↑

Properties

Label 552.2.b.c
Level $552$
Weight $2$
Character orbit 552.b
Analytic conductor $4.408$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [552,2,Mod(413,552)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(552, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("552.413");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 552.b (of order \(2\), degree \(1\), 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: \(80\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 12 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 12 q^{6} - 4 q^{9} + 16 q^{12} + 8 q^{16} + 20 q^{18} - 8 q^{24} - 144 q^{25} - 24 q^{31} + 40 q^{36} + 68 q^{39} - 24 q^{46} + 92 q^{48} - 160 q^{49} - 48 q^{52} - 32 q^{54} + 32 q^{55} - 40 q^{58} + 48 q^{64} + 72 q^{70} + 68 q^{72} - 8 q^{73} + 64 q^{78} + 12 q^{81} - 48 q^{82} + 92 q^{87} - 144 q^{94} + 68 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
413.1 −1.41196 − 0.0797643i −0.807372 − 1.53237i 1.98728 + 0.225248i − 2.83704i 1.01775 + 2.22805i 1.51036i −2.78799 − 0.476556i −1.69630 + 2.47438i −0.226295 + 4.00580i
413.2 −1.41196 − 0.0797643i −0.807372 − 1.53237i 1.98728 + 0.225248i 2.83704i 1.01775 + 2.22805i − 1.51036i −2.78799 − 0.476556i −1.69630 + 2.47438i 0.226295 − 4.00580i
413.3 −1.41196 + 0.0797643i −0.807372 + 1.53237i 1.98728 − 0.225248i − 2.83704i 1.01775 − 2.22805i 1.51036i −2.78799 + 0.476556i −1.69630 − 2.47438i 0.226295 + 4.00580i
413.4 −1.41196 + 0.0797643i −0.807372 + 1.53237i 1.98728 − 0.225248i 2.83704i 1.01775 − 2.22805i − 1.51036i −2.78799 + 0.476556i −1.69630 − 2.47438i −0.226295 − 4.00580i
413.5 −1.36401 − 0.373449i 1.13395 − 1.30925i 1.72107 + 1.01878i − 0.395927i −2.03567 + 1.36237i − 4.71612i −1.96710 − 2.03236i −0.428296 − 2.96927i −0.147859 + 0.540050i
413.6 −1.36401 − 0.373449i 1.13395 − 1.30925i 1.72107 + 1.01878i 0.395927i −2.03567 + 1.36237i 4.71612i −1.96710 − 2.03236i −0.428296 − 2.96927i 0.147859 − 0.540050i
413.7 −1.36401 + 0.373449i 1.13395 + 1.30925i 1.72107 − 1.01878i − 0.395927i −2.03567 − 1.36237i − 4.71612i −1.96710 + 2.03236i −0.428296 + 2.96927i 0.147859 + 0.540050i
413.8 −1.36401 + 0.373449i 1.13395 + 1.30925i 1.72107 − 1.01878i 0.395927i −2.03567 − 1.36237i 4.71612i −1.96710 + 2.03236i −0.428296 + 2.96927i −0.147859 − 0.540050i
413.9 −1.32616 − 0.491215i 1.50954 + 0.849295i 1.51742 + 1.30286i − 3.96539i −1.58470 − 1.86781i − 0.287329i −1.37235 − 2.47319i 1.55740 + 2.56408i −1.94786 + 5.25875i
413.10 −1.32616 − 0.491215i 1.50954 + 0.849295i 1.51742 + 1.30286i 3.96539i −1.58470 − 1.86781i 0.287329i −1.37235 − 2.47319i 1.55740 + 2.56408i 1.94786 − 5.25875i
413.11 −1.32616 + 0.491215i 1.50954 − 0.849295i 1.51742 − 1.30286i − 3.96539i −1.58470 + 1.86781i − 0.287329i −1.37235 + 2.47319i 1.55740 − 2.56408i 1.94786 + 5.25875i
413.12 −1.32616 + 0.491215i 1.50954 − 0.849295i 1.51742 − 1.30286i 3.96539i −1.58470 + 1.86781i 0.287329i −1.37235 + 2.47319i 1.55740 − 2.56408i −1.94786 − 5.25875i
