# Properties

 Label 354.6.a.h Level 354 Weight 6 Character orbit 354.a Self dual Yes Analytic conductor 56.776 Analytic rank 0 Dimension 8 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$354 = 2 \cdot 3 \cdot 59$$ Weight: $$k$$ = $$6$$ Character orbit: $$[\chi]$$ = 354.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$56.7758722138$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -4 q^{2} + 9 q^{3} + 16 q^{4} + ( 5 + \beta_{1} ) q^{5} -36 q^{6} + ( 23 + \beta_{1} - \beta_{3} ) q^{7} -64 q^{8} + 81 q^{9} +O(q^{10})$$ $$q -4 q^{2} + 9 q^{3} + 16 q^{4} + ( 5 + \beta_{1} ) q^{5} -36 q^{6} + ( 23 + \beta_{1} - \beta_{3} ) q^{7} -64 q^{8} + 81 q^{9} + ( -20 - 4 \beta_{1} ) q^{10} + ( -44 + \beta_{1} + \beta_{2} - \beta_{4} - \beta_{7} ) q^{11} + 144 q^{12} + ( 15 + \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{13} + ( -92 - 4 \beta_{1} + 4 \beta_{3} ) q^{14} + ( 45 + 9 \beta_{1} ) q^{15} + 256 q^{16} + ( 59 + 5 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{17} -324 q^{18} + ( 167 + 11 \beta_{1} + 2 \beta_{2} - 7 \beta_{3} - \beta_{4} + 6 \beta_{5} + \beta_{7} ) q^{19} + ( 80 + 16 \beta_{1} ) q^{20} + ( 207 + 9 \beta_{1} - 9 \beta_{3} ) q^{21} + ( 176 - 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{4} + 4 \beta_{7} ) q^{22} + ( 141 + 21 \beta_{1} + 8 \beta_{2} + 11 \beta_{3} + 15 \beta_{4} + \beta_{6} ) q^{23} -576 q^{24} + ( 1206 + 24 \beta_{1} - 13 \beta_{2} - 2 \beta_{3} - \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{25} + ( -60 - 4 \beta_{2} + 8 \beta_{3} + 8 \beta_{5} - 4 \beta_{6} ) q^{26} + 729 q^{27} + ( 368 + 16 \beta_{1} - 16 \beta_{3} ) q^{28} + ( 654 + 30 \beta_{1} + 8 \beta_{2} + 11 \beta_{3} - 11 \beta_{4} + \beta_{6} ) q^{29} + ( -180 - 36 \beta_{1} ) q^{30} + ( 2296 - \beta_{1} + \beta_{2} - 3 \beta_{3} - 21 \beta_{4} - 16 \beta_{5} - 5 \beta_{6} ) q^{31} -1024 q^{32} + ( -396 + 9 \beta_{1} + 9 \beta_{2} - 9 \beta_{4} - 9 \beta_{7} ) q^{33} + ( -236 - 20 \beta_{1} + 16 \beta_{2} + 20 \beta_{3} + 4 \beta_{4} + 8 \beta_{6} ) q^{34} + ( 2473 + 30 \beta_{1} - 41 \beta_{2} - 11 \beta_{3} - 36 \beta_{4} - 4 \beta_{5} - \beta_{6} ) q^{35} + 1296 q^{36} + ( 3748 - 59 \beta_{1} + 31 \beta_{2} + 43 \beta_{3} + 58 \beta_{4} + 14 \beta_{5} - 8 \beta_{6} ) q^{37} + ( -668 - 44 \beta_{1} - 8 \beta_{2} + 28 \beta_{3} + 4 \beta_{4} - 24 \beta_{5} - 4 \beta_{7} ) q^{38} + ( 135 + 9 \beta_{2} - 18 \beta_{3} - 18 \beta_{5} + 9 \beta_{6} ) q^{39} + ( -320 - 64 \beta_{1} ) q^{40} + ( 1030 + 33 \beta_{1} + 9 \beta_{2} + 27 \beta_{3} - 8 \beta_{4} + 21 \beta_{5} + 13 \beta_{6} + 10 \beta_{7} ) q^{41} + ( -828 - 36 \beta_{1} + 