# Properties

 Label 1568.2.i.f Level $1568$ Weight $2$ Character orbit 1568.i Analytic conductor $12.521$ Analytic rank $1$ Dimension $2$ CM discriminant -4 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 32) Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q -2 \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} -6 q^{13} + ( 2 - 2 \zeta_{6} ) q^{17} + ( 1 - \zeta_{6} ) q^{25} -10 q^{29} + 2 \zeta_{6} q^{37} -10 q^{41} + ( 6 - 6 \zeta_{6} ) q^{45} + ( -14 + 14 \zeta_{6} ) q^{53} -10 \zeta_{6} q^{61} + 12 \zeta_{6} q^{65} + ( -6 + 6 \zeta_{6} ) q^{73} + ( -9 + 9 \zeta_{6} ) q^{81} -4 q^{85} + 10 \zeta_{6} q^{89} -18 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{5} + 3q^{9} + O(q^{10})$$ $$2q - 2q^{5} + 3q^{9} - 12q^{13} + 2q^{17} + q^{25} - 20q^{29} + 2q^{37} - 20q^{41} + 6q^{45} - 14q^{53} - 10q^{61} + 12q^{65} - 6q^{73} - 9q^{81} - 8q^{85} + 10q^{89} - 36q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$197$$ $$1471$$ $$1473$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −1.00000 + 1.73205i 0 0 0 1.50000 2.59808i 0
1537.1 0 0 0 −1.00000 1.73205i 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.c even 3 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.f 2
4.b odd 2 1 CM 1568.2.i.f 2
7.b odd 2 1 1568.2.i.g 2
7.c even 3 1 1568.2.a.e 1
7.c even 3 1 inner 1568.2.i.f 2
7.d odd 6 1 32.2.a.a 1
7.d odd 6 1 1568.2.i.g 2
21.g even 6 1 288.2.a.d 1
28.d even 2 1 1568.2.i.g 2
28.f even 6 1 32.2.a.a 1
28.f even 6 1 1568.2.i.g 2
28.g odd 6 1 1568.2.a.e 1
28.g odd 6 1 inner 1568.2.i.f 2
35.i odd 6 1 800.2.a.d 1
35.k even 12 2 800.2.c.e 2
56.j odd 6 1 64.2.a.a 1
56.k odd 6 1 3136.2.a.m 1
56.m even 6 1 64.2.a.a 1
56.p even 6 1 3136.2.a.m 1
63.i even 6 1 2592.2.i.e 2
63.k odd 6 1 2592.2.i.t 2
63.s even 6 1 2592.2.i.e 2
63.t odd 6 1 2592.2.i.t 2
77.i even 6 1 3872.2.a.f 1
84.j odd 6 1 288.2.a.d 1
91.s odd 6 1 5408.2.a.g 1
105.p even 6 1 7200.2.a.v 1
105.w odd 12 2 7200.2.f.m 2
112.v even 12 2 256.2.b.b 2
112.x odd 12 2 256.2.b.b 2
119.h odd 6 1 9248.2.a.f 1
140.s even 6 1 800.2.a.d 1
140.x odd 12 2 800.2.c.e 2
168.ba even 6 1 576.2.a.c 1
168.be odd 6 1 576.2.a.c 1
224.bc odd 24 4 1024.2.e.j 4
224.be even 24 4 1024.2.e.j 4
252.n even 6 1 2592.2.i.t 2
252.r odd 6 1 2592.2.i.e 2
252.bj even 6 1 2592.2.i.t 2
252.bn odd 6 1 2592.2.i.e 2
280.ba even 6 1 1600.2.a.n 1
280.bk odd 6 1 1600.2.a.n 1
280.bp odd 12 2 1600.2.c.l 2
280.bv even 12 2 1600.2.c.l 2
308.m odd 6 1 3872.2.a.f 1
336.bo even 12 2 2304.2.d.j 2
336.br odd 12 2 2304.2.d.j 2
364.x even 6 1 5408.2.a.g 1
420.be odd 6 1 7200.2.a.v 1
420.br even 12 2 7200.2.f.m 2
476.q even 6 1 9248.2.a.f 1
616.s even 6 1 7744.2.a.v 1
616.z odd 6 1 7744.2.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 7.d odd 6 1
32.2.a.a 1 28.f even 6 1
64.2.a.a 1 56.j odd 6 1
64.2.a.a 1 56.m even 6 1
256.2.b.b 2 112.v even 12 2
256.2.b.b 2 112.x odd 12 2
288.2.a.d 1 21.g even 6 1
288.2.a.d 1 84.j odd 6 1
576.2.a.c 1 168.ba even 6 1
576.2.a.c 1 168.be odd 6 1
800.2.a.d 1 35.i odd 6 1
800.2.a.d 1 140.s even 6 1
800.2.c.e 2 35.k even 12 2
800.2.c.e 2 140.x odd 12 2
1024.2.e.j 4 224.bc odd 24 4
1024.2.e.j 4 224.be even 24 4
1568.2.a.e 1 7.c even 3 1
1568.2.a.e 1 28.g odd 6 1
1568.2.i.f 2 1.a even 1 1 trivial
1568.2.i.f 2 4.b odd 2 1 CM
1568.2.i.f 2 7.c even 3 1 inner
1568.2.i.f 2 28.g odd 6 1 inner
1568.2.i.g 2 7.b odd 2 1
1568.2.i.g 2 7.d odd 6 1
1568.2.i.g 2 28.d even 2 1
1568.2.i.g 2 28.f even 6 1
1600.2.a.n 1 280.ba even 6 1
1600.2.a.n 1 280.bk odd 6 1
1600.2.c.l 2 280.bp odd 12 2
1600.2.c.l 2 280.bv even 12 2
2304.2.d.j 2 336.bo even 12 2
2304.2.d.j 2 336.br odd 12 2
2592.2.i.e 2 63.i even 6 1
2592.2.i.e 2 63.s even 6 1
2592.2.i.e 2 252.r odd 6 1
2592.2.i.e 2 252.bn odd 6 1
2592.2.i.t 2 63.k odd 6 1
2592.2.i.t 2 63.t odd 6 1
2592.2.i.t 2 252.n even 6 1
2592.2.i.t 2 252.bj even 6 1
3136.2.a.m 1 56.k odd 6 1
3136.2.a.m 1 56.p even 6 1
3872.2.a.f 1 77.i even 6 1
3872.2.a.f 1 308.m odd 6 1
5408.2.a.g 1 91.s odd 6 1
5408.2.a.g 1 364.x even 6 1
7200.2.a.v 1 105.p even 6 1
7200.2.a.v 1 420.be odd 6 1
7200.2.f.m 2 105.w odd 12 2
7200.2.f.m 2 420.br even 12 2
7744.2.a.v 1 616.s even 6 1
7744.2.a.v 1 616.z odd 6 1
9248.2.a.f 1 119.h odd 6 1
9248.2.a.f 1 476.q even 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$4 + 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$( 6 + T )^{2}$$
$17$ $$4 - 2 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 10 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$4 - 2 T + T^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$196 + 14 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$100 + 10 T + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$36 + 6 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$100 - 10 T + T^{2}$$
$97$ $$( 18 + T )^{2}$$