# Properties

 Label 4014.2.a.p Level 4014 Weight 2 Character orbit 4014.a Self dual Yes Analytic conductor 32.052 Analytic rank 0 Dimension 3 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4014 = 2 \cdot 3^{2} \cdot 223$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4014.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$32.0519513713$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.473.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + \beta_{2} q^{5} + ( 3 - \beta_{1} ) q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} + \beta_{2} q^{5} + ( 3 - \beta_{1} ) q^{7} + q^{8} + \beta_{2} q^{10} + ( -2 + \beta_{1} ) q^{11} + ( 4 - \beta_{2} ) q^{13} + ( 3 - \beta_{1} ) q^{14} + q^{16} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{17} + ( 2 + \beta_{1} - \beta_{2} ) q^{19} + \beta_{2} q^{20} + ( -2 + \beta_{1} ) q^{22} + ( -\beta_{1} + 2 \beta_{2} ) q^{23} + ( 1 + \beta_{1} - \beta_{2} ) q^{25} + ( 4 - \beta_{2} ) q^{26} + ( 3 - \beta_{1} ) q^{28} + ( 3 + \beta_{1} ) q^{29} + ( -2 - 3 \beta_{1} + \beta_{2} ) q^{31} + q^{32} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{34} + ( -1 - 2 \beta_{1} + 3 \beta_{2} ) q^{35} + ( -\beta_{1} + 4 \beta_{2} ) q^{37} + ( 2 + \beta_{1} - \beta_{2} ) q^{38} + \beta_{2} q^{40} + ( -3 - 5 \beta_{1} + \beta_{2} ) q^{41} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{43} + ( -2 + \beta_{1} ) q^{44} + ( -\beta_{1} + 2 \beta_{2} ) q^{46} + ( 4 - 2 \beta_{1} - \beta_{2} ) q^{47} + ( 5 - 6 \beta_{1} + \beta_{2} ) q^{49} + ( 1 + \beta_{1} - \beta_{2} ) q^{50} + ( 4 - \beta_{2} ) q^{52} + ( \beta_{1} - 3 \beta_{2} ) q^{53} + ( 1 + 2 \beta_{1} - 2 \beta_{2} ) q^{55} + ( 3 - \beta_{1} ) q^{56} + ( 3 + \beta_{1} ) q^{58} + ( 3 - 3 \beta_{2} ) q^{59} + ( -1 - 5 \beta_{1} + 2 \beta_{2} ) q^{61} + ( -2 - 3 \beta_{1} + \beta_{2} ) q^{62} + q^{64} + ( -6 - \beta_{1} + 5 \beta_{2} ) q^{65} + ( 4 + 3 \beta_{2} ) q^{67} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{68} + ( -1 - 2 \beta_{1} + 3 \beta_{2} ) q^{70} + ( 5 + 2 \beta_{1} ) q^{71} + ( -1 + 6 \beta_{1} - \beta_{2} ) q^{73} + ( -\beta_{1} + 4 \beta_{2} ) q^{74} + ( 2 + \beta_{1} - \beta_{2} ) q^{76} + ( -9 + 5 \beta_{1} - \beta_{2} ) q^{77} + ( 8 + 5 \beta_{1} - 2 \beta_{2} ) q^{79} + \beta_{2} q^{80} + ( -3 - 5 \beta_{1} + \beta_{2} ) q^{82} + ( -6 + 3 \beta_{1} + 3 \beta_{2} ) q^{83} + ( -4 + 3 \beta_{1} + 2 \beta_{2} ) q^{85} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{86} + ( -2 + \beta_{1} ) q^{88} + ( 1 + \beta_{1} - 6 \beta_{2} ) q^{89} + ( 13 - 2 \beta_{1} - 3 \beta_{2} ) q^{91} + ( -\beta_{1} + 2 \beta_{2} ) q^{92} + ( 4 - 2 \beta_{1} - \beta_{2} ) q^{94} + ( -5 + \beta_{1} + 3 \beta_{2} ) q^{95} + ( 5 \beta_{1} - \beta_{2} ) q^{97} + ( 5 - 6 \beta_{1} + \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} + 3q^{4} + q^{5} + 9q^{7} + 3q^{8} + O(q^{10})$$ $$3q + 3q^{2} + 3q^{4} + q^{5} + 9q^{7} + 3q^{8} + q^{10} - 6q^{11} + 11q^{13} + 9q^{14} + 3q^{16} + 2q^{17} + 5q^{19} + q^{20} - 6q^{22} + 2q^{23} + 2q^{25} + 11q^{26} + 9q^{28} + 9q^{29} - 5q^{31} + 3q^{32} + 2q^{34} + 4q^{37} + 5q^{38} + q^{40} - 8q^{41} + 2q^{43} - 6q^{44} + 2q^{46} + 11q^{47} + 16q^{49} + 2q^{50} + 11q^{52} - 3q^{53} + q^{55} + 9q^{56} + 9q^{58} + 6q^{59} - q^{61} - 5q^{62} + 3q^{64} - 13q^{65} + 15q^{67} + 2q^{68} + 15q^{71} - 4q^{73} + 4q^{74} + 5q^{76} - 28q^{77} + 22q^{79} + q^{80} - 8q^{82} - 15q^{83} - 10q^{85} + 2q^{86} - 6q^{88} - 3q^{89} + 36q^{91} + 2q^{92} + 11q^{94} - 12q^{95} - q^{97} + 16q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 5 x - 1$$:

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

## 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
 −0.201640 −2.12842 2.33006
1.00000 0 1.00000 −2.95934 0 3.20164 1.00000 0 −2.95934
1.2 1.00000 0 1.00000 1.53017 0 5.12842 1.00000 0 1.53017
1.3 1.00000 0 1.00000 2.42917 0 0.669941 1.00000 0 2.42917
 $$n$$: 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$223$$ $$-1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4014))$$:

 $$T_{5}^{3} - T_{5}^{2} - 8 T_{5} + 11$$ $$T_{7}^{3} - 9 T_{7}^{2} + 22 T_{7} - 11$$ $$T_{11}^{3} + 6 T_{11}^{2} + 7 T_{11} - 3$$