# Properties

 Label 354.6.a.b Level 354 Weight 6 Character orbit 354.a Self dual Yes Analytic conductor 56.776 Analytic rank 1 Dimension 4 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: $$4$$ Coefficient field: 4.4.32832.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{4}$$ Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + 9 q^{3} + 16 q^{4} + ( -25 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{5} + 36 q^{6} + ( -42 - 3 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{7} + 64 q^{8} + 81 q^{9} +O(q^{10})$$ $$q + 4 q^{2} + 9 q^{3} + 16 q^{4} + ( -25 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{5} + 36 q^{6} + ( -42 - 3 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{7} + 64 q^{8} + 81 q^{9} + ( -100 + 8 \beta_{1} - 8 \beta_{2} + 12 \beta_{3} ) q^{10} + ( -175 - 12 \beta_{1} + 20 \beta_{2} + 3 \beta_{3} ) q^{11} + 144 q^{12} + ( -194 + 8 \beta_{1} - 17 \beta_{2} - 26 \beta_{3} ) q^{13} + ( -168 - 12 \beta_{1} + 16 \beta_{2} - 24 \beta_{3} ) q^{14} + ( -225 + 18 \beta_{1} - 18 \beta_{2} + 27 \beta_{3} ) q^{15} + 256 q^{16} + ( -634 - 31 \beta_{1} - 27 \beta_{2} - 35 \beta_{3} ) q^{17} + 324 q^{18} + ( -1158 - 47 \beta_{1} - 15 \beta_{2} + 20 \beta_{3} ) q^{19} + ( -400 + 32 \beta_{1} - 32 \beta_{2} + 48 \beta_{3} ) q^{20} + ( -378 - 27 \beta_{1} + 36 \beta_{2} - 54 \beta_{3} ) q^{21} + ( -700 - 48 \beta_{1} + 80 \beta_{2} + 12 \beta_{3} ) q^{22} + ( -298 + 23 \beta_{1} + 13 \beta_{2} - 43 \beta_{3} ) q^{23} + 576 q^{24} + ( -196 + 24 \beta_{1} + 120 \beta_{2} - 22 \beta_{3} ) q^{25} + ( -776 + 32 \beta_{1} - 68 \beta_{2} - 104 \beta_{3} ) q^{26} + 729 q^{27} + ( -672 - 48 \beta_{1} + 64 \beta_{2} - 96 \beta_{3} ) q^{28} + ( -1143 + 193 \beta_{1} + 21 \beta_{2} + 122 \beta_{3} ) q^{29} + ( -900 + 72 \beta_{1} - 72 \beta_{2} + 108 \beta_{3} ) q^{30} + ( -1792 - 15 \beta_{1} + 220 \beta_{2} + 21 \beta_{3} ) q^{31} + 1024 q^{32} + ( -1575 - 108 \beta_{1} + 180 \beta_{2} + 27 \beta_{3} ) q^{33} + ( -2536 - 124 \beta_{1} - 108 \beta_{2} - 140 \beta_{3} ) q^{34} + ( -3511 - 245 \beta_{1} - 69 \beta_{2} - 209 \beta_{3} ) q^{35} + 1296 q^{36} + ( -3628 - 471 \beta_{1} - 77 \beta_{2} + 288 \beta_{3} ) q^{37} + ( -4632 - 188 \beta_{1} - 60 \beta_{2} + 80 \beta_{3} ) q^{38} + ( -1746 + 72 \beta_{1} - 153 \beta_{2} - 234 \beta_{3} ) q^{39} + ( -1600 + 128 \beta_{1} - 128 \beta_{2} + 192 \beta_{3} ) q^{40} + ( -3409 + 95 \beta_{1} - 145 \beta_{2} - 354 \beta_{3} ) q^{41} + ( -1512 - 108 \beta_{1} + 144 \beta_{2} - 216 \beta_{3} ) q^{42} + ( 45 + 708 \beta_{1} - 78 \beta_{2} + 424 \beta_{3} ) q^{43} + ( -2800 - 192 \beta_{1} + 320 \beta_{2} + 48 \beta_{3} ) q^{44} + ( -2025 + 162 \beta_{1} - 162 \beta_{2} + 243 \beta_{3} ) q^{45} + ( -1192 + 92 \beta_{1} + 52 \beta_{2} - 172 \beta_{3} ) q^{46} + ( -2823 + 559 \beta_{1} - 183 \beta_{2} - 530 \beta_{3} ) q^{47} + 2304 q^{48} + ( -5814 + 683 \beta_{1} - 206 \beta_{2} + 923 \beta_{3} ) q^{49} + ( -784 + 96 \beta_{1} + 480 \beta_{2} - 88 \beta_{3} ) q^{50} + ( -5706 - 279 \beta_{1} - 243 \beta_{2} - 315 \beta_{3} ) q^{51} + ( -3104 + 128 \beta_{1} - 272 \beta_{2} - 416 \beta_{3} ) q^{52} + ( -13089 - 726 \beta_{1} + 228 \beta_{2} + 237 \beta_{3} ) q^{53} + 2916 q^{54} + ( -5287 - 138 \beta_{1} - 916 \beta_{2} - 772 \beta_{3} ) q^{55} + ( -2688 - 192 \beta_{1} + 256 \beta_{2} - 384 \beta_{3} ) q^{56} + ( -10422 - 423 \beta_{1} - 135 \beta_{2} + 180 \beta_{3} ) q^{57} + ( -4572 + 772 \beta_{1} + 84 \beta_{2} + 488 \beta_{3} ) q^{58} -3481 q^{59} + ( -3600 + 288 \beta_{1} - 288 \beta_{2} + 432 \beta_{3} ) q^{60} + ( -17508 + 295 \beta_{1} - 616 \beta_{2} + 2144 \beta_{3} ) q^{61} + ( -7168 - 60 \beta_{1} + 880 \beta_{2} + 84 \beta_{3} ) q^{62} + ( -3402 - 243 \beta_{1} + 324 \beta_{2} - 486 \beta_{3} ) q^{63} + 4096 q^{64} + ( 3720 - 1290 \beta_{1} + 1821 \beta_{2} - 493 \beta_{3} ) q^{65} + ( -6300 - 432 \beta_{1} + 720 \beta_{2} + 108 \beta_{3} ) q^{66} + ( -11641 + 1724 \beta_{1} - 1980 \beta_{2} + 3344 \beta_{3} ) q^{67} + ( -10144 - 496 \beta_{1} - 432 \beta_{2} - 560 \beta_{3} ) q^{68} + ( -2682 + 207 \beta_{1} + 117 \beta_{2} - 387 \beta_{3} ) q^{69} + ( -14044 - 980 \beta_{1} - 276 \beta_{2} - 836 \beta_{3} ) q^{70} + ( 6380 + 2164 \beta_{1} - 413 \beta_{2} + 2437 \beta_{3} ) q^{71} + 5184 q^{72} + ( -1548 + 3583 \beta_{1} - 780 \beta_{2} - 370 \beta_{3} ) q^{73} + ( -14512 - 1884 \beta_{1} - 308 \beta_{2} + 1152 \beta_{3} ) q^{74} + ( -1764 + 216 \beta_{1} + 1080 \beta_{2} - 198 \beta_{3} ) q^{75} + ( -18528 - 752 \beta_{1} - 240 \beta_{2} + 320 \beta_{3} ) q^{76} + ( 25271 + 1213 \beta_{1} + 139 \beta_{2} + 1327 \beta_{3} ) q^{77} + ( -6984 + 288 \beta_{1} - 612 \beta_{2} - 936 \beta_{3} ) q^{78} + ( 268 - 2696 \beta_{1} - 1729 \beta_{2} - 478 \beta_{3} ) q^{79} + ( -6400 + 512 \beta_{1} - 512 \beta_{2} + 768 \beta_{3} ) q^{80} + 6561 q^{81} + ( -13636 + 380 \beta_{1} - 580 \beta_{2} - 1416 \beta_{3} ) q^{82} + ( 10601 - 4551 \beta_{1} - 1601 \beta_{2} - 4274 \beta_{3} ) q^{83} + ( -6048 - 432 \beta_{1} + 576 \beta_{2} - 864 \beta_{3} ) q^{84} + ( 14481 - 1931 \beta_{1} + 4024 \beta_{2} - 2747 \beta_{3} ) q^{85} + ( 180 + 2832 \beta_{1} - 312 \beta_{2} + 1696 \beta_{3} ) q^{86} + ( -10287 + 1737 \beta_{1} + 189 \beta_{2} + 1098 \beta_{3} ) q^{87} + ( -11200 - 768 \beta_{1} + 1280 \beta_{2} + 192 \beta_{3} ) q^{88} + ( 21091 - 3187 \beta_{1} + 2229 \beta_{2} - 5399 \beta_{3} ) q^{89} + ( -8100 + 648 \beta_{1} - 648 \beta_{2} + 972 \beta_{3} ) q^{90} + ( 12964 + 1493 \beta_{1} - 2222 \beta_{2} + 3119 \beta_{3} ) q^{91} + ( -4768 + 368 \beta_{1} + 208 \beta_{2} - 688 \beta_{3} ) q^{92} + ( -16128 - 135 \beta_{1} + 1980 \beta_{2} + 189 \beta_{3} ) q^{93} + ( -11292 + 2236 \beta_{1} - 732 \beta_{2} - 2120 \beta_{3} ) q^{94} + ( 42331 - 1035 \beta_{1} + 3682 \beta_{2} - 4510 \beta_{3} ) q^{95} + 9216 q^{96} + ( -68238 - 3442 \beta_{1} + 957 \beta_{2} + 486 \beta_{3} ) q^{97} + ( -23256 + 2732 \beta_{1} - 824 \beta_{2} + 3692 \beta_{3} ) q^{98} + ( -14175 - 972 \beta_{1} + 1620 \beta_{2} + 243 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 16q^{2} + 36q^{3} + 64q^{4} - 104q^{5} + 144q^{6} - 162q^{7} + 256q^{8} + 324q^{9} + O(q^{10})$$ $$4q + 16q^{2} + 36q^{3} + 64q^{4} - 104q^{5} + 144q^{6} - 162q^{7} + 256q^{8} + 324q^{9} - 416q^{10} - 676q^{11} + 576q^{12} - 792q^{13} - 648q^{14} - 936q^{15} + 1024q^{16} - 2474q^{17} + 1296q^{18} - 4538q^{19} - 1664q^{20} - 1458q^{21} - 2704q^{22} - 1238q^{23} + 2304q^{24} - 832q^{25} - 3168q^{26} + 2916q^{27} - 2592q^{28} - 4958q^{29} - 3744q^{30} - 7138q^{31} + 4096q^{32} - 6084q^{33} - 9896q^{34} - 13554q^{35} + 5184q^{36} - 13570q^{37} - 18152q^{38} - 7128q^{39} - 6656q^{40} - 13826q^{41} - 5832q^{42} - 1236q^{43} - 10816q^{44} - 8424q^{45} - 4952q^{46} - 12410q^{47} + 9216q^{48} - 24622q^{49} - 3328q^{50} - 22266q^{51} - 12672q^{52} - 50904q^{53} + 11664q^{54} - 20872q^{55} - 10368q^{56} - 40842q^{57} - 19832q^{58} - 13924q^{59} - 14976q^{60} - 70622q^{61} - 28552q^{62} - 13122q^{63} + 16384q^{64} + 17460q^{65} - 24336q^{66} - 50012q^{67} - 39584q^{68} - 11142q^{69} - 54216q^{70} + 21192q^{71} + 20736q^{72} - 13358q^{73} - 54280q^{74} - 7488q^{75} - 72608q^{76} + 98658q^{77} - 28512q^{78} + 6464q^{79} - 26624q^{80} + 26244q^{81} - 55304q^{82} + 51506q^{83} - 23328q^{84} + 61786q^{85} - 4944q^{86} - 44622q^{87} - 43264q^{88} + 90738q^{89} - 33696q^{90} + 48870q^{91} - 19808q^{92} - 64242q^{93} - 49640q^{94} + 171394q^{95} + 36864q^{96} - 266068q^{97} - 98488q^{98} - 54756q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 9 x^{2} + 10 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$3 \nu^{3} - 3 \nu^{2} - 30 \nu + 7$$ $$\beta_{2}$$ $$=$$ $$3 \nu^{2} + 3 \nu - 18$$ $$\beta_{3}$$ $$=$$ $$-3 \nu^{3} + 6 \nu^{2} + 24 \nu - 21$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} - \beta_{1} + 4$$$$)/9$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{2} + \beta_{1} + 50$$$$)/9$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{3} + 4 \beta_{2} - 2 \beta_{1} + 23$$$$)/3$$

## 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
 3.68575 −0.0924698 1.09247 −2.68575
4.00000 9.00000 16.0000 −84.5867 36.0000 83.0564 64.0000 81.0000 −338.347
1.2 4.00000 9.00000 16.0000 −38.5011 36.0000 −5.25164 64.0000 81.0000 −154.005
1.3 4.00000 9.00000 16.0000 −28.1958 36.0000 −61.0514 64.0000 81.0000 −112.783
1.4 4.00000 9.00000 16.0000 47.2837 36.0000 −178.753 64.0000 81.0000 189.135
 $$n$$: e.g. 2-40 or 990-1000 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 $$T_{5}^{4} + 104 T_{5}^{3} - 426 T_{5}^{2} - 226264 T_{5} - 4341815$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(354))$$.