# Properties

 Label 1216.2.a.c Level $1216$ Weight $2$ Character orbit 1216.a Self dual yes Analytic conductor $9.710$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.70980888579$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{3} + q^{5} - 3q^{7} + q^{9} + O(q^{10})$$ $$q - 2q^{3} + q^{5} - 3q^{7} + q^{9} - 5q^{11} + 4q^{13} - 2q^{15} - 3q^{17} + q^{19} + 6q^{21} + 8q^{23} - 4q^{25} + 4q^{27} + 2q^{29} + 4q^{31} + 10q^{33} - 3q^{35} - 10q^{37} - 8q^{39} + 10q^{41} - q^{43} + q^{45} - q^{47} + 2q^{49} + 6q^{51} + 4q^{53} - 5q^{55} - 2q^{57} - 6q^{59} + 13q^{61} - 3q^{63} + 4q^{65} + 12q^{67} - 16q^{69} + 2q^{71} + 9q^{73} + 8q^{75} + 15q^{77} + 8q^{79} - 11q^{81} + 12q^{83} - 3q^{85} - 4q^{87} + 12q^{89} - 12q^{91} - 8q^{93} + q^{95} - 8q^{97} - 5q^{99} + O(q^{100})$$

## 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 −2.00000 0 1.00000 0 −3.00000 0 1.00000 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
$$2$$ $$1$$
$$19$$ $$-1$$

## 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 1216.2.a.c 1
4.b odd 2 1 1216.2.a.q 1
8.b even 2 1 76.2.a.a 1
8.d odd 2 1 304.2.a.a 1
24.f even 2 1 2736.2.a.q 1
24.h odd 2 1 684.2.a.b 1
40.e odd 2 1 7600.2.a.p 1
40.f even 2 1 1900.2.a.b 1
40.i odd 4 2 1900.2.c.b 2
56.h odd 2 1 3724.2.a.a 1
88.b odd 2 1 9196.2.a.f 1
152.b even 2 1 5776.2.a.p 1
152.g odd 2 1 1444.2.a.a 1
152.l odd 6 2 1444.2.e.c 2
152.p even 6 2 1444.2.e.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.a.a 1 8.b even 2 1
304.2.a.a 1 8.d odd 2 1
684.2.a.b 1 24.h odd 2 1
1216.2.a.c 1 1.a even 1 1 trivial
1216.2.a.q 1 4.b odd 2 1
1444.2.a.a 1 152.g odd 2 1
1444.2.e.a 2 152.p even 6 2
1444.2.e.c 2 152.l odd 6 2
1900.2.a.b 1 40.f even 2 1
1900.2.c.b 2 40.i odd 4 2
2736.2.a.q 1 24.f even 2 1
3724.2.a.a 1 56.h odd 2 1
5776.2.a.p 1 152.b even 2 1
7600.2.a.p 1 40.e odd 2 1
9196.2.a.f 1 88.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1216))$$:

 $$T_{3} + 2$$ $$T_{5} - 1$$ $$T_{7} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$2 + T$$
$5$ $$-1 + T$$
$7$ $$3 + T$$
$11$ $$5 + T$$
$13$ $$-4 + T$$
$17$ $$3 + T$$
$19$ $$-1 + T$$
$23$ $$-8 + T$$
$29$ $$-2 + T$$
$31$ $$-4 + T$$
$37$ $$10 + T$$
$41$ $$-10 + T$$
$43$ $$1 + T$$
$47$ $$1 + T$$
$53$ $$-4 + T$$
$59$ $$6 + T$$
$61$ $$-13 + T$$
$67$ $$-12 + T$$
$71$ $$-2 + T$$
$73$ $$-9 + T$$
$79$ $$-8 + T$$
$83$ $$-12 + T$$
$89$ $$-12 + T$$
$97$ $$8 + T$$