# Properties

 Label 3249.2.a.bf Level $3249$ Weight $2$ Character orbit 3249.a Self dual yes Analytic conductor $25.943$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3249 = 3^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3249.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.9433956167$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.13068.1 Defining polynomial: $$x^{4} - x^{3} - 6x^{2} - x + 1$$ x^4 - x^3 - 6*x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 171) 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{3} + 2) q^{4} + ( - \beta_{2} - \beta_1) q^{5} + ( - \beta_{3} + 1) q^{7} + ( - 2 \beta_{2} - 3 \beta_1) q^{8}+O(q^{10})$$ q - b1 * q^2 + (b3 + 2) * q^4 + (-b2 - b1) * q^5 + (-b3 + 1) * q^7 + (-2*b2 - 3*b1) * q^8 $$q - \beta_1 q^{2} + (\beta_{3} + 2) q^{4} + ( - \beta_{2} - \beta_1) q^{5} + ( - \beta_{3} + 1) q^{7} + ( - 2 \beta_{2} - 3 \beta_1) q^{8} + (2 \beta_{3} + 2) q^{10} + (\beta_{2} - \beta_1) q^{11} - 2 q^{13} + (2 \beta_{2} + 2 \beta_1) q^{14} + (3 \beta_{3} + 4) q^{16} + ( - \beta_{2} - \beta_1) q^{17} + ( - 2 \beta_{2} - 6 \beta_1) q^{20} + 6 q^{22} + 2 \beta_{2} q^{23} + (\beta_{3} + 2) q^{25} + 2 \beta_1 q^{26} + ( - 2 \beta_{3} - 6) q^{28} - 2 \beta_{2} q^{29} + ( - 2 \beta_{3} + 2) q^{31} + ( - 2 \beta_{2} - 7 \beta_1) q^{32} + (2 \beta_{3} + 2) q^{34} + ( - \beta_{2} + 3 \beta_1) q^{35} + ( - 2 \beta_{3} - 4) q^{37} + (4 \beta_{3} + 16) q^{40} + 2 \beta_1 q^{41} + ( - 3 \beta_{3} - 1) q^{43} + ( - 2 \beta_{2} - 4 \beta_1) q^{44} + ( - 2 \beta_{3} + 4) q^{46} + ( - 3 \beta_{2} - \beta_1) q^{47} + ( - \beta_{3} + 2) q^{49} + ( - 2 \beta_{2} - 5 \beta_1) q^{50} + ( - 2 \beta_{3} - 4) q^{52} + (2 \beta_{2} - 4 \beta_1) q^{53} + (3 \beta_{3} - 3) q^{55} + 8 \beta_1 q^{56} + (2 \beta_{3} - 4) q^{58} - 4 \beta_1 q^{59} + ( - 3 \beta_{3} + 5) q^{61} + (4 \beta_{2} + 4 \beta_1) q^{62} + (3 \beta_{3} + 16) q^{64} + (2 \beta_{2} + 2 \beta_1) q^{65} + 4 q^{67} + ( - 2 \beta_{2} - 6 \beta_1) q^{68} + ( - 2 \beta_{3} - 14) q^{70} - 4 \beta_{2} q^{71} + ( - 3 \beta_{3} + 5) q^{73} + (4 \beta_{2} + 10 \beta_1) q^{74} + (5 \beta_{2} + \beta_1) q^{77} + 4 q^{79} + ( - 4 \beta_{2} - 16 \beta_1) q^{80} + ( - 2 \beta_{3} - 8) q^{82} + ( - 2 \beta_{2} - 4 \beta_1) q^{83} + (\beta_{3} + 7) q^{85} + (6 \beta_{2} + 10 \beta_1) q^{86} + 6 \beta_{3} q^{88} + (2 \beta_{2} - 4 \beta_1) q^{89} + (2 \beta_{3} - 2) q^{91} + 2 \beta_1 q^{92} + (4 \beta_{3} - 2) q^{94} + ( - 4 \beta_{3} + 6) q^{97} + (2 \beta_{2} + \beta_1) q^{98}+O(q^{100})$$ q - b1 * q^2 + (b3 + 2) * q^4 + (-b2 - b1) * q^5 + (-b3 + 1) * q^7 + (-2*b2 - 3*b1) * q^8 + (2*b3 + 2) * q^10 + (b2 - b1) * q^11 - 2 * q^13 + (2*b2 + 2*b1) * q^14 + (3*b3 + 4) * q^16 + (-b2 - b1) * q^17 + (-2*b2 - 6*b1) * q^20 + 6 * q^22 + 2*b2 * q^23 + (b3 + 2) * q^25 + 2*b1 * q^26 + (-2*b3 - 6) * q^28 - 2*b2 * q^29 + (-2*b3 + 2) * q^31 + (-2*b2 - 7*b1) * q^32 + (2*b3 + 2) * q^34 + (-b2 + 3*b1) * q^35 + (-2*b3 - 4) * q^37 + (4*b3 + 16) * q^40 + 2*b1 * q^41 + (-3*b3 - 1) * q^43 + (-2*b2 - 4*b1) * q^44 + (-2*b3 + 4) * q^46 + (-3*b2 - b1) * q^47 + (-b3 + 2) * q^49 + (-2*b2 - 5*b1) * q^50 + (-2*b3 - 4) * q^52 + (2*b2 - 4*b1) * q^53 + (3*b3 - 3) * q^55 + 8*b1 * q^56 + (2*b3 - 4) * q^58 - 4*b1 * q^59 + (-3*b3 + 5) * q^61 + (4*b2 + 4*b1) * q^62 + (3*b3 + 16) * q^64 + (2*b2 + 2*b1) * q^65 + 4 * q^67 + (-2*b2 - 6*b1) * q^68 + (-2*b3 - 14) * q^70 - 4*b2 * q^71 + (-3*b3 + 5) * q^73 + (4*b2 + 10*b1) * q^74 + (5*b2 + b1) * q^77 + 4 * q^79 + (-4*b2 - 16*b1) * q^80 + (-2*b3 - 8) * q^82 + (-2*b2 - 4*b1) * q^83 + (b3 + 7) * q^85 + (6*b2 + 10*b1) * q^86 + 6*b3 * q^88 + (2*b2 - 4*b1) * q^89 + (2*b3 - 2) * q^91 + 2*b1 * q^92 + (4*b3 - 2) * q^94 + (-4*b3 + 6) * q^97 + (2*b2 + b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 10 q^{4} + 2 q^{7}+O(q^{10})$$ 4 * q + 10 * q^4 + 2 * q^7 $$4 q + 10 q^{4} + 2 q^{7} + 12 q^{10} - 8 q^{13} + 22 q^{16} + 24 q^{22} + 10 q^{25} - 28 q^{28} + 4 q^{31} + 12 q^{34} - 20 q^{37} + 72 q^{40} - 10 q^{43} + 12 q^{46} + 6 q^{49} - 20 q^{52} - 6 q^{55} - 12 q^{58} + 14 q^{61} + 70 q^{64} + 16 q^{67} - 60 q^{70} + 14 q^{73} + 16 q^{79} - 36 q^{82} + 30 q^{85} + 12 q^{88} - 4 q^{91} + 16 q^{97}+O(q^{100})$$ 4 * q + 10 * q^4 + 2 * q^7 + 12 * q^10 - 8 * q^13 + 22 * q^16 + 24 * q^22 + 10 * q^25 - 28 * q^28 + 4 * q^31 + 12 * q^34 - 20 * q^37 + 72 * q^40 - 10 * q^43 + 12 * q^46 + 6 * q^49 - 20 * q^52 - 6 * q^55 - 12 * q^58 + 14 * q^61 + 70 * q^64 + 16 * q^67 - 60 * q^70 + 14 * q^73 + 16 * q^79 - 36 * q^82 + 30 * q^85 + 12 * q^88 - 4 * q^91 + 16 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 6x^{2} - x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} - \nu^{2} - 5\nu - 1$$ v^3 - v^2 - 5*v - 1 $$\beta_{2}$$ $$=$$ $$-\nu^{3} + 2\nu^{2} + 4\nu - 2$$ -v^3 + 2*v^2 + 4*v - 2 $$\beta_{3}$$ $$=$$ $$-\nu^{3} + \nu^{2} + 7\nu + 1$$ -v^3 + v^2 + 7*v + 1
