# Properties

 Label 91.10.h.a Level $91$ Weight $10$ Character orbit 91.h Analytic conductor $46.868$ Analytic rank $0$ Dimension $164$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 91.h (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$46.8682610909$$ Analytic rank: $$0$$ Dimension: $$164$$ Relative dimension: $$82$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$164 q - 2 q^{2} + 163 q^{3} + 40958 q^{4} - 2 q^{5} - 1026 q^{6} - 913 q^{7} - 1032 q^{8} - 512605 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$164 q - 2 q^{2} + 163 q^{3} + 40958 q^{4} - 2 q^{5} - 1026 q^{6} - 913 q^{7} - 1032 q^{8} - 512605 q^{9} + 38977 q^{10} - 59285 q^{11} + 44088 q^{12} + 232454 q^{13} + 17108 q^{14} - 407531 q^{15} + 10497654 q^{16} - 1395086 q^{17} - 76575 q^{18} + 1207545 q^{19} + 713618 q^{20} + 1790137 q^{21} + 199216 q^{22} + 2148062 q^{23} + 3435832 q^{24} - 29660780 q^{25} - 6096660 q^{26} - 9655538 q^{27} - 5996877 q^{28} + 996812 q^{29} + 4910674 q^{30} + 7678612 q^{31} - 13584366 q^{32} - 3082415 q^{33} + 32705872 q^{34} + 11865077 q^{35} - 134627249 q^{36} - 25237130 q^{37} + 25519874 q^{38} + 30627737 q^{39} + 45727182 q^{40} - 13667582 q^{41} - 81628563 q^{42} + 26867814 q^{43} - 75673614 q^{44} + 29593550 q^{45} - 70893356 q^{46} - 23074040 q^{47} + 29166348 q^{48} - 13541447 q^{49} + 77715962 q^{50} - 94065066 q^{51} + 139792121 q^{52} + 46529776 q^{53} - 14474550 q^{54} + 218800681 q^{55} - 649788015 q^{56} - 143710238 q^{57} + 1142701 q^{58} + 439131230 q^{59} - 557690411 q^{60} + 508694743 q^{61} + 307731320 q^{62} - 355730647 q^{63} + 1713678376 q^{64} + 148754944 q^{65} + 278425259 q^{66} + 37418955 q^{67} - 1290484830 q^{68} + 290468024 q^{69} - 182370322 q^{70} - 415622020 q^{71} + 720832393 q^{72} + 514120432 q^{73} + 128989762 q^{74} + 241348444 q^{75} + 985961470 q^{76} - 1189530907 q^{77} + 2167273673 q^{78} - 376448286 q^{79} - 482394712 q^{80} - 3131306974 q^{81} - 971561072 q^{82} + 971157612 q^{83} + 3735517629 q^{84} + 538436691 q^{85} + 1470174876 q^{86} - 912549686 q^{87} - 629673387 q^{88} + 699403942 q^{89} - 8083448244 q^{90} + 2740572083 q^{91} + 2309185554 q^{92} + 820704894 q^{93} + 419356886 q^{94} - 5283246 q^{95} - 4707829686 q^{96} + 1339771120 q^{97} - 1961676608 q^{98} + 1482300132 q^{99} + O(q^{100})$$

