# Properties

 Label 3969.2.a.z Level $3969$ Weight $2$ Character orbit 3969.a Self dual yes Analytic conductor $31.693$ Analytic rank $1$ Dimension $5$ 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: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.574857.1 Defining polynomial: $$x^{5} - 2 x^{4} - 5 x^{3} + 9 x^{2} + 3 x - 3$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 - \beta_{4} ) q^{5} + ( -1 - \beta_{3} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 - \beta_{4} ) q^{5} + ( -1 - \beta_{3} ) q^{8} + ( 2 + \beta_{3} + \beta_{4} ) q^{10} + ( -1 + \beta_{2} - \beta_{3} ) q^{11} + ( 1 + \beta_{1} - \beta_{4} ) q^{13} + ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{16} + ( -3 + \beta_{1} - \beta_{2} ) q^{17} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{19} + ( -2 - \beta_{2} - 2 \beta_{3} ) q^{20} + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{22} + ( -2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{23} + ( \beta_{1} - \beta_{2} + 2 \beta_{4} ) q^{25} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{26} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{29} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{31} + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{32} + ( -2 + 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{34} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{37} + ( -5 + 2 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{38} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{40} + ( -\beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{41} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{43} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{44} + ( -4 + 5 \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{46} + ( -4 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{47} + ( -6 + 3 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{50} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{52} + ( 5 - \beta_{1} + 2 \beta_{4} ) q^{53} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{55} + ( 3 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{58} + ( -6 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{59} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{61} + ( -3 + 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{62} + ( -6 + \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{64} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{65} + ( -1 + 4 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{67} + ( -7 + 2 \beta_{1} - 3 \beta_{2} - \beta_{4} ) q^{68} + ( 2 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{71} + ( -4 + \beta_{2} - 3 \beta_{3} ) q^{73} + ( 10 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{74} + ( -3 + 5 \beta_{1} - \beta_{2} + \beta_{4} ) q^{76} + ( 3 - 4 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{79} + ( -3 - 2 \beta_{1} + \beta_{3} - \beta_{4} ) q^{80} + ( -2 + 4 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} ) q^{82} + ( -1 - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} ) q^{83} + ( 2 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{85} + ( -1 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - 4 \beta_{4} ) q^{86} + ( 4 - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{88} + ( -7 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{89} + ( -5 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{92} + ( -3 + 4 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} ) q^{94} + ( 2 - 2 \beta_{2} - \beta_{3} ) q^{95} + ( 2 + \beta_{1} + 4 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q - 2q^{2} + 4q^{4} - 4q^{5} - 3q^{8} + O(q^{10})$$ $$5q - 2q^{2} + 4q^{4} - 4q^{5} - 3q^{8} + 7q^{10} - 4q^{11} + 8q^{13} - 2q^{16} - 12q^{17} - q^{19} - 5q^{20} + q^{22} - 3q^{23} + q^{25} - 11q^{26} - 7q^{29} + 3q^{31} + 2q^{32} - 3q^{34} - 20q^{38} + 3q^{40} - 5q^{41} + 7q^{43} + 10q^{44} - 3q^{46} - 27q^{47} - 19q^{50} + 10q^{52} + 21q^{53} + 2q^{55} + 10q^{58} - 30q^{59} + 14q^{61} - 6q^{62} - 25q^{64} + 11q^{65} + 2q^{67} - 27q^{68} - 3q^{71} - 15q^{73} + 36q^{74} - 5q^{76} + 4q^{79} - 20q^{80} + 5q^{82} - 9q^{83} + 6q^{85} + 8q^{86} + 18q^{88} - 28q^{89} - 27q^{92} + 3q^{94} + 14q^{95} + 12q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 5 x^{3} + 9 x^{2} + 3 x - 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4 \nu - 1$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 5 \nu^{2} + 4 \nu + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 5 \beta_{2} + 14$$

## 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
 2.38687 1.84124 0.495868 −0.670333 −2.05365
−2.38687 0 3.69714 −2.92087 0 0 −4.05086 0 6.97172
1.2 −1.84124 0 1.39017 1.33475 0 0 1.12285 0 −2.45760
1.3 −0.495868 0 −1.75411 −3.69258 0 0 1.86155 0 1.83103
1.4 0.670333 0 −1.55065 1.42494 0 0 −2.38012 0 0.955182
1.5 2.05365 0 2.21746 −0.146246 0 0 0.446582 0 −0.300337
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.z 5
3.b odd 2 1 3969.2.a.bc 5
7.b odd 2 1 3969.2.a.ba 5
7.c even 3 2 567.2.e.f 10
9.c even 3 2 441.2.f.e 10
9.d odd 6 2 1323.2.f.e 10
21.c even 2 1 3969.2.a.bb 5
21.h odd 6 2 567.2.e.e 10
63.g even 3 2 63.2.g.b 10
63.h even 3 2 63.2.h.b yes 10
63.i even 6 2 1323.2.h.f 10
63.j odd 6 2 189.2.h.b 10
63.k odd 6 2 441.2.g.f 10
63.l odd 6 2 441.2.f.f 10
63.n odd 6 2 189.2.g.b 10
63.o even 6 2 1323.2.f.f 10
63.s even 6 2 1323.2.g.f 10
63.t odd 6 2 441.2.h.f 10
252.o even 6 2 3024.2.t.i 10
252.u odd 6 2 1008.2.q.i 10
252.bb even 6 2 3024.2.q.i 10
252.bl odd 6 2 1008.2.t.i 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.b 10 63.g even 3 2
63.2.h.b yes 10 63.h even 3 2
189.2.g.b 10 63.n odd 6 2
189.2.h.b 10 63.j odd 6 2
441.2.f.e 10 9.c even 3 2
441.2.f.f 10 63.l odd 6 2
441.2.g.f 10 63.k odd 6 2
441.2.h.f 10 63.t odd 6 2
567.2.e.e 10 21.h odd 6 2
567.2.e.f 10 7.c even 3 2
1008.2.q.i 10 252.u odd 6 2
1008.2.t.i 10 252.bl odd 6 2
1323.2.f.e 10 9.d odd 6 2
1323.2.f.f 10 63.o even 6 2
1323.2.g.f 10 63.s even 6 2
1323.2.h.f 10 63.i even 6 2
3024.2.q.i 10 252.bb even 6 2
3024.2.t.i 10 252.o even 6 2
3969.2.a.z 5 1.a even 1 1 trivial
3969.2.a.ba 5 7.b odd 2 1
3969.2.a.bb 5 21.c even 2 1
3969.2.a.bc 5 3.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{5} + 2 T_{2}^{4} - 5 T_{2}^{3} - 9 T_{2}^{2} + 3 T_{2} + 3$$ $$T_{5}^{5} + 4 T_{5}^{4} - 5 T_{5}^{3} - 18 T_{5}^{2} + 18 T_{5} + 3$$ $$T_{11}^{5} + 4 T_{11}^{4} - 8 T_{11}^{3} - 15 T_{11}^{2} + 12 T_{11} + 15$$ $$T_{13}^{5} - 8 T_{13}^{4} + 13 T_{13}^{3} + 13 T_{13}^{2} - 23 T_{13} - 5$$