# Properties

 Label 1296.2.i.i Level $1296$ Weight $2$ Character orbit 1296.i Analytic conductor $10.349$ Analytic rank $0$ 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.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 27) Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{7}+O(q^{10})$$ q + (z - 1) * q^7 $$q + (\zeta_{6} - 1) q^{7} - 5 \zeta_{6} q^{13} + 7 q^{19} + ( - 5 \zeta_{6} + 5) q^{25} - 4 \zeta_{6} q^{31} + 11 q^{37} + ( - 8 \zeta_{6} + 8) q^{43} + 6 \zeta_{6} q^{49} + ( - \zeta_{6} + 1) q^{61} + 5 \zeta_{6} q^{67} - 7 q^{73} + ( - 17 \zeta_{6} + 17) q^{79} + 5 q^{91} + ( - 19 \zeta_{6} + 19) q^{97} +O(q^{100})$$ q + (z - 1) * q^7 - 5*z * q^13 + 7 * q^19 + (-5*z + 5) * q^25 - 4*z * q^31 + 11 * q^37 + (-8*z + 8) * q^43 + 6*z * q^49 + (-z + 1) * q^61 + 5*z * q^67 - 7 * q^73 + (-17*z + 17) * q^79 + 5 * q^91 + (-19*z + 19) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{7}+O(q^{10})$$ 2 * q - q^7 $$2 q - q^{7} - 5 q^{13} + 14 q^{19} + 5 q^{25} - 4 q^{31} + 22 q^{37} + 8 q^{43} + 6 q^{49} + q^{61} + 5 q^{67} - 14 q^{73} + 17 q^{79} + 10 q^{91} + 19 q^{97}+O(q^{100})$$ 2 * q - q^7 - 5 * q^13 + 14 * q^19 + 5 * q^25 - 4 * q^31 + 22 * q^37 + 8 * q^43 + 6 * q^49 + q^61 + 5 * q^67 - 14 * q^73 + 17 * q^79 + 10 * q^91 + 19 * 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 0 0 −0.500000 0.866025i 0 0 0
865.1 0 0 0 0 0 −0.500000 + 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 $$\Q(\sqrt{-3})$$
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.2.i.i 2
3.b odd 2 1 CM 1296.2.i.i 2
4.b odd 2 1 81.2.c.a 2
9.c even 3 1 432.2.a.e 1
9.c even 3 1 inner 1296.2.i.i 2
9.d odd 6 1 432.2.a.e 1
9.d odd 6 1 inner 1296.2.i.i 2
12.b even 2 1 81.2.c.a 2
36.f odd 6 1 27.2.a.a 1
36.f odd 6 1 81.2.c.a 2
36.h even 6 1 27.2.a.a 1
36.h even 6 1 81.2.c.a 2
72.j odd 6 1 1728.2.a.o 1
72.l even 6 1 1728.2.a.n 1
72.n even 6 1 1728.2.a.o 1
72.p odd 6 1 1728.2.a.n 1
108.j odd 18 6 729.2.e.f 6
108.l even 18 6 729.2.e.f 6
180.n even 6 1 675.2.a.e 1
180.p odd 6 1 675.2.a.e 1
180.v odd 12 2 675.2.b.f 2
180.x even 12 2 675.2.b.f 2
252.s odd 6 1 1323.2.a.i 1
252.bi even 6 1 1323.2.a.i 1
396.k even 6 1 3267.2.a.f 1
396.o odd 6 1 3267.2.a.f 1
468.x even 6 1 4563.2.a.e 1
468.bg odd 6 1 4563.2.a.e 1
612.n even 6 1 7803.2.a.k 1
612.q odd 6 1 7803.2.a.k 1
684.w even 6 1 9747.2.a.f 1
684.bh odd 6 1 9747.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.a.a 1 36.f odd 6 1
27.2.a.a 1 36.h even 6 1
81.2.c.a 2 4.b odd 2 1
81.2.c.a 2 12.b even 2 1
81.2.c.a 2 36.f odd 6 1
81.2.c.a 2 36.h even 6 1
432.2.a.e 1 9.c even 3 1
432.2.a.e 1 9.d odd 6 1
675.2.a.e 1 180.n even 6 1
675.2.a.e 1 180.p odd 6 1
675.2.b.f 2 180.v odd 12 2
675.2.b.f 2 180.x even 12 2
729.2.e.f 6 108.j odd 18 6
729.2.e.f 6 108.l even 18 6
1296.2.i.i 2 1.a even 1 1 trivial
1296.2.i.i 2 3.b odd 2 1 CM
1296.2.i.i 2 9.c even 3 1 inner
1296.2.i.i 2 9.d odd 6 1 inner
1323.2.a.i 1 252.s odd 6 1
1323.2.a.i 1 252.bi even 6 1
1728.2.a.n 1 72.l even 6 1
1728.2.a.n 1 72.p odd 6 1
1728.2.a.o 1 72.j odd 6 1
1728.2.a.o 1 72.n even 6 1
3267.2.a.f 1 396.k even 6 1
3267.2.a.f 1 396.o odd 6 1
4563.2.a.e 1 468.x even 6 1
4563.2.a.e 1 468.bg odd 6 1
7803.2.a.k 1 612.n even 6 1
7803.2.a.k 1 612.q odd 6 1
9747.2.a.f 1 684.w even 6 1
9747.2.a.f 1 684.bh 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 $$T_{7}^{2} + T_{7} + 1$$ T7^2 + T7 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + T + 1$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 5T + 25$$
$17$ $$T^{2}$$
$19$ $$(T - 7)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 4T + 16$$
$37$ $$(T - 11)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 8T + 64$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} - 5T + 25$$
$71$ $$T^{2}$$
$73$ $$(T + 7)^{2}$$
$79$ $$T^{2} - 17T + 289$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 19T + 361$$