# Properties

 Label 552.2.f.a Level $552$ Weight $2$ Character orbit 552.f 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.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.40774219157$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + i ) q^{2} + i q^{3} + 2 i q^{4} + 2 i q^{5} + ( -1 + i ) q^{6} + 4 q^{7} + ( -2 + 2 i ) q^{8} - q^{9} +O(q^{10})$$ $$q + ( 1 + i ) q^{2} + i q^{3} + 2 i q^{4} + 2 i q^{5} + ( -1 + i ) q^{6} + 4 q^{7} + ( -2 + 2 i ) q^{8} - q^{9} + ( -2 + 2 i ) q^{10} + 2 i q^{11} -2 q^{12} -4 i q^{13} + ( 4 + 4 i ) q^{14} -2 q^{15} -4 q^{16} + ( -1 - i ) q^{18} -4 i q^{19} -4 q^{20} + 4 i q^{21} + ( -2 + 2 i ) q^{22} + q^{23} + ( -2 - 2 i ) q^{24} + q^{25} + ( 4 - 4 i ) q^{26} -i q^{27} + 8 i q^{28} -2 i q^{29} + ( -2 - 2 i ) q^{30} -6 q^{31} + ( -4 - 4 i ) q^{32} -2 q^{33} + 8 i q^{35} -2 i q^{36} + 6 i q^{37} + ( 4 - 4 i ) q^{38} + 4 q^{39} + ( -4 - 4 i ) q^{40} + 2 q^{41} + ( -4 + 4 i ) q^{42} -8 i q^{43} -4 q^{44} -2 i q^{45} + ( 1 + i ) q^{46} -8 q^{47} -4 i q^{48} + 9 q^{49} + ( 1 + i ) q^{50} + 8 q^{52} + 6 i q^{53} + ( 1 - i ) q^{54} -4 q^{55} + ( -8 + 8 i ) q^{56} + 4 q^{57} + ( 2 - 2 i ) q^{58} -4 i q^{59} -4 i q^{60} + 6 i q^{61} + ( -6 - 6 i ) q^{62} -4 q^{63} -8 i q^{64} + 8 q^{65} + ( -2 - 2 i ) q^{66} -4 i q^{67} + i q^{69} + ( -8 + 8 i ) q^{70} + 12 q^{71} + ( 2 - 2 i ) q^{72} + 14 q^{73} + ( -6 + 6 i ) q^{74} + i q^{75} + 8 q^{76} + 8 i q^{77} + ( 4 + 4 i ) q^{78} + 8 q^{79} -8 i q^{80} + q^{81} + ( 2 + 2 i ) q^{82} -14 i q^{83} -8 q^{84} + ( 8 - 8 i ) q^{86} + 2 q^{87} + ( -4 - 4 i ) q^{88} + 12 q^{89} + ( 2 - 2 i ) q^{90} -16 i q^{91} + 2 i q^{92} -6 i q^{93} + ( -8 - 8 i ) q^{94} + 8 q^{95} + ( 4 - 4 i ) q^{96} -14 q^{97} + ( 9 + 9 i ) q^{98} -2 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{6} + 8q^{7} - 4q^{8} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{6} + 8q^{7} - 4q^{8} - 2q^{9} - 4q^{10} - 4q^{12} + 8q^{14} - 4q^{15} - 8q^{16} - 2q^{18} - 8q^{20} - 4q^{22} + 2q^{23} - 4q^{24} + 2q^{25} + 8q^{26} - 4q^{30} - 12q^{31} - 8q^{32} - 4q^{33} + 8q^{38} + 8q^{39} - 8q^{40} + 4q^{41} - 8q^{42} - 8q^{44} + 2q^{46} - 16q^{47} + 18q^{49} + 2q^{50} + 16q^{52} + 2q^{54} - 8q^{55} - 16q^{56} + 8q^{57} + 4q^{58} - 12q^{62} - 8q^{63} + 16q^{65} - 4q^{66} - 16q^{70} + 24q^{71} + 4q^{72} + 28q^{73} - 12q^{74} + 16q^{76} + 8q^{78} + 16q^{79} + 2q^{81} + 4q^{82} - 16q^{84} + 16q^{86} + 4q^{87} - 8q^{88} + 24q^{89} + 4q^{90} - 16q^{94} + 16q^{95} + 8q^{96} - 28q^{97} + 18q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
277.1
 − 1.00000i 1.00000i
1.00000 1.00000i 1.00000i 2.00000i 2.00000i −1.00000 1.00000i 4.00000 −2.00000 2.00000i −1.00000 −2.00000 2.00000i
277.2 1.00000 + 1.00000i 1.00000i 2.00000i 2.00000i −1.00000 + 1.00000i 4.00000 −2.00000 + 2.00000i −1.00000 −2.00000 + 2.00000i
 $$n$$: 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
8.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.f.a 2
4.b odd 2 1 2208.2.f.a 2
8.b even 2 1 inner 552.2.f.a 2
8.d odd 2 1 2208.2.f.a 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 - 2 T + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$4 + T^{2}$$
$7$ $$( -4 + T )^{2}$$
$11$ $$4 + T^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$16 + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$4 + T^{2}$$
$31$ $$( 6 + T )^{2}$$
$37$ $$36 + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$( 8 + T )^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$16 + T^{2}$$
$61$ $$36 + T^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$( -14 + T )^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$196 + T^{2}$$
$89$ $$( -12 + T )^{2}$$
$97$ $$( 14 + T )^{2}$$