# Properties

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

# 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: 4.4.14013.1 Defining polynomial: $$x^{4} - x^{3} - 6 x^{2} + 6 x + 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 567) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{5} + ( -1 + \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{5} + ( -1 + \beta_{1} + \beta_{3} ) q^{8} + ( -1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{10} + ( 1 + \beta_{2} + 2 \beta_{3} ) q^{11} + ( -1 - \beta_{2} ) q^{13} + ( -\beta_{1} + \beta_{3} ) q^{16} + ( 1 + 2 \beta_{2} - \beta_{3} ) q^{17} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{19} + ( -1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{20} + ( -3 + 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{22} + ( -3 + \beta_{1} - \beta_{2} ) q^{23} + ( 1 + 3 \beta_{1} + \beta_{2} ) q^{25} + ( 1 - 3 \beta_{1} - \beta_{3} ) q^{26} + ( -3 + 2 \beta_{2} ) q^{29} + ( -5 + 2 \beta_{1} + \beta_{3} ) q^{31} + ( -2 - 2 \beta_{1} - \beta_{3} ) q^{32} + ( -1 + 5 \beta_{1} - \beta_{2} + \beta_{3} ) q^{34} + ( 1 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{37} + ( -5 + 2 \beta_{3} ) q^{38} + ( -4 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{40} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -2 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{43} + ( 2 + \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{44} + ( 4 - 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{46} + ( -5 - \beta_{1} - \beta_{3} ) q^{47} + ( 8 + 3 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{50} + ( -6 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{52} + ( -2 - 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{53} + ( -5 - 5 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{55} + ( -2 + \beta_{1} + 2 \beta_{3} ) q^{58} + ( 2 + \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{59} + ( -6 + 4 \beta_{1} - \beta_{3} ) q^{61} + ( 5 - 5 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{62} + ( -5 - 3 \beta_{2} - 3 \beta_{3} ) q^{64} + ( 1 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{65} + ( -2 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{67} + ( 13 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{68} + ( -3 + 3 \beta_{1} ) q^{71} + ( -1 - 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{73} + ( 2 - 5 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} ) q^{74} + ( 2 - 3 \beta_{1} ) q^{76} + ( -1 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{79} + ( -1 + 2 \beta_{2} + 4 \beta_{3} ) q^{80} + ( -7 - 4 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{82} + ( 2 - \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{83} + ( -1 - 4 \beta_{1} - \beta_{2} - 5 \beta_{3} ) q^{85} + ( -6 + 2 \beta_{1} + 3 \beta_{3} ) q^{86} + ( 4 + 4 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{88} + ( -6 + \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{89} + ( -9 + 4 \beta_{1} - 4 \beta_{2} ) q^{92} + ( -2 - 5 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{94} + ( -5 + 4 \beta_{2} + 5 \beta_{3} ) q^{95} + ( -3 + 4 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} + 5 q^{4} + 2 q^{5} - 3 q^{8} + O(q^{10})$$ $$4 q + q^{2} + 5 q^{4} + 2 q^{5} - 3 q^{8} - 7 q^{10} + 5 q^{11} - 5 q^{13} - q^{16} + 6 q^{17} - 8 q^{19} - 8 q^{20} - 7 q^{22} - 12 q^{23} + 8 q^{25} + q^{26} - 10 q^{29} - 18 q^{31} - 10 q^{32} - 20 q^{38} - 18 q^{40} - 5 q^{41} - 7 q^{43} + 13 q^{44} + 12 q^{46} - 21 q^{47} + 38 q^{50} - 25 q^{52} - 12 q^{53} - 26 q^{55} - 7 q^{58} + 6 q^{59} - 20 q^{61} + 18 q^{62} - 23 q^{64} + 8 q^{65} - 5 q^{67} + 51 q^{68} - 9 q^{71} - 6 q^{73} + 5 q^{76} - 10 q^{79} - 2 q^{80} - 35 q^{82} + 9 q^{83} - 9 q^{85} - 22 q^{86} + 18 q^{88} - 22 q^{89} - 36 q^{92} - 15 q^{94} - 16 q^{95} - 9 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 6 x^{2} + 6 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5 \beta_{1} - 1$$

