# Properties

 Label 552.2.q.b 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30q - 3q^{3} - 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q - 3q^{3} - 3q^{9} + 9q^{11} + 13q^{13} + 17q^{17} - 9q^{19} - 11q^{21} - 12q^{23} - 23q^{25} - 3q^{27} - q^{29} - 37q^{31} + 9q^{33} + 10q^{35} + 7q^{37} + 13q^{39} + 16q^{41} + 20q^{43} - 22q^{45} + 22q^{47} + 19q^{49} + 17q^{51} + 25q^{53} + 10q^{55} + 24q^{57} + 7q^{59} + 8q^{61} - 28q^{65} - 23q^{67} - q^{69} + 5q^{71} + 34q^{73} - 23q^{75} + 62q^{77} + 20q^{79} - 3q^{81} + 29q^{83} - 46q^{85} + 10q^{87} - 67q^{89} - 118q^{91} - 26q^{93} - 99q^{95} - 41q^{97} + 9q^{99} + O(q^{100})$$

## 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 −0.911030 0.267503i 0 −1.46467 + 1.69032i 0 −0.959493 + 0.281733i 0
25.2 0 −0.142315 0.989821i 0 −0.451280 0.132508i 0 −0.325189 + 0.375288i 0 −0.959493 + 0.281733i 0
25.3 0 −0.142315 0.989821i 0 2.48412 + 0.729404i 0 3.07806 3.55227i 0 −0.959493 + 0.281733i 0
49.1 0 0.415415 0.909632i 0 −1.20489 + 1.39052i 0 −0.921355 + 0.592119i 0 −0.654861 0.755750i 0
49.2 0 0.415415 0.909632i 0 1.31805 1.52111i 0 4.20804 2.70434i 0 −0.654861 0.755750i 0
49.3 0 0.415415 0.909632i 0 2.45323 2.83118i 0 −2.34830 + 1.50916i 0 −0.654861 0.755750i 0
73.1 0 −0.959493 + 0.281733i 0 −2.59066 1.66492i 0 0.565805 + 3.93526i 0 0.841254 0.540641i 0
73.2 0 −0.959493 + 0.281733i 0 −2.20233 1.41535i 0 −0.514962 3.58164i 0 0.841254 0.540641i 0
73.3 0 −0.959493 + 0.281733i 0 2.00868 + 1.29090i 0 0.340497 + 2.36821i 0 0.841254 0.540641i 0
121.1 0 −0.959493 0.281733i 0 −2.59066 + 1.66492i 0 0.565805 3.93526i 0 0.841254 + 0.540641i 0
121.2 0 −0.959493 0.281733i 0 −2.20233 + 1.41535i 0 −0.514962 + 3.58164i 0 0.841254 + 0.540641i 0
121.3 0 −0.959493 0.281733i 0 2.00868 1.29090i 0 0.340497 2.36821i 0 0.841254 + 0.540641i 0
169.1 0 0.415415 + 0.909632i 0 −1.20489 1.39052i 0 −0.921355 0.592119i 0 −0.654861 + 0.755750i 0
169.2 0 0.415415 + 0.909632i 0 1.31805 + 1.52111i 0 4.20804 + 2.70434i 0 −0.654861 + 0.755750i 0
169.3 0 0.415415 + 0.909632i 0 2.45323 + 2.83118i 0 −2.34830 1.50916i 0 −0.654861 + 0.755750i 0
193.1 0 −0.654861 0.755750i 0 −0.412187 2.86682i 0 1.09223 2.39166i 0 −0.142315 + 0.989821i 0
193.2 0 −0.654861 0.755750i 0 −0.000386898 0.00269093i 0 −0.858275 + 1.87936i 0 −0.142315 + 0.989821i 0
193.3 0 −0.654861 0.755750i 0 0.457758 + 3.18377i 0 0.0191394 0.0419095i 0 −0.142315 + 0.989821i 0
265.1 0 −0.142315 + 0.989821i 0 −0.911030 + 0.267503i 0 −1.46467 1.69032i 0 −0.959493 0.281733i 0
265.2 0 −0.142315 + 0.989821i 0 −0.451280 + 0.132508i 0 −0.325189 0.375288i 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.b 30
23.c even 11 1 inner 552.2.q.b 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.q.b 30 1.a even 1 1 trivial
552.2.q.b 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 $$S_{2}^{\mathrm{new}}(552, [\chi])$$.