# Properties

 Label 143.4.e.b Level $143$ Weight $4$ Character orbit 143.e Analytic conductor $8.437$ Analytic rank $0$ Dimension $34$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$143 = 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 143.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.43727313082$$ Analytic rank: $$0$$ Dimension: $$34$$ Relative dimension: $$17$$ over $$\Q(\zeta_{3})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$34 q + 6 q^{3} - 50 q^{4} - 48 q^{5} - 16 q^{6} + 62 q^{7} - 42 q^{8} - 135 q^{9}+O(q^{10})$$ 34 * q + 6 * q^3 - 50 * q^4 - 48 * q^5 - 16 * q^6 + 62 * q^7 - 42 * q^8 - 135 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$34 q + 6 q^{3} - 50 q^{4} - 48 q^{5} - 16 q^{6} + 62 q^{7} - 42 q^{8} - 135 q^{9} - 2 q^{10} - 187 q^{11} - 254 q^{12} + 76 q^{13} + 148 q^{15} - 126 q^{16} + 74 q^{17} + 180 q^{18} + 159 q^{19} + 222 q^{20} - 368 q^{21} + 215 q^{23} - 214 q^{24} + 190 q^{25} + 123 q^{26} - 384 q^{27} + 358 q^{28} + 157 q^{29} - 829 q^{30} - 788 q^{31} + 553 q^{32} + 66 q^{33} - 1404 q^{34} - 58 q^{35} + 700 q^{36} - 88 q^{37} - 2636 q^{38} + 798 q^{39} + 1466 q^{40} + 512 q^{41} - 337 q^{42} - 927 q^{43} + 1100 q^{44} + 1482 q^{45} + 1361 q^{46} - 286 q^{47} + 178 q^{48} - 1835 q^{49} + 583 q^{50} - 1136 q^{51} + 2306 q^{52} + 212 q^{53} + 67 q^{54} + 264 q^{55} - 2059 q^{56} + 2596 q^{57} + 1690 q^{58} + 266 q^{59} + 74 q^{60} + 624 q^{61} - 643 q^{62} + 2360 q^{63} - 3178 q^{64} + 470 q^{65} + 352 q^{66} + 676 q^{67} + 413 q^{68} - 764 q^{69} - 2122 q^{70} + 763 q^{71} + 1366 q^{72} - 4748 q^{73} + 1649 q^{74} - 2420 q^{75} + 2101 q^{76} - 1364 q^{77} - 5848 q^{78} + 4328 q^{79} + 1013 q^{80} - 537 q^{81} - 3152 q^{82} + 1554 q^{83} + 3381 q^{84} + 1690 q^{85} + 5788 q^{86} + 4200 q^{87} + 231 q^{88} + 1687 q^{89} - 10798 q^{90} - 3380 q^{91} + 11084 q^{92} + 4310 q^{93} - 1777 q^{94} - 1124 q^{95} - 6930 q^{96} + 2047 q^{97} - 1553 q^{98} + 2970 q^{99}+O(q^{100})$$ 34 * q + 6 * q^3 - 50 * q^4 - 48 * q^5 - 16 * q^6 + 62 * q^7 - 42 * q^8 - 135 * q^9 - 2 * q^10 - 187 * q^11 - 254 * q^12 + 76 * q^13 + 148 * q^15 - 126 * q^16 + 74 * q^17 + 180 * q^18 + 159 * q^19 + 222 * q^20 - 368 * q^21 + 215 * q^23 - 214 * q^24 + 190 * q^25 + 123 * q^26 - 384 * q^27 + 358 * q^28 + 157 * q^29 - 829 * q^30 - 788 * q^31 + 553 * q^32 + 66 * q^33 - 1404 * q^34 - 58 * q^35 + 700 * q^36 - 88 * q^37 - 2636 * q^38 + 798 * q^39 + 1466 * q^40 + 512 * q^41 - 337 * q^42 - 927 * q^43 + 1100 * q^44 + 1482 * q^45 + 1361 * q^46 - 286 * q^47 + 178 * q^48 - 1835 * q^49 + 583 * q^50 - 1136 * q^51 + 2306 * q^52 + 212 * q^53 + 67 * q^54 + 264 * q^55 - 2059 * q^56 + 2596 * q^57 + 1690 * q^58 + 266 * q^59 + 74 * q^60 + 624 * q^61 - 643 * q^62 + 2360 * q^63 - 3178 * q^64 + 470 * q^65 + 352 * q^66 + 676 * q^67 + 413 * q^68 - 764 * q^69 - 2122 * q^70 + 763 * q^71 + 1366 * q^72 - 4748 * q^73 + 1649 * q^74 - 2420 * q^75 + 2101 * q^76 - 1364 * q^77 - 5848 * q^78 + 4328 * q^79 + 1013 * q^80 - 537 * q^81 - 3152 * q^82 + 1554 * q^83 + 3381 * q^84 + 1690 * q^85 + 5788 * q^86 + 4200 * q^87 + 231 * q^88 + 1687 * q^89 - 10798 * q^90 - 3380 * q^91 + 11084 * q^92 + 4310 * q^93 - 1777 * q^94 - 1124 * q^95 - 6930 * q^96 + 2047 * q^97 - 1553 * q^98 + 2970 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
100.1 −2.55759 + 4.42988i 1.79098 3.10206i −9.08257 15.7315i −2.16776 9.16117 + 15.8676i −10.4904 18.1698i 51.9966 7.08482 + 12.2713i 5.54425 9.60293i
100.2 −2.09541 + 3.62936i −2.35272 + 4.07503i −4.78153 8.28185i 14.4259 −9.85985 17.0778i 17.3506 + 30.0521i 6.55048 2.42942 + 4.20788i −30.2283 + 52.3570i
