# Properties

 Label 1224.2.bq.a Level $1224$ Weight $2$ Character orbit 1224.bq Analytic conductor $9.774$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1224 = 2^{3} \cdot 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1224.bq (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.77368920740$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 136) Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{8} ) q^{5} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{7} +O(q^{10})$$ $$q + ( -1 + \zeta_{8} ) q^{5} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{7} + ( -1 + \zeta_{8}^{3} ) q^{11} + ( -1 + 4 \zeta_{8}^{2} ) q^{17} + ( -3 + 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{19} + ( -1 - 4 \zeta_{8} + 4 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{23} + ( 1 + 3 \zeta_{8} + \zeta_{8}^{2} ) q^{25} + ( 3 - 3 \zeta_{8} ) q^{29} + ( 3 + 3 \zeta_{8} ) q^{31} + ( 2 + \zeta_{8} - \zeta_{8}^{3} ) q^{35} + ( 1 + \zeta_{8} ) q^{37} + ( 3 + 4 \zeta_{8} + 4 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{41} + ( 3 - 6 \zeta_{8} + 3 \zeta_{8}^{2} ) q^{43} + ( -2 \zeta_{8} + 10 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{47} + ( -7 + 7 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{49} + ( -5 + 5 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{53} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{55} + ( -7 - 2 \zeta_{8} - 7 \zeta_{8}^{2} ) q^{59} + ( -7 - 4 \zeta_{8} - 4 \zeta_{8}^{2} - 7 \zeta_{8}^{3} ) q^{61} + ( 4 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{67} + ( -3 - 3 \zeta_{8} + 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{71} + ( 1 - \zeta_{8} + 8 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{73} + ( 3 + 4 \zeta_{8} + 3 \zeta_{8}^{2} ) q^{77} + ( 9 - 4 \zeta_{8} + 4 \zeta_{8}^{2} - 9 \zeta_{8}^{3} ) q^{79} + ( 3 - 3 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{83} + ( 1 - \zeta_{8} - 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{85} + ( -4 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{89} + ( 1 - 3 \zeta_{8} - 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{95} + ( -7 + 7 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{5} - 4 q^{7} + O(q^{10})$$ $$4 q - 4 q^{5} - 4 q^{7} - 4 q^{11} - 4 q^{17} - 12 q^{19} - 4 q^{23} + 4 q^{25} + 12 q^{29} + 12 q^{31} + 8 q^{35} + 4 q^{37} + 12 q^{41} + 12 q^{43} - 28 q^{49} - 20 q^{53} - 28 q^{59} - 28 q^{61} + 16 q^{67} - 12 q^{71} + 4 q^{73} + 12 q^{77} + 36 q^{79} + 12 q^{83} + 4 q^{85} + 4 q^{95} - 28 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$$.

 $$n$$ $$137$$ $$613$$ $$649$$ $$919$$ $$\chi(n)$$ $$1$$ $$1$$ $$\zeta_{8}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 0.707107 + 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
0 0 0 −0.292893 + 0.707107i 0 −1.70711 4.12132i 0 0 0
433.1 0 0 0 −1.70711 0.707107i 0 −0.292893 + 0.121320i 0 0 0
865.1 0 0 0 −1.70711 + 0.707107i 0 −0.292893 0.121320i 0 0 0
937.1 0 0 0 −0.292893 0.707107i 0 −1.70711 + 4.12132i 0 0 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
17.d even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.2.bq.a 4
3.b odd 2 1 136.2.n.a 4
12.b even 2 1 272.2.v.e 4
17.d even 8 1 inner 1224.2.bq.a 4
51.g odd 8 1 136.2.n.a 4
51.i even 16 2 2312.2.a.s 4
51.i even 16 2 2312.2.b.j 4
204.p even 8 1 272.2.v.e 4
204.t odd 16 2 4624.2.a.bm 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.n.a 4 3.b odd 2 1
136.2.n.a 4 51.g odd 8 1
272.2.v.e 4 12.b even 2 1
272.2.v.e 4 204.p even 8 1
1224.2.bq.a 4 1.a even 1 1 trivial
1224.2.bq.a 4 17.d even 8 1 inner
2312.2.a.s 4 51.i even 16 2
2312.2.b.j 4 51.i even 16 2
4624.2.a.bm 4 204.t odd 16 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 4 T_{5}^{3} + 6 T_{5}^{2} + 4 T_{5} + 2$$ acting on $$S_{2}^{\mathrm{new}}(1224, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$2 + 4 T + 6 T^{2} + 4 T^{3} + T^{4}$$
$7$ $$2 + 12 T + 22 T^{2} + 4 T^{3} + T^{4}$$
$11$ $$2 + 4 T + 6 T^{2} + 4 T^{3} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$( 17 + 2 T + T^{2} )^{2}$$
$19$ $$196 + 168 T + 72 T^{2} + 12 T^{3} + T^{4}$$
$23$ $$1058 + 276 T + 22 T^{2} + 4 T^{3} + T^{4}$$
$29$ $$162 - 108 T + 54 T^{2} - 12 T^{3} + T^{4}$$
$31$ $$162 - 108 T + 54 T^{2} - 12 T^{3} + T^{4}$$
$37$ $$2 - 4 T + 6 T^{2} - 4 T^{3} + T^{4}$$
$41$ $$578 - 476 T + 134 T^{2} - 12 T^{3} + T^{4}$$
$43$ $$324 + 216 T + 72 T^{2} - 12 T^{3} + T^{4}$$
$47$ $$8464 + 216 T^{2} + T^{4}$$
$53$ $$1156 + 680 T + 200 T^{2} + 20 T^{3} + T^{4}$$
$59$ $$8836 + 2632 T + 392 T^{2} + 28 T^{3} + T^{4}$$
$61$ $$15842 + 3916 T + 438 T^{2} + 28 T^{3} + T^{4}$$
$67$ $$( -16 - 8 T + T^{2} )^{2}$$
$71$ $$578 + 476 T + 134 T^{2} + 12 T^{3} + T^{4}$$
$73$ $$12482 - 2212 T + 102 T^{2} - 4 T^{3} + T^{4}$$
$79$ $$37538 - 7124 T + 662 T^{2} - 36 T^{3} + T^{4}$$
$83$ $$324 + 216 T + 72 T^{2} - 12 T^{3} + T^{4}$$
$89$ $$256 + 96 T^{2} + T^{4}$$
$97$ $$1058 - 276 T + 214 T^{2} + 28 T^{3} + T^{4}$$