# Properties

 Label 3969.2.a.t Level $3969$ Weight $2$ Character orbit 3969.a Self dual yes Analytic conductor $31.693$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.14013.1 Defining polynomial: $$x^{4} - x^{3} - 6x^{2} + 6x + 3$$ x^4 - x^3 - 6*x^2 + 6*x + 3 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{5} + ( - \beta_{3} - \beta_1 + 1) q^{8}+O(q^{10})$$ q - b1 * q^2 + (b2 + 1) * q^4 + (-b3 - b2 - b1 + 1) * q^5 + (-b3 - b1 + 1) * q^8 $$q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{5} + ( - \beta_{3} - \beta_1 + 1) q^{8} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{10} + ( - 2 \beta_{3} - \beta_{2} - 1) q^{11} + (\beta_{2} + 1) q^{13} + (\beta_{3} - \beta_1) q^{16} + ( - \beta_{3} + 2 \beta_{2} + 1) q^{17} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{19} + ( - 2 \beta_{3} - \beta_{2} - 3 \beta_1 - 1) q^{20} + (3 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 3) q^{22} + (\beta_{2} - \beta_1 + 3) q^{23} + (\beta_{2} + 3 \beta_1 + 1) q^{25} + ( - \beta_{3} - 3 \beta_1 + 1) q^{26} + ( - 2 \beta_{2} + 3) q^{29} + ( - \beta_{3} - 2 \beta_1 + 5) q^{31} + (\beta_{3} + 2 \beta_1 + 2) q^{32} + ( - \beta_{3} + \beta_{2} - 5 \beta_1 + 1) q^{34} + ( - 2 \beta_{3} - 3 \beta_{2} - \beta_1 + 1) q^{37} + (2 \beta_{3} - 5) q^{38} + ( - \beta_{3} + \beta_{2} + \beta_1 + 4) q^{40} + ( - 2 \beta_{2} - 3 \beta_1) q^{41} + (\beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{43} + ( - \beta_{3} - 4 \beta_{2} - \beta_1 - 2) q^{44} + ( - \beta_{3} + \beta_{2} - 5 \beta_1 + 4) q^{46} + ( - \beta_{3} - \beta_1 - 5) q^{47} + ( - \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 8) q^{50} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 6) q^{52} + (\beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{53} + ( - 2 \beta_{3} + \beta_{2} + 5 \beta_1 + 5) q^{55} + (2 \beta_{3} + \beta_1 - 2) q^{58} + (\beta_{3} - 3 \beta_{2} + \beta_1 + 2) q^{59} + (\beta_{3} - 4 \beta_1 + 6) q^{61} + (\beta_{3} + 3 \beta_{2} - 5 \beta_1 + 5) q^{62} + ( - 3 \beta_{3} - 3 \beta_{2} - 5) q^{64} + ( - 2 \beta_{3} - \beta_{2} - 3 \beta_1 - 1) q^{65} + ( - \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{67} + (2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 13) q^{68} + ( - 3 \beta_1 + 3) q^{71} + ( - 3 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{73} + (5 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 2) q^{74} + (3 \beta_1 - 2) q^{76} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 1) q^{79} + (4 \beta_{3} + 2 \beta_{2} - 1) q^{80} + (2 \beta_{3} + 3 \beta_{2} + 4 \beta_1 + 7) q^{82} + (4 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{83} + ( - 5 \beta_{3} - \beta_{2} - 4 \beta_1 - 1) q^{85} + ( - 3 \beta_{3} - 2 \beta_1 + 6) q^{86} + ( - \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 4) q^{88} + (3 \beta_{3} + \beta_{2} + \beta_1 - 6) q^{89} + (4 \beta_{2} - 4 \beta_1 + 