# Properties

 Label 1296.2.c.a Level $1296$ Weight $2$ Character orbit 1296.c Analytic conductor $10.349$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1296.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.3486121020$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 144) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-3}$$. 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 $$q - \beta q^{5} - \beta q^{7} - 3 q^{11} - 5 q^{13} + 4 \beta q^{17} - 2 \beta q^{19} - 9 q^{23} + 2 q^{25} + \beta q^{29} + 3 \beta q^{31} - 3 q^{35} + 2 q^{37} + 3 \beta q^{41} + 3 \beta q^{43} - 3 q^{47} + 4 q^{49} + 3 \beta q^{55} + 3 q^{59} - q^{61} + 5 \beta q^{65} - 5 \beta q^{67} - 12 q^{71} - 2 q^{73} + 3 \beta q^{77} - 5 \beta q^{79} - 15 q^{83} + 12 q^{85} - 4 \beta q^{89} + 5 \beta q^{91} - 6 q^{95} - 5 q^{97} +O(q^{100})$$ q - b * q^5 - b * q^7 - 3 * q^11 - 5 * q^13 + 4*b * q^17 - 2*b * q^19 - 9 * q^23 + 2 * q^25 + b * q^29 + 3*b * q^31 - 3 * q^35 + 2 * q^37 + 3*b * q^41 + 3*b * q^43 - 3 * q^47 + 4 * q^49 + 3*b * q^55 + 3 * q^59 - q^61 + 5*b * q^65 - 5*b * q^67 - 12 * q^71 - 2 * q^73 + 3*b * q^77 - 5*b * q^79 - 15 * q^83 + 12 * q^85 - 4*b * q^89 + 5*b * q^91 - 6 * q^95 - 5 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 6 q^{11} - 10 q^{13} - 18 q^{23} + 4 q^{25} - 6 q^{35} + 4 q^{37} - 6 q^{47} + 8 q^{49} + 6 q^{59} - 2 q^{61} - 24 q^{71} - 4 q^{73} - 30 q^{83} + 24 q^{85} - 12 q^{95} - 10 q^{97}+O(q^{100})$$ 2 * q - 6 * q^11 - 10 * q^13 - 18 * q^23 + 4 * q^25 - 6 * q^35 + 4 * q^37 - 6 * q^47 + 8 * q^49 + 6 * q^59 - 2 * q^61 - 24 * q^71 - 4 * q^73 - 30 * q^83 + 24 * q^85 - 12 * q^95 - 10 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1135$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1295.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 1.73205i 0 1.73205i 0 0 0
1295.2 0 0 0 1.73205i 0 1.73205i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.2.c.a 2
3.b odd 2 1 1296.2.c.c 2
4.b odd 2 1 1296.2.c.c 2
8.b even 2 1 5184.2.c.d 2
8.d odd 2 1 5184.2.c.b 2
9.c even 3 1 144.2.s.b 2
9.c even 3 1 432.2.s.b 2
9.d odd 6 1 144.2.s.c yes 2
9.d odd 6 1 432.2.s.a 2
12.b even 2 1 inner 1296.2.c.a 2
24.f even 2 1 5184.2.c.d 2
24.h odd 2 1 5184.2.c.b 2
36.f odd 6 1 144.2.s.c yes 2
36.f odd 6 1 432.2.s.a 2
36.h even 6 1 144.2.s.b 2
36.h even 6 1 432.2.s.b 2
72.j odd 6 1 576.2.s.c 2
72.j odd 6 1 1728.2.s.c 2
72.l even 6 1 576.2.s.b 2
72.l even 6 1 1728.2.s.d 2
72.n even 6 1 576.2.s.b 2
72.n even 6 1 1728.2.s.d 2
72.p odd 6 1 576.2.s.c 2
72.p odd 6 1 1728.2.s.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.s.b 2 9.c even 3 1
144.2.s.b 2 36.h even 6 1
144.2.s.c yes 2 9.d odd 6 1
144.2.s.c yes 2 36.f odd 6 1
432.2.s.a 2 9.d odd 6 1
432.2.s.a 2 36.f odd 6 1
432.2.s.b 2 9.c even 3 1
432.2.s.b 2 36.h even 6 1
576.2.s.b 2 72.l even 6 1
576.2.s.b 2 72.n even 6 1
576.2.s.c 2 72.j odd 6 1
576.2.s.c 2 72.p odd 6 1
1296.2.c.a 2 1.a even 1 1 trivial
1296.2.c.a 2 12.b even 2 1 inner
1296.2.c.c 2 3.b odd 2 1
1296.2.c.c 2 4.b odd 2 1
1728.2.s.c 2 72.j odd 6 1
1728.2.s.c 2 72.p odd 6 1
1728.2.s.d 2 72.l even 6 1
1728.2.s.d 2 72.n even 6 1
5184.2.c.b 2 8.d odd 2 1
5184.2.c.b 2 24.h odd 2 1
5184.2.c.d 2 8.b even 2 1
5184.2.c.d 2 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1296, [\chi])$$:

 $$T_{5}^{2} + 3$$ T5^2 + 3 $$T_{7}^{2} + 3$$ T7^2 + 3 $$T_{11} + 3$$ T11 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 3$$
$7$ $$T^{2} + 3$$
$11$ $$(T + 3)^{2}$$
$13$ $$(T + 5)^{2}$$
$17$ $$T^{2} + 48$$
$19$ $$T^{2} + 12$$
$23$ $$(T + 9)^{2}$$
$29$ $$T^{2} + 3$$
$31$ $$T^{2} + 27$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2} + 27$$
$43$ $$T^{2} + 27$$
$47$ $$(T + 3)^{2}$$
$53$ $$T^{2}$$
$59$ $$(T - 3)^{2}$$
$61$ $$(T + 1)^{2}$$
$67$ $$T^{2} + 75$$
$71$ $$(T + 12)^{2}$$
$73$ $$(T + 2)^{2}$$
$79$ $$T^{2} + 75$$
$83$ $$(T + 15)^{2}$$
$89$ $$T^{2} + 48$$
$97$ $$(T + 5)^{2}$$