# Properties

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

# 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: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 224) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{5} + ( 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{5} + ( 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{9} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{11} + ( -3 + \beta_{3} ) q^{13} -4 q^{15} -2 \beta_{2} q^{17} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{19} -4 \beta_{1} q^{23} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{25} + ( -10 + 2 \beta_{3} ) q^{27} -2 \beta_{3} q^{29} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{31} + 8 \beta_{1} q^{33} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{37} + ( -8 - 8 \beta_{1} - 4 \beta_{2} ) q^{39} + ( 4 + 2 \beta_{3} ) q^{41} + ( -2 - 2 \beta_{3} ) q^{43} + ( -7 - 7 \beta_{1} - \beta_{2} ) q^{45} + ( 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{47} + ( -10 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{51} + ( 10 + 10 \beta_{1} ) q^{53} + ( -12 - 4 \beta_{3} ) q^{55} + ( -6 + 2 \beta_{3} ) q^{57} + ( 7 + 7 \beta_{1} - \beta_{2} ) q^{59} + ( -9 \beta_{1} + \beta_{2} + \beta_{3} ) q^{61} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{65} + ( 4 + 4 \beta_{1} ) q^{67} + ( 4 - 4 \beta_{3} ) q^{69} + ( -4 + 4 \beta_{3} ) q^{71} + ( 6 + 6 \beta_{1} + 4 \beta_{2} ) q^{73} + ( 9 \beta_{1} + \beta_{2} + \beta_{3} ) q^{75} + ( -4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{79} + ( -11 - 11 \beta_{1} - 6 \beta_{2} ) q^{81} + ( -7 - \beta_{3} ) q^{83} + ( 10 + 2 \beta_{3} ) q^{85} + ( 10 + 10 \beta_{1} + 2 \beta_{2} ) q^{87} + 6 \beta_{1} q^{89} + ( -12 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{93} + ( -4 - 4 \beta_{1} ) q^{95} + ( -8 - 2 \beta_{3} ) q^{97} + ( -14 + 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} + 2q^{5} - 6q^{9} + O(q^{10})$$ $$4q + 2q^{3} + 2q^{5} - 6q^{9} - 4q^{11} - 12q^{13} - 16q^{15} - 2q^{19} + 8q^{23} - 2q^{25} - 40q^{27} - 4q^{31} - 16q^{33} - 16q^{39} + 16q^{41} - 8q^{43} - 14q^{45} - 12q^{47} + 20q^{51} + 20q^{53} - 48q^{55} - 24q^{57} + 14q^{59} + 18q^{61} + 4q^{65} + 8q^{67} + 16q^{69} - 16q^{71} + 12q^{73} - 18q^{75} + 8q^{79} - 22q^{81} - 28q^{83} + 40q^{85} + 20q^{87} - 12q^{89} + 24q^{93} - 8q^{95} - 32q^{97} - 56q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu^{2} - 2 \nu - 1$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 2 \nu^{2} + 6 \nu - 1$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 3 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
961.1
 −0.309017 − 0.535233i 0.809017 + 1.40126i −0.309017 + 0.535233i 0.809017 − 1.40126i
0 −0.618034 1.07047i 0 1.61803 2.80252i 0 0 0 0.736068 1.27491i 0
961.2 0 1.61803 + 2.80252i 0 −0.618034 + 1.07047i 0 0 0 −3.73607 + 6.47106i 0
1537.1 0 −0.618034 + 1.07047i 0 1.61803 + 2.80252i 0 0 0 0.736068 + 1.27491i 0
1537.2 0 1.61803 2.80252i 0 −0.618034 1.07047i 0 0 0 −3.73607 6.47106i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.2.i.w 4
4.b odd 2 1 1568.2.i.n 4
7.b odd 2 1 1568.2.i.m 4
7.c even 3 1 1568.2.a.k 2
7.c even 3 1 inner 1568.2.i.w 4
7.d odd 6 1 224.2.a.d yes 2
7.d odd 6 1 1568.2.i.m 4
21.g even 6 1 2016.2.a.o 2
28.d even 2 1 1568.2.i.v 4
28.f even 6 1 224.2.a.c 2
28.f even 6 1 1568.2.i.v 4
28.g odd 6 1 1568.2.a.v 2
28.g odd 6 1 1568.2.i.n 4
35.i odd 6 1 5600.2.a.z 2
56.j odd 6 1 448.2.a.i 2
56.k odd 6 1 3136.2.a.bf 2
56.m even 6 1 448.2.a.j 2
56.p even 6 1 3136.2.a.by 2
84.j odd 6 1 2016.2.a.r 2
112.v even 12 2 1792.2.b.k 4
112.x odd 12 2 1792.2.b.m 4
140.s even 6 1 5600.2.a.bk 2
168.ba even 6 1 4032.2.a.bv 2
168.be odd 6 1 4032.2.a.bw 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.c 2 28.f even 6 1
224.2.a.d yes 2 7.d odd 6 1
448.2.a.i 2 56.j odd 6 1
448.2.a.j 2 56.m even 6 1
1568.2.a.k 2 7.c even 3 1
1568.2.a.v 2 28.g odd 6 1
1568.2.i.m 4 7.b odd 2 1
1568.2.i.m 4 7.d odd 6 1
1568.2.i.n 4 4.b odd 2 1
1568.2.i.n 4 28.g odd 6 1
1568.2.i.v 4 28.d even 2 1
1568.2.i.v 4 28.f even 6 1
1568.2.i.w 4 1.a even 1 1 trivial
1568.2.i.w 4 7.c even 3 1 inner
1792.2.b.k 4 112.v even 12 2
1792.2.b.m 4 112.x odd 12 2
2016.2.a.o 2 21.g even 6 1
2016.2.a.r 2 84.j odd 6 1
3136.2.a.bf 2 56.k odd 6 1
3136.2.a.by 2 56.p even 6 1
4032.2.a.bv 2 168.ba even 6 1
4032.2.a.bw 2 168.be odd 6 1
5600.2.a.z 2 35.i odd 6 1
5600.2.a.bk 2 140.s even 6 1

