# Properties

 Label 552.2.b.b Level $552$ Weight $2$ Character orbit 552.b Analytic conductor $4.408$ Analytic rank $0$ Dimension $6$ CM discriminant -23 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$552 = 2^{3} \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 552.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.40774219157$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.8869743.1 Defining polynomial: $$x^{6} - 3 x^{3} + 8$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + \beta_{2} q^{3} + ( \beta_{2} + \beta_{5} ) q^{4} + ( 1 - \beta_{3} ) q^{6} + ( -1 - \beta_{3} ) q^{8} + ( -2 \beta_{1} + \beta_{4} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + \beta_{2} q^{3} + ( \beta_{2} + \beta_{5} ) q^{4} + ( 1 - \beta_{3} ) q^{6} + ( -1 - \beta_{3} ) q^{8} + ( -2 \beta_{1} + \beta_{4} ) q^{9} + ( -\beta_{1} + 2 \beta_{4} ) q^{12} + ( 2 \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{13} + ( \beta_{1} + 2 \beta_{4} ) q^{16} + ( \beta_{2} + 3 \beta_{5} ) q^{18} + ( -1 + 2 \beta_{3} ) q^{23} + ( -\beta_{2} + 3 \beta_{5} ) q^{24} + 5 q^{25} + ( 5 - 3 \beta_{2} - \beta_{3} - \beta_{5} ) q^{26} + ( -1 - 2 \beta_{3} ) q^{27} + ( -2 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} - 2 \beta_{5} ) q^{29} + ( -4 \beta_{1} - \beta_{2} + \beta_{4} - 4 \beta_{5} ) q^{31} + ( -3 \beta_{2} + \beta_{5} ) q^{32} + ( -5 - \beta_{3} ) q^{36} + ( -5 - 4 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{39} + ( 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{4} - 2 \beta_{5} ) q^{41} + ( \beta_{1} - 4 \beta_{4} ) q^{46} + ( 4 \beta_{1} - 3 \beta_{2} - 3 \beta_{4} - 4 \beta_{5} ) q^{47} + ( -7 + \beta_{3} ) q^{48} + 7 q^{49} -5 \beta_{1} q^{50} + ( -1 - 5 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} ) q^{52} + ( \beta_{1} + 4 \beta_{4} ) q^{54} + ( 1 - \beta_{2} + 3 \beta_{3} + 5 \beta_{5} ) q^{58} -12 q^{59} + ( 7 + 3 \beta_{2} + \beta_{3} + 5 \beta_{5} ) q^{62} + ( -5 + 3 \beta_{3} ) q^{64} + ( -\beta_{2} - 6 \beta_{5} ) q^{69} + ( 4 \beta_{1} + 3 \beta_{2} + 3 \beta_{4} - 4 \beta_{5} ) q^{71} + ( 5 \beta_{1} + 2 \beta_{4} ) q^{72} + ( 2 \beta_{1} + 5 \beta_{2} - 5 \beta_{4} + 2 \beta_{5} ) q^{73} + 5 \beta_{2} q^{75} + ( 5 \beta_{1} + 5 \beta_{2} - 4 \beta_{4} + 3 \beta_{5} ) q^{78} + ( -\beta_{2} + 6 \beta_{5} ) q^{81} + ( 7 - 5 \beta_{2} - 3 \beta_{3} + \beta_{5} ) q^{82} + ( -7 + 4 \beta_{1} - 2 \beta_{3} - 5 \beta_{4} ) q^{87} + ( 3 \beta_{2} - 5 \beta_{5} ) q^{92} + ( 1 - 2 \beta_{1} - 4 \beta_{3} - 5 \beta_{4} ) q^{93} + ( 5 - \beta_{2} + 3 \beta_{3} - 7 \beta_{5} ) q^{94} + ( 7 \beta_{1} - 2 \beta_{4} ) q^{96} -7 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 3q^{6} - 9q^{8} + O(q^{10})$$ $$6q + 3q^{6} - 9q^{8} + 30q^{25} + 27q^{26} - 12q^{27} - 33q^{36} - 24q^{39} - 39q^{48} + 42q^{49} + 3q^{52} + 15q^{58} - 72q^{59} + 45q^{62} - 21q^{64} + 33q^{82} - 48q^{87} - 6q^{93} + 39q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{3} + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + \nu^{2}$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 1$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} - \nu$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{5} + 3 \nu^{2}$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 1$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{4} + \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-\beta_{5} + 3 \beta_{2}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/552\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$185$$ $$277$$ $$415$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
