# Properties

 Label 354.4.a.f Level 354 Weight 4 Character orbit 354.a Self dual Yes Analytic conductor 20.887 Analytic rank 0 Dimension 3 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$354 = 2 \cdot 3 \cdot 59$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 354.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$20.886676142$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.18989.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -2 q^{2} + 3 q^{3} + 4 q^{4} + ( -9 + \beta_{1} + \beta_{2} ) q^{5} -6 q^{6} + ( 8 + \beta_{1} - 2 \beta_{2} ) q^{7} -8 q^{8} + 9 q^{9} +O(q^{10})$$ $$q -2 q^{2} + 3 q^{3} + 4 q^{4} + ( -9 + \beta_{1} + \beta_{2} ) q^{5} -6 q^{6} + ( 8 + \beta_{1} - 2 \beta_{2} ) q^{7} -8 q^{8} + 9 q^{9} + ( 18 - 2 \beta_{1} - 2 \beta_{2} ) q^{10} + ( -3 + 2 \beta_{1} - 4 \beta_{2} ) q^{11} + 12 q^{12} + ( -5 + 4 \beta_{2} ) q^{13} + ( -16 - 2 \beta_{1} + 4 \beta_{2} ) q^{14} + ( -27 + 3 \beta_{1} + 3 \beta_{2} ) q^{15} + 16 q^{16} + ( 9 + 4 \beta_{1} + 7 \beta_{2} ) q^{17} -18 q^{18} + ( 41 - \beta_{1} + 2 \beta_{2} ) q^{19} + ( -36 + 4 \beta_{1} + 4 \beta_{2} ) q^{20} + ( 24 + 3 \beta_{1} - 6 \beta_{2} ) q^{21} + ( 6 - 4 \beta_{1} + 8 \beta_{2} ) q^{22} + ( 1 - 7 \beta_{1} + 10 \beta_{2} ) q^{23} -24 q^{24} + ( 120 - 9 \beta_{1} - 11 \beta_{2} ) q^{25} + ( 10 - 8 \beta_{2} ) q^{26} + 27 q^{27} + ( 32 + 4 \beta_{1} - 8 \beta_{2} ) q^{28} + ( -46 - 12 \beta_{1} - 17 \beta_{2} ) q^{29} + ( 54 - 6 \beta_{1} - 6 \beta_{2} ) q^{30} + ( 214 + 2 \beta_{1} - 9 \beta_{2} ) q^{31} -32 q^{32} + ( -9 + 6 \beta_{1} - 12 \beta_{2} ) q^{33} + ( -18 - 8 \beta_{1} - 14 \beta_{2} ) q^{34} + ( -82 - 10 \beta_{1} + 45 \beta_{2} ) q^{35} + 36 q^{36} + ( 205 + 14 \beta_{1} - 3 \beta_{2} ) q^{37} + ( -82 + 2 \beta_{1} - 4 \beta_{2} ) q^{38} + ( -15 + 12 \beta_{2} ) q^{39} + ( 72 - 8 \beta_{1} - 8 \beta_{2} ) q^{40} + ( 98 + 9 \beta_{1} - 20 \beta_{2} ) q^{41} + ( -48 - 6 \beta_{1} + 12 \beta_{2} ) q^{42} + ( -79 - 14 \beta_{1} + 28 \beta_{2} ) q^{43} + ( -12 + 8 \beta_{1} - 16 \beta_{2} ) q^{44} + ( -81 + 9 \beta_{1} + 9 \beta_{2} ) q^{45} + ( -2 + 14 \beta_{1} - 20 \beta_{2} ) q^{46} + ( 48 + 28 \beta_{1} - 35 \beta_{2} ) q^{47} + 48 q^{48} + ( 68 - 2 \beta_{1} - 55 \beta_{2} ) q^{49} + ( -240 + 18 \beta_{1} + 22 \beta_{2} ) q^{50} + ( 27 + 12 \beta_{1} + 21 \beta_{2} ) q^{51} + ( -20 + 16 \beta_{2} ) q^{52} + ( 93 - 33 \beta_{1} + 27 \beta_{2} ) q^{53} -54 q^{54} + ( 7 - 39 \beta_{1} + 71 \beta_{2} ) q^{55} + ( -64 - 8 \beta_{1} + 16 \beta_{2} ) q^{56} + ( 123 - 3 \beta_{1} + 6 \beta_{2} ) q^{57} + ( 92 + 24 \beta_{1} + 34 \beta_{2} ) q^{58} + 59 q^{59} + ( -108 + 12 \beta_{1} + 12 \beta_{2} ) q^{60} + ( 324 + 6 \beta_{1} - 25 \beta_{2} ) q^{61} + ( -428 - 4 \beta_{1} + 18 \beta_{2} ) q^{62} + ( 72 + 9 \beta_{1} - 18 \beta_{2} ) q^{63} + 64 q^{64} + ( 277 + 19 \beta_{1} - 57 \beta_{2} ) q^{65} + ( 18 - 12 \beta_{1} + 24 \beta_{2} ) q^{66} + ( 137 + 47 \beta_{1} - 49 \beta_{2} ) q^{67} + ( 36 + 16 \beta_{1} + 28 \beta_{2} ) q^{68} + ( 3 - 21 \beta_{1} + 30 \beta_{2} ) q^{69} + ( 164 + 20 \beta_{1} - 90 \beta_{2} ) q^{70} + ( 409 - 2 \beta_{1} - 36 \beta_{2} ) q^{71} -72 