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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [552,2,Mod(25,552)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(552, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("552.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 25.3
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 \( T_{5}^{30} + 2 T_{5}^{29} + T_{5}^{28} + 7 T_{5}^{27} + 176 T_{5}^{26} + 287 T_{5}^{25} + \cdots + 17912342569 \) acting on \(S_{2}^{\mathrm{new}}(552, [\chi])\). Copy content Toggle raw display