# Properties

 Label 1296.2.s.e Level $1296$ Weight $2$ Character orbit 1296.s 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.s (of order $$6$$, 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: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 48) 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.

 $$f(q)$$ $$=$$ $$q + ( 2 + 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 2 + 2 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{13} + ( -2 + 4 \zeta_{6} ) q^{19} + ( -5 + 5 \zeta_{6} ) q^{25} + ( 12 - 6 \zeta_{6} ) q^{31} -10 q^{37} + ( 6 + 6 \zeta_{6} ) q^{43} + 5 \zeta_{6} q^{49} + ( -14 + 14 \zeta_{6} ) q^{61} + ( 4 - 2 \zeta_{6} ) q^{67} + 10 q^{73} + ( 10 + 10 \zeta_{6} ) q^{79} + ( -4 + 8 \zeta_{6} ) q^{91} + ( 14 - 14 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{7} + O(q^{10})$$ $$2 q + 6 q^{7} + 2 q^{13} - 5 q^{25} + 18 q^{31} - 20 q^{37} + 18 q^{43} + 5 q^{49} - 14 q^{61} + 6 q^{67} + 20 q^{73} + 30 q^{79} + 14 q^{97} + O(q^{100})$$

## 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 3.00000 + 1.73205i 0 0 0
863.1 0 0 0 0 0 3.00000 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
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.e 2
3.b odd 2 1 CM 1296.2.s.e 2
4.b odd 2 1 1296.2.s.b 2
9.c even 3 1 48.2.c.a 2
9.c even 3 1 1296.2.s.b 2
9.d odd 6 1 48.2.c.a 2
9.d odd 6 1 1296.2.s.b 2
12.b even 2 1 1296.2.s.b 2
36.f odd 6 1 48.2.c.a 2
36.f odd 6 1 inner 1296.2.s.e 2
36.h even 6 1 48.2.c.a 2
36.h even 6 1 inner 1296.2.s.e 2
45.h odd 6 1 1200.2.h.e 2
45.j even 6 1 1200.2.h.e 2
45.k odd 12 2 1200.2.o.i 4
45.l even 12 2 1200.2.o.i 4
63.l odd 6 1 2352.2.h.c 2
63.o even 6 1 2352.2.h.c 2
72.j odd 6 1 192.2.c.a 2
72.l even 6 1 192.2.c.a 2
72.n even 6 1 192.2.c.a 2
72.p odd 6 1 192.2.c.a 2
144.u even 12 2 768.2.f.d 4
144.v odd 12 2 768.2.f.d 4
144.w odd 12 2 768.2.f.d 4
144.x even 12 2 768.2.f.d 4
180.n even 6 1 1200.2.h.e 2
180.p odd 6 1 1200.2.h.e 2
180.v odd 12 2 1200.2.o.i 4
180.x even 12 2 1200.2.o.i 4
252.s odd 6 1 2352.2.h.c 2
252.bi even 6 1 2352.2.h.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.c.a 2 9.c even 3 1
48.2.c.a 2 9.d odd 6 1
48.2.c.a 2 36.f odd 6 1
48.2.c.a 2 36.h even 6 1
192.2.c.a 2 72.j odd 6 1
192.2.c.a 2 72.l even 6 1
192.2.c.a 2 72.n even 6 1
192.2.c.a 2 72.p odd 6 1
768.2.f.d 4 144.u even 12 2
768.2.f.d 4 144.v odd 12 2
768.2.f.d 4 144.w odd 12 2
768.2.f.d 4 144.x even 12 2
1200.2.h.e 2 45.h odd 6 1
1200.2.h.e 2 45.j even 6 1
1200.2.h.e 2 180.n even 6 1
1200.2.h.e 2 180.p odd 6 1
1200.2.o.i 4 45.k odd 12 2
1200.2.o.i 4 45.l even 12 2
1200.2.o.i 4 180.v odd 12 2
1200.2.o.i 4 180.x even 12 2
1296.2.s.b 2 4.b odd 2 1
1296.2.s.b 2 9.c even 3 1
1296.2.s.b 2 9.d odd 6 1
1296.2.s.b 2 12.b even 2 1
1296.2.s.e 2 1.a even 1 1 trivial
1296.2.s.e 2 3.b odd 2 1 CM
1296.2.s.e 2 36.f odd 6 1 inner
1296.2.s.e 2 36.h even 6 1 inner
2352.2.h.c 2 63.l odd 6 1
2352.2.h.c 2 63.o even 6 1
2352.2.h.c 2 252.s odd 6 1
2352.2.h.c 2 252.bi even 6 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{7}^{2} - 6 T_{7} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$12 - 6 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$4 - 2 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$12 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$108 - 18 T + T^{2}$$
$37$ $$( 10 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$108 - 18 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$196 + 14 T + T^{2}$$
$67$ $$12 - 6 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -10 + T )^{2}$$
$79$ $$300 - 30 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$196 - 14 T + T^{2}$$