Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [37,7,Mod(6,37)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(37, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("37.6");
S:= CuspForms(chi, 7);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 37 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 37.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: | \(8.51200109393\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(18\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
6.1 | −10.4127 | + | 10.4127i | − | 29.1399i | − | 152.847i | −98.7683 | − | 98.7683i | 303.424 | + | 303.424i | −196.570 | 925.130 | + | 925.130i | −120.135 | 2056.88 | ||||||||
6.2 | −10.2905 | + | 10.2905i | 12.1509i | − | 147.790i | 92.7424 | + | 92.7424i | −125.039 | − | 125.039i | 599.396 | 862.241 | + | 862.241i | 581.356 | −1908.74 | |||||||||
6.3 | −8.48840 | + | 8.48840i | 25.9060i | − | 80.1058i | 25.7278 | + | 25.7278i | −219.900 | − | 219.900i | −623.749 | 136.713 | + | 136.713i | 57.8805 | −436.776 | |||||||||
6.4 | −7.66813 | + | 7.66813i | 49.4380i | − | 53.6006i | −147.246 | − | 147.246i | −379.097 | − | 379.097i | 416.828 | −79.7443 | − | 79.7443i | −1715.12 | 2258.21 | |||||||||
6.5 | −6.59379 | + | 6.59379i | − | 40.5120i | − | 22.9560i | 83.9037 | + | 83.9037i | 267.127 | + | 267.127i | −26.0039 | −270.635 | − | 270.635i | −912.222 | −1106.49 | ||||||||
6.6 | −5.58722 | + | 5.58722i | − | 13.0839i | 1.56600i | −61.8977 | − | 61.8977i | 73.1027 | + | 73.1027i | 184.379 | −366.331 | − | 366.331i | 557.811 | 691.671 | |||||||||
6.7 | −3.76006 | + | 3.76006i | 16.7537i | 35.7238i | 120.059 | + | 120.059i | −62.9949 | − | 62.9949i | 144.567 | −374.968 | − | 374.968i | 448.314 | −902.857 | ||||||||||
6.8 | −1.40537 | + | 1.40537i | 9.49587i | 60.0499i | −104.306 | − | 104.306i | −13.3452 | − | 13.3452i | −165.435 | −174.336 | − | 174.336i | 638.828 | 293.177 | ||||||||||
6.9 | −0.186040 | + | 0.186040i | 44.0184i | 63.9308i | 56.4752 | + | 56.4752i | −8.18919 | − | 8.18919i | 89.8430 | −23.8003 | − | 23.8003i | −1208.62 | −21.0133 | ||||||||||
6.10 | 1.01641 | − | 1.01641i | − | 50.0255i | 61.9338i | −128.165 | − | 128.165i | −50.8466 | − | 50.8466i | 143.000 | 128.001 | + | 128.001i | −1773.55 | −260.537 | |||||||||
6.11 | 1.13198 | − | 1.13198i | − | 20.4994i | 61.4373i | 84.2468 | + | 84.2468i | −23.2048 | − | 23.2048i | −640.570 | 141.992 | + | 141.992i | 308.775 | 190.731 | |||||||||
6.12 | 2.77875 | − | 2.77875i | − | 27.0135i | 48.5571i | 96.7657 | + | 96.7657i | −75.0636 | − | 75.0636i | 312.208 | 312.768 | + | 312.768i | −0.728696 | 537.775 | |||||||||
6.13 | 5.04935 | − | 5.04935i | 11.9325i | 13.0081i | −47.3379 | − | 47.3379i | 60.2515 | + | 60.2515i | 421.079 | 388.841 | + | 388.841i | 586.615 | −478.051 | ||||||||||
6.14 | 5.77519 | − | 5.77519i | 42.6591i | − | 2.70569i | −53.2840 | − | 53.2840i | 246.364 | + | 246.364i | −452.267 | 353.986 | + | 353.986i | −1090.80 | −615.451 | |||||||||
6.15 | 7.44064 | − | 7.44064i | − | 17.9951i | − | 46.7261i | −84.4779 | − | 84.4779i | −133.895 | − | 133.895i | −514.613 | 128.529 | + | 128.529i | 405.176 | −1257.14 | ||||||||
6.16 | 8.60959 | − | 8.60959i | 22.2519i | − | 84.2501i | 144.668 | + | 144.668i | 191.579 | + | 191.579i | −94.2990 | −174.345 | − | 174.345i | 233.855 | 2491.07 | |||||||||
6.17 | 9.39156 | − | 9.39156i | − | 33.9178i | − | 112.403i | 19.7999 | + | 19.7999i | −318.541 | − | 318.541i | 150.778 | −454.578 | − | 454.578i | −421.419 | 371.904 | ||||||||
6.18 | 11.1987 | − | 11.1987i | 29.5809i | − | 186.823i | −126.906 | − | 126.906i | 331.268 | + | 331.268i | 249.429 | −1375.46 | − | 1375.46i | −146.027 | −2842.36 | |||||||||
31.1 | −10.4127 | − | 10.4127i | 29.1399i | 152.847i | −98.7683 | + | 98.7683i | 303.424 | − | 303.424i | −196.570 | 925.130 | − | 925.130i | −120.135 | 2056.88 | ||||||||||
31.2 | −10.2905 | − | 10.2905i | − | 12.1509i | 147.790i | 92.7424 | − | 92.7424i | −125.039 | + | 125.039i | 599.396 | 862.241 | − | 862.241i | 581.356 | −1908.74 | |||||||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
37.d | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 37.7.d.a | ✓ | 36 |
37.d | odd | 4 | 1 | inner | 37.7.d.a | ✓ | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
37.7.d.a | ✓ | 36 | 1.a | even | 1 | 1 | trivial |
37.7.d.a | ✓ | 36 | 37.d | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(37, [\chi])\).