# Properties

 Label 1296.2.s.h Level $1296$ Weight $2$ Character orbit 1296.s Analytic conductor $10.349$ Analytic rank $0$ Dimension $4$ CM discriminant -4 Inner twists $8$

# 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 144) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{5}+O(q^{10})$$ q + b1 * q^5 $$q + \beta_1 q^{5} + (4 \beta_{2} - 4) q^{13} + \beta_{3} q^{17} + 13 \beta_{2} q^{25} + ( - \beta_{3} + \beta_1) q^{29} + 2 q^{37} - 3 \beta_1 q^{41} + (7 \beta_{2} - 7) q^{49} - 3 \beta_{3} q^{53} + 10 \beta_{2} q^{61} + (4 \beta_{3} - 4 \beta_1) q^{65} + 16 q^{73} + (18 \beta_{2} - 18) q^{85} - \beta_{3} q^{89} + 8 \beta_{2} q^{97}+O(q^{100})$$ q + b1 * q^5 + (4*b2 - 4) * q^13 + b3 * q^17 + 13*b2 * q^25 + (-b3 + b1) * q^29 + 2 * q^37 - 3*b1 * q^41 + (7*b2 - 7) * q^49 - 3*b3 * q^53 + 10*b2 * q^61 + (4*b3 - 4*b1) * q^65 + 16 * q^73 + (18*b2 - 18) * q^85 - b3 * q^89 + 8*b2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 8 q^{13} + 26 q^{25} + 8 q^{37} - 14 q^{49} + 20 q^{61} + 64 q^{73} - 36 q^{85} + 16 q^{97}+O(q^{100})$$ 4 * q - 8 * q^13 + 26 * q^25 + 8 * q^37 - 14 * q^49 + 20 * q^61 + 64 * q^73 - 36 * q^85 + 16 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$3\nu$$ 3*v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( 3\nu^{3} ) / 2$$ (3*v^3) / 2
 $$\nu$$ $$=$$ $$( \beta_1 ) / 3$$ (b1) / 3 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{3} ) / 3$$ (2*b3) / 3

## Character values

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

 $$n$$ $$325$$ $$1135$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1 - \beta_{2}$$

## 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
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
0 0 0 −3.67423 + 2.12132i 0 0 0 0 0
431.2 0 0 0 3.67423 2.12132i 0 0 0 0 0
863.1 0 0 0 −3.67423 2.12132i 0 0 0 0 0
863.2 0 0 0 3.67423 + 2.12132i 0 0 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
12.b even 2 1 inner
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.h 4
3.b odd 2 1 inner 1296.2.s.h 4
4.b odd 2 1 CM 1296.2.s.h 4
9.c even 3 1 144.2.c.a 2
9.c even 3 1 inner 1296.2.s.h 4
9.d odd 6 1 144.2.c.a 2
9.d odd 6 1 inner 1296.2.s.h 4
12.b even 2 1 inner 1296.2.s.h 4
36.f odd 6 1 144.2.c.a 2
36.f odd 6 1 inner 1296.2.s.h 4
36.h even 6 1 144.2.c.a 2
36.h even 6 1 inner 1296.2.s.h 4
45.h odd 6 1 3600.2.h.b 2
45.j even 6 1 3600.2.h.b 2
45.k odd 12 2 3600.2.o.a 4
45.l even 12 2 3600.2.o.a 4
63.l odd 6 1 7056.2.h.b 2
63.o even 6 1 7056.2.h.b 2
72.j odd 6 1 576.2.c.a 2
72.l even 6 1 576.2.c.a 2
72.n even 6 1 576.2.c.a 2
72.p odd 6 1 576.2.c.a 2
144.u even 12 2 2304.2.f.f 4
144.v odd 12 2 2304.2.f.f 4
144.w odd 12 2 2304.2.f.f 4
144.x even 12 2 2304.2.f.f 4
180.n even 6 1 3600.2.h.b 2
180.p odd 6 1 3600.2.h.b 2
180.v odd 12 2 3600.2.o.a 4
180.x even 12 2 3600.2.o.a 4
252.s odd 6 1 7056.2.h.b 2
252.bi even 6 1 7056.2.h.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.c.a 2 9.c even 3 1
144.2.c.a 2 9.d odd 6 1
144.2.c.a 2 36.f odd 6 1
144.2.c.a 2 36.h even 6 1
576.2.c.a 2 72.j odd 6 1
576.2.c.a 2 72.l even 6 1
576.2.c.a 2 72.n even 6 1
576.2.c.a 2 72.p odd 6 1
1296.2.s.h 4 1.a even 1 1 trivial
1296.2.s.h 4 3.b odd 2 1 inner
1296.2.s.h 4 4.b odd 2 1 CM
1296.2.s.h 4 9.c even 3 1 inner
1296.2.s.h 4 9.d odd 6 1 inner
1296.2.s.h 4 12.b even 2 1 inner
1296.2.s.h 4 36.f odd 6 1 inner
1296.2.s.h 4 36.h even 6 1 inner
2304.2.f.f 4 144.u even 12 2
2304.2.f.f 4 144.v odd 12 2
2304.2.f.f 4 144.w odd 12 2
2304.2.f.f 4 144.x even 12 2
3600.2.h.b 2 45.h odd 6 1
3600.2.h.b 2 45.j even 6 1
3600.2.h.b 2 180.n even 6 1
3600.2.h.b 2 180.p odd 6 1
3600.2.o.a 4 45.k odd 12 2
3600.2.o.a 4 45.l even 12 2
3600.2.o.a 4 180.v odd 12 2
3600.2.o.a 4 180.x even 12 2
7056.2.h.b 2 63.l odd 6 1
7056.2.h.b 2 63.o even 6 1
7056.2.h.b 2 252.s odd 6 1
7056.2.h.b 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 $$S_{2}^{\mathrm{new}}(1296, [\chi])$$:

 $$T_{5}^{4} - 18T_{5}^{2} + 324$$ T5^4 - 18*T5^2 + 324 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 18T^{2} + 324$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$(T^{2} + 4 T + 16)^{2}$$
$17$ $$(T^{2} + 18)^{2}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4} - 18T^{2} + 324$$
$31$ $$T^{4}$$
$37$ $$(T - 2)^{4}$$
$41$ $$T^{4} - 162 T^{2} + 26244$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$(T^{2} + 162)^{2}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} - 10 T + 100)^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$(T - 16)^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$(T^{2} + 18)^{2}$$
$97$ $$(T^{2} - 8 T + 64)^{2}$$