# Properties

 Label 3969.2.a.q 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: $$\Q(\zeta_{18})^+$$ Defining polynomial: $$x^{3} - 3 x - 1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta ) q^{2} + ( -1 - 2 \beta + \beta^{2} ) q^{4} + ( -3 - \beta + \beta^{2} ) q^{5} + ( -4 - 2 \beta + 3 \beta^{2} ) q^{8} +O(q^{10})$$ $$q + ( 1 - \beta ) q^{2} + ( -1 - 2 \beta + \beta^{2} ) q^{4} + ( -3 - \beta + \beta^{2} ) q^{5} + ( -4 - 2 \beta + 3 \beta^{2} ) q^{8} + ( -4 - \beta + 2 \beta^{2} ) q^{10} + ( -\beta + \beta^{2} ) q^{11} + ( -7 - 2 \beta + 4 \beta^{2} ) q^{13} + ( -5 - 3 \beta + 3 \beta^{2} ) q^{16} + ( -4 + \beta^{2} ) q^{17} + ( 3 + 2 \beta - \beta^{2} ) q^{19} + ( -\beta + \beta^{2} ) q^{20} + ( -1 - 4 \beta + 2 \beta^{2} ) q^{22} + ( \beta + 2 \beta^{2} ) q^{23} + ( 2 + \beta - 2 \beta^{2} ) q^{25} + ( -11 - 7 \beta + 6 \beta^{2} ) q^{26} + ( 11 + 5 \beta - 4 \beta^{2} ) q^{29} + ( 7 - 3 \beta - 3 \beta^{2} ) q^{31} -3 \beta q^{32} + ( -5 + \beta + \beta^{2} ) q^{34} + ( -7 + 3 \beta + 3 \beta^{2} ) q^{37} + ( 4 + 2 \beta - 3 \beta^{2} ) q^{38} + ( 7 - 2 \beta - 2 \beta^{2} ) q^{40} + ( 2 - \beta - \beta^{2} ) q^{41} + ( 1 - \beta - \beta^{2} ) q^{43} + ( -3 - 7 \beta + 4 \beta^{2} ) q^{44} + ( -2 - 5 \beta + \beta^{2} ) q^{46} + ( 5 - 2 \beta - 3 \beta^{2} ) q^{47} + ( 4 + 5 \beta - 3 \beta^{2} ) q^{50} + ( -3 - 10 \beta + 5 \beta^{2} ) q^{52} + ( -2 - 3 \beta + 2 \beta^{2} ) q^{53} + ( -2 - 2 \beta + \beta^{2} ) q^{55} + ( 15 + 6 \beta - 9 \beta^{2} ) q^{58} + ( 1 + 5 \beta ) q^{59} + ( -2 - 3 \beta ) q^{61} + ( 10 - \beta ) q^{62} + ( 10 + 3 \beta - 3 \beta^{2} ) q^{64} + ( 15 - \beta - 5 \beta^{2} ) q^{65} + ( -10 + 3 \beta^{2} ) q^{67} + ( 2 + 3 \beta - 2 \beta^{2} ) q^{68} + ( 15 + 3 \beta - 6 \beta^{2} ) q^{71} + ( 9 - 4 \beta - \beta^{2} ) q^{73} + ( -10 + \beta ) q^{74} + ( 1 + 3 \beta - 3 \beta^{2} ) q^{76} + ( -13 + 3 \beta^{2} ) q^{79} + ( 9 - \beta - 2 \beta^{2} ) q^{80} + 3 q^{82} + ( 4 + 4 \beta + \beta^{2} ) q^{83} + ( 11 + 2 \beta - 4 \beta^{2} ) q^{85} + ( 2 + \beta ) q^{86} + ( -5 - 8 \beta + 7 \beta^{2} ) q^{88} + ( 10 + 3 \beta - 7 \beta^{2} ) q^{89} + ( -3 - 8 \beta + 2 \beta^{2} ) q^{92} + ( 8 + 2 \beta - \beta^{2} ) q^{94} + ( -6 - \beta + \beta^{2} ) q^{95} + ( 17 + 7 \beta - 8 \beta^{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} + 3q^{4} - 3q^{5} + 6q^{8} + O(q^{10})$$ $$3q + 3q^{2} + 3q^{4} - 3q^{5} + 6q^{8} + 6q^{11} + 3q^{13} + 3q^{16} - 6q^{17} + 3q^{19} + 6q^{20} + 9q^{22} + 12q^{23} - 6q^{25} + 3q^{26} + 9q^{29} + 3q^{31} - 9q^{34} - 3q^{37} - 6q^{38} + 9q^{40} - 3q^{43} + 15q^{44} - 3q^{47} - 6q^{50} + 21q^{52} + 6q^{53} - 9q^{58} + 3q^{59} - 6q^{61} + 30q^{62} + 12q^{64} + 15q^{65} - 12q^{67} - 6q^{68} + 9q^{71} + 21q^{73} - 30q^{74} - 15q^{76} - 21q^{79} + 15q^{80} + 9q^{82} + 18q^{83} + 9q^{85} + 6q^{86} + 27q^{88} - 12q^{89} + 3q^{92} + 18q^{94} - 12q^{95} + 3q^{97} + O(q^{100})$$

