# Properties

 Label 3360.2.a.bi Level $3360$ Weight $2$ Character orbit 3360.a Self dual yes Analytic conductor $26.830$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3360.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.8297350792$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} - q^{7} + q^{9}+O(q^{10})$$ q - q^3 + q^5 - q^7 + q^9 $$q - q^{3} + q^{5} - q^{7} + q^{9} + (\beta_{2} - 1) q^{11} + (\beta_1 + 1) q^{13} - q^{15} + ( - \beta_{2} - \beta_1) q^{17} + ( - \beta_{2} - 3) q^{19} + q^{21} + (\beta_{2} - \beta_1) q^{23} + q^{25} - q^{27} + ( - \beta_{2} - \beta_1) q^{29} + ( - 2 \beta_{2} + \beta_1 - 3) q^{31} + ( - \beta_{2} + 1) q^{33} - q^{35} + (\beta_{2} + \beta_1) q^{37} + ( - \beta_1 - 1) q^{39} + 2 \beta_{2} q^{41} + ( - \beta_{2} - \beta_1 - 2) q^{43} + q^{45} + ( - \beta_{2} + \beta_1) q^{47} + q^{49} + (\beta_{2} + \beta_1) q^{51} + (\beta_{2} - 3) q^{53} + (\beta_{2} - 1) q^{55} + (\beta_{2} + 3) q^{57} - 8 q^{59} + (\beta_{2} - \beta_1 - 2) q^{61} - q^{63} + (\beta_1 + 1) q^{65} + (3 \beta_{2} + \beta_1) q^{67} + ( - \beta_{2} + \beta_1) q^{69} + (\beta_1 - 1) q^{71} + ( - \beta_1 + 3) q^{73} - q^{75} + ( - \beta_{2} + 1) q^{77} - 4 q^{79} + q^{81} + ( - 2 \beta_{2} - 2) q^{83} + ( - \beta_{2} - \beta_1) q^{85} + (\beta_{2} + \beta_1) q^{87} + 2 \beta_1 q^{89} + ( - \beta_1 - 1) q^{91} + (2 \beta_{2} - \beta_1 + 3) q^{93} + ( - \beta_{2} - 3) q^{95} + (\beta_1 + 1) q^{97} + (\beta_{2} - 1) q^{99}+O(q^{100})$$ q - q^3 + q^5 - q^7 + q^9 + (b2 - 1) * q^11 + (b1 + 1) * q^13 - q^15 + (-b2 - b1) * q^17 + (-b2 - 3) * q^19 + q^21 + (b2 - b1) * q^23 + q^25 - q^27 + (-b2 - b1) * q^29 + (-2*b2 + b1 - 3) * q^31 + (-b2 + 1) * q^33 - q^35 + (b2 + b1) * q^37 + (-b1 - 1) * q^39 + 2*b2 * q^41 + (-b2 - b1 - 2) * q^43 + q^45 + (-b2 + b1) * q^47 + q^49 + (b2 + b1) * q^51 + (b2 - 3) * q^53 + (b2 - 1) * q^55 + (b2 + 3) * q^57 - 8 * q^59 + (b2 - b1 - 2) * q^61 - q^63 + (b1 + 1) * q^65 + (3*b2 + b1) * q^67 + (-b2 + b1) * q^69 + (b1 - 1) * q^71 + (-b1 + 3) * q^73 - q^75 + (-b2 + 1) * q^77 - 4 * q^79 + q^81 + (-2*b2 - 2) * q^83 + (-b2 - b1) * q^85 + (b2 + b1) * q^87 + 2*b1 * q^89 + (-b1 - 1) * q^91 + (2*b2 - b1 + 3) * q^93 + (-b2 - 3) * q^95 + (b1 + 1) * q^97 + (b2 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9 $$3 q - 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9} - 4 q^{11} + 2 q^{13} - 3 q^{15} + 2 q^{17} - 8 q^{19} + 3 q^{21} + 3 q^{25} - 3 q^{27} + 2 q^{29} - 8 q^{31} + 4 q^{33} - 3 q^{35} - 2 q^{37} - 2 q^{39} - 2 q^{41} - 4 q^{43} + 3 q^{45} + 3 q^{49} - 2 q^{51} - 10 q^{53} - 4 q^{55} + 8 q^{57} - 24 q^{59} - 6 q^{61} - 3 q^{63} + 2 q^{65} - 4 q^{67} - 4 q^{71} + 10 q^{73} - 3 q^{75} + 4 q^{77} - 12 q^{79} + 3 q^{81} - 4 q^{83} + 2 q^{85} - 2 q^{87} - 2 q^{89} - 2 q^{91} + 8 q^{93} - 8 q^{95} + 2 q^{97} - 4 q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9 - 4 * q^11 + 2 * q^13 - 3 * q^15 + 2 * q^17 - 8 * q^19 + 3 * q^21 + 3 * q^25 - 3 * q^27 + 2 * q^29 - 8 * q^31 + 4 * q^33 - 3 * q^35 - 2 * q^37 - 2 * q^39 - 2 * q^41 - 4 * q^43 + 3 * q^45 + 3 * q^49 - 2 * q^51 - 10 * q^53 - 4 * q^55 + 8 * q^57 - 24 * q^59 - 6 * q^61 - 3 * q^63 + 2 * q^65 - 4 * q^67 - 4 * q^71 + 10 * q^73 - 3 * q^75 + 4 * q^77 - 12 * q^79 + 3 * q^81 - 4 * q^83 + 2 * q^85 - 2 * q^87 - 2 * q^89 - 2 * q^91 + 8 * q^93 - 8 * q^95 + 2 * q^97 - 4 * q^99