36 \beta_{3} ) q^{42} + ( 3828 - 101 \beta_{1} - 11 \beta_{2} + 32 \beta_{3} + 91 \beta_{4} + 20 \beta_{5} + 13 \beta_{6} ) q^{43} + ( -704 + 16 \beta_{1} + 16 \beta_{2} - 16 \beta_{4} - 16 \beta_{7} ) q^{44} + ( 405 + 81 \beta_{1} ) q^{45} + ( -564 - 84 \beta_{1} - 32 \beta_{2} - 44 \beta_{3} - 60 \beta_{4} - 4 \beta_{6} ) q^{46} + ( -704 - 97 \beta_{1} - 21 \beta_{2} - 15 \beta_{3} + 16 \beta_{4} - 9 \beta_{5} + 10 \beta_{7} ) q^{47} + 2304 q^{48} + ( 4051 - 25 \beta_{1} + 35 \beta_{2} - 45 \beta_{3} - 53 \beta_{4} + \beta_{5} - 21 \beta_{6} - 10 \beta_{7} ) q^{49} + ( -4824 - 96 \beta_{1} + 52 \beta_{2} + 8 \beta_{3} + 4 \beta_{4} + 12 \beta_{5} - 8 \beta_{6} - 4 \beta_{7} ) q^{50} + ( 531 + 45 \beta_{1} - 36 \beta_{2} - 45 \beta_{3} - 9 \beta_{4} - 18 \beta_{6} ) q^{51} + ( 240 + 16 \beta_{2} - 32 \beta_{3} - 32 \beta_{5} + 16 \beta_{6} ) q^{52} + ( -4271 - 55 \beta_{1} + 96 \beta_{2} + 40 \beta_{3} + 110 \beta_{4} + 80 \beta_{5} ) q^{53} -2916 q^{54} + ( 3088 - 65 \beta_{1} - 107 \beta_{2} - 100 \beta_{3} - 179 \beta_{4} - 82 \beta_{5} - 26 \beta_{6} + 14 \beta_{7} ) q^{55} + ( -1472 - 64 \beta_{1} + 64 \beta_{3} ) q^{56} + ( 1503 + 99 \beta_{1} + 18 \beta_{2} - 63 \beta_{3} - 9 \beta_{4} + 54 \beta_{5} + 9 \beta_{7} ) q^{57} + ( -2616 - 120 \beta_{1} - 32 \beta_{2} - 44 \beta_{3} + 44 \beta_{4} - 4 \beta_{6} ) q^{58} -3481 q^{59} + ( 720 + 144 \beta_{1} ) q^{60} + ( 457 - 54 \beta_{1} - 161 \beta_{2} - 9 \beta_{3} - 105 \beta_{4} + 97 \beta_{5} + 4 \beta_{6} - 10 \beta_{7} ) q^{61} + ( -9184 + 4 \beta_{1} - 4 \beta_{2} + 12 \beta_{3} + 84 \beta_{4} + 64 \beta_{5} + 20 \beta_{6} ) q^{62} + ( 1863 + 81 \beta_{1} - 81 \beta_{3} ) q^{63} + 4096 q^{64} + ( -7501 + 105 \beta_{1} - 24 \beta_{2} + 42 \beta_{3} - 65 \beta_{4} - 219 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{65} + ( 1584 - 36 \beta_{1} - 36 \beta_{2} + 36 \beta_{4} + 36 \beta_{7} ) q^{66} + ( 6940 - 191 \beta_{1} + 3 \beta_{2} + 142 \beta_{3} + 171 \beta_{4} + 52 \beta_{5} - 4 \beta_{6} ) q^{67} + ( 944 + 80 \beta_{1} - 64 \beta_{2} - 80 \beta_{3} - 16 \beta_{4} - 32 \beta_{6} ) q^{68} + ( 1269 + 189 \beta_{1} + 72 \beta_{2} + 99 \beta_{3} + 135 \beta_{4} + 9 \beta_{6} ) q^{69} + ( -9892 - 120 \beta_{1} + 164 \beta_{2} + 44 \beta_{3} + 144 \beta_{4} + 16 \beta_{5} + 4 \beta_{6} ) q^{70} + ( -6158 + 55 \beta_{1} + 129 \beta_{2} + 62 \beta_{3} + 10 \beta_{4} - 105 \beta_{5} + 34 \beta_{6} + 70 \beta_{7} ) q^{71} -5184 q^{72} + ( -4027 - 113 \beta_{1} + 262 \beta_{2} + 95 \beta_{3} + 10 \beta_{4} - 60 \beta_{5} + 7 \beta_{6} ) q^{73} + ( -14992 + 236 \beta_{1} - 124 \beta_{2} - 172 \beta_{3} - 232 \beta_{4} - 56 \beta_{5} + 32 \beta_{6} ) q^{74} + ( 10854 + 216 \beta_{1} - 117 \beta_{2} - 18 \beta_{3} - 9 \beta_{4} - 27 \beta_{5} + 18 \beta_{6} + 9 \beta_{7} ) q^{75} + ( 2672 + 176 \beta_{1} + 32 \beta_{2} - 112 \beta_{3} - 16 \beta_{4} + 96 \beta_{5} + 16 \beta_{7} ) q^{76} + ( 7731 - 377 \beta_{2} - 243 \beta_{3} - 12 \beta_{4} - 22 \beta_{5} - 24 \beta_{6} - 98 \beta_{7} ) q^{77} + ( -540 - 36 \beta_{2} + 72 \beta_{3} + 72 \beta_{5} - 36 \beta_{6} ) q^{78} + ( 16799 + 91 \beta_{1} + 92 \beta_{2} + 206 \beta_{3} - 9 \beta_{4} - 5 \beta_{5} + 78 \beta_{6} - 70 \beta_{7} ) q^{79} + ( 1280 + 256 \beta_{1} ) q^{80} + 6561 q^{81} + ( -4120 - 132 \beta_{1} - 36 \beta_{2} - 108 \beta_{3} + 32 \beta_{4} - 84 \beta_{5} - 52 \beta_{6} - 40 \beta_{7} ) q^{82} + ( -11242 - 87 \beta_{1} + 315 \beta_{2} + 89 \beta_{3} + 146 \beta_{4} - 5 \beta_{5} + 71 \beta_{6} + 70 \beta_{7} ) q^{83} + ( 3312 + 144 \beta_{1} - 144 \beta_{3} ) q^{84} + ( 14454 + 215 \beta_{1} - 49 \beta_{2} - 149 \beta_{3} - 251 \beta_{4} + 175 \beta_{5} + 84 \beta_{6} - 22 \beta_{7} ) q^{85} + ( -15312 + 404 \beta_{1} + 44 \beta_{2} - 128 \beta_{3} - 364 \beta_{4} - 80 \beta_{5} - 52 \beta_{6} ) q^{86} + ( 5886 + 270 \beta_{1} + 72 \beta_{2} + 99 \beta_{3} - 99 \beta_{4} + 9 \beta_{6} ) q^{87} + ( 2816 - 64 \beta_{1} - 64 \beta_{2} + 64 \beta_{4} + 64 \beta_{7} ) q^{88} + ( -14192 - 437 \beta_{1} + 179 \beta_{2} + 67 \beta_{3} + 246 \beta_{4} + 103 \beta_{5} + 33 \beta_{6} - \beta_{7} ) q^{89} + ( -1620 - 324 \beta_{1} ) q^{90} + ( 46948 - 404 \beta_{1} + 324 \beta_{2} + 139 \beta_{3} + 80 \beta_{4} - 49 \beta_{5} - 95 \beta_{6} - 22 \beta_{7} ) q^{91} + ( 2256 + 336 \beta_{1} + 128 \beta_{2} + 176 \beta_{3} + 240 \beta_{4} + 16 \beta_{6} ) q^{92} + ( 20664 - 9 \beta_{1} + 9 \beta_{2} - 27 \beta_{3} - 189 \beta_{4} - 144 \beta_{5} - 45 \beta_{6} ) q^{93} + ( 2816 + 388 \beta_{1} + 84 \beta_{2} + 60 \beta_{3} - 64 \beta_{4} + 36 \beta_{5} - 40 \beta_{7} ) q^{94} + ( 41513 - 188 \beta_{1} - 645 \beta_{2} - 201 \beta_{3} - 123 \beta_{4} + 380 \beta_{5} - 163 \beta_{6} - 14 \beta_{7} ) q^{95} -9216 q^{96} + ( 68947 + 253 \beta_{1} - 44 \beta_{2} - 66 \beta_{3} - 39 \beta_{4} - 237 \beta_{5} - 53 \beta_{6} - 70 \beta_{7} ) q^{97} + ( -16204 + 100 \beta_{1} - 140 \beta_{2} + 180 \beta_{3} + 212 \beta_{4} - 4 \beta_{5} + 84 \beta_{6} + 40 \beta_{7} ) q^{98} + ( -3564 + 81 \beta_{1} + 81 \beta_{2} - 81 \beta_{4} - 81 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 32q^{2} + 72q^{3} + 128q^{4} + 40q^{5} - 288q^{6} + 181q^{7} - 512q^{8} + 648q^{9} + O(q^{10})$$ $$8q - 32q^{2} + 