## 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
 1.87939 −0.347296 −1.53209
−0.879385 0 −1.22668 −1.34730 0 0 2.83750 0 1.18479
1.2 1.34730 0 −0.184793 −2.53209 0 0 −2.94356 0 −3.41147
1.3 2.53209 0 4.41147 0.879385 0 0 6.10607 0 2.22668
 $$n$$: 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.q 3
3.b odd 2 1 3969.2.a.l 3
7.b odd 2 1 567.2.a.h 3
9.c even 3 2 441.2.f.c 6
9.d odd 6 2 1323.2.f.d 6
21.c even 2 1 567.2.a.c 3
28.d even 2 1 9072.2.a.ca 3
63.g even 3 2 441.2.g.b 6
63.h even 3 2 441.2.h.e 6
63.i even 6 2 1323.2.h.c 6
63.j odd 6 2 1323.2.h.b 6
63.k odd 6 2 441.2.g.c 6
63.l odd 6 2 63.2.f.a 6
63.n odd 6 2 1323.2.g.e 6
63.o even 6 2 189.2.f.b 6
63.s even 6 2 1323.2.g.d 6
63.t odd 6 2 441.2.h.d 6
84.h odd 2 1 9072.2.a.bs 3
252.s odd 6 2 3024.2.r.k 6
252.bi even 6 2 1008.2.r.h 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.a 6 63.l odd 6 2
189.2.f.b 6 63.o even 6 2
441.2.f.c 6 9.c even 3 2
441.2.g.b 6 63.g even 3 2
441.2.g.c 6 63.k odd 6 2
441.2.h.d 6 63.t odd 6 2
441.2.h.e 6 63.h even 3 2
567.2.a.c 3 21.c even 2 1
567.2.a.h 3 7.b odd 2 1
1008.2.r.h 6 252.bi even 6 2
1323.2.f.d 6 9.d odd 6 2
1323.2.g.d 6 63.s even 6 2
1323.2.g.e 6 63.n odd 6 2
1323.2.h.b 6 63.j odd 6 2
1323.2.h.c 6 63.i even 6 2
3024.2.r.k 6 252.s odd 6 2
3969.2.a.l 3 3.b odd 2 1
3969.2.a.q 3 1.a even 1 1 trivial
9072.2.a.bs 3 84.h odd 2 1
9072.2.a.ca 3 28.d even 2 1

