# Properties

 Label 552.2.f.b Level $552$ Weight $2$ Character orbit 552.f Analytic conductor $4.408$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{2} + \zeta_{8}^{2} q^{3} + 2 q^{4} + ( \zeta_{8} + 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{5} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{6} -2 q^{7} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} - q^{9} +O(q^{10})$$ $$q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{2} + \zeta_{8}^{2} q^{3} + 2 q^{4} + ( \zeta_{8} + 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{5} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{6} -2 q^{7} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} - q^{9} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{10} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{11} + 2 \zeta_{8}^{2} q^{12} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{13} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{14} + ( -2 - \zeta_{8} + \zeta_{8}^{3} ) q^{15} + 4 q^{16} + ( -4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{17} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{18} + ( 3 \zeta_{8} - 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{19} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{20} -2 \zeta_{8}^{2} q^{21} + 6 \zeta_{8}^{2} q^{22} - q^{23} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{24} + ( -1 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{25} + ( -4 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{26} -\zeta_{8}^{2} q^{27} -4 q^{28} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{29} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{30} + ( 8 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{31} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{32} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{33} + ( 4 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{34} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{35} -2 q^{36} + ( -5 \zeta_{8} + 4 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{37} + ( -2 \zeta_{8} + 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{38} + ( 4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{39} + ( 4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{40} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{41} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{42} + ( 3 \zeta_{8} - 6 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{43} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{44} + ( -\zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{45} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{46} + 6 q^{47} + 4 \zeta_{8}^{2} q^{48} -3 q^{49} + ( -8 - \zeta_{8} + \zeta_{8}^{3} ) q^{50} + ( 2 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{51} + ( -4 \zeta_{8} - 8 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{52} + ( -\zeta_{8} - 10 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{53} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{54} + ( -6 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{55} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{56} + ( 2 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{57} + 4 \zeta_{8}^{2} q^{58} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{59} + ( -4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{60} + ( 3 \zeta_{8} - 8 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{61} + ( 4 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{62} + 2 q^{63} + 8 q^{64} + ( 12 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{65} -6 q^{66} + ( 5 \zeta_{8} + 2 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{67} + ( -8 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{68} -\zeta_{8}^{2} q^{69} + ( -4 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{70} + ( -6 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{71} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{72} + ( 4 \zeta_{8} - 10 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{74} + ( -4 \zeta_{8} - \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{75} + ( 6 \zeta_{8} - 4 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{76} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{77} + ( 4 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{78} + ( -2 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{79} + ( 4 \zeta_{8} + 8 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{80} + q^{81} + 12 q^{82} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{83} -4 \zeta_{8}^{2} q^{84} -4 \zeta_{8}^{2} q^{85} + ( -6 \zeta_{8} + 6 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{86} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{87} + 12 \zeta_{8}^{2} q^{88} + ( 8 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{89} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{90} + ( 4 \zeta_{8} + 8 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{91} -2 q^{92} + ( 2 \zeta_{8} + 8 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{93} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{94} + ( -2 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{95} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{96} + ( -2 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{97} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{98} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{4} - 8q^{7} - 4q^{9} + O(q^{10})$$ $$4q + 8q^{4} - 8q^{7} - 4q^{9} - 8q^{15} + 16q^{16} - 16q^{17} - 4q^{23} - 4q^{25} - 16q^{28} - 8q^{30} + 32q^{31} + 16q^{34} - 8q^{36} + 16q^{39} + 24q^{47} - 12q^{49} - 32q^{50} - 24q^{55} + 8q^{57} - 16q^{60} + 16q^{62} + 8q^{63} + 32q^{64} + 48q^{65} - 24q^{66} - 32q^{68} - 24q^{71} + 16q^{78} - 8q^{79} + 4q^{81} + 48q^{82} + 32q^{89} - 8q^{92} - 8q^{95} - 8q^{97} + O(q^{100})$$

## 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
 −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
−1.41421 1.00000i 2.00000 0.585786i 1.41421i −2.00000 −2.82843 −1.00000 0.828427i
277.2 −1.41421 1.00000i 2.00000 0.585786i 1.41421i −2.00000 −2.82843 −1.00000 0.828427i
277.3 1.41421 1.00000i 2.00000 3.41421i 1.41421i −2.00000 2.82843 −1.00000 4.82843i
277.4 1.41421 1.00000i 2.00000 3.41421i 1.41421i −2.00000 2.82843 −1.00000 4.82843i
 $$n$$: 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.b 4
4.b odd 2 1 2208.2.f.b 4
8.b even 2 1 inner 552.2.f.b 4
8.d odd 2 1 2208.2.f.b 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 12 T_{5}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(552, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$4 + 12 T^{2} + T^{4}$$
$7$ $$( 2 + T )^{4}$$
$11$ $$( 18 + T^{2} )^{2}$$
$13$ $$64 + 48 T^{2} + T^{4}$$
$17$ $$( 8 + 8 T + T^{2} )^{2}$$
$19$ $$196 + 44 T^{2} + T^{4}$$
$23$ $$( 1 + T )^{4}$$
$29$ $$( 8 + T^{2} )^{2}$$
$31$ $$( 56 - 16 T + T^{2} )^{2}$$
$37$ $$1156 + 132 T^{2} + T^{4}$$
$41$ $$( -72 + T^{2} )^{2}$$
$43$ $$324 + 108 T^{2} + T^{4}$$
$47$ $$( -6 + T )^{4}$$
$53$ $$9604 + 204 T^{2} + T^{4}$$
$59$ $$( 32 + T^{2} )^{2}$$
$61$ $$2116 + 164 T^{2} + T^{4}$$
$67$ $$2116 + 108 T^{2} + T^{4}$$
$71$ $$( 4 + 12 T + T^{2} )^{2}$$
$73$ $$T^{4}$$
$79$ $$( -124 + 4 T + T^{2} )^{2}$$
$83$ $$( 18 + T^{2} )^{2}$$
$89$ $$( 56 - 16 T + T^{2} )^{2}$$
$97$ $$( -28 + 4 T + T^{2} )^{2}$$