100.3 −2.07785 + 3.59895i −3.58804 + 6.21467i −4.63495 8.02797i −11.1721 −14.9108 25.8263i 3.89211 + 6.74134i 5.27734 −12.2481 21.2143i 23.2141 40.2080i
100.4 −2.06912 + 3.58383i 2.73933 4.74466i −4.56254 7.90254i −3.23279 11.3360 + 19.6346i −1.53684 2.66188i 4.65582 −1.50786 2.61170i 6.68904 11.5858i
100.5 −1.39209 + 2.41118i 4.47320 7.74782i 0.124146 + 0.215028i −17.0433 12.4542 + 21.5714i 15.4676 + 26.7907i −22.9648 −26.5191 45.9324i 23.7259 41.0944i
100.6 −0.944521 + 1.63596i −0.783201 + 1.35654i 2.21576 + 3.83781i 2.20557 −1.47950 2.56257i 0.902360 + 1.56293i −23.4837 12.2732 + 21.2578i −2.08320 + 3.60821i
100.7 −0.706890 + 1.22437i −2.88255 + 4.99273i 3.00061 + 5.19721i −4.58681 −4.07530 7.05862i −18.2120 31.5441i −19.7947 −3.11823 5.40093i 3.24237 5.61595i
100.8 −0.495908 + 0.858937i 2.68049 4.64275i 3.50815 + 6.07630i 18.5970 2.65856 + 4.60475i 6.70244 + 11.6090i −14.8934 −0.870106 1.50707i −9.22237 + 15.9736i
100.9 0.315613 0.546657i −4.62136 + 8.00442i 3.80078 + 6.58314i −19.0524 2.91712 + 5.05260i 12.7218 + 22.0348i 9.84810 −29.2139 50.5999i −6.01317 + 10.4151i
100.10 0.332082 0.575182i 0.423352 0.733267i 3.77944 + 6.54619i −8.92433 −0.281175 0.487009i −1.76528 3.05755i 10.3336 13.1415 + 22.7618i −2.96360 + 5.13311i
100.11 0.534214 0.925286i 4.92554 8.53128i 3.42923 + 5.93960i 2.35929 −5.26259 9.11507i −6.12528 10.6093i 15.8752 −35.0219 60.6597i 1.26037 2.18302i
100.12 0.856937 1.48426i −3.79109 + 6.56636i 2.53132 + 4.38437i 12.0244 6.49745 + 11.2539i −0.574325 0.994761i 22.3877 −15.2447 26.4047i 10.3041 17.8473i
100.13 1.56350 2.70806i 0.268802 0.465578i −0.889055 1.53989i −0.516265 −0.840543 1.45586i 17.3210 + 30.0008i 19.4558 13.3555 + 23.1324i −0.807180 + 1.39808i
100.14 1.58752 2.74967i 0.300230 0.520013i −1.04047 1.80214i 12.4679 −0.953244 1.65107i −5.31270 9.20186i 18.7933 13.3197 + 23.0704i 19.7931 34.2827i
100.15 2.05606 3.56120i 2.00775 3.47753i −4.45478 7.71591i −12.3424 −8.25614 14.3001i −14.8185 25.6664i −3.74025 5.43784 + 9.41862i −25.3768 + 43.9540i
100.16 2.54066 4.40055i 3.69680 6.40305i −8.90987 15.4324i 8.39690 −18.7846 32.5359i 7.09626 + 12.2911i −49.8972 −13.8327 23.9589i 21.3336 36.9509i
100.17 2.55281 4.42160i −2.28752 + 3.96210i −9.03368 15.6468i −15.4387 11.6792 + 20.2290i 8.38111 + 14.5165i −51.4001 3.03450 + 5.25591i −39.4122 + 68.2639i
133.1 −2.55759 4.42988i 1.79098 + 3.10206i −9.08257 + 15.7315i −2.16776 9.16117 15.8676i −10.4904 + 18.1698i 51.9966 7.08482 12.2713i 5.54425 + 9.60293i
133.2 −2.09541 3.62936i −2.35272 4.07503i −4.78153 + 8.28185i 14.4259 −9.85985 + 17.0778i 17.3506 30.0521i 6.55048 2.42942 4.20788i −30.2283 52.3570i
133.3 −2.07785 3.59895i −3.58804 6.21467i −4.63495 + 8.02797i −11.1721 −14.9108 + 25.8263i 3.89211 6.74134i 5.27734 −12.2481 + 21.2143i 23.2141 + 40.2080i
See all 34 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 133.17 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 143.4.e.b 34
13.c even 3 1 inner 143.4.e.b 34
13.c even 3 1 1859.4.a.g 17
13.e even 6 1 1859.4.a.h 17

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.e.b 34 1.a even 1 1 trivial
143.4.e.b 34 13.c even 3 1 inner
1859.4.a.g 17 13.c even 3 1
1859.4.a.h 17 13.e even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{34} + 93 T_{2}^{32} + 14 T_{2}^{31} + 5200 T_{2}^{30} + 1057 T_{2}^{29} + 190322 T_{2}^{28} + 43431 T_{2}^{27} + 5161774 T_{2}^{26} + 1106448 T_{2}^{25} + 104327879 T_{2}^{24} + \cdots + 4887141433344$$ acting on $$S_{4}^{\mathrm{new}}(143, [\chi])$$.