# Properties

 Label 4014.2.a.q Level 4014 Weight 2 Character orbit 4014.a Self dual yes Analytic conductor 32.052 Analytic rank 1 Dimension 4 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.10273.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1338) 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + ( -1 + \beta_{1} ) q^{5} + ( -1 - \beta_{3} ) q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} + ( -1 + \beta_{1} ) q^{5} + ( -1 - \beta_{3} ) q^{7} + q^{8} + ( -1 + \beta_{1} ) q^{10} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{11} + ( -2 - \beta_{2} + 2 \beta_{3} ) q^{13} + ( -1 - \beta_{3} ) q^{14} + q^{16} + ( 2 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{17} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{19} + ( -1 + \beta_{1} ) q^{20} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{22} + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{23} + ( -1 + \beta_{2} - \beta_{3} ) q^{25} + ( -2 - \beta_{2} + 2 \beta_{3} ) q^{26} + ( -1 - \beta_{3} ) q^{28} + ( -3 - 2 \beta_{2} + \beta_{3} ) q^{29} + ( -1 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{31} + q^{32} + ( 2 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{34} + ( -\beta_{1} + 2 \beta_{3} ) q^{35} + ( -4 + 2 \beta_{1} + \beta_{3} ) q^{37} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{38} + ( -1 + \beta_{1} ) q^{40} + ( -3 + 3 \beta_{2} + \beta_{3} ) q^{41} + ( -5 + \beta_{2} - \beta_{3} ) q^{43} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{44} + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{46} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{47} + ( -1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{49} + ( -1 + \beta_{2} - \beta_{3} ) q^{50} + ( -2 - \beta_{2} + 2 \beta_{3} ) q^{52} + ( -3 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{53} + ( -2 - 3 \beta_{1} - \beta_{2} ) q^{55} + ( -1 - \beta_{3} ) q^{56} + ( -3 - 2 \beta_{2} + \beta_{3} ) q^{58} + ( 2 + \beta_{1} - 4 \beta_{3} ) q^{59} + ( -1 - 2 \beta_{1} - \beta_{3} ) q^{61} + ( -1 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{62} + q^{64} + ( 3 - 3 \beta_{1} - 3 \beta_{3} ) q^{65} + ( -5 - \beta_{1} ) q^{67} + ( 2 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{68} + ( -\beta_{1} + 2 \beta_{3} ) q^{70} + ( -3 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{71} + ( -1 - 3 \beta_{2} - 2 \beta_{3} ) q^{73} + ( -4 + 2 \beta_{1} + \beta_{3} ) q^{74} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{76} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{77} + ( -10 + 4 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{79} + ( -1 + \beta_{1} ) q^{80} + ( -3 + 3 \beta_{2} + \beta_{3} ) q^{82} + ( 2 - \beta_{2} + 3 \beta_{3} ) q^{83} + ( -5 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{85} + ( -5 + \beta_{2} - \beta_{3} ) q^{86} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{88} + ( -5 + 4 \beta_{1} + 3 \beta_{3} ) q^{89} + ( -6 + 3 \beta_{1} ) q^{91} + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{92} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{94} + ( 1 - \beta_{2} - 3 \beta_{3} ) q^{95} + ( -9 + 3 \beta_{1} + \beta_{3} ) q^{97} + ( -1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} + 4q^{4} - 2q^{5} - 5q^{7} + 4q^{8} + O(q^{10})$$ $$4q + 4q^{2} + 4q^{4} - 2q^{5} - 5q^{7} + 4q^{8} - 2q^{10} - 4q^{11} - 5q^{13} - 5q^{14} + 4q^{16} + 2q^{17} - 7q^{19} - 2q^{20} - 4q^{22} + 4q^{23} - 6q^{25} - 5q^{26} - 5q^{28} - 9q^{29} - q^{31} + 4q^{32} + 2q^{34} - 11q^{37} - 7q^{38} - 2q^{40} - 14q^{41} - 22q^{43} - 4q^{44} + 4q^{46} + 8q^{47} - 7q^{49} - 6q^{50} - 5q^{52} - 3q^{53} - 13q^{55} - 5q^{56} - 9q^{58} + 6q^{59} - 9q^{61} - q^{62} + 4q^{64} + 3q^{65} - 22q^{67} + 2q^{68} - 5q^{71} - 3q^{73} - 11q^{74} - 7q^{76} - 2q^{77} - 29q^{79} - 2q^{80} - 14q^{82} + 12q^{83} - 10q^{85} - 22q^{86} - 4q^{88} - 9q^{89} - 18q^{91} + 4q^{92} + 8q^{94} + 2q^{95} - 29q^{97} - 7q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 5 x^{2} + x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 4 \nu$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3 \nu^{2} - 2 \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} + 2 \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{3} + 3 \beta_{2} + 8 \beta_{1} + 6$$

## 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
 −1.38266 −0.641043 0.673533 3.35017
1.00000 0 1.00000 −2.38266 0 1.61326 1.00000 0 −2.38266
1.2 1.00000 0 1.00000 −1.64104 0 −3.78585 1.00000 0 −1.64104
1.3 1.00000 0 1.00000 −0.326467 0 −1.59754 1.00000 0 −0.326467
1.4 1.00000 0 1.00000 2.35017 0 −1.22988 1.00000 0 2.35017
 $$n$$: 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.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4014.2.a.q 4
3.b odd 2 1 1338.2.a.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1338.2.a.g 4 3.b odd 2 1
4014.2.a.q 4 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$223$$ $$1$$

## Hecke kernels

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

 $$T_{5}^{4} + 2 T_{5}^{3} - 5 T_{5}^{2} - 11 T_{5} - 3$$ $$T_{7}^{4} + 5 T_{7}^{3} + 2 T_{7}^{2} - 13 T_{7} - 12$$ $$T_{11}^{4} + 4 T_{11}^{3} - 13 T_{11}^{2} - 9 T_{11} + 16$$