# Properties

 Label 3249.2.a.t Level $3249$ Weight $2$ Character orbit 3249.a Self dual yes Analytic conductor $25.943$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.564.1 Defining polynomial: $$x^{3} - x^{2} - 5x + 3$$ x^3 - x^2 - 5*x + 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 57) 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{2} - 1) q^{5} + \beta_{2} q^{7} + ( - \beta_{2} - \beta_1 - 1) q^{8}+O(q^{10})$$ q - b1 * q^2 + (b2 + 2) * q^4 + (-b2 - 1) * q^5 + b2 * q^7 + (-b2 - b1 - 1) * q^8 $$q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{2} - 1) q^{5} + \beta_{2} q^{7} + ( - \beta_{2} - \beta_1 - 1) q^{8} + (\beta_{2} + 2 \beta_1 + 1) q^{10} + ( - \beta_{2} + 2 \beta_1 - 1) q^{11} + ( - 2 \beta_1 + 1) q^{13} + ( - \beta_{2} - \beta_1 - 1) q^{14} + (2 \beta_1 + 1) q^{16} + ( - \beta_{2} - 2 \beta_1 - 7) q^{20} + ( - \beta_{2} + 2 \beta_1 - 7) q^{22} + (\beta_{2} + 2 \beta_1 - 5) q^{23} + (2 \beta_1 + 1) q^{25} + (2 \beta_{2} - \beta_1 + 8) q^{26} + (2 \beta_1 + 5) q^{28} + (2 \beta_{2} + 2) q^{29} + (\beta_{2} - 2 \beta_1 - 4) q^{31} + (\beta_1 - 6) q^{32} + (\beta_{2} - 2 \beta_1 - 5) q^{35} + q^{37} + (\beta_{2} + 4 \beta_1 + 7) q^{40} + ( - 2 \beta_{2} - 2) q^{41} + ( - \beta_{2} + 2 \beta_1 - 2) q^{43} + (\beta_{2} + 4 \beta_1 - 5) q^{44} + ( - 3 \beta_{2} + 4 \beta_1 - 9) q^{46} + 6 q^{47} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{49} + ( - 2 \beta_{2} - \beta_1 - 8) q^{50} + ( - \beta_{2} - 6 \beta_1) q^{52} + ( - 3 \beta_{2} - 3) q^{53} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{55} + ( - 3 \beta_1 - 6) q^{56} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{58} + ( - \beta_{2} + 2 \beta_1 - 1) q^{59} + ( - 2 \beta_{2} + 2 \beta_1 + 3) q^{61} + (\beta_{2} + 3 \beta_1 + 7) q^{62} + ( - \beta_{2} + 2 \beta_1 - 6) q^{64} + (\beta_{2} + 4 \beta_1 + 1) q^{65} + (\beta_{2} + 4 \beta_1 - 4) q^{67} + (\beta_{2} + 4 \beta_1 + 7) q^{70} + (2 \beta_{2} - 4 \beta_1 - 4) q^{71} + (2 \beta_{2} + 7) q^{73} - \beta_1 q^{74} + (3 \beta_{2} - 3) q^{77} + (\beta_{2} + 2 \beta_1 - 4) q^{79} + ( - 3 \beta_{2} - 4 \beta_1 - 3) q^{80} + (2 \beta_{2} + 4 \beta_1 + 2) q^{82} + 4 \beta_1 q^{83} + ( - \beta_{2} + 3 \beta_1 - 7) q^{86} + ( - 3 \beta_{2} - 3) q^{88} + (\beta_{2} - 5) q^{89} + ( - \beta_{2} - 2 \beta_1 - 2) q^{91} + ( - 3 \beta_{2} + 8 \beta_1 - 3) q^{92} - 6 \beta_1 q^{94} + ( - 