## 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
 −2.37167 −0.372845 1.53652 2.20800
−2.37167 0 3.62484 1.22875 0 0 −3.85358 0 −2.91418
1.2 −0.372845 0 −1.86099 1.42143 0 0 1.43955 0 −0.529976
1.3 1.53652 0 0.360904 3.15761 0 0 −2.51851 0 4.85173
1.4 2.20800 0 2.87525 −3.80779 0 0 1.93254 0 −8.40758
 $$n$$: 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

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 3969.2.a.x 4
3.b odd 2 1 3969.2.a.s 4
7.b odd 2 1 3969.2.a.w 4
7.c even 3 2 567.2.e.c 8
21.c even 2 1 3969.2.a.t 4
21.h odd 6 2 567.2.e.d yes 8
63.g even 3 2 567.2.g.j 8
63.h even 3 2 567.2.h.k 8
63.j odd 6 2 567.2.h.j 8
63.n odd 6 2 567.2.g.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.2.e.c 8 7.c even 3 2
567.2.e.d yes 8 21.h odd 6 2
567.2.g.j 8 63.g even 3 2
567.2.g.k 8 63.n odd 6 2
567.2.h.j 8 63.j odd 6 2
567.2.h.k 8 63.h even 3 2
3969.2.a.s 4 3.b odd 2 1
3969.2.a.t 4 21.c even 2 1
3969.2.a.w 4 7.b odd 2 1
3969.2.a.x 4 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{2}^{4} - T_{2}^{3} - 6 T_{2}^{2} + 6 T_{2} + 3$$ $$T_{5}^{4} - 2 T_{5}^{3} - 12 T_{5}^{2} + 33 T_{5} - 21$$ $$T_{11}^{4} - 5 T_{11}^{3} - 24 T_{11}^{2} + 174 T_{11} - 249$$ $$T_{13}^{4} + 5 T_{13}^{3} - 20 T_{13} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + 6 T - 6 T^{2} - T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$-21 + 33 T - 12 T^{2} - 2 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$-249 + 174 T - 24 T^{2} - 5 T^{3} + T^{4}$$
$13$ $$-7 - 20 T + 5 T^{3} + T^{4}$$
$17$ $$-567 + 354 T - 45 T^{2} - 6 T^{3} + T^{4}$$
$19$ $$-49 - 47 T + 8 T^{3} + T^{4}$$
$23$ $$9 + 33 T + 36 T^{2} + 12 T^{3} + T^{4}$$
$29$ $$63 - 90 T + 10 T^{3} + T^{4}$$
$31$ $$-21 + 136 T + 93 T^{2} + 18 T^{3} + T^{4}$$
$37$ $$951 + 37 T - 78 T^{2} + T^{4}$$
$41$ $$441 - 126 T - 72 T^{2} + 5 T^{3} + T^{4}$$
$43$ $$49 - 176 T - 30 T^{2} + 7 T^{3} + T^{4}$$
$47$ $$441 + 447 T + 153 T^{2} + 21 T^{3} + T^{4}$$
$53$ $$-81 - 105 T + 12 T^{3} + T^{4}$$
$59$ $$189 + 201 T - 108 T^{2} - 6 T^{3} + T^{4}$$
$61$ $$1043 - 650 T + 27 T^{2} + 20 T^{3} + T^{4}$$
$67$ $$353 - 74 T - 72 T^{2} + 5 T^{3} + T^{4}$$
$71$ $$243 - 135 T - 27 T^{2} + 9 T^{3} + T^{4}$$
$73$ $$2289 + 73 T - 150 T^{2} + 6 T^{3} + T^{4}$$
$79$ $$7 - 5 T - 84 T^{2} + 10 T^{3} + T^{4}$$
$83$ $$-5103 + 1710 T - 126 T^{2} - 9 T^{3} + T^{4}$$
$89$ $$-21 + 51 T + 114 T^{2} + 22 T^{3} + T^{4}$$
$97$ $$2877 - 2456 T - 258 T^{2} + 9 T^{3} + T^{4}$$