# Properties

 Label 1568.2.i.u 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: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + ( - \beta_1 + 1) q^{5}+O(q^{10})$$ q - b2 * q^3 + (-b1 + 1) * q^5 $$q - \beta_{2} q^{3} + ( - \beta_1 + 1) q^{5} + 3 \beta_{2} q^{11} - \beta_{3} q^{15} + 5 \beta_1 q^{17} + ( - \beta_{3} + \beta_{2}) q^{19} + ( - \beta_{3} + \beta_{2}) q^{23} + 4 \beta_1 q^{25} - 3 \beta_{3} q^{27} + 8 q^{29} + 5 \beta_{2} q^{31} + ( - 9 \beta_1 + 9) q^{33} + ( - 5 \beta_1 + 5) q^{37} - 4 q^{41} - 4 \beta_{3} q^{43} + ( - 5 \beta_{3} + 5 \beta_{2}) q^{47} + (5 \beta_{3} - 5 \beta_{2}) q^{51} + \beta_1 q^{53} + 3 \beta_{3} q^{55} + 3 q^{57} - \beta_{2} q^{59} + ( - 11 \beta_1 + 11) q^{61} - 7 \beta_{2} q^{67} + 3 q^{69} + 8 \beta_{3} q^{71} + 15 \beta_1 q^{73} + (4 \beta_{3} - 4 \beta_{2}) q^{75} + ( - \beta_{3} + \beta_{2}) q^{79} + 9 \beta_1 q^{81} + 4 \beta_{3} q^{83} + 5 q^{85} - 8 \beta_{2} q^{87} + ( - 7 \beta_1 + 7) q^{89} + ( - 15 \beta_1 + 15) q^{93} + \beta_{2} q^{95} - 12 q^{97}+O(q^{100})$$ q - b2 * q^3 + (-b1 + 1) * q^5 + 3*b2 * q^11 - b3 * q^15 + 5*b1 * q^17 + (-b3 + b2) * q^19 + (-b3 + b2) * q^23 + 4*b1 * q^25 - 3*b3 * q^27 + 8 * q^29 + 5*b2 * q^31 + (-9*b1 + 9) * q^33 + (-5*b1 + 5) * q^37 - 4 * q^41 - 4*b3 * q^43 + (-5*b3 + 5*b2) * q^47 + (5*b3 - 5*b2) * q^51 + b1 * q^53 + 3*b3 * q^55 + 3 * q^57 - b2 * q^59 + (-11*b1 + 11) * q^61 - 7*b2 * q^67 + 3 * q^69 + 8*b3 * q^71 + 15*b1 * q^73 + (4*b3 - 4*b2) * q^75 + (-b3 + b2) * q^79 + 9*b1 * q^81 + 4*b3 * q^83 + 5 * q^85 - 8*b2 * q^87 + (-7*b1 + 7) * q^89 + (-15*b1 + 15) * q^93 + b2 * q^95 - 12 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{5}+O(q^{10})$$ 4 * q + 2 * q^5 $$4 q + 2 q^{5} + 10 q^{17} + 8 q^{25} + 32 q^{29} + 18 q^{33} + 10 q^{37} - 16 q^{41} + 2 q^{53} + 12 q^{57} + 22 q^{61} + 12 q^{69} + 30 q^{73} + 18 q^{81} + 20 q^{85} + 14 q^{89} + 30 q^{93} - 48 q^{97}+O(q^{100})$$ 4 * q + 2 * q^5 + 10 * q^17 + 8 * q^25 + 32 * q^29 + 18 * q^33 + 10 * q^37 - 16 * q^41 + 2 * q^53 + 12 * q^57 + 22 * q^61 + 12 * q^69 + 30 * q^73 + 18 * q^81 + 20 * q^85 + 14 * q^89 + 30 * q^93 - 48 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{3} + \zeta_{12}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{12}^{3}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} ) / 3$$ (-b3 + 2*b2) / 3

