# 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: $$x^{4} - 6x^{2} + 4$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - 1) q^{2} - \beta_{2} q^{4} - \beta_1 q^{5} + ( - 2 \beta_{2} + 1) q^{8}+O(q^{10})$$ q + (b2 - 1) * q^2 - b2 * q^4 - b1 * q^5 + (-2*b2 + 1) * q^8 $$q + (\beta_{2} - 1) q^{2} - \beta_{2} q^{4} - \beta_1 q^{5} + ( - 2 \beta_{2} + 1) q^{8} - \beta_{3} q^{10} - q^{11} + 2 \beta_1 q^{13} + (3 \beta_{2} - 3) q^{16} + ( - \beta_{3} + 2 \beta_1) q^{17} + (2 \beta_{3} - 3 \beta_1) q^{19} + (\beta_{3} + \beta_1) q^{20} + ( - \beta_{2} + 1) q^{22} + (2 \beta_{2} + 4) q^{23} + (2 \beta_{2} - 3) q^{25} + 2 \beta_{3} q^{26} + ( - 2 \beta_{2} + 2) q^{29} + ( - \beta_{3} - 2 \beta_1) q^{31} + (\beta_{2} + 4) q^{32} + (3 \beta_{3} - \beta_1) q^{34} + ( - 6 \beta_{2} - 1) q^{37} + ( - 5 \beta_{3} + 2 \beta_1) q^{38} + (2 \beta_{3} + \beta_1) q^{40} + ( - 4 \beta_{3} + 5 \beta_1) q^{41} + (8 \beta_{2} - 7) q^{43} + \beta_{2} q^{44} + (4 \beta_{2} - 2) q^{46} + (7 \beta_{3} - 3 \beta_1) q^{47} + ( - 3 \beta_{2} + 5) q^{50} + ( - 2 \beta_{3} - 2 \beta_1) q^{52} + ( - 6 \beta_{2} - 1) q^{53} + \beta_1 q^{55} + (2 \beta_{2} - 4) q^{58} + (7 \beta_{3} + 2 \beta_1) q^{59} + ( - 5 \beta_{3} + 7 \beta_1) q^{61} + ( - \beta_{3} - \beta_1) q^{62} + ( - 2 \beta_{2} + 3) q^{64} + ( - 4 \beta_{2} - 4) q^{65} + (4 \beta_{2} - 5) q^{67} + ( - 2 \beta_{3} - \beta_1) q^{68} + ( - 4 \beta_{2} + 1) q^{71} + ( - 8 \beta_{3} + 2 \beta_1) q^{73} + ( - \beta_{2} - 5) q^{74} + (3 \beta_{3} + \beta_1) q^{76} + (2 \beta_{2} - 11) q^{79} - 3 \beta_{3} q^{80} + (9 \beta_{3} - 4 \beta_1) q^{82} + ( - 2 \beta_{3} + 5 \beta_1) q^{83} + ( - 2 \beta_{2} - 4) q^{85} + ( - 7 \beta_{2} + 15) q^{86} + (2 \beta_{2} - 1) q^{88} + (2 \beta_{3} - 4 \beta_1) q^{89} + ( - 6 \beta_{2} - 2) q^{92} + ( - 10 \beta_{3} + 7 \beta_1) q^{94} + (2 \beta_{2} + 6) q^{95} + (6 \beta_{3} - \beta_1) q^{97}+O(q^{100})$$ q + (b2 - 1) * q^2 - b2 * q^4 - b1 * q^5 + (-2*b2 + 1) * q^8 - b3 * q^10 - q^11 + 2*b1 * q^13 + (3*b2 - 3) * q^16 + (-b3 + 2*b1) * q^17 + (2*b3 - 3*b1) * q^19 + (b3 + b1) * q^20 + (-b2 + 1) * q^22 + (2*b2 + 4) * q^23 + (2*b2 - 3) * q^25 + 2*b3 * q^26 + (-2*b2 + 2) * q^29 + (-b3 - 2*b1) * q^31 + (b2 + 4) * q^32 + (3*b3 - b1) * q^34 + (-6*b2 - 1) * q^37 + (-5*b3 + 2*b1) * q^38 + (2*b3 + b1) * q^40 + (-4*b3 + 5*b1) * q^41 + (8*b2 - 7) * q^43 + b2 * q^44 + (4*b2 - 2) * q^46 + (7*b3 - 3*b1) * q^47 + (-3*b2 + 5) * q^50 + (-2*b3 - 2*b1) * q^52 + (-6*b2 - 1) * q^53 + b1 * q^55 + (2*b2 - 4) * q^58 + (7*b3 + 2*b1) * q^59 + (-5*b3 + 7*b1) * q^61 + (-b3 - b1) * q^62 + (-2*b2 + 3) * q^64 + (-4*b2 - 4) * q^65 + (4*b2 - 5) * q^67 + (-2*b3 - b1) * q^68 + (-4*b2 + 1) * q^71 + (-8*b3 + 2*b1) * q^73 + (-b2 - 5) * q^74 + (3*b3 + b1) * q^76 + (2*b2 - 11) * q^79 - 3*b3 * q^80 + (9*b3 - 4*b1) * q^82 + (-2*b3 + 5*b1) * q^83 + (-2*b2 - 4) * q^85 + (-7*b2 + 15) * q^86 + (2*b2 - 1) * q^88 + (2*b3 - 4*b1) * q^89 + (-6*b2 - 2) * q^92 + (-10*b3 + 7*b1) * q^94 + (2*b2 + 6) * q^95 + (6*b3 - b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 2 q^{4}+O(q^{10})$$ 4 * q - 2 * q^2 - 2 * q^4 $$4 q - 2 q^{2} - 2 q^{4} - 4 q^{11} - 6 q^{16} + 2 q^{22} + 20 q^{23} - 8 q^{25} + 4 q^{29} + 18 q^{32} - 16 q^{37} - 12 q^{43} + 2 q^{44} + 14 q^{50} - 16 q^{53} - 12 q^{58} + 8 q^{64} - 24 q^{65} - 12 q^{67} - 4 q^{71} - 22 q^{74} - 40 q^{79} - 20 q^{85} + 46 q^{86} - 20 q^{92} + 28 q^{95}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} - 2 ) / 2$$ (v^2 - 2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} - 4\nu ) / 2$$ (v^3 - 4*v) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2} + 2$$ 2*b2 + 2 $$\nu^{3}$$ $$=$$ $$2\beta_{3} + 4\beta_1$$ 2*b3 + 4*b1

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3969))$$:

 $$T_{2}^{2} + T_{2} - 1$$ T2^2 + T2 - 1 $$T_{5}^{4} - 6T_{5}^{2} + 4$$ T5^4 - 6*T5^2 + 4 $$T_{11} + 1$$ T11 + 1 $$T_{13}^{4} - 24T_{13}^{2} + 64$$ T13^4 - 24*T13^2 + 64

## Hecke characteristic polynomials

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