# Properties

 Label 900.6.a.l Level $900$ Weight $6$ Character orbit 900.a Self dual yes Analytic conductor $144.345$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 900.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$144.345437832$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{94})$$ Defining polynomial: $$x^{2} - 94$$ x^2 - 94 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 3\cdot 5$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 60\sqrt{94}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 71 q^{7}+O(q^{10})$$ q - 71 * q^7 $$q - 71 q^{7} + \beta q^{11} - 137 q^{13} + \beta q^{17} - 1087 q^{19} + 7 \beta q^{23} - 7 \beta q^{29} - 2269 q^{31} + 6010 q^{37} + 3 \beta q^{41} - 4283 q^{43} + 2 \beta q^{47} - 11766 q^{49} - 43 \beta q^{53} + 50 \beta q^{59} + 12719 q^{61} - 6899 q^{67} - 69 \beta q^{71} + 24010 q^{73} - 71 \beta q^{77} + 12236 q^{79} + 46 \beta q^{83} - 96 \beta q^{89} + 9727 q^{91} + 19651 q^{97} +O(q^{100})$$ q - 71 * q^7 + b * q^11 - 137 * q^13 + b * q^17 - 1087 * q^19 + 7*b * q^23 - 7*b * q^29 - 2269 * q^31 + 6010 * q^37 + 3*b * q^41 - 4283 * q^43 + 2*b * q^47 - 11766 * q^49 - 43*b * q^53 + 50*b * q^59 + 12719 * q^61 - 6899 * q^67 - 69*b * q^71 + 24010 * q^73 - 71*b * q^77 + 12236 * q^79 + 46*b * q^83 - 96*b * q^89 + 9727 * q^91 + 19651 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 142 q^{7}+O(q^{10})$$ 2 * q - 142 * q^7 $$2 q - 142 q^{7} - 274 q^{13} - 2174 q^{19} - 4538 q^{31} + 12020 q^{37} - 8566 q^{43} - 23532 q^{49} + 25438 q^{61} - 13798 q^{67} + 48020 q^{73} + 24472 q^{79} + 19454 q^{91} + 39302 q^{97}+O(q^{100})$$ 2 * q - 142 * q^7 - 274 * q^13 - 2174 * q^19 - 4538 * q^31 + 12020 * q^37 - 8566 * q^43 - 23532 * q^49 + 25438 * q^61 - 13798 * q^67 + 48020 * q^73 + 24472 * q^79 + 19454 * q^91 + 39302 * 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
 −9.69536 9.69536
0 0 0 0 0 −71.0000 0 0 0
1.2 0 0 0 0 0 −71.0000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$

## 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 900.6.a.l 2
3.b odd 2 1 inner 900.6.a.l 2
5.b even 2 1 900.6.a.v yes 2
5.c odd 4 2 900.6.d.l 4
15.d odd 2 1 900.6.a.v yes 2
15.e even 4 2 900.6.d.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.6.a.l 2 1.a even 1 1 trivial
900.6.a.l 2 3.b odd 2 1 inner
900.6.a.v yes 2 5.b even 2 1
900.6.a.v yes 2 15.d odd 2 1
900.6.d.l 4 5.c odd 4 2
900.6.d.l 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(900))$$:

 $$T_{7} + 71$$ T7 + 71 $$T_{11}^{2} - 338400$$ T11^2 - 338400

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 71)^{2}$$
$11$ $$T^{2} - 338400$$
$13$ $$(T + 137)^{2}$$
$17$ $$T^{2} - 338400$$
$19$ $$(T + 1087)^{2}$$
$23$ $$T^{2} - 16581600$$
$29$ $$T^{2} - 16581600$$
$31$ $$(T + 2269)^{2}$$
$37$ $$(T - 6010)^{2}$$
$41$ $$T^{2} - 3045600$$
$43$ $$(T + 4283)^{2}$$
$47$ $$T^{2} - 1353600$$
$53$ $$T^{2} - 625701600$$
$59$ $$T^{2} - 846000000$$
$61$ $$(T - 12719)^{2}$$
$67$ $$(T + 6899)^{2}$$
$71$ $$T^{2} - 1611122400$$
$73$ $$(T - 24010)^{2}$$
$79$ $$(T - 12236)^{2}$$
$83$ $$T^{2} - 716054400$$
$89$ $$T^{2} - 3118694400$$
$97$ $$(T - 19651)^{2}$$