q^{72} + ( 548 - 16 \beta_{1} + 101 \beta_{2} ) q^{73} + ( -410 - 28 \beta_{1} + 6 \beta_{2} ) q^{74} + ( 360 - 27 \beta_{1} - 33 \beta_{2} ) q^{75} + ( 164 - 4 \beta_{1} + 8 \beta_{2} ) q^{76} + ( 670 - 23 \beta_{1} - 72 \beta_{2} ) q^{77} + ( 30 - 24 \beta_{2} ) q^{78} + ( 607 + 8 \beta_{1} + 102 \beta_{2} ) q^{79} + ( -144 + 16 \beta_{1} + 16 \beta_{2} ) q^{80} + 81 q^{81} + ( -196 - 18 \beta_{1} + 40 \beta_{2} ) q^{82} + ( -78 + 5 \beta_{1} - 10 \beta_{2} ) q^{83} + ( 96 + 12 \beta_{1} - 24 \beta_{2} ) q^{84} + ( 749 + 27 \beta_{1} - 38 \beta_{2} ) q^{85} + ( 158 + 28 \beta_{1} - 56 \beta_{2} ) q^{86} + ( -138 - 36 \beta_{1} - 51 \beta_{2} ) q^{87} + ( 24 - 16 \beta_{1} + 32 \beta_{2} ) q^{88} + ( 452 - 16 \beta_{1} + 9 \beta_{2} ) q^{89} + ( 162 - 18 \beta_{1} - 18 \beta_{2} ) q^{90} + ( -516 + 7 \beta_{1} + 98 \beta_{2} ) q^{91} + ( 4 - 28 \beta_{1} + 40 \beta_{2} ) q^{92} + ( 642 + 6 \beta_{1} - 27 \beta_{2} ) q^{93} + ( -96 - 56 \beta_{1} + 70 \beta_{2} ) q^{94} + ( -359 + 59 \beta_{1} + 4 \beta_{2} ) q^{95} -96 q^{96} + ( 401 + 39 \beta_{1} + 3 \beta_{2} ) q^{97} + ( -136 + 4 \beta_{1} + 110 \beta_{2} ) q^{98} + ( -27 + 18 \beta_{1} - 36 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 6q^{2} + 9q^{3} + 12q^{4} - 28q^{5} - 18q^{6} + 26q^{7} - 24q^{8} + 27q^{9} + O(q^{10})$$ $$3q - 6q^{2} + 9q^{3} + 12q^{4} - 28q^{5} - 18q^{6} + 26q^{7} - 24q^{8} + 27q^{9} + 56q^{10} - 5q^{11} + 36q^{12} - 19q^{13} - 52q^{14} - 84q^{15} + 48q^{16} + 20q^{17} - 54q^{18} + 121q^{19} - 112q^{20} + 78q^{21} + 10q^{22} - 7q^{23} - 72q^{24} + 371q^{25} + 38q^{26} + 81q^{27} + 104q^{28} - 121q^{29} + 168q^{30} + 651q^{31} - 96q^{32} - 15q^{33} - 40q^{34} - 291q^{35} + 108q^{36} + 618q^{37} - 242q^{38} - 57q^{39} + 224q^{40} + 314q^{41} - 156q^{42} - 265q^{43} - 20q^{44} - 252q^{45} + 14q^{46} + 179q^{47} + 144q^{48} + 259q^{49} - 742q^{50} + 60q^{51} - 76q^{52} + 252q^{53} - 162q^{54} - 50q^{55} - 208q^{56} + 363q^{57} + 242q^{58} + 177q^{59} - 336q^{60} + 997q^{61} - 1302q^{62} + 234q^{63} + 192q^{64} + 888q^{65} + 30q^{66} + 460q^{67} + 80q^{68} - 21q^{69} + 582q^{70} + 1263q^{71} - 216q^{72} + 1543q^{73} - 1236q^{74} + 1113q^{75} + 484q^{76} + 2082q^{77} + 114q^{78} + 1719q^{79} - 448q^{80} + 243q^{81} - 628q^{82} - 224q^{83} + 312q^{84} + 2285q^{85} + 530q^{86} - 363q^{87} + 40q^{88} + 1347q^{89} + 504q^{90} - 1646q^{91} - 28q^{92} + 1953q^{93} - 358q^{94} - 1081q^{95} - 288q^{96} + 1200q^{97} - 518q^{98} - 45q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 17 x + 14$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$3 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 12$$

## 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
 0.816330 −4.05043 4.23410
−2.00000 3.00000 4.00000 −18.8846 −6.00000 32.1162 −8.00000 9.00000 37.7692
1.2 −2.00000 3.00000 4.00000 −17.7453 −6.00000 −13.9633 −8.00000 9.00000 35.4906
1.3 −2.00000 3.00000 4.00000 8.62992 −6.00000 7.84707 −8.00000 9.00000 −17.2598
 $$n$$: 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 $$T_{5}^{3} + 28 T_{5}^{2} + 19 T_{5} - 2892$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(354))$$.