# Properties

 Label 1216.1.p.a Level $1216$ Weight $1$ Character orbit 1216.p Analytic conductor $0.607$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -8 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.606863055362$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.1267762688.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{3} + ( - \zeta_{12}^{4} - \zeta_{12}^{2} - 1) q^{9}+O(q^{10})$$ q + (-z^5 - z^3) * q^3 + (-z^4 - z^2 - 1) * q^9 $$q + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{3} + ( - \zeta_{12}^{4} - \zeta_{12}^{2} - 1) q^{9} - \zeta_{12}^{3} q^{11} + \zeta_{12}^{4} q^{17} - \zeta_{12}^{5} q^{19} - \zeta_{12}^{2} q^{25} + (\zeta_{12}^{5} + \zeta_{12}) q^{27} + ( - \zeta_{12}^{2} - 1) q^{33} + (\zeta_{12}^{2} + 1) q^{41} + \zeta_{12} q^{43} - q^{49} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{51} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{57} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{59} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{67} + \zeta_{12}^{4} q^{73} + (\zeta_{12}^{5} - \zeta_{12}) q^{75} + (\zeta_{12}^{4} + 1) q^{81} - \zeta_{12}^{3} q^{83} + (\zeta_{12}^{2} + 1) q^{97} + (\zeta_{12}^{5} + \zeta_{12}^{3} - \zeta_{12}) q^{99} +O(q^{100})$$ q + (-z^5 - z^3) * q^3 + (-z^4 - z^2 - 1) * q^9 - z^3 * q^11 + z^4 * q^17 - z^5 * q^19 - z^2 * q^25 + (z^5 + z) * q^27 + (-z^2 - 1) * q^33 + (z^2 + 1) * q^41 + z * q^43 - q^49 + (2*z^3 + 2*z) * q^51 + (-z^4 - z^2) * q^57 + (z^5 + z^3) * q^59 + (-z^3 - z) * q^67 + z^4 * q^73 + (z^5 - z) * q^75 + (z^4 + 1) * q^81 - z^3 * q^83 + (z^2 + 1) * q^97 + (z^5 + z^3 - z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} - 4 q^{17} - 2 q^{25} - 6 q^{33} + 6 q^{41} - 4 q^{49} - 2 q^{73} - 2 q^{81} + 6 q^{97}+O(q^{100})$$ 4 * q - 4 * q^9 - 4 * q^17 - 2 * q^25 - 6 * q^33 + 6 * q^41 - 4 * q^49 - 2 * q^73 - 2 * q^81 + 6 * q^97

## Character values

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

 $$n$$ $$191$$ $$705$$ $$837$$ $$\chi(n)$$ $$1$$ $$\zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
673.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 −0.866025 + 1.50000i 0 0 0 0 0 −1.00000 1.73205i 0
673.2 0 0.866025 1.50000i 0 0 0 0 0 −1.00000 1.73205i 0
1057.1 0 −0.866025 1.50000i 0 0 0 0 0 −1.00000 + 1.73205i 0
1057.2 0 0.866025 + 1.50000i 0 0 0 0 0 −1.00000 + 1.73205i 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 $$\Q(\sqrt{-2})$$
4.b odd 2 1 inner
8.b even 2 1 inner
19.d odd 6 1 inner
76.f even 6 1 inner
152.l odd 6 1 inner
152.o even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.1.p.a 4
4.b odd 2 1 inner 1216.1.p.a 4
8.b even 2 1 inner 1216.1.p.a 4
8.d odd 2 1 CM 1216.1.p.a 4
19.d odd 6 1 inner 1216.1.p.a 4
76.f even 6 1 inner 1216.1.p.a 4
152.l odd 6 1 inner 1216.1.p.a 4
152.o even 6 1 inner 1216.1.p.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.1.p.a 4 1.a even 1 1 trivial
1216.1.p.a 4 4.b odd 2 1 inner
1216.1.p.a 4 8.b even 2 1 inner
1216.1.p.a 4 8.d odd 2 1 CM
1216.1.p.a 4 19.d odd 6 1 inner
1216.1.p.a 4 76.f even 6 1 inner
1216.1.p.a 4 152.l odd 6 1 inner
1216.1.p.a 4 152.o even 6 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1216, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 1)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 2 T + 4)^{2}$$
$19$ $$T^{4} - T^{2} + 1$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} - 3 T + 3)^{2}$$
$43$ $$T^{4} - 4T^{2} + 16$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4} + 3T^{2} + 9$$
$61$ $$T^{4}$$
$67$ $$T^{4} + 3T^{2} + 9$$
$71$ $$T^{4}$$
$73$ $$(T^{2} + T + 1)^{2}$$
$79$ $$T^{4}$$
$83$ $$(T^{2} + 1)^{2}$$
$89$ $$T^{4}$$
$97$ $$(T^{2} - 3 T + 3)^{2}$$