# Properties

 Label 354.4.a.h Level 354 Weight 4 Character orbit 354.a Self dual Yes Analytic conductor 20.887 Analytic rank 0 Dimension 4 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: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + 2 q^{2} -3 q^{3} + 4 q^{4} + ( 5 + \beta_{1} - \beta_{3} ) q^{5} -6 q^{6} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + 8 q^{8} + 9 q^{9} +O(q^{10})$$ $$q + 2 q^{2} -3 q^{3} + 4 q^{4} + ( 5 + \beta_{1} - \beta_{3} ) q^{5} -6 q^{6} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + 8 q^{8} + 9 q^{9} + ( 10 + 2 \beta_{1} - 2 \beta_{3} ) q^{10} + ( 5 + 2 \beta_{1} + \beta_{3} ) q^{11} -12 q^{12} + ( 5 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{13} + ( 6 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{14} + ( -15 - 3 \beta_{1} + 3 \beta_{3} ) q^{15} + 16 q^{16} + ( 26 - 7 \beta_{1} - \beta_{2} ) q^{17} + 18 q^{18} + ( 38 - 5 \beta_{1} + \beta_{2} + 6 \beta_{3} ) q^{19} + ( 20 + 4 \beta_{1} - 4 \beta_{3} ) q^{20} + ( -9 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{21} + ( 10 + 4 \beta_{1} + 2 \beta_{3} ) q^{22} + ( 8 - 9 \beta_{1} + \beta_{2} ) q^{23} -24 q^{24} + ( 64 + 10 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{25} + ( 10 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{26} -27 q^{27} + ( 12 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{28} + ( 71 + 6 \beta_{1} + \beta_{2} + \beta_{3} ) q^{29} + ( -30 - 6 \beta_{1} + 6 \beta_{3} ) q^{30} + ( 71 + 10 \beta_{1} - 7 \beta_{2} + 3 \beta_{3} ) q^{31} + 32 q^{32} + ( -15 - 6 \beta_{1} - 3 \beta_{3} ) q^{33} + ( 52 - 14 \beta_{1} - 2 \beta_{2} ) q^{34} + ( 133 + 15 \beta_{1} + 11 \beta_{2} + 9 \beta_{3} ) q^{35} + 36 q^{36} + ( 160 - 18 \beta_{1} - 5 \beta_{2} - 10 \beta_{3} ) q^{37} + ( 76 - 10 \beta_{1} + 2 \beta_{2} + 12 \beta_{3} ) q^{38} + ( -15 + 3 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} ) q^{39} + ( 40 + 8 \beta_{1} - 8 \beta_{3} ) q^{40} + ( 167 + 7 \beta_{1} + 5 \beta_{2} + 9 \beta_{3} ) q^{41} + ( -18 - 6 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{42} + ( 47 - 6 \beta_{1} + 6 \beta_{2} - 13 \beta_{3} ) q^{43} + ( 20 + 8 \beta_{1} + 4 \beta_{3} ) q^{44} + ( 45 + 9 \beta_{1} - 9 \beta_{3} ) q^{45} + ( 16 - 18 \beta_{1} + 2 \beta_{2} ) q^{46} + ( 23 - 37 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{47} -48 q^{48} + ( 155 + 19 \beta_{1} + 7 \beta_{2} - 6 \beta_{3} ) q^{49} + ( 128 + 20 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{50} + ( -78 + 21 \beta_{1} + 3 \beta_{2} ) q^{51} + ( 20 - 4 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{52} + ( 59 - 37 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{53} -54 q^{54} + ( 11 + 36 \beta_{1} + 8 \beta_{2} - 17 \beta_{3} ) q^{55} + ( 24 + 8 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{56} + ( -114 + 15 \beta_{1} - 3 \beta_{2} - 18 \beta_{3} ) q^{57} + ( 142 + 12 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{58} + 59 q^{59} + ( -60 - 12 \beta_{1} + 12 \beta_{3} ) q^{60} + ( 165 - 4 \beta_{1} + \beta_{2} + 7 \beta_{3} ) q^{61} + ( 142 + 20 \beta_{1} - 14 \beta_{2} + 6 \beta_{3} ) q^{62} + ( 27 + 9 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} ) q^{63} + 64 q^{64} + ( 181 - 28 \beta_{1} - 32 \beta_{2} - 7 \beta_{3} ) q^{65} + ( -30 - 12 \beta_{1} - 6 \beta_{3} ) q^{66} + ( -233 + 12 \beta_{1} - 14 \beta_{2} - 5 \beta_{3} ) q^{67} + ( 104 - 28 \beta_{1} - 4 \beta_{2} ) q^{68} + ( -24 + 27 \beta_{1} - 3 \beta_{2} ) q^{69} + ( 266 + 30 \beta_{1} + 22 \beta_{2} + 18 \beta_{3} ) q^{70} + ( 51 + 35 \beta_{1} - 24 \beta_{2} + 11 \beta_{3} ) q^{71} + 72 q^{72} + ( 215 + 27 \beta_{1} + 27 \beta_{2} - 7 \beta_{3} ) q^{73} + ( 320 - 36 \beta_{1} - 10 \beta_{2} - 20 \beta_{3} ) q^{74} + ( -192 - 30 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{75} + ( 152 - 20 \beta_{1} + 4 \beta_{2} + 24 \beta_{3} ) q^{76} + ( -55 + 73 \beta_{1} + 5 \beta_{2} - 19 \beta_{3} ) q^{77} + ( -30 + 6 \beta_{1} + 12 \beta_{2} + 6 \beta_{3} ) q^{78} + ( -179 - 48 \beta_{1} - 28 \beta_{2} + 9 \beta_{3} ) q^{79} + ( 80 + 16 \beta_{1} - 16 \beta_{3} ) q^{80} + 81 q^{81} + ( 334 + 14 \beta_{1} + 10 \beta_{2} + 18 \beta_{3} ) q^{82} + ( 251 - 13 \beta_{1} + 7 \beta_{2} + 11 \beta_{3} ) q^{83} + ( -36 - 12 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} ) q^{84} + ( -174 - 65 \beta_{1} - 27 \beta_{2} - 18 \beta_{3} ) q^{85} + ( 94 - 12 \beta_{1} + 12 \beta_{2} - 26 \beta_{3} ) q^{86} + ( -213 - 18 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{87} + ( 40 + 16 \beta_{1} + 8 \beta_{3} ) q^{88} + ( 319 + 21 \beta_{1} - 3 \beta_{2} + 57 \beta_{3} ) q^{89} + ( 90 + 18 \beta_{1} - 18 \beta_{3} ) q^{90} + ( -641 - 49 \beta_{1} + 11 \beta_{2} + 43 \beta_{3} ) q^{91} + ( 32 - 36 \beta_{1} + 4 \beta_{2} ) q^{92} + ( -213 - 30 \beta_{1} + 21 \beta_{2} - 9 \beta_{3} ) q^{93} + ( 46 - 74 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} ) q^{94} + ( -790 + 27 \beta_{1} + 27 \beta_{2} - 70 \beta_{3} ) q^{95} -96 q^{96} + ( -481 - 2 \beta_{2} + 57 \beta_{3} ) q^{97} + ( 310 + 38 \beta_{1} + 14 \beta_{2} - 12 \beta_{3} ) q^{98} + ( 45 + 18 \beta_{1} + 9 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{2} - 12q^{3} + 16q^{4} + 22q^{5} - 24q^{6} + 13q^{7} + 32q^{8} + 36q^{9} + O(q^{10})$$ $$4q + 8q^{2} - 12q^{3} + 16q^{4} + 22q^{5} - 24q^{6} + 13q^{7} + 32q^{8} + 36q^{9} + 44q^{10} + 24q^{11} - 48q^{12} + 20q^{13} + 26q^{14} - 66q^{15} + 64q^{16} + 91q^{17} + 72q^{18} + 141q^{19} + 88q^{20} - 39q^{21} + 48q^{22} + 13q^{23} - 96q^{24} + 278q^{25} + 40q^{26} - 108q^{27} + 52q^{28} + 295q^{29} - 132q^{30} + 311q^{31} + 128q^{32} - 72q^{33} + 182q^{34} + 551q^{35} + 144q^{36} + 609q^{37} + 282q^{38} - 60q^{39} + 176q^{40} + 677q^{41} - 78q^{42} + 170q^{43} + 96q^{44} + 198q^{45} + 26q^{46} + 17q^{47} - 192q^{48} + 651q^{49} + 556q^{50} - 273q^{51} + 80q^{52} + 166q^{53} - 216q^{54} + 108q^{55} + 104q^{56} - 423q^{57} + 590q^{58} + 236q^{59} - 264q^{60} + 651q^{61} + 622q^{62} + 117q^{63} + 256q^{64} + 700q^{65} - 144q^{66} - 894q^{67} + 364q^{68} - 39q^{69} + 1102q^{70} + 298q^{71} + 288q^{72} + 887q^{73} + 1218q^{74} - 834q^{75} + 564q^{76} - 79q^{77} - 120q^{78} - 784q^{79} + 352q^{80} + 324q^{81} + 1354q^{82} + 971q^{83} - 156q^{84} - 799q^{85} + 340q^{86} - 885q^{87} + 192q^{88} + 1321q^{89} + 396q^{90} - 2673q^{91} + 52q^{92} - 933q^{93} + 34q^{94} - 3133q^{95} - 384q^{96} - 1922q^{97} + 1302q^{98} + 216q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 44 x^{2} + 19 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 45 \nu - 8$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 43 \nu + 14$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} + \beta_{1} + 22$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{2} + 45 \beta_{1} + 16$$$$)/2$$

## 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
 −0.122980 −6.36731 0.552190 6.93810
2.00000 −3.00000 4.00000 −14.5171 −6.00000 −18.9849 8.00000 9.00000 −29.0343
1.2 2.00000 −3.00000 4.00000 3.16153 −6.00000 21.5427 8.00000 9.00000 6.32306
1.3 2.00000 −3.00000 4.00000 15.9851 −6.00000 −18.6951 8.00000 9.00000 31.9702
1.4 2.00000 −3.00000 4.00000 17.3705 −6.00000 29.1373 8.00000 9.00000 34.7410
 $$n$$: 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 $$T_{5}^{4} - 22 T_{5}^{3} - 147 T_{5}^{2} + 4684 T_{5} - 12744$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(354))$$.