# Properties

 Label 91.8.e.b Level $91$ Weight $8$ Character orbit 91.e Analytic conductor $28.427$ Analytic rank $0$ Dimension $58$ CM no Inner twists $2$

# Learn more

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$28.4270373191$$ Analytic rank: $$0$$ Dimension: $$58$$ Relative dimension: $$29$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$58 q - 31 q^{2} - 1893 q^{4} + 48 q^{5} - 4602 q^{6} + 85 q^{7} + 11436 q^{8} - 25535 q^{9}+O(q^{10})$$ 58 * q - 31 * q^2 - 1893 * q^4 + 48 * q^5 - 4602 * q^6 + 85 * q^7 + 11436 * q^8 - 25535 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$58 q - 31 q^{2} - 1893 q^{4} + 48 q^{5} - 4602 q^{6} + 85 q^{7} + 11436 q^{8} - 25535 q^{9} + 3029 q^{10} - 15843 q^{11} + 10995 q^{12} - 127426 q^{13} - 65720 q^{14} + 115668 q^{15} - 154909 q^{16} + 43475 q^{17} - 67288 q^{18} + 64641 q^{19} + 40146 q^{20} - 321354 q^{21} + 170964 q^{22} - 96894 q^{23} + 501382 q^{24} - 637413 q^{25} + 68107 q^{26} + 156816 q^{27} - 877133 q^{28} + 155846 q^{29} - 335448 q^{30} + 393806 q^{31} - 1827215 q^{32} + 299052 q^{33} - 38692 q^{34} - 133704 q^{35} + 5417220 q^{36} - 597180 q^{37} - 502034 q^{38} + 1223016 q^{40} - 1436280 q^{41} - 2542432 q^{42} + 3496372 q^{43} - 2679210 q^{44} - 2509582 q^{45} + 2358719 q^{46} + 2368643 q^{47} - 843698 q^{48} + 995403 q^{49} + 3451796 q^{50} - 2396956 q^{51} + 4158921 q^{52} - 1919163 q^{53} - 4029674 q^{54} + 6075808 q^{55} + 3653181 q^{56} + 11452960 q^{57} - 4051049 q^{58} + 48733 q^{59} - 2792030 q^{60} + 1417921 q^{61} - 10244784 q^{62} + 7636291 q^{63} + 30707792 q^{64} - 105456 q^{65} - 22397813 q^{66} - 7025007 q^{67} + 5107354 q^{68} - 19729612 q^{69} - 13744304 q^{70} + 29267486 q^{71} - 7267524 q^{72} - 2417490 q^{73} + 2601511 q^{74} + 17567528 q^{75} - 19814476 q^{76} - 824982 q^{77} + 10110594 q^{78} - 7837082 q^{79} - 7428483 q^{80} - 24976533 q^{81} - 11520562 q^{82} - 38573688 q^{83} - 14286428 q^{84} + 41189436 q^{85} + 17943598 q^{86} - 37502020 q^{87} - 2949879 q^{88} + 15969588 q^{89} - 114635026 q^{90} - 186745 q^{91} + 83440638 q^{92} - 14525204 q^{93} - 30315990 q^{94} - 8678760 q^{95} + 23478593 q^{96} - 14169276 q^{97} + 14221151 q^{98} + 83443706 q^{99}+O(q^{100})$$ 58 * q - 31 * q^2 - 1893 * q^4 + 48 * q^5 - 4602 * q^6 + 85 * q^7 + 11436 * q^8 - 25535 * q^9 + 3029 * q^10 - 15843 * q^11 + 10995 * q^12 - 127426 * q^13 - 65720 * q^14 + 115668 * q^15 - 154909 * q^16 + 43475 * q^17 - 67288 * q^18 + 64641 * q^19 + 40146 * q^20 - 321354 * q^21 + 170964 * q^22 - 96894 * q^23 + 501382 * q^24 - 637413 * q^25 + 68107 * q^26 + 156816 * q^27 - 877133 * q^28 + 155846 * q^29 - 335448 * q^30 + 393806 * q^31 - 1827215 * q^32 + 299052 * q^33 - 38692 * q^34 - 133704 * q^35 + 5417220 * q^36 - 597180 * q^37 - 502034 * q^38 + 1223016 * q^40 - 1436280 * q^41 - 2542432 * q^42 + 3496372 * q^43 - 2679210 * q^44 - 2509582 * q^45 + 2358719 * q^46 + 2368643 * q^47 - 843698 * q^48 + 995403 * q^49 + 3451796 * q^50 - 2396956 * q^51 + 4158921 * q^52 - 1919163 * q^53 - 4029674 * q^54 + 6075808 * q^55 + 3653181 * q^56 + 11452960 * q^57 - 4051049 * q^58 + 48733 * q^59 - 2792030 * q^60 + 1417921 * q^61 - 10244784 * q^62 + 7636291 * q^63 + 30707792 * q^64 - 105456 * q^65 - 22397813 * q^66 - 7025007 * q^67 + 5107354 * q^68 - 19729612 * q^69 - 13744304 * q^70 + 29267486 * q^71 - 7267524 * q^72 - 2417490 * q^73 + 2601511 * q^74 + 17567528 * q^75 - 19814476 * q^76 - 824982 * q^77 + 10110594 * q^78 - 7837082 * q^79 - 7428483 * q^80 - 24976533 * q^81 - 11520562 * q^82 - 38573688 * q^83 - 14286428 * q^84 + 41189436 * q^85 + 17943598 * q^86 - 37502020 * q^87 - 2949879 * q^88 + 15969588 * q^89 - 114635026 * q^90 - 186745 * q^91 + 83440638 * q^92 - 14525204 * q^93 - 30315990 * q^94 - 8678760 * q^95 + 23478593 * q^96 - 14169276 * q^97 + 14221151 * q^98 + 83443706 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1 −11.1978 19.3951i 34.3559 59.5061i −186.780 + 323.513i 189.321 + 327.914i −1538.84 −105.148 901.381i 5499.45 −1267.15 2194.77i 4239.95 7343.80i
