# Properties

 Label 1216.2.t.a Level $1216$ Weight $2$ Character orbit 1216.t Analytic conductor $9.710$ Analytic rank $0$ Dimension $8$ CM discriminant -8 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$9.70980888579$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{3} + ( 2 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{9} +O(q^{10})$$ $$q + ( -2 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{3} + ( 2 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{9} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 3 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{11} + ( 6 - 6 \zeta_{24}^{4} ) q^{17} + ( -3 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{19} -5 \zeta_{24}^{4} q^{25} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 13 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{27} + ( 3 + 4 \zeta_{24} - 2 \zeta_{24}^{3} - 3 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{33} + ( -3 + 8 \zeta_{24} - 4 \zeta_{24}^{3} + 3 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{41} -10 \zeta_{24}^{2} q^{43} -7 q^{49} + ( -6 \zeta_{24} + 6 \zeta_{24}^{2} - 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} - 6 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{51} + ( -5 + 2 \zeta_{24} - 4 \zeta_{24}^{3} + 12 \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{57} + ( -10 \zeta_{24} - 3 \zeta_{24}^{2} - 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 5 \zeta_{24}^{7} ) q^{59} + ( -3 \zeta_{24} + 7 \zeta_{24}^{2} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 7 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{67} + ( 1 - 12 \zeta_{24} + 6 \zeta_{24}^{3} - \zeta_{24}^{4} + 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{73} + ( 5 \zeta_{24} - 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 5 \zeta_{24}^{6} + 10 \zeta_{24}^{7} ) q^{75} + ( -19 + 20 \zeta_{24} - 10 \zeta_{24}^{3} + 19 \zeta_{24}^{4} - 10 \zeta_{24}^{5} - 10 \zeta_{24}^{7} ) q^{81} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 9 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{83} -18 \zeta_{24}^{4} q^{89} + ( 5 + 12 \zeta_{24} - 6 \zeta_{24}^{3} - 5 \zeta_{24}^{4} - 6 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{97} + ( 2 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 16q^{9} + O(q^{10})$$ $$8q + 16q^{9} + 24q^{17} - 20q^{25} + 12q^{33} - 12q^{41} - 56q^{49} + 8q^{57} + 4q^{73} - 76q^{81} - 72q^{89} + 20q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$191$$ $$705$$ $$837$$ $$\chi(n)$$ $$1$$ $$-\zeta_{24}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
353.1
 −0.258819 + 0.965926i 0.965926 + 0.258819i 0.258819 − 0.965926i −0.965926 − 0.258819i −0.258819 − 0.965926i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.965926 + 0.258819i
0 −2.98735 1.72474i 0 0 0 0 0 4.44949 + 7.70674i 0
353.2 0 −1.25529 0.724745i 0 0 0 0 0 −0.449490 0.778539i 0
353.3 0 1.25529 + 0.724745i 0 0 0 0 0 −0.449490 0.778539i 0
353.4 0 2.98735 + 1.72474i 0 0 0 0 0 4.44949 + 7.70674i 0
1185.1 0 −2.98735 + 1.72474i 0 0 0 0 0 4.44949 7.70674i 0
1185.2 0 −1.25529 + 0.724745i 0 0 0 0 0 −0.449490 + 0.778539i 0
1185.3 0 1.25529 0.724745i 0 0 0 0 0 −0.449490 + 0.778539i 0
1185.4 0 2.98735 1.72474i 0 0 0 0 0 4.44949 7.70674i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1185.4 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.c even 3 1 inner
76.g odd 6 1 inner
152.k odd 6 1 inner
152.p even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.t.a 8
4.b odd 2 1 inner 1216.2.t.a 8
8.b even 2 1 inner 1216.2.t.a 8
8.d odd 2 1 CM 1216.2.t.a 8
19.c even 3 1 inner 1216.2.t.a 8
76.g odd 6 1 inner 1216.2.t.a 8
152.k odd 6 1 inner 1216.2.t.a 8
152.p even 6 1 inner 1216.2.t.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.t.a 8 1.a even 1 1 trivial
1216.2.t.a 8 4.b odd 2 1 inner
1216.2.t.a 8 8.b even 2 1 inner
1216.2.t.a 8 8.d odd 2 1 CM
1216.2.t.a 8 19.c even 3 1 inner
1216.2.t.a 8 76.g odd 6 1 inner
1216.2.t.a 8 152.k odd 6 1 inner
1216.2.t.a 8 152.p even 6 1 inner

## Hecke kernels

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

 $$T_{3}^{8} - 14 T_{3}^{6} + 171 T_{3}^{4} - 350 T_{3}^{2} + 625$$ $$T_{5}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$625 - 350 T^{2} + 171 T^{4} - 14 T^{6} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8}$$
$11$ $$( 9 + 30 T^{2} + T^{4} )^{2}$$
$13$ $$T^{8}$$
$17$ $$( 36 - 6 T + T^{2} )^{4}$$
$19$ $$130321 + 12274 T^{2} + 795 T^{4} + 34 T^{6} + T^{8}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$( 7569 - 522 T + 123 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$43$ $$( 10000 - 100 T^{2} + T^{4} )^{2}$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$395254161 - 6322158 T^{2} + 81243 T^{4} - 318 T^{6} + T^{8}$$
$61$ $$T^{8}$$
$67$ $$625 - 5150 T^{2} + 42411 T^{4} - 206 T^{6} + T^{8}$$
$71$ $$T^{8}$$
$73$ $$( 46225 + 430 T + 219 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$79$ $$T^{8}$$
$83$ $$( 5625 + 174 T^{2} + T^{4} )^{2}$$
$89$ $$( 324 + 18 T + T^{2} )^{4}$$
$97$ $$( 36481 + 1910 T + 291 T^{2} - 10 T^{3} + T^{4} )^{2}$$