# Properties

 Label 354.6.a.e Level 354 Weight 6 Character orbit 354.a Self dual Yes Analytic conductor 56.776 Analytic rank 0 Dimension 5 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: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{4}\cdot 5$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 q^{2} -9 q^{3} + 16 q^{4} + ( 33 - \beta_{2} ) q^{5} -36 q^{6} + ( -40 + \beta_{1} ) q^{7} + 64 q^{8} + 81 q^{9} +O(q^{10})$$ $$q + 4 q^{2} -9 q^{3} + 16 q^{4} + ( 33 - \beta_{2} ) q^{5} -36 q^{6} + ( -40 + \beta_{1} ) q^{7} + 64 q^{8} + 81 q^{9} + ( 132 - 4 \beta_{2} ) q^{10} + ( -105 - 5 \beta_{2} - 2 \beta_{3} ) q^{11} -144 q^{12} + ( 205 + 3 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{13} + ( -160 + 4 \beta_{1} ) q^{14} + ( -297 + 9 \beta_{2} ) q^{15} + 256 q^{16} + ( 597 + \beta_{1} - 11 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{17} + 324 q^{18} + ( -25 - 4 \beta_{1} - 19 \beta_{2} + \beta_{3} - \beta_{4} ) q^{19} + ( 528 - 16 \beta_{2} ) q^{20} + ( 360 - 9 \beta_{1} ) q^{21} + ( -420 - 20 \beta_{2} - 8 \beta_{3} ) q^{22} + ( -251 - 7 \beta_{1} + 11 \beta_{2} + 5 \beta_{3} + 6 \beta_{4} ) q^{23} -576 q^{24} + ( 1776 - 7 \beta_{1} - 47 \beta_{2} - 22 \beta_{3} - 5 \beta_{4} ) q^{25} + ( 820 + 12 \beta_{1} + 20 \beta_{2} + 12 \beta_{3} + 4 \beta_{4} ) q^{26} -729 q^{27} + ( -640 + 16 \beta_{1} ) q^{28} + ( 3284 + \beta_{1} + 24 \beta_{2} + 21 \beta_{3} - 8 \beta_{4} ) q^{29} + ( -1188 + 36 \beta_{2} ) q^{30} + ( -944 - 14 \beta_{1} + 34 \beta_{2} + 6 \beta_{3} - 5 \beta_{4} ) q^{31} + 1024 q^{32} + ( 945 + 45 \beta_{2} + 18 \beta_{3} ) q^{33} + ( 2388 + 4 \beta_{1} - 44 \beta_{2} - 4 \beta_{3} - 8 \beta_{4} ) q^{34} + ( -1918 + 58 \beta_{1} + 88 \beta_{2} + 57 \beta_{3} + 13 \beta_{4} ) q^{35} + 1296 q^{36} + ( 1003 - 10 \beta_{1} + 31 \beta_{2} - 41 \beta_{3} + 15 \beta_{4} ) q^{37} + ( -100 - 16 \beta_{1} - 76 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} ) q^{38} + ( -1845 - 27 \beta_{1} - 45 \beta_{2} - 27 \beta_{3} - 9 \beta_{4} ) q^{39} + ( 2112 - 64 \beta_{2} ) q^{40} + ( 1076 - 5 \beta_{1} + 52 \beta_{2} - 81 \beta_{3} + 20 \beta_{4} ) q^{41} + ( 1440 - 36 \beta_{1} ) q^{42} + ( 1097 - 41 \beta_{1} - 61 \beta_{2} - 26 \beta_{3} + 11 \beta_{4} ) q^{43} + ( -1680 - 80 \beta_{2} - 32 \beta_{3} ) q^{44} + ( 2673 - 81 \beta_{2} ) q^{45} + ( -1004 - 28 \beta_{1} + 44 \beta_{2} + 20 \beta_{3} + 24 \beta_{4} ) q^{46} + ( 3906 - 3 \beta_{1} + 132 \beta_{2} - 25 \beta_{3} - 14 \beta_{4} ) q^{47} -2304 q^{48} + ( 16130 - 21 \beta_{1} - 67 \beta_{2} + 204 \beta_{3} - 28 \beta_{4} ) q^{49} + ( 7104 - 28 \beta_{1} - 188 \beta_{2} - 88 \beta_{3} - 20 \beta_{4} ) q^{50} + ( -5373 - 9 \beta_{1} + 99 \beta_{2} + 9 \beta_{3} + 18 \beta_{4} ) q^{51} + ( 3280 + 48 \beta_{1} + 80 \beta_{2} + 48 \beta_{3} + 16 \beta_{4} ) q^{52} + ( 12651 + 32 \beta_{1} + 45 \beta_{2} + 38 \beta_{3} - 56 \beta_{4} ) q^{53} -2916 q^{54} + ( 12427 - 55 \beta_{1} - 73 \beta_{2} - 108 \beta_{3} - 5 \beta_{4} ) q^{55} + ( -2560 + 64 \beta_{1} ) q^{56} + ( 