Properties

 Label 3969.2.a.v Level $3969$ Weight $2$ Character orbit 3969.a Self dual yes Analytic conductor $31.693$ Analytic rank $1$ Dimension $4$ CM discriminant -7 Inner twists $4$

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: $$\Q(\sqrt{3}, \sqrt{7})$$ Defining polynomial: $$x^{4} - 5 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

$q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} -\beta_{2} q^{4} + ( \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{3} q^{2} -\beta_{2} q^{4} + ( \beta_{1} + \beta_{3} ) q^{8} + ( -\beta_{1} - 3 \beta_{3} ) q^{11} + ( 1 + \beta_{2} ) q^{16} + ( -5 + 3 \beta_{2} ) q^{22} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{23} -5 q^{25} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{29} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{32} + ( -1 + 4 \beta_{2} ) q^{37} + ( -5 + 2 \beta_{2} ) q^{43} + ( -\beta_{1} - 8 \beta_{3} ) q^{44} + ( -6 + 2 \beta_{2} ) q^{46} -5 \beta_{3} q^{50} + ( 7 \beta_{1} + 3 \beta_{3} ) q^{53} + ( 12 - 4 \beta_{2} ) q^{58} + ( -7 + 2 \beta_{2} ) q^{64} + ( -5 - 6 \beta_{2} ) q^{67} + ( 9 \beta_{1} + 7 \beta_{3} ) q^{71} + ( -4 \beta_{1} - 13 \beta_{3} ) q^{74} + ( -1 + 6 \beta_{2} ) q^{79} + ( -2 \beta_{1} - 11 \beta_{3} ) q^{86} + ( -5 + 2 \beta_{2} ) q^{88} + ( -6 \beta_{1} - 8 \beta_{3} ) q^{92} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} + O(q^{10})$$ $$4 q + 2 q^{4} + 2 q^{16} - 26 q^{22} - 20 q^{25} - 12 q^{37} - 24 q^{43} - 28 q^{46} + 56 q^{58} - 32 q^{64} - 8 q^{67} - 16 q^{79} - 24 q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} + 1$$:

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

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
 0.456850 2.18890 −2.18890 −0.456850
−2.18890 0 2.79129 0 0 0 −1.73205 0 0
1.2 −0.456850 0 −1.79129 0 0 0 1.73205 0 0
1.3 0.456850 0 −1.79129 0 0 0 −1.73205 0 0
1.4 2.18890 0 2.79129 0 0 0 1.73205 0 0
 $$n$$: 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 CM by $$\Q(\sqrt{-7})$$
3.b odd 2 1 inner
21.c even 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.v 4
3.b odd 2 1 inner 3969.2.a.v 4
7.b odd 2 1 CM 3969.2.a.v 4
21.c even 2 1 inner 3969.2.a.v 4

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

Hecke kernels

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

 $$T_{2}^{4} - 5 T_{2}^{2} + 1$$ $$T_{5}$$ $$T_{11}^{4} - 38 T_{11}^{2} + 25$$ $$T_{13}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 5 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$25 - 38 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$( -28 + T^{2} )^{2}$$
$29$ $$( -112 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( -75 + 6 T + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( 15 + 12 T + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$2209 - 206 T^{2} + T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$( -185 + 4 T + T^{2} )^{2}$$
$71$ $$34225 - 398 T^{2} + T^{4}$$
$73$ $$T^{4}$$
$79$ $$( -173 + 8 T + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$