# Properties

 Label 1568.2.i.r 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: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - \beta_{2} q^{9}+O(q^{10})$$ q + b1 * q^3 - b2 * q^9 $$q + \beta_1 q^{3} - \beta_{2} q^{9} + (2 \beta_{2} + 2) q^{11} - 2 \beta_{3} q^{13} - 3 \beta_1 q^{17} + (3 \beta_{3} + 3 \beta_1) q^{19} - 8 \beta_{2} q^{23} + (5 \beta_{2} + 5) q^{25} - 4 \beta_{3} q^{27} + 6 q^{29} + 6 \beta_1 q^{31} + (2 \beta_{3} + 2 \beta_1) q^{33} - 2 \beta_{2} q^{37} + (4 \beta_{2} + 4) q^{39} + 3 \beta_{3} q^{41} - 6 q^{43} + (2 \beta_{3} + 2 \beta_1) q^{47} - 6 \beta_{2} q^{51} + ( - 6 \beta_{2} - 6) q^{53} - 6 q^{57} + 9 \beta_1 q^{59} + ( - 4 \beta_{3} - 4 \beta_1) q^{61} + (12 \beta_{2} + 12) q^{67} - 8 \beta_{3} q^{69} + 4 q^{71} + \beta_1 q^{73} + (5 \beta_{3} + 5 \beta_1) q^{75} - 12 \beta_{2} q^{79} + (5 \beta_{2} + 5) q^{81} - 7 \beta_{3} q^{83} + 6 \beta_1 q^{87} + (3 \beta_{3} + 3 \beta_1) q^{89} + 12 \beta_{2} q^{93} + 13 \beta_{3} q^{97} + 2 q^{99}+O(q^{100})$$ q + b1 * q^3 - b2 * q^9 + (2*b2 + 2) * q^11 - 2*b3 * q^13 - 3*b1 * q^17 + (3*b3 + 3*b1) * q^19 - 8*b2 * q^23 + (5*b2 + 5) * q^25 - 4*b3 * q^27 + 6 * q^29 + 6*b1 * q^31 + (2*b3 + 2*b1) * q^33 - 2*b2 * q^37 + (4*b2 + 4) * q^39 + 3*b3 * q^41 - 6 * q^43 + (2*b3 + 2*b1) * q^47 - 6*b2 * q^51 + (-6*b2 - 6) * q^53 - 6 * q^57 + 9*b1 * q^59 + (-4*b3 - 4*b1) * q^61 + (12*b2 + 12) * q^67 - 8*b3 * q^69 + 4 * q^71 + b1 * q^73 + (5*b3 + 5*b1) * q^75 - 12*b2 * q^79 + (5*b2 + 5) * q^81 - 7*b3 * q^83 + 6*b1 * q^87 + (3*b3 + 3*b1) * q^89 + 12*b2 * q^93 + 13*b3 * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^9 $$4 q + 2 q^{9} + 4 q^{11} + 16 q^{23} + 10 q^{25} + 24 q^{29} + 4 q^{37} + 8 q^{39} - 24 q^{43} + 12 q^{51} - 12 q^{53} - 24 q^{57} + 24 q^{67} + 16 q^{71} + 24 q^{79} + 10 q^{81} - 24 q^{93} + 8 q^{99}+O(q^{100})$$ 4 * q + 2 * q^9 + 4 * q^11 + 16 * q^23 + 10 * q^25 + 24 * q^29 + 4 * q^37 + 8 * q^39 - 24 * q^43 + 12 * q^51 - 12 * q^53 - 24 * q^57 + 24 * q^67 + 16 * q^71 + 24 * q^79 + 10 * q^81 - 24 * q^93 + 8 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

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

## 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 −0.707107 1.22474i 0 0 0 0 0 0.500000 0.866025i 0
961.2 0 0.707107 + 1.22474i 0 0 0 0 0 0.500000 0.866025i 0
1537.1 0 −0.707107 + 1.22474i 0 0 0 0 0 0.500000 + 0.866025i 0
1537.2 0 0.707107 1.22474i 0 0 0 0 0 0.500000 + 0.866025i 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.b odd 2 1 inner
7.c even 3 1 inner
7.d 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.r 4
4.b odd 2 1 1568.2.i.q 4
7.b odd 2 1 inner 1568.2.i.r 4
7.c even 3 1 1568.2.a.q 2
7.c even 3 1 inner 1568.2.i.r 4
7.d odd 6 1 1568.2.a.q 2
7.d odd 6 1 inner 1568.2.i.r 4
28.d even 2 1 1568.2.i.q 4
28.f even 6 1 1568.2.a.r yes 2
28.f even 6 1 1568.2.i.q 4
28.g odd 6 1 1568.2.a.r yes 2
28.g odd 6 1 1568.2.i.q 4
56.j odd 6 1 3136.2.a.bo 2
56.k odd 6 1 3136.2.a.bl 2
56.m even 6 1 3136.2.a.bl 2
56.p even 6 1 3136.2.a.bo 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1568.2.a.q 2 7.c even 3 1
1568.2.a.q 2 7.d odd 6 1
1568.2.a.r yes 2 28.f even 6 1
1568.2.a.r yes 2 28.g odd 6 1
1568.2.i.q 4 4.b odd 2 1
1568.2.i.q 4 28.d even 2 1
1568.2.i.q 4 28.f even 6 1
1568.2.i.q 4 28.g odd 6 1
1568.2.i.r 4 1.a even 1 1 trivial
1568.2.i.r 4 7.b odd 2 1 inner
1568.2.i.r 4 7.c even 3 1 inner
1568.2.i.r 4 7.d odd 6 1 inner
3136.2.a.bl 2 56.k odd 6 1
3136.2.a.bl 2 56.m even 6 1
3136.2.a.bo 2 56.j odd 6 1
3136.2.a.bo 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 $$S_{2}^{\mathrm{new}}(1568, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 2T^{2} + 4$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 2 T + 4)^{2}$$
$13$ $$(T^{2} - 8)^{2}$$
$17$ $$T^{4} + 18T^{2} + 324$$
$19$ $$T^{4} + 18T^{2} + 324$$
$23$ $$(T^{2} - 8 T + 64)^{2}$$
$29$ $$(T - 6)^{4}$$
$31$ $$T^{4} + 72T^{2} + 5184$$
$37$ $$(T^{2} - 2 T + 4)^{2}$$
$41$ $$(T^{2} - 18)^{2}$$
$43$ $$(T + 6)^{4}$$
$47$ $$T^{4} + 8T^{2} + 64$$
$53$ $$(T^{2} + 6 T + 36)^{2}$$
$59$ $$T^{4} + 162 T^{2} + 26244$$
$61$ $$T^{4} + 32T^{2} + 1024$$
$67$ $$(T^{2} - 12 T + 144)^{2}$$
$71$ $$(T - 4)^{4}$$
$73$ $$T^{4} + 2T^{2} + 4$$
$79$ $$(T^{2} - 12 T + 144)^{2}$$
$83$ $$(T^{2} - 98)^{2}$$
$89$ $$T^{4} + 18T^{2} + 324$$
$97$ $$(T^{2} - 338)^{2}$$