225 + 36 \beta_{1} + 171 \beta_{2} - 9 \beta_{3} + 9 \beta_{4} ) q^{57} + ( 13136 + 4 \beta_{1} + 96 \beta_{2} + 84 \beta_{3} - 32 \beta_{4} ) q^{58} -3481 q^{59} + ( -4752 + 144 \beta_{2} ) q^{60} + ( 14364 - 153 \beta_{1} + 130 \beta_{2} - 192 \beta_{3} - 4 \beta_{4} ) q^{61} + ( -3776 - 56 \beta_{1} + 136 \beta_{2} + 24 \beta_{3} - 20 \beta_{4} ) q^{62} + ( -3240 + 81 \beta_{1} ) q^{63} + 4096 q^{64} + ( -7835 + 280 \beta_{1} + 311 \beta_{2} + 187 \beta_{3} + 96 \beta_{4} ) q^{65} + ( 3780 + 180 \beta_{2} + 72 \beta_{3} ) q^{66} + ( 15893 + 91 \beta_{1} - 33 \beta_{2} - 172 \beta_{3} + 31 \beta_{4} ) q^{67} + ( 9552 + 16 \beta_{1} - 176 \beta_{2} - 16 \beta_{3} - 32 \beta_{4} ) q^{68} + ( 2259 + 63 \beta_{1} - 99 \beta_{2} - 45 \beta_{3} - 54 \beta_{4} ) q^{69} + ( -7672 + 232 \beta_{1} + 352 \beta_{2} + 228 \beta_{3} + 52 \beta_{4} ) q^{70} + ( 1861 + 28 \beta_{1} + 59 \beta_{2} - 253 \beta_{3} - 60 \beta_{4} ) q^{71} + 5184 q^{72} + ( 18464 + 89 \beta_{1} - 358 \beta_{2} + 28 \beta_{3} + 14 \beta_{4} ) q^{73} + ( 4012 - 40 \beta_{1} + 124 \beta_{2} - 164 \beta_{3} + 60 \beta_{4} ) q^{74} + ( -15984 + 63 \beta_{1} + 423 \beta_{2} + 198 \beta_{3} + 45 \beta_{4} ) q^{75} + ( -400 - 64 \beta_{1} - 304 \beta_{2} + 16 \beta_{3} - 16 \beta_{4} ) q^{76} + ( 8950 - 130 \beta_{1} + 772 \beta_{2} + 381 \beta_{3} + 103 \beta_{4} ) q^{77} + ( -7380 - 108 \beta_{1} - 180 \beta_{2} - 108 \beta_{3} - 36 \beta_{4} ) q^{78} + ( -3455 + 33 \beta_{1} - 37 \beta_{2} + 579 \beta_{3} + 57 \beta_{4} ) q^{79} + ( 8448 - 256 \beta_{2} ) q^{80} + 6561 q^{81} + ( 4304 - 20 \beta_{1} + 208 \beta_{2} - 324 \beta_{3} + 80 \beta_{4} ) q^{82} + ( 20988 - 51 \beta_{1} - 162 \beta_{2} - 367 \beta_{3} - 148 \beta_{4} ) q^{83} + ( 5760 - 144 \beta_{1} ) q^{84} + ( 56447 - 111 \beta_{1} - 1037 \beta_{2} - 2 \beta_{3} - 156 \beta_{4} ) q^{85} + ( 4388 - 164 \beta_{1} - 244 \beta_{2} - 104 \beta_{3} + 44 \beta_{4} ) q^{86} + ( -29556 - 9 \beta_{1} - 216 \beta_{2} - 189 \beta_{3} + 72 \beta_{4} ) q^{87} + ( -6720 - 320 \beta_{2} - 128 \beta_{3} ) q^{88} + ( 63272 - 26 \beta_{1} - 94 \beta_{2} + 179 \beta_{3} + 47 \beta_{4} ) q^{89} + ( 10692 - 324 \beta_{2} ) q^{90} + ( 70242 + 204 \beta_{1} - 1654 \beta_{2} - 156 \beta_{3} - 247 \beta_{4} ) q^{91} + ( -4016 - 112 \beta_{1} + 176 \beta_{2} + 80 \beta_{3} + 96 \beta_{4} ) q^{92} + ( 8496 + 126 \beta_{1} - 306 \beta_{2} - 54 \beta_{3} + 45 \beta_{4} ) q^{93} + ( 15624 - 12 \beta_{1} + 528 \beta_{2} - 100 \beta_{3} - 56 \beta_{4} ) q^{94} + ( 74077 - 396 \beta_{1} - 519 \beta_{2} - 556 \beta_{3} - 219 \beta_{4} ) q^{95} -9216 q^{96} + ( 53651 - 41 \beta_{1} - 953 \beta_{2} + 413 \beta_{3} + 111 \beta_{4} ) q^{97} + ( 64520 - 84 \beta_{1} - 268 \beta_{2} + 816 \beta_{3} - 112 \beta_{4} ) q^{98} + ( -8505 - 405 \beta_{2} - 162 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q + 20q^{2} - 45q^{3} + 80q^{4} + 166q^{5} - 180q^{6} - 198q^{7} + 320q^{8} + 405q^{9} + O(q^{10})$$ $$5q + 20q^{2} - 45q^{3} + 80q^{4} + 166q^{5} - 180q^{6} - 