Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [29,7,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, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("29.12");
S:= CuspForms(chi, 7);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 29 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
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: | \(6.67156842498\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(14\) 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 | −11.1386 | − | 11.1386i | 25.3142 | + | 25.3142i | 184.138i | − | 151.054i | − | 563.931i | −332.137 | 1338.17 | − | 1338.17i | 552.618i | −1682.54 | + | 1682.54i | ||||||||
12.2 | −9.26981 | − | 9.26981i | −17.8548 | − | 17.8548i | 107.859i | 107.938i | 331.022i | −104.064 | 406.562 | − | 406.562i | − | 91.4089i | 1000.56 | − | 1000.56i | |||||||||
12.3 | −6.90454 | − | 6.90454i | 11.2332 | + | 11.2332i | 31.3454i | 10.9400i | − | 155.120i | 617.401 | −225.465 | + | 225.465i | − | 476.632i | 75.5356 | − | 75.5356i | ||||||||
12.4 | −6.25438 | − | 6.25438i | −11.2134 | − | 11.2134i | 14.2346i | − | 192.776i | 140.266i | −54.6116 | −311.252 | + | 311.252i | − | 477.517i | −1205.69 | + | 1205.69i | ||||||||
12.5 | −5.36218 | − | 5.36218i | 27.0129 | + | 27.0129i | − | 6.49405i | 130.759i | − | 289.696i | −330.754 | −378.002 | + | 378.002i | 730.397i | 701.154 | − | 701.154i | ||||||||
12.6 | −1.83652 | − | 1.83652i | −32.7814 | − | 32.7814i | − | 57.2544i | − | 4.82891i | 120.407i | 52.1989 | −222.686 | + | 222.686i | 1420.24i | −8.86838 | + | 8.86838i | ||||||||
12.7 | −0.624225 | − | 0.624225i | 1.83796 | + | 1.83796i | − | 63.2207i | 41.8066i | − | 2.29460i | −304.112 | −79.4144 | + | 79.4144i | − | 722.244i | 26.0967 | − | 26.0967i | |||||||
12.8 | 0.260765 | + | 0.260765i | 29.0542 | + | 29.0542i | − | 63.8640i | − | 194.251i | 15.1526i | 282.863 | 33.3424 | − | 33.3424i | 959.292i | 50.6540 | − | 50.6540i | ||||||||
12.9 | 2.37438 | + | 2.37438i | −6.54119 | − | 6.54119i | − | 52.7246i | 220.621i | − | 31.0626i | 573.636 | 277.149 | − | 277.149i | − | 643.426i | −523.840 | + | 523.840i | |||||||
12.10 | 4.72749 | + | 4.72749i | −5.84368 | − | 5.84368i | − | 19.3016i | − | 122.935i | − | 55.2519i | −317.611 | 393.808 | − | 393.808i | − | 660.703i | 581.174 | − | 581.174i | ||||||
12.11 | 6.98171 | + | 6.98171i | 24.6261 | + | 24.6261i | 33.4886i | 68.1677i | 343.865i | −10.7616 | 213.022 | − | 213.022i | 483.892i | −475.927 | + | 475.927i | ||||||||||
12.12 | 7.44805 | + | 7.44805i | −22.6252 | − | 22.6252i | 46.9470i | − | 114.512i | − | 337.027i | 512.680 | 127.012 | − | 127.012i | 294.796i | 852.893 | − | 852.893i | ||||||||
12.13 | 8.70689 | + | 8.70689i | −29.1706 | − | 29.1706i | 87.6199i | 207.022i | − | 507.971i | −652.893 | −205.655 | + | 205.655i | 972.849i | −1802.52 | + | 1802.52i | |||||||||
12.14 | 10.8910 | + | 10.8910i | 5.95173 | + | 5.95173i | 173.227i | − | 50.8966i | 129.640i | 66.1657 | −1189.59 | + | 1189.59i | − | 658.154i | 554.314 | − | 554.314i | ||||||||
17.1 | −11.1386 | + | 11.1386i | 25.3142 | − | 25.3142i | − | 184.138i | 151.054i | 563.931i | −332.137 | 1338.17 | + | 1338.17i | − | 552.618i | −1682.54 | − | 1682.54i | ||||||||
17.2 | −9.26981 | + | 9.26981i | −17.8548 | + | 17.8548i | − | 107.859i | − | 107.938i | − | 331.022i | −104.064 | 406.562 | + | 406.562i | 91.4089i | 1000.56 | + | 1000.56i | |||||||
17.3 | −6.90454 | + | 6.90454i | 11.2332 | − | 11.2332i | − | 31.3454i | − | 10.9400i | 155.120i | 617.401 | −225.465 | − | 225.465i | 476.632i | 75.5356 | + | 75.5356i | ||||||||
17.4 | −6.25438 | + | 6.25438i | −11.2134 | + | 11.2134i | − | 14.2346i | 192.776i | − | 140.266i | −54.6116 | −311.252 | − | 311.252i | 477.517i | −1205.69 | − | 1205.69i | ||||||||
17.5 | −5.36218 | + | 5.36218i | 27.0129 | − | 27.0129i | 6.49405i | − | 130.759i | 289.696i | −330.754 | −378.002 | − | 378.002i | − | 730.397i | 701.154 | + | 701.154i | ||||||||
17.6 | −1.83652 | + | 1.83652i | −32.7814 | + | 32.7814i | 57.2544i | 4.82891i | − | 120.407i | 52.1989 | −222.686 | − | 222.686i | − | 1420.24i | −8.86838 | − | 8.86838i | ||||||||
See all 28 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 | 29.7.c.a | ✓ | 28 |
29.c | odd | 4 | 1 | inner | 29.7.c.a | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
29.7.c.a | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
29.7.c.a | ✓ | 28 | 29.c | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(29, [\chi])\).