# Properties

 Label 354.6.a.d Level 354 Weight 6 Character orbit 354.a Self dual Yes Analytic conductor 56.776 Analytic rank 1 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: $$1$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + 4 q^{2} -9 q^{3} + 16 q^{4} + ( -5 + \beta_{3} + \beta_{4} ) q^{5} -36 q^{6} + ( -21 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{7} + 64 q^{8} + 81 q^{9} +O(q^{10})$$ $$q + 4 q^{2} -9 q^{3} + 16 q^{4} + ( -5 + \beta_{3} + \beta_{4} ) q^{5} -36 q^{6} + ( -21 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{7} + 64 q^{8} + 81 q^{9} + ( -20 + 4 \beta_{3} + 4 \beta_{4} ) q^{10} + ( -38 - 10 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} - 7 \beta_{4} ) q^{11} -144 q^{12} + ( -20 + 14 \beta_{1} - \beta_{2} - 13 \beta_{3} - 10 \beta_{4} ) q^{13} + ( -84 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{14} + ( 45 - 9 \beta_{3} - 9 \beta_{4} ) q^{15} + 256 q^{16} + ( -185 - 37 \beta_{1} + 12 \beta_{2} - 15 \beta_{3} + 17 \beta_{4} ) q^{17} + 324 q^{18} + ( -51 + 63 \beta_{1} - 36 \beta_{2} - 16 \beta_{3} + 10 \beta_{4} ) q^{19} + ( -80 + 16 \beta_{3} + 16 \beta_{4} ) q^{20} + ( 189 - 9 \beta_{1} - 9 \beta_{2} - 9 \beta_{3} ) q^{21} + ( -152 - 40 \beta_{1} - 16 \beta_{2} - 24 \beta_{3} - 28 \beta_{4} ) q^{22} + ( -346 - 77 \beta_{1} + 22 \beta_{2} + 14 \beta_{3} - 35 \beta_{4} ) q^{23} -576 q^{24} + ( 291 + 80 \beta_{1} + 60 \beta_{2} - 5 \beta_{3} - 80 \beta_{4} ) q^{25} + ( -80 + 56 \beta_{1} - 4 \beta_{2} - 52 \beta_{3} - 40 \beta_{4} ) q^{26} -729 q^{27} + ( -336 + 16 \beta_{1} + 16 \beta_{2} + 16 \beta_{3} ) q^{28} + ( -1259 - 95 \beta_{1} - 56 \beta_{2} + 23 \beta_{3} - 18 \beta_{4} ) q^{29} + ( 180 - 36 \beta_{3} - 36 \beta_{4} ) q^{30} + ( -1853 - 33 \beta_{1} - 43 \beta_{2} + 64 \beta_{3} + 71 \beta_{4} ) q^{31} + 1024 q^{32} + ( 342 + 90 \beta_{1} + 36 \beta_{2} + 54 \beta_{3} + 63 \beta_{4} ) q^{33} + ( -740 - 148 \beta_{1} + 48 \beta_{2} - 60 \beta_{3} + 68 \beta_{4} ) q^{34} + ( 2378 + 31 \beta_{1} + 76 \beta_{2} + 7 \beta_{3} - 19 \beta_{4} ) q^{35} + 1296 q^{36} + ( -1510 - 47 \beta_{1} - 86 \beta_{2} + 121 \beta_{3} + 44 \beta_{4} ) q^{37} + ( -204 + 252 \beta_{1} - 144 \beta_{2} - 64 \beta_{3} + 40 \beta_{4} ) q^{38} + ( 180 - 126 \beta_{1} + 9 \beta_{2} + 117 \beta_{3} + 90 \beta_{4} ) q^{39} + ( -320 + 64 \beta_{3} + 64 \beta_{4} ) q^{40} + ( -7818 + 15 \beta_{1} - 180 \beta_{2} + 30 \beta_{3} + 152 \beta_{4} ) q^{41} + ( 756 - 36 \beta_{1} - 36 \beta_{2} - 36 \beta_{3} ) q^{42} + ( -1813 + 460 \beta_{1} + 426 \beta_{2} - 50 \beta_{3} - 4 \beta_{4} ) q^{43} + ( -608 - 160 \beta_{1} - 64 \beta_{2} - 96 \beta_{3} - 112 \beta_{4} ) q^{44} + ( -405 + 81 \beta_{3} + 81 \beta_{4} ) q^{45} + ( -1384 - 308 \beta_{1} + 88 \beta_{2} + 56 \beta_{3} - 140 \beta_{4} ) q^{46} + ( -7291 - 199 \beta_{1} + 286 \beta_{2} - 101 \beta_{3} + 190 \beta_{4} ) q^{47} -2304 q^{48} + ( -12645 - 33 \beta_{1} + 75 \beta_{2} + 27 \beta_{4} ) q^{49} + ( 1164 + 320 \beta_{1} + 240 \beta_{2} - 20 \beta_{3} - 320 \beta_{4} ) q^{50} + ( 1665 + 333 \beta_{1} - 108 \beta_{2} + 135 \beta_{3} - 153 \beta_{4} ) q^{51} + ( -320 + 224 \beta_{1} - 16 \beta_{2} - 208 \beta_{3} - 160 \beta_{4} ) q^{52} + ( -5825 + 452 \beta_{1} - 366 \beta_{2} + 173 \beta_{3} - 189 \beta_{4} ) q^{53} -2916 q^{54} + ( -23950 - 430 \beta_{1} - 542 \beta_{2} - 143 \beta_{3} + 430 \beta_{4} ) q^{55} + ( -1344 + 64 \beta_{1} + 64 \beta_{2} + 64 \beta_{3} ) q^{56} + ( 459 - 567 \beta_{1} + 324 \beta_{2} + 144 \beta_{3} - 90 \beta_{4} ) q^{57} + ( -5036 - 380 \beta_{1} - 224 \beta_{2} + 92 \beta_{3} - 72 \beta_{4} ) q^{58} + 3481 q^{59} + ( 720 - 144 \beta_{3} - 144 \beta_{4} ) q^{60} + ( -150 + 645 \beta_{1} + 553 \beta_{2} - 298 \beta_{3} - 560 \beta_{4} ) q^{61} + ( -7412 - 132 \beta_{1} - 172 \beta_{2} + 256 \beta_{3} + 284 \beta_{4} ) q^{62} + ( -1701 + 81 \beta_{1} + 81 \beta_{2} + 81 \beta_{3} ) q^{63} + 4096 q^{64} + ( -39546 - 1046 \beta_{1} - 837 \beta_{2} + 50 \beta_{3} + 861 \beta_{4} ) q^{65} + ( 1368 + 360 \beta_{1} + 144 \beta_{2} + 216 \beta_{3} + 252 \beta_{4} ) q^{66} + ( -4102 - 300 \beta_{1} - 612 \beta_{2} + 27 \beta_{3} + 38 \beta_{4} ) q^{67} + ( -2960 - 592 \beta_{1} + 192 \beta_{2} - 240 \beta_{3} + 272 \beta_{4} ) q^{68} + ( 3114 + 693 \beta_{1} - 198 \beta_{2} - 126 \beta_{3} + 315 \beta_{4} ) q^{69} + ( 9512 + 124 \beta_{1} + 304 \beta_{2} + 28 \beta_{3} - 76 \beta_{4} ) q^{70} + ( -14659 + 104 \beta_{1} + 1181 \beta_{2} + 167 \beta_{3} + 191 \beta_{4} ) q^{71} + 5184 q^{72} + ( -11786 - 883 \beta_{1} - 775 \beta_{2} - 160 \beta_{3} - 794 \beta_{4} ) q^{73} + ( -6040 - 188 \beta_{1} - 344 \beta_{2} + 484 \beta_{3} + 176 \beta_{4} ) q^{74} + ( -2619 - 720 \beta_{1} - 540 \beta_{2} + 45 \beta_{3} + 720 \beta_{4} ) q^{75} + ( -816 + 1008 \beta_{1} - 576 \beta_{2} - 256 \beta_{3} + 160 \beta_{4} ) q^{76} + ( -20585 + 339 \beta_{1} - 666 \beta_{2} - 322 \beta_{3} + 145 \beta_{4} ) q^{77} + ( 720 - 504 \beta_{1} + 36 \beta_{2} + 468 \beta_{3} + 360 \beta_{4} ) q^{78} + ( -2800 + 222 \beta_{1} + 1273 \beta_{2} - 485 \beta_{3} - 738 \beta_{4} ) q^{79} + ( -1280 + 256 \beta_{3} + 256 \beta_{4} ) q^{80} + 6561 q^{81} + ( -31272 + 60 \beta_{1} - 720 \beta_{2} + 120 \beta_{3} + 608 \beta_{4} ) q^{82} + ( -14990 + 1019 \beta_{1} + 206 \beta_{2} + 352 \beta_{3} - 596 \beta_{4} ) q^{83} + ( 3024 - 144 \beta_{1} - 144 \beta_{2} - 144 \beta_{3} ) q^{84} + ( 9283 + 413 \beta_{1} + 1043 \beta_{2} - 725 \beta_{3} - 1921 \beta_{4} ) q^{85} + ( -7252 + 1840 \beta_{1} + 1704 \beta_{2} - 200 \beta_{3} - 16 \beta_{4} ) q^{86} + ( 11331 + 855 \beta_{1} + 504 \beta_{2} - 207 \beta_{3} + 162 \beta_{4} ) q^{87} + ( -2432 - 640 \beta_{1} - 256 \beta_{2} - 384 \beta_{3} - 448 \beta_{4} ) q^{88} + ( -18346 - 617 \beta_{1} + 2128 \beta_{2} + 379 \beta_{3} - 1115 \beta_{4} ) q^{89} + ( -1620 + 324 \beta_{3} + 324 \beta_{4} ) q^{90} + ( -25978 - 1369 \beta_{1} - 1093 \beta_{2} - 39 \beta_{3} + 115 \beta_{4} ) q^{91} + ( -5536 - 1232 \beta_{1} + 352 \beta_{2} + 224 \beta_{3} - 560 \beta_{4} ) q^{92} + ( 16677 + 297 \beta_{1} + 387 \beta_{2} - 576 \beta_{3} - 639 \beta_{4} ) q^{93} + ( -29164 - 796 \beta_{1} + 1144 \beta_{2} - 404 \beta_{3} + 760 \beta_{4} ) q^{94} + ( -30174 - 807 \beta_{1} - 2549 \beta_{2} - 157 \beta_{3} + 28 \beta_{4} ) q^{95} -9216 q^{96} + ( -79839 - 2992 \beta_{1} - 265 \beta_{2} - 504 \beta_{3} + 400 \beta_{4} ) q^{97} + ( -50580 - 132 \beta_{1} + 300 \beta_{2} + 108 \beta_{4} ) q^{98} + ( -3078 - 810 \beta_{1} - 324 \beta_{2} - 486 \beta_{3} - 567 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q + 20q^{2} - 45q^{3} + 80q^{4} - 24q^{5} - 180q^{6} - 103q^{7} + 320q^{8} + 405q^{9} + O(q^{10})$$ $$5q + 20q^{2} - 45q^{3} + 80q^{4} - 24q^{5} - 180q^{6} - 103q^{7} + 320q^{8} + 405q^{9} - 96q^{10} - 211q^{11} - 720q^{12} - 97q^{13} - 412q^{14} + 216q^{15} + 1280q^{16} - 933q^{17} + 1620q^{18} - 218q^{19} - 384q^{20} + 927q^{21} - 844q^{22} - 1820q^{23} - 2880q^{24} + 1515q^{25} - 388q^{26} - 3645q^{27} - 1648q^{28} - 6464q^{29} + 864q^{30} - 9270q^{31} + 5120q^{32} + 1899q^{33} - 3732q^{34} + 11978q^{35} + 6480q^{36} - 7639q^{37} - 872q^{38} + 873q^{39} - 1536q^{40} - 39103q^{41} + 3708q^{42} - 8183q^{43} - 3376q^{44} - 1944q^{45} - 7280q^{46} - 36178q^{47} - 11520q^{48} - 63156q^{49} + 6060q^{50} + 8397q^{51} - 1552q^{52} - 29228q^{53} - 14580q^{54} - 120292q^{55} - 6592q^{56} + 1962q^{57} - 25856q^{58} + 17405q^{59} + 3456q^{60} - 112q^{61} - 37080q^{62} - 8343q^{63} + 20480q^{64} - 198752q^{65} + 7596q^{66} - 21384q^{67} - 14928q^{68} + 16380q^{69} + 47912q^{70} - 71819q^{71} + 25920q^{72} - 61382q^{73} - 30556q^{74} - 13635q^{75} - 3488q^{76} - 103107q^{77} + 3492q^{78} - 13243q^{79} - 6144q^{80} + 32805q^{81} - 156412q^{82} - 74321q^{83} + 14832q^{84} + 45950q^{85} - 32732q^{86} + 58176q^{87} - 13504q^{88} - 91334q^{89} - 7776q^{90} - 132237q^{91} - 29120q^{92} + 83430q^{93} - 144712q^{94} - 154198q^{95} - 46080q^{96} - 402052q^{97} - 252624q^{98} - 17091q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 835 x^{3} + 14269 x^{2} - 82497 x + 143433$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-14 \nu^{4} - 115 \nu^{3} + 10397 \nu^{2} - 107621 \nu + 298110$$$$)/1089$$ $$\beta_{3}$$ $$=$$ $$($$$$29 \nu^{4} + 316 \nu^{3} - 20681 \nu^{2} + 165290 \nu - 264522$$$$)/1089$$ $$\beta_{4}$$ $$=$$ $$($$$$-70 \nu^{4} - 575 \nu^{3} + 53074 \nu^{2} - 510880 \nu + 1120290$$$$)/1089$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} - 5 \beta_{2} - 25 \beta_{1} + 340$$ $$\nu^{3}$$ $$=$$ $$-11 \beta_{4} + 14 \beta_{3} + 84 \beta_{2} + 1016 \beta_{1} - 8278$$ $$\nu^{4}$$ $$=$$ $$833 \beta_{4} - 115 \beta_{3} - 4481 \beta_{2} - 34599 \beta_{1} + 341790$$

## 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
 8.70328 10.1932 −35.6129 14.6087 3.10767
4.00000 −9.00000 16.0000 −100.786 −36.0000 −35.1732 64.0000 81.0000 −403.143
1.2 4.00000 −9.00000 16.0000 −21.6596 −36.0000 −77.1916 64.0000 81.0000 −86.6382
1.3 4.00000 −9.00000 16.0000 −9.50125 −36.0000 −69.2606 64.0000 81.0000 −38.0050
1.4 4.00000 −9.00000 16.0000 36.7026 −36.0000 −14.5171 64.0000 81.0000 146.810
1.5 4.00000 −9.00000 16.0000 71.2441 −36.0000 93.1425 64.0000 81.0000 284.976
 $$n$$: 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 $$T_{5}^{5} + 24 T_{5}^{4} - 8282 T_{5}^{3} + 4532 T_{5}^{2} + 6511281 T_{5} + 54234444$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(354))$$.