# Properties

 Label 552.2.q.a Level $552$ Weight $2$ Character orbit 552.q Analytic conductor $4.408$ Analytic rank $0$ Dimension $30$ 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.q (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.40774219157$$ Analytic rank: $$0$$ Dimension: $$30$$ Relative dimension: $$3$$ over $$\Q(\zeta_{11})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$30q - 3q^{3} - 2q^{7} - 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q - 3q^{3} - 2q^{7} - 3q^{9} - 15q^{11} + 5q^{13} - 15q^{17} + 5q^{19} + 9q^{21} + 14q^{23} + 5q^{25} - 3q^{27} + 13q^{29} - 29q^{31} - 15q^{33} + 8q^{35} - 3q^{37} + 5q^{39} + 24q^{41} + 2q^{43} + 22q^{45} + 38q^{47} + 15q^{49} + 7q^{51} - 7q^{53} + 46q^{55} - 6q^{57} - 43q^{59} - 22q^{61} - 2q^{63} + 44q^{65} + 31q^{67} + 3q^{69} + 19q^{71} - 50q^{73} + 5q^{75} + 16q^{77} - 84q^{79} - 3q^{81} - 73q^{83} - 8q^{85} + 2q^{87} + 19q^{89} + 18q^{91} + 26q^{93} - 67q^{95} - 29q^{97} - 15q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1 0 −0.142315 0.989821i 0 −2.98780 0.877298i 0 0.0902354 0.104137i 0 −0.959493 + 0.281733i 0
25.2 0 −0.142315 0.989821i 0 −0.768760 0.225728i 0 −0.186229 + 0.214919i 0 −0.959493 + 0.281733i 0
25.3 0 −0.142315 0.989821i 0 2.63475 + 0.773633i 0 −2.50193 + 2.88738i 0 −0.959493 + 0.281733i 0
49.1 0 0.415415 0.909632i 0 −1.96611 + 2.26901i 0 3.59431 2.30992i 0 −0.654861 0.755750i 0
49.2 0 0.415415 0.909632i 0 −1.13129 + 1.30558i 0 −2.90759 + 1.86859i 0 −0.654861 0.755750i 0
49.3 0 0.415415 0.909632i 0 0.531006 0.612813i 0 0.0574024 0.0368903i 0 −0.654861 0.755750i 0
73.1 0 −0.959493 + 0.281733i 0 −1.03173 0.663053i 0 −0.0639556 0.444821i 0 0.841254 0.540641i 0
73.2 0 −0.959493 + 0.281733i 0 0.191143 + 0.122840i 0 0.0348746 + 0.242558i 0 0.841254 0.540641i 0
73.3 0 −0.959493 + 0.281733i 0 3.62490 + 2.32958i 0 −0.646888 4.49921i 0 0.841254 0.540641i 0
121.1 0 −0.959493 0.281733i 0 −1.03173 + 0.663053i 0 −0.0639556 + 0.444821i 0 0.841254 + 0.540641i 0
121.2 0 −0.959493 0.281733i 0 0.191143 0.122840i 0 0.0348746 0.242558i 0 0.841254 + 0.540641i 0
121.3 0 −0.959493 0.281733i 0 3.62490 2.32958i 0 −0.646888 + 4.49921i 0 0.841254 + 0.540641i 0
169.1 0 0.415415 + 0.909632i 0 −1.96611 2.26901i 0 3.59431 + 2.30992i 0 −0.654861 + 0.755750i 0
169.2 0 0.415415 + 0.909632i 0 −1.13129 1.30558i 0 −2.90759 1.86859i 0 −0.654861 + 0.755750i 0
169.3 0 0.415415 + 0.909632i 0 0.531006 + 0.612813i 0 0.0574024 + 0.0368903i 0 −0.654861 + 0.755750i 0
193.1 0 −0.654861 0.755750i 0 −0.494929 3.44231i 0 0.813772 1.78191i 0 −0.142315 + 0.989821i 0
193.2 0 −0.654861 0.755750i 0 0.140637 + 0.978155i 0 −1.46137 + 3.19994i 0 −0.142315 + 0.989821i 0
193.3 0 −0.654861 0.755750i 0 0.309108 + 2.14989i 0 1.22533 2.68309i 0 −0.142315 + 0.989821i 0
265.1 0 −0.142315 + 0.989821i 0 −2.98780 + 0.877298i 0 0.0902354 + 0.104137i 0 −0.959493 0.281733i 0
265.2 0 −0.142315 + 0.989821i 0 −0.768760 + 0.225728i 0 −0.186229 0.214919i 0 −0.959493 0.281733i 0
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 409.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 552.2.q.a 30
23.c even 11 1 inner 552.2.q.a 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.q.a 30 1.a even 1 1 trivial
552.2.q.a 30 23.c even 11 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{30} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(552, [\chi])$$.