# Properties

 Label 3969.2.a.u 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: $$\Q(\sqrt{3}, \sqrt{7})$$ Defining polynomial: $$x^{4} - 5 x^{2} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} -\beta_{2} q^{4} + 2 \beta_{1} q^{5} + ( \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{3} q^{2} -\beta_{2} q^{4} + 2 \beta_{1} q^{5} + ( \beta_{1} + \beta_{3} ) q^{8} -2 q^{10} + ( -\beta_{1} + \beta_{3} ) q^{11} -4 q^{13} + ( 1 + \beta_{2} ) q^{16} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{17} + 2 \beta_{2} q^{19} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{20} + ( 3 - \beta_{2} ) q^{22} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{23} + ( 7 + 4 \beta_{2} ) q^{25} -4 \beta_{3} q^{26} -4 \beta_{3} q^{29} + ( -2 - 4 \beta_{2} ) q^{31} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{32} + ( -2 + 2 \beta_{2} ) q^{34} + 3 q^{37} + ( -2 \beta_{1} - 6 \beta_{3} ) q^{38} + ( 4 + 2 \beta_{2} ) q^{40} -2 \beta_{1} q^{41} + ( -5 - 2 \beta_{2} ) q^{43} + ( 3 \beta_{1} + 4 \beta_{3} ) q^{44} + ( -2 + 2 \beta_{2} ) q^{46} + 6 \beta_{3} q^{47} + ( -4 \beta_{1} - 5 \beta_{3} ) q^{50} + 4 \beta_{2} q^{52} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{53} + ( -8 - 2 \beta_{2} ) q^{55} + ( -8 + 4 \beta_{2} ) q^{58} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{59} + ( -6 + 2 \beta_{2} ) q^{61} + ( 4 \beta_{1} + 10 \beta_{3} ) q^{62} + ( -7 + 2 \beta_{2} ) q^{64} -8 \beta_{1} q^{65} + ( 3 - 2 \beta_{2} ) q^{67} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{68} + ( 5 \beta_{1} - \beta_{3} ) q^{71} + ( -4 + 4 \beta_{2} ) q^{73} + 3 \beta_{3} q^{74} + ( -10 + 2 \beta_{2} ) q^{76} + ( -5 - 2 \beta_{2} ) q^{79} + ( 6 \beta_{1} + 2 \beta_{3} ) q^{80} + 2 q^{82} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{83} + ( -8 - 4 \beta_{2} ) q^{85} + ( 2 \beta_{1} + \beta_{3} ) q^{86} + ( -1 - 2 \beta_{2} ) q^{88} -4 \beta_{1} q^{89} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{92} + ( 12 - 6 \beta_{2} ) q^{94} + ( 8 \beta_{1} + 4 \beta_{3} ) q^{95} + ( -4 - 2 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + O(q^{10})$$ $$4q + 2q^{4} - 8q^{10} - 16q^{13} + 2q^{16} - 4q^{19} + 14q^{22} + 20q^{25} - 12q^{34} + 12q^{37} + 12q^{40} - 16q^{43} - 12q^{46} - 8q^{52} - 28q^{55} - 40q^{58} - 28q^{61} - 32q^{64} + 16q^{67} - 24q^{73} - 44q^{76} - 16q^{79} + 8q^{82} - 24q^{85} + 60q^{94} - 12q^{97} + O(q^{100})$$

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 0.913701 0 0 −1.73205 0 −2.00000
1.2 −0.456850 0 −1.79129 4.37780 0 0 1.73205 0 −2.00000
1.3 0.456850 0 −1.79129 −4.37780 0 0 −1.73205 0 −2.00000
1.4 2.18890 0 2.79129 −0.913701 0 0 1.73205 0 −2.00000
 $$n$$: 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
3.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.u 4
3.b odd 2 1 inner 3969.2.a.u 4
7.b odd 2 1 567.2.a.i 4
21.c even 2 1 567.2.a.i 4
28.d even 2 1 9072.2.a.ci 4
63.l odd 6 2 567.2.f.n 8
63.o even 6 2 567.2.f.n 8
84.h odd 2 1 9072.2.a.ci 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.2.a.i 4 7.b odd 2 1
567.2.a.i 4 21.c even 2 1
567.2.f.n 8 63.l odd 6 2
567.2.f.n 8 63.o even 6 2
3969.2.a.u 4 1.a even 1 1 trivial
3969.2.a.u 4 3.b odd 2 1 inner
9072.2.a.ci 4 28.d even 2 1
9072.2.a.ci 4 84.h odd 2 1

## Hecke kernels

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

 $$T_{2}^{4} - 5 T_{2}^{2} + 1$$ $$T_{5}^{4} - 20 T_{5}^{2} + 16$$ $$T_{11}^{2} - 7$$ $$T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 5 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$16 - 20 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -7 + T^{2} )^{2}$$
$13$ $$( 4 + T )^{4}$$
$17$ $$( -12 + T^{2} )^{2}$$
$19$ $$( -20 + 2 T + T^{2} )^{2}$$
$23$ $$( -12 + T^{2} )^{2}$$
$29$ $$256 - 80 T^{2} + T^{4}$$
$31$ $$( -84 + T^{2} )^{2}$$
$37$ $$( -3 + T )^{4}$$
$41$ $$16 - 20 T^{2} + T^{4}$$
$43$ $$( -5 + 8 T + T^{2} )^{2}$$
$47$ $$1296 - 180 T^{2} + T^{4}$$
$53$ $$( -75 + T^{2} )^{2}$$
$59$ $$( -12 + T^{2} )^{2}$$
$61$ $$( 28 + 14 T + T^{2} )^{2}$$
$67$ $$( -5 - 8 T + T^{2} )^{2}$$
$71$ $$2601 - 150 T^{2} + T^{4}$$
$73$ $$( -48 + 12 T + T^{2} )^{2}$$
$79$ $$( -5 + 8 T + T^{2} )^{2}$$
$83$ $$3600 - 132 T^{2} + T^{4}$$
$89$ $$256 - 80 T^{2} + T^{4}$$
$97$ $$( -12 + 6 T + T^{2} )^{2}$$