# Properties

 Label 2736.2.a.z Level $2736$ Weight $2$ Character orbit 2736.a Self dual yes Analytic conductor $21.847$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 456) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{5} - \beta q^{7} +O(q^{10})$$ q - b * q^5 - b * q^7 $$q - \beta q^{5} - \beta q^{7} + ( - \beta + 4) q^{11} - 2 \beta q^{13} + (3 \beta - 2) q^{17} - q^{19} + ( - 2 \beta + 6) q^{23} + (\beta - 1) q^{25} + ( - 4 \beta + 2) q^{29} + 2 q^{31} + (\beta + 4) q^{35} - 8 q^{37} + ( - 2 \beta + 2) q^{41} - \beta q^{43} + (3 \beta - 2) q^{47} + (\beta - 3) q^{49} + (4 \beta + 2) q^{53} + ( - 3 \beta + 4) q^{55} + 12 q^{59} + ( - 3 \beta + 2) q^{61} + (2 \beta + 8) q^{65} + 4 \beta q^{67} + (7 \beta - 6) q^{73} + ( - 3 \beta + 4) q^{77} + (6 \beta - 2) q^{79} + 4 q^{83} + ( - \beta - 12) q^{85} + 6 q^{89} + (2 \beta + 8) q^{91} + \beta q^{95} + ( - 4 \beta - 2) q^{97} +O(q^{100})$$ q - b * q^5 - b * q^7 + (-b + 4) * q^11 - 2*b * q^13 + (3*b - 2) * q^17 - q^19 + (-2*b + 6) * q^23 + (b - 1) * q^25 + (-4*b + 2) * q^29 + 2 * q^31 + (b + 4) * q^35 - 8 * q^37 + (-2*b + 2) * q^41 - b * q^43 + (3*b - 2) * q^47 + (b - 3) * q^49 + (4*b + 2) * q^53 + (-3*b + 4) * q^55 + 12 * q^59 + (-3*b + 2) * q^61 + (2*b + 8) * q^65 + 4*b * q^67 + (7*b - 6) * q^73 + (-3*b + 4) * q^77 + (6*b - 2) * q^79 + 4 * q^83 + (-b - 12) * q^85 + 6 * q^89 + (2*b + 8) * q^91 + b * q^95 + (-4*b - 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{5} - q^{7}+O(q^{10})$$ 2 * q - q^5 - q^7 $$2 q - q^{5} - q^{7} + 7 q^{11} - 2 q^{13} - q^{17} - 2 q^{19} + 10 q^{23} - q^{25} + 4 q^{31} + 9 q^{35} - 16 q^{37} + 2 q^{41} - q^{43} - q^{47} - 5 q^{49} + 8 q^{53} + 5 q^{55} + 24 q^{59} + q^{61} + 18 q^{65} + 4 q^{67} - 5 q^{73} + 5 q^{77} + 2 q^{79} + 8 q^{83} - 25 q^{85} + 12 q^{89} + 18 q^{91} + q^{95} - 8 q^{97}+O(q^{100})$$ 2 * q - q^5 - q^7 + 7 * q^11 - 2 * q^13 - q^17 - 2 * q^19 + 10 * q^23 - q^25 + 4 * q^31 + 9 * q^35 - 16 * q^37 + 2 * q^41 - q^43 - q^47 - 5 * q^49 + 8 * q^53 + 5 * q^55 + 24 * q^59 + q^61 + 18 * q^65 + 4 * q^67 - 5 * q^73 + 5 * q^77 + 2 * q^79 + 8 * q^83 - 25 * q^85 + 12 * q^89 + 18 * q^91 + q^95 - 8 * 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
 2.56155 −1.56155
0 0 0 −2.56155 0 −2.56155 0 0 0
1.2 0 0 0 1.56155 0 1.56155 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$19$$ $$1$$

## 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 2736.2.a.z 2
3.b odd 2 1 912.2.a.m 2
4.b odd 2 1 1368.2.a.k 2
12.b even 2 1 456.2.a.f 2
24.f even 2 1 3648.2.a.bl 2
24.h odd 2 1 3648.2.a.br 2
228.b odd 2 1 8664.2.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.a.f 2 12.b even 2 1
912.2.a.m 2 3.b odd 2 1
1368.2.a.k 2 4.b odd 2 1
2736.2.a.z 2 1.a even 1 1 trivial
3648.2.a.bl 2 24.f even 2 1
3648.2.a.br 2 24.h odd 2 1
8664.2.a.r 2 228.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2736))$$:

 $$T_{5}^{2} + T_{5} - 4$$ T5^2 + T5 - 4 $$T_{7}^{2} + T_{7} - 4$$ T7^2 + T7 - 4 $$T_{11}^{2} - 7T_{11} + 8$$ T11^2 - 7*T11 + 8 $$T_{13}^{2} + 2T_{13} - 16$$ T13^2 + 2*T13 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T - 4$$
$7$ $$T^{2} + T - 4$$
$11$ $$T^{2} - 7T + 8$$
$13$ $$T^{2} + 2T - 16$$
$17$ $$T^{2} + T - 38$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} - 10T + 8$$
$29$ $$T^{2} - 68$$
$31$ $$(T - 2)^{2}$$
$37$ $$(T + 8)^{2}$$
$41$ $$T^{2} - 2T - 16$$
$43$ $$T^{2} + T - 4$$
$47$ $$T^{2} + T - 38$$
$53$ $$T^{2} - 8T - 52$$
$59$ $$(T - 12)^{2}$$
$61$ $$T^{2} - T - 38$$
$67$ $$T^{2} - 4T - 64$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 5T - 202$$
$79$ $$T^{2} - 2T - 152$$
$83$ $$(T - 4)^{2}$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 8T - 52$$