# Properties

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

# Learn more about

## 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.30645.1 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.

 $$f(q)$$ $$=$$ $$q -2 q^{2} -3 q^{3} + 4 q^{4} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{5} + 6 q^{6} + ( 2 + \beta_{2} ) q^{7} -8 q^{8} + 9 q^{9} +O(q^{10})$$ $$q -2 q^{2} -3 q^{3} + 4 q^{4} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{5} + 6 q^{6} + ( 2 + \beta_{2} ) q^{7} -8 q^{8} + 9 q^{9} + ( 4 + 2 \beta_{1} + 4 \beta_{2} ) q^{10} + ( -15 + 6 \beta_{1} + 2 \beta_{2} ) q^{11} -12 q^{12} + ( 13 - 4 \beta_{1} + 6 \beta_{2} ) q^{13} + ( -4 - 2 \beta_{2} ) q^{14} + ( 6 + 3 \beta_{1} + 6 \beta_{2} ) q^{15} + 16 q^{16} + ( 24 - \beta_{1} - \beta_{2} ) q^{17} -18 q^{18} + ( -1 + 4 \beta_{1} + 7 \beta_{2} ) q^{19} + ( -8 - 4 \beta_{1} - 8 \beta_{2} ) q^{20} + ( -6 - 3 \beta_{2} ) q^{21} + ( 30 - 12 \beta_{1} - 4 \beta_{2} ) q^{22} + ( -39 + 8 \beta_{1} - \beta_{2} ) q^{23} + 24 q^{24} + ( 121 + 21 \beta_{1} - 14 \beta_{2} ) q^{25} + ( -26 + 8 \beta_{1} - 12 \beta_{2} ) q^{26} -27 q^{27} + ( 8 + 4 \beta_{2} ) q^{28} + ( -1 - 47 \beta_{1} + \beta_{2} ) q^{29} + ( -12 - 6 \beta_{1} - 12 \beta_{2} ) q^{30} + ( 13 - 9 \beta_{1} + 13 \beta_{2} ) q^{31} -32 q^{32} + ( 45 - 18 \beta_{1} - 6 \beta_{2} ) q^{33} + ( -48 + 2 \beta_{1} + 2 \beta_{2} ) q^{34} + ( -111 - 5 \beta_{1} + 5 \beta_{2} ) q^{35} + 36 q^{36} + ( -8 - 15 \beta_{1} + \beta_{2} ) q^{37} + ( 2 - 8 \beta_{1} - 14 \beta_{2} ) q^{38} + ( -39 + 12 \beta_{1} - 18 \beta_{2} ) q^{39} + ( 16 + 8 \beta_{1} + 16 \beta_{2} ) q^{40} + ( -168 + 38 \beta_{1} - 7 \beta_{2} ) q^{41} + ( 12 + 6 \beta_{2} ) q^{42} + ( 67 + 84 \beta_{1} - 28 \beta_{2} ) q^{43} + ( -60 + 24 \beta_{1} + 8 \beta_{2} ) q^{44} + ( -18 - 9 \beta_{1} - 18 \beta_{2} ) q^{45} + ( 78 - 16 \beta_{1} + 2 \beta_{2} ) q^{46} + ( -221 - 47 \beta_{1} + 23 \beta_{2} ) q^{47} -48 q^{48} + ( -287 - \beta_{1} - \beta_{2} ) q^{49} + ( -242 - 42 \beta_{1} + 28 \beta_{2} ) q^{50} + ( -72 + 3 \beta_{1} + 3 \beta_{2} ) q^{51} + ( 52 - 16 \beta_{1} + 24 \beta_{2} ) q^{52} + ( -366 + 69 \beta_{1} + 6 \beta_{2} ) q^{53} + 54 q^{54} + ( -352 - 69 \beta_{1} + 48 \beta_{2} ) q^{55} + ( -16 - 8 \beta_{2} ) q^{56} + ( 3 - 12 \beta_{1} - 21 \beta_{2} ) q^{57} + ( 2 + 94 \beta_{1} - 2 \beta_{2} ) q^{58} + 59 q^{59} + ( 24 + 12 \beta_{1} + 24 \beta_{2} ) q^{60} + ( -81 - 77 \beta_{1} - 3 \beta_{2} ) q^{61} + ( -26 + 18 \beta_{1} - 26 \beta_{2} ) q^{62} + ( 18 + 9 \beta_{2} ) q^{63} + 64 q^{64} + ( -556 + 21 \beta_{1} + 28 \beta_{2} ) q^{65} + ( -90 + 36 \beta_{1} + 12 \beta_{2} ) q^{66} + ( -110 + 119 \beta_{1} - 14 \beta_{2} ) q^{67} + ( 96 - 4 \beta_{1} - 4 \beta_{2} ) q^{68} + ( 117 - 24 \beta_{1} + 3 \beta_{2} ) q^{69} + ( 222 + 10 \beta_{1} - 10 \beta_{2} ) q^{70} + ( -757 - 38 \beta_{1} + 24 \beta_{2} ) q^{71} -72 q^{72} + ( -127 - 131 \beta_{1} - 11 \beta_{2} ) q^{73} + ( 16 + 30 \beta_{1} - 2 \beta_{2} ) q^{74} + ( -363 - 63 \beta_{1} + 