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 2\beta_{2} + 3\beta _1 + 6 ) / 2$$ (b3 + 2*b2 + 3*b1 + 6) / 2 $$\nu^{3}$$ $$=$$ $$3\beta_{3} + \beta_{2} + 5\beta _1 + 4$$ 3*b3 + b2 + 5*b1 + 4

## 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
 3.04374 −0.548230 −1.82405 0.328543
−2.71519 0 5.37228 −3.22060 0 −2.37228 −9.15640 0 8.74456
1.2 −1.27582 0 −0.372281 2.15121 0 3.37228 3.02661 0 −2.74456
1.3 1.27582 0 −0.372281 −2.15121 0 3.37228 −3.02661 0 −2.74456
1.4 2.71519 0 5.37228 3.22060 0 −2.37228 9.15640 0 8.74456
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$19$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3249.2.a.bf 4
3.b odd 2 1 inner 3249.2.a.bf 4
19.b odd 2 1 171.2.a.e 4
57.d even 2 1 171.2.a.e 4
76.d even 2 1 2736.2.a.bf 4
95.d odd 2 1 4275.2.a.bp 4
133.c even 2 1 8379.2.a.bw 4
228.b odd 2 1 2736.2.a.bf 4
285.b even 2 1 4275.2.a.bp 4
399.h odd 2 1 8379.2.a.bw 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.a.e 4 19.b odd 2 1
171.2.a.e 4 57.d even 2 1
2736.2.a.bf 4 76.d even 2 1
2736.2.a.bf 4 228.b odd 2 1
3249.2.a.bf 4 1.a even 1 1 trivial
3249.2.a.bf 4 3.b odd 2 1 inner
4275.2.a.bp 4 95.d odd 2 1
4275.2.a.bp 4 285.b even 2 1
8379.2.a.bw 4 133.c even 2 1
8379.2.a.bw 4 399.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3249))$$:

 $$T_{2}^{4} - 9T_{2}^{2} + 12$$ T2^4 - 9*T2^2 + 12 $$T_{5}^{4} - 15T_{5}^{2} + 48$$ T5^4 - 15*T5^2 + 48 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 9T^{2} + 12$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 15T^{2} + 48$$
$7$ $$(T^{2} - T - 8)^{2}$$
$11$ $$T^{4} - 27T^{2} + 108$$
$13$ $$(T + 2)^{4}$$
$17$ $$T^{4} - 15T^{2} + 48$$
$19$ $$T^{4}$$
$23$ $$T^{4} - 48T^{2} + 48$$
$29$ $$T^{4} - 48T^{2} + 48$$
$31$ $$(T^{2} - 2 T - 32)^{2}$$
$37$ $$(T^{2} + 10 T - 8)^{2}$$
$41$ $$T^{4} - 36T^{2} + 192$$
$43$ $$(T^{2} + 5 T - 68)^{2}$$
$47$ $$T^{4} - 99T^{2} + 1452$$
$53$ $$T^{4} - 240 T^{2} + 13872$$
$59$ $$T^{4} - 144T^{2} + 3072$$
$61$ $$(T^{2} - 7 T - 62)^{2}$$
$67$ $$(T - 4)^{4}$$
$71$ $$T^{4} - 192T^{2} + 768$$
$73$ $$(T^{2} - 7 T - 62)^{2}$$
$79$ $$(T - 4)^{4}$$
$83$ $$T^{4} - 144T^{2} + 432$$
$89$ $$T^{4} - 240 T^{2} + 13872$$
$97$ $$(T^{2} - 8 T - 116)^{2}$$