# Properties

 Label 48.8.k.a Level $48$ Weight $8$ Character orbit 48.k Analytic conductor $14.994$ Analytic rank $0$ Dimension $108$ CM no Inner twists $4$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 48.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.9944812232$$ Analytic rank: $$0$$ Dimension: $$108$$ Relative dimension: $$54$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$108 q - 2 q^{3} - 4 q^{4} + 172 q^{6} - 8 q^{7}+O(q^{10})$$ 108 * q - 2 * q^3 - 4 * q^4 + 172 * q^6 - 8 * q^7 $$\operatorname{Tr}(f)(q) =$$ $$108 q - 2 q^{3} - 4 q^{4} + 172 q^{6} - 8 q^{7} - 6504 q^{10} + 12016 q^{12} - 4 q^{13} - 55280 q^{16} + 21172 q^{18} - 60588 q^{19} - 4376 q^{21} - 212744 q^{22} - 317304 q^{24} + 238042 q^{27} + 219736 q^{28} - 503852 q^{30} - 4 q^{33} - 364856 q^{34} + 867620 q^{36} - 4 q^{37} - 283948 q^{39} + 390392 q^{40} - 3022344 q^{42} + 752844 q^{43} - 156252 q^{45} - 1166824 q^{46} + 358320 q^{48} + 9882508 q^{49} + 1121592 q^{51} - 4458688 q^{52} + 203816 q^{54} + 4191000 q^{55} + 4975544 q^{58} - 4390240 q^{60} - 2279892 q^{61} + 4781432 q^{64} + 1457476 q^{66} + 1552540 q^{67} + 4372 q^{69} - 9366152 q^{70} - 1660280 q^{72} - 886066 q^{75} - 6569272 q^{76} - 13616772 q^{78} - 4 q^{81} + 11560256 q^{82} - 10944176 q^{84} + 7846496 q^{85} - 40789116 q^{87} + 24614816 q^{88} + 25746032 q^{90} - 1590680 q^{91} - 14495396 q^{93} - 6995856 q^{94} - 13927736 q^{96} - 8 q^{97} - 4701076 q^{99}+O(q^{100})$$ 108 * q - 2 * q^3 - 4 * q^4 + 172 * q^6 - 8 * q^7 - 6504 * q^10 + 12016 * q^12 - 4 * q^13 - 55280 * q^16 + 21172 * q^18 - 60588 * q^19 - 4376 * q^21 - 212744 * q^22 - 317304 * q^24 + 238042 * q^27 + 219736 * q^28 - 503852 * q^30 - 4 * q^33 - 364856 * q^34 + 867620 * q^36 - 4 * q^37 - 283948 * q^39 + 390392 * q^40 - 3022344 * q^42 + 752844 * q^43 - 156252 * q^45 - 1166824 * q^46 + 358320 * q^48 + 9882508 * q^49 + 1121592 * q^51 - 4458688 * q^52 + 203816 * q^54 + 4191000 * q^55 + 4975544 * q^58 - 4390240 * q^60 - 2279892 * q^61 + 4781432 * q^64 + 1457476 * q^66 + 1552540 * q^67 + 4372 * q^69 - 9366152 * q^70 - 1660280 * q^72 - 886066 * q^75 - 6569272 * q^76 - 13616772 * q^78 - 4 * q^81 + 11560256 * q^82 - 10944176 * q^84 + 7846496 * q^85 - 40789116 * q^87 + 24614816 * q^88 + 25746032 * q^90 - 1590680 * q^91 - 14495396 * q^93 - 6995856 * q^94 - 13927736 * q^96 - 8 * q^97 - 4701076 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −11.3125 + 0.167474i 4.08471 + 46.5866i 127.944 3.78909i −280.037 + 280.037i −54.0103 526.326i −532.864 −1446.73 + 64.2912i −2153.63 + 380.586i 3121.01 3214.81i
11.2 −11.1336 + 2.01052i 1.90114 + 46.7267i 119.916 44.7688i 277.365 277.365i −115.112 516.416i 418.647 −1245.09 + 739.532i −2179.77 + 177.668i −2530.44 + 3645.73i
11.3 −11.0402 2.47286i −21.9431 41.2977i 115.770 + 54.6016i 47.3788 47.3788i 140.132 + 510.195i −321.346 −1143.10 889.093i −1224.00 + 1812.40i −640.231 + 405.908i
11.4 −10.9905 2.68503i 42.7767 18.8985i 113.581 + 59.0195i 312.645 312.645i −520.879 + 92.8472i 1303.49 −1089.84 953.621i 1472.69 1616.83i −4275.58 + 2596.66i