42 \beta_{2} ) q^{75} + ( -4 + 16 \beta_{1} + 28 \beta_{2} ) q^{76} + ( 92 + 40 \beta_{1} - 27 \beta_{2} ) q^{77} + ( 78 - 24 \beta_{1} + 36 \beta_{2} ) q^{78} + ( -371 - 14 \beta_{1} - 66 \beta_{2} ) q^{79} + ( -32 - 16 \beta_{1} - 32 \beta_{2} ) q^{80} + 81 q^{81} + ( 336 - 76 \beta_{1} + 14 \beta_{2} ) q^{82} + ( -754 + 114 \beta_{1} + 15 \beta_{2} ) q^{83} + ( -24 - 12 \beta_{2} ) q^{84} + ( 87 - 8 \beta_{1} - 57 \beta_{2} ) q^{85} + ( -134 - 168 \beta_{1} + 56 \beta_{2} ) q^{86} + ( 3 + 141 \beta_{1} - 3 \beta_{2} ) q^{87} + ( 120 - 48 \beta_{1} - 16 \beta_{2} ) q^{88} + ( -685 - 31 \beta_{1} - \beta_{2} ) q^{89} + ( 36 + 18 \beta_{1} + 36 \beta_{2} ) q^{90} + ( 326 - 34 \beta_{1} - \beta_{2} ) q^{91} + ( -156 + 32 \beta_{1} - 4 \beta_{2} ) q^{92} + ( -39 + 27 \beta_{1} - 39 \beta_{2} ) q^{93} + ( 442 + 94 \beta_{1} - 46 \beta_{2} ) q^{94} + ( -859 - 72 \beta_{1} + 65 \beta_{2} ) q^{95} + 96 q^{96} + ( -6 + 35 \beta_{1} - 150 \beta_{2} ) q^{97} + ( 574 + 2 \beta_{1} + 2 \beta_{2} ) q^{98} + ( -135 + 54 \beta_{1} + 18 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 6q^{2} - 9q^{3} + 12q^{4} - 6q^{5} + 18q^{6} + 6q^{7} - 24q^{8} + 27q^{9} + O(q^{10})$$ $$3q - 6q^{2} - 9q^{3} + 12q^{4} - 6q^{5} + 18q^{6} + 6q^{7} - 24q^{8} + 27q^{9} + 12q^{10} - 45q^{11} - 36q^{12} + 39q^{13} - 12q^{14} + 18q^{15} + 48q^{16} + 72q^{17} - 54q^{18} - 3q^{19} - 24q^{20} - 18q^{21} + 90q^{22} - 117q^{23} + 72q^{24} + 363q^{25} - 78q^{26} - 81q^{27} + 24q^{28} - 3q^{29} - 36q^{30} + 39q^{31} - 96q^{32} + 135q^{33} - 144q^{34} - 333q^{35} + 108q^{36} - 24q^{37} + 6q^{38} - 117q^{39} + 48q^{40} - 504q^{41} + 36q^{42} + 201q^{43} - 180q^{44} - 54q^{45} + 234q^{46} - 663q^{47} - 144q^{48} - 861q^{49} - 726q^{50} - 216q^{51} + 156q^{52} - 1098q^{53} + 162q^{54} - 1056q^{55} - 48q^{56} + 9q^{57} + 6q^{58} + 177q^{59} + 72q^{60} - 243q^{61} - 78q^{62} + 54q^{63} + 192q^{64} - 1668q^{65} - 270q^{66} - 330q^{67} + 288q^{68} + 351q^{69} + 666q^{70} - 2271q^{71} - 216q^{72} - 381q^{73} + 48q^{74} - 1089q^{75} - 12q^{76} + 276q^{77} + 234q^{78} - 1113q^{79} - 96q^{80} + 243q^{81} + 1008q^{82} - 2262q^{83} - 72q^{84} + 261q^{85} - 402q^{86} + 9q^{87} + 360q^{88} - 2055q^{89} + 108q^{90} + 978q^{91} - 468q^{92} - 117q^{93} + 1326q^{94} - 2577q^{95} + 288q^{96} - 18q^{97} + 1722q^{98} - 405q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 33 x - 28$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - \nu - 22$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2} + \beta_{1} + 22$$

## 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
 6.12929 −5.26097 −0.868324
−2.00000 −3.00000 4.00000 −17.5682 6.00000 6.71947 −8.00000 9.00000 35.1365
1.2 −2.00000 −3.00000 4.00000 −7.67779 6.00000 7.46938 −8.00000 9.00000 15.3556
1.3 −2.00000 −3.00000 4.00000 19.2460 6.00000 −8.18884 −8.00000 9.00000 −38.4920
 $$n$$: 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 $$T_{5}^{3} + 6 T_{5}^{2} - 351 T_{5} - 2596$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(354))$$.