# Properties

 Label 6900.2.a.x Level $6900$ Weight $2$ Character orbit 6900.a Self dual yes Analytic conductor $55.097$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6900.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$55.0967773947$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.3144.1 Defining polynomial: $$x^{3} - x^{2} - 16x - 8$$ x^3 - x^2 - 16*x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1380) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + (\beta_1 - 1) q^{7} + q^{9}+O(q^{10})$$ q - q^3 + (b1 - 1) * q^7 + q^9 $$q - q^{3} + (\beta_1 - 1) q^{7} + q^{9} + (\beta_{2} + 1) q^{11} + (\beta_{2} - 1) q^{13} + (\beta_{2} - \beta_1) q^{17} + (\beta_{2} + 3) q^{19} + ( - \beta_1 + 1) q^{21} - q^{23} - q^{27} + (\beta_{2} - \beta_1) q^{29} + (\beta_1 + 3) q^{31} + ( - \beta_{2} - 1) q^{33} + (2 \beta_{2} - \beta_1 - 1) q^{37} + ( - \beta_{2} + 1) q^{39} + ( - \beta_{2} + 3 \beta_1 + 2) q^{41} + (2 \beta_1 - 6) q^{43} + ( - \beta_{2} + 2 \beta_1 + 1) q^{47} + (2 \beta_{2} - \beta_1 + 4) q^{49} + ( - \beta_{2} + \beta_1) q^{51} + (\beta_{2} - 3 \beta_1 - 2) q^{53} + ( - \beta_{2} - 3) q^{57} + ( - \beta_{2} + \beta_1) q^{59} + ( - \beta_{2} + 2 \beta_1 + 3) q^{61} + (\beta_1 - 1) q^{63} + ( - \beta_1 - 3) q^{67} + q^{69} + (\beta_{2} - 3 \beta_1 - 2) q^{71} + ( - 3 \beta_{2} - 5) q^{73} + ( - \beta_{2} + 4 \beta_1 + 3) q^{77} + (2 \beta_1 + 4) q^{79} + q^{81} + (\beta_{2} + \beta_1 - 4) q^{83} + ( - \beta_{2} + \beta_1) q^{87} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{89} + ( - \beta_{2} + 2 \beta_1 + 5) q^{91} + ( - \beta_1 - 3) q^{93} + ( - 2 \beta_{2} + 4 \beta_1 + 6) q^{97} + (\beta_{2} + 1) q^{99}+O(q^{100})$$ q - q^3 + (b1 - 1) * q^7 + q^9 + (b2 + 1) * q^11 + (b2 - 1) * q^13 + (b2 - b1) * q^17 + (b2 + 3) * q^19 + (-b1 + 1) * q^21 - q^23 - q^27 + (b2 - b1) * q^29 + (b1 + 3) * q^31 + (-b2 - 1) * q^33 + (2*b2 - b1 - 1) * q^37 + (-b2 + 1) * q^39 + (-b2 + 3*b1 + 2) * q^41 + (2*b1 - 6) * q^43 + (-b2 + 2*b1 + 1) * q^47 + (2*b2 - b1 + 4) * q^49 + (-b2 + b1) * q^51 + (b2 - 3*b1 - 2) * q^53 + (-b2 - 3) * q^57 + (-b2 + b1) * q^59 + (-b2 + 2*b1 + 3) * q^61 + (b1 - 1) * q^63 + (-b1 - 3) * q^67 + q^69 + (b2 - 3*b1 - 2) * q^71 + (-3*b2 - 5) * q^73 + (-b2 + 4*b1 + 3) * q^77 + (2*b1 + 4) * q^79 + q^81 + (b2 + b1 - 4) * q^83 + (-b2 + b1) * q^87 + (-2*b2 - 2*b1 + 2) * q^89 + (-b2 + 2*b1 + 5) * q^91 + (-b1 - 3) * q^93 + (-2*b2 + 4*b1 + 6) * q^97 + (b2 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 - 2 * q^7 + 3 * q^9 $$3 q - 3 q^{3} - 2 q^{7} + 3 q^{9} + 4 q^{11} - 2 q^{13} + 10 q^{19} + 2 q^{21} - 3 q^{23} - 3 q^{27} + 10 q^{31} - 4 q^{33} - 2 q^{37} + 2 q^{39} + 8 q^{41} - 16 q^{43} + 4 q^{47} + 13 q^{49} - 8 q^{53} - 10 q^{57} + 10 q^{61} - 2 q^{63} - 10 q^{67} + 3 q^{69} - 8 q^{71} - 18 q^{73} + 12 q^{77} + 14 q^{79} + 3 q^{81} - 10 q^{83} + 2 q^{89} + 16 q^{91} - 10 q^{93} + 20 q^{97} + 4 q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 - 2 * q^7 + 3 * q^9 + 4 * q^11 - 2 * q^13 + 10 * q^19 + 2 * q^21 - 3 * q^23 - 3 * q^27 + 10 * q^31 - 4 * q^33 - 2 * q^37 + 2 * q^39 + 8 * q^41 - 16 * q^43 + 4 * q^47 + 13 * q^49 - 8 * q^53 - 10 * q^57 + 10 * q^61 - 2 * q^63 - 10 * q^67 + 3 * q^69 - 8 * q^71 - 18 * q^73 + 12 * q^77 + 14 * q^79 + 3 * q^81 - 10 * q^83 + 2 * q^89 + 16 * q^91 - 10 * q^93 + 20 * q^97 + 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 16x - 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} - \nu - 10 ) / 2$$ (v^2 - v - 10) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2} + \beta _1 + 10$$ 2*b2 + b1 + 10