## Hecke kernels

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

 $$T_{3}^{4} - 2 T_{3}^{3} + 8 T_{3}^{2} + 8 T_{3} + 16$$ $$T_{5}^{4} - 2 T_{5}^{3} + 8 T_{5}^{2} + 8 T_{5} + 16$$ $$T_{11}^{4} + 4 T_{11}^{3} + 32 T_{11}^{2} - 64 T_{11} + 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$16 + 8 T + 8 T^{2} - 2 T^{3} + T^{4}$$
$5$ $$16 + 8 T + 8 T^{2} - 2 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$256 - 64 T + 32 T^{2} + 4 T^{3} + T^{4}$$
$13$ $$( 4 + 6 T + T^{2} )^{2}$$
$17$ $$400 + 20 T^{2} + T^{4}$$
$19$ $$16 - 8 T + 8 T^{2} + 2 T^{3} + T^{4}$$
$23$ $$( 16 - 4 T + T^{2} )^{2}$$
$29$ $$( -20 + T^{2} )^{2}$$
$31$ $$256 - 64 T + 32 T^{2} + 4 T^{3} + T^{4}$$
$37$ $$400 + 20 T^{2} + T^{4}$$
$41$ $$( -4 - 8 T + T^{2} )^{2}$$
$43$ $$( -16 + 4 T + T^{2} )^{2}$$
$47$ $$256 + 192 T + 128 T^{2} + 12 T^{3} + T^{4}$$
$53$ $$( 100 - 10 T + T^{2} )^{2}$$
$59$ $$1936 - 616 T + 152 T^{2} - 14 T^{3} + T^{4}$$
$61$ $$5776 - 1368 T + 248 T^{2} - 18 T^{3} + T^{4}$$
$67$ $$( 16 - 4 T + T^{2} )^{2}$$
$71$ $$( -64 + 8 T + T^{2} )^{2}$$
$73$ $$1936 + 528 T + 188 T^{2} - 12 T^{3} + T^{4}$$
$79$ $$4096 + 512 T + 128 T^{2} - 8 T^{3} + T^{4}$$
$83$ $$( 44 + 14 T + T^{2} )^{2}$$
$89$ $$( 36 + 6 T + T^{2} )^{2}$$
$97$ $$( 44 + 16 T + T^{2} )^{2}$$