413.13 −1.25126 − 0.659042i −1.59997 − 0.663406i 1.13133 + 1.64927i − 2.68496i 1.56477 + 1.88454i − 3.86988i −0.328648 − 2.80927i 2.11979 + 2.12285i −1.76950 + 3.35959i
413.14 −1.25126 − 0.659042i −1.59997 − 0.663406i 1.13133 + 1.64927i 2.68496i 1.56477 + 1.88454i 3.86988i −0.328648 − 2.80927i 2.11979 + 2.12285i 1.76950 − 3.35959i
413.15 −1.25126 + 0.659042i −1.59997 + 0.663406i 1.13133 − 1.64927i − 2.68496i 1.56477 − 1.88454i − 3.86988i −0.328648 + 2.80927i 2.11979 − 2.12285i 1.76950 + 3.35959i
413.16 −1.25126 + 0.659042i −1.59997 + 0.663406i 1.13133 − 1.64927i 2.68496i 1.56477 − 1.88454i 3.86988i −0.328648 + 2.80927i 2.11979 − 2.12285i −1.76950 − 3.35959i
413.17 −1.02836 − 0.970807i 1.71832 − 0.217636i 0.115069 + 1.99669i − 1.34112i −1.97835 − 1.44435i 2.80797i 1.82006 − 2.16503i 2.90527 − 0.747937i −1.30196 + 1.37916i
413.18 −1.02836 − 0.970807i 1.71832 − 0.217636i 0.115069 + 1.99669i 1.34112i −1.97835 − 1.44435i − 2.80797i 1.82006 − 2.16503i 2.90527 − 0.747937i 1.30196 − 1.37916i
413.19 −1.02836 + 0.970807i 1.71832 + 0.217636i 0.115069 − 1.99669i − 1.34112i −1.97835 + 1.44435i 2.80797i 1.82006 + 2.16503i 2.90527 + 0.747937i 1.30196 + 1.37916i
413.20 −1.02836 + 0.970807i 1.71832 + 0.217636i 0.115069 − 1.99669i 1.34112i −1.97835 + 1.44435i − 2.80797i 1.82006 + 2.16503i 2.90527 + 0.747937i −1.30196 − 1.37916i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 413.80
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
23.b odd 2 1 inner
24.h odd 2 1 inner
69.c even 2 1 inner
184.e odd 2 1 inner
552.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 552.2.b.c ✓ 80
3.b odd 2 1 inner 552.2.b.c ✓ 80
4.b odd 2 1 2208.2.b.c 80
8.b even 2 1 inner 552.2.b.c ✓ 80
8.d odd 2 1 2208.2.b.c 80
12.b even 2 1 2208.2.b.c 80
23.b odd 2 1 inner 552.2.b.c ✓ 80
24.f even 2 1 2208.2.b.c 80
24.h odd 2 1 inner 552.2.b.c ✓ 80
69.c even 2 1 inner 552.2.b.c ✓ 80
92.b even 2 1 2208.2.b.c 80
184.e odd 2 1 inner 552.2.b.c ✓ 80
184.h even 2 1 2208.2.b.c 80
276.h odd 2 1 2208.2.b.c 80
552.b even 2 1 inner 552.2.b.c ✓ 80
552.h odd 2 1 2208.2.b.c 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.b.c ✓ 80 1.a even 1 1 trivial
552.2.b.c ✓ 80 3.b odd 2 1 inner
552.2.b.c ✓ 80 8.b even 2 1 inner
552.2.b.c ✓ 80 23.b odd 2 1 inner
552.2.b.c ✓ 80 24.h odd 2 1 inner
552.2.b.c ✓ 80 69.c even 2 1 inner
552.2.b.c ✓ 80 184.e odd 2 1 inner
552.2.b.c ✓ 80 552.b even 2 1 inner
2208.2.b.c 80 4.b odd 2 1
2208.2.b.c 80 8.d odd 2 1
2208.2.b.c 80 12.b even 2 1
2208.2.b.c 80 24.f even 2 1
2208.2.b.c 80 92.b even 2 1
2208.2.b.c 80 184.h even 2 1
2208.2.b.c 80 276.h odd 2 1
2208.2.b.c 80 552.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(552, [\chi])\):

\( T_{5}^{20} + 68 T_{5}^{18} + 1942 T_{5}^{16} + 30552 T_{5}^{14} + 292000 T_{5}^{12} + 1759816 T_{5}^{10} + \cdots + 1470208 \) Copy content Toggle raw display
\( T_{29}^{20} - 219 T_{29}^{18} + 19775 T_{29}^{16} - 969353 T_{29}^{14} + 28631892 T_{29}^{12} + \cdots + 36691771392 \) Copy content Toggle raw display