# Properties

 Label 1568.2.i.t Level $1568$ Weight $2$ Character orbit 1568.i Analytic conductor $12.521$ Analytic rank $0$ Dimension $4$ CM discriminant -4 Inner twists $8$

# 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: $\mathrm{U}(1)[D_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{5} + ( 3 + 3 \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{5} + ( 3 + 3 \beta_{2} ) q^{9} + \beta_{3} q^{13} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{17} -3 \beta_{2} q^{25} -4 q^{29} + ( 12 + 12 \beta_{2} ) q^{37} -9 \beta_{3} q^{41} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{45} -14 \beta_{2} q^{53} -11 \beta_{1} q^{61} + ( 2 + 2 \beta_{2} ) q^{65} + ( -11 \beta_{1} - 11 \beta_{3} ) q^{73} + 9 \beta_{2} q^{81} -10 q^{85} -3 \beta_{1} q^{89} -5 \beta_{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{9} + O(q^{10})$$ $$4 q + 6 q^{9} + 6 q^{25} - 16 q^{29} + 24 q^{37} + 28 q^{53} + 4 q^{65} - 18 q^{81} - 40 q^{85} + O(q^{100})$$

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 $$\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$$.

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

## 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 0 −0.707107 + 1.22474i 0 0 0 1.50000 2.59808i 0
961.2 0 0 0 0.707107 1.22474i 0 0 0 1.50000 2.59808i 0
1537.1 0 0 0 −0.707107 1.22474i 0 0 0 1.50000 + 2.59808i 0
1537.2 0 0 0 0.707107 + 1.22474i 0 0 0 1.50000 + 2.59808i 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g 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.t 4
4.b odd 2 1 CM 1568.2.i.t 4
7.b odd 2 1 inner 1568.2.i.t 4
7.c even 3 1 1568.2.a.o 2
7.c even 3 1 inner 1568.2.i.t 4
7.d odd 6 1 1568.2.a.o 2
7.d odd 6 1 inner 1568.2.i.t 4
28.d even 2 1 inner 1568.2.i.t 4
28.f even 6 1 1568.2.a.o 2
28.f even 6 1 inner 1568.2.i.t 4
28.g odd 6 1 1568.2.a.o 2
28.g odd 6 1 inner 1568.2.i.t 4
56.j odd 6 1 3136.2.a.bi 2
56.k odd 6 1 3136.2.a.bi 2
56.m even 6 1 3136.2.a.bi 2
56.p even 6 1 3136.2.a.bi 2

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$4 + 2 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$( -2 + T^{2} )^{2}$$
$17$ $$2500 + 50 T^{2} + T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( 4 + T )^{4}$$
$31$ $$T^{4}$$
$37$ $$( 144 - 12 T + T^{2} )^{2}$$
$41$ $$( -162 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$( 196 - 14 T + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$58564 + 242 T^{2} + T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$58564 + 242 T^{2} + T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$324 + 18 T^{2} + T^{4}$$
$97$ $$( -50 + T^{2} )^{2}$$