## 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
 −3.20905 −0.526440 4.73549
0 −1.00000 0 0 0 −4.20905 0 1.00000 0
1.2 0 −1.00000 0 0 0 −1.52644 0 1.00000 0
1.3 0 −1.00000 0 0 0 3.73549 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$23$$ $$1$$

## 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 6900.2.a.x 3
5.b even 2 1 1380.2.a.j 3
5.c odd 4 2 6900.2.f.r 6
15.d odd 2 1 4140.2.a.s 3
20.d odd 2 1 5520.2.a.bv 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.a.j 3 5.b even 2 1
4140.2.a.s 3 15.d odd 2 1
5520.2.a.bv 3 20.d odd 2 1
6900.2.a.x 3 1.a even 1 1 trivial
6900.2.f.r 6 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6900))$$:

 $$T_{7}^{3} + 2T_{7}^{2} - 15T_{7} - 24$$ T7^3 + 2*T7^2 - 15*T7 - 24 $$T_{11}^{3} - 4T_{11}^{2} - 14T_{11} + 48$$ T11^3 - 4*T11^2 - 14*T11 + 48

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T + 1)^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} + 2 T^{2} - 15 T - 24$$
$11$ $$T^{3} - 4 T^{2} - 14 T + 48$$
$13$ $$T^{3} + 2 T^{2} - 18 T + 12$$
$17$ $$T^{3} - 21T - 18$$
$19$ $$T^{3} - 10 T^{2} + 14 T + 52$$
$23$ $$(T + 1)^{3}$$
$29$ $$T^{3} - 21T - 18$$
$31$ $$T^{3} - 10 T^{2} + 17 T + 4$$
$37$ $$T^{3} + 2 T^{2} - 63 T + 108$$
$41$ $$T^{3} - 8 T^{2} - 101 T + 582$$
$43$ $$T^{3} + 16 T^{2} + 20 T - 304$$
$47$ $$T^{3} - 4 T^{2} - 50 T + 216$$
$53$ $$T^{3} + 8 T^{2} - 101 T - 582$$
$59$ $$T^{3} - 21T + 18$$
$61$ $$T^{3} - 10 T^{2} - 22 T + 292$$
$67$ $$T^{3} + 10 T^{2} + 17 T - 4$$
$71$ $$T^{3} + 8 T^{2} - 101 T - 582$$
$73$ $$T^{3} + 18 T^{2} - 66 T - 1492$$
$79$ $$T^{3} - 14T^{2} + 96$$
$83$ $$T^{3} + 10 T^{2} - 17 T - 228$$
$89$ $$T^{3} - 2 T^{2} - 200 T + 912$$
$97$ $$T^{3} - 20 T^{2} - 88 T + 2336$$