11.5 −10.8871 + 3.07743i 22.4120 41.0451i 109.059 67.0087i −78.5897 + 78.5897i −117.689 + 515.835i −861.367 −981.123 + 1065.15i −1182.40 1839.81i 613.761 1097.47i
11.6 −10.7935 3.39113i −42.3573 + 19.8208i 105.000 + 73.2046i −65.9576 + 65.9576i 524.399 70.2967i 652.374 −885.079 1146.21i 1401.28 1679.11i 935.586 488.244i
11.7 −10.5420 + 4.10672i −46.5860 4.09257i 94.2696 86.5866i 254.666 254.666i 507.918 148.172i −665.648 −638.208 + 1299.94i 2153.50 + 381.312i −1638.86 + 3730.54i
11.8 −10.3588 + 4.54910i 45.2039 + 11.9836i 86.6113 94.2469i −93.8822 + 93.8822i −522.775 + 81.5006i 809.729 −468.455 + 1370.29i 1899.79 + 1083.41i 545.432 1399.59i
11.9 −10.3435 4.58397i 45.5791 + 10.4665i 85.9744 + 94.8283i 25.0262 25.0262i −423.468 317.193i −1569.84 −454.583 1374.96i 1967.91 + 954.104i −373.577 + 144.138i
11.10 −9.75422 + 5.73194i −37.4127 28.0587i 62.2898 111.821i −287.248 + 287.248i 525.763 + 59.2434i 1269.27 33.3635 + 1447.77i 612.421 + 2099.50i 1155.40 4448.37i
11.11 −9.27322 6.48131i 27.2746 37.9881i 43.9852 + 120.205i −379.549 + 379.549i −499.136 + 175.497i 864.981 371.203 1399.77i −699.194 2072.22i 5979.61 1059.67i
11.12 −8.00843 + 7.99156i −34.6593 + 31.3963i 0.269890 128.000i −153.366 + 153.366i 26.6610 528.418i −1061.45 1020.76 + 1027.23i 215.539 2176.35i 2.58700 2453.85i
11.13 −7.71273 8.27730i −22.7540 + 40.8565i −9.02745 + 127.681i 277.521 277.521i 513.677 126.773i −1218.23 1126.48 910.049i −1151.51 1859.30i −4437.56 156.679i
11.14 −7.70062 8.28857i 25.6221 + 39.1217i −9.40086 + 127.654i −45.8713 + 45.8713i 126.957 513.632i 943.511 1130.46 905.098i −874.018 + 2004.76i 733.445 + 26.9701i
11.15 −7.46649 + 8.50009i 0.872512 46.7572i −16.5031 126.932i 294.772 294.772i 390.926 + 356.529i 694.446 1202.15 + 807.456i −2185.48 81.5925i 304.678 + 4706.51i
11.16 −7.22422 8.70693i −28.4787 37.0940i −23.6212 + 125.802i 118.895 118.895i −117.238 + 515.937i 553.079 1265.99 703.151i −564.924 + 2112.78i −1894.14 176.286i
11.17 −6.86971 + 8.98928i 34.0729 + 32.0318i −33.6142 123.507i 201.123 201.123i −522.014 + 86.2416i −1025.82 1341.16 + 546.292i 134.926 + 2182.83i 426.294 + 3189.61i
11.18 −6.30926 9.39113i −45.6917 9.96321i −48.3865 + 118.502i −316.866 + 316.866i 194.715 + 491.957i −1511.21 1418.15 293.257i 1988.47 + 910.472i 4974.91 + 976.537i
11.19 −5.66532 + 9.79307i −17.1135 + 43.5216i −63.8083 110.962i 27.9730 27.9730i −329.256 414.157i 1719.05 1448.15 + 3.75424i −1601.26 1489.61i 115.466 + 432.418i
11.20 −5.44512 + 9.91719i 40.1781 23.9317i −68.7013 108.001i −101.752 + 101.752i 18.5603 + 528.764i −33.7333 1445.15 93.2472i 1041.55 1923.05i −455.042 1563.15i
See next 80 embeddings (of 108 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 35.54 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.8.k.a 108
3.b odd 2 1 inner 48.8.k.a 108
16.f odd 4 1 inner 48.8.k.a 108
48.k even 4 1 inner 48.8.k.a 108

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.8.k.a 108 1.a even 1 1 trivial
48.8.k.a 108 3.b odd 2 1 inner
48.8.k.a 108 16.f odd 4 1 inner
48.8.k.a 108 48.k even 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{8}^{\mathrm{new}}(48, [\chi])$$.