53.2 −10.9318 18.9344i −36.9978 + 64.0821i −175.008 + 303.123i −66.9050 115.883i 1617.81 901.581 103.412i 4854.07 −1644.18 2847.80i −1462.78 + 2533.61i
53.3 −10.7904 18.6896i 26.1873 45.3577i −168.867 + 292.486i −237.704 411.715i −1130.29 −163.395 + 892.662i 4526.24 −278.045 481.589i −5129.85 + 8885.17i
53.4 −9.22341 15.9754i 3.04138 5.26783i −106.143 + 183.845i 108.094 + 187.224i −112.208 −906.844 34.3060i 1554.80 1075.00 + 1861.95i 1993.99 3453.70i
53.5 −8.46022 14.6535i −15.0269 + 26.0273i −79.1507 + 137.093i 198.041 + 343.017i 508.523 843.909 333.707i 512.713 641.886 + 1111.78i 3350.94 5804.00i
53.6 −7.61807 13.1949i 39.7374 68.8273i −52.0700 + 90.1878i 179.532 + 310.959i −1210.89 568.450 + 707.395i −363.535 −2064.63 3576.04i 2735.38 4737.82i
53.7 −7.20296 12.4759i −36.3036 + 62.8796i −39.7654 + 68.8756i −62.6575 108.526i 1045.97 −607.927 + 673.771i −698.245 −1542.40 2671.51i −902.640 + 1563.42i
53.8 −6.60583 11.4416i 33.9784 58.8524i −23.2740 + 40.3117i −146.782 254.233i −897.823 420.880 803.992i −1076.12 −1215.57 2105.43i −1939.23 + 3358.84i
53.9 −6.25043 10.8261i 2.85376 4.94285i −14.1357 + 24.4837i −229.621 397.716i −71.3488 873.653 + 245.508i −1246.69 1077.21 + 1865.79i −2870.46 + 4971.78i
53.10 −5.99820 10.3892i 7.37665 12.7767i −7.95676 + 13.7815i 30.9424 + 53.5939i −176.986 −376.848 + 825.547i −1344.63 984.670 + 1705.50i 371.198 642.933i
53.11 −4.21209 7.29555i −9.89272 + 17.1347i 28.5167 49.3923i 32.7323 + 56.6940i 166.676 −264.708 868.028i −1558.75 897.768 + 1554.98i 275.742 477.600i
53.12 −2.36379 4.09421i −41.6183 + 72.0850i 52.8250 91.4955i 95.2956 + 165.057i 393.508 739.487 + 526.025i −1104.60 −2370.66 4106.11i 450.518 780.320i
53.13 −2.14228 3.71054i 31.8589 55.1812i 54.8212 94.9532i −6.65070 11.5193i −273.003 −200.823 884.993i −1018.20 −936.475 1622.02i −28.4954 + 49.3554i
53.14 −1.65473 2.86607i −41.7818 + 72.3681i 58.5238 101.366i −271.168 469.677i 276.550 −86.3153 903.378i −810.974 −2397.93 4153.34i −897.419 + 1554.38i
53.15 −0.413075 0.715467i −4.44124 + 7.69245i 63.6587 110.260i −30.3570 52.5798i 7.33826 888.005 + 187.058i −210.931 1054.05 + 1825.67i −25.0794 + 43.4388i
53.16 −0.0355890 0.0616420i −26.4772 + 45.8599i 63.9975 110.847i 184.974 + 320.384i 3.76919 −905.689 57.1831i −18.2212 −308.586 534.486i 13.1661 22.8043i
53.17 0.305464 + 0.529079i 39.4641 68.3538i 63.8134 110.528i 214.121 + 370.869i 48.2194 −779.036 + 465.452i 156.169 −2021.33 3501.05i −130.813 + 226.574i
53.18 1.72165 + 2.98198i 25.0145 43.3264i 58.0719 100.583i −175.303 303.634i 172.265 −837.808 348.741i 840.659 −157.950 273.577i 603.621 1045.50i
53.19 2.17186 + 3.76177i 8.57877 14.8589i 54.5660 94.5112i 225.848 + 391.181i 74.5276 886.944 192.023i 1030.04 946.309 + 1639.06i −981.022 + 1699.18i
53.20 2.78054 + 4.81603i −15.5186 + 26.8790i 48.5372 84.0689i −150.624 260.889i −172.600 −256.901 + 870.370i 1251.66 611.846 + 1059.75i 837.634 1450.82i
See all 58 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 79.29 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.8.e.b 58
7.c even 3 1 inner 91.8.e.b 58

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.8.e.b 58 1.a even 1 1 trivial
91.8.e.b 58 7.c even 3 1 inner