# Properties

 Label 91.8.f.a Level $91$ Weight $8$ Character orbit 91.f Analytic conductor $28.427$ Analytic rank $0$ Dimension $48$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$28.4270373191$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 8 q^{2} + 41 q^{3} - 1472 q^{4} + 526 q^{5} - 519 q^{6} + 8232 q^{7} + 474 q^{8} - 15427 q^{9}+O(q^{10})$$ 48 * q - 8 * q^2 + 41 * q^3 - 1472 * q^4 + 526 * q^5 - 519 * q^6 + 8232 * q^7 + 474 * q^8 - 15427 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 8 q^{2} + 41 q^{3} - 1472 q^{4} + 526 q^{5} - 519 q^{6} + 8232 q^{7} + 474 q^{8} - 15427 q^{9} + 17800 q^{10} - 11735 q^{11} + 30484 q^{12} - 5622 q^{13} - 5488 q^{14} + 3465 q^{15} - 110364 q^{16} - 3506 q^{17} - 196096 q^{18} - 25291 q^{19} - 13351 q^{20} + 28126 q^{21} + 91579 q^{22} + 82970 q^{23} - 97143 q^{24} + 1263842 q^{25} - 51951 q^{26} - 1080760 q^{27} + 504896 q^{28} + 209315 q^{29} + 1063844 q^{30} + 948356 q^{31} - 495024 q^{32} - 143209 q^{33} + 436128 q^{34} + 90209 q^{35} - 469485 q^{36} - 377142 q^{37} - 198346 q^{38} + 647900 q^{39} - 4780340 q^{40} - 1895174 q^{41} + 178017 q^{42} - 1366459 q^{43} + 8096352 q^{44} - 2154733 q^{45} + 2854528 q^{46} + 2557720 q^{47} - 4102289 q^{48} - 2823576 q^{49} - 2378593 q^{50} - 4369338 q^{51} + 4270295 q^{52} - 4295420 q^{53} + 1447259 q^{54} - 560241 q^{55} + 81291 q^{56} + 15936670 q^{57} - 384326 q^{58} + 6714194 q^{59} + 5267974 q^{60} - 987536 q^{61} + 8520899 q^{62} + 5291461 q^{63} - 8211866 q^{64} - 11284709 q^{65} - 20326376 q^{66} - 1296682 q^{67} - 8400264 q^{68} - 1147530 q^{69} + 12210800 q^{70} - 7723282 q^{71} + 26270290 q^{72} + 15769600 q^{73} + 58033 q^{74} - 928389 q^{75} - 9095520 q^{76} - 8050210 q^{77} - 30641513 q^{78} - 24665908 q^{79} + 16716608 q^{80} - 5846292 q^{81} - 9977079 q^{82} + 32429868 q^{83} + 5228006 q^{84} + 17821519 q^{85} + 89024148 q^{86} - 16967309 q^{87} + 28910709 q^{88} - 27957029 q^{89} - 155228046 q^{90} - 5184102 q^{91} - 5073264 q^{92} + 11673603 q^{93} - 9670145 q^{94} + 38478411 q^{95} + 111942976 q^{96} - 28714467 q^{97} - 941192 q^{98} + 42173094 q^{99}+O(q^{100})$$ 48 * q - 8 * q^2 + 41 * q^3 - 1472 * q^4 + 526 * q^5 - 519 * q^6 + 8232 * q^7 + 474 * q^8 - 15427 * q^9 + 17800 * q^10 - 11735 * q^11 + 30484 * q^12 - 5622 * q^13 - 5488 * q^14 + 3465 * q^15 - 110364 * q^16 - 3506 * q^17 - 196096 * q^18 - 25291 * q^19 - 13351 * q^20 + 28126 * q^21 + 91579 * q^22 + 82970 * q^23 - 97143 * q^24 + 1263842 * q^25 - 51951 * q^26 - 1080760 * q^27 + 504896 * q^28 + 209315 * q^29 + 1063844 * q^30 + 948356 * q^31 - 495024 * q^32 - 143209 * q^33 + 436128 * q^34 + 90209 * q^35 - 469485 * q^36 - 377142 * q^37 - 198346 * q^38 + 647900 * q^39 - 4780340 * q^40 - 1895174 * q^41 + 178017 * q^42 - 1366459 * q^43 + 8096352 * q^44 - 2154733 * q^45 + 2854528 * q^46 + 2557720 * q^47 - 4102289 * q^48 - 2823576 * q^49 - 2378593 * q^50 - 4369338 * q^51 + 4270295 * q^52 - 4295420 * q^53 + 1447259 * q^54 - 560241 * q^55 + 81291 * q^56 + 15936670 * q^57 - 384326 * q^58 + 6714194 * q^59 + 5267974 * q^60 - 987536 * q^61 + 8520899 * q^62 + 5291461 * q^63 - 8211866 * q^64 - 11284709 * q^65 - 20326376 * q^66 - 1296682 * q^67 - 8400264 * q^68 - 1147530 * q^69 + 12210800 * q^70 - 7723282 * q^71 + 26270290 * q^72 + 15769600 * q^73 + 58033 * q^74 - 928389 * q^75 - 9095520 * q^76 - 8050210 * q^77 - 30641513 * q^78 - 24665908 * q^79 + 16716608 * q^80 - 5846292 * q^81 - 9977079 * q^82 + 32429868 * q^83 + 5228006 * q^84 + 17821519 * q^85 + 89024148 * q^86 - 16967309 * q^87 + 28910709 * q^88 - 27957029 * q^89 - 155228046 * q^90 - 5184102 * q^91 - 5073264 * q^92 + 11673603 * q^93 - 9670145 * q^94 + 38478411 * q^95 + 111942976 * q^96 - 28714467 * q^97 - 941192 * q^98 + 42173094 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1 −11.0832 + 19.1967i −25.9723 + 44.9854i −181.677 314.673i −486.993 −575.715 997.168i 171.500 + 297.047i 5216.95 −255.623 442.751i 5397.46 9348.68i
