# Properties

 Label 1296.2.i.n Level $1296$ Weight $2$ Character orbit 1296.i Analytic conductor $10.349$ Analytic rank $0$ 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.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.3486121020$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 162) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{6} q^{5} + (4 \zeta_{6} - 4) q^{7}+O(q^{10})$$ q + 3*z * q^5 + (4*z - 4) * q^7 $$q + 3 \zeta_{6} q^{5} + (4 \zeta_{6} - 4) q^{7} + \zeta_{6} q^{13} - 3 q^{17} + 4 q^{19} + (4 \zeta_{6} - 4) q^{25} + (9 \zeta_{6} - 9) q^{29} - 4 \zeta_{6} q^{31} - 12 q^{35} - q^{37} - 6 \zeta_{6} q^{41} + ( - 8 \zeta_{6} + 8) q^{43} + (12 \zeta_{6} - 12) q^{47} - 9 \zeta_{6} q^{49} - 6 q^{53} + ( - \zeta_{6} + 1) q^{61} + (3 \zeta_{6} - 3) q^{65} - 4 \zeta_{6} q^{67} + 12 q^{71} + 11 q^{73} + (16 \zeta_{6} - 16) q^{79} + (12 \zeta_{6} - 12) q^{83} - 9 \zeta_{6} q^{85} - 3 q^{89} - 4 q^{91} + 12 \zeta_{6} q^{95} + (2 \zeta_{6} - 2) q^{97} +O(q^{100})$$ q + 3*z * q^5 + (4*z - 4) * q^7 + z * q^13 - 3 * q^17 + 4 * q^19 + (4*z - 4) * q^25 + (9*z - 9) * q^29 - 4*z * q^31 - 12 * q^35 - q^37 - 6*z * q^41 + (-8*z + 8) * q^43 + (12*z - 12) * q^47 - 9*z * q^49 - 6 * q^53 + (-z + 1) * q^61 + (3*z - 3) * q^65 - 4*z * q^67 + 12 * q^71 + 11 * q^73 + (16*z - 16) * q^79 + (12*z - 12) * q^83 - 9*z * q^85 - 3 * q^89 - 4 * q^91 + 12*z * q^95 + (2*z - 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{5} - 4 q^{7}+O(q^{10})$$ 2 * q + 3 * q^5 - 4 * q^7 $$2 q + 3 q^{5} - 4 q^{7} + q^{13} - 6 q^{17} + 8 q^{19} - 4 q^{25} - 9 q^{29} - 4 q^{31} - 24 q^{35} - 2 q^{37} - 6 q^{41} + 8 q^{43} - 12 q^{47} - 9 q^{49} - 12 q^{53} + q^{61} - 3 q^{65} - 4 q^{67} + 24 q^{71} + 22 q^{73} - 16 q^{79} - 12 q^{83} - 9 q^{85} - 6 q^{89} - 8 q^{91} + 12 q^{95} - 2 q^{97}+O(q^{100})$$ 2 * q + 3 * q^5 - 4 * q^7 + q^13 - 6 * q^17 + 8 * q^19 - 4 * q^25 - 9 * q^29 - 4 * q^31 - 24 * q^35 - 2 * q^37 - 6 * q^41 + 8 * q^43 - 12 * q^47 - 9 * q^49 - 12 * q^53 + q^61 - 3 * q^65 - 4 * q^67 + 24 * q^71 + 22 * q^73 - 16 * q^79 - 12 * q^83 - 9 * q^85 - 6 * q^89 - 8 * q^91 + 12 * q^95 - 2 * q^97

## Character values

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

 $$n$$ $$325$$ $$1135$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
433.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 1.50000 2.59808i 0 −2.00000 3.46410i 0 0 0
865.1 0 0 0 1.50000 + 2.59808i 0 −2.00000 + 3.46410i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.2.i.n 2
3.b odd 2 1 1296.2.i.b 2
4.b odd 2 1 162.2.c.d 2
9.c even 3 1 1296.2.a.c 1
9.c even 3 1 inner 1296.2.i.n 2
9.d odd 6 1 1296.2.a.l 1
9.d odd 6 1 1296.2.i.b 2
12.b even 2 1 162.2.c.a 2
36.f odd 6 1 162.2.a.a 1
36.f odd 6 1 162.2.c.d 2
36.h even 6 1 162.2.a.d yes 1
36.h even 6 1 162.2.c.a 2
72.j odd 6 1 5184.2.a.h 1
72.l even 6 1 5184.2.a.c 1
72.n even 6 1 5184.2.a.bd 1
72.p odd 6 1 5184.2.a.y 1
180.n even 6 1 4050.2.a.r 1
180.p odd 6 1 4050.2.a.bh 1
180.v odd 12 2 4050.2.c.n 2
180.x even 12 2 4050.2.c.g 2
252.s odd 6 1 7938.2.a.s 1
252.bi even 6 1 7938.2.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.2.a.a 1 36.f odd 6 1
162.2.a.d yes 1 36.h even 6 1
162.2.c.a 2 12.b even 2 1
162.2.c.a 2 36.h even 6 1
162.2.c.d 2 4.b odd 2 1
162.2.c.d 2 36.f odd 6 1
1296.2.a.c 1 9.c even 3 1
1296.2.a.l 1 9.d odd 6 1
1296.2.i.b 2 3.b odd 2 1
1296.2.i.b 2 9.d odd 6 1
1296.2.i.n 2 1.a even 1 1 trivial
1296.2.i.n 2 9.c even 3 1 inner
4050.2.a.r 1 180.n even 6 1
4050.2.a.bh 1 180.p odd 6 1
4050.2.c.g 2 180.x even 12 2
4050.2.c.n 2 180.v odd 12 2
5184.2.a.c 1 72.l even 6 1
5184.2.a.h 1 72.j odd 6 1
5184.2.a.y 1 72.p odd 6 1
5184.2.a.bd 1 72.n even 6 1
7938.2.a.n 1 252.bi even 6 1
7938.2.a.s 1 252.s odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 3T_{5} + 9$$ T5^2 - 3*T5 + 9 $$T_{7}^{2} + 4T_{7} + 16$$ T7^2 + 4*T7 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2} + 4T + 16$$
$11$ $$T^{2}$$
$13$ $$T^{2} - T + 1$$
$17$ $$(T + 3)^{2}$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 9T + 81$$
$31$ $$T^{2} + 4T + 16$$
$37$ $$(T + 1)^{2}$$
$41$ $$T^{2} + 6T + 36$$
$43$ $$T^{2} - 8T + 64$$
$47$ $$T^{2} + 12T + 144$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$(T - 12)^{2}$$
$73$ $$(T - 11)^{2}$$
$79$ $$T^{2} + 16T + 256$$
$83$ $$T^{2} + 12T + 144$$
$89$ $$(T + 3)^{2}$$
$97$ $$T^{2} + 2T + 4$$