9) q^{92} + (\beta_{3} + 2 \beta_{2} + 5 \beta_1 + 2) q^{94} + ( - 5 \beta_{3} - 4 \beta_{2} + 5) q^{95} + (4 \beta_{3} + \beta_{2} - 4 \beta_1 + 3) q^{97}+O(q^{100})$$ q - b1 * q^2 + (b2 + 1) * q^4 + (-b3 - b2 - b1 + 1) * q^5 + (-b3 - b1 + 1) * q^8 + (2*b3 + 2*b2 + b1 + 1) * q^10 + (-2*b3 - b2 - 1) * q^11 + (b2 + 1) * q^13 + (b3 - b1) * q^16 + (-b3 + 2*b2 + 1) * q^17 + (-b3 - b2 + b1 + 2) * q^19 + (-2*b3 - b2 - 3*b1 - 1) * q^20 + (3*b3 + 2*b2 + 3*b1 - 3) * q^22 + (b2 - b1 + 3) * q^23 + (b2 + 3*b1 + 1) * q^25 + (-b3 - 3*b1 + 1) * q^26 + (-2*b2 + 3) * q^29 + (-b3 - 2*b1 + 5) * q^31 + (b3 + 2*b1 + 2) * q^32 + (-b3 + b2 - 5*b1 + 1) * q^34 + (-2*b3 - 3*b2 - b1 + 1) * q^37 + (2*b3 - 5) * q^38 + (-b3 + b2 + b1 + 4) * q^40 + (-2*b2 - 3*b1) * q^41 + (b3 + 2*b2 - b1 - 2) * q^43 + (-b3 - 4*b2 - b1 - 2) * q^44 + (-b3 + b2 - 5*b1 + 4) * q^46 + (-b3 - b1 - 5) * q^47 + (-b3 - 3*b2 - 3*b1 - 8) * q^50 + (b3 + 2*b2 - b1 + 6) * q^52 + (b3 + b2 + 3*b1 + 2) * q^53 + (-2*b3 + b2 + 5*b1 + 5) * q^55 + (2*b3 + b1 - 2) * q^58 + (b3 - 3*b2 + b1 + 2) * q^59 + (b3 - 4*b1 + 6) * q^61 + (b3 + 3*b2 - 5*b1 + 5) * q^62 + (-3*b3 - 3*b2 - 5) * q^64 + (-2*b3 - b2 - 3*b1 - 1) * q^65 + (-b3 - 3*b2 - 2*b1) * q^67 + (2*b3 + 2*b2 - 3*b1 + 13) * q^68 + (-3*b1 + 3) * q^71 + (-3*b3 - b2 + 3*b1 + 1) * q^73 + (5*b3 + 3*b2 + 5*b1 - 2) * q^74 + (3*b1 - 2) * q^76 + (-3*b3 - 3*b2 - 3*b1 - 1) * q^79 + (4*b3 + 2*b2 - 1) * q^80 + (2*b3 + 3*b2 + 4*b1 + 7) * q^82 + (4*b3 + 2*b2 - b1 + 2) * q^83 + (-5*b3 - b2 - 4*b1 - 1) * q^85 + (-3*b3 - 2*b1 + 6) * q^86 + (-b3 - 2*b2 + 4*b1 + 4) * q^88 + (3*b3 + b2 + b1 - 6) * q^89 + (4*b2 - 4*b1 + 9) * q^92 + (b3 + 2*b2 + 5*b1 + 2) * q^94 + (-5*b3 - 4*b2 + 5) * q^95 + (4*b3 + b2 - 4*b1 + 3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{2} + 5 q^{4} + 2 q^{5} + 3 q^{8}+O(q^{10})$$ 4 * q - q^2 + 5 * q^4 + 2 * q^5 + 3 * q^8 $$4 q - q^{2} + 5 q^{4} + 2 q^{5} + 3 q^{8} + 7 q^{10} - 5 q^{11} + 5 q^{13} - q^{16} + 6 q^{17} + 8 q^{19} - 8 q^{20} - 7 q^{22} + 12 q^{23} + 8 q^{25} + q^{26} + 10 q^{29} + 18 q^{31} + 10 q^{32} - 20 q^{38} + 18 q^{40} - 5 q^{41} - 7 q^{43} - 13 q^{44} + 12 q^{46} - 21 q^{47} - 38 q^{50} + 25 q^{52} + 12 q^{53} + 26 q^{55} - 7 q^{58} + 6 q^{59} + 20 q^{61} + 18 q^{62} - 23 q^{64} - 8 q^{65} - 5 q^{67} + 51 q^{68} + 9 q^{71} + 6 q^{73} - 5 q^{76} - 10 q^{79} - 2 q^{80} + 35 q^{82} + 9 q^{83} - 9 q^{85} + 22 q^{86} + 18 q^{88} - 22 q^{89} + 36 q^{92} + 15 q^{94} + 16 q^{95} + 9 q^{97}+O(q^{100})$$ 4 * q - q^2 + 5 * q^4 + 2 * q^5 + 3 * q^8 + 7 * q^10 - 5 * q^11 + 5 * q^13 - q^16 + 6 * q^17 + 8 * q^19 - 8 * q^20 - 7 * q^22 + 12 * q^23 + 8 * q^25 + q^26 + 10 * q^29 + 18 * q^31 + 10 * q^32 - 20 * q^38 + 18 * q^40 - 5 * q^41 - 7 * q^43 - 13 * q^44 + 12 * q^46 - 21 * q^47 - 38 * q^50 + 25 * q^52 + 12 * q^53 + 26 * q^55 - 7 * q^58 + 6 * q^59 + 20 * q^61 + 18 * q^62 - 23 * q^64 - 8 * q^65 - 5 * q^67 + 51 * q^68 + 9 * q^71 + 6 * q^73 - 5 * q^76 - 10 * q^79 - 2 * q^80 + 35 * q^82 + 9 * q^83 - 9 * q^85 + 22 * q^86 + 18 * q^88 - 22 * q^89 + 36 * q^92 + 15 * q^94 + 16 * q^95 + 9 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 6x^{2} + 6x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5\nu + 1$$ v^3 - 5*v + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5\beta _1 - 1$$ b3 + 5*b1 - 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