## Hecke kernels

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

 $$T_{2}^{3} - 3 T_{2}^{2} + 3$$ $$T_{5}^{3} + 3 T_{5}^{2} - 3$$ $$T_{11}^{3} - 6 T_{11}^{2} + 9 T_{11} - 3$$ $$T_{13}^{3} - 3 T_{13}^{2} - 33 T_{13} + 107$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T + 6 T^{2} - 9 T^{3} + 12 T^{4} - 12 T^{5} + 8 T^{6}$$
$3$ 1
$5$ $$1 + 3 T + 15 T^{2} + 27 T^{3} + 75 T^{4} + 75 T^{5} + 125 T^{6}$$
$7$ 1
$11$ $$1 - 6 T + 42 T^{2} - 135 T^{3} + 462 T^{4} - 726 T^{5} + 1331 T^{6}$$
$13$ $$1 - 3 T + 6 T^{2} + 29 T^{3} + 78 T^{4} - 507 T^{5} + 2197 T^{6}$$
$17$ $$1 + 6 T + 60 T^{2} + 207 T^{3} + 1020 T^{4} + 1734 T^{5} + 4913 T^{6}$$
$19$ $$1 - 3 T + 51 T^{2} - 97 T^{3} + 969 T^{4} - 1083 T^{5} + 6859 T^{6}$$
$23$ $$1 - 12 T + 96 T^{2} - 549 T^{3} + 2208 T^{4} - 6348 T^{5} + 12167 T^{6}$$
$29$ $$1 - 9 T + 51 T^{2} - 189 T^{3} + 1479 T^{4} - 7569 T^{5} + 24389 T^{6}$$
$31$ $$1 - 3 T + 15 T^{2} + 137 T^{3} + 465 T^{4} - 2883 T^{5} + 29791 T^{6}$$
$37$ $$1 + 3 T + 33 T^{2} - 101 T^{3} + 1221 T^{4} + 4107 T^{5} + 50653 T^{6}$$
$41$ $$1 + 114 T^{2} + 9 T^{3} + 4674 T^{4} + 68921 T^{6}$$
$43$ $$1 + 3 T + 123 T^{2} + 259 T^{3} + 5289 T^{4} + 5547 T^{5} + 79507 T^{6}$$
$47$ $$1 + 3 T + 87 T^{2} + 333 T^{3} + 4089 T^{4} + 6627 T^{5} + 103823 T^{6}$$
$53$ $$1 - 6 T + 150 T^{2} - 639 T^{3} + 7950 T^{4} - 16854 T^{5} + 148877 T^{6}$$
$59$ $$1 - 3 T + 105 T^{2} - 405 T^{3} + 6195 T^{4} - 10443 T^{5} + 205379 T^{6}$$
$61$ $$1 + 6 T + 168 T^{2} + 713 T^{3} + 10248 T^{4} + 22326 T^{5} + 226981 T^{6}$$
$67$ $$1 + 12 T + 222 T^{2} + 1591 T^{3} + 14874 T^{4} + 53868 T^{5} + 300763 T^{6}$$
$71$ $$1 - 9 T + 159 T^{2} - 1305 T^{3} + 11289 T^{4} - 45369 T^{5} + 357911 T^{6}$$
$73$ $$1 - 21 T + 303 T^{2} - 2797 T^{3} + 22119 T^{4} - 111909 T^{5} + 389017 T^{6}$$
$79$ $$1 + 21 T + 357 T^{2} + 3499 T^{3} + 28203 T^{4} + 131061 T^{5} + 493039 T^{6}$$
$83$ $$1 - 18 T + 294 T^{2} - 2997 T^{3} + 24402 T^{4} - 124002 T^{5} + 571787 T^{6}$$
$89$ $$1 + 12 T + 204 T^{2} + 1323 T^{3} + 18156 T^{4} + 95052 T^{5} + 704969 T^{6}$$
$97$ $$1 - 3 T + 123 T^{2} - 259 T^{3} + 11931 T^{4} - 28227 T^{5} + 912673 T^{6}$$