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Properties

Label 440.2.bo.a
Level $440$
Weight $2$
Character orbit 440.bo
Analytic conductor $3.513$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [440,2,Mod(17,440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(440, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 0, 5, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("440.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 440.bo (of order \(20\), degree \(8\), 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: \(3.51341768894\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 4 q^{11} + 24 q^{23} + 24 q^{25} - 12 q^{27} - 16 q^{31} + 16 q^{37} - 60 q^{41} + 32 q^{45} - 28 q^{47} - 120 q^{51} + 12 q^{53} + 12 q^{55} + 80 q^{57} + 112 q^{67} - 48 q^{71} - 60 q^{73} + 28 q^{75} + 40 q^{77} + 4 q^{81} + 28 q^{91} - 96 q^{93} - 40 q^{95} - 60 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −3.07078 + 0.486364i 0 0.822548 + 2.07928i 0 0.177088 − 1.11809i 0 6.33997 − 2.05998i 0
17.2 0 −2.79493 + 0.442673i 0 −2.20117 + 0.393489i 0 0.497524 − 3.14124i 0 4.76250 − 1.54743i 0
17.3 0 −2.29680 + 0.363777i 0 2.22957 + 0.170403i 0 0.297474 − 1.87817i 0 2.28977 − 0.743992i 0
17.4 0 −2.23194 + 0.353504i 0 0.381841 − 2.20322i 0 −0.549238 + 3.46775i 0 2.00341 − 0.650948i 0
17.5 0 −1.82695 + 0.289360i 0 −1.60655 − 1.55531i 0 −0.108281 + 0.683657i 0 0.400833 − 0.130238i 0
17.6 0 −1.25415 + 0.198637i 0 2.22614 − 0.210458i 0 −0.717997 + 4.53325i 0 −1.31974 + 0.428811i 0
17.7 0 −1.19632 + 0.189478i 0 −1.57534 + 1.58692i 0 0.126665 − 0.799733i 0 −1.45790 + 0.473701i 0
17.8 0 −0.714723 + 0.113201i 0 0.307816 − 2.21478i 0 0.657321 − 4.15016i 0 −2.35515 + 0.765236i 0
17.9 0 −0.354205 + 0.0561005i 0 1.39688 + 1.74606i 0 −0.0554986 + 0.350404i 0 −2.73086 + 0.887309i 0
17.10 0 0.321542 − 0.0509273i 0 −1.79879 − 1.32828i 0 −0.416196 + 2.62776i 0 −2.75237 + 0.894300i 0
17.11 0 0.607781 − 0.0962630i 0 −2.23225 + 0.130664i 0 0.715219 − 4.51572i 0 −2.49304 + 0.810037i 0
17.12 0 1.23761 − 0.196018i 0 −0.429213 + 2.19449i 0 −0.400295 + 2.52736i 0 −1.35992 + 0.441865i 0
17.13 0 1.32916 − 0.210518i 0 1.06397 − 1.96671i 0 0.118323 − 0.747061i 0 −1.13082 + 0.367425i 0
17.14 0 1.48878 − 0.235799i 0 1.65604 + 1.50251i 0 0.727126 − 4.59089i 0 −0.692319 + 0.224948i 0
17.15 0 2.01660 − 0.319399i 0 1.30684 − 1.81444i 0 −0.248728 + 1.57041i 0 1.11151 − 0.361150i 0
17.16 0 2.68972 − 0.426010i 0 −1.57034 + 1.59187i 0 −0.542693 + 3.42643i 0 4.19994 − 1.36464i 0
17.17 0 2.72966 − 0.432336i 0 2.19660 + 0.418294i 0 0.102777 − 0.648911i 0 4.41098 − 1.43321i 0
17.18 0 3.31992 − 0.525824i 0 −2.17458 − 0.520773i 0 0.461024 − 2.91079i 0 7.89224 − 2.56434i 0
57.1 0 −0.486364 + 3.07078i 0 −1.88763 − 1.19869i 0 −1.11809 + 0.177088i 0 −6.33997 − 2.05998i 0
57.2 0 −0.442673 + 2.79493i 0 1.54950 − 1.61216i 0 −3.14124 + 0.497524i 0 −4.76250 − 1.54743i 0
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.d odd 10 1 inner
55.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 440.2.bo.a ✓ 144
4.b odd 2 1 880.2.cm.d 144
5.c odd 4 1 inner 440.2.bo.a ✓ 144
11.d odd 10 1 inner 440.2.bo.a ✓ 144
20.e even 4 1 880.2.cm.d 144
44.g even 10 1 880.2.cm.d 144
55.l even 20 1 inner 440.2.bo.a ✓ 144
220.w odd 20 1 880.2.cm.d 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.bo.a ✓ 144 1.a even 1 1 trivial
440.2.bo.a ✓ 144 5.c odd 4 1 inner
440.2.bo.a ✓ 144 11.d odd 10 1 inner
440.2.bo.a ✓ 144 55.l even 20 1 inner
880.2.cm.d 144 4.b odd 2 1
880.2.cm.d 144 20.e even 4 1
880.2.cm.d 144 44.g even 10 1
880.2.cm.d 144 220.w odd 20 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(440, [\chi])\).