198q^{7} + 320q^{8} + 405q^{9} + 664q^{10} - 516q^{11} - 720q^{12} + 1018q^{13} - 792q^{14} - 1494q^{15} + 1280q^{16} + 3004q^{17} + 1620q^{18} - 114q^{19} + 2656q^{20} + 1782q^{21} - 2064q^{22} - 1302q^{23} - 2880q^{24} + 8967q^{25} + 4072q^{26} - 3645q^{27} - 3168q^{28} + 16372q^{29} - 5976q^{30} - 4784q^{31} + 5120q^{32} + 4644q^{33} + 12016q^{34} - 9702q^{35} + 6480q^{36} + 5016q^{37} - 456q^{38} - 9162q^{39} + 10624q^{40} + 5440q^{41} + 7128q^{42} + 5494q^{43} - 8256q^{44} + 13446q^{45} - 5208q^{46} + 19470q^{47} - 11520q^{48} + 80323q^{49} + 35868q^{50} - 27036q^{51} + 16288q^{52} + 63310q^{53} - 14580q^{54} + 62324q^{55} - 12672q^{56} + 1026q^{57} + 65488q^{58} - 17405q^{59} - 23904q^{60} + 71776q^{61} - 19136q^{62} - 16038q^{63} + 20480q^{64} - 39492q^{65} + 18576q^{66} + 79962q^{67} + 48064q^{68} + 11718q^{69} - 38808q^{70} + 9928q^{71} + 25920q^{72} + 92772q^{73} + 20064q^{74} - 80703q^{75} - 1824q^{76} + 42750q^{77} - 36648q^{78} - 18444q^{79} + 42496q^{80} + 32805q^{81} + 21760q^{82} + 106030q^{83} + 28512q^{84} + 283366q^{85} + 21976q^{86} - 147348q^{87} - 33024q^{88} + 315950q^{89} + 53784q^{90} + 354078q^{91} - 20832q^{92} + 43056q^{93} + 77880q^{94} + 371662q^{95} - 46080q^{96} + 268078q^{97} + 321292q^{98} - 41796q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 1060 x^{3} + 10546 x^{2} + 78584 x - 857860$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$24 \nu^{4} + 7095 \nu^{3} + 67029 \nu^{2} - 5641890 \nu + 4060528$$$$)/202474$$ $$\beta_{2}$$ $$=$$ $$($$$$-1154 \nu^{4} - 12131 \nu^{3} + 1104904 \nu^{2} - 186138 \nu - 100418398$$$$)/303711$$ $$\beta_{3}$$ $$=$$ $$($$$$-1817 \nu^{4} - 18311 \nu^{3} + 1644952 \nu^{2} - 664164 \nu - 113378224$$$$)/303711$$ $$\beta_{4}$$ $$=$$ $$($$$$-10142 \nu^{4} - 87665 \nu^{3} + 10170031 \nu^{2} + 391662 \nu - 969727276$$$$)/607422$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} - 6 \beta_{2} - \beta_{1} + 6$$$$)/30$$ $$\nu^{2}$$ $$=$$ $$($$$$4 \beta_{4} - 126 \beta_{3} + 181 \beta_{2} + 6 \beta_{1} + 19074$$$$)/45$$ $$\nu^{3}$$ $$=$$ $$($$$$79 \beta_{4} + 193 \beta_{3} - 651 \beta_{2} + \beta_{1} - 17096$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-8869 \beta_{4} - 151314 \beta_{3} + 265559 \beta_{2} + 5829 \beta_{1} + 17040996$$$$)/45$$

## 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
 −35.2304 12.0064 −9.01128 23.7650 9.47027
4.00000 −9.00000 16.0000 −66.4665 −36.0000 22.9671 64.0000 81.0000 −265.866
1.2 4.00000 −9.00000 16.0000 −5.34525 −36.0000 −243.667 64.0000 81.0000 −21.3810
1.3 4.00000 −9.00000 16.0000 58.5245 −36.0000 233.174 64.0000 81.0000 234.098
1.4 4.00000 −9.00000 16.0000 71.6358 −36.0000 12.9515 64.0000 81.0000 286.543
1.5 4.00000 −9.00000 16.0000 107.651 −36.0000 −223.426 64.0000 81.0000 430.606
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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 $$T_{5}^{5} - 166 T_{5}^{4} + 1482 T_{5}^{3} + 771476 T_{5}^{2} - 25942643 T_{5} - 160346070$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(354))$$.