# Properties

 Label 91.10.q.a Level $91$ Weight $10$ Character orbit 91.q Analytic conductor $46.868$ Analytic rank $0$ Dimension $124$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 91.q (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$46.8682610909$$ Analytic rank: $$0$$ Dimension: $$124$$ Relative dimension: $$62$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$124 q + 15360 q^{4} + 6570 q^{6} - 381386 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$124 q + 15360 q^{4} + 6570 q^{6} - 381386 q^{9} - 47368 q^{10} + 18420 q^{11} + 434316 q^{12} - 103730 q^{13} - 307328 q^{14} - 994308 q^{15} - 4327304 q^{16} + 826226 q^{17} - 832512 q^{20} - 2238874 q^{22} - 5178348 q^{23} + 13060272 q^{24} - 56263016 q^{25} + 12486810 q^{26} - 16794384 q^{27} + 8409710 q^{29} - 18091000 q^{30} + 30354048 q^{32} + 36508140 q^{33} + 8902908 q^{35} + 42350446 q^{36} + 26865678 q^{37} - 7473860 q^{38} - 21470652 q^{39} - 112860988 q^{40} - 109829778 q^{41} + 12446784 q^{42} - 53384856 q^{43} - 84576450 q^{45} + 160042176 q^{46} - 39524126 q^{48} + 357417662 q^{49} + 35767422 q^{50} + 121877584 q^{51} + 61751010 q^{52} - 252868460 q^{53} + 951692088 q^{54} - 414833028 q^{55} - 118013952 q^{56} - 342666540 q^{58} - 440088588 q^{59} + 264060146 q^{61} + 995887736 q^{62} - 383747028 q^{63} - 2239569444 q^{64} - 501903354 q^{65} + 304284804 q^{66} + 392937636 q^{67} - 3665660 q^{68} - 249969892 q^{69} - 375620748 q^{71} - 2829560808 q^{72} + 1336479018 q^{74} + 1306893216 q^{75} + 1652559990 q^{76} + 823754288 q^{77} + 857091302 q^{78} - 2357982776 q^{79} + 1107773136 q^{80} - 1492992434 q^{81} - 1984241462 q^{82} - 2156016366 q^{84} + 3060837654 q^{85} - 1286987624 q^{87} + 1904443442 q^{88} + 887446752 q^{89} + 7601525656 q^{90} - 86522436 q^{91} - 8365653336 q^{92} - 776657196 q^{93} + 6270763420 q^{94} - 6325568332 q^{95} - 5522676732 q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
36.1 −38.9200 + 22.4705i −55.4073 95.9683i 753.843 1305.69i 740.456i 4312.90 + 2490.06i −2079.33 1200.50i 44747.0i 3701.56 6411.29i −16638.4 28818.5i
36.2 −38.5034 + 22.2300i −91.4804 158.449i 732.343 1268.46i 2102.73i 7044.62 + 4067.21i 2079.33 + 1200.50i 42356.4i −6895.84 + 11943.9i 46743.6 + 80962.4i
36.3 −35.6614 + 20.5891i 116.392 + 201.597i 591.821 1025.06i 877.811i −8301.40 4792.82i −2079.33 1200.50i 27657.0i −17252.8 + 29882.7i 18073.3 + 31303.9i
36.4 −35.5029 + 20.4976i 54.8657 + 95.0301i 584.302 1012.04i 455.216i −3895.78 2249.23i 2079.33 + 1200.50i 26917.6i 3821.02 6618.20i −9330.84 16161.5i
36.5 −35.0192 + 20.2183i 52.5461 + 91.0125i 561.562 972.654i 1911.46i −3680.24 2124.79i −2079.33 1200.50i 24711.8i 4319.32 7481.28i −38646.5 66937.7i
