# Properties

 Label 1296.2.a.g Level $1296$ Weight $2$ Character orbit 1296.a Self dual yes Analytic conductor $10.349$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1296.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$10.3486121020$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 18) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{7} + O(q^{10})$$ $$q - 2q^{7} + 3q^{11} + 2q^{13} - 3q^{17} + q^{19} + 6q^{23} - 5q^{25} + 6q^{29} + 4q^{31} - 4q^{37} + 9q^{41} + q^{43} + 6q^{47} - 3q^{49} + 12q^{53} - 3q^{59} + 8q^{61} - 5q^{67} + 12q^{71} + 11q^{73} - 6q^{77} + 4q^{79} - 12q^{83} + 6q^{89} - 4q^{91} + 5q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 0 0 −2.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.2.a.g 1
3.b odd 2 1 1296.2.a.f 1
4.b odd 2 1 162.2.a.c 1
8.b even 2 1 5184.2.a.o 1
8.d odd 2 1 5184.2.a.r 1
9.c even 3 2 144.2.i.c 2
9.d odd 6 2 432.2.i.b 2
12.b even 2 1 162.2.a.b 1
20.d odd 2 1 4050.2.a.c 1
20.e even 4 2 4050.2.c.c 2
24.f even 2 1 5184.2.a.q 1
24.h odd 2 1 5184.2.a.p 1
28.d even 2 1 7938.2.a.x 1
36.f odd 6 2 18.2.c.a 2
36.h even 6 2 54.2.c.a 2
60.h even 2 1 4050.2.a.v 1
60.l odd 4 2 4050.2.c.r 2
72.j odd 6 2 1728.2.i.f 2
72.l even 6 2 1728.2.i.e 2
72.n even 6 2 576.2.i.a 2
72.p odd 6 2 576.2.i.g 2
84.h odd 2 1 7938.2.a.i 1
180.n even 6 2 1350.2.e.c 2
180.p odd 6 2 450.2.e.i 2
180.v odd 12 4 1350.2.j.a 4
180.x even 12 4 450.2.j.e 4
252.n even 6 2 882.2.h.b 2
252.o even 6 2 2646.2.h.h 2
252.r odd 6 2 2646.2.e.c 2
252.s odd 6 2 2646.2.f.g 2
252.u odd 6 2 882.2.e.i 2
252.bb even 6 2 2646.2.e.b 2
252.bi even 6 2 882.2.f.d 2
252.bj even 6 2 882.2.e.g 2
252.bl odd 6 2 882.2.h.c 2
252.bn odd 6 2 2646.2.h.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.2.c.a 2 36.f odd 6 2
54.2.c.a 2 36.h even 6 2
144.2.i.c 2 9.c even 3 2
162.2.a.b 1 12.b even 2 1
162.2.a.c 1 4.b odd 2 1
432.2.i.b 2 9.d odd 6 2
450.2.e.i 2 180.p odd 6 2
450.2.j.e 4 180.x even 12 4
576.2.i.a 2 72.n even 6 2
576.2.i.g 2 72.p odd 6 2
882.2.e.g 2 252.bj even 6 2
882.2.e.i 2 252.u odd 6 2
882.2.f.d 2 252.bi even 6 2
882.2.h.b 2 252.n even 6 2
882.2.h.c 2 252.bl odd 6 2
1296.2.a.f 1 3.b odd 2 1
1296.2.a.g 1 1.a even 1 1 trivial
1350.2.e.c 2 180.n even 6 2
1350.2.j.a 4 180.v odd 12 4
1728.2.i.e 2 72.l even 6 2
1728.2.i.f 2 72.j odd 6 2
2646.2.e.b 2 252.bb even 6 2
2646.2.e.c 2 252.r odd 6 2
2646.2.f.g 2 252.s odd 6 2
2646.2.h.h 2 252.o even 6 2
2646.2.h.i 2 252.bn odd 6 2
4050.2.a.c 1 20.d odd 2 1
4050.2.a.v 1 60.h even 2 1
4050.2.c.c 2 20.e even 4 2
4050.2.c.r 2 60.l odd 4 2
5184.2.a.o 1 8.b even 2 1
5184.2.a.p 1 24.h odd 2 1
5184.2.a.q 1 24.f even 2 1
5184.2.a.r 1 8.d odd 2 1
7938.2.a.i 1 84.h odd 2 1
7938.2.a.x 1 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1296))$$:

 $$T_{5}$$ $$T_{7} + 2$$ $$T_{11} - 3$$

## Hecke characteristic polynomials

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