# Properties

 Label 4014.2.a.r Level 4014 Weight 2 Character orbit 4014.a Self dual yes Analytic conductor 32.052 Analytic rank 1 Dimension 5 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: $$5$$ Coefficient field: 5.5.356173.1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + ( -1 + \beta_{1} - \beta_{2} ) q^{5} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} + ( -1 + \beta_{1} - \beta_{2} ) q^{5} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{7} + q^{8} + ( -1 + \beta_{1} - \beta_{2} ) q^{10} + ( -3 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{11} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{14} + q^{16} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{17} + ( -1 + \beta_{2} + \beta_{3} ) q^{19} + ( -1 + \beta_{1} - \beta_{2} ) q^{20} + ( -3 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{22} + ( -2 - \beta_{1} + 2 \beta_{4} ) q^{23} + ( \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{25} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{26} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{28} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{29} + ( -1 + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{31} + q^{32} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{34} + ( -3 - \beta_{1} + 3 \beta_{3} + \beta_{4} ) q^{35} + ( -\beta_{1} + 4 \beta_{3} - 2 \beta_{4} ) q^{37} + ( -1 + \beta_{2} + \beta_{3} ) q^{38} + ( -1 + \beta_{1} - \beta_{2} ) q^{40} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{41} + ( 4 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{43} + ( -3 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{44} + ( -2 - \beta_{1} + 2 \beta_{4} ) q^{46} + ( -4 - \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{47} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{49} + ( \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{50} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{52} + ( -5 + \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{53} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{55} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{56} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{58} + ( -5 + \beta_{1} + 4 \beta_{2} + 3 \beta_{4} ) q^{59} + ( -5 + 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{61} + ( -1 + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{62} + q^{64} + ( -1 + 3 \beta_{2} - 2 \beta_{3} ) q^{65} + ( -3 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 4 \beta_{4} ) q^{67} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{68} + ( -3 - \beta_{1} + 3 \beta_{3} + \beta_{4} ) q^{70} + ( -2 - \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{71} + ( 3 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{73} + ( -\beta_{1} + 4 \beta_{3} - 2 \beta_{4} ) q^{74} + ( -1 + \beta_{2} + \beta_{3} ) q^{76} + ( -1 - 2 \beta_{2} + 3 \beta_{3} - 5 \beta_{4} ) q^{77} + ( -3 - \beta_{1} + \beta_{2} - 4 \beta_{3} + 5 \beta_{4} ) q^{79} + ( -1 + \beta_{1} - \beta_{2} ) q^{80} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{82} + ( -3 + \beta_{2} - 6 \beta_{3} + 2 \beta_{4} ) q^{83} + ( -2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{4} ) q^{85} + ( 4 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{86} + ( -3 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{88} + ( \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{89} + ( -3 + 3 \beta_{1} - \beta_{3} + \beta_{4} ) q^{91} + ( -2 - \beta_{1} + 2 \beta_{4} ) q^{92} + ( -4 - \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{94} + ( -1 - 4 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{95} + ( -3 + 4 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{97} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q + 5q^{2} + 5q^{4} - 5q^{5} - q^{7} + 5q^{8} + O(q^{10})$$ $$5q + 5q^{2} + 5q^{4} - 5q^{5} - q^{7} + 5q^{8} - 5q^{10} - 9q^{11} - q^{14} + 5q^{16} - 6q^{17} - 4q^{19} - 5q^{20} - 9q^{22} - 16q^{23} + 8q^{25} - q^{28} - 8q^{29} - q^{31} + 5q^{32} - 6q^{34} - 22q^{35} - 2q^{37} - 4q^{38} - 5q^{40} - 4q^{41} + 3q^{43} - 9q^{44} - 16q^{46} - 18q^{47} + 2q^{49} + 8q^{50} - 26q^{53} + q^{55} - q^{56} - 8q^{58} - 21q^{59} - 20q^{61} - q^{62} + 5q^{64} + 3q^{65} - 5q^{67} - 6q^{68} - 22q^{70} - 17q^{71} + 5q^{73} - 2q^{74} - 4q^{76} - 2q^{77} - 21q^{79} - 5q^{80} - 4q^{82} - 11q^{83} - 12q^{85} + 3q^{86} - 9q^{88} + 5q^{89} - 10q^{91} - 16q^{92} - 18q^{94} - 10q^{95} - 11q^{97} + 2q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 7 x^{3} + 9 x^{2} + 14 x - 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4 \nu + 1$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{4} - 2 \nu^{3} - 3 \nu^{2} + 5 \nu - 2$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 7 \nu + 4$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{4} + \beta_{3} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$-\beta_{4} + \beta_{3} + \beta_{2} + 5 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{4} + 7 \beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 15$$

## 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
 −1.38363 2.75496 −1.85688 0.253142 2.23241
1.00000 0 1.00000 −4.35484 0 3.05676 1.00000 0 −4.35484
1.2 1.00000 0 1.00000 −1.54501 0 −1.28984 1.00000 0 −1.54501
1.3 1.00000 0 1.00000 −1.43386 0 −1.87103 1.00000 0 −1.43386
1.4 1.00000 0 1.00000 −0.686428 0 2.87549 1.00000 0 −0.686428
1.5 1.00000 0 1.00000 3.02013 0 −3.77138 1.00000 0 3.02013
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.r 5
3.b odd 2 1 1338.2.a.h 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1338.2.a.h 5 3.b odd 2 1
4014.2.a.r 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4014))$$:

 $$T_{5}^{5} + 5 T_{5}^{4} - 4 T_{5}^{3} - 41 T_{5}^{2} - 54 T_{5} - 20$$ $$T_{7}^{5} + T_{7}^{4} - 18 T_{7}^{3} - 15 T_{7}^{2} + 72 T_{7} + 80$$ $$T_{11}^{5} + 9 T_{11}^{4} + 6 T_{11}^{3} - 82 T_{11}^{2} - 38 T_{11} + 67$$