# Properties

 Label 552.2.m.a Level $552$ Weight $2$ Character orbit 552.m Analytic conductor $4.408$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.40774219157$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.40960000.1 Defining polynomial: $$x^{8} + 7 x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{2} ) q^{3} + \beta_{3} q^{5} + \beta_{5} q^{7} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{4} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{2} ) q^{3} + \beta_{3} q^{5} + \beta_{5} q^{7} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{4} ) q^{9} + ( -\beta_{3} + 3 \beta_{6} ) q^{11} + ( -2 - \beta_{2} - \beta_{4} ) q^{13} + ( -2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{15} + ( 4 \beta_{3} - \beta_{6} ) q^{17} + ( \beta_{5} + 3 \beta_{7} ) q^{19} + ( \beta_{3} - \beta_{7} ) q^{21} + ( -\beta_{1} - \beta_{2} + \beta_{4} - 3 \beta_{6} ) q^{23} + ( -3 - \beta_{2} - \beta_{4} ) q^{25} + ( -3 - 4 \beta_{1} + \beta_{4} ) q^{27} + ( 4 \beta_{1} + \beta_{2} - \beta_{4} ) q^{29} -6 q^{31} + ( 3 \beta_{3} - \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{33} + ( \beta_{2} - \beta_{4} ) q^{35} + ( 6 \beta_{5} + \beta_{7} ) q^{37} + ( 4 + 2 \beta_{1} + \beta_{2} + \beta_{4} ) q^{39} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} ) q^{41} + ( -\beta_{5} + 5 \beta_{7} ) q^{43} + ( -\beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{45} + ( 8 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} ) q^{47} + 5 q^{49} + ( -\beta_{3} - 7 \beta_{5} + 5 \beta_{6} + 3 \beta_{7} ) q^{51} -\beta_{3} q^{53} + ( -2 - 2 \beta_{2} - 2 \beta_{4} ) q^{55} + ( \beta_{3} - 3 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} ) q^{57} -8 \beta_{1} q^{59} + ( 4 \beta_{5} - 3 \beta_{7} ) q^{61} + ( -\beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{63} + 2 \beta_{6} q^{65} + ( -3 \beta_{5} - \beta_{7} ) q^{67} + ( -2 + 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{69} + ( -6 \beta_{1} - 5 \beta_{2} + 5 \beta_{4} ) q^{71} + ( -6 - \beta_{2} - \beta_{4} ) q^{73} + ( 5 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{75} + ( -6 \beta_{1} - \beta_{2} + \beta_{4} ) q^{77} + ( 5 \beta_{5} - 6 \beta_{7} ) q^{79} + ( 1 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{4} ) q^{81} + ( \beta_{3} - 5 \beta_{6} ) q^{83} + ( 8 - 3 \beta_{2} - 3 \beta_{4} ) q^{85} + ( 2 - 6 \beta_{1} - \beta_{2} + 3 \beta_{4} ) q^{87} + ( -2 \beta_{3} - \beta_{6} ) q^{89} -2 \beta_{7} q^{91} + ( 6 + 6 \beta_{2} ) q^{93} + ( 6 \beta_{1} + \beta_{2} - \beta_{4} ) q^{95} + ( 8 \beta_{5} - 6 \beta_{7} ) q^{97} + ( -5 \beta_{3} - 4 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{3} + O(q^{10})$$ $$8q - 4q^{3} - 8q^{13} - 16q^{25} - 28q^{27} - 48q^{31} + 24q^{39} + 40q^{49} - 20q^{69} - 40q^{73} + 28q^{75} + 8q^{81} + 88q^{85} + 8q^{87} + 24q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 7 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} + 8 \nu^{2}$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + \nu^{4} + 5 \nu^{2} + 2$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} - 8 \nu^{3} + 3 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} + \nu^{4} - 5 \nu^{2} + 2$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{7} + \nu^{5} + 13 \nu^{3} + 5 \nu$$$$)/3$$ $$\beta_{6}$$ $$=$$ $$($$$$2 \nu^{7} - \nu^{5} + 13 \nu^{3} - 5 \nu$$$$)/3$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{7} + \nu^{5} + 21 \nu^{3} + 8 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{5} + \beta_{3}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} - \beta_{2} + 2 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} - \beta_{6} - 3 \beta_{5} - 2 \beta_{3}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{4} + 3 \beta_{2} - 4$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{7} - 3 \beta_{6} + 8 \beta_{5} - 5 \beta_{3}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-4 \beta_{4} + 4 \beta_{2} - 5 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$($$$$-13 \beta_{7} + 8 \beta_{6} + 21 \beta_{5} + 13 \beta_{3}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
137.1
 −0.437016 − 0.437016i 0.437016 + 0.437016i −0.437016 + 0.437016i 0.437016 − 0.437016i −1.14412 + 1.14412i 1.14412 − 1.14412i −1.14412 − 1.14412i 1.14412 + 1.14412i
0 −1.61803 0.618034i 0 −0.874032 0 1.41421i 0 2.23607 + 2.00000i 0
137.2 0 −1.61803 0.618034i 0 0.874032 0 1.41421i 0 2.23607 + 2.00000i 0
137.3 0 −1.61803 + 0.618034i 0 −0.874032 0 1.41421i 0 2.23607 2.00000i 0
137.4 0 −1.61803 + 0.618034i 0 0.874032 0 1.41421i 0 2.23607 2.00000i 0
137.5 0 0.618034 1.61803i 0 −2.28825 0 1.41421i 0 −2.23607 2.00000i 0
137.6 0 0.618034 1.61803i 0 2.28825 0 1.41421i 0 −2.23607 2.00000i 0
137.7 0 0.618034 + 1.61803i 0 −2.28825 0 1.41421i 0 −2.23607 + 2.00000i 0
137.8 0 0.618034 + 1.61803i 0 2.28825 0 1.41421i 0 −2.23607 + 2.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 137.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.b odd 2 1 inner
69.c 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.m.a 8
3.b odd 2 1 inner 552.2.m.a 8
4.b odd 2 1 1104.2.m.c 8
12.b even 2 1 1104.2.m.c 8
23.b odd 2 1 inner 552.2.m.a 8
69.c even 2 1 inner 552.2.m.a 8
92.b even 2 1 1104.2.m.c 8
276.h odd 2 1 1104.2.m.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.m.a 8 1.a even 1 1 trivial
552.2.m.a 8 3.b odd 2 1 inner
552.2.m.a 8 23.b odd 2 1 inner
552.2.m.a 8 69.c even 2 1 inner
1104.2.m.c 8 4.b odd 2 1
1104.2.m.c 8 12.b even 2 1
1104.2.m.c 8 92.b even 2 1
1104.2.m.c 8 276.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 6 T_{5}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(552, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 9 + 6 T + 2 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$5$ $$( 4 - 6 T^{2} + T^{4} )^{2}$$
$7$ $$( 2 + T^{2} )^{4}$$
$11$ $$( 100 - 30 T^{2} + T^{4} )^{2}$$
$13$ $$( -4 + 2 T + T^{2} )^{4}$$
$17$ $$( 1444 - 84 T^{2} + T^{4} )^{2}$$
$19$ $$( 100 + 70 T^{2} + T^{4} )^{2}$$
$23$ $$( 529 - 26 T^{2} + T^{4} )^{2}$$
$29$ $$( 16 + 28 T^{2} + T^{4} )^{2}$$
$31$ $$( 6 + T )^{8}$$
$37$ $$( 6724 + 174 T^{2} + T^{4} )^{2}$$
$41$ $$( 16 + 72 T^{2} + T^{4} )^{2}$$
$43$ $$( 3364 + 134 T^{2} + T^{4} )^{2}$$
$47$ $$( 400 + 140 T^{2} + T^{4} )^{2}$$
$53$ $$( 4 - 6 T^{2} + T^{4} )^{2}$$
$59$ $$( 64 + T^{2} )^{4}$$
$61$ $$( 100 + 70 T^{2} + T^{4} )^{2}$$
$67$ $$( 484 + 54 T^{2} + T^{4} )^{2}$$
$71$ $$( 15376 + 252 T^{2} + T^{4} )^{2}$$
$73$ $$( 20 + 10 T + T^{2} )^{4}$$
$79$ $$( 6724 + 196 T^{2} + T^{4} )^{2}$$
$83$ $$( 1444 - 86 T^{2} + T^{4} )^{2}$$
$89$ $$( 4 - 36 T^{2} + T^{4} )^{2}$$
$97$ $$( 1600 + 280 T^{2} + T^{4} )^{2}$$