# Properties

 Label 3969.2.a.m Level 3969 Weight 2 Character orbit 3969.a Self dual yes Analytic conductor 31.693 Analytic rank 0 Dimension 3 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.321.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} + ( 2 + \beta_{2} ) q^{5} + ( -2 - \beta_{1} - \beta_{2} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} + ( 2 + \beta_{2} ) q^{5} + ( -2 - \beta_{1} - \beta_{2} ) q^{8} + ( 1 - 3 \beta_{1} ) q^{10} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{11} - q^{13} + ( 2 \beta_{1} - \beta_{2} ) q^{16} + ( 4 - \beta_{1} - \beta_{2} ) q^{17} + ( 1 + \beta_{1} + \beta_{2} ) q^{19} + ( 5 + 2 \beta_{1} + \beta_{2} ) q^{20} + ( -5 - 2 \beta_{1} - 2 \beta_{2} ) q^{22} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{23} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{25} + \beta_{1} q^{26} + \beta_{1} q^{29} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{31} + ( -3 + \beta_{1} ) q^{32} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{34} + 3 \beta_{2} q^{37} + ( -2 - 3 \beta_{1} - \beta_{2} ) q^{38} + ( -7 - 2 \beta_{1} - 2 \beta_{2} ) q^{40} + ( 7 - \beta_{2} ) q^{41} + ( -4 \beta_{1} - \beta_{2} ) q^{43} + ( 6 + 5 \beta_{1} ) q^{44} + ( 5 - 2 \beta_{1} + \beta_{2} ) q^{46} + ( 3 + 3 \beta_{1} + 3 \beta_{2} ) q^{47} + ( 5 - 4 \beta_{1} + \beta_{2} ) q^{50} + ( -1 - \beta_{1} - \beta_{2} ) q^{52} + ( -5 - \beta_{1} + 2 \beta_{2} ) q^{53} + ( 5 \beta_{1} - \beta_{2} ) q^{55} + ( -3 - \beta_{1} - \beta_{2} ) q^{58} + ( 4 - \beta_{1} + 2 \beta_{2} ) q^{59} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{61} + ( 7 - \beta_{1} + 2 \beta_{2} ) q^{62} + ( -3 - 2 \beta_{1} + \beta_{2} ) q^{64} + ( -2 - \beta_{2} ) q^{65} + ( 2 - 7 \beta_{1} - \beta_{2} ) q^{67} + ( -1 + \beta_{1} + 4 \beta_{2} ) q^{68} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{71} + ( 1 - \beta_{1} + 5 \beta_{2} ) q^{73} + ( 3 - 3 \beta_{1} ) q^{74} + ( 6 + 4 \beta_{1} + \beta_{2} ) q^{76} + ( 3 + 5 \beta_{1} - \beta_{2} ) q^{79} + ( -6 + 7 \beta_{1} ) q^{80} + ( -1 - 6 \beta_{1} ) q^{82} + ( 4 - \beta_{1} - \beta_{2} ) q^{83} + ( 5 - 2 \beta_{1} + 4 \beta_{2} ) q^{85} + ( 11 + 5 \beta_{1} + 4 \beta_{2} ) q^{86} + ( -5 - 7 \beta_{1} - \beta_{2} ) q^{88} + ( -1 + 6 \beta_{1} + \beta_{2} ) q^{89} + ( 5 - 2 \beta_{1} - 2 \beta_{2} ) q^{92} + ( -6 - 9 \beta_{1} - 3 \beta_{2} ) q^{94} + ( 5 + 2 \beta_{1} + \beta_{2} ) q^{95} + ( -2 + 5 \beta_{1} + 2 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - q^{2} + 3q^{4} + 5q^{5} - 6q^{8} + O(q^{10})$$ $$3q - q^{2} + 3q^{4} + 5q^{5} - 6q^{8} - 2q^{11} - 3q^{13} + 3q^{16} + 12q^{17} + 3q^{19} + 16q^{20} - 15q^{22} + 6q^{25} + q^{26} + q^{29} + 3q^{31} - 8q^{32} + 3q^{34} - 3q^{37} - 8q^{38} - 21q^{40} + 22q^{41} - 3q^{43} + 23q^{44} + 12q^{46} + 9q^{47} + 10q^{50} - 3q^{52} - 18q^{53} + 6q^{55} - 9q^{58} + 9q^{59} + 6q^{61} + 18q^{62} - 12q^{64} - 5q^{65} - 6q^{68} + 9q^{71} - 3q^{73} + 6q^{74} + 21q^{76} + 15q^{79} - 11q^{80} - 9q^{82} + 12q^{83} + 9q^{85} + 34q^{86} - 21q^{88} + 2q^{89} + 15q^{92} - 24q^{94} + 16q^{95} - 3q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 1$$:

