Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [13,13,Mod(5,13)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3]))
N = Newforms(chi, 13, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13.5");
S:= CuspForms(chi, 13);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 13 \) |
Weight: | \( k \) | \(=\) | \( 13 \) |
Character orbit: | \([\chi]\) | \(=\) | 13.d (of order \(4\), degree \(2\), 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: | \(11.8819196246\) |
Analytic rank: | \(0\) |
Dimension: | \(26\) |
Relative dimension: | \(13\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −82.3157 | + | 82.3157i | 532.405 | − | 9455.73i | −1258.93 | + | 1258.93i | −43825.3 | + | 43825.3i | 123092. | + | 123092.i | 441190. | + | 441190.i | −247986. | − | 207260.i | ||||||
5.2 | −73.6638 | + | 73.6638i | −1197.88 | − | 6756.71i | 2452.94 | − | 2452.94i | 88240.2 | − | 88240.2i | −37699.0 | − | 37699.0i | 195998. | + | 195998.i | 903471. | 361386.i | |||||||
5.3 | −55.4726 | + | 55.4726i | −123.365 | − | 2058.41i | −10153.2 | + | 10153.2i | 6843.39 | − | 6843.39i | −69327.0 | − | 69327.0i | −113030. | − | 113030.i | −516222. | − | 1.12645e6i | ||||||
5.4 | −51.8539 | + | 51.8539i | 843.934 | − | 1281.66i | 16935.0 | − | 16935.0i | −43761.3 | + | 43761.3i | −94929.1 | − | 94929.1i | −145935. | − | 145935.i | 180783. | 1.75629e6i | |||||||
5.5 | −23.7102 | + | 23.7102i | −547.334 | 2971.66i | 18861.9 | − | 18861.9i | 12977.4 | − | 12977.4i | 68016.7 | + | 68016.7i | −167575. | − | 167575.i | −231867. | 894440.i | ||||||||
5.6 | −20.6340 | + | 20.6340i | 1133.68 | 3244.47i | −11267.3 | + | 11267.3i | −23392.4 | + | 23392.4i | 41508.2 | + | 41508.2i | −151463. | − | 151463.i | 753788. | − | 464977.i | |||||||
5.7 | −11.9679 | + | 11.9679i | −712.886 | 3809.54i | −8528.30 | + | 8528.30i | 8531.76 | − | 8531.76i | 71048.2 | + | 71048.2i | −94612.8 | − | 94612.8i | −23234.8 | − | 204132.i | |||||||
5.8 | 21.2021 | − | 21.2021i | 104.714 | 3196.95i | −2687.41 | + | 2687.41i | 2220.14 | − | 2220.14i | −129238. | − | 129238.i | 154625. | + | 154625.i | −520476. | 113957.i | ||||||||
5.9 | 35.7844 | − | 35.7844i | 720.183 | 1534.96i | 8189.78 | − | 8189.78i | 25771.3 | − | 25771.3i | 69804.0 | + | 69804.0i | 201500. | + | 201500.i | −12777.9 | − | 586132.i | |||||||
5.10 | 40.0541 | − | 40.0541i | −1363.29 | 887.334i | 2683.01 | − | 2683.01i | −54605.4 | + | 54605.4i | −74989.8 | − | 74989.8i | 199603. | + | 199603.i | 1.32712e6 | − | 214931.i | |||||||
5.11 | 65.2537 | − | 65.2537i | −339.809 | − | 4420.09i | −17964.9 | + | 17964.9i | −22173.8 | + | 22173.8i | 90738.1 | + | 90738.1i | −21147.8 | − | 21147.8i | −415971. | 2.34455e6i | |||||||
5.12 | 77.2713 | − | 77.2713i | −352.569 | − | 7845.71i | 14032.6 | − | 14032.6i | −27243.5 | + | 27243.5i | −2786.27 | − | 2786.27i | −289745. | − | 289745.i | −407136. | − | 2.16863e6i | ||||||
5.13 | 79.0525 | − | 79.0525i | 1300.22 | − | 8402.59i | −8722.22 | + | 8722.22i | 102785. | − | 102785.i | −112440. | − | 112440.i | −340447. | − | 340447.i | 1.15912e6 | 1.37903e6i | |||||||
8.1 | −82.3157 | − | 82.3157i | 532.405 | 9455.73i | −1258.93 | − | 1258.93i | −43825.3 | − | 43825.3i | 123092. | − | 123092.i | 441190. | − | 441190.i | −247986. | 207260.i | ||||||||
8.2 | −73.6638 | − | 73.6638i | −1197.88 | 6756.71i | 2452.94 | + | 2452.94i | 88240.2 | + | 88240.2i | −37699.0 | + | 37699.0i | 195998. | − | 195998.i | 903471. | − | 361386.i | |||||||
8.3 | −55.4726 | − | 55.4726i | −123.365 | 2058.41i | −10153.2 | − | 10153.2i | 6843.39 | + | 6843.39i | −69327.0 | + | 69327.0i | −113030. | + | 113030.i | −516222. | 1.12645e6i | ||||||||
8.4 | −51.8539 | − | 51.8539i | 843.934 | 1281.66i | 16935.0 | + | 16935.0i | −43761.3 | − | 43761.3i | −94929.1 | + | 94929.1i | −145935. | + | 145935.i | 180783. | − | 1.75629e6i | |||||||
8.5 | −23.7102 | − | 23.7102i | −547.334 | − | 2971.66i | 18861.9 | + | 18861.9i | 12977.4 | + | 12977.4i | 68016.7 | − | 68016.7i | −167575. | + | 167575.i | −231867. | − | 894440.i | ||||||
8.6 | −20.6340 | − | 20.6340i | 1133.68 | − | 3244.47i | −11267.3 | − | 11267.3i | −23392.4 | − | 23392.4i | 41508.2 | − | 41508.2i | −151463. | + | 151463.i | 753788. | 464977.i | |||||||
8.7 | −11.9679 | − | 11.9679i | −712.886 | − | 3809.54i | −8528.30 | − | 8528.30i | 8531.76 | + | 8531.76i | 71048.2 | − | 71048.2i | −94612.8 | + | 94612.8i | −23234.8 | 204132.i | |||||||
See all 26 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.d | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 13.13.d.a | ✓ | 26 |
13.d | odd | 4 | 1 | inner | 13.13.d.a | ✓ | 26 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
13.13.d.a | ✓ | 26 | 1.a | even | 1 | 1 | trivial |
13.13.d.a | ✓ | 26 | 13.d | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(13, [\chi])\).