Properties

Label 552.2.q.c
Level $552$
Weight $2$
Character orbit 552.q
Analytic conductor $4.408$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 30q + 3q^{3} - 2q^{5} - 2q^{7} - 3q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 30q + 3q^{3} - 2q^{5} - 2q^{7} - 3q^{9} + 13q^{11} - 13q^{13} + 2q^{15} - 11q^{17} + 11q^{19} - 9q^{21} - 12q^{23} + 17q^{25} + 3q^{27} + 9q^{29} + 33q^{31} + 9q^{33} + 62q^{35} + 11q^{37} + 13q^{39} + 2q^{41} - 40q^{43} - 2q^{45} - 38q^{47} - 13q^{49} + 11q^{51} - 25q^{53} + 62q^{55} + 22q^{57} + 73q^{59} + 4q^{61} - 2q^{63} - 12q^{65} - 29q^{67} - 21q^{69} - 19q^{71} - 38q^{73} + 27q^{75} + 4q^{77} - 12q^{79} - 3q^{81} - 65q^{83} + 8q^{85} + 2q^{87} - 61q^{89} - 150q^{91} - 22q^{93} - 9q^{95} - 9q^{97} - 9q^{99} + O(q^{100}) \)

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 −3.30472 0.970352i 0 1.20819 1.39433i 0 −0.959493 + 0.281733i 0
25.2 0 0.142315 + 0.989821i 0 −0.991724 0.291196i 0 −2.31042 + 2.66637i 0 −0.959493 + 0.281733i 0
25.3 0 0.142315 + 0.989821i 0 3.49926 + 1.02748i 0 1.49653 1.72709i 0 −0.959493 + 0.281733i 0
49.1 0 −0.415415 + 0.909632i 0 −2.29838 + 2.65247i 0 −2.05854 + 1.32294i 0 −0.654861 0.755750i 0
49.2 0 −0.415415 + 0.909632i 0 1.04288 1.20354i 0 −1.78166 + 1.14501i 0 −0.654861 0.755750i 0
49.3 0 −0.415415 + 0.909632i 0 2.51217 2.89920i 0 2.61719 1.68197i 0 −0.654861 0.755750i 0
73.1 0 0.959493 0.281733i 0 −2.87348 1.84667i 0 0.703532 + 4.89317i 0 0.841254 0.540641i 0
73.2 0 0.959493 0.281733i 0 −0.873450 0.561333i 0 −0.322387 2.24225i 0 0.841254 0.540641i 0
73.3 0 0.959493 0.281733i 0 2.64512 + 1.69992i 0 0.0583451 + 0.405799i 0 0.841254 0.540641i 0
121.1 0 0.959493 + 0.281733i 0 −2.87348 + 1.84667i 0 0.703532 4.89317i 0 0.841254 + 0.540641i 0
121.2 0 0.959493 + 0.281733i 0 −0.873450 + 0.561333i 0 −0.322387 + 2.24225i 0 0.841254 + 0.540641i 0
121.3 0 0.959493 + 0.281733i 0 2.64512 1.69992i 0 0.0583451 0.405799i 0 0.841254 + 0.540641i 0
169.1 0 −0.415415 0.909632i 0 −2.29838 2.65247i 0 −2.05854 1.32294i 0 −0.654861 + 0.755750i 0
169.2 0 −0.415415 0.909632i 0 1.04288 + 1.20354i 0 −1.78166 1.14501i 0 −0.654861 + 0.755750i 0
169.3 0 −0.415415 0.909632i 0 2.51217 + 2.89920i 0 2.61719 + 1.68197i 0 −0.654861 + 0.755750i 0
193.1 0 0.654861 + 0.755750i 0 −0.321356 2.23508i 0 0.387169 0.847782i 0 −0.142315 + 0.989821i 0
193.2 0 0.654861 + 0.755750i 0 −0.291779 2.02937i 0 −2.05584 + 4.50166i 0 −0.142315 + 0.989821i 0
193.3 0 0.654861 + 0.755750i 0 0.373689 + 2.59907i 0 −0.503412 + 1.10232i 0 −0.142315 + 0.989821i 0
265.1 0 0.142315 0.989821i 0 −3.30472 + 0.970352i 0 1.20819 + 1.39433i 0 −0.959493 0.281733i 0
265.2 0 0.142315 0.989821i 0 −0.991724 + 0.291196i 0 −2.31042 2.66637i 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:

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.c 30
23.c even 11 1 inner 552.2.q.c 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.q.c 30 1.a even 1 1 trivial
552.2.q.c 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 \(S_{2}^{\mathrm{new}}(552, [\chi])\).