## Character values

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

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

## 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
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 −0.866025 1.50000i 0 0.500000 0.866025i 0 0 0 0 0
961.2 0 0.866025 + 1.50000i 0 0.500000 0.866025i 0 0 0 0 0
1537.1 0 −0.866025 + 1.50000i 0 0.500000 + 0.866025i 0 0 0 0 0
1537.2 0 0.866025 1.50000i 0 0.500000 + 0.866025i 0 0 0 0 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 inner
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.u 4
4.b odd 2 1 inner 1568.2.i.u 4
7.b odd 2 1 224.2.i.b 4
7.c even 3 1 1568.2.a.n 2
7.c even 3 1 inner 1568.2.i.u 4
7.d odd 6 1 224.2.i.b 4
7.d odd 6 1 1568.2.a.s 2
21.c even 2 1 2016.2.s.r 4
21.g even 6 1 2016.2.s.r 4
28.d even 2 1 224.2.i.b 4
28.f even 6 1 224.2.i.b 4
28.f even 6 1 1568.2.a.s 2
28.g odd 6 1 1568.2.a.n 2
28.g odd 6 1 inner 1568.2.i.u 4
56.e even 2 1 448.2.i.i 4
56.h odd 2 1 448.2.i.i 4
56.j odd 6 1 448.2.i.i 4
56.j odd 6 1 3136.2.a.bh 2
56.k odd 6 1 3136.2.a.bu 2
56.m even 6 1 448.2.i.i 4
56.m even 6 1 3136.2.a.bh 2
56.p even 6 1 3136.2.a.bu 2
84.h odd 2 1 2016.2.s.r 4
84.j odd 6 1 2016.2.s.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.b 4 7.b odd 2 1
224.2.i.b 4 7.d odd 6 1
224.2.i.b 4 28.d even 2 1
224.2.i.b 4 28.f even 6 1
448.2.i.i 4 56.e even 2 1
448.2.i.i 4 56.h odd 2 1
448.2.i.i 4 56.j odd 6 1
448.2.i.i 4 56.m even 6 1
1568.2.a.n 2 7.c even 3 1
1568.2.a.n 2 28.g odd 6 1
1568.2.a.s 2 7.d odd 6 1
1568.2.a.s 2 28.f even 6 1
1568.2.i.u 4 1.a even 1 1 trivial
1568.2.i.u 4 4.b odd 2 1 inner
1568.2.i.u 4 7.c even 3 1 inner
1568.2.i.u 4 28.g odd 6 1 inner
2016.2.s.r 4 21.c even 2 1
2016.2.s.r 4 21.g even 6 1
2016.2.s.r 4 84.h odd 2 1
2016.2.s.r 4 84.j odd 6 1
3136.2.a.bh 2 56.j odd 6 1
3136.2.a.bh 2 56.m even 6 1
3136.2.a.bu 2 56.k odd 6 1
3136.2.a.bu 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 $$S_{2}^{\mathrm{new}}(1568, [\chi])$$:

 $$T_{3}^{4} + 3T_{3}^{2} + 9$$ T3^4 + 3*T3^2 + 9 $$T_{5}^{2} - T_{5} + 1$$ T5^2 - T5 + 1 $$T_{11}^{4} + 27T_{11}^{2} + 729$$ T11^4 + 27*T11^2 + 729

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$(T^{2} - T + 1)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 27T^{2} + 729$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 5 T + 25)^{2}$$
$19$ $$T^{4} + 3T^{2} + 9$$
$23$ $$T^{4} + 3T^{2} + 9$$
$29$ $$(T - 8)^{4}$$
$31$ $$T^{4} + 75T^{2} + 5625$$
$37$ $$(T^{2} - 5 T + 25)^{2}$$
$41$ $$(T + 4)^{4}$$
$43$ $$(T^{2} - 48)^{2}$$
$47$ $$T^{4} + 75T^{2} + 5625$$
$53$ $$(T^{2} - T + 1)^{2}$$
$59$ $$T^{4} + 3T^{2} + 9$$
$61$ $$(T^{2} - 11 T + 121)^{2}$$
$67$ $$T^{4} + 147 T^{2} + 21609$$
$71$ $$(T^{2} - 192)^{2}$$
$73$ $$(T^{2} - 15 T + 225)^{2}$$
$79$ $$T^{4} + 3T^{2} + 9$$
$83$ $$(T^{2} - 48)^{2}$$
$89$ $$(T^{2} - 7 T + 49)^{2}$$
$97$ $$(T + 12)^{4}$$