# Properties

 Label 91.8.u.a Level $91$ Weight $8$ Character orbit 91.u Analytic conductor $28.427$ Analytic rank $0$ Dimension $126$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$28.4270373191$$ Analytic rank: $$0$$ Dimension: $$126$$ Relative dimension: $$63$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$126 q - 3 q^{2} - 164 q^{3} + 3841 q^{4} - 6 q^{6} + 754 q^{7} + 85710 q^{9}+O(q^{10})$$ 126 * q - 3 * q^2 - 164 * q^3 + 3841 * q^4 - 6 * q^6 + 754 * q^7 + 85710 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$126 q - 3 q^{2} - 164 q^{3} + 3841 q^{4} - 6 q^{6} + 754 q^{7} + 85710 q^{9} + 7486 q^{10} - 18328 q^{12} + 1816 q^{13} - 6668 q^{14} - 11817 q^{15} - 237219 q^{16} - 28969 q^{17} + 75423 q^{18} + 12342 q^{20} - 54000 q^{21} + 51388 q^{22} + 17067 q^{23} + 810665 q^{25} + 144300 q^{26} - 528170 q^{27} + 18464 q^{28} + 191096 q^{29} - 432160 q^{30} - 259995 q^{31} + 49149 q^{32} + 87303 q^{35} + 1965131 q^{36} - 571857 q^{37} - 1061954 q^{38} - 1352875 q^{39} + 345362 q^{40} - 636567 q^{41} - 3473694 q^{42} - 659976 q^{43} + 2519268 q^{44} + 1980717 q^{45} - 4068600 q^{46} - 294840 q^{47} + 4726652 q^{48} + 541120 q^{49} - 2908992 q^{50} - 502094 q^{51} + 7809224 q^{52} - 434723 q^{53} + 2119563 q^{54} + 2152196 q^{55} - 858015 q^{56} + 1376949 q^{59} - 3768141 q^{60} - 4854916 q^{61} - 1713436 q^{62} + 10091940 q^{63} - 40270472 q^{64} - 4876182 q^{65} + 8270835 q^{66} + 531379 q^{68} - 311689 q^{69} + 5298534 q^{70} + 1505877 q^{71} - 13207461 q^{73} + 3730527 q^{74} + 5624951 q^{75} + 25297146 q^{76} + 11107478 q^{77} - 9362803 q^{78} + 5012226 q^{79} + 45989766 q^{81} - 34046788 q^{82} - 35428032 q^{84} + 19854117 q^{85} + 5226210 q^{86} + 17331733 q^{87} + 24348310 q^{88} - 43738596 q^{89} + 90517324 q^{90} + 9582031 q^{91} - 8502078 q^{92} - 30517344 q^{93} - 7710416 q^{94} + 14810909 q^{95} + 41499936 q^{96} - 60232128 q^{97} - 65830509 q^{98}+O(q^{100})$$ 126 * q - 3 * q^2 - 164 * q^3 + 3841 * q^4 - 6 * q^6 + 754 * q^7 + 85710 * q^9 + 7486 * q^10 - 18328 * q^12 + 1816 * q^13 - 6668 * q^14 - 11817 * q^15 - 237219 * q^16 - 28969 * q^17 + 75423 * q^18 + 12342 * q^20 - 54000 * q^21 + 51388 * q^22 + 17067 * q^23 + 810665 * q^25 + 144300 * q^26 - 528170 * q^27 + 18464 * q^28 + 191096 * q^29 - 432160 * q^30 - 259995 * q^31 + 49149 * q^32 + 87303 * q^35 + 1965131 * q^36 - 571857 * q^37 - 1061954 * q^38 - 1352875 * q^39 + 345362 * q^40 - 636567 * q^41 - 3473694 * q^42 - 659976 * q^43 + 2519268 * q^44 + 1980717 * q^45 - 4068600 * q^46 - 294840 * q^47 + 4726652 * q^48 + 541120 * q^49 - 2908992 * q^50 - 502094 * q^51 + 7809224 * q^52 - 434723 * q^53 + 2119563 * q^54 + 2152196 * q^55 - 858015 * q^56 + 1376949 * q^59 - 3768141 * q^60 - 4854916 * q^61 - 1713436 * q^62 + 10091940 * q^63 - 40270472 * q^64 - 4876182 * q^65 + 8270835 * q^66 + 531379 * q^68 - 311689 * q^69 + 5298534 * q^70 + 1505877 * q^71 - 13207461 * q^73 + 3730527 * q^74 + 5624951 * q^75 + 25297146 * q^76 + 11107478 * q^77 - 9362803 * q^78 + 5012226 * q^79 + 45989766 * q^81 - 34046788 * q^82 - 35428032 * q^84 + 19854117 * q^85 + 5226210 * q^86 + 17331733 * q^87 + 24348310 * q^88 - 43738596 * q^89 + 90517324 * q^90 + 9582031 * q^91 - 8502078 * q^92 - 30517344 * q^93 - 7710416 * q^94 + 14810909 * q^95 + 41499936 * q^96 - 60232128 * q^97 - 65830509 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