 2.20800 1.53652 −0.372845 −2.37167
−2.20800 0 2.87525 −3.80779 0 0 −1.93254 0 8.40758
1.2 −1.53652 0 0.360904 3.15761 0 0 2.51851 0 −4.85173
1.3 0.372845 0 −1.86099 1.42143 0 0 −1.43955 0 0.529976
1.4 2.37167 0 3.62484 1.22875 0 0 3.85358 0 2.91418
 $$n$$: 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.t 4
3.b odd 2 1 3969.2.a.w 4
7.b odd 2 1 3969.2.a.s 4
7.d odd 6 2 567.2.e.d yes 8
21.c even 2 1 3969.2.a.x 4
21.g even 6 2 567.2.e.c 8
63.i even 6 2 567.2.h.k 8
63.k odd 6 2 567.2.g.k 8
63.s even 6 2 567.2.g.j 8
63.t odd 6 2 567.2.h.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.2.e.c 8 21.g even 6 2
567.2.e.d yes 8 7.d odd 6 2
567.2.g.j 8 63.s even 6 2
567.2.g.k 8 63.k odd 6 2
567.2.h.j 8 63.t odd 6 2
567.2.h.k 8 63.i even 6 2
3969.2.a.s 4 7.b odd 2 1
3969.2.a.t 4 1.a even 1 1 trivial
3969.2.a.w 4 3.b odd 2 1
3969.2.a.x 4 21.c even 2 1

## Hecke kernels

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

 $$T_{2}^{4} + T_{2}^{3} - 6T_{2}^{2} - 6T_{2} + 3$$ T2^4 + T2^3 - 6*T2^2 - 6*T2 + 3 $$T_{5}^{4} - 2T_{5}^{3} - 12T_{5}^{2} + 33T_{5} - 21$$ T5^4 - 2*T5^3 - 12*T5^2 + 33*T5 - 21 $$T_{11}^{4} + 5T_{11}^{3} - 24T_{11}^{2} - 174T_{11} - 249$$ T11^4 + 5*T11^3 - 24*T11^2 - 174*T11 - 249 $$T_{13}^{4} - 5T_{13}^{3} + 20T_{13} - 7$$ T13^4 - 5*T13^3 + 20*T13 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} - 6 T^{2} - 6 T + 3$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 2 T^{3} - 12 T^{2} + 33 T - 21$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 5 T^{3} - 24 T^{2} - 174 T - 249$$
$13$ $$T^{4} - 5 T^{3} + 20 T - 7$$
$17$ $$T^{4} - 6 T^{3} - 45 T^{2} + 354 T - 567$$
$19$ $$T^{4} - 8 T^{3} + 47 T - 49$$
$23$ $$T^{4} - 12 T^{3} + 36 T^{2} - 33 T + 9$$
$29$ $$T^{4} - 10 T^{3} + 90 T + 63$$
$31$ $$T^{4} - 18 T^{3} + 93 T^{2} - 136 T - 21$$
$37$ $$T^{4} - 78 T^{2} + 37 T + 951$$
$41$ $$T^{4} + 5 T^{3} - 72 T^{2} - 126 T + 441$$
$43$ $$T^{4} + 7 T^{3} - 30 T^{2} - 176 T + 49$$
$47$ $$T^{4} + 21 T^{3} + 153 T^{2} + \cdots + 441$$
$53$ $$T^{4} - 12 T^{3} + 105 T - 81$$
$59$ $$T^{4} - 6 T^{3} - 108 T^{2} + \cdots + 189$$
$61$ $$T^{4} - 20 T^{3} + 27 T^{2} + \cdots + 1043$$
$67$ $$T^{4} + 5 T^{3} - 72 T^{2} - 74 T + 353$$
$71$ $$T^{4} - 9 T^{3} - 27 T^{2} + 135 T + 243$$
$73$ $$T^{4} - 6 T^{3} - 150 T^{2} + \cdots + 2289$$
$79$ $$T^{4} + 10 T^{3} - 84 T^{2} - 5 T + 7$$
$83$ $$T^{4} - 9 T^{3} - 126 T^{2} + \cdots - 5103$$
$89$ $$T^{4} + 22 T^{3} + 114 T^{2} + \cdots - 21$$
$97$ $$T^{4} - 9 T^{3} - 258 T^{2} + \cdots + 2877$$