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 $2$

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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.

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

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:

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 \(S_{2}^{\mathrm{new}}(552, [\chi])\).