Properties

Label 91.10.e.a
Level $91$
Weight $10$
Character orbit 91.e
Analytic conductor $46.868$
Analytic rank $0$
Dimension $70$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(46.8682610909\)
Analytic rank: \(0\)
Dimension: \(70\)
Relative dimension: \(35\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 70 q + 65 q^{2} - 8533 q^{4} - 2118 q^{5} + 17358 q^{6} - 8764 q^{7} - 106980 q^{8} - 215597 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 70 q + 65 q^{2} - 8533 q^{4} - 2118 q^{5} + 17358 q^{6} - 8764 q^{7} - 106980 q^{8} - 215597 q^{9} + 16725 q^{10} + 117246 q^{11} + 79155 q^{12} - 1999270 q^{13} + 155800 q^{14} - 1075914 q^{15} - 1801693 q^{16} + 534494 q^{17} + 2034224 q^{18} - 1070554 q^{19} - 213462 q^{20} + 252483 q^{21} - 656508 q^{22} + 3982482 q^{23} - 8713082 q^{24} - 10791923 q^{25} - 1856465 q^{26} - 4695462 q^{27} + 43151747 q^{28} - 8248096 q^{29} + 7494360 q^{30} - 18116139 q^{31} + 23284165 q^{32} - 18685134 q^{33} - 35031796 q^{34} + 41739462 q^{35} + 69822756 q^{36} + 12174052 q^{37} - 86227130 q^{38} - 45752120 q^{40} - 14899440 q^{41} + 226294688 q^{42} - 100872292 q^{43} + 55709994 q^{44} - 128782765 q^{45} + 86862955 q^{46} - 15243502 q^{47} - 251600258 q^{48} + 168431298 q^{49} - 23415484 q^{50} + 89493887 q^{51} + 243711013 q^{52} + 42576150 q^{53} - 466420190 q^{54} - 72659096 q^{55} + 237491061 q^{56} - 533277320 q^{57} + 302514895 q^{58} + 40301350 q^{59} - 191686454 q^{60} + 9519912 q^{61} - 257610720 q^{62} - 292394291 q^{63} + 453888112 q^{64} + 60492198 q^{65} + 1445199979 q^{66} + 919692072 q^{67} - 220295462 q^{68} - 1381257496 q^{69} + 860352188 q^{70} - 811959580 q^{71} + 2266117080 q^{72} + 924540004 q^{73} - 727436321 q^{74} + 445426010 q^{75} + 649412196 q^{76} - 1151937348 q^{77} - 495761838 q^{78} + 1174788834 q^{79} + 2332520349 q^{80} - 800698791 q^{81} + 1124625342 q^{82} + 830514540 q^{83} + 1448184376 q^{84} - 1969339174 q^{85} + 1497823546 q^{86} + 721368044 q^{87} - 808482999 q^{88} - 1968216306 q^{89} + 777819470 q^{90} + 250308604 q^{91} - 4887921618 q^{92} + 853972744 q^{93} + 4664723138 q^{94} + 1357929150 q^{95} - 3736344295 q^{96} + 3467931764 q^{97} - 1557173329 q^{98} - 8781494080 q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −21.7445 37.6626i −31.2108 + 54.0587i −689.647 + 1194.50i 146.760 + 254.196i 2714.65 −5982.50 + 2136.18i 37717.8 7893.27 + 13671.5i 6382.45 11054.7i
53.2 −19.4314 33.6561i −91.9975 + 159.344i −499.156 + 864.564i 723.891 + 1253.82i 7150.55 13.7108 6352.43i 18899.4 −7085.57 12272.6i 28132.4 48726.7i
53.3 −18.4208 31.9058i 63.9657 110.792i −422.654 + 732.058i 169.293 + 293.224i −4713.21 1350.37 6207.26i 12279.6 1658.27 + 2872.20i 6237.03 10802.9i
53.4 −18.1742 31.4787i −22.4027 + 38.8026i −404.604 + 700.795i −449.239 778.105i 1628.61 2896.03 + 5653.90i 10803.1 8837.74 + 15307.4i −16329.1 + 28282.9i
