# Properties

 Label 143.4.h.a Level $143$ Weight $4$ Character orbit 143.h Analytic conductor $8.437$ Analytic rank $0$ Dimension $68$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$143 = 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 143.h (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.43727313082$$ Analytic rank: $$0$$ Dimension: $$68$$ Relative dimension: $$17$$ over $$\Q(\zeta_{5})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$68 q + 4 q^{2} + 12 q^{3} - 16 q^{4} + 24 q^{5} + 7 q^{6} + 8 q^{7} - 34 q^{8} - 55 q^{9}+O(q^{10})$$ 68 * q + 4 * q^2 + 12 * q^3 - 16 * q^4 + 24 * q^5 + 7 * q^6 + 8 * q^7 - 34 * q^8 - 55 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$68 q + 4 q^{2} + 12 q^{3} - 16 q^{4} + 24 q^{5} + 7 q^{6} + 8 q^{7} - 34 q^{8} - 55 q^{9} - 36 q^{10} - 51 q^{11} - 524 q^{12} - 221 q^{13} + 133 q^{14} + 178 q^{15} - 140 q^{16} + 302 q^{17} + 575 q^{18} - 59 q^{19} + 73 q^{20} - 136 q^{21} - 196 q^{22} - 1264 q^{23} + 224 q^{24} - 603 q^{25} - 13 q^{26} - 45 q^{27} + 948 q^{28} + 916 q^{29} - 90 q^{30} + 160 q^{31} + 1752 q^{32} + 713 q^{33} - 1204 q^{34} - 582 q^{35} + 373 q^{36} - 692 q^{37} + 663 q^{38} + 221 q^{39} + 1293 q^{40} - 2 q^{41} - 1782 q^{42} + 698 q^{43} + 897 q^{44} - 2048 q^{45} - 3385 q^{46} + 1754 q^{47} - 3944 q^{48} - 1579 q^{49} + 1061 q^{50} + 1103 q^{51} - 208 q^{52} + 1354 q^{53} + 6262 q^{54} + 3556 q^{55} - 3350 q^{56} + 1765 q^{57} + 819 q^{58} - 217 q^{59} + 228 q^{60} - 1632 q^{61} - 823 q^{62} + 352 q^{63} - 6388 q^{64} - 728 q^{65} + 6541 q^{66} - 6426 q^{67} + 1688 q^{68} + 486 q^{69} - 6242 q^{70} + 2988 q^{71} - 2073 q^{72} + 2116 q^{73} - 3120 q^{74} - 5631 q^{75} + 4008 q^{76} + 5450 q^{77} + 936 q^{78} + 4520 q^{79} + 2955 q^{80} - 6210 q^{81} + 6592 q^{82} - 641 q^{83} + 517 q^{84} - 1856 q^{85} - 3869 q^{86} + 1924 q^{87} + 4998 q^{88} - 7266 q^{89} + 6420 q^{90} + 104 q^{91} - 1429 q^{92} + 7114 q^{93} + 1427 q^{94} - 708 q^{95} + 8925 q^{96} - 3725 q^{97} - 2974 q^{98} + 7833 q^{99}+O(q^{100})$$ 68 * q + 4 * q^2 + 12 * q^3 - 16 * q^4 + 24 * q^5 + 7 * q^6 + 8 * q^7 - 34 * q^8 - 55 * q^9 - 36 * q^10 - 51 * q^11 - 524 * q^12 - 221 * q^13 + 133 * q^14 + 178 * q^15 - 140 * q^16 + 302 * q^17 + 575 * q^18 - 59 * q^19 + 73 * q^20 - 136 * q^21 - 196 * q^22 - 1264 * q^23 + 224 * q^24 - 603 * q^25 - 13 * q^26 - 45 * q^27 + 948 * q^28 + 916 * q^29 - 90 * q^30 + 160 * q^31 + 1752 * q^32 + 713 * q^33 - 1204 * q^34 - 582 * q^35 + 373 * q^36 - 692 * q^37 + 663 * q^38 + 221 * q^39 + 1293 * q^40 - 2 * q^41 - 1782 * q^42 + 698 * q^43 + 897 * q^44 - 2048 * q^45 - 3385 * q^46 + 1754 * q^47 - 3944 * q^48 - 1579 * q^49 + 1061 * q^50 + 1103 * q^51 - 208 * q^52 + 1354 * q^53 + 6262 * q^54 + 3556 * q^55 - 3350 * q^56 + 1765 * q^57 + 819 * q^58 - 217 * q^59 + 228 * q^60 - 1632 * q^61 - 823 * q^62 + 352 * q^63 - 6388 * q^64 - 728 * q^65 + 6541 * q^66 - 6426 * q^67 + 1688 * q^68 + 486 * q^69 - 6242 * q^70 + 2988 * q^71 - 2073 * q^72 + 2116 * q^73 - 3120 * q^74 - 5631 * q^75 + 4008 * q^76 + 5450 * q^77 + 936 * q^78 + 4520 * q^79 + 2955 * q^80 - 6210 * q^81 + 6592 * q^82 - 641 * q^83 + 517 * q^84 - 1856 * q^85 - 3869 * q^86 + 1924 * q^87 + 4998 * q^88 - 7266 * q^89 + 6420 * q^90 + 104 * q^91 - 1429 * q^92 + 7114 * q^93 + 1427 * q^94 - 708 * q^95 + 8925 * q^96 - 3725 * q^97 - 2974 * q^98 + 7833 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
14.1 −4.43083 + 3.21918i −1.61795 4.97954i 6.79695 20.9189i −1.77922 1.29268i 23.1989 + 16.8550i −7.95857 + 24.4940i 23.6862 + 72.8985i −0.334627 + 0.243121i 12.0448
14.2 −3.47986 + 2.52827i −0.669520 2.06057i 3.24516 9.98758i −8.81271 6.40281i 7.53951 + 5.47777i 1.33129 4.09730i 3.32506 + 10.2335i 18.0458 13.1110i 46.8550
14.3 −3.01508 + 2.19058i 2.64240 + 8.13246i 1.81991 5.60111i 9.33768 + 6.78422i −25.7819 18.7316i −3.65701 + 11.2551i −2.43074 7.48105i −37.3112 + 27.1082i −43.0152
14.4 −2.45816 + 1.78596i 0.944462 + 2.90676i 0.380774 1.17190i 6.23533 + 4.53024i −7.51299 5.45850i 9.62984 29.6376i −6.35451 19.5572i 14.2862 10.3796i −23.4183
14.5 −2.23637 + 1.62482i −2.82465 8.69339i −0.110814 + 0.341050i 3.43743 + 2.49744i 20.4422 + 14.8521i −5.09482 + 15.6802i −7.14007 21.9749i −45.7528 + 33.2414i −11.7453
14.6 −1.95153 + 1.41787i 1.17190 + 3.60673i −0.674012 + 2.07439i −13.9051 10.1026i −7.40089 5.37706i −2.26259 + 6.96352i −7.58923 23.3573i 10.2083 7.41675i 41.4606
14.7 −1.92631 + 1.39955i −1.97595 6.08135i −0.720198 + 2.21654i 4.29934 + 3.12365i 12.3174 + 8.94913i 1.64214 5.05400i −7.60110 23.3938i −11.2350 + 8.16270i −12.6535
14.8 −0.400584 + 0.291041i 0.225252 + 0.693255i −2.39637 + 7.37528i 16.7143 + 12.1437i −0.291998 0.212149i −2.65028 + 8.15671i −2.41064 7.41918i 21.4136 15.5579i −10.2298
14.9 −0.0253892 + 0.0184463i 1.76760 + 5.44013i −2.47183 + 7.60752i −6.11234 4.44088i −0.145228 0.105515i −1.91554 + 5.89541i −0.155155 0.477519i −4.62709 + 3.36178i 0.237105