22.2 −10.1427 + 17.5677i −14.5450 + 25.1926i −141.749 245.517i 326.304 −295.051 511.043i 171.500 + 297.047i 3154.36 670.388 + 1161.15i −3309.61 + 5732.41i
22.3 −9.35205 + 16.1982i 31.2473 54.1220i −110.922 192.122i −263.837 584.454 + 1012.30i 171.500 + 297.047i 1755.26 −859.293 1488.34i 2467.42 4273.70i
22.4 −8.31541 + 14.4027i 18.6917 32.3751i −74.2921 128.678i 517.036 310.859 + 538.424i 171.500 + 297.047i 342.331 394.737 + 683.705i −4299.36 + 7446.72i
22.5 −8.31356 + 14.3995i 3.55199 6.15223i −74.2305 128.571i −126.313 59.0594 + 102.294i 171.500 + 297.047i 340.208 1068.27 + 1850.29i 1050.11 1818.85i
22.6 −6.42253 + 11.1242i −13.9142 + 24.1002i −18.4978 32.0392i −93.0462 −178.729 309.568i 171.500 + 297.047i −1168.96 706.288 + 1223.33i 597.592 1035.06i
22.7 −6.02827 + 10.4413i −33.3343 + 57.7367i −8.68004 15.0343i −353.391 −401.896 696.105i 171.500 + 297.047i −1333.93 −1128.85 1955.23i 2130.33 3689.85i
22.8 −5.90267 + 10.2237i 36.7895 63.7213i −5.68308 9.84339i −76.4660 434.313 + 752.252i 171.500 + 297.047i −1376.90 −1613.43 2794.55i 451.354 781.768i
22.9 −3.64211 + 6.30832i −23.2226 + 40.2227i 37.4701 + 64.9001i 324.801 −169.158 292.991i 171.500 + 297.047i −1478.26 14.9215 + 25.8448i −1182.96 + 2048.95i
22.10 −1.96989 + 3.41195i 18.4244 31.9119i 56.2391 + 97.4089i 280.776 72.5879 + 125.726i 171.500 + 297.047i −947.431 414.587 + 718.085i −553.099 + 957.995i
22.11 −0.951312 + 1.64772i −41.0916 + 71.1728i 62.1900 + 107.716i 94.6646 −78.1819 135.415i 171.500 + 297.047i −480.185 −2283.54 3955.21i −90.0557 + 155.981i
22.12 −0.463694 + 0.803142i −3.61457 + 6.26061i 63.5700 + 110.106i −525.610 −3.35211 5.80602i 171.500 + 297.047i −236.614 1067.37 + 1848.74i 243.723 422.140i
22.13 −0.408038 + 0.706743i 34.4693 59.7025i 63.6670 + 110.274i −447.050 28.1296 + 48.7218i 171.500 + 297.047i −208.372 −1282.76 2221.80i 182.414 315.950i
22.14 0.786346 1.36199i 46.3303 80.2465i 62.7633 + 108.709i 412.529 −72.8633 126.203i 171.500 + 297.047i 398.719 −3199.50 5541.69i 324.391 561.861i
22.15 0.904301 1.56630i −10.7648 + 18.6452i 62.3645 + 108.018i −102.650 19.4692 + 33.7217i 171.500 + 297.047i 457.086 861.739 + 1492.58i −92.8265 + 160.780i
22.16 3.76119 6.51457i −16.5227 + 28.6182i 35.7070 + 61.8463i 415.555 124.290 + 215.276i 171.500 + 297.047i 1500.07 547.501 + 948.299i 1562.98 2707.16i
22.17 4.98688 8.63753i 13.2496 22.9490i 14.2620 + 24.7026i 86.0031 −132.148 228.888i 171.500 + 297.047i 1561.13 742.397 + 1285.87i 428.887 742.854i
22.18 5.17651 8.96598i 22.2003 38.4521i 10.4074 + 18.0262i −162.299 −229.841 398.096i 171.500 + 297.047i 1540.68 107.790 + 186.698i −840.144 + 1455.17i
22.19 6.11392 10.5896i −32.6404 + 56.5348i −10.7599 18.6367i −72.0010 399.121 + 691.298i 171.500 + 297.047i 1302.02 −1037.29 1796.63i −440.208 + 762.462i
22.20 8.46862 14.6681i 17.8769 30.9637i −79.4351 137.586i 457.543 −302.785 524.440i 171.500 + 297.047i −522.854 454.332 + 786.927i 3874.76 6711.28i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.24 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.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.f.a 48
13.c even 3 1 inner 91.8.f.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.8.f.a 48 1.a even 1 1 trivial
91.8.f.a 48 13.c even 3 1 inner