Properties

 Label 552.2.n.a 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 4q^{2} + 24q^{3} + 4q^{4} - 4q^{6} - 4q^{8} + 24q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 4q^{2} + 24q^{3} + 4q^{4} - 4q^{6} - 4q^{8} + 24q^{9} + 4q^{12} + 4q^{16} - 4q^{18} - 4q^{24} + 24q^{25} + 24q^{27} - 4q^{32} + 4q^{36} + 4q^{46} + 4q^{48} - 8q^{49} - 44q^{50} - 4q^{54} + 48q^{58} - 40q^{62} + 4q^{64} - 4q^{72} - 32q^{73} + 24q^{75} + 24q^{81} - 40q^{82} - 52q^{92} + 40q^{94} - 4q^{96} - 20q^{98} + O(q^{100})$$

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 −1.39844 0.210663i 1.00000 1.91124 + 0.589197i −0.707233 −1.39844 0.210663i −4.06441 −2.54863 1.22658i 1.00000 0.989020 + 0.148988i
91.2 −1.39844 0.210663i 1.00000 1.91124 + 0.589197i 0.707233 −1.39844 0.210663i 4.06441 −2.54863 1.22658i 1.00000 −0.989020 0.148988i
91.3 −1.39844 + 0.210663i 1.00000 1.91124 0.589197i −0.707233 −1.39844 + 0.210663i −4.06441 −2.54863 + 1.22658i 1.00000 0.989020 0.148988i
91.4 −1.39844 + 0.210663i 1.00000 1.91124 0.589197i 0.707233 −1.39844 + 0.210663i 4.06441 −2.54863 + 1.22658i 1.00000 −0.989020 + 0.148988i
91.5 −1.21506 0.723626i 1.00000 0.952732 + 1.75849i −4.31142 −1.21506 0.723626i 1.64666 0.114868 2.82609i 1.00000 5.23862 + 3.11985i
91.6 −1.21506 0.723626i 1.00000 0.952732 + 1.75849i 4.31142 −1.21506 0.723626i −1.64666 0.114868 2.82609i 1.00000 −5.23862 3.11985i
91.7 −1.21506 + 0.723626i 1.00000 0.952732 1.75849i −4.31142 −1.21506 + 0.723626i 1.64666 0.114868 + 2.82609i 1.00000 5.23862 3.11985i
91.8 −1.21506 + 0.723626i 1.00000 0.952732 1.75849i 4.31142 −1.21506 + 0.723626i −1.64666 0.114868 + 2.82609i 1.00000 −5.23862 + 3.11985i
91.9 −0.695292 1.23149i 1.00000 −1.03314 + 1.71249i −1.12657 −0.695292 1.23149i −2.04137 2.82725 + 0.0816198i 1.00000 0.783294 + 1.38736i
91.10 −0.695292 1.23149i 1.00000 −1.03314 + 1.71249i 1.12657 −0.695292 1.23149i 2.04137 2.82725 + 0.0816198i 1.00000 −0.783294 1.38736i
91.11 −0.695292 + 1.23149i 1.00000 −1.03314 1.71249i −1.12657 −0.695292 + 1.23149i −2.04137 2.82725 0.0816198i 1.00000 0.783294 1.38736i
91.12 −0.695292 + 1.23149i 1.00000 −1.03314 1.71249i 1.12657 −0.695292 + 1.23149i 2.04137 2.82725 0.0816198i 1.00000 −0.783294 + 1.38736i
91.13 0.0902148 1.41133i 1.00000 −1.98372 0.254646i −3.06098 0.0902148 1.41133i −0.892124 −0.538352 + 2.77672i 1.00000 −0.276146 + 4.32007i
91.14 0.0902148 1.41133i 1.00000 −1.98372 0.254646i 3.06098 0.0902148 1.41133i 0.892124 −0.538352 + 2.77672i 1.00000 0.276146 4.32007i
91.15 0.0902148 + 1.41133i 1.00000 −1.98372 + 0.254646i −3.06098 0.0902148 + 1.41133i −0.892124 −0.538352 2.77672i 1.00000 −0.276146 4.32007i
91.16 0.0902148 + 1.41133i 1.00000 −1.98372 + 0.254646i 3.06098 0.0902148 + 1.41133i 0.892124 −0.538352 2.77672i 1.00000 0.276146 + 4.32007i
91.17 0.869059 1.11568i 1.00000 −0.489471 1.93918i −1.54212 0.869059 1.11568i −2.92435 −2.58888 1.13917i 1.00000 −1.34019 + 1.72051i
91.18 0.869059 1.11568i 1.00000 −0.489471 1.93918i 1.54212 0.869059 1.11568i 2.92435 −2.58888 1.13917i 1.00000 1.34019 1.72051i
91.19 0.869059 + 1.11568i 1.00000 −0.489471 + 1.93918i −1.54212 0.869059 + 1.11568i −2.92435 −2.58888 + 1.13917i 1.00000 −1.34019 1.72051i
91.20 0.869059 + 1.11568i 1.00000 −0.489471 + 1.93918i 1.54212 0.869059 + 1.11568i 2.92435 −2.58888 + 1.13917i 1.00000 1.34019 + 1.72051i
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.a 24
4.b odd 2 1 2208.2.n.a 24
8.b even 2 1 2208.2.n.a 24
8.d odd 2 1 inner 552.2.n.a 24
23.b odd 2 1 inner 552.2.n.a 24
92.b even 2 1 2208.2.n.a 24
184.e odd 2 1 2208.2.n.a 24
184.h even 2 1 inner 552.2.n.a 24

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{12} - 36 T_{5}^{10} + 420 T_{5}^{8} - 2008 T_{5}^{6} + 4232 T_{5}^{4} - 3712 T_{5}^{2} + 1024$$ acting on $$S_{2}^{\mathrm{new}}(552, [\chi])$$.