53.5 −17.8760 30.9621i 96.3548 166.891i −383.101 + 663.550i −1261.17 2184.41i −6889.74 −4557.75 4424.99i 9088.22 −8726.99 15115.6i −45089.4 + 78097.1i
53.6 −16.2746 28.1884i 60.6201 104.997i −273.726 + 474.107i 1179.89 + 2043.63i −3946.27 −2105.73 + 5993.29i 1153.91 2491.91 + 4316.12i 38404.6 66518.7i
53.7 −15.4941 26.8366i −87.7424 + 151.974i −224.136 + 388.215i −994.442 1722.42i 5437.97 5647.80 2907.91i −1974.83 −5555.97 9623.22i −30816.0 + 53374.9i
53.8 −11.6654 20.2051i −114.632 + 198.549i −16.1632 + 27.9955i −540.667 936.463i 5348.93 −5257.21 + 3565.86i −11191.2 −16439.7 28474.3i −12614.2 + 21848.4i
53.9 −10.6060 18.3701i 42.6958 73.9513i 31.0274 53.7410i 422.676 + 732.096i −1811.32 6099.93 1773.28i −12176.8 6195.64 + 10731.2i 8965.77 15529.2i
53.10 −9.68351 16.7723i −16.9452 + 29.3499i 68.4592 118.575i −732.091 1268.02i 656.356 −5194.70 3656.32i −12567.6 9267.22 + 16051.3i −14178.4 + 24557.7i
53.11 −9.60109 16.6296i −128.933 + 223.318i 71.6381 124.081i 679.560 + 1177.03i 4951.58 6297.72 832.067i −12582.7 −23405.8 40540.0i 13049.0 22601.6i
53.12 −8.75534 15.1647i 126.100 218.411i 102.688 177.861i −351.181 608.263i −4416.19 −5379.03 + 3379.30i −12561.7 −21960.9 38037.3i −6149.41 + 10651.1i
53.13 −7.79810 13.5067i 70.3015 121.766i 134.379 232.752i 740.523 + 1282.62i −2192.87 −6071.29 1868.97i −12176.9 −43.0895 74.6332i 11549.3 20004.1i
53.14 −4.24872 7.35900i 43.1429 74.7257i 219.897 380.872i −1055.39 1827.99i −733.208 4469.67 + 4513.94i −8087.81 6118.88 + 10598.2i −8968.10 + 15533.2i
53.15 −4.19848 7.27198i −49.8365 + 86.3193i 220.745 382.342i 1320.17 + 2286.60i 836.950 −1027.59 6268.79i −8006.43 4874.15 + 8442.27i 11085.4 19200.5i
53.16 −3.17713 5.50295i −43.2245 + 74.8670i 235.812 408.438i 309.299 + 535.721i 549.319 2332.15 + 5908.87i −6250.20 6104.79 + 10573.8i 1965.36 3404.11i
53.17 1.23518 + 2.13940i 123.516 213.936i 252.949 438.120i 7.89996 + 13.6831i 610.260 6152.39 + 1581.67i 2514.58 −20671.0 35803.2i −19.5158 + 33.8023i
53.18 3.40903 + 5.90461i 60.0224 103.962i 232.757 403.147i −1037.25 1796.57i 818.472 −3655.86 + 5195.02i 6664.75 2636.12 + 4565.90i 7072.05 12249.2i
53.19 3.45631 + 5.98650i 85.5848 148.237i 232.108 402.023i 432.273 + 748.719i 1183.23 3520.91 5287.42i 6748.21 −4808.01 8327.72i −2988.14 + 5175.61i
53.20 3.62786 + 6.28364i −95.9625 + 166.212i 229.677 397.813i −40.4745 70.1038i −1392.55 −5859.16 2454.35i 7047.87 −8576.10 14854.2i 293.671 508.654i
See all 70 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.35
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c 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.e.a 70
7.c even 3 1 inner 91.10.e.a 70
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.10.e.a 70 1.a even 1 1 trivial
91.10.e.a 70 7.c even 3 1 inner