# Properties

 Label 900.6.a.n Level $900$ Weight $6$ Character orbit 900.a Self dual yes Analytic conductor $144.345$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3\cdot 5$$ Twist minimal: no (minimal twist has level 300) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 60\sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 11) q^{7}+O(q^{10})$$ q + (b - 11) * q^7 $$q + (\beta - 11) q^{7} + ( - 3 \beta - 186) q^{11} + (4 \beta - 5) q^{13} + (3 \beta + 606) q^{17} + ( - 11 \beta - 775) q^{19} + (3 \beta + 1326) q^{23} + ( - 45 \beta + 906) q^{29} + (\beta + 5459) q^{31} + (18 \beta - 1658) q^{37} + ( - 3 \beta - 9540) q^{41} + ( - 29 \beta - 6131) q^{43} + (72 \beta - 7626) q^{47} + ( - 22 \beta + 8514) q^{49} + ( - 165 \beta + 10392) q^{53} + (48 \beta - 9354) q^{59} + ( - 42 \beta + 17867) q^{61} + ( - 3 \beta + 49081) q^{67} + ( - 75 \beta - 57060) q^{71} + ( - 162 \beta - 54938) q^{73} + ( - 153 \beta - 73554) q^{77} + (216 \beta + 47768) q^{79} + (528 \beta + 1866) q^{83} + (132 \beta - 46800) q^{89} + ( - 49 \beta + 100855) q^{91} + ( - 352 \beta + 45943) q^{97}+O(q^{100})$$ q + (b - 11) * q^7 + (-3*b - 186) * q^11 + (4*b - 5) * q^13 + (3*b + 606) * q^17 + (-11*b - 775) * q^19 + (3*b + 1326) * q^23 + (-45*b + 906) * q^29 + (b + 5459) * q^31 + (18*b - 1658) * q^37 + (-3*b - 9540) * q^41 + (-29*b - 6131) * q^43 + (72*b - 7626) * q^47 + (-22*b + 8514) * q^49 + (-165*b + 10392) * q^53 + (48*b - 9354) * q^59 + (-42*b + 17867) * q^61 + (-3*b + 49081) * q^67 + (-75*b - 57060) * q^71 + (-162*b - 54938) * q^73 + (-153*b - 73554) * q^77 + (216*b + 47768) * q^79 + (528*b + 1866) * q^83 + (132*b - 46800) * q^89 + (-49*b + 100855) * q^91 + (-352*b + 45943) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 22 q^{7}+O(q^{10})$$ 2 * q - 22 * q^7 $$2 q - 22 q^{7} - 372 q^{11} - 10 q^{13} + 1212 q^{17} - 1550 q^{19} + 2652 q^{23} + 1812 q^{29} + 10918 q^{31} - 3316 q^{37} - 19080 q^{41} - 12262 q^{43} - 15252 q^{47} + 17028 q^{49} + 20784 q^{53} - 18708 q^{59} + 35734 q^{61} + 98162 q^{67} - 114120 q^{71} - 109876 q^{73} - 147108 q^{77} + 95536 q^{79} + 3732 q^{83} - 93600 q^{89} + 201710 q^{91} + 91886 q^{97}+O(q^{100})$$ 2 * q - 22 * q^7 - 372 * q^11 - 10 * q^13 + 1212 * q^17 - 1550 * q^19 + 2652 * q^23 + 1812 * q^29 + 10918 * q^31 - 3316 * q^37 - 19080 * q^41 - 12262 * q^43 - 15252 * q^47 + 17028 * q^49 + 20784 * q^53 - 18708 * q^59 + 35734 * q^61 + 98162 * q^67 - 114120 * q^71 - 109876 * q^73 - 147108 * q^77 + 95536 * q^79 + 3732 * q^83 - 93600 * q^89 + 201710 * q^91 + 91886 * 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
 −2.64575 2.64575
0 0 0 0 0 −169.745 0 0 0
1.2 0 0 0 0 0 147.745 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.n 2
3.b odd 2 1 300.6.a.g 2
5.b even 2 1 900.6.a.r 2
5.c odd 4 2 900.6.d.j 4
15.d odd 2 1 300.6.a.h yes 2
15.e even 4 2 300.6.d.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.6.a.g 2 3.b odd 2 1
300.6.a.h yes 2 15.d odd 2 1
300.6.d.f 4 15.e even 4 2
900.6.a.n 2 1.a even 1 1 trivial
900.6.a.r 2 5.b even 2 1
900.6.d.j 4 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(900))$$:

 $$T_{7}^{2} + 22T_{7} - 25079$$ T7^2 + 22*T7 - 25079 $$T_{11}^{2} + 372T_{11} - 192204$$ T11^2 + 372*T11 - 192204

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 22T - 25079$$
$11$ $$T^{2} + 372T - 192204$$
$13$ $$T^{2} + 10T - 403175$$
$17$ $$T^{2} - 1212 T + 140436$$
$19$ $$T^{2} + 1550 T - 2448575$$
$23$ $$T^{2} - 2652 T + 1531476$$
$29$ $$T^{2} - 1812 T - 50209164$$
$31$ $$T^{2} - 10918 T + 29775481$$
$37$ $$T^{2} + 3316 T - 5415836$$
$41$ $$T^{2} + 19080 T + 90784800$$
$43$ $$T^{2} + 12262 T + 16395961$$
$47$ $$T^{2} + 15252 T - 72480924$$
$53$ $$T^{2} - 20784 T - 578076336$$
$59$ $$T^{2} + 18708 T + 29436516$$
$61$ $$T^{2} - 35734 T + 274776889$$
$67$ $$T^{2} - 98162 T + 2408717761$$
$71$ $$T^{2} + 114120 T + 3114093600$$
$73$ $$T^{2} + 109876 T + 2356835044$$
$79$ $$T^{2} - 95536 T + 1106050624$$
$83$ $$T^{2} - 3732 T - 7021874844$$
$89$ $$T^{2} + 93600 T + 1751155200$$
$97$ $$T^{2} - 91886 T - 1011621551$$