30.1 −19.4750 + 11.2439i −9.82037 188.851 327.100i −268.738 155.156i 191.252 110.419i −444.698 791.067i 5615.28i −2090.56 6978.26
30.2 −18.7037 + 10.7986i −81.8286 169.220 293.097i 42.3342 + 24.4417i 1530.50 883.635i 907.195 23.2380i 4544.89i 4508.93 −1055.74
30.3 −18.4889 + 10.6746i 37.9468 163.893 283.870i 411.268 + 237.445i −701.593 + 405.065i 874.632 + 241.997i 4265.24i −747.044 −10138.5
30.4 −17.4054 + 10.0490i 67.2867 137.965 238.962i −13.7956 7.96491i −1171.15 + 676.165i −799.780 + 428.830i 2973.10i 2340.51 320.158
30.5 −16.7071 + 9.64583i −37.2922 122.084 211.456i 25.2662 + 14.5875i 623.044 359.715i −138.861 + 896.806i 2241.08i −796.290 −562.833
30.6 −16.6633 + 9.62058i 50.6980 121.111 209.771i −428.668 247.491i −844.797 + 487.744i 696.474 + 581.779i 2197.77i 383.285 9524.05
30.7 −16.4215 + 9.48093i −29.8348 115.776 200.530i 406.721 + 234.820i 489.930 282.861i −907.087 27.1366i 1963.54i −1296.89 −8905.26
30.8 −15.6734 + 9.04906i −13.5023 99.7708 172.808i −99.0313 57.1757i 211.628 122.183i −118.523 899.720i 1294.77i −2004.69 2069.55
30.9 −15.1682 + 8.75734i 15.9696 89.3819 154.814i −59.9286 34.5998i −242.229 + 139.851i 907.068 + 27.7649i 889.110i −1931.97 1212.01
30.10 −15.1610 + 8.75321i 67.9628 89.2375 154.564i 154.359 + 89.1194i −1030.39 + 594.893i −48.6239 906.189i 883.636i 2431.95 −3120.32
30.11 −14.7550 + 8.51878i −62.4377 81.1393 140.537i −428.751 247.539i 921.266 531.893i −516.024 + 746.500i 584.023i 1711.47 8434.94
30.12 −13.4632 + 7.77296i −86.2083 56.8378 98.4459i 122.410 + 70.6734i 1160.64 670.094i −741.564 523.093i 222.687i 5244.87 −2197.36
30.13 −12.3706 + 7.14217i −51.8106 38.0211 65.8544i 189.087 + 109.169i 640.928 370.040i 746.771 515.632i 742.183i 497.341 −3118.82
30.14 −11.2351 + 6.48656i 29.5960 20.1509 34.9025i −169.575 97.9040i −332.513 + 191.977i −849.576 + 319.005i 1137.72i −1311.07 2540.24
30.15 −11.0003 + 6.35102i −56.5248 16.6710 28.8751i −371.359 214.404i 621.789 358.990i 801.191 426.188i 1202.35i 1008.05 5446.75
30.16 −10.8046 + 6.23801i 1.75115 13.8256 23.9467i 129.006 + 74.4818i −18.9204 + 10.9237i 201.857 + 884.758i 1251.95i −2183.93 −1858.47
30.17 −10.5632 + 6.09867i 77.3789 10.3875 17.9917i −388.438 224.265i −817.369 + 471.908i 310.115 852.861i 1307.86i 3800.50 5470.87
30.18 −9.96940 + 5.75584i 88.5355 2.25936 3.91332i 146.660 + 84.6740i −882.646 + 509.596i 353.859 + 835.659i 1421.48i 5651.53 −1949.48
30.19 −9.60240 + 5.54395i 17.1242 −2.52922 + 4.38073i 357.050 + 206.143i −164.434 + 94.9360i 152.798 894.537i 1475.34i −1893.76 −4571.38
30.20 −8.37044 + 4.83268i 37.7504 −17.2905 + 29.9480i 377.122 + 217.731i −315.987 + 182.435i −808.021 + 413.093i 1571.40i −761.911 −4208.90
See next 80 embeddings (of 126 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 88.63 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.u 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.u.a yes 126
7.c even 3 1 91.8.k.a 126
13.e even 6 1 91.8.k.a 126
91.u even 6 1 inner 91.8.u.a yes 126

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.8.k.a 126 7.c even 3 1
91.8.k.a 126 13.e even 6 1
91.8.u.a yes 126 1.a even 1 1 trivial
91.8.u.a yes 126 91.u even 6 1 inner

## Hecke kernels

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