# Properties

 Label 3969.2.a.r Level $3969$ Weight $2$ Character orbit 3969.a Self dual yes Analytic conductor $31.693$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3969 = 3^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3969.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$31.6926245622$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{5})$$ Defining polynomial: $$x^{4} - 6 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{2} ) q^{2} -\beta_{2} q^{4} -\beta_{1} q^{5} + ( 1 - 2 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta_{2} ) q^{2} -\beta_{2} q^{4} -\beta_{1} q^{5} + ( 1 - 2 \beta_{2} ) q^{8} -\beta_{3} q^{10} - q^{11} + 2 \beta_{1} q^{13} + ( -3 + 3 \beta_{2} ) q^{16} + ( 2 \beta_{1} - \beta_{3} ) q^{17} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{19} + ( \beta_{1} + \beta_{3} ) q^{20} + ( 1 - \beta_{2} ) q^{22} + ( 4 + 2 \beta_{2} ) q^{23} + ( -3 + 2 \beta_{2} ) q^{25} + 2 \beta_{3} q^{26} + ( 2 - 2 \beta_{2} ) q^{29} + ( -2 \beta_{1} - \beta_{3} ) q^{31} + ( 4 + \beta_{2} ) q^{32} + ( -\beta_{1} + 3 \beta_{3} ) q^{34} + ( -1 - 6 \beta_{2} ) q^{37} + ( 2 \beta_{1} - 5 \beta_{3} ) q^{38} + ( \beta_{1} + 2 \beta_{3} ) q^{40} + ( 5 \beta_{1} - 4 \beta_{3} ) q^{41} + ( -7 + 8 \beta_{2} ) q^{43} + \beta_{2} q^{44} + ( -2 + 4 \beta_{2} ) q^{46} + ( -3 \beta_{1} + 7 \beta_{3} ) q^{47} + ( 5 - 3 \beta_{2} ) q^{50} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{52} + ( -1 - 6 \beta_{2} ) q^{53} + \beta_{1} q^{55} + ( -4 + 2 \beta_{2} ) q^{58} + ( 2 \beta_{1} + 7 \beta_{3} ) q^{59} + ( 7 \beta_{1} - 5 \beta_{3} ) q^{61} + ( -\beta_{1} - \beta_{3} ) q^{62} + ( 3 - 2 \beta_{2} ) q^{64} + ( -4 - 4 \beta_{2} ) q^{65} + ( -5 + 4 \beta_{2} ) q^{67} + ( -\beta_{1} - 2 \beta_{3} ) q^{68} + ( 1 - 4 \beta_{2} ) q^{71} + ( 2 \beta_{1} - 8 \beta_{3} ) q^{73} + ( -5 - \beta_{2} ) q^{74} + ( \beta_{1} + 3 \beta_{3} ) q^{76} + ( -11 + 2 \beta_{2} ) q^{79} -3 \beta_{3} q^{80} + ( -4 \beta_{1} + 9 \beta_{3} ) q^{82} + ( 5 \beta_{1} - 2 \beta_{3} ) q^{83} + ( -4 - 2 \beta_{2} ) q^{85} + ( 15 - 7 \beta_{2} ) q^{86} + ( -1 + 2 \beta_{2} ) q^{88} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{89} + ( -2 - 6 \beta_{2} ) q^{92} + ( 7 \beta_{1} - 10 \beta_{3} ) q^{94} + ( 6 + 2 \beta_{2} ) q^{95} + ( -\beta_{1} + 6 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 2 q^{4} + O(q^{10})$$ $$4 q - 2 q^{2} - 2 q^{4} - 4 q^{11} - 6 q^{16} + 2 q^{22} + 20 q^{23} - 8 q^{25} + 4 q^{29} + 18 q^{32} - 16 q^{37} - 12 q^{43} + 2 q^{44} + 14 q^{50} - 16 q^{53} - 12 q^{58} + 8 q^{64} - 24 q^{65} - 12 q^{67} - 4 q^{71} - 22 q^{74} - 40 q^{79} - 20 q^{85} + 46 q^{86} - 20 q^{92} + 28 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 6 x^{2} + 4$$:

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

## 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
 0.874032 −0.874032 2.28825 −2.28825
−1.61803 0 0.618034 −0.874032 0 0 2.23607 0 1.41421
1.2 −1.61803 0 0.618034 0.874032 0 0 2.23607 0 −1.41421
1.3 0.618034 0 −1.61803 −2.28825 0 0 −2.23607 0 −1.41421
1.4 0.618034 0 −1.61803 2.28825 0 0 −2.23607 0 1.41421
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3969.2.a.r 4
3.b odd 2 1 3969.2.a.y yes 4
7.b odd 2 1 inner 3969.2.a.r 4
21.c even 2 1 3969.2.a.y yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3969.2.a.r 4 1.a even 1 1 trivial
3969.2.a.r 4 7.b odd 2 1 inner
3969.2.a.y yes 4 3.b odd 2 1
3969.2.a.y yes 4 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3969))$$:

 $$T_{2}^{2} + T_{2} - 1$$ $$T_{5}^{4} - 6 T_{5}^{2} + 4$$ $$T_{11} + 1$$ $$T_{13}^{4} - 24 T_{13}^{2} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$4 - 6 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 1 + T )^{4}$$
$13$ $$64 - 24 T^{2} + T^{4}$$
$17$ $$( -10 + T^{2} )^{2}$$
$19$ $$484 - 46 T^{2} + T^{4}$$
$23$ $$( 20 - 10 T + T^{2} )^{2}$$
$29$ $$( -4 - 2 T + T^{2} )^{2}$$
$31$ $$4 - 36 T^{2} + T^{4}$$
$37$ $$( -29 + 8 T + T^{2} )^{2}$$
$41$ $$3364 - 134 T^{2} + T^{4}$$
$43$ $$( -71 + 6 T + T^{2} )^{2}$$
$47$ $$1444 - 166 T^{2} + T^{4}$$
$53$ $$( -29 + 8 T + T^{2} )^{2}$$
$59$ $$13924 - 276 T^{2} + T^{4}$$
$61$ $$13924 - 254 T^{2} + T^{4}$$
$67$ $$( -11 + 6 T + T^{2} )^{2}$$
$71$ $$( -19 + 2 T + T^{2} )^{2}$$
$73$ $$7744 - 216 T^{2} + T^{4}$$
$79$ $$( 95 + 20 T + T^{2} )^{2}$$
$83$ $$3844 - 126 T^{2} + T^{4}$$
$89$ $$( -40 + T^{2} )^{2}$$
$97$ $$3364 - 126 T^{2} + T^{4}$$