# Properties

 Label 143.4.h.b Level $143$ Weight $4$ Character orbit 143.h Analytic conductor $8.437$ Analytic rank $0$ Dimension $76$ 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: $$76$$ Relative dimension: $$19$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$76 q + 4 q^{2} - 12 q^{3} - 148 q^{4} - 16 q^{5} + 77 q^{6} + 56 q^{7} - 34 q^{8} - 241 q^{9}+O(q^{10})$$ 76 * q + 4 * q^2 - 12 * q^3 - 148 * q^4 - 16 * q^5 + 77 * q^6 + 56 * q^7 - 34 * q^8 - 241 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$76 q + 4 q^{2} - 12 q^{3} - 148 q^{4} - 16 q^{5} + 77 q^{6} + 56 q^{7} - 34 q^{8} - 241 q^{9} + 60 q^{10} - 47 q^{11} + 244 q^{12} + 247 q^{13} + 117 q^{14} - 2 q^{15} + 212 q^{16} - 344 q^{17} - 461 q^{18} - 59 q^{19} + 55 q^{20} - 296 q^{21} - 76 q^{22} + 1680 q^{23} + 784 q^{24} - 89 q^{25} + 13 q^{26} - 309 q^{27} - 654 q^{28} - 306 q^{29} + 606 q^{30} + 344 q^{31} - 2408 q^{32} + 1265 q^{33} - 116 q^{34} + 934 q^{35} - 3571 q^{36} + 188 q^{37} - 9 q^{38} + 91 q^{39} - 1803 q^{40} + 1518 q^{41} + 1734 q^{42} - 966 q^{43} + 1513 q^{44} + 2272 q^{45} - 165 q^{46} - 2146 q^{47} + 2320 q^{48} - 2085 q^{49} - 71 q^{50} - 501 q^{51} + 1924 q^{52} + 814 q^{53} - 6546 q^{54} - 1924 q^{55} + 6538 q^{56} - 1099 q^{57} - 4169 q^{58} - 41 q^{59} - 2812 q^{60} + 1788 q^{61} - 3931 q^{62} + 4980 q^{63} + 4256 q^{64} - 1352 q^{65} - 1543 q^{66} + 9878 q^{67} + 648 q^{68} - 2994 q^{69} + 6094 q^{70} - 612 q^{71} + 2965 q^{72} + 3436 q^{73} + 2616 q^{74} + 5089 q^{75} - 6632 q^{76} - 2604 q^{77} + 2704 q^{78} - 7688 q^{79} - 11093 q^{80} - 3972 q^{81} + 1076 q^{82} + 2007 q^{83} - 5729 q^{84} - 2128 q^{85} + 2433 q^{86} + 1812 q^{87} - 9006 q^{88} + 7454 q^{89} - 2204 q^{90} - 728 q^{91} + 2143 q^{92} - 8158 q^{93} + 4055 q^{94} + 824 q^{95} + 811 q^{96} + 5711 q^{97} - 4086 q^{98} - 11439 q^{99}+O(q^{100})$$ 76 * q + 4 * q^2 - 12 * q^3 - 148 * q^4 - 16 * q^5 + 77 * q^6 + 56 * q^7 - 34 * q^8 - 241 * q^9 + 60 * q^10 - 47 * q^11 + 244 * q^12 + 247 * q^13 + 117 * q^14 - 2 * q^15 + 212 * q^16 - 344 * q^17 - 461 * q^18 - 59 * q^19 + 55 * q^20 - 296 * q^21 - 76 * q^22 + 1680 * q^23 + 784 * q^24 - 89 * q^25 + 13 * q^26 - 309 * q^27 - 654 * q^28 - 306 * q^29 + 606 * q^30 + 344 * q^31 - 2408 * q^32 + 1265 * q^33 - 116 * q^34 + 934 * q^35 - 3571 * q^36 + 188 * q^37 - 9 * q^38 + 91 * q^39 - 1803 * q^40 + 1518 * q^41 + 1734 * q^42 - 966 * q^43 + 1513 * q^44 + 2272 * q^45 - 165 * q^46 - 2146 * q^47 + 2320 * q^48 - 2085 * q^49 - 71 * q^50 - 501 * q^51 + 1924 * q^52 + 814 * q^53 - 6546 * q^54 - 1924 * q^55 + 6538 * q^56 - 1099 * q^57 - 4169 * q^58 - 41 * q^59 - 2812 * q^60 + 1788 * q^61 - 3931 * q^62 + 4980 * q^63 + 4256 * q^64 - 1352 * q^65 - 1543 * q^66 + 9878 * q^67 + 648 * q^68 - 2994 * q^69 + 6094 * q^70 - 612 * q^71 + 2965 * q^72 + 3436 * q^73 + 2616 * q^74 + 5089 * q^75 - 6632 * q^76 - 2604 * q^77 + 2704 * q^78 - 7688 * q^79 - 11093 * q^80 - 3972 * q^81 + 1076 * q^82 + 2007 * q^83 - 5729 * q^84 - 2128 * q^85 + 2433 * q^86 + 1812 * q^87 - 9006 * q^88 + 7454 * q^89 - 2204 * q^90 - 728 * q^91 + 2143 * q^92 - 8158 * q^93 + 4055 * q^94 + 824 * q^95 + 811 * q^96 + 5711 * q^97 - 4086 * q^98 - 11439 * 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 −3.97940 + 2.89120i 1.95471 + 6.01598i 5.00442 15.4020i −11.7650 8.54779i −25.1720 18.2885i 3.34037 10.2806i 12.4558 + 38.3350i −10.5277 + 7.64881i 71.5310
14.2 −3.86396 + 2.80733i −2.15852 6.64323i 4.57694 14.0864i 15.1953 + 11.0400i 26.9902 + 19.6095i 5.98138 18.4088i 10.0528 + 30.9394i −17.6298 + 12.8088i −89.7068
