# Properties

 Label 1568.2.i.q Level $1568$ Weight $2$ Character orbit 1568.i Analytic conductor $12.521$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1568 = 2^{5} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1568.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.5205430369$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -\beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -\beta_{2} q^{9} + ( -2 - 2 \beta_{2} ) q^{11} + 2 \beta_{3} q^{13} + 3 \beta_{1} q^{17} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{19} + 8 \beta_{2} q^{23} + ( 5 + 5 \beta_{2} ) q^{25} -4 \beta_{3} q^{27} + 6 q^{29} + 6 \beta_{1} q^{31} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{33} -2 \beta_{2} q^{37} + ( -4 - 4 \beta_{2} ) q^{39} -3 \beta_{3} q^{41} + 6 q^{43} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{47} + 6 \beta_{2} q^{51} + ( -6 - 6 \beta_{2} ) q^{53} -6 q^{57} + 9 \beta_{1} q^{59} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{61} + ( -12 - 12 \beta_{2} ) q^{67} + 8 \beta_{3} q^{69} -4 q^{71} -\beta_{1} q^{73} + ( 5 \beta_{1} + 5 \beta_{3} ) q^{75} + 12 \beta_{2} q^{79} + ( 5 + 5 \beta_{2} ) q^{81} -7 \beta_{3} q^{83} + 6 \beta_{1} q^{87} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{89} + 12 \beta_{2} q^{93} -13 \beta_{3} q^{97} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{9} + O(q^{10})$$ $$4q + 2q^{9} - 4q^{11} - 16q^{23} + 10q^{25} + 24q^{29} + 4q^{37} - 8q^{39} + 24q^{43} - 12q^{51} - 12q^{53} - 24q^{57} - 24q^{67} - 16q^{71} - 24q^{79} + 10q^{81} - 24q^{93} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$1471$$ $$1473$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
961.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0 −0.707107 1.22474i 0 0 0 0 0 0.500000 0.866025i 0
961.2 0 0.707107 + 1.22474i 0 0 0 0 0 0.500000 0.866025i 0
1537.1 0 −0.707107 + 1.22474i 0 0 0 0 0 0.500000 + 0.866025i 0
1537.2 0 0.707107 1.22474i 0 0 0 0 0 0.500000 + 0.866025i 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
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.2.i.q 4
4.b odd 2 1 1568.2.i.r 4
7.b odd 2 1 inner 1568.2.i.q 4
7.c even 3 1 1568.2.a.r yes 2
7.c even 3 1 inner 1568.2.i.q 4
7.d odd 6 1 1568.2.a.r yes 2
7.d odd 6 1 inner 1568.2.i.q 4
28.d even 2 1 1568.2.i.r 4
28.f even 6 1 1568.2.a.q 2
28.f even 6 1 1568.2.i.r 4
28.g odd 6 1 1568.2.a.q 2
28.g odd 6 1 1568.2.i.r 4
56.j odd 6 1 3136.2.a.bl 2
56.k odd 6 1 3136.2.a.bo 2
56.m even 6 1 3136.2.a.bo 2
56.p even 6 1 3136.2.a.bl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1568.2.a.q 2 28.f even 6 1
1568.2.a.q 2 28.g odd 6 1
1568.2.a.r yes 2 7.c even 3 1
1568.2.a.r yes 2 7.d odd 6 1
1568.2.i.q 4 1.a even 1 1 trivial
1568.2.i.q 4 7.b odd 2 1 inner
1568.2.i.q 4 7.c even 3 1 inner
1568.2.i.q 4 7.d odd 6 1 inner
1568.2.i.r 4 4.b odd 2 1
1568.2.i.r 4 28.d even 2 1
1568.2.i.r 4 28.f even 6 1
1568.2.i.r 4 28.g odd 6 1
3136.2.a.bl 2 56.j odd 6 1
3136.2.a.bl 2 56.p even 6 1
3136.2.a.bo 2 56.k odd 6 1
3136.2.a.bo 2 56.m even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1568, [\chi])$$:

 $$T_{3}^{4} + 2 T_{3}^{2} + 4$$ $$T_{5}$$ $$T_{11}^{2} + 2 T_{11} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$4 + 2 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 4 + 2 T + T^{2} )^{2}$$
$13$ $$( -8 + T^{2} )^{2}$$
$17$ $$324 + 18 T^{2} + T^{4}$$
$19$ $$324 + 18 T^{2} + T^{4}$$
$23$ $$( 64 + 8 T + T^{2} )^{2}$$
$29$ $$( -6 + T )^{4}$$
$31$ $$5184 + 72 T^{2} + T^{4}$$
$37$ $$( 4 - 2 T + T^{2} )^{2}$$
$41$ $$( -18 + T^{2} )^{2}$$
$43$ $$( -6 + T )^{4}$$
$47$ $$64 + 8 T^{2} + T^{4}$$
$53$ $$( 36 + 6 T + T^{2} )^{2}$$
$59$ $$26244 + 162 T^{2} + T^{4}$$
$61$ $$1024 + 32 T^{2} + T^{4}$$
$67$ $$( 144 + 12 T + T^{2} )^{2}$$
$71$ $$( 4 + T )^{4}$$
$73$ $$4 + 2 T^{2} + T^{4}$$
$79$ $$( 144 + 12 T + T^{2} )^{2}$$
$83$ $$( -98 + T^{2} )^{2}$$
$89$ $$324 + 18 T^{2} + T^{4}$$
$97$ $$( -338 + T^{2} )^{2}$$