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Properties

Label 124.6.d.b
Level $124$
Weight $6$
Character orbit 124.d
Analytic conductor $19.888$
Analytic rank $0$
Dimension $72$
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] = [124,6,Mod(123,124)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(124, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("124.123");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 124.d (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: \(19.8875936568\)
Analytic rank: \(0\)
Dimension: \(72\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 2 q^{2} - 22 q^{4} - 4 q^{5} - 1394 q^{8} + 7448 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 2 q^{2} - 22 q^{4} - 4 q^{5} - 1394 q^{8} + 7448 q^{9} - 1612 q^{10} - 4040 q^{14} - 462 q^{16} - 78 q^{18} - 20156 q^{20} + 27724 q^{25} + 23616 q^{28} + 1318 q^{32} + 8456 q^{33} - 41810 q^{36} + 17748 q^{38} - 25828 q^{40} - 4 q^{41} + 22548 q^{45} - 119916 q^{49} + 58474 q^{50} + 45976 q^{56} + 41930 q^{62} + 78866 q^{64} + 10500 q^{66} + 33736 q^{69} - 63840 q^{70} - 198774 q^{72} + 100020 q^{76} + 150676 q^{78} - 292180 q^{80} + 51512 q^{81} - 87640 q^{82} + 303380 q^{90} + 290232 q^{93} + 20136 q^{94} + 111204 q^{97} - 503822 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
123.1 −5.64223 − 0.406501i −16.1083 31.6695 + 4.58714i −76.5186 90.8869 + 6.54805i 181.773i −176.822 − 38.7554i 16.4781 431.736 + 31.1049i
123.2 −5.64223 − 0.406501i 16.1083 31.6695 + 4.58714i −76.5186 −90.8869 − 6.54805i 181.773i −176.822 − 38.7554i 16.4781 431.736 + 31.1049i
123.3 −5.64223 + 0.406501i −16.1083 31.6695 − 4.58714i −76.5186 90.8869 − 6.54805i − 181.773i −176.822 + 38.7554i 16.4781 431.736 − 31.1049i
123.4 −5.64223 + 0.406501i 16.1083 31.6695 − 4.58714i −76.5186 −90.8869 + 6.54805i − 181.773i −176.822 + 38.7554i 16.4781 431.736 − 31.1049i
123.5 −5.49754 − 1.33305i −8.71587 28.4460 + 14.6570i 52.4122 47.9159 + 11.6187i − 109.826i −136.844 − 118.497i −167.034 −288.138 − 69.8680i
123.6 −5.49754 − 1.33305i 8.71587 28.4460 + 14.6570i 52.4122 −47.9159 − 11.6187i − 109.826i −136.844 − 118.497i −167.034 −288.138 − 69.8680i
123.7 −5.49754 + 1.33305i −8.71587 28.4460 − 14.6570i 52.4122 47.9159 − 11.6187i 109.826i −136.844 + 118.497i −167.034 −288.138 + 69.8680i
123.8 −5.49754 + 1.33305i 8.71587 28.4460 − 14.6570i 52.4122 −47.9159 + 11.6187i 109.826i −136.844 + 118.497i −167.034 −288.138 + 69.8680i
123.9 −5.39481 − 1.70176i −26.5536 26.2080 + 18.3614i 34.4435 143.252 + 45.1879i 146.519i −110.141 − 143.656i 462.093 −185.816 − 58.6146i
123.10 −5.39481 − 1.70176i 26.5536 26.2080 + 18.3614i 34.4435 −143.252 − 45.1879i 146.519i −110.141 − 143.656i 462.093 −185.816 − 58.6146i
123.11 −5.39481 + 1.70176i −26.5536 26.2080 − 18.3614i 34.4435 143.252 − 45.1879i − 146.519i −110.141 + 143.656i 462.093 −185.816 + 58.6146i
123.12 −5.39481 + 1.70176i 26.5536 26.2080 − 18.3614i 34.4435 −143.252 + 45.1879i − 146.519i −110.141 + 143.656i 462.093 −185.816 + 58.6146i
123.13 −4.71237 − 3.12947i −8.27255 12.4129 + 29.4944i −38.9756 38.9833 + 25.8887i − 32.3704i 33.8076 − 177.834i −174.565 183.667 + 121.973i
123.14 −4.71237 − 3.12947i 8.27255 12.4129 + 29.4944i −38.9756 −38.9833 − 25.8887i − 32.3704i 33.8076 − 177.834i −174.565 183.667 + 121.973i
123.15 −4.71237 + 3.12947i −8.27255 12.4129 − 29.4944i −38.9756 38.9833 − 25.8887i 32.3704i 33.8076 + 177.834i −174.565 183.667 − 121.973i
123.16 −4.71237 + 3.12947i 8.27255 12.4129 − 29.4944i −38.9756 −38.9833 + 25.8887i 32.3704i 33.8076 + 177.834i −174.565 183.667 − 121.973i
123.17 −3.93584 − 4.06314i −26.3801 −1.01829 + 31.9838i −62.0164 103.828 + 107.186i − 89.2806i 133.963 − 121.746i 452.908 244.087 + 251.982i
123.18 −3.93584 − 4.06314i 26.3801 −1.01829 + 31.9838i −62.0164 −103.828 − 107.186i − 89.2806i 133.963 − 121.746i 452.908 244.087 + 251.982i
123.19 −3.93584 + 4.06314i −26.3801 −1.01829 − 31.9838i −62.0164 103.828 − 107.186i 89.2806i 133.963 + 121.746i 452.908 244.087 − 251.982i
123.20 −3.93584 + 4.06314i 26.3801 −1.01829 − 31.9838i −62.0164 −103.828 + 107.186i 89.2806i 133.963 + 121.746i 452.908 244.087 − 251.982i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 123.72
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
31.b odd 2 1 inner
124.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.6.d.b ✓ 72
4.b odd 2 1 inner 124.6.d.b ✓ 72
31.b odd 2 1 inner 124.6.d.b ✓ 72
124.d even 2 1 inner 124.6.d.b ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.6.d.b ✓ 72 1.a even 1 1 trivial
124.6.d.b ✓ 72 4.b odd 2 1 inner
124.6.d.b ✓ 72 31.b odd 2 1 inner
124.6.d.b ✓ 72 124.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} - 6236 T_{3}^{34} + 17695828 T_{3}^{32} - 30287606528 T_{3}^{30} + 34945069994944 T_{3}^{28} + \cdots + 69\!\cdots\!00 \) acting on \(S_{6}^{\mathrm{new}}(124, [\chi])\). Copy content Toggle raw display