14.3 −3.52735 + 2.56277i −2.18335 6.71965i 3.40227 10.4711i −10.8529 7.88510i 24.9223 + 18.1071i 8.14177 25.0578i 4.05543 + 12.4813i −18.5432 + 13.4724i 58.4897
14.4 −2.91090 + 2.11489i −0.259273 0.797959i 1.52843 4.70403i 11.0738 + 8.04557i 2.44232 + 1.77445i −4.53569 + 13.9594i −3.39551 10.4503i 21.2739 15.4564i −49.2501
14.5 −2.72658 + 1.98097i 0.202439 + 0.623042i 1.03782 3.19409i −5.78677 4.20434i −1.78619 1.29775i −10.4846 + 32.2682i −4.83397 14.8774i 21.4963 15.6179i 24.1068
14.6 −1.83784 + 1.33527i 3.01069 + 9.26595i −0.877420 + 2.70042i −5.97680 4.34240i −17.9057 13.0093i 4.55364 14.0147i −7.60918 23.4186i −54.9500 + 39.9235i 16.7827
14.7 −1.16976 + 0.849884i 0.145177 + 0.446808i −1.82609 + 5.62012i −0.581588 0.422548i −0.549558 0.399277i 9.00894 27.7267i −6.21484 19.1273i 21.6649 15.7405i 1.03944
14.8 −1.00764 + 0.732094i −2.29028 7.04876i −1.99276 + 6.13307i −17.2161 12.5082i 7.46814 + 5.42592i −3.51372 + 10.8141i −5.56108 17.1153i −22.5962 + 16.4171i 26.5048
14.9 −0.471643 + 0.342668i −1.46721 4.51561i −2.36711 + 7.28522i 1.58259 + 1.14982i 2.23936 + 1.62699i −0.116501 + 0.358552i −2.82120 8.68275i 3.60544 2.61951i −1.14042
14.10 −0.367022 + 0.266657i 2.05644 + 6.32908i −2.40854 + 7.41271i 4.33326 + 3.14829i −2.44245 1.77455i −8.57944 + 26.4048i −2.21419 6.81457i −13.9848 + 10.1606i −2.42992
14.11 0.778995 0.565973i 2.31474 + 7.12403i −2.18563 + 6.72667i 14.4838 + 10.5231i 5.83518 + 4.23951i 7.91022 24.3451i 4.48492 + 13.8032i −23.5504 + 17.1103i 17.2386
14.12 1.29329 0.939630i −2.72994 8.40189i −1.68244 + 5.17803i 12.6344 + 9.17944i −11.4253 8.30094i −8.06287 + 24.8150i 6.64148 + 20.4404i −41.2957 + 30.0031i 24.9652
14.13 1.57062 1.14112i 0.990304 + 3.04784i −1.30744 + 4.02390i −16.0734 11.6780i 5.03336 + 3.65695i 8.77337 27.0017i 7.33766 + 22.5830i 13.5348 9.83362i −38.5714
14.14 2.37777 1.72755i 0.497021 + 1.52967i 0.197213 0.606958i 2.32261 + 1.68747i 3.82439 + 2.77858i −3.54970 + 10.9248i 6.68618 + 20.5780i 19.7506 14.3496i 8.43781
14.15 2.81215 2.04314i 2.92761 + 9.01025i 1.26159 3.88278i −15.8411 11.5093i 26.6421 + 19.3566i −7.78834 + 23.9700i 4.20786 + 12.9505i −50.7703 + 36.8868i −68.0627
14.16 3.22629 2.34404i −0.854707 2.63052i 2.44231 7.51667i 11.3312 + 8.23261i −8.92357 6.48335i 6.77683 20.8569i 0.118933 + 0.366039i 15.6544 11.3736i 55.8554
14.17 3.35922 2.44061i −2.65195 8.16188i 2.85561 8.78865i −4.97791 3.61667i −28.8285 20.9451i −0.679316 + 2.09072i −1.59225 4.90045i −37.7399 + 27.4196i −25.5488
14.18 4.28022 3.10976i 2.69766 + 8.30254i 6.17752 19.0125i 8.55907 + 6.21853i 37.3655 + 27.1476i 1.51239 4.65465i −19.6039 60.3345i −39.8113 + 28.9246i 55.9728
14.19 4.28158 3.11075i 0.388610 + 1.19602i 6.18303 19.0294i −9.86077 7.16427i 5.38439 + 3.91199i −3.63304 + 11.1813i −19.6393 60.4435i 20.5640 14.9406i −64.5059
27.1 −1.72974 5.32359i 3.50563 + 2.54699i −18.8765 + 13.7146i −2.86043 + 8.80351i 7.49531 23.0682i 15.0972 10.9687i 69.4343 + 50.4470i −2.54117 7.82093i 51.8141
See all 76 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 92.19 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.b 76
11.c even 5 1 inner 143.4.h.b 76
11.c even 5 1 1573.4.a.q 38
11.d odd 10 1 1573.4.a.r 38

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.h.b 76 1.a even 1 1 trivial
143.4.h.b 76 11.c even 5 1 inner
1573.4.a.q 38 11.c even 5 1
1573.4.a.r 38 11.d odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{76} - 4 T_{2}^{75} + 158 T_{2}^{74} - 594 T_{2}^{73} + 13147 T_{2}^{72} - 45392 T_{2}^{71} + 765812 T_{2}^{70} - 2395859 T_{2}^{69} + 35734212 T_{2}^{68} - 101491716 T_{2}^{67} + \cdots + 12\!\cdots\!96$$ acting on $$S_{4}^{\mathrm{new}}(143, [\chi])$$.