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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 $2$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{3} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{3} - 3 q^{9} + 9 q^{11} + 13 q^{13} + 17 q^{17} - 9 q^{19} - 11 q^{21} - 12 q^{23} - 23 q^{25} - 3 q^{27} - q^{29} - 37 q^{31} + 9 q^{33} + 10 q^{35} + 7 q^{37} + 13 q^{39} + 16 q^{41} + 20 q^{43} - 22 q^{45} + 22 q^{47} + 19 q^{49} + 17 q^{51} + 25 q^{53} + 10 q^{55} + 24 q^{57} + 7 q^{59} + 8 q^{61} - 28 q^{65} - 23 q^{67} - q^{69} + 5 q^{71} + 34 q^{73} - 23 q^{75} + 62 q^{77} + 20 q^{79} - 3 q^{81} + 29 q^{83} - 46 q^{85} + 10 q^{87} - 67 q^{89} - 118 q^{91} - 26 q^{93} - 99 q^{95} - 41 q^{97} + 9 q^{99}+O(q^{100}) \) 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 −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 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.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 \( T_{5}^{30} + 19 T_{5}^{28} - 25 T_{5}^{27} + 284 T_{5}^{26} - 37 T_{5}^{25} + 2116 T_{5}^{24} + \cdots + 529 \) acting on \(S_{2}^{\mathrm{new}}(552, [\chi])\). Copy content Toggle raw display