# Properties

 Label 354.4.a.e 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: 3.3.45581.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q -2 q^{2} -3 q^{3} + 4 q^{4} + ( 2 - \beta_{1} + \beta_{2} ) q^{5} + 6 q^{6} + ( 6 - 3 \beta_{1} + 2 \beta_{2} ) q^{7} -8 q^{8} + 9 q^{9} +O(q^{10})$$ $$q -2 q^{2} -3 q^{3} + 4 q^{4} + ( 2 - \beta_{1} + \beta_{2} ) q^{5} + 6 q^{6} + ( 6 - 3 \beta_{1} + 2 \beta_{2} ) q^{7} -8 q^{8} + 9 q^{9} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{10} + ( 20 - 3 \beta_{1} + 3 \beta_{2} ) q^{11} -12 q^{12} + ( -14 - 7 \beta_{1} + 3 \beta_{2} ) q^{13} + ( -12 + 6 \beta_{1} - 4 \beta_{2} ) q^{14} + ( -6 + 3 \beta_{1} - 3 \beta_{2} ) q^{15} + 16 q^{16} + ( 32 - 4 \beta_{1} + 3 \beta_{2} ) q^{17} -18 q^{18} + ( -58 + 18 \beta_{1} + 7 \beta_{2} ) q^{19} + ( 8 - 4 \beta_{1} + 4 \beta_{2} ) q^{20} + ( -18 + 9 \beta_{1} - 6 \beta_{2} ) q^{21} + ( -40 + 6 \beta_{1} - 6 \beta_{2} ) q^{22} + ( -6 - 22 \beta_{1} - 9 \beta_{2} ) q^{23} + 24 q^{24} + ( -1 - 13 \beta_{1} - 7 \beta_{2} ) q^{25} + ( 28 + 14 \beta_{1} - 6 \beta_{2} ) q^{26} -27 q^{27} + ( 24 - 12 \beta_{1} + 8 \beta_{2} ) q^{28} + ( -52 + 37 \beta_{1} + 8 \beta_{2} ) q^{29} + ( 12 - 6 \beta_{1} + 6 \beta_{2} ) q^{30} + ( -22 + 7 \beta_{1} - 4 \beta_{2} ) q^{31} -32 q^{32} + ( -60 + 9 \beta_{1} - 9 \beta_{2} ) q^{33} + ( -64 + 8 \beta_{1} - 6 \beta_{2} ) q^{34} + ( 290 - 37 \beta_{1} - 14 \beta_{2} ) q^{35} + 36 q^{36} + ( -124 - 22 \beta_{1} + 7 \beta_{2} ) q^{37} + ( 116 - 36 \beta_{1} - 14 \beta_{2} ) q^{38} + ( 42 + 21 \beta_{1} - 9 \beta_{2} ) q^{39} + ( -16 + 8 \beta_{1} - 8 \beta_{2} ) q^{40} + ( 72 + 59 \beta_{1} + 20 \beta_{2} ) q^{41} + ( 36 - 18 \beta_{1} + 12 \beta_{2} ) q^{42} + ( -136 + 39 \beta_{1} + 31 \beta_{2} ) q^{43} + ( 80 - 12 \beta_{1} + 12 \beta_{2} ) q^{44} + ( 18 - 9 \beta_{1} + 9 \beta_{2} ) q^{45} + ( 12 + 44 \beta_{1} + 18 \beta_{2} ) q^{46} + ( 418 - 23 \beta_{1} - 16 \beta_{2} ) q^{47} -48 q^{48} + ( 351 - 102 \beta_{1} - 27 \beta_{2} ) q^{49} + ( 2 + 26 \beta_{1} + 14 \beta_{2} ) q^{50} + ( -96 + 12 \beta_{1} - 9 \beta_{2} ) q^{51} + ( -56 - 28 \beta_{1} + 12 \beta_{2} ) q^{52} + ( 154 + 65 \beta_{1} + 31 \beta_{2} ) q^{53} + 54 q^{54} + ( 400 - 53 \beta_{1} - 7 \beta_{2} ) q^{55} + ( -48 + 24 \beta_{1} - 16 \beta_{2} ) q^{56} + ( 174 - 54 \beta_{1} - 21 \beta_{2} ) q^{57} + ( 104 - 74 \beta_{1} - 16 \beta_{2} ) q^{58} -59 q^{59} + ( -24 + 12 \beta_{1} - 12 \beta_{2} ) q^{60} + ( 344 + 23 \beta_{1} - 8 \beta_{2} ) q^{61} + ( 44 - 14 \beta_{1} + 8 \beta_{2} ) q^{62} + ( 54 - 27 \beta_{1} + 18 \beta_{2} ) q^{63} + 64 q^{64} + ( 484 - 55 \beta_{1} - 49 \beta_{2} ) q^{65} + ( 120 - 18 \beta_{1} + 18 \beta_{2} ) q^{66} + ( -40 + 59 \beta_{1} + 11 \beta_{2} ) q^{67} + ( 128 - 16 \beta_{1} + 12 \beta_{2} ) q^{68} + ( 18 + 66 \beta_{1} + 27 \beta_{2} ) q^{69} + ( -580 + 74 \beta_{1} + 28 \beta_{2} ) q^{70} + ( 764 - \beta_{1} + 57 \beta_{2} ) q^{71} -72 q^{72} + ( -224 + 103 \beta_{1} + 8 \beta_{2} ) q^{73} + ( 248 + 44 \beta_{1} - 14 \beta_{2} ) q^{74} + ( 3 + 39 \beta_{1} + 