# Properties

 Label 900.6.a.t Level $900$ Weight $6$ Character orbit 900.a Self dual yes Analytic conductor $144.345$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# 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: $$\Q(\sqrt{241})$$ Defining polynomial: $$x^{2} - x - 60$$ x^2 - x - 60 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ 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 $$\beta = 6\sqrt{241}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 40) q^{7}+O(q^{10})$$ q + (-b + 40) * q^7 $$q + ( - \beta + 40) q^{7} + ( - 5 \beta - 120) q^{11} + (8 \beta + 340) q^{13} + ( - 20 \beta + 270) q^{17} + (20 \beta + 548) q^{19} + (10 \beta + 1740) q^{23} + ( - 40 \beta - 4710) q^{29} + ( - 70 \beta + 1496) q^{31} + ( - 60 \beta + 3760) q^{37} + (60 \beta - 13560) q^{41} + ( - 88 \beta - 2360) q^{43} + (110 \beta + 19140) q^{47} + ( - 80 \beta - 6531) q^{49} + ( - 40 \beta + 26790) q^{53} + (275 \beta - 23400) q^{59} + (120 \beta + 10754) q^{61} + ( - 54 \beta + 5440) q^{67} + ( - 330 \beta - 22920) q^{71} + (240 \beta - 4790) q^{73} + ( - 80 \beta + 38580) q^{77} + (150 \beta + 45536) q^{79} + ( - 170 \beta - 7080) q^{83} + ( - 420 \beta - 17580) q^{89} + ( - 20 \beta - 55808) q^{91} + (1096 \beta + 49390) q^{97}+O(q^{100})$$ q + (-b + 40) * q^7 + (-5*b - 120) * q^11 + (8*b + 340) * q^13 + (-20*b + 270) * q^17 + (20*b + 548) * q^19 + (10*b + 1740) * q^23 + (-40*b - 4710) * q^29 + (-70*b + 1496) * q^31 + (-60*b + 3760) * q^37 + (60*b - 13560) * q^41 + (-88*b - 2360) * q^43 + (110*b + 19140) * q^47 + (-80*b - 6531) * q^49 + (-40*b + 26790) * q^53 + (275*b - 23400) * q^59 + (120*b + 10754) * q^61 + (-54*b + 5440) * q^67 + (-330*b - 22920) * q^71 + (240*b - 4790) * q^73 + (-80*b + 38580) * q^77 + (150*b + 45536) * q^79 + (-170*b - 7080) * q^83 + (-420*b - 17580) * q^89 + (-20*b - 55808) * q^91 + (1096*b + 49390) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 80 q^{7}+O(q^{10})$$ 2 * q + 80 * q^7 $$2 q + 80 q^{7} - 240 q^{11} + 680 q^{13} + 540 q^{17} + 1096 q^{19} + 3480 q^{23} - 9420 q^{29} + 2992 q^{31} + 7520 q^{37} - 27120 q^{41} - 4720 q^{43} + 38280 q^{47} - 13062 q^{49} + 53580 q^{53} - 46800 q^{59} + 21508 q^{61} + 10880 q^{67} - 45840 q^{71} - 9580 q^{73} + 77160 q^{77} + 91072 q^{79} - 14160 q^{83} - 35160 q^{89} - 111616 q^{91} + 98780 q^{97}+O(q^{100})$$ 2 * q + 80 * q^7 - 240 * q^11 + 680 * q^13 + 540 * q^17 + 1096 * q^19 + 3480 * q^23 - 9420 * q^29 + 2992 * q^31 + 7520 * q^37 - 27120 * q^41 - 4720 * q^43 + 38280 * q^47 - 13062 * q^49 + 53580 * q^53 - 46800 * q^59 + 21508 * q^61 + 10880 * q^67 - 45840 * q^71 - 9580 * q^73 + 77160 * q^77 + 91072 * q^79 - 14160 * q^83 - 35160 * q^89 - 111616 * q^91 + 98780 * 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
 8.26209 −7.26209
0 0 0 0 0 −53.1450 0 0 0
1.2 0 0 0 0 0 133.145 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

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 900.6.a.t 2
3.b odd 2 1 900.6.a.u 2
5.b even 2 1 180.6.a.g yes 2
5.c odd 4 2 900.6.d.k 4
15.d odd 2 1 180.6.a.f 2
15.e even 4 2 900.6.d.n 4
20.d odd 2 1 720.6.a.bg 2
60.h even 2 1 720.6.a.bc 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.6.a.f 2 15.d odd 2 1
180.6.a.g yes 2 5.b even 2 1
720.6.a.bc 2 60.h even 2 1
720.6.a.bg 2 20.d odd 2 1
900.6.a.t 2 1.a even 1 1 trivial
900.6.a.u 2 3.b odd 2 1
900.6.d.k 4 5.c odd 4 2
900.6.d.n 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(900))$$:

 $$T_{7}^{2} - 80T_{7} - 7076$$ T7^2 - 80*T7 - 7076 $$T_{11}^{2} + 240T_{11} - 202500$$ T11^2 + 240*T11 - 202500

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 80T - 7076$$
$11$ $$T^{2} + 240T - 202500$$
$13$ $$T^{2} - 680T - 439664$$
$17$ $$T^{2} - 540 T - 3397500$$
$19$ $$T^{2} - 1096 T - 3170096$$
$23$ $$T^{2} - 3480 T + 2160000$$
$29$ $$T^{2} + 9420 T + 8302500$$
$31$ $$T^{2} - 2992 T - 40274384$$
$37$ $$T^{2} - 7520 T - 17096000$$
$41$ $$T^{2} + 27120 T + 152640000$$
$43$ $$T^{2} + 4720 T - 61617344$$
$47$ $$T^{2} - 38280 T + 261360000$$
$53$ $$T^{2} - 53580 T + 703822500$$
$59$ $$T^{2} + 46800 T - 108562500$$
$61$ $$T^{2} - 21508 T - 9285884$$
$67$ $$T^{2} - 10880 T + 4294384$$
$71$ $$T^{2} + 45840 T - 419490000$$
$73$ $$T^{2} + 9580 T - 476793500$$
$79$ $$T^{2} - 91072 T + 1878317296$$
$83$ $$T^{2} + 14160 T - 200610000$$
$89$ $$T^{2} + 35160 T - 1221390000$$
$97$ $$T^{2} - 98780 T - 7982377916$$