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Properties

Label 440.2.g.c
Level $440$
Weight $2$
Character orbit 440.g
Analytic conductor $3.513$
Analytic rank $0$
Dimension $24$
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] = [440,2,Mod(221,440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(440, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("440.221");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 440.g (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: \(3.51341768894\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{4} - 4 q^{6} + 6 q^{8} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{4} - 4 q^{6} + 6 q^{8} - 40 q^{9} - 2 q^{10} + 4 q^{12} + 10 q^{14} - 4 q^{15} - 10 q^{16} + 28 q^{17} - 18 q^{18} + 2 q^{22} + 12 q^{23} - 32 q^{24} - 24 q^{25} + 4 q^{26} + 10 q^{28} + 10 q^{30} + 8 q^{31} - 10 q^{32} + 4 q^{33} - 26 q^{34} + 28 q^{36} + 16 q^{38} - 24 q^{39} + 8 q^{40} - 40 q^{41} - 68 q^{42} + 12 q^{46} - 20 q^{47} - 52 q^{48} + 64 q^{49} + 60 q^{52} + 64 q^{54} + 24 q^{55} - 38 q^{56} - 56 q^{57} - 64 q^{58} + 18 q^{60} + 26 q^{62} + 40 q^{63} - 22 q^{64} + 20 q^{65} - 10 q^{66} + 38 q^{68} + 8 q^{70} + 24 q^{71} - 108 q^{72} - 36 q^{73} - 28 q^{74} + 44 q^{76} + 92 q^{78} - 8 q^{79} + 96 q^{81} - 44 q^{82} + 132 q^{84} + 56 q^{86} - 8 q^{88} - 72 q^{89} - 10 q^{90} + 72 q^{92} + 72 q^{94} - 112 q^{96} + 48 q^{97} - 98 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1 −1.39008 − 0.260144i − 0.308537i 1.86465 + 0.723244i 1.00000i −0.0802642 + 0.428891i −3.26920 −2.40387 − 1.49045i 2.90480 0.260144 − 1.39008i
221.2 −1.39008 + 0.260144i 0.308537i 1.86465 − 0.723244i − 1.00000i −0.0802642 − 0.428891i −3.26920 −2.40387 + 1.49045i 2.90480 0.260144 + 1.39008i
221.3 −1.31035 − 0.531950i − 1.41634i 1.43406 + 1.39409i − 1.00000i −0.753424 + 1.85591i 4.34669 −1.13754 − 2.58960i 0.993975 −0.531950 + 1.31035i
221.4 −1.31035 + 0.531950i 1.41634i 1.43406 − 1.39409i 1.00000i −0.753424 − 1.85591i 4.34669 −1.13754 + 2.58960i 0.993975 −0.531950 − 1.31035i
221.5 −0.934802 − 1.06120i − 0.324436i −0.252291 + 1.98402i − 1.00000i −0.344292 + 0.303284i −2.45816 2.34129 − 1.58694i 2.89474 −1.06120 + 0.934802i
221.6 −0.934802 + 1.06120i 0.324436i −0.252291 − 1.98402i 1.00000i −0.344292 − 0.303284i −2.45816 2.34129 + 1.58694i 2.89474 −1.06120 − 0.934802i
221.7 −0.918693 − 1.07518i − 2.73124i −0.312008 + 1.97551i 1.00000i −2.93657 + 2.50917i 4.25690 2.41066 − 1.47943i −4.45970 1.07518 − 0.918693i
221.8 −0.918693 + 1.07518i 2.73124i −0.312008 − 1.97551i − 1.00000i −2.93657 − 2.50917i 4.25690 2.41066 + 1.47943i −4.45970 1.07518 + 0.918693i
221.9 −0.820353 − 1.15196i 2.93797i −0.654042 + 1.89003i 1.00000i 3.38444 − 2.41017i −4.84749 2.71380 − 0.797062i −5.63167 1.15196 − 0.820353i
221.10 −0.820353 + 1.15196i − 2.93797i −0.654042 − 1.89003i − 1.00000i 3.38444 + 2.41017i −4.84749 2.71380 + 0.797062i −5.63167 1.15196 + 0.820353i
221.11 −0.152145 − 1.40601i − 3.34174i −1.95370 + 0.427833i − 1.00000i −4.69850 + 0.508428i 0.333306 0.898781 + 2.68183i −8.16722 −1.40601 + 0.152145i
221.12 −0.152145 + 1.40601i 3.34174i −1.95370 − 0.427833i 1.00000i −4.69850 − 0.508428i 0.333306 0.898781 − 2.68183i −8.16722 −1.40601 − 0.152145i
221.13 0.0308432 − 1.41388i 1.82409i −1.99810 − 0.0872170i − 1.00000i 2.57904 + 0.0562609i −3.30914 −0.184942 + 2.82237i −0.327312 −1.41388 − 0.0308432i
221.14 0.0308432 + 1.41388i − 1.82409i −1.99810 + 0.0872170i 1.00000i 2.57904 − 0.0562609i −3.30914 −0.184942 − 2.82237i −0.327312 −1.41388 + 0.0308432i
221.15 0.581017 − 1.28935i 2.08501i −1.32484 − 1.49827i 1.00000i 2.68831 + 1.21143i 2.28761 −2.70154 + 0.837663i −1.34727 1.28935 + 0.581017i
221.16 0.581017 + 1.28935i − 2.08501i −1.32484 + 1.49827i − 1.00000i 2.68831 − 1.21143i 2.28761 −2.70154 − 0.837663i −1.34727 1.28935 − 0.581017i
221.17 1.05426 − 0.942617i 1.29586i 0.222948 − 1.98753i − 1.00000i 1.22150 + 1.36618i 1.35877 −1.63844 − 2.30554i 1.32075 −0.942617 − 1.05426i
221.18 1.05426 + 0.942617i − 1.29586i 0.222948 + 1.98753i 1.00000i 1.22150 − 1.36618i 1.35877 −1.63844 + 2.30554i 1.32075 −0.942617 + 1.05426i
221.19 1.17972 − 0.779909i − 0.894414i 0.783484 − 1.84015i 1.00000i −0.697562 − 1.05516i 2.43095 −0.510858 − 2.78191i 2.20002 0.779909 + 1.17972i
221.20 1.17972 + 0.779909i 0.894414i 0.783484 + 1.84015i − 1.00000i −0.697562 + 1.05516i 2.43095 −0.510858 + 2.78191i 2.20002 0.779909 − 1.17972i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 221.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 440.2.g.c ✓ 24
4.b odd 2 1 1760.2.g.c 24
8.b even 2 1 inner 440.2.g.c ✓ 24
8.d odd 2 1 1760.2.g.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.g.c ✓ 24 1.a even 1 1 trivial
440.2.g.c ✓ 24 8.b even 2 1 inner
1760.2.g.c 24 4.b odd 2 1
1760.2.g.c 24 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 56 T_{3}^{22} + 1346 T_{3}^{20} + 18208 T_{3}^{18} + 152849 T_{3}^{16} + 828472 T_{3}^{14} + \cdots + 16384 \) acting on \(S_{2}^{\mathrm{new}}(440, [\chi])\). Copy content Toggle raw display