# Properties

 Label 3969.2.a.be Level $3969$ Weight $2$ Character orbit 3969.a Self dual yes Analytic conductor $31.693$ Analytic rank $0$ Dimension $6$ 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: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.59351616.1 Defining polynomial: $$x^{6} - 12 x^{4} + 21 x^{2} - 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 441) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( 1 + \beta_{2} + \beta_{4} ) q^{4} + \beta_{1} q^{5} + ( 2 + \beta_{2} + \beta_{4} ) q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( 1 + \beta_{2} + \beta_{4} ) q^{4} + \beta_{1} q^{5} + ( 2 + \beta_{2} + \beta_{4} ) q^{8} + ( 2 \beta_{1} + \beta_{5} ) q^{10} + ( 1 - \beta_{4} ) q^{11} + \beta_{3} q^{13} + ( 2 \beta_{2} - \beta_{4} ) q^{16} + ( -\beta_{1} + \beta_{3} - \beta_{5} ) q^{17} + ( \beta_{1} - \beta_{5} ) q^{19} + ( 3 \beta_{1} + \beta_{3} + 2 \beta_{5} ) q^{20} + q^{22} + ( 1 + \beta_{2} + 2 \beta_{4} ) q^{23} + ( 1 + 3 \beta_{2} ) q^{25} + ( -\beta_{3} + \beta_{5} ) q^{26} + ( 4 + \beta_{2} + 2 \beta_{4} ) q^{29} + ( -\beta_{1} - 2 \beta_{5} ) q^{31} + ( 3 - \beta_{2} ) q^{32} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{34} + ( -2 + 2 \beta_{2} - \beta_{4} ) q^{37} + ( \beta_{1} - \beta_{3} + \beta_{5} ) q^{38} + ( 4 \beta_{1} + \beta_{3} + 2 \beta_{5} ) q^{40} + ( -\beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{41} + ( 2 - 2 \beta_{2} + \beta_{4} ) q^{43} + ( -2 + \beta_{2} + 2 \beta_{4} ) q^{44} + ( 1 + 4 \beta_{2} + \beta_{4} ) q^{46} + ( \beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{47} + ( 9 + 4 \beta_{2} + 3 \beta_{4} ) q^{50} + ( \beta_{1} - \beta_{5} ) q^{52} + ( 5 - 3 \beta_{2} - 2 \beta_{4} ) q^{53} + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{55} + ( 1 + 7 \beta_{2} + \beta_{4} ) q^{58} + ( -2 \beta_{1} - \beta_{3} + \beta_{5} ) q^{59} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{61} + ( -4 \beta_{1} - 2 \beta_{3} - \beta_{5} ) q^{62} + ( -3 - 2 \beta_{2} + \beta_{4} ) q^{64} + 3 \beta_{4} q^{65} + ( -2 + 3 \beta_{2} - 3 \beta_{4} ) q^{67} + ( -4 \beta_{1} - 3 \beta_{5} ) q^{68} + ( 6 + \beta_{2} ) q^{71} + ( -\beta_{1} + \beta_{5} ) q^{73} + ( 7 - \beta_{2} + 2 \beta_{4} ) q^{74} + ( \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{76} + ( -1 + 3 \beta_{2} + 3 \beta_{4} ) q^{79} + ( 4 \beta_{1} - \beta_{3} + \beta_{5} ) q^{80} + ( -4 \beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{82} + ( -3 \beta_{1} + \beta_{3} - \beta_{5} ) q^{83} + ( -3 - 6 \beta_{2} ) q^{85} + ( -7 + \beta_{2} - 2 \beta_{4} ) q^{86} + ( -1 + \beta_{2} + \beta_{4} ) q^{88} + ( -3 \beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{89} + ( 9 + 4 \beta_{2} ) q^{92} + ( 3 \beta_{1} - \beta_{3} + 3 \beta_{5} ) q^{94} + ( 9 - 3 \beta_{4} ) q^{95} + ( 2 \beta_{1} + \beta_{3} + \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 2q^{2} + 6q^{4} + 12q^{8} + O(q^{10})$$ $$6q + 2q^{2} + 6q^{4} + 12q^{8} + 8q^{11} + 6q^{16} + 6q^{22} + 4q^{23} + 12q^{25} + 22q^{29} + 16q^{32} - 6q^{37} + 6q^{43} - 14q^{44} + 12q^{46} + 56q^{50} + 28q^{53} + 18q^{58} - 24q^{64} - 6q^{65} + 38q^{71} + 36q^{74} - 6q^{79} - 30q^{85} - 36q^{86} - 6q^{88} + 62q^{92} + 60q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 12 x^{4} + 21 x^{2} - 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} - 12 \nu^{3} + 21 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} - 12 \nu^{2} + 15$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{5} - 21 \nu^{3} + 15 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{4} - 21 \nu^{2} + 15$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - 11 \nu^{3} + 9 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{5} + \beta_{3} + \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} - 2 \beta_{2} + 5$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{5} + 4 \beta_{3} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$12 \beta_{4} - 21 \beta_{2} + 45$$ $$\nu^{5}$$ $$=$$ $$-29 \beta_{5} + 41 \beta_{3} + 8 \beta_{1}$$

