# Properties

 Label 1216.1.g.a Level $1216$ Weight $1$ Character orbit 1216.g Analytic conductor $0.607$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -8, -19, 152 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1216.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.606863055362$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{-2}, \sqrt{-19})$$ Artin image $D_4:C_2$ Artin field Galois closure of 8.0.378535936.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - q^{9} +O(q^{10})$$ $$q - q^{9} -2 i q^{11} + 2 q^{17} -i q^{19} + q^{25} + 2 i q^{43} - q^{49} -2 q^{73} + q^{81} -2 i q^{83} + 2 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 4q^{17} + 2q^{25} - 2q^{49} - 4q^{73} + 2q^{81} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$705$$ $$837$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
417.1
 1.00000i − 1.00000i
0 0 0 0 0 0 0 −1.00000 0
417.2 0 0 0 0 0 0 0 −1.00000 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
152.b even 2 1 RM by $$\Q(\sqrt{38})$$
4.b odd 2 1 inner
8.b even 2 1 inner
76.d even 2 1 inner
152.g odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.1.g.a 2
4.b odd 2 1 inner 1216.1.g.a 2
8.b even 2 1 inner 1216.1.g.a 2
8.d odd 2 1 CM 1216.1.g.a 2
19.b odd 2 1 CM 1216.1.g.a 2
76.d even 2 1 inner 1216.1.g.a 2
152.b even 2 1 RM 1216.1.g.a 2
152.g odd 2 1 inner 1216.1.g.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.1.g.a 2 1.a even 1 1 trivial
1216.1.g.a 2 4.b odd 2 1 inner
1216.1.g.a 2 8.b even 2 1 inner
1216.1.g.a 2 8.d odd 2 1 CM
1216.1.g.a 2 19.b odd 2 1 CM
1216.1.g.a 2 76.d even 2 1 inner
1216.1.g.a 2 152.b even 2 1 RM
1216.1.g.a 2 152.g odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{1}^{\mathrm{new}}(1216, [\chi])$$.

## Hecke characteristic polynomials

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