2 \beta_{2} - 6 \beta_1 + 2) q^{97} + (4 \beta_1 - 6) q^{98}+O(q^{100})$$ q - b1 * q^2 + (b2 + 2) * q^4 + (-b2 - 1) * q^5 + b2 * q^7 + (-b2 - b1 - 1) * q^8 + (b2 + 2*b1 + 1) * q^10 + (-b2 + 2*b1 - 1) * q^11 + (-2*b1 + 1) * q^13 + (-b2 - b1 - 1) * q^14 + (2*b1 + 1) * q^16 + (-b2 - 2*b1 - 7) * q^20 + (-b2 + 2*b1 - 7) * q^22 + (b2 + 2*b1 - 5) * q^23 + (2*b1 + 1) * q^25 + (2*b2 - b1 + 8) * q^26 + (2*b1 + 5) * q^28 + (2*b2 + 2) * q^29 + (b2 - 2*b1 - 4) * q^31 + (b1 - 6) * q^32 + (b2 - 2*b1 - 5) * q^35 + q^37 + (b2 + 4*b1 + 7) * q^40 + (-2*b2 - 2) * q^41 + (-b2 + 2*b1 - 2) * q^43 + (b2 + 4*b1 - 5) * q^44 + (-3*b2 + 4*b1 - 9) * q^46 + 6 * q^47 + (-2*b2 + 2*b1 - 2) * q^49 + (-2*b2 - b1 - 8) * q^50 + (-b2 - 6*b1) * q^52 + (-3*b2 - 3) * q^53 + (-2*b2 - 2*b1 + 4) * q^55 + (-3*b1 - 6) * q^56 + (-2*b2 - 4*b1 - 2) * q^58 + (-b2 + 2*b1 - 1) * q^59 + (-2*b2 + 2*b1 + 3) * q^61 + (b2 + 3*b1 + 7) * q^62 + (-b2 + 2*b1 - 6) * q^64 + (b2 + 4*b1 + 1) * q^65 + (b2 + 4*b1 - 4) * q^67 + (b2 + 4*b1 + 7) * q^70 + (2*b2 - 4*b1 - 4) * q^71 + (2*b2 + 7) * q^73 - b1 * q^74 + (3*b2 - 3) * q^77 + (b2 + 2*b1 - 4) * q^79 + (-3*b2 - 4*b1 - 3) * q^80 + (2*b2 + 4*b1 + 2) * q^82 + 4*b1 * q^83 + (-b2 + 3*b1 - 7) * q^86 + (-3*b2 - 3) * q^88 + (b2 - 5) * q^89 + (-b2 - 2*b1 - 2) * q^91 + (-3*b2 + 8*b1 - 3) * q^92 - 6*b1 * q^94 + (-2*b2 - 6*b1 + 2) * q^97 + (4*b1 - 6) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 5 q^{4} - 2 q^{5} - q^{7} - 3 q^{8}+O(q^{10})$$ 3 * q - q^2 + 5 * q^4 - 2 * q^5 - q^7 - 3 * q^8 $$3 q - q^{2} + 5 q^{4} - 2 q^{5} - q^{7} - 3 q^{8} + 4 q^{10} + q^{13} - 3 q^{14} + 5 q^{16} - 22 q^{20} - 18 q^{22} - 14 q^{23} + 5 q^{25} + 21 q^{26} + 17 q^{28} + 4 q^{29} - 15 q^{31} - 17 q^{32} - 18 q^{35} + 3 q^{37} + 24 q^{40} - 4 q^{41} - 3 q^{43} - 12 q^{44} - 20 q^{46} + 18 q^{47} - 2 q^{49} - 23 q^{50} - 5 q^{52} - 6 q^{53} + 12 q^{55} - 21 q^{56} - 8 q^{58} + 13 q^{61} + 23 q^{62} - 15 q^{64} + 6 q^{65} - 9 q^{67} + 24 q^{70} - 18 q^{71} + 19 q^{73} - q^{74} - 12 q^{77} - 11 q^{79} - 10 q^{80} + 8 q^{82} + 4 q^{83} - 17 q^{86} - 6 q^{88} - 16 q^{89} - 7 q^{91} + 2 q^{92} - 6 q^{94} + 2 q^{97} - 14 q^{98}+O(q^{100})$$ 3 * q - q^2 + 5 * q^4 - 2 * q^5 - q^7 - 3 * q^8 + 4 * q^10 + q^13 - 3 * q^14 + 5 * q^16 - 22 * q^20 - 18 * q^22 - 14 * q^23 + 5 * q^25 + 21 * q^26 + 17 * q^28 + 4 * q^29 - 15 * q^31 - 17 * q^32 - 18 * q^35 + 3 * q^37 + 24 * q^40 - 