↑

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 $2$

Related objects

Downloads

Learn more

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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{3} - 2 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{3} - 2 q^{7} - 3 q^{9} - 15 q^{11} + 5 q^{13} - 15 q^{17} + 5 q^{19} + 9 q^{21} + 14 q^{23} + 5 q^{25} - 3 q^{27} + 13 q^{29} - 29 q^{31} - 15 q^{33} + 8 q^{35} - 3 q^{37} + 5 q^{39} + 24 q^{41} + 2 q^{43} + 22 q^{45} + 38 q^{47} + 15 q^{49} + 7 q^{51} - 7 q^{53} + 46 q^{55} - 6 q^{57} - 43 q^{59} - 22 q^{61} - 2 q^{63} + 44 q^{65} + 31 q^{67} + 3 q^{69} + 19 q^{71} - 50 q^{73} + 5 q^{75} + 16 q^{77} - 84 q^{79} - 3 q^{81} - 73 q^{83} - 8 q^{85} + 2 q^{87} + 19 q^{89} + 18 q^{91} + 26 q^{93} - 67 q^{95} - 29 q^{97} - 15 q^{99}+O(q^{100}) \) 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 −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 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.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 \( T_{5}^{30} + 5 T_{5}^{28} + 17 T_{5}^{27} + 80 T_{5}^{26} + 511 T_{5}^{25} + 2218 T_{5}^{24} + \cdots + 214369 \) acting on \(S_{2}^{\mathrm{new}}(552, [\chi])\). Copy content Toggle raw display