# Properties

 Label 91.8.h.a Level $91$ Weight $8$ Character orbit 91.h Analytic conductor $28.427$ Analytic rank $0$ Dimension $126$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 91.h (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.4270373191$$ Analytic rank: $$0$$ Dimension: $$126$$ Relative dimension: $$63$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$126 q - 2 q^{2} - 80 q^{3} + 7678 q^{4} - 2 q^{5} - 258 q^{6} + 953 q^{7} - 264 q^{8} - 41707 q^{9}+O(q^{10})$$ 126 * q - 2 * q^2 - 80 * q^3 + 7678 * q^4 - 2 * q^5 - 258 * q^6 + 953 * q^7 - 264 * q^8 - 41707 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$126 q - 2 q^{2} - 80 q^{3} + 7678 q^{4} - 2 q^{5} - 258 q^{6} + 953 q^{7} - 264 q^{8} - 41707 q^{9} - 4255 q^{10} + 7423 q^{11} - 18072 q^{12} - 1824 q^{13} - 15916 q^{14} + 11047 q^{15} + 410294 q^{16} + 99274 q^{17} - 21111 q^{18} - 4844 q^{19} + 36626 q^{20} + 73198 q^{21} - 43360 q^{22} + 34130 q^{23} - 130304 q^{24} - 810669 q^{25} + 390708 q^{26} + 416614 q^{27} + 69267 q^{28} + 191096 q^{29} + 70666 q^{30} + 183609 q^{31} + 695346 q^{32} + 178591 q^{33} - 742000 q^{34} + 579437 q^{35} - 1732409 q^{36} - 1362898 q^{37} - 877438 q^{38} + 459227 q^{39} - 1502882 q^{40} + 593701 q^{41} + 1442877 q^{42} - 592188 q^{43} + 1254066 q^{44} + 1954226 q^{45} - 1385452 q^{46} + 98278 q^{47} - 374916 q^{48} - 1497637 q^{49} + 289562 q^{50} - 281250 q^{51} - 8455 q^{52} - 1256681 q^{53} - 293814 q^{54} - 3015554 q^{55} - 1646151 q^{56} + 2743072 q^{57} + 1851469 q^{58} - 2279458 q^{59} + 1031485 q^{60} - 1127636 q^{61} - 7724680 q^{62} + 5142530 q^{63} + 11827496 q^{64} - 2789048 q^{65} + 435059 q^{66} + 4373348 q^{67} + 8737122 q^{68} - 7610377 q^{69} - 8084578 q^{70} - 60067 q^{71} - 11660951 q^{72} - 5060877 q^{73} + 1184002 q^{74} + 22637254 q^{75} + 637438 q^{76} + 21469082 q^{77} - 7230943 q^{78} - 2446486 q^{79} - 123568 q^{80} - 30756895 q^{81} + 8626952 q^{82} - 18679806 q^{83} + 11302461 q^{84} + 5330093 q^{85} - 8512356 q^{86} - 16508762 q^{87} - 6438219 q^{88} - 6176684 q^{89} + 53481084 q^{90} + 29236777 q^{91} + 2460546 q^{92} + 27682404 q^{93} - 17421794 q^{94} + 39553398 q^{95} + 26667066 q^{96} + 7705262 q^{97} + 14093080 q^{98} - 60531432 q^{99}+O(q^{100})$$ 126 * q - 2 * q^2 - 80 * q^3 + 7678 * q^4 - 2 * q^5 - 258 * q^6 + 953 * q^7 - 264 * q^8 - 41707 * q^9 - 4255 * q^10 + 7423 * q^11 - 18072 * q^12 - 1824 * q^13 - 15916 * q^14 + 11047 * q^15 + 410294 * q^16 + 99274 * q^17 - 21111 * q^18 - 4844 * q^19 + 36626 * q^20 + 73198 * q^21 - 43360 * q^22 + 34130 * q^23 - 130304 * q^24 - 810669 * q^25 + 390708 * q^26 + 416614 * q^27 + 69267 * q^28 + 191096 * q^29 + 70666 * q^30 + 183609 * q^31 + 695346 * q^32 + 178591 * q^33 - 742000 * q^34 + 579437 * q^35 - 1732409 * q^36 - 1362898 * q^37 - 877438 * q^38 + 459227 * q^39 - 1502882 * q^40 + 593701 * q^41 + 1442877 * q^42 - 592188 * q^43 + 1254066 * q^44 + 1954226 * q^45 - 1385452 * q^46 + 98278 * q^47 - 374916 * q^48 - 1497637 * q^49 + 289562 * q^50 - 281250 * q^51 - 8455 * q^52 - 1256681 * q^53 - 293814 * q^54 - 3015554 * q^55 - 1646151 * q^56 + 2743072 * q^57 + 1851469 * q^58 - 2279458 * q^59 + 1031485 * q^60 - 1127636 * q^61 - 7724680 * q^62 + 5142530 * q^63 + 11827496 * q^64 - 2789048 * q^65 + 435059 * q^66 + 4373348 * q^67 + 8737122 * q^68 - 7610377 * q^69 - 8084578 * q^70 - 60067 * q^71 - 11660951 * q^72 - 5060877 * q^73 + 1184002 * q^74 + 22637254 * q^75 + 637438 * q^76 + 21469082 * q^77 - 7230943 * q^78 - 2446486 * q^79 - 123568 * q^80 - 30756895 * q^81 + 8626952 * q^82 - 18679806 * q^83 + 11302461 * q^84 + 5330093 * q^85 - 8512356 * q^86 - 16508762 * q^87 - 6438219 * q^88 - 6176684 * q^89 + 53481084 * q^90 + 29236777 * q^91 + 2460546 * q^92 + 27682404 * q^93 - 17421794 * q^94 + 39553398 * q^95 + 26667066 * q^96 + 7705262 * q^97 + 14093080 * q^98 - 60531432 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1 −21.8777 −23.9756 + 41.5269i 350.632 −21.7436 + 37.6610i 524.529 908.511i −34.8107 906.825i −4870.67 −56.1551 97.2635i 475.699 823.934i
