# Properties

 Label 552.2.j.b Level $552$ Weight $2$ Character orbit 552.j Analytic conductor $4.408$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.40774219157$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( -1 + \beta ) q^{3} -2 q^{4} + 4 q^{5} + ( -2 - \beta ) q^{6} -2 \beta q^{7} -2 \beta q^{8} + ( -1 - 2 \beta ) q^{9} +O(q^{10})$$ $$q + \beta q^{2} + ( -1 + \beta ) q^{3} -2 q^{4} + 4 q^{5} + ( -2 - \beta ) q^{6} -2 \beta q^{7} -2 \beta q^{8} + ( -1 - 2 \beta ) q^{9} + 4 \beta q^{10} + 4 \beta q^{11} + ( 2 - 2 \beta ) q^{12} + 4 \beta q^{13} + 4 q^{14} + ( -4 + 4 \beta ) q^{15} + 4 q^{16} + 2 \beta q^{17} + ( 4 - \beta ) q^{18} -2 q^{19} -8 q^{20} + ( 4 + 2 \beta ) q^{21} -8 q^{22} - q^{23} + ( 4 + 2 \beta ) q^{24} + 11 q^{25} -8 q^{26} + ( 5 + \beta ) q^{27} + 4 \beta q^{28} + ( -8 - 4 \beta ) q^{30} + 4 \beta q^{31} + 4 \beta q^{32} + ( -8 - 4 \beta ) q^{33} -4 q^{34} -8 \beta q^{35} + ( 2 + 4 \beta ) q^{36} -2 \beta q^{37} -2 \beta q^{38} + ( -8 - 4 \beta ) q^{39} -8 \beta q^{40} -4 \beta q^{41} + ( -4 + 4 \beta ) q^{42} -6 q^{43} -8 \beta q^{44} + ( -4 - 8 \beta ) q^{45} -\beta q^{46} + ( -4 + 4 \beta ) q^{48} - q^{49} + 11 \beta q^{50} + ( -4 - 2 \beta ) q^{51} -8 \beta q^{52} + 12 q^{53} + ( -2 + 5 \beta ) q^{54} + 16 \beta q^{55} -8 q^{56} + ( 2 - 2 \beta ) q^{57} -2 \beta q^{59} + ( 8 - 8 \beta ) q^{60} + 2 \beta q^{61} -8 q^{62} + ( -8 + 2 \beta ) q^{63} -8 q^{64} + 16 \beta q^{65} + ( 8 - 8 \beta ) q^{66} + 2 q^{67} -4 \beta q^{68} + ( 1 - \beta ) q^{69} + 16 q^{70} + 8 q^{71} + ( -8 + 2 \beta ) q^{72} -6 q^{73} + 4 q^{74} + ( -11 + 11 \beta ) q^{75} + 4 q^{76} + 16 q^{77} + ( 8 - 8 \beta ) q^{78} -2 \beta q^{79} + 16 q^{80} + ( -7 + 4 \beta ) q^{81} + 8 q^{82} -4 \beta q^{83} + ( -8 - 4 \beta ) q^{84} + 8 \beta q^{85} -6 \beta q^{86} + 16 q^{88} -6 \beta q^{89} + ( 16 - 4 \beta ) q^{90} + 16 q^{91} + 2 q^{92} + ( -8 - 4 \beta ) q^{93} -8 q^{95} + ( -8 - 4 \beta ) q^{96} -2 q^{97} -\beta q^{98} + ( 16 - 4 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 4q^{4} + 8q^{5} - 4q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 4q^{4} + 8q^{5} - 4q^{6} - 2q^{9} + 4q^{12} + 8q^{14} - 8q^{15} + 8q^{16} + 8q^{18} - 4q^{19} - 16q^{20} + 8q^{21} - 16q^{22} - 2q^{23} + 8q^{24} + 22q^{25} - 16q^{26} + 10q^{27} - 16q^{30} - 16q^{33} - 8q^{34} + 4q^{36} - 16q^{39} - 8q^{42} - 12q^{43} - 8q^{45} - 8q^{48} - 2q^{49} - 8q^{51} + 24q^{53} - 4q^{54} - 16q^{56} + 4q^{57} + 16q^{60} - 16q^{62} - 16q^{63} - 16q^{64} + 16q^{66} + 4q^{67} + 2q^{69} + 32q^{70} + 16q^{71} - 16q^{72} - 12q^{73} + 8q^{74} - 22q^{75} + 8q^{76} + 32q^{77} + 16q^{78} + 32q^{80} - 14q^{81} + 16q^{82} - 16q^{84} + 32q^{88} + 32q^{90} + 32q^{91} + 4q^{92} - 16q^{93} - 16q^{95} - 16q^{96} - 4q^{97} + 32q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
323.1
 − 1.41421i 1.41421i
1.41421i −1.00000 1.41421i −2.00000 4.00000 −2.00000 + 1.41421i 2.82843i 2.82843i −1.00000 + 2.82843i 5.65685i
323.2 1.41421i −1.00000 + 1.41421i −2.00000 4.00000 −2.00000 1.41421i 2.82843i 2.82843i −1.00000 2.82843i 5.65685i
 $$n$$: 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
24.f 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.j.b yes 2
3.b odd 2 1 552.2.j.a 2
4.b odd 2 1 2208.2.j.b 2
8.b even 2 1 2208.2.j.a 2
8.d odd 2 1 552.2.j.a 2
12.b even 2 1 2208.2.j.a 2
24.f even 2 1 inner 552.2.j.b yes 2
24.h odd 2 1 2208.2.j.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.j.a 2 3.b odd 2 1
552.2.j.a 2 8.d odd 2 1
552.2.j.b yes 2 1.a even 1 1 trivial
552.2.j.b yes 2 24.f even 2 1 inner
2208.2.j.a 2 8.b even 2 1
2208.2.j.a 2 12.b even 2 1
2208.2.j.b 2 4.b odd 2 1
2208.2.j.b 2 24.h odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T^{2}$$
$3$ $$3 + 2 T + T^{2}$$
$5$ $$( -4 + T )^{2}$$
$7$ $$8 + T^{2}$$
$11$ $$32 + T^{2}$$
$13$ $$32 + T^{2}$$
$17$ $$8 + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$T^{2}$$
$31$ $$32 + T^{2}$$
$37$ $$8 + T^{2}$$
$41$ $$32 + T^{2}$$
$43$ $$( 6 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$( -12 + T )^{2}$$
$59$ $$8 + T^{2}$$
$61$ $$8 + T^{2}$$
$67$ $$( -2 + T )^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$( 6 + T )^{2}$$
$79$ $$8 + T^{2}$$
$83$ $$32 + T^{2}$$
$89$ $$72 + T^{2}$$
$97$ $$( 2 + T )^{2}$$