# Properties

 Label 552.2.q.d 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30q + 3q^{3} + 2q^{5} + 4q^{7} - 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q + 3q^{3} + 2q^{5} + 4q^{7} - 3q^{9} - 15q^{11} - 5q^{13} - 2q^{15} + 9q^{17} - 3q^{19} + 7q^{21} + 18q^{23} - 19q^{25} + 3q^{27} - 21q^{29} + 17q^{31} - 7q^{33} - 36q^{35} + 9q^{37} + 5q^{39} + 18q^{41} + 50q^{43} + 2q^{45} + 74q^{47} - 17q^{49} + 13q^{51} + 43q^{53} - 42q^{55} - 8q^{57} + 7q^{59} - 10q^{61} + 4q^{63} - 4q^{65} + 33q^{67} + 15q^{69} + 3q^{71} + 30q^{73} - 25q^{75} - 82q^{77} - 40q^{79} - 3q^{81} + 9q^{83} - 54q^{85} + 10q^{87} + 25q^{89} - 30q^{91} + 38q^{93} - 49q^{95} - 69q^{97} + 7q^{99} + O(q^{100})$$

## 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 −2.56277 0.752498i 0 1.17327 1.35403i 0 −0.959493 + 0.281733i 0
25.2 0 0.142315 + 0.989821i 0 0.832695 + 0.244501i 0 1.47208 1.69888i 0 −0.959493 + 0.281733i 0
25.3 0 0.142315 + 0.989821i 0 2.52725 + 0.742069i 0 −1.72994 + 1.99646i 0 −0.959493 + 0.281733i 0
49.1 0 −0.415415 + 0.909632i 0 −2.35936 + 2.72285i 0 1.95097 1.25381i 0 −0.654861 0.755750i 0
49.2 0 −0.415415 + 0.909632i 0 0.182341 0.210432i 0 1.45078 0.932358i 0 −0.654861 0.755750i 0
49.3 0 −0.415415 + 0.909632i 0 0.920354 1.06214i 0 −3.86124 + 2.48147i 0 −0.654861 0.755750i 0
73.1 0 0.959493 0.281733i 0 −0.776140 0.498795i 0 0.172807 + 1.20190i 0 0.841254 0.540641i 0
73.2 0 0.959493 0.281733i 0 −0.0432483 0.0277940i 0 −0.706094 4.91099i 0 0.841254 0.540641i 0
73.3 0 0.959493 0.281733i 0 1.92120 + 1.23468i 0 0.378426 + 2.63201i 0 0.841254 0.540641i 0
121.1 0 0.959493 + 0.281733i 0 −0.776140 + 0.498795i 0 0.172807 1.20190i 0 0.841254 + 0.540641i 0
121.2 0 0.959493 + 0.281733i 0 −0.0432483 + 0.0277940i 0 −0.706094 + 4.91099i 0 0.841254 + 0.540641i 0
121.3 0 0.959493 + 0.281733i 0 1.92120 1.23468i 0 0.378426 2.63201i 0 0.841254 + 0.540641i 0
169.1 0 −0.415415 0.909632i 0 −2.35936 2.72285i 0 1.95097 + 1.25381i 0 −0.654861 + 0.755750i 0
169.2 0 −0.415415 0.909632i 0 0.182341 + 0.210432i 0 1.45078 + 0.932358i 0 −0.654861 + 0.755750i 0
169.3 0 −0.415415 0.909632i 0 0.920354 + 1.06214i 0 −3.86124 2.48147i 0 −0.654861 + 0.755750i 0
193.1 0 0.654861 + 0.755750i 0 −0.280570 1.95141i 0 1.41601 3.10064i 0 −0.142315 + 0.989821i 0
193.2 0 0.654861 + 0.755750i 0 −0.0891792 0.620255i 0 0.0603595 0.132169i 0 −0.142315 + 0.989821i 0
193.3 0 0.654861 + 0.755750i 0 0.609195 + 4.23705i 0 −0.135120 + 0.295872i 0 −0.142315 + 0.989821i 0
265.1 0 0.142315 0.989821i 0 −2.56277 + 0.752498i 0 1.17327 + 1.35403i 0 −0.959493 0.281733i 0
265.2 0 0.142315 0.989821i 0 0.832695 0.244501i 0 1.47208 + 1.69888i 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.d 30
23.c even 11 1 inner 552.2.q.d 30

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