## 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 −44.9267 −48.7021 + 84.3546i 1506.41 −218.727 + 378.846i 2188.03 3789.78i −3419.91 + 5353.30i −44675.7 5097.70 + 8829.48i 9826.67 17020.3i
16.2 −42.8048 136.894 237.108i 1320.25 98.6739 170.908i −5859.74 + 10149.4i 4801.69 + 4159.02i −34597.0 −27638.7 47871.6i −4223.72 + 7315.69i
16.3 −41.4944 79.2639 137.289i 1209.79 −958.800 + 1660.69i −3289.01 + 5696.74i −3727.25 5144.04i −28954.3 −2724.04 4718.18i 39784.9 68909.4i
16.4 −41.2731 64.2311 111.252i 1191.47 877.691 1520.21i −2651.02 + 4591.70i −5795.09 2602.03i −28044.0 1590.23 + 2754.36i −36225.1 + 62743.7i
16.5 −41.2552 −8.92868 + 15.4649i 1189.99 −842.207 + 1458.75i 368.355 638.009i 6320.66 + 634.750i −27970.8 9682.06 + 16769.8i 34745.4 60180.9i
16.6 −40.0257 −36.1738 + 62.6549i 1090.06 319.890 554.065i 1447.88 2507.80i 2743.86 5729.30i −23137.1 7224.41 + 12513.0i −12803.8 + 22176.8i
16.7 −39.7623 42.2099 73.1097i 1069.04 915.875 1586.34i −1678.36 + 2907.01i 6254.10 1113.51i −22149.3 6278.15 + 10874.1i −36417.3 + 63076.7i
16.8 −39.6408 −110.934 + 192.143i 1059.40 854.067 1479.29i 4397.50 7616.70i −6023.08 2018.95i −21699.3 −14771.1 25584.2i −33855.9 + 58640.2i
16.9 −39.3517 −134.908 + 233.667i 1036.55 −1292.21 + 2238.17i 5308.84 9195.18i 797.607 6302.18i −20642.1 −26558.7 46000.9i 50850.6 88075.8i
16.10 −36.4163 −91.6449 + 158.734i 814.145 939.046 1626.48i 3337.36 5780.49i 6277.96 + 969.960i −11003.0 −6956.07 12048.3i −34196.6 + 59230.2i
16.11 −35.7216 56.1841 97.3137i 764.034 568.907 985.376i −2006.99 + 3476.20i −1487.09 + 6175.94i −9003.08 3528.20 + 6111.01i −20322.3 + 35199.2i
16.12 −35.3278 −104.968 + 181.810i 736.050 −141.598 + 245.255i 3708.29 6422.95i 2929.99 + 5636.38i −7915.19 −12195.2 21122.6i 5002.34 8664.31i
16.13 −33.6954 −7.73692 + 13.4007i 623.381 −406.981 + 704.911i 260.699 451.543i −6248.30 + 1145.56i −3753.02 9721.78 + 16838.6i 13713.4 23752.3i
16.14 −33.5094 52.0738 90.1944i 610.880 −1097.77 + 1901.40i −1744.96 + 3022.36i 2263.61 + 5935.46i −3313.40 4418.14 + 7652.45i 36785.7 63714.7i
16.15 −32.2619 79.9269 138.437i 528.833 −412.767 + 714.934i −2578.60 + 4466.26i 2253.08 5939.46i −543.049 −2935.12 5083.77i 13316.7 23065.2i
16.16 −29.7445 −60.8004 + 105.309i 372.734 −912.935 + 1581.25i 1808.48 3132.37i −6325.30 + 586.682i 4142.40 2448.13 + 4240.28i 27154.8 47033.4i
16.17 −29.7072 −48.0812 + 83.2791i 370.519 −95.8890 + 166.085i 1428.36 2473.99i −1720.94 6114.90i 4203.00 5217.89 + 9037.66i 2848.60 4933.91i
16.18 −29.4349 122.315 211.855i 354.414 −310.503 + 537.807i −3600.32 + 6235.94i −5780.64 + 2633.98i 4638.52 −20080.3 34780.0i 9139.63 15830.3i
16.19 −26.7467 112.136 194.225i 203.387 1156.03 2002.30i −2999.27 + 5194.89i −2358.40 5898.44i 8254.38 −15307.5 26513.3i −30920.0 + 53555.0i
16.20 −26.4661 2.88206 4.99188i 188.453 778.057 1347.63i −76.2769 + 132.115i 442.842 + 6336.99i 8563.02 9824.89 + 17017.2i −20592.1 + 35666.6i
See next 80 embeddings (of 164 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 74.82 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 91.10.h.a yes 164
7.c even 3 1 91.10.g.a 164
13.c even 3 1 91.10.g.a 164
91.h even 3 1 inner 91.10.h.a yes 164

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.10.g.a 164 7.c even 3 1
91.10.g.a 164 13.c even 3 1
91.10.h.a yes 164 1.a even 1 1 trivial
91.10.h.a yes 164 91.h even 3 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{10}^{\mathrm{new}}(91, [\chi])$$.