↑

Properties

Label 141.6.c.c
Level $141$
Weight $6$
Character orbit 141.c
Analytic conductor $22.614$
Analytic rank $0$
Dimension $68$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [141,6,Mod(140,141)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(141, 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("141.140");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 141 = 3 \cdot 47 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 141.c (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: \(22.6141185936\)
Analytic rank: \(0\)
Dimension: \(68\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 4 q^{3} - 920 q^{4} + 28 q^{6} - 4 q^{7} - 664 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 4 q^{3} - 920 q^{4} + 28 q^{6} - 4 q^{7} - 664 q^{9} - 4932 q^{12} + 5184 q^{16} + 1468 q^{18} - 1028 q^{21} + 22028 q^{24} + 68384 q^{25} + 8084 q^{27} - 12264 q^{28} - 25016 q^{34} + 17088 q^{36} - 24364 q^{37} - 157972 q^{42} + 180724 q^{48} - 35960 q^{49} - 211976 q^{51} + 83320 q^{54} + 62388 q^{55} - 20056 q^{61} + 63004 q^{63} + 485552 q^{64} + 18848 q^{72} - 238600 q^{75} - 27172 q^{79} - 43888 q^{81} - 208056 q^{84} + 299072 q^{94} - 570876 q^{96} + 228068 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
140.1 − 2.41118i 5.33411 − 14.6474i 26.1862 −104.404 −35.3176 − 12.8615i −98.5678 − 140.298i −186.095 − 156.262i 251.738i
140.2 2.41118i 5.33411 + 14.6474i 26.1862 −104.404 −35.3176 + 12.8615i −98.5678 140.298i −186.095 + 156.262i − 251.738i
140.3 − 9.43549i −6.23357 + 14.2878i −57.0285 90.0016 134.813 + 58.8168i −172.204 236.156i −165.285 − 178.129i − 849.210i
140.4 9.43549i −6.23357 − 14.2878i −57.0285 90.0016 134.813 − 58.8168i −172.204 − 236.156i −165.285 + 178.129i 849.210i
140.5 − 6.99168i −14.5335 − 5.63709i −16.8835 −88.0587 −39.4127 + 101.614i −141.690 − 105.689i 179.447 + 163.853i 615.678i
140.6 6.99168i −14.5335 + 5.63709i −16.8835 −88.0587 −39.4127 − 101.614i −141.690 105.689i 179.447 − 163.853i − 615.678i
140.7 − 3.69901i −6.46680 + 14.1838i 18.3173 −86.2668 52.4660 + 23.9207i 171.088 − 186.124i −159.361 − 183.448i 319.102i
140.8 3.69901i −6.46680 − 14.1838i 18.3173 −86.2668 52.4660 − 23.9207i 171.088 186.124i −159.361 + 183.448i − 319.102i
140.9 − 8.78598i −9.64769 − 12.2443i −45.1934 −80.3906 −107.578 + 84.7644i 244.457 115.917i −56.8442 + 236.258i 706.310i
140.10 8.78598i −9.64769 + 12.2443i −45.1934 −80.3906 −107.578 − 84.7644i 244.457 − 115.917i −56.8442 − 236.258i − 706.310i
140.11 − 5.01148i 15.5715 + 0.727000i 6.88509 −73.2231 3.64334 − 78.0362i 74.3619 − 194.872i 241.943 + 22.6409i 366.956i
140.12 5.01148i 15.5715 − 0.727000i 6.88509 −73.2231 3.64334 + 78.0362i 74.3619 194.872i 241.943 − 22.6409i − 366.956i
140.13 − 10.1803i 0.931326 − 15.5606i −71.6387 −65.9715 −158.412 − 9.48119i −84.2827 403.535i −241.265 − 28.9840i 671.611i
140.14 10.1803i 0.931326 + 15.5606i −71.6387 −65.9715 −158.412 + 9.48119i −84.2827 − 403.535i −241.265 + 28.9840i − 671.611i
140.15 − 10.3242i 14.9005 − 4.57972i −74.5889 −60.0886 −47.2819 − 153.836i 88.5938 439.695i 201.052 − 136.481i 620.366i
140.16 10.3242i 14.9005 + 4.57972i −74.5889 −60.0886 −47.2819 + 153.836i 88.5938 − 439.695i 201.052 + 136.481i − 620.366i
140.17 − 7.86697i 8.28947 + 13.2017i −29.8893 59.9134 103.857 − 65.2130i 116.113 − 16.6049i −105.569 + 218.870i − 471.337i
140.18 7.86697i 8.28947 − 13.2017i −29.8893 59.9134 103.857 + 65.2130i 116.113 16.6049i −105.569 − 218.870i 471.337i
140.19 − 2.80714i −15.1312 − 3.74793i 24.1199 −58.0863 −10.5210 + 42.4754i 100.144 − 157.537i 214.906 + 113.421i 163.057i
140.20 2.80714i −15.1312 + 3.74793i 24.1199 −58.0863 −10.5210 − 42.4754i 100.144 157.537i 214.906 − 113.421i − 163.057i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 140.68
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
47.b odd 2 1 inner
141.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 141.6.c.c ✓ 68
3.b odd 2 1 inner 141.6.c.c ✓ 68
47.b odd 2 1 inner 141.6.c.c ✓ 68
141.c even 2 1 inner 141.6.c.c ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.6.c.c ✓ 68 1.a even 1 1 trivial
141.6.c.c ✓ 68 3.b odd 2 1 inner
141.6.c.c ✓ 68 47.b odd 2 1 inner
141.6.c.c ✓ 68 141.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{34} + 774 T_{2}^{32} + 270442 T_{2}^{30} + 56424144 T_{2}^{28} + 7835793585 T_{2}^{26} + \cdots + 57\!\cdots\!00 \) acting on \(S_{6}^{\mathrm{new}}(141, [\chi])\). Copy content Toggle raw display