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

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

## 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
 0.311108 −1.48119 2.17009
0 −1.00000 0 1.00000 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −1.00000 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 −1.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## 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 3360.2.a.bi 3
4.b odd 2 1 3360.2.a.bj yes 3
8.b even 2 1 6720.2.a.db 3
8.d odd 2 1 6720.2.a.da 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.2.a.bi 3 1.a even 1 1 trivial
3360.2.a.bj yes 3 4.b odd 2 1
6720.2.a.da 3 8.d odd 2 1
6720.2.a.db 3 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3360))$$:

 $$T_{11}^{3} + 4T_{11}^{2} - 16T_{11} - 32$$ T11^3 + 4*T11^2 - 16*T11 - 32 $$T_{13}^{3} - 2T_{13}^{2} - 36T_{13} + 104$$ T13^3 - 2*T13^2 - 36*T13 + 104 $$T_{17}^{3} - 2T_{17}^{2} - 52T_{17} + 40$$ T17^3 - 2*T17^2 - 52*T17 + 40 $$T_{19}^{3} + 8T_{19}^{2} - 32$$ T19^3 + 8*T19^2 - 32 $$T_{23}^{3} - 64T_{23} + 128$$ T23^3 - 64*T23 + 128

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T + 1)^{3}$$
$5$ $$(T - 1)^{3}$$
$7$ $$(T + 1)^{3}$$
$11$ $$T^{3} + 4 T^{2} - 16 T - 32$$
$13$ $$T^{3} - 2 T^{2} - 36 T + 104$$
$17$ $$T^{3} - 2 T^{2} - 52 T + 40$$
$19$ $$T^{3} + 8T^{2} - 32$$
$23$ $$T^{3} - 64T + 128$$
$29$ $$T^{3} - 2 T^{2} - 52 T + 40$$
$31$ $$T^{3} + 8 T^{2} - 112 T - 928$$
$37$ $$T^{3} + 2 T^{2} - 52 T - 40$$
$41$ $$T^{3} + 2 T^{2} - 84 T - 104$$
$43$ $$T^{3} + 4 T^{2} - 48 T - 64$$
$47$ $$T^{3} - 64T - 128$$
$53$ $$T^{3} + 10 T^{2} + 12 T - 40$$
$59$ $$(T + 8)^{3}$$
$61$ $$T^{3} + 6 T^{2} - 52 T + 8$$
$67$ $$T^{3} + 4 T^{2} - 208 T - 1472$$
$71$ $$T^{3} + 4 T^{2} - 32 T + 32$$
$73$ $$T^{3} - 10 T^{2} - 4 T + 8$$
$79$ $$(T + 4)^{3}$$
$83$ $$T^{3} + 4 T^{2} - 80 T - 64$$
$89$ $$T^{3} + 2 T^{2} - 148 T + 536$$
$97$ $$T^{3} - 2 T^{2} - 36 T + 104$$