# Properties

 Label 552.2.n.b Level $552$ Weight $2$ Character orbit 552.n Analytic conductor $4.408$ Analytic rank $0$ Dimension $24$ 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.n (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.40774219157$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 24q^{3} + 4q^{4} + 24q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 24q^{3} + 4q^{4} + 24q^{9} - 4q^{12} + 4q^{16} + 24q^{25} - 24q^{27} + 4q^{36} - 44q^{46} - 4q^{48} + 56q^{49} - 40q^{50} - 48q^{58} - 40q^{62} + 4q^{64} + 32q^{73} - 24q^{75} + 24q^{81} - 40q^{82} + 40q^{92} - 48q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1 −1.38507 0.285614i −1.00000 1.83685 + 0.791193i −2.03317 1.38507 + 0.285614i 1.21988 −2.31819 1.62049i 1.00000 2.81608 + 0.580701i
91.2 −1.38507 0.285614i −1.00000 1.83685 + 0.791193i 2.03317 1.38507 + 0.285614i −1.21988 −2.31819 1.62049i 1.00000 −2.81608 0.580701i
91.3 −1.38507 + 0.285614i −1.00000 1.83685 0.791193i −2.03317 1.38507 0.285614i 1.21988 −2.31819 + 1.62049i 1.00000 2.81608 0.580701i
91.4 −1.38507 + 0.285614i −1.00000 1.83685 0.791193i 2.03317 1.38507 0.285614i −1.21988 −2.31819 + 1.62049i 1.00000 −2.81608 + 0.580701i
91.5 −1.07870 0.914558i −1.00000 0.327167 + 1.97306i −3.32574 1.07870 + 0.914558i −3.62001 1.45156 2.42754i 1.00000 3.58746 + 3.04158i
91.6 −1.07870 0.914558i −1.00000 0.327167 + 1.97306i 3.32574 1.07870 + 0.914558i 3.62001 1.45156 2.42754i 1.00000 −3.58746 3.04158i
91.7 −1.07870 + 0.914558i −1.00000 0.327167 1.97306i −3.32574 1.07870 0.914558i −3.62001 1.45156 + 2.42754i 1.00000 3.58746 3.04158i
91.8 −1.07870 + 0.914558i −1.00000 0.327167 1.97306i 3.32574 1.07870 0.914558i 3.62001 1.45156 + 2.42754i 1.00000 −3.58746 + 3.04158i
91.9 −0.409868 1.35352i −1.00000 −1.66402 + 1.10953i −3.34475 0.409868 + 1.35352i 4.25021 2.18379 + 1.79751i 1.00000 1.37091 + 4.52717i
91.10 −0.409868 1.35352i −1.00000 −1.66402 + 1.10953i 3.34475 0.409868 + 1.35352i −4.25021 2.18379 + 1.79751i 1.00000 −1.37091 4.52717i
91.11 −0.409868 + 1.35352i −1.00000 −1.66402 1.10953i −3.34475 0.409868 1.35352i 4.25021 2.18379 1.79751i 1.00000 1.37091 4.52717i
91.12 −0.409868 + 1.35352i −1.00000 −1.66402 1.10953i 3.34475 0.409868 1.35352i −4.25021 2.18379 1.79751i 1.00000 −1.37091 + 4.52717i
91.13 0.409868 1.35352i −1.00000 −1.66402 1.10953i −0.901487 −0.409868 + 1.35352i −1.56211 −2.18379 + 1.79751i 1.00000 −0.369491 + 1.22018i
91.14 0.409868 1.35352i −1.00000 −1.66402 1.10953i 0.901487 −0.409868 + 1.35352i 1.56211 −2.18379 + 1.79751i 1.00000 0.369491 1.22018i
91.15 0.409868 + 1.35352i −1.00000 −1.66402 + 1.10953i −0.901487 −0.409868 1.35352i −1.56211 −2.18379 1.79751i 1.00000 −0.369491 1.22018i
91.16 0.409868 + 1.35352i −1.00000 −1.66402 + 1.10953i 0.901487 −0.409868 1.35352i 1.56211 −2.18379 1.79751i 1.00000 0.369491 + 1.22018i
91.17 1.07870 0.914558i −1.00000 0.327167 1.97306i −0.969269 −1.07870 + 0.914558i −4.55308 −1.45156 2.42754i 1.00000 −1.04555 + 0.886452i
91.18 1.07870 0.914558i −1.00000 0.327167 1.97306i 0.969269 −1.07870 + 0.914558i 4.55308 −1.45156 2.42754i 1.00000 1.04555 0.886452i
91.19 1.07870 + 0.914558i −1.00000 0.327167 + 1.97306i −0.969269 −1.07870 0.914558i −4.55308 −1.45156 + 2.42754i 1.00000 −1.04555 0.886452i
91.20 1.07870 + 0.914558i −1.00000 0.327167 + 1.97306i 0.969269 −1.07870 0.914558i 4.55308 −1.45156 + 2.42754i 1.00000 1.04555 + 0.886452i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.24 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.d odd 2 1 inner
23.b odd 2 1 inner
184.h 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.n.b 24
4.b odd 2 1 2208.2.n.b 24
8.b even 2 1 2208.2.n.b 24
8.d odd 2 1 inner 552.2.n.b 24
23.b odd 2 1 inner 552.2.n.b 24
92.b even 2 1 2208.2.n.b 24
184.e odd 2 1 2208.2.n.b 24
184.h even 2 1 inner 552.2.n.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.n.b 24 1.a even 1 1 trivial
552.2.n.b 24 8.d odd 2 1 inner
552.2.n.b 24 23.b odd 2 1 inner
552.2.n.b 24 184.h even 2 1 inner
2208.2.n.b 24 4.b odd 2 1
2208.2.n.b 24 8.b even 2 1
2208.2.n.b 24 92.b even 2 1
2208.2.n.b 24 184.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{12} - 36 T_{5}^{10} + 484 T_{5}^{8} - 2976 T_{5}^{6} + 8216 T_{5}^{4} - 8736 T_{5}^{2} + 3072$$ acting on $$S_{2}^{\mathrm{new}}(552, [\chi])$$.