# Properties

 Label 1568.3.g.f Level 1568 Weight 3 Character orbit 1568.g Analytic conductor 42.725 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1568 = 2^{5} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1568.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$42.7249054517$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( -3 + 6 \zeta_{6} ) q^{5} -8 q^{9} +O(q^{10})$$ $$q + q^{3} + ( -3 + 6 \zeta_{6} ) q^{5} -8 q^{9} -17 q^{11} + ( -8 + 16 \zeta_{6} ) q^{13} + ( -3 + 6 \zeta_{6} ) q^{15} + 25 q^{17} -7 q^{19} + ( 3 - 6 \zeta_{6} ) q^{23} -2 q^{25} -17 q^{27} + ( -8 + 16 \zeta_{6} ) q^{29} + ( 19 - 38 \zeta_{6} ) q^{31} -17 q^{33} + ( -5 + 10 \zeta_{6} ) q^{37} + ( -8 + 16 \zeta_{6} ) q^{39} -26 q^{41} -14 q^{43} + ( 24 - 48 \zeta_{6} ) q^{45} + ( 29 - 58 \zeta_{6} ) q^{47} + 25 q^{51} + ( 53 - 106 \zeta_{6} ) q^{53} + ( 51 - 102 \zeta_{6} ) q^{55} -7 q^{57} -55 q^{59} + ( 13 - 26 \zeta_{6} ) q^{61} -72 q^{65} -17 q^{67} + ( 3 - 6 \zeta_{6} ) q^{69} -119 q^{73} -2 q^{75} + ( 43 - 86 \zeta_{6} ) q^{79} + 55 q^{81} + 110 q^{83} + ( -75 + 150 \zeta_{6} ) q^{85} + ( -8 + 16 \zeta_{6} ) q^{87} -71 q^{89} + ( 19 - 38 \zeta_{6} ) q^{93} + ( 21 - 42 \zeta_{6} ) q^{95} + 22 q^{97} + 136 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 16q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 16q^{9} - 34q^{11} + 50q^{17} - 14q^{19} - 4q^{25} - 34q^{27} - 34q^{33} - 52q^{41} - 28q^{43} + 50q^{51} - 14q^{57} - 110q^{59} - 144q^{65} - 34q^{67} - 238q^{73} - 4q^{75} + 110q^{81} + 220q^{83} - 142q^{89} + 44q^{97} + 272q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$197$$ $$1471$$ $$1473$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
687.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.00000 0 5.19615i 0 0 0 −8.00000 0
687.2 0 1.00000 0 5.19615i 0 0 0 −8.00000 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
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.3.g.f 2
4.b odd 2 1 392.3.g.d 2
7.b odd 2 1 1568.3.g.c 2
7.d odd 6 1 224.3.o.a 2
7.d odd 6 1 224.3.o.b 2
8.b even 2 1 392.3.g.d 2
8.d odd 2 1 inner 1568.3.g.f 2
28.d even 2 1 392.3.g.e 2
28.f even 6 1 56.3.k.a 2
28.f even 6 1 56.3.k.b yes 2
28.g odd 6 1 392.3.k.a 2
28.g odd 6 1 392.3.k.c 2
56.e even 2 1 1568.3.g.c 2
56.h odd 2 1 392.3.g.e 2
56.j odd 6 1 56.3.k.a 2
56.j odd 6 1 56.3.k.b yes 2
56.m even 6 1 224.3.o.a 2
56.m even 6 1 224.3.o.b 2
56.p even 6 1 392.3.k.a 2
56.p even 6 1 392.3.k.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.k.a 2 28.f even 6 1
56.3.k.a 2 56.j odd 6 1
56.3.k.b yes 2 28.f even 6 1
56.3.k.b yes 2 56.j odd 6 1
224.3.o.a 2 7.d odd 6 1
224.3.o.a 2 56.m even 6 1
224.3.o.b 2 7.d odd 6 1
224.3.o.b 2 56.m even 6 1
392.3.g.d 2 4.b odd 2 1
392.3.g.d 2 8.b even 2 1
392.3.g.e 2 28.d even 2 1
392.3.g.e 2 56.h odd 2 1
392.3.k.a 2 28.g odd 6 1
392.3.k.a 2 56.p even 6 1
392.3.k.c 2 28.g odd 6 1
392.3.k.c 2 56.p even 6 1
1568.3.g.c 2 7.b odd 2 1
1568.3.g.c 2 56.e even 2 1
1568.3.g.f 2 1.a even 1 1 trivial
1568.3.g.f 2 8.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 1$$ acting on $$S_{3}^{\mathrm{new}}(1568, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - T + 9 T^{2} )^{2}$$
$5$ $$1 - 23 T^{2} + 625 T^{4}$$
$7$ 1
$11$ $$( 1 + 17 T + 121 T^{2} )^{2}$$
$13$ $$( 1 - 22 T + 169 T^{2} )( 1 + 22 T + 169 T^{2} )$$
$17$ $$( 1 - 25 T + 289 T^{2} )^{2}$$
$19$ $$( 1 + 7 T + 361 T^{2} )^{2}$$
$23$ $$1 - 1031 T^{2} + 279841 T^{4}$$
$29$ $$1 - 1490 T^{2} + 707281 T^{4}$$
$31$ $$1 - 839 T^{2} + 923521 T^{4}$$
$37$ $$1 - 2663 T^{2} + 1874161 T^{4}$$
$41$ $$( 1 + 26 T + 1681 T^{2} )^{2}$$
$43$ $$( 1 + 14 T + 1849 T^{2} )^{2}$$
$47$ $$1 - 1895 T^{2} + 4879681 T^{4}$$
$53$ $$( 1 - 53 T + 2809 T^{2} )( 1 + 53 T + 2809 T^{2} )$$
$59$ $$( 1 + 55 T + 3481 T^{2} )^{2}$$
$61$ $$1 - 6935 T^{2} + 13845841 T^{4}$$
$67$ $$( 1 + 17 T + 4489 T^{2} )^{2}$$
$71$ $$( 1 - 71 T )^{2}( 1 + 71 T )^{2}$$
$73$ $$( 1 + 119 T + 5329 T^{2} )^{2}$$
$79$ $$1 - 6935 T^{2} + 38950081 T^{4}$$
$83$ $$( 1 - 110 T + 6889 T^{2} )^{2}$$
$89$ $$( 1 + 71 T + 7921 T^{2} )^{2}$$
$97$ $$( 1 - 22 T + 9409 T^{2} )^{2}$$