Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [29,13,Mod(12,29)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(29, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 13, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("29.12");
S:= CuspForms(chi, 13);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 29 \) |
Weight: | \( k \) | \(=\) | \( 13 \) |
Character orbit: | \([\chi]\) | \(=\) | 29.c (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: | \(26.5058207010\) |
Analytic rank: | \(0\) |
Dimension: | \(58\) |
Relative dimension: | \(29\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
12.1 | −88.1500 | − | 88.1500i | 347.445 | + | 347.445i | 11444.8i | 14434.6i | − | 61254.6i | −35450.1 | 647800. | − | 647800.i | − | 290005.i | 1.27241e6 | − | 1.27241e6i | ||||||||
12.2 | −79.7284 | − | 79.7284i | −520.777 | − | 520.777i | 8617.22i | − | 29504.0i | 83041.4i | −164320. | 360470. | − | 360470.i | 10976.2i | −2.35231e6 | + | 2.35231e6i | |||||||||
12.3 | −76.2293 | − | 76.2293i | −132.533 | − | 132.533i | 7525.81i | − | 2767.06i | 20205.8i | 132887. | 261452. | − | 261452.i | − | 496311.i | −210931. | + | 210931.i | ||||||||
12.4 | −70.5170 | − | 70.5170i | −767.125 | − | 767.125i | 5849.31i | 9904.63i | 108191.i | 71097.8 | 123638. | − | 123638.i | 645520.i | 698445. | − | 698445.i | ||||||||||
12.5 | −69.9680 | − | 69.9680i | 814.355 | + | 814.355i | 5695.05i | − | 19209.4i | − | 113958.i | −6147.85 | 111882. | − | 111882.i | 794908.i | −1.34405e6 | + | 1.34405e6i | ||||||||
12.6 | −58.0743 | − | 58.0743i | 433.237 | + | 433.237i | 2649.26i | 10278.3i | − | 50319.9i | −72219.2 | −84018.7 | + | 84018.7i | − | 156053.i | 596905. | − | 596905.i | ||||||||
12.7 | −54.5113 | − | 54.5113i | −434.460 | − | 434.460i | 1846.95i | 19548.9i | 47365.9i | −201373. | −122598. | + | 122598.i | − | 153930.i | 1.06564e6 | − | 1.06564e6i | |||||||||
12.8 | −44.8874 | − | 44.8874i | 80.4158 | + | 80.4158i | − | 66.2505i | − | 12643.6i | − | 7219.30i | 209784. | −186832. | + | 186832.i | − | 518508.i | −567536. | + | 567536.i | ||||||
12.9 | −44.4183 | − | 44.4183i | 834.913 | + | 834.913i | − | 150.028i | 28014.5i | − | 74170.9i | 122123. | −188601. | + | 188601.i | 862719.i | 1.24436e6 | − | 1.24436e6i | ||||||||
12.10 | −33.9863 | − | 33.9863i | −37.2586 | − | 37.2586i | − | 1785.87i | − | 14007.5i | 2532.56i | −78700.5 | −199903. | + | 199903.i | − | 528665.i | −476064. | + | 476064.i | |||||||
12.11 | −29.6845 | − | 29.6845i | −878.412 | − | 878.412i | − | 2333.66i | − | 15619.8i | 52150.4i | 10196.8 | −190861. | + | 190861.i | 1.01178e6i | −463667. | + | 463667.i | ||||||||
12.12 | −15.4217 | − | 15.4217i | −236.320 | − | 236.320i | − | 3620.34i | 24342.5i | 7288.92i | 119306. | −118999. | + | 118999.i | − | 419746.i | 375403. | − | 375403.i | ||||||||
12.13 | −12.2710 | − | 12.2710i | 762.089 | + | 762.089i | − | 3794.84i | − | 6703.86i | − | 18703.2i | −134602. | −96828.8 | + | 96828.8i | 630118.i | −82263.3 | + | 82263.3i | |||||||
12.14 | −9.03178 | − | 9.03178i | −704.167 | − | 704.167i | − | 3932.85i | 8770.94i | 12719.8i | 37099.6 | −72514.8 | + | 72514.8i | 460261.i | 79217.2 | − | 79217.2i | |||||||||
12.15 | 2.50807 | + | 2.50807i | 773.536 | + | 773.536i | − | 4083.42i | − | 10269.1i | 3880.17i | 192386. | 20514.6 | − | 20514.6i | 665275.i | 25755.5 | − | 25755.5i | ||||||||
12.16 | 8.76721 | + | 8.76721i | 327.395 | + | 327.395i | − | 3942.27i | 12032.8i | 5740.68i | −63314.5 | 70473.2 | − | 70473.2i | − | 317066.i | −105494. | + | 105494.i | ||||||||
12.17 | 18.1237 | + | 18.1237i | −105.386 | − | 105.386i | − | 3439.06i | − | 24861.6i | − | 3819.95i | 4681.32 | 136563. | − | 136563.i | − | 509229.i | 450583. | − | 450583.i | ||||||
12.18 | 26.4284 | + | 26.4284i | −670.941 | − | 670.941i | − | 2699.08i | 5810.81i | − | 35463.8i | −231201. | 179583. | − | 179583.i | 368882.i | −153570. | + | 153570.i | ||||||||
12.19 | 33.1274 | + | 33.1274i | −631.679 | − | 631.679i | − | 1901.16i | − | 13469.1i | − | 41851.7i | 113718. | 198670. | − | 198670.i | 266596.i | 446197. | − | 446197.i | |||||||
12.20 | 43.6471 | + | 43.6471i | −94.5951 | − | 94.5951i | − | 285.862i | 15326.9i | − | 8257.61i | 49832.3 | 191256. | − | 191256.i | − | 513545.i | −668974. | + | 668974.i | |||||||
See all 58 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
29.c | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 29.13.c.a | ✓ | 58 |
29.c | odd | 4 | 1 | inner | 29.13.c.a | ✓ | 58 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
29.13.c.a | ✓ | 58 | 1.a | even | 1 | 1 | trivial |
29.13.c.a | ✓ | 58 | 29.c | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(29, [\chi])\).