413.1
 1.33454 + 0.467979i 1.33454 − 0.467979i −0.261988 + 1.38973i −0.261988 − 1.38973i −1.07255 + 0.921756i −1.07255 − 0.921756i
−1.33454 0.467979i 0.227452 + 1.71705i 1.56199 + 1.24907i 0 0.500000 2.39792i 0 −1.50000 2.39792i −2.89653 + 0.781094i 0
413.2 −1.33454 + 0.467979i 0.227452 1.71705i 1.56199 1.24907i 0 0.500000 + 2.39792i 0 −1.50000 + 2.39792i −2.89653 0.781094i 0
413.3 0.261988 1.38973i −1.60074 + 0.661546i −1.86272 0.728188i 0 0.500000 + 2.39792i 0 −1.50000 + 2.39792i 2.12471 2.11792i 0
413.4 0.261988 + 1.38973i −1.60074 0.661546i −1.86272 + 0.728188i 0 0.500000 2.39792i 0 −1.50000 2.39792i 2.12471 + 2.11792i 0
413.5 1.07255 0.921756i 1.37328 1.05550i 0.300733 1.97726i 0 0.500000 2.39792i 0 −1.50000 2.39792i 0.771819 2.89902i 0
413.6 1.07255 + 0.921756i 1.37328 + 1.05550i 0.300733 + 1.97726i 0 0.500000 + 2.39792i 0 −1.50000 + 2.39792i 0.771819 + 2.89902i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 413.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by $$\Q(\sqrt{-23})$$
24.h odd 2 1 inner
552.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 552.2.b.b yes 6
3.b odd 2 1 552.2.b.a 6
4.b odd 2 1 2208.2.b.b 6
8.b even 2 1 552.2.b.a 6
8.d odd 2 1 2208.2.b.a 6
12.b even 2 1 2208.2.b.a 6
23.b odd 2 1 CM 552.2.b.b yes 6
24.f even 2 1 2208.2.b.b 6
24.h odd 2 1 inner 552.2.b.b yes 6
69.c even 2 1 552.2.b.a 6
92.b even 2 1 2208.2.b.b 6
184.e odd 2 1 552.2.b.a 6
184.h even 2 1 2208.2.b.a 6
276.h odd 2 1 2208.2.b.a 6
552.b even 2 1 inner 552.2.b.b yes 6
552.h odd 2 1 2208.2.b.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.b.a 6 3.b odd 2 1
552.2.b.a 6 8.b even 2 1
552.2.b.a 6 69.c even 2 1
552.2.b.a 6 184.e odd 2 1
552.2.b.b yes 6 1.a even 1 1 trivial
552.2.b.b yes 6 23.b odd 2 1 CM
552.2.b.b yes 6 24.h odd 2 1 inner
552.2.b.b yes 6 552.b even 2 1 inner
2208.2.b.a 6 8.d odd 2 1
2208.2.b.a 6 12.b even 2 1
2208.2.b.a 6 184.h even 2 1
2208.2.b.a 6 276.h odd 2 1
2208.2.b.b 6 4.b odd 2 1
2208.2.b.b 6 24.f even 2 1
2208.2.b.b 6 92.b even 2 1
2208.2.b.b 6 552.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(552, [\chi])$$:

 $$T_{5}$$ $$T_{29}^{3} - 87 T_{29} - 282$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$8 + 3 T^{3} + T^{6}$$
$3$ $$27 + 4 T^{3} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$T^{6}$$
$11$ $$T^{6}$$
$13$ $$3312 + 1521 T^{2} + 78 T^{4} + T^{6}$$
$17$ $$T^{6}$$
$19$ $$T^{6}$$
$23$ $$( 23 + T^{2} )^{3}$$
$29$ $$( -282 - 87 T + T^{3} )^{2}$$
$31$ $$( 344 - 93 T + T^{3} )^{2}$$
$37$ $$T^{6}$$
$41$ $$94208 + 15129 T^{2} + 246 T^{4} + T^{6}$$
$43$ $$T^{6}$$
$47$ $$412988 + 19881 T^{2} + 282 T^{4} + T^{6}$$
$53$ $$T^{6}$$
$59$ $$( 12 + T )^{6}$$
$61$ $$T^{6}$$
$67$ $$T^{6}$$
$71$ $$48668 + 45369 T^{2} + 426 T^{4} + T^{6}$$
$73$ $$( 1226 - 219 T + T^{3} )^{2}$$
$79$ $$T^{6}$$
$83$ $$T^{6}$$
$89$ $$T^{6}$$
$97$ $$T^{6}$$