72q^{3} + 128q^{4} + 40q^{5} - 288q^{6} + 181q^{7} - 512q^{8} + 648q^{9} - 160q^{10} - 349q^{11} + 1152q^{12} + 121q^{13} - 724q^{14} + 360q^{15} + 2048q^{16} + 437q^{17} - 2592q^{18} + 1314q^{19} + 640q^{20} + 1629q^{21} + 1396q^{22} + 1224q^{23} - 4608q^{24} + 9592q^{25} - 484q^{26} + 5832q^{27} + 2896q^{28} + 5276q^{29} - 1440q^{30} + 18332q^{31} - 8192q^{32} - 3141q^{33} - 1748q^{34} + 19518q^{35} + 10368q^{36} + 30331q^{37} - 5256q^{38} + 1089q^{39} - 2560q^{40} + 8323q^{41} - 6516q^{42} + 30851q^{43} - 5584q^{44} + 3240q^{45} - 4896q^{46} - 5730q^{47} + 18432q^{48} + 32295q^{49} - 38368q^{50} + 3933q^{51} + 1936q^{52} - 33524q^{53} - 23328q^{54} + 23660q^{55} - 11584q^{56} + 11826q^{57} - 21104q^{58} - 27848q^{59} + 5760q^{60} + 2692q^{61} - 73328q^{62} + 14661q^{63} + 32768q^{64} - 59892q^{65} + 12564q^{66} + 56244q^{67} + 6992q^{68} + 11016q^{69} - 78072q^{70} - 48473q^{71} - 41472q^{72} - 30796q^{73} - 121324q^{74} + 86328q^{75} + 21024q^{76} + 59683q^{77} - 4356q^{78} + 135513q^{79} + 10240q^{80} + 52488q^{81} - 33292q^{82} - 88111q^{83} + 26064q^{84} + 114418q^{85} - 123404q^{86} + 47484q^{87} + 22336q^{88} - 112196q^{89} - 12960q^{90} + 377433q^{91} + 19584q^{92} + 164988q^{93} + 22920q^{94} + 328146q^{95} - 73728q^{96} + 551378q^{97} - 129180q^{98} - 28269q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 17196 x^{6} - 154000 x^{5} + 98085975 x^{4} + 1816612536 x^{3} - 184506058580 x^{2} - 5340414471536 x - 7060184373200$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$3531716616097 \nu^{7} - 639912128181572 \nu^{6} - 59395676919722840 \nu^{5} + 7662854129567508840 \nu^{4} + 413818596650176083735 \nu^{3} - 24015765812473884854868 \nu^{2} - 1090983151757535031607492 \nu + 3757903959923072022194600$$$$)/$$$$25\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$4331671669311 \nu^{7} + 339562400775494 \nu^{6} - 45540275145935855 \nu^{5} - 4450296145865676045 \nu^{4} + 40881078223701439170 \nu^{3} + 14020655241938427706501 \nu^{2} + 447903627862793835852124 \nu + 2466030296544154766061300$$$$)/$$$$83\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$18941466661253 \nu^{7} + 2929018216048652 \nu^{6} - 187936906375918270 \nu^{5} - 37470742214234503830 \nu^{4} - 77726975016093988395 \nu^{3} + 118882081408474805243778 \nu^{2} + 3100320863871597998825912 \nu - 1246432753351265153998100$$$$)/$$$$25\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$7432150279107 \nu^{7} + 249623166066313 \nu^{6} - 87615891526001305 \nu^{5} - 4165604419827313895 \nu^{4} + 212054457767145037920 \nu^{3} + 15858474912144450807232 \nu^{2} + 279549793018908407947528 \nu + 961370624180832490921100$$$$)/$$$$83\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$4967625572903 \nu^{7} - 294881418090673 \nu^{6} - 55517818438958185 \nu^{5} + 2575360308409497765 \nu^{4} + 201230858717170732080 \nu^{3} - 4284446168296475413242 \nu^{2} - 290245484235187120648048 \nu - 2881296845168889228464600$$$$)/$$$$50\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$108056909469313 \nu^{7} + 1842957629844727 \nu^{6} - 1466687196552360215 \nu^{5} - 27799458267591747885 \nu^{4} + 5443382783511001755660 \nu^{3} + 101381882227450008974508 \nu^{2} - 3326776608798912127855508 \nu - 7825638820126873133659600$$$$)/$$$$25\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + 2 \beta_{6} - 3 \beta_{5} - \beta_{4} - 2 \beta_{3} - 13 \beta_{2} + 14 \beta_{1} + 4306$$ $$\nu^{3}$$ $$=$$ $$-7 \beta_{7} + 45 \beta_{6} - 118 \beta_{5} + 196 \beta_{4} - 198 \beta_{3} + 320 \beta_{2} + 5937 \beta_{1} + 57594$$ $$\nu^{4}$$ $$=$$ $$7588 \beta_{7} + 13622 \beta_{6} - 32590 \beta_{5} - 10207 \beta_{4} + 5106 \beta_{3} - 86263 \beta_{2} + 120603 \beta_{1} + 24921557$$ $$\nu^{5}$$ $$=$$ $$-98944 \beta_{7} + 664035 \beta_{6} - 1607631 \beta_{5} + 1790372 \beta_{4} - 1061246 \beta_{3} + 2809200 \beta_{2} + 36897508 \beta_{1} + 517981613$$ $$\nu^{6}$$ $$=$$ $$54289545 \beta_{7} + 92992622 \beta_{6} - 295325547 \beta_{5} - 49185113 \beta_{4} + 91231074 \beta_{3} - 522500173 \beta_{2} + 923618362 \beta_{1} + 153680491878$$ $$\nu^{7}$$ $$=$$ $$-670933991 \beta_{7} + 6788260045 \beta_{6} - 16409305734 \beta_{5} + 13578877668 \beta_{4} - 2796242614 \beta_{3} + 20925831340 \beta_{2} + 234669656229 \beta_{1} + 3952224957842$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −79.5001 −74.0889 −46.8085 −36.8040 −1.38959 75.7365 77.0538 85.8007
−4.00000 9.00000 16.0000 −74.5001 −36.0000 −124.731 −64.0000 81.0000 298.000
1.2 −4.00000 9.00000 16.0000 −69.0889 −36.0000 −83.5010 −64.0000 81.0000 276.356
1.3 −4.00000 9.00000 16.0000 −41.8085 −36.0000 175.859 −64.0000 81.0000 167.234
1.4 −4.00000 9.00000 16.0000 −31.8040 −36.0000 191.018 −64.0000 81.0000 127.216
1.5 −4.00000 9.00000 16.0000 3.61041 −36.0000 −202.768 −64.0000 81.0000 −14.4417
1.6 −4.00000 9.00000 16.0000 80.7365 −36.0000 18.8441 −64.0000 81.0000 −322.946
1.7 −4.00000 9.00000 16.0000 82.0538 −36.0000 186.988 −64.0000 81.0000 −328.215
1.8 −4.00000 9.00000 16.0000 90.8007 −36.0000 19.2910 −64.0000 81.0000 −363.203
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$59$$ $$1$$

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{5}^{8} - \cdots$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(354))$$.