Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [60,4,Mod(11,60)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(60, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("60.11");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 60 = 2^{2} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 60.e (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: | \(3.54011460034\) |
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.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −2.81775 | − | 0.245521i | −1.56674 | − | 4.95432i | 7.87944 | + | 1.38363i | − | 5.00000i | 3.19829 | + | 14.3447i | − | 5.80602i | −21.8626 | − | 5.83329i | −22.0907 | + | 15.5243i | −1.22760 | + | 14.0888i | ||
11.2 | −2.81775 | + | 0.245521i | −1.56674 | + | 4.95432i | 7.87944 | − | 1.38363i | 5.00000i | 3.19829 | − | 14.3447i | 5.80602i | −21.8626 | + | 5.83329i | −22.0907 | − | 15.5243i | −1.22760 | − | 14.0888i | ||||
11.3 | −2.72356 | − | 0.763049i | 4.56698 | + | 2.47844i | 6.83551 | + | 4.15641i | − | 5.00000i | −10.5473 | − | 10.2350i | 20.9745i | −15.4454 | − | 16.5360i | 14.7146 | + | 22.6380i | −3.81524 | + | 13.6178i | |||
11.4 | −2.72356 | + | 0.763049i | 4.56698 | − | 2.47844i | 6.83551 | − | 4.15641i | 5.00000i | −10.5473 | + | 10.2350i | − | 20.9745i | −15.4454 | + | 16.5360i | 14.7146 | − | 22.6380i | −3.81524 | − | 13.6178i | |||
11.5 | −2.29793 | − | 1.64910i | −5.00938 | − | 1.38062i | 2.56093 | + | 7.57903i | 5.00000i | 9.23440 | + | 11.4335i | − | 0.228949i | 6.61376 | − | 21.6393i | 23.1878 | + | 13.8321i | 8.24551 | − | 11.4896i | |||
11.6 | −2.29793 | + | 1.64910i | −5.00938 | + | 1.38062i | 2.56093 | − | 7.57903i | − | 5.00000i | 9.23440 | − | 11.4335i | 0.228949i | 6.61376 | + | 21.6393i | 23.1878 | − | 13.8321i | 8.24551 | + | 11.4896i | |||
11.7 | −1.91554 | − | 2.08103i | −2.09257 | + | 4.75617i | −0.661378 | + | 7.97261i | − | 5.00000i | 13.9062 | − | 4.75594i | − | 32.2690i | 17.8582 | − | 13.8955i | −18.2423 | − | 19.9053i | −10.4052 | + | 9.57772i | ||
11.8 | −1.91554 | + | 2.08103i | −2.09257 | − | 4.75617i | −0.661378 | − | 7.97261i | 5.00000i | 13.9062 | + | 4.75594i | 32.2690i | 17.8582 | + | 13.8955i | −18.2423 | + | 19.9053i | −10.4052 | − | 9.57772i | ||||
11.9 | −0.622346 | − | 2.75911i | 1.95519 | + | 4.81427i | −7.22537 | + | 3.43424i | 5.00000i | 12.0663 | − | 8.39074i | 20.3642i | 13.9721 | + | 17.7983i | −19.3544 | + | 18.8257i | 13.7955 | − | 3.11173i | ||||
11.10 | −0.622346 | + | 2.75911i | 1.95519 | − | 4.81427i | −7.22537 | − | 3.43424i | − | 5.00000i | 12.0663 | + | 8.39074i | − | 20.3642i | 13.9721 | − | 17.7983i | −19.3544 | − | 18.8257i | 13.7955 | + | 3.11173i | ||
11.11 | −0.235442 | − | 2.81861i | −5.18580 | − | 0.327893i | −7.88913 | + | 1.32724i | − | 5.00000i | 0.296752 | + | 14.6939i | 26.3120i | 5.59840 | + | 21.9239i | 26.7850 | + | 3.40077i | −14.0931 | + | 1.17721i | |||