## 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
 −3.16032 3.16032 −1.15750 1.15750 −0.820103 0.820103
−1.69963 0 0.888736 −0.949271 0 0 1.88874 0 1.61341
1.2 −1.69963 0 0.888736 0.949271 0 0 1.88874 0 −1.61341
1.3 0.239123 0 −1.94282 −2.59179 0 0 −0.942820 0 −0.619757
1.4 0.239123 0 −1.94282 2.59179 0 0 −0.942820 0 0.619757
1.5 2.46050 0 4.05408 −3.65808 0 0 5.05408 0 −9.00071
1.6 2.46050 0 4.05408 3.65808 0 0 5.05408 0 9.00071
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.be 6
3.b odd 2 1 3969.2.a.bd 6
7.b odd 2 1 inner 3969.2.a.be 6
9.c even 3 2 441.2.f.g 12
9.d odd 6 2 1323.2.f.g 12
21.c even 2 1 3969.2.a.bd 6
63.g even 3 2 441.2.g.g 12
63.h even 3 2 441.2.h.g 12
63.i even 6 2 1323.2.h.g 12
63.j odd 6 2 1323.2.h.g 12
63.k odd 6 2 441.2.g.g 12
63.l odd 6 2 441.2.f.g 12
63.n odd 6 2 1323.2.g.g 12
63.o even 6 2 1323.2.f.g 12
63.s even 6 2 1323.2.g.g 12
63.t odd 6 2 441.2.h.g 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.f.g 12 9.c even 3 2
441.2.f.g 12 63.l odd 6 2
441.2.g.g 12 63.g even 3 2
441.2.g.g 12 63.k odd 6 2
441.2.h.g 12 63.h even 3 2
441.2.h.g 12 63.t odd 6 2
1323.2.f.g 12 9.d odd 6 2
1323.2.f.g 12 63.o even 6 2
1323.2.g.g 12 63.n odd 6 2
1323.2.g.g 12 63.s even 6 2
1323.2.h.g 12 63.i even 6 2
1323.2.h.g 12 63.j odd 6 2
3969.2.a.bd 6 3.b odd 2 1
3969.2.a.bd 6 21.c even 2 1
3969.2.a.be 6 1.a even 1 1 trivial
3969.2.a.be 6 7.b odd 2 1 inner

## Hecke kernels

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

 $$T_{2}^{3} - T_{2}^{2} - 4 T_{2} + 1$$ $$T_{5}^{6} - 21 T_{5}^{4} + 108 T_{5}^{2} - 81$$ $$T_{11}^{3} - 4 T_{11}^{2} - T_{11} + 1$$ $$T_{13}^{6} - 39 T_{13}^{4} + 351 T_{13}^{2} - 81$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 4 T - T^{2} + T^{3} )^{2}$$
$3$ $$T^{6}$$
$5$ $$-81 + 108 T^{2} - 21 T^{4} + T^{6}$$
$7$ $$T^{6}$$
$11$ $$( 1 - T - 4 T^{2} + T^{3} )^{2}$$
$13$ $$-81 + 351 T^{2} - 39 T^{4} + T^{6}$$
$17$ $$-3969 + 1593 T^{2} - 84 T^{4} + T^{6}$$
$19$ $$-3969 + 1296 T^{2} - 75 T^{4} + T^{6}$$
$23$ $$( 59 - 25 T - 2 T^{2} + T^{3} )^{2}$$
$29$ $$( 89 + 14 T - 11 T^{2} + T^{3} )^{2}$$
$31$ $$-77841 + 5508 T^{2} - 129 T^{4} + T^{6}$$
$37$ $$( 27 - 24 T + 3 T^{2} + T^{3} )^{2}$$
$41$ $$-6561 + 6075 T^{2} - 162 T^{4} + T^{6}$$
$43$ $$( -27 - 24 T - 3 T^{2} + T^{3} )^{2}$$
$47$ $$-194481 + 10584 T^{2} - 183 T^{4} + T^{6}$$
$53$ $$( 263 + 11 T - 14 T^{2} + T^{3} )^{2}$$
$59$ $$-149769 + 9450 T^{2} - 183 T^{4} + T^{6}$$
$61$ $$-558009 + 21519 T^{2} - 264 T^{4} + T^{6}$$
$67$ $$( 353 - 111 T + T^{3} )^{2}$$
$71$ $$( -227 + 116 T - 19 T^{2} + T^{3} )^{2}$$
$73$ $$-3969 + 1296 T^{2} - 75 T^{4} + T^{6}$$
$79$ $$( -107 - 78 T + 3 T^{2} + T^{3} )^{2}$$
$83$ $$-227529 + 13365 T^{2} - 228 T^{4} + T^{6}$$
$89$ $$-3969 + 5751 T^{2} - 246 T^{4} + T^{6}$$
$97$ $$-9801 + 1998 T^{2} - 111 T^{4} + T^{6}$$