# Properties

 Label 91.8.r.a Level $91$ Weight $8$ Character orbit 91.r Analytic conductor $28.427$ Analytic rank $0$ Dimension $128$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 91.r (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.4270373191$$ Analytic rank: $$0$$ Dimension: $$128$$ Relative dimension: $$64$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$128 q + 52 q^{3} + 4094 q^{4} - 44052 q^{9}+O(q^{10})$$ 128 * q + 52 * q^3 + 4094 * q^4 - 44052 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$128 q + 52 q^{3} + 4094 q^{4} - 44052 q^{9} - 4514 q^{10} + 2090 q^{12} + 11132 q^{13} + 18748 q^{14} - 230074 q^{16} - 28972 q^{17} + 179920 q^{22} + 20508 q^{23} + 920040 q^{25} + 144684 q^{26} - 357584 q^{27} + 44600 q^{29} + 46580 q^{30} - 1216824 q^{35} - 4529248 q^{36} - 129032 q^{38} + 104546 q^{39} + 2082020 q^{40} + 2426496 q^{42} - 1434416 q^{43} - 3783940 q^{48} - 34428 q^{49} - 756008 q^{51} - 2348612 q^{52} + 1103824 q^{53} - 4216216 q^{55} + 8135574 q^{56} + 531568 q^{61} + 26627120 q^{62} - 27402496 q^{64} - 7495542 q^{65} + 2808402 q^{66} + 531760 q^{68} + 24238760 q^{69} - 7462590 q^{74} - 1547944 q^{75} - 1811116 q^{77} - 31233508 q^{78} + 2961020 q^{79} - 34639872 q^{81} - 10292872 q^{82} + 43455484 q^{87} + 21936790 q^{88} - 58442348 q^{90} + 16479848 q^{91} + 24834900 q^{92} + 26036992 q^{94} + 89048 q^{95}+O(q^{100})$$ 128 * q + 52 * q^3 + 4094 * q^4 - 44052 * q^9 - 4514 * q^10 + 2090 * q^12 + 11132 * q^13 + 18748 * q^14 - 230074 * q^16 - 28972 * q^17 + 179920 * q^22 + 20508 * q^23 + 920040 * q^25 + 144684 * q^26 - 357584 * q^27 + 44600 * q^29 + 46580 * q^30 - 1216824 * q^35 - 4529248 * q^36 - 129032 * q^38 + 104546 * q^39 + 2082020 * q^40 + 2426496 * q^42 - 1434416 * q^43 - 3783940 * q^48 - 34428 * q^49 - 756008 * q^51 - 2348612 * q^52 + 1103824 * q^53 - 4216216 * q^55 + 8135574 * q^56 + 531568 * q^61 + 26627120 * q^62 - 27402496 * q^64 - 7495542 * q^65 + 2808402 * q^66 + 531760 * q^68 + 24238760 * q^69 - 7462590 * q^74 - 1547944 * q^75 - 1811116 * q^77 - 31233508 * q^78 + 2961020 * q^79 - 34639872 * q^81 - 10292872 * q^82 + 43455484 * q^87 + 21936790 * q^88 - 58442348 * q^90 + 16479848 * q^91 + 24834900 * q^92 + 26036992 * q^94 + 89048 * q^95

