# Properties

 Label 354.2.a.h Level 354 Weight 2 Character orbit 354.a Self dual Yes Analytic conductor 2.827 Analytic rank 0 Dimension 3 CM No Inner twists 1

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: Yes Analytic conductor: $$2.82670423155$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + q^{6} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{7} + q^{8} + q^{9} +O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + q^{6} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{7} + q^{8} + q^{9} + ( 1 - \beta_{1} + \beta_{2} ) q^{10} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{11} + q^{12} + \beta_{1} q^{13} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{14} + ( 1 - \beta_{1} + \beta_{2} ) q^{15} + q^{16} + ( -1 + \beta_{2} ) q^{17} + q^{18} -2 \beta_{2} q^{19} + ( 1 - \beta_{1} + \beta_{2} ) q^{20} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{21} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{22} + ( -4 + 2 \beta_{1} ) q^{23} + q^{24} + ( 1 - 2 \beta_{1} ) q^{25} + \beta_{1} q^{26} + q^{27} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{28} + ( -3 + \beta_{1} + \beta_{2} ) q^{29} + ( 1 - \beta_{1} + \beta_{2} ) q^{30} + ( -1 + \beta_{1} + \beta_{2} ) q^{31} + q^{32} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{33} + ( -1 + \beta_{2} ) q^{34} + ( -8 + 4 \beta_{1} ) q^{35} + q^{36} + ( 4 - 3 \beta_{1} + 4 \beta_{2} ) q^{37} -2 \beta_{2} q^{38} + \beta_{1} q^{39} + ( 1 - \beta_{1} + \beta_{2} ) q^{40} + ( 1 - 4 \beta_{1} - \beta_{2} ) q^{41} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{42} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{43} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{44} + ( 1 - \beta_{1} + \beta_{2} ) q^{45} + ( -4 + 2 \beta_{1} ) q^{46} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{47} + q^{48} + ( 6 - 6 \beta_{1} + \beta_{2} ) q^{49} + ( 1 - 2 \beta_{1} ) q^{50} + ( -1 + \beta_{2} ) q^{51} + \beta_{1} q^{52} + ( -7 + 3 \beta_{1} + \beta_{2} ) q^{53} + q^{54} + ( 2 - 6 \beta_{1} + 2 \beta_{2} ) q^{55} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{56} -2 \beta_{2} q^{57} + ( -3 + \beta_{1} + \beta_{2} ) q^{58} + q^{59} + ( 1 - \beta_{1} + \beta_{2} ) q^{60} + ( -1 + 3 \beta_{1} - 5 \beta_{2} ) q^{61} + ( -1 + \beta_{1} + \beta_{2} ) q^{62} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{63} + q^{64} + ( -2 + 2 \beta_{1} ) q^{65} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{66} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{67} + ( -1 + \beta_{2} ) q^{68} + ( -4 + 2 \beta_{1} ) q^{69} + ( -8 + 4 \beta_{1} ) q^{70} + ( -2 - 3 \beta_{1} ) q^{71} + q^{72} + ( 8 \beta_{1} - 2 \beta_{2} ) q^{73} + ( 4 - 3 \beta_{1} + 4 \beta_{2} ) q^{74} + ( 1 - 2 \beta_{1} ) q^{75} -2 \beta_{2} q^{76} + ( -9 + 6 \beta_{1} - 5 \beta_{2} ) q^{77} + \beta_{1} q^{78} + ( -7 + 2 \beta_{1} - 3 \beta_{2} ) q^{79} + ( 1 - \beta_{1} + \beta_{2} ) q^{80} + q^{81} + ( 1 - 4 \beta_{1} - \beta_{2} ) q^{82} + ( -3 + 2 \beta_{1} - 3 \beta_{2} ) q^{83} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{84} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{85} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{86} + ( -3 + \beta_{1} + \beta_{2} ) q^{87} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{88} + ( 2 - 6 \beta_{1} + 6 \beta_{2} ) q^{89} + ( 1 - \beta_{1} + \beta_{2} ) q^{90} + ( 5 - 2 \beta_{1} + \beta_{2} ) q^{91} + ( -4 + 2 \beta_{1} ) q^{92} + ( -1 + \beta_{1} + \beta_{2} ) q^{93} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{94} + ( -6 - 2 \beta_{1} + 2 \beta_{2} ) q^{95} + q^{96} + ( 10 - 2 \beta_{1} ) q^{97} + ( 6 - 6 \beta_{1} + \beta_{2} ) q^{98} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} + 3q^{3} + 3q^{4} + 2q^{5} + 3q^{6} - q^{7} + 3q^{8} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{2} + 3q^{3} + 3q^{4} + 2q^{5} + 3q^{6} - q^{7} + 3q^{8} + 3q^{9} + 2q^{10} + q^{11} + 3q^{12} + q^{13} - q^{14} + 2q^{15} + 3q^{16} - 3q^{17} + 3q^{18} + 2q^{20} - q^{21} + q^{22} - 10q^{23} + 3q^{24} + q^{25} + q^{26} + 3q^{27} - q^{28} - 8q^{29} + 2q^{30} - 2q^{31} + 3q^{32} + q^{33} - 3q^{34} - 20q^{35} + 3q^{36} + 9q^{37} + q^{39} + 2q^{40} - q^{41} - q^{42} - 7q^{43} + q^{44} + 2q^{45} - 10q^{46} - 8q^{47} + 3q^{48} + 12q^{49} + q^{50} - 3q^{51} + q^{52} - 18q^{53} + 3q^{54} - q^{56} - 8q^{58} + 3q^{59} + 2q^{60} - 2q^{62} - q^{63} + 3q^{64} - 4q^{65} + q^{66} + 8q^{67} - 3q^{68} - 10q^{69} - 20q^{70} - 9q^{71} + 3q^{72} + 8q^{73} + 9q^{74} + q^{75} - 21q^{77} + q^{78} - 19q^{79} + 2q^{80} + 3q^{81} - q^{82} - 7q^{83} - q^{84} + 8q^{85} - 7q^{86} - 8q^{87} + q^{88} + 2q^{90} + 13q^{91} - 10q^{92} - 2q^{93} - 8q^{94} - 20q^{95} + 3q^{96} + 28q^{97} + 12q^{98} + q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$

## 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
 0.470683 2.34292 −1.81361
1.00000 1.00000 1.00000 −2.24914 1.00000 2.71982 1.00000 1.00000 −2.24914
1.2 1.00000 1.00000 1.00000 1.14637 1.00000 1.19656 1.00000 1.00000 1.14637
1.3 1.00000 1.00000 1.00000 3.10278 1.00000 −4.91638 1.00000 1.00000 3.10278
 $$n$$: 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 intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(354))$$:

 $$T_{5}^{3} - 2 T_{5}^{2} - 6 T_{5} + 8$$ $$T_{7}^{3} + T_{7}^{2} - 16 T_{7} + 16$$ $$T_{11}^{3} - T_{11}^{2} - 32 T_{11} + 76$$