# Properties

 Label 1296.2.s.c Level $1296$ Weight $2$ Character orbit 1296.s Analytic conductor $10.349$ Analytic rank $1$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1296.s (of order $$6$$, degree $$2$$, not 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 432) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} - 1) q^{7}+O(q^{10})$$ q + (-z - 1) * q^7 $$q + ( - \zeta_{6} - 1) q^{7} - 7 \zeta_{6} q^{13} + (10 \zeta_{6} - 5) q^{19} + (5 \zeta_{6} - 5) q^{25} + (6 \zeta_{6} - 12) q^{31} - q^{37} + ( - 6 \zeta_{6} - 6) q^{43} - 4 \zeta_{6} q^{49} + ( - 13 \zeta_{6} + 13) q^{61} + (7 \zeta_{6} - 14) q^{67} - 17 q^{73} + (7 \zeta_{6} + 7) q^{79} + (14 \zeta_{6} - 7) q^{91} + ( - 5 \zeta_{6} + 5) q^{97} +O(q^{100})$$ q + (-z - 1) * q^7 - 7*z * q^13 + (10*z - 5) * q^19 + (5*z - 5) * q^25 + (6*z - 12) * q^31 - q^37 + (-6*z - 6) * q^43 - 4*z * q^49 + (-13*z + 13) * q^61 + (7*z - 14) * q^67 - 17 * q^73 + (7*z + 7) * q^79 + (14*z - 7) * q^91 + (-5*z + 5) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{7}+O(q^{10})$$ 2 * q - 3 * q^7 $$2 q - 3 q^{7} - 7 q^{13} - 5 q^{25} - 18 q^{31} - 2 q^{37} - 18 q^{43} - 4 q^{49} + 13 q^{61} - 21 q^{67} - 34 q^{73} + 21 q^{79} + 5 q^{97}+O(q^{100})$$ 2 * q - 3 * q^7 - 7 * q^13 - 5 * q^25 - 18 * q^31 - 2 * q^37 - 18 * q^43 - 4 * q^49 + 13 * q^61 - 21 * q^67 - 34 * q^73 + 21 * q^79 + 5 * 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
431.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 0 0 −1.50000 0.866025i 0 0 0
863.1 0 0 0 0 0 −1.50000 + 0.866025i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
36.f odd 6 1 inner
36.h even 6 1 inner

## Twists

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

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 3T + 3$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 7T + 49$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 75$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 18T + 108$$
$37$ $$(T + 1)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 18T + 108$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 13T + 169$$
$67$ $$T^{2} + 21T + 147$$
$71$ $$T^{2}$$
$73$ $$(T + 17)^{2}$$
$79$ $$T^{2} - 21T + 147$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 5T + 25$$