11.12 | −0.235442 | + | 2.81861i | −5.18580 | + | 0.327893i | −7.88913 | − | 1.32724i | 5.00000i | 0.296752 | − | 14.6939i | − | 26.3120i | 5.59840 | − | 21.9239i | 26.7850 | − | 3.40077i | −14.0931 | − | 1.17721i | |||
11.13 | 0.235442 | − | 2.81861i | 5.18580 | + | 0.327893i | −7.88913 | − | 1.32724i | − | 5.00000i | 2.14516 | − | 14.5395i | − | 26.3120i | −5.59840 | + | 21.9239i | 26.7850 | + | 3.40077i | −14.0931 | − | 1.17721i | ||
11.14 | 0.235442 | + | 2.81861i | 5.18580 | − | 0.327893i | −7.88913 | + | 1.32724i | 5.00000i | 2.14516 | + | 14.5395i | 26.3120i | −5.59840 | − | 21.9239i | 26.7850 | − | 3.40077i | −14.0931 | + | 1.17721i | ||||
11.15 | 0.622346 | − | 2.75911i | −1.95519 | − | 4.81427i | −7.22537 | − | 3.43424i | 5.00000i | −14.4999 | + | 2.39845i | − | 20.3642i | −13.9721 | + | 17.7983i | −19.3544 | + | 18.8257i | 13.7955 | + | 3.11173i | |||
11.16 | 0.622346 | + | 2.75911i | −1.95519 | + | 4.81427i | −7.22537 | + | 3.43424i | − | 5.00000i | −14.4999 | − | 2.39845i | 20.3642i | −13.9721 | − | 17.7983i | −19.3544 | − | 18.8257i | 13.7955 | − | 3.11173i | |||
11.17 | 1.91554 | − | 2.08103i | 2.09257 | − | 4.75617i | −0.661378 | − | 7.97261i | − | 5.00000i | −5.88931 | − | 13.4654i | 32.2690i | −17.8582 | − | 13.8955i | −18.2423 | − | 19.9053i | −10.4052 | − | 9.57772i | |||
11.18 | 1.91554 | + | 2.08103i | 2.09257 | + | 4.75617i | −0.661378 | + | 7.97261i | 5.00000i | −5.88931 | + | 13.4654i | − | 32.2690i | −17.8582 | + | 13.8955i | −18.2423 | + | 19.9053i | −10.4052 | + | 9.57772i | |||
11.19 | 2.29793 | − | 1.64910i | 5.00938 | + | 1.38062i | 2.56093 | − | 7.57903i | 5.00000i | 13.7880 | − | 5.08841i | 0.228949i | −6.61376 | − | 21.6393i | 23.1878 | + | 13.8321i | 8.24551 | + | 11.4896i | ||||
11.20 | 2.29793 | + | 1.64910i | 5.00938 | − | 1.38062i | 2.56093 | + | 7.57903i | − | 5.00000i | 13.7880 | + | 5.08841i | − | 0.228949i | −6.61376 | + | 21.6393i | 23.1878 | − | 13.8321i | 8.24551 | − | 11.4896i | ||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 60.4.e.a | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 60.4.e.a | ✓ | 24 |
4.b | odd | 2 | 1 | inner | 60.4.e.a | ✓ | 24 |
8.b | even | 2 | 1 | 960.4.h.d | 24 | ||
8.d | odd | 2 | 1 | 960.4.h.d | 24 | ||
12.b | even | 2 | 1 | inner | 60.4.e.a | ✓ | 24 |
24.f | even | 2 | 1 | 960.4.h.d | 24 | ||
24.h | odd | 2 | 1 | 960.4.h.d | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
60.4.e.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
60.4.e.a | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
60.4.e.a | ✓ | 24 | 4.b | odd | 2 | 1 | inner |
60.4.e.a | ✓ | 24 | 12.b | even | 2 | 1 | inner |
960.4.h.d | 24 | 8.b | even | 2 | 1 | ||
960.4.h.d | 24 | 8.d | odd | 2 | 1 | ||
960.4.h.d | 24 | 24.f | even | 2 | 1 | ||
960.4.h.d | 24 | 24.h | odd | 2 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(60, [\chi])\).