36.6 −32.9663 + 19.0331i −108.110 187.253i 468.516 811.494i 2130.68i 7127.99 + 4115.35i 2079.33 + 1200.50i 16179.3i −13534.2 + 23442.0i −40553.5 70240.7i
36.7 −32.0069 + 18.4792i −34.1506 59.1506i 426.960 739.517i 1335.08i 2186.11 + 1262.15i 2079.33 + 1200.50i 12636.8i 7508.97 13005.9i −24671.1 42731.6i
36.8 −31.2009 + 18.0138i 134.462 + 232.896i 392.997 680.690i 2021.81i −8390.69 4844.37i 2079.33 + 1200.50i 9871.33i −26318.8 + 45585.5i −36420.6 63082.3i
36.9 −30.7856 + 17.7740i −47.1433 81.6546i 375.833 650.963i 1486.53i 2902.67 + 1675.85i −2079.33 1200.50i 8519.70i 5396.52 9347.04i 26421.7 + 45763.7i
36.10 −30.3753 + 17.5372i 13.0681 + 22.6346i 359.107 621.991i 1205.75i −793.895 458.355i 2079.33 + 1200.50i 7232.81i 9499.95 16454.4i 21145.4 + 36625.0i
36.11 −29.7427 + 17.1720i −63.0297 109.171i 333.753 578.077i 894.218i 3749.35 + 2164.69i −2079.33 1200.50i 5340.70i 1896.02 3284.00i −15355.5 26596.5i
36.12 −26.9451 + 15.5567i 20.0545 + 34.7353i 228.024 394.949i 2230.19i −1080.74 623.963i −2079.33 1200.50i 1740.87i 9037.14 15652.8i 34694.4 + 60092.5i
36.13 −25.2553 + 14.5812i −95.5625 165.519i 169.221 293.099i 1073.46i 4826.92 + 2786.83i 2079.33 + 1200.50i 5061.36i −8422.88 + 14588.9i 15652.4 + 27110.7i
36.14 −23.8670 + 13.7796i 110.756 + 191.835i 123.755 214.349i 2578.04i −5286.82 3052.35i 2079.33 + 1200.50i 7289.15i −14692.3 + 25447.8i 35524.4 + 61530.0i
36.15 −22.8497 + 13.1923i 61.9983 + 107.384i 92.0727 159.475i 1099.16i −2833.29 1635.80i −2079.33 1200.50i 8650.30i 2153.91 3730.69i −14500.4 25115.4i
36.16 −21.5135 + 12.4208i 94.2787 + 163.295i 52.5525 91.0236i 179.352i −4056.52 2342.03i 2079.33 + 1200.50i 10107.9i −7935.43 + 13744.6i −2227.69 3858.47i
36.17 −21.0351 + 12.1446i 102.233 + 177.072i 38.9839 67.5221i 705.619i −4300.95 2483.16i −2079.33 1200.50i 10542.3i −11061.6 + 19159.2i 8569.48 + 14842.8i
36.18 −18.9950 + 10.9668i −12.3077 21.3175i −15.4603 + 26.7780i 364.739i 467.569 + 269.951i 2079.33 + 1200.50i 11908.2i 9538.54 16521.2i −4000.01 6928.22i
36.19 −17.4096 + 10.0514i −79.7414 138.116i −53.9368 + 93.4213i 2021.81i 2776.53 + 1603.03i 2079.33 + 1200.50i 12461.3i −2875.88 + 4981.18i 20322.1 + 35198.9i
36.20 −14.7416 + 8.51109i −92.7073 160.574i −111.123 + 192.470i 2393.29i 2733.32 + 1578.08i −2079.33 1200.50i 12498.5i −7347.78 + 12726.7i −20369.6 35281.1i
See next 80 embeddings (of 124 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 43.62 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.10.q.a 124
13.e even 6 1 inner 91.10.q.a 124

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.10.q.a 124 1.a even 1 1 trivial
91.10.q.a 124 13.e even 6 1 inner

## Hecke kernels

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