14.10 1.06952 0.777051i −2.87269 8.84124i −1.93207 + 5.94631i −4.88097 3.54623i −9.94250 7.22365i 11.1335 34.2654i 5.82236 + 17.9194i −48.0717 + 34.9261i −7.97590
14.11 1.64125 1.19244i −1.40236 4.31601i −1.20034 + 3.69426i 9.87094 + 7.17166i −7.44820 5.41143i 3.82371 11.7682i 7.45035 + 22.9298i 5.18214 3.76505i 24.7525
14.12 1.88993 1.37311i −1.38946 4.27631i −0.785744 + 2.41827i −10.3395 7.51211i −8.49785 6.17405i −10.5881 + 32.5869i 7.61067 + 23.4232i 5.48720 3.98669i −29.8560
14.13 2.24227 1.62910i 2.35875 + 7.25948i −0.0983427 + 0.302668i 3.65722 + 2.65713i 17.1154 + 12.4351i 0.863031 2.65614i 7.12433 + 21.9264i −25.2928 + 18.3763i 12.5292
14.14 3.57314 2.59604i −1.21298 3.73318i 3.55578 10.9436i −10.5709 7.68024i −14.0256 10.1902i 2.06707 6.36178i −4.78609 14.7301i 9.37817 6.81364i −57.7097
14.15 3.57791 2.59950i 0.693568 + 2.13458i 3.57187 10.9931i 11.6383 + 8.45575i 8.03038 + 5.83441i −10.3864 + 31.9661i −4.86361 14.9687i 17.7680 12.9092i 63.6217
14.16 3.62344 2.63258i 1.38748 + 4.27021i 3.72667 11.4695i −3.97822 2.89035i 16.2691 + 11.8202i 8.08076 24.8700i −5.61882 17.2930i 5.53385 4.02058i −22.0239
14.17 4.42470 3.21473i −2.05208 6.31565i 6.77132 20.8400i 10.1327 + 7.36185i −29.3829 21.3480i −1.00225 + 3.08462i −23.5132 72.3663i −13.8329 + 10.0502i 68.5006
27.1 −1.47502 4.53963i −3.70948 2.69510i −11.9604 + 8.68976i −2.80173 + 8.62283i −6.76320 + 20.8150i −9.62173 + 6.99059i 26.1970 + 19.0332i −1.84675 5.68371i 43.2770
27.2 −1.38998 4.27791i 7.41235 + 5.38539i −9.89632 + 7.19010i 4.49878 13.8458i 12.7352 39.1949i 11.1788 8.12190i 15.4022 + 11.1903i 17.5971 + 54.1583i −65.4843
27.3 −1.34438 4.13757i 0.689758 + 0.501138i −8.84001 + 6.42264i 1.46848 4.51952i 1.14620 3.52764i 7.20065 5.23158i 10.3014 + 7.48443i −8.11883 24.9872i −20.6740
See all 68 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 92.17 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 143.4.h.a 68
11.c even 5 1 inner 143.4.h.a 68
11.c even 5 1 1573.4.a.o 34
11.d odd 10 1 1573.4.a.p 34

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.h.a 68 1.a even 1 1 trivial
143.4.h.a 68 11.c even 5 1 inner
1573.4.a.o 34 11.c even 5 1
1573.4.a.p 34 11.d odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{68} - 4 T_{2}^{67} + 84 T_{2}^{66} - 298 T_{2}^{65} + 4505 T_{2}^{64} - 14468 T_{2}^{63} + 195412 T_{2}^{62} - 572111 T_{2}^{61} + 7310250 T_{2}^{60} - 19938338 T_{2}^{59} + 227463970 T_{2}^{58} + \cdots + 27\!\cdots\!00$$ acting on $$S_{4}^{\mathrm{new}}(143, [\chi])$$.