16.2 −21.6162 −8.26223 + 14.3106i 339.261 269.902 467.485i 178.598 309.341i 280.582 + 863.028i −4566.66 956.971 + 1657.52i −5834.27 + 10105.3i
16.3 −21.0975 26.1888 45.3604i 317.105 −66.6229 + 115.394i −552.519 + 956.990i 658.312 624.634i −3989.64 −278.208 481.870i 1405.58 2434.53i
16.4 −20.9336 26.1734 45.3336i 310.215 −90.5560 + 156.848i −547.902 + 948.995i −623.340 + 659.538i −3814.40 −276.592 479.071i 1895.66 3283.38i
16.5 −18.7319 −26.1441 + 45.2829i 222.884 −97.2612 + 168.461i 489.729 848.235i 880.970 + 217.796i −1777.37 −273.527 473.763i 1821.89 3155.60i
16.6 −18.3767 −24.0823 + 41.7118i 209.703 −251.617 + 435.813i 442.554 766.525i −798.848 + 430.564i −1501.43 −66.4166 115.037i 4623.89 8008.81i
16.7 −18.3303 −41.3019 + 71.5369i 208.000 68.8190 119.198i 757.075 1311.29i −737.556 + 528.729i −1466.42 −2318.19 4015.22i −1261.47 + 2184.93i
16.8 −18.1736 6.23452 10.7985i 202.279 −132.496 + 229.490i −113.303 + 196.247i 592.599 + 687.291i −1349.91 1015.76 + 1759.35i 2407.93 4170.66i
16.9 −18.0124 2.20243 3.81472i 196.447 111.005 192.267i −39.6710 + 68.7122i −798.622 430.983i −1232.89 1083.80 + 1877.19i −1999.47 + 3463.19i
16.10 −17.9775 33.7545 58.4645i 195.192 176.342 305.434i −606.823 + 1051.05i 544.696 725.844i −1207.95 −1185.23 2052.88i −3170.20 + 5490.95i
16.11 −15.9608 38.5973 66.8525i 126.746 172.468 298.723i −616.042 + 1067.02i −37.4390 + 906.720i 20.0123 −1886.00 3266.65i −2752.72 + 4767.84i
16.12 −14.8162 −42.5519 + 73.7021i 91.5204 187.452 324.677i 630.458 1091.99i 581.575 696.644i 540.490 −2527.83 4378.33i −2777.33 + 4810.48i
16.13 −14.8112 4.92210 8.52533i 91.3725 67.2870 116.545i −72.9024 + 126.271i −792.466 442.199i 542.499 1045.05 + 1810.07i −996.603 + 1726.17i
16.14 −14.2960 −8.02646 + 13.9022i 76.3766 107.975 187.018i 114.747 198.747i 868.950 + 261.665i 738.010 964.652 + 1670.83i −1543.61 + 2673.61i
16.15 −14.2658 42.8483 74.2155i 75.5130 −151.993 + 263.260i −611.265 + 1058.74i −754.932 503.607i 748.769 −2578.46 4466.02i 2168.30 3755.61i
16.16 −13.3993 9.37130 16.2316i 51.5416 −257.461 + 445.935i −125.569 + 217.492i 237.569 875.845i 1024.49 917.858 + 1589.78i 3449.80 5975.23i
16.17 −13.0238 −26.2436 + 45.4552i 41.6200 −63.3839 + 109.784i 341.792 592.001i 185.431 888.346i 1125.00 −283.950 491.816i 825.501 1429.81i
16.18 −11.4816 28.8130 49.9056i 3.82665 −60.6077 + 104.976i −330.819 + 572.995i 720.487 + 551.762i 1425.71 −566.877 981.860i 695.873 1205.29i
16.19 −11.4327 −25.3886 + 43.9743i 2.70595 116.445 201.688i 290.259 502.743i −337.374 + 842.450i 1432.45 −195.658 338.890i −1331.27 + 2305.83i
16.20 −9.50049 9.63027 16.6801i −37.7406 −115.203 + 199.537i −91.4923 + 158.469i −595.376 + 684.888i 1574.62 908.016 + 1572.73i 1094.48 1895.70i
See next 80 embeddings (of 126 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 74.63 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.8.h.a yes 126
7.c even 3 1 91.8.g.a 126
13.c even 3 1 91.8.g.a 126
91.h even 3 1 inner 91.8.h.a yes 126

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.8.g.a 126 7.c even 3 1
91.8.g.a 126 13.c even 3 1
91.8.h.a yes 126 1.a even 1 1 trivial
91.8.h.a yes 126 91.h even 3 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{8}^{\mathrm{new}}(91, [\chi])$$.