# Properties

 Label 6300.2.a.bg Level $6300$ Weight $2$ Character orbit 6300.a Self dual yes Analytic conductor $50.306$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6300.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$50.3057532734$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 3\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{7}+O(q^{10})$$ q - q^7 $$q - q^{7} + \beta q^{11} - 2 q^{13} + 2 q^{19} - \beta q^{23} + \beta q^{29} + 2 q^{31} + q^{37} - 5 q^{43} + 2 \beta q^{47} + q^{49} - 2 \beta q^{59} + 8 q^{61} + 7 q^{67} - \beta q^{71} - 8 q^{73} - \beta q^{77} + 5 q^{79} + 2 \beta q^{83} + 2 \beta q^{89} + 2 q^{91} + 10 q^{97} +O(q^{100})$$ q - q^7 + b * q^11 - 2 * q^13 + 2 * q^19 - b * q^23 + b * q^29 + 2 * q^31 + q^37 - 5 * q^43 + 2*b * q^47 + q^49 - 2*b * q^59 + 8 * q^61 + 7 * q^67 - b * q^71 - 8 * q^73 - b * q^77 + 5 * q^79 + 2*b * q^83 + 2*b * q^89 + 2 * q^91 + 10 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7}+O(q^{10})$$ 2 * q - 2 * q^7 $$2 q - 2 q^{7} - 4 q^{13} + 4 q^{19} + 4 q^{31} + 2 q^{37} - 10 q^{43} + 2 q^{49} + 16 q^{61} + 14 q^{67} - 16 q^{73} + 10 q^{79} + 4 q^{91} + 20 q^{97}+O(q^{100})$$ 2 * q - 2 * q^7 - 4 * q^13 + 4 * q^19 + 4 * q^31 + 2 * q^37 - 10 * q^43 + 2 * q^49 + 16 * q^61 + 14 * q^67 - 16 * q^73 + 10 * q^79 + 4 * q^91 + 20 * q^97

## 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
 −1.73205 1.73205
0 0 0 0 0 −1.00000 0 0 0
1.2 0 0 0 0 0 −1.00000 0 0 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

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 6300.2.a.bg 2
3.b odd 2 1 inner 6300.2.a.bg 2
5.b even 2 1 6300.2.a.bj yes 2
5.c odd 4 2 6300.2.k.s 4
15.d odd 2 1 6300.2.a.bj yes 2
15.e even 4 2 6300.2.k.s 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6300.2.a.bg 2 1.a even 1 1 trivial
6300.2.a.bg 2 3.b odd 2 1 inner
6300.2.a.bj yes 2 5.b even 2 1
6300.2.a.bj yes 2 15.d odd 2 1
6300.2.k.s 4 5.c odd 4 2
6300.2.k.s 4 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6300))$$:

 $$T_{11}^{2} - 27$$ T11^2 - 27 $$T_{13} + 2$$ T13 + 2 $$T_{17}$$ T17 $$T_{37} - 1$$ T37 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} - 27$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2}$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} - 27$$
$29$ $$T^{2} - 27$$
$31$ $$(T - 2)^{2}$$
$37$ $$(T - 1)^{2}$$
$41$ $$T^{2}$$
$43$ $$(T + 5)^{2}$$
$47$ $$T^{2} - 108$$
$53$ $$T^{2}$$
$59$ $$T^{2} - 108$$
$61$ $$(T - 8)^{2}$$
$67$ $$(T - 7)^{2}$$
$71$ $$T^{2} - 27$$
$73$ $$(T + 8)^{2}$$
$79$ $$(T - 5)^{2}$$
$83$ $$T^{2} - 108$$
$89$ $$T^{2} - 108$$
$97$ $$(T - 10)^{2}$$