4 * q^41 - 3 * q^43 - 12 * q^44 - 20 * q^46 + 18 * q^47 - 2 * q^49 - 23 * q^50 - 5 * q^52 - 6 * q^53 + 12 * q^55 - 21 * q^56 - 8 * q^58 + 13 * q^61 + 23 * q^62 - 15 * q^64 + 6 * q^65 - 9 * q^67 + 24 * q^70 - 18 * q^71 + 19 * q^73 - q^74 - 12 * q^77 - 11 * q^79 - 10 * q^80 + 8 * q^82 + 4 * q^83 - 17 * q^86 - 6 * q^88 - 16 * q^89 - 7 * q^91 + 2 * q^92 - 6 * q^94 + 2 * q^97 - 14 * q^98

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 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
 2.51414 0.571993 −2.08613
−2.51414 0 4.32088 −3.32088 0 2.32088 −5.83502 0 8.34916
1.2 −0.571993 0 −1.67282 2.67282 0 −3.67282 2.10083 0 −1.52884
1.3 2.08613 0 2.35194 −1.35194 0 0.351939 0.734191 0 −2.82032
 $$n$$: 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

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 3249.2.a.t 3
3.b odd 2 1 1083.2.a.o 3
19.b odd 2 1 3249.2.a.y 3
19.d odd 6 2 171.2.f.b 6
57.d even 2 1 1083.2.a.l 3
57.f even 6 2 57.2.e.b 6
76.f even 6 2 2736.2.s.z 6
228.n odd 6 2 912.2.q.l 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.e.b 6 57.f even 6 2
171.2.f.b 6 19.d odd 6 2
912.2.q.l 6 228.n odd 6 2
1083.2.a.l 3 57.d even 2 1
1083.2.a.o 3 3.b odd 2 1
2736.2.s.z 6 76.f even 6 2
3249.2.a.t 3 1.a even 1 1 trivial
3249.2.a.y 3 19.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3249))$$:

 $$T_{2}^{3} + T_{2}^{2} - 5T_{2} - 3$$ T2^3 + T2^2 - 5*T2 - 3 $$T_{5}^{3} + 2T_{5}^{2} - 8T_{5} - 12$$ T5^3 + 2*T5^2 - 8*T5 - 12 $$T_{13}^{3} - T_{13}^{2} - 21T_{13} - 3$$ T13^3 - T13^2 - 21*T13 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + T^{2} - 5T - 3$$
$3$ $$T^{3}$$
$5$ $$T^{3} + 2 T^{2} - 8 T - 12$$
$7$ $$T^{3} + T^{2} - 9T + 3$$
$11$ $$T^{3} - 24T + 36$$
$13$ $$T^{3} - T^{2} - 21T - 3$$
$17$ $$T^{3}$$
$19$ $$T^{3}$$
$23$ $$T^{3} + 14 T^{2} + 28 T - 156$$
$29$ $$T^{3} - 4 T^{2} - 32 T + 96$$
$31$ $$T^{3} + 15 T^{2} + 51 T - 31$$
$37$ $$(T - 1)^{3}$$
$41$ $$T^{3} + 4 T^{2} - 32 T - 96$$
$43$ $$T^{3} + 3 T^{2} - 21 T + 13$$
$47$ $$(T - 6)^{3}$$
$53$ $$T^{3} + 6 T^{2} - 72 T - 324$$
$59$ $$T^{3} - 24T + 36$$
$61$ $$T^{3} - 13 T^{2} + 11 T + 73$$
$67$ $$T^{3} + 9 T^{2} - 81 T - 541$$
$71$ $$T^{3} + 18 T^{2} + 12 T - 648$$
$73$ $$T^{3} - 19 T^{2} + 83 T + 31$$
$79$ $$T^{3} + 11 T^{2} + 3 T - 171$$
$83$ $$T^{3} - 4 T^{2} - 80 T + 192$$
$89$ $$T^{3} + 16 T^{2} + 76 T + 108$$
$97$ $$T^{3} - 2 T^{2} - 268 T + 1448$$