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1 −19.2835 + 11.1333i 26.6444 46.1495i 183.903 318.529i −122.796 + 70.8965i 1186.57i 646.881 636.466i 5339.67i −326.351 565.257i 1578.63 2734.27i
25.2 −18.4363 + 10.6442i −3.90686 + 6.76688i 162.597 281.627i 454.211 262.239i 166.341i −231.201 877.547i 4197.94i 1062.97 + 1841.12i −5582.63 + 9669.41i
25.3 −18.3178 + 10.5758i −26.3747 + 45.6824i 159.695 276.600i −32.1615 + 18.5685i 1115.74i 767.468 + 484.289i 4048.20i −297.754 515.725i 392.752 680.267i
25.4 −17.5746 + 10.1467i −9.13036 + 15.8143i 141.910 245.795i −149.843 + 86.5119i 370.571i −905.314 62.8521i 3162.10i 926.773 + 1605.22i 1755.62 3040.82i
25.5 −17.0172 + 9.82489i 40.5593 70.2508i 129.057 223.533i 227.842 131.545i 1593.96i −454.805 + 785.299i 2556.71i −2196.61 3804.65i −2584.82 + 4477.05i
25.6 −16.7709 + 9.68269i −45.1459 + 78.1950i 123.509 213.924i 25.4254 14.6793i 1748.54i −897.233 136.071i 2304.83i −2982.81 5166.37i −284.271 + 492.372i
25.7 −16.7440 + 9.66716i 14.6384 25.3545i 122.908 212.883i −318.778 + 184.047i 566.049i −46.7306 + 906.289i 2277.90i 664.932 + 1151.70i 3558.42 6163.36i
25.8 −16.2488 + 9.38127i −13.0103 + 22.5344i 112.016 194.018i 334.391 193.061i 488.211i −212.350 + 882.298i 1801.81i 754.966 + 1307.64i −3622.31 + 6274.02i
25.9 −15.3790 + 8.87907i −27.8080 + 48.1649i 93.6757 162.251i −158.175 + 91.3222i 987.638i 870.250 257.311i 1053.97i −453.074 784.748i 1621.71 2808.89i
25.10 −15.3472 + 8.86073i 35.5987 61.6587i 93.0250 161.124i −94.1334 + 54.3479i 1261.72i −632.542 650.717i 1028.73i −1441.03 2495.94i 963.125 1668.18i
25.11 −14.1249 + 8.15502i 19.6075 33.9612i 69.0088 119.527i 255.306 147.401i 639.600i 784.629 + 455.961i 163.389i 324.589 + 562.205i −2404.11 + 4164.05i
25.12 −13.5528 + 7.82474i −9.72553 + 16.8451i 58.4531 101.244i −434.311 + 250.749i 304.399i −299.078 856.794i 173.614i 904.328 + 1566.34i 3924.10 6796.74i
25.13 −13.0889 + 7.55687i 9.02507 15.6319i 50.2126 86.9708i 23.7868 13.7333i 272.805i 635.750 647.584i 416.758i 930.596 + 1611.84i −207.562 + 359.507i
25.14 −12.1057 + 6.98926i 20.9404 36.2699i 33.6994 58.3691i 235.809 136.145i 585.432i −735.146 532.075i 847.114i 216.498 + 374.985i −1903.10 + 3296.27i
25.15 −11.6400 + 6.72038i −33.7165 + 58.3987i 26.3270 45.5996i 196.222 113.289i 906.350i 90.8149 902.937i 1012.71i −1180.10 2044.00i −1522.69 + 2637.37i
25.16 −11.0974 + 6.40707i 40.2137 69.6521i 18.1011 31.3520i −402.415 + 232.335i 1030.61i 893.492 + 158.790i 1176.31i −2140.78 3707.94i 2977.17 5156.61i
25.17 −10.9207 + 6.30508i −7.65371 + 13.2566i 15.5082 26.8609i 37.8419 21.8480i 193.029i −534.194 + 733.608i 1222.98i 976.341 + 1691.07i −275.507 + 477.192i
25.18 −10.4074 + 6.00872i −40.8107 + 70.6862i 8.20937 14.2190i −437.809 + 252.769i 980.880i −207.112 + 883.543i 1340.92i −2237.53 3875.52i 3037.64 5261.34i
25.19 −9.58245 + 5.53243i −34.1390 + 59.1305i −2.78448 + 4.82285i 414.916 239.552i 755.486i 670.273 + 611.781i 1477.92i −1237.44 2143.31i −2650.61 + 4590.99i
25.20 −8.90167 + 5.13938i 28.1429 48.7449i −11.1735 + 19.3530i −295.835 + 170.800i 578.549i −852.238 + 311.822i 1545.38i −490.546 849.650i 1755.62 3040.82i
See next 80 embeddings (of 128 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 51.64 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
13.b even 2 1 inner
91.r even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.8.r.a 128
7.c even 3 1 inner 91.8.r.a 128
13.b even 2 1 inner 91.8.r.a 128
91.r even 6 1 inner 91.8.r.a 128

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.8.r.a 128 1.a even 1 1 trivial
91.8.r.a 128 7.c even 3 1 inner
91.8.r.a 128 13.b even 2 1 inner
91.8.r.a 128 91.r even 6 1 inner

## Hecke kernels

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