# Properties

 Label 900.6.a.o Level $900$ Weight $6$ Character orbit 900.a Self dual yes Analytic conductor $144.345$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

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## 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{61})$$ Defining polynomial: $$x^{2} - x - 15$$ x^2 - x - 15 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 180) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{61}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 8 \beta q^{7} +O(q^{10})$$ q - 8*b * q^7 $$q - 8 \beta q^{7} - 80 q^{11} + 24 \beta q^{13} + 35 \beta q^{17} - 12 q^{19} - 130 \beta q^{23} + 4560 q^{29} - 344 q^{31} + 280 \beta q^{37} - 14240 q^{41} + 1216 \beta q^{43} + 1570 \beta q^{47} - 1191 q^{49} - 1755 \beta q^{53} + 38000 q^{59} - 8206 q^{61} + 848 \beta q^{67} - 48480 q^{71} - 2720 \beta q^{73} + 640 \beta q^{77} - 9264 q^{79} - 2140 \beta q^{83} - 24320 q^{89} - 46848 q^{91} - 8752 \beta q^{97} +O(q^{100})$$ q - 8*b * q^7 - 80 * q^11 + 24*b * q^13 + 35*b * q^17 - 12 * q^19 - 130*b * q^23 + 4560 * q^29 - 344 * q^31 + 280*b * q^37 - 14240 * q^41 + 1216*b * q^43 + 1570*b * q^47 - 1191 * q^49 - 1755*b * q^53 + 38000 * q^59 - 8206 * q^61 + 848*b * q^67 - 48480 * q^71 - 2720*b * q^73 + 640*b * q^77 - 9264 * q^79 - 2140*b * q^83 - 24320 * q^89 - 46848 * q^91 - 8752*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 160 q^{11} - 24 q^{19} + 9120 q^{29} - 688 q^{31} - 28480 q^{41} - 2382 q^{49} + 76000 q^{59} - 16412 q^{61} - 96960 q^{71} - 18528 q^{79} - 48640 q^{89} - 93696 q^{91}+O(q^{100})$$ 2 * q - 160 * q^11 - 24 * q^19 + 9120 * q^29 - 688 * q^31 - 28480 * q^41 - 2382 * q^49 + 76000 * q^59 - 16412 * q^61 - 96960 * q^71 - 18528 * q^79 - 48640 * q^89 - 93696 * q^91

## 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
 4.40512 −3.40512
0 0 0 0 0 −124.964 0 0 0
1.2 0 0 0 0 0 124.964 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
5.b even 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.o 2
3.b odd 2 1 900.6.a.p 2
5.b even 2 1 inner 900.6.a.o 2
5.c odd 4 2 180.6.d.a 2
15.d odd 2 1 900.6.a.p 2
15.e even 4 2 180.6.d.c yes 2
20.e even 4 2 720.6.f.b 2
60.l odd 4 2 720.6.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.6.d.a 2 5.c odd 4 2
180.6.d.c yes 2 15.e even 4 2
720.6.f.b 2 20.e even 4 2
720.6.f.e 2 60.l odd 4 2
900.6.a.o 2 1.a even 1 1 trivial
900.6.a.o 2 5.b even 2 1 inner
900.6.a.p 2 3.b odd 2 1
900.6.a.p 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(900))$$:

 $$T_{7}^{2} - 15616$$ T7^2 - 15616 $$T_{11} + 80$$ T11 + 80

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 15616$$
$11$ $$(T + 80)^{2}$$
$13$ $$T^{2} - 140544$$
$17$ $$T^{2} - 298900$$
$19$ $$(T + 12)^{2}$$
$23$ $$T^{2} - 4123600$$
$29$ $$(T - 4560)^{2}$$
$31$ $$(T + 344)^{2}$$
$37$ $$T^{2} - 19129600$$
$41$ $$(T + 14240)^{2}$$
$43$ $$T^{2} - 360792064$$
$47$ $$T^{2} - 601435600$$
$53$ $$T^{2} - 751526100$$
$59$ $$(T - 38000)^{2}$$
$61$ $$(T + 8206)^{2}$$
$67$ $$T^{2} - 175461376$$
$71$ $$(T + 48480)^{2}$$
$73$ $$T^{2} - 1805209600$$
$79$ $$(T + 9264)^{2}$$
$83$ $$T^{2} - 1117422400$$
$89$ $$(T + 24320)^{2}$$
$97$ $$T^{2} - 18689790976$$
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