21 \beta_{2} ) q^{75} + ( -232 + 72 \beta_{1} + 28 \beta_{2} ) q^{76} + ( 954 - 153 \beta_{1} - 14 \beta_{2} ) q^{77} + ( -84 - 42 \beta_{1} + 18 \beta_{2} ) q^{78} + ( 324 + 171 \beta_{1} - 21 \beta_{2} ) q^{79} + ( 32 - 16 \beta_{1} + 16 \beta_{2} ) q^{80} + 81 q^{81} + ( -144 - 118 \beta_{1} - 40 \beta_{2} ) q^{82} + ( -30 + 117 \beta_{1} - 58 \beta_{2} ) q^{83} + ( -72 + 36 \beta_{1} - 24 \beta_{2} ) q^{84} + ( 462 - 74 \beta_{1} + 3 \beta_{2} ) q^{85} + ( 272 - 78 \beta_{1} - 62 \beta_{2} ) q^{86} + ( 156 - 111 \beta_{1} - 24 \beta_{2} ) q^{87} + ( -160 + 24 \beta_{1} - 24 \beta_{2} ) q^{88} + ( -148 - 41 \beta_{1} - 18 \beta_{2} ) q^{89} + ( -36 + 18 \beta_{1} - 18 \beta_{2} ) q^{90} + ( 1158 - 139 \beta_{1} - 94 \beta_{2} ) q^{91} + ( -24 - 88 \beta_{1} - 36 \beta_{2} ) q^{92} + ( 66 - 21 \beta_{1} + 12 \beta_{2} ) q^{93} + ( -836 + 46 \beta_{1} + 32 \beta_{2} ) q^{94} + ( -226 + 206 \beta_{1} - 71 \beta_{2} ) q^{95} + 96 q^{96} + ( -1178 + 29 \beta_{1} - 7 \beta_{2} ) q^{97} + ( -702 + 204 \beta_{1} + 54 \beta_{2} ) q^{98} + ( 180 - 27 \beta_{1} + 27 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 6q^{2} - 9q^{3} + 12q^{4} + 4q^{5} + 18q^{6} + 13q^{7} - 24q^{8} + 27q^{9} + O(q^{10})$$ $$3q - 6q^{2} - 9q^{3} + 12q^{4} + 4q^{5} + 18q^{6} + 13q^{7} - 24q^{8} + 27q^{9} - 8q^{10} + 54q^{11} - 36q^{12} - 52q^{13} - 26q^{14} - 12q^{15} + 48q^{16} + 89q^{17} - 54q^{18} - 163q^{19} + 16q^{20} - 39q^{21} - 108q^{22} - 31q^{23} + 72q^{24} - 9q^{25} + 104q^{26} - 81q^{27} + 52q^{28} - 127q^{29} + 24q^{30} - 55q^{31} - 96q^{32} - 162q^{33} - 178q^{34} + 847q^{35} + 108q^{36} - 401q^{37} + 326q^{38} + 156q^{39} - 32q^{40} + 255q^{41} + 78q^{42} - 400q^{43} + 216q^{44} + 36q^{45} + 62q^{46} + 1247q^{47} - 144q^{48} + 978q^{49} + 18q^{50} - 267q^{51} - 208q^{52} + 496q^{53} + 162q^{54} + 1154q^{55} - 104q^{56} + 489q^{57} + 254q^{58} - 177q^{59} - 48q^{60} + 1063q^{61} + 110q^{62} + 117q^{63} + 192q^{64} + 1446q^{65} + 324q^{66} - 72q^{67} + 356q^{68} + 93q^{69} - 1694q^{70} + 2234q^{71} - 216q^{72} - 577q^{73} + 802q^{74} + 27q^{75} - 652q^{76} + 2723q^{77} - 312q^{78} + 1164q^{79} + 64q^{80} + 243q^{81} - 510q^{82} + 85q^{83} - 156q^{84} + 1309q^{85} + 800q^{86} + 381q^{87} - 432q^{88} - 467q^{89} - 72q^{90} + 3429q^{91} - 124q^{92} + 165q^{93} - 2494q^{94} - 401q^{95} + 288q^{96} - 3498q^{97} - 1956q^{98} + 486q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 37 x + 90$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2 \nu - 26$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2 \beta_{1} + 26$$

## 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
 2.82712 4.80213 −6.62925
−2.00000 −3.00000 4.00000 −13.1803 6.00000 −27.1877 −8.00000 9.00000 26.3605
1.2 −2.00000 −3.00000 4.00000 3.86257 6.00000 4.92301 −8.00000 9.00000 −7.72514
1.3 −2.00000 −3.00000 4.00000 13.3177 6.00000 35.2646 −8.00000 9.00000 −26.6354
 $$n$$: 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 $$T_{5}^{3} - 4 T_{5}^{2} - 175 T_{5} + 678$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(354))$$.