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

## 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
 2.46050 0.239123 −1.69963
−2.46050 0 4.05408 2.59358 0 0 −5.05408 0 −6.38151
1.2 −0.239123 0 −1.94282 −1.18194 0 0 0.942820 0 0.282630
1.3 1.69963 0 0.888736 3.58836 0 0 −1.88874 0 6.09888
 $$n$$: 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.m 3
3.b odd 2 1 3969.2.a.p 3
7.b odd 2 1 567.2.a.d 3
9.c even 3 2 441.2.f.d 6
9.d odd 6 2 1323.2.f.c 6
21.c even 2 1 567.2.a.g 3
28.d even 2 1 9072.2.a.bq 3
63.g even 3 2 441.2.g.d 6
63.h even 3 2 441.2.h.b 6
63.i even 6 2 1323.2.h.d 6
63.j odd 6 2 1323.2.h.e 6
63.k odd 6 2 441.2.g.e 6
63.l odd 6 2 63.2.f.b 6
63.n odd 6 2 1323.2.g.b 6
63.o even 6 2 189.2.f.a 6
63.s even 6 2 1323.2.g.c 6
63.t odd 6 2 441.2.h.c 6
84.h odd 2 1 9072.2.a.cd 3
252.s odd 6 2 3024.2.r.g 6
252.bi even 6 2 1008.2.r.k 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.b 6 63.l odd 6 2
189.2.f.a 6 63.o even 6 2
441.2.f.d 6 9.c even 3 2
441.2.g.d 6 63.g even 3 2
441.2.g.e 6 63.k odd 6 2
441.2.h.b 6 63.h even 3 2
441.2.h.c 6 63.t odd 6 2
567.2.a.d 3 7.b odd 2 1
567.2.a.g 3 21.c even 2 1
1008.2.r.k 6 252.bi even 6 2
1323.2.f.c 6 9.d odd 6 2
1323.2.g.b 6 63.n odd 6 2
1323.2.g.c 6 63.s even 6 2
1323.2.h.d 6 63.i even 6 2
1323.2.h.e 6 63.j odd 6 2
3024.2.r.g 6 252.s odd 6 2
3969.2.a.m 3 1.a even 1 1 trivial
3969.2.a.p 3 3.b odd 2 1
9072.2.a.bq 3 28.d even 2 1
9072.2.a.cd 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3969))$$:

 $$T_{2}^{3} + T_{2}^{2} - 4 T_{2} - 1$$ $$T_{5}^{3} - 5 T_{5}^{2} + 2 T_{5} + 11$$ $$T_{11}^{3} + 2 T_{11}^{2} - 19 T_{11} - 47$$ $$T_{13} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 2 T^{2} + 3 T^{3} + 4 T^{4} + 4 T^{5} + 8 T^{6}$$
$3$ 1
$5$ $$1 - 5 T + 17 T^{2} - 39 T^{3} + 85 T^{4} - 125 T^{5} + 125 T^{6}$$
$7$ 1
$11$ $$1 + 2 T + 14 T^{2} - 3 T^{3} + 154 T^{4} + 242 T^{5} + 1331 T^{6}$$
$13$ $$( 1 + T + 13 T^{2} )^{3}$$
$17$ $$1 - 12 T + 90 T^{2} - 435 T^{3} + 1530 T^{4} - 3468 T^{5} + 4913 T^{6}$$
$19$ $$1 - 3 T + 51 T^{2} - 107 T^{3} + 969 T^{4} - 1083 T^{5} + 6859 T^{6}$$
$23$ $$1 + 36 T^{2} - 9 T^{3} + 828 T^{4} + 12167 T^{6}$$
$29$ $$1 - T + 83 T^{2} - 57 T^{3} + 2407 T^{4} - 841 T^{5} + 24389 T^{6}$$
$31$ $$1 - 3 T + 69 T^{2} - 213 T^{3} + 2139 T^{4} - 2883 T^{5} + 29791 T^{6}$$
$37$ $$1 + 3 T + 57 T^{2} + 303 T^{3} + 2109 T^{4} + 4107 T^{5} + 50653 T^{6}$$
$41$ $$1 - 22 T + 278 T^{2} - 2157 T^{3} + 11398 T^{4} - 36982 T^{5} + 68921 T^{6}$$
$43$ $$1 + 3 T + 63 T^{2} + 379 T^{3} + 2709 T^{4} + 5547 T^{5} + 79507 T^{6}$$
$47$ $$1 - 9 T + 87 T^{2} - 657 T^{3} + 4089 T^{4} - 19881 T^{5} + 103823 T^{6}$$
$53$ $$1 + 18 T + 234 T^{2} + 1917 T^{3} + 12402 T^{4} + 50562 T^{5} + 148877 T^{6}$$
$59$ $$1 - 9 T + 171 T^{2} - 999 T^{3} + 10089 T^{4} - 31329 T^{5} + 205379 T^{6}$$
$61$ $$1 - 6 T + 162 T^{2} - 665 T^{3} + 9882 T^{4} - 22326 T^{5} + 226981 T^{6}$$
$67$ $$1 - 6 T^{2} + 683 T^{3} - 402 T^{4} + 300763 T^{6}$$
$71$ $$1 - 9 T + 207 T^{2} - 1197 T^{3} + 14697 T^{4} - 45369 T^{5} + 357911 T^{6}$$
$73$ $$1 + 3 T + 51 T^{2} + 681 T^{3} + 3723 T^{4} + 15987 T^{5} + 389017 T^{6}$$
$79$ $$1 - 15 T + 189 T^{2} - 1601 T^{3} + 14931 T^{4} - 93615 T^{5} + 493039 T^{6}$$
$83$ $$1 - 12 T + 288 T^{2} - 2019 T^{3} + 23904 T^{4} - 82668 T^{5} + 571787 T^{6}$$
$89$ $$1 - 2 T + 116 T^{2} - 735 T^{3} + 10324 T^{4} - 15842 T^{5} + 704969 T^{6}$$
$97$ $$1 + 3 T + 177 T^{2} - 21 T^{3} + 17169 T^{4} + 28227 T^{5} + 912673 T^{6}$$