80595 27.1019i 2.65043 + 25.2171i 17.7605 + 30.7620i
3.5 −4.23464 1.88538i 0.592838 0.658414i 9.02446 + 10.0227i 14.0266 + 10.1910i −3.75182 + 1.67042i 6.56606 + 7.29235i −7.85943 24.1888i 2.74022 + 26.0714i −40.1839 69.6006i
3.6 −4.06062 1.80790i −3.81539 + 4.23742i 7.86706 + 8.73726i −15.6413 11.3641i 23.1537 10.3087i 19.4235 + 21.5720i −5.16062 15.8827i −0.576259 5.48274i 42.9683 + 74.4232i
3.7 −3.96685 1.76615i −6.00386 + 6.66796i 7.26353 + 8.06697i −4.50154 3.27056i 35.5930 15.8470i −18.3573 20.3879i −3.83117 11.7911i −5.59310 53.2148i 12.0806 + 20.9242i
3.8 −3.83132 1.70581i 3.43033 3.80977i 6.41615 + 7.12585i −11.3449 8.24252i −19.6414 + 8.74492i −6.22750 6.91634i −2.05903 6.33706i 0.0751056 + 0.714582i 29.4055 + 50.9319i
3.9 −3.42768 1.52610i 5.60413 6.22401i 4.06695 + 4.51680i 12.9835 + 9.43308i −28.7076 + 12.7815i −18.5875 20.6435i 2.22852 + 6.85867i −4.50984 42.9083i −30.1075 52.1477i
3.10 −3.20952 1.42897i −1.99079 + 2.21099i 2.90603 + 3.22747i 4.33588 + 3.15020i 9.54892 4.25145i −17.9899 19.9798i 3.97026 + 12.2192i 1.89701 + 18.0489i −9.41455 16.3065i
3.11 −2.70930 1.20626i 3.54938 3.94198i 0.532200 + 0.591068i 2.89861 + 2.10596i −14.3714 + 6.39855i 13.2416 + 14.7062i 6.60270 + 20.3210i −0.118883 1.13109i −5.31287 9.20216i
3.12 −2.22111 0.988904i −0.129116 + 0.143398i −1.39763 1.55222i −4.65106 3.37919i 0.428589 0.190820i 13.4245 + 14.9094i 7.57983 + 23.3283i 2.81838 + 26.8151i 6.98884 + 12.1050i
3.13 −2.18439 0.972553i −3.69451 + 4.10317i −1.52734 1.69629i 5.70526 + 4.14511i 12.0608 5.36982i −1.74948 1.94300i 7.59774 + 23.3834i −0.364326 3.46633i −8.43117 14.6032i
3.14 −2.05383 0.914426i 4.63025 5.14241i −1.97098 2.18900i −5.67493 4.12307i −14.2121 + 6.32764i −11.4861 12.7566i 7.60426 + 23.4035i −2.18293 20.7692i 7.88511 + 13.6574i
3.15 −1.47923 0.658595i −5.15589 + 5.72620i −3.59867 3.99673i 6.90200 + 5.01460i 11.3980 5.07471i 17.3187 + 19.2343i 6.69396 + 20.6019i −3.38386 32.1953i −6.90705 11.9634i
3.16 −1.34373 0.598266i −0.820388 + 0.911134i −3.90536 4.33735i −16.3994 11.9149i 1.64748 0.733505i −14.7799 16.4148i 6.28910 + 19.3559i 2.66514 + 25.3571i 14.9081 + 25.8216i
3.17 −0.978817 0.435797i −5.19277 + 5.76716i −4.58488 5.09203i −11.4291 8.30375i 7.59609 3.38200i −1.86862 2.07532i 4.91743 + 15.1343i −3.47296 33.0430i 7.56827 + 13.1086i
3.18 −0.868356 0.386617i 2.30554 2.56056i −4.74848 5.27372i 13.8209 + 10.0415i −2.99199 + 1.33212i 9.48972 + 10.5394i 4.43431 + 13.6474i 1.58131 + 15.0451i −8.11928 14.0630i
3.19 −0.437809 0.194925i −1.51751 + 1.68536i −5.19936 5.77448i 16.7692 + 12.1836i 0.992898 0.442067i −15.7902 17.5368i 2.33549 + 7.18789i 2.28465 + 21.7370i −4.96684 8.60281i
3.20 −0.257195 0.114511i 6.30663 7.00422i −5.30001 5.88626i −15.3363 11.1425i −2.42409 + 1.07927i 15.1685 + 16.8463i 1.38509 + 4.26287i −6.46326 61.4939i 2.66849 + 4.62197i
See next 80 embeddings (of 320 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 126.40 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.c even 5 1 inner
13.c even 3 1 inner
143.q even 15 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 143.4.q.a 320
11.c even 5 1 inner 143.4.q.a 320
13.c even 3 1 inner 143.4.q.a 320
143.q even 15 1 inner 143.4.q.a 320

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.q.a 320 1.a even 1 1 trivial
143.4.q.a 320 11.c even 5 1 inner
143.4.q.a 320 13.c even 3 1 inner
143.4.q.a 320 143.q even 15 1 inner

Hecke kernels

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