Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [630,4,Mod(89,630)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(630, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 5]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("630.89");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 630.bo (of order \(6\), 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: | \(37.1712033036\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
89.1 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | −11.0128 | − | 1.92807i | 0 | −18.5106 | − | 0.596872i | −8.00000 | 0 | −14.3524 | + | 17.1467i | ||||||||
89.2 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | −10.1071 | + | 4.77971i | 0 | 1.14987 | − | 18.4845i | −8.00000 | 0 | −1.82845 | + | 22.2858i | ||||||||
89.3 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | −10.0230 | − | 4.95368i | 0 | −13.6176 | − | 12.5523i | −8.00000 | 0 | −18.6031 | + | 12.4067i | ||||||||
89.4 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | −9.99303 | + | 5.01392i | 0 | 15.1516 | − | 10.6503i | −8.00000 | 0 | −1.30866 | + | 22.3224i | ||||||||
89.5 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | −9.30153 | − | 6.20335i | 0 | 13.6176 | + | 12.5523i | −8.00000 | 0 | −20.0460 | + | 9.90737i | ||||||||
89.6 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | −8.97207 | + | 6.67098i | 0 | 8.88443 | + | 16.2501i | −8.00000 | 0 | 2.58241 | + | 22.2111i | ||||||||
89.7 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | −8.42518 | + | 7.34959i | 0 | −12.3205 | + | 13.8277i | −8.00000 | 0 | 4.30468 | + | 21.9424i | ||||||||
89.8 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | −7.17618 | − | 8.57336i | 0 | 18.5106 | + | 0.596872i | −8.00000 | 0 | −22.0257 | + | 3.85614i | ||||||||
89.9 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | −3.68683 | + | 10.5550i | 0 | −16.1725 | − | 9.02502i | −8.00000 | 0 | 14.5949 | + | 16.9407i | ||||||||
89.10 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | −0.914226 | − | 11.1429i | 0 | −1.14987 | + | 18.4845i | −8.00000 | 0 | −20.2143 | − | 9.55941i | ||||||||
89.11 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | −0.654331 | − | 11.1612i | 0 | −15.1516 | + | 10.6503i | −8.00000 | 0 | −19.9861 | − | 10.0278i | ||||||||
89.12 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | 1.29121 | − | 11.1055i | 0 | −8.88443 | − | 16.2501i | −8.00000 | 0 | −17.9441 | − | 13.3420i | ||||||||
89.13 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | 2.03180 | + | 10.9942i | 0 | 15.7564 | − | 9.73321i | −8.00000 | 0 | 21.0743 | + | 7.47498i | ||||||||
89.14 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | 2.15234 | − | 10.9712i | 0 | 12.3205 | − | 13.8277i | −8.00000 | 0 | −16.8504 | − | 14.6992i | ||||||||
89.15 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | 3.37521 | + | 10.6587i | 0 | −3.57853 | + | 18.1712i | −8.00000 | 0 | 21.8366 | + | 4.81268i | ||||||||
89.16 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | 4.72065 | + | 10.1349i | 0 | −14.6848 | − | 11.2852i | −8.00000 | 0 | 22.2747 | + | 1.95846i | ||||||||
89.17 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | 5.44876 | + | 9.76274i | 0 | 17.7270 | − | 5.36235i | −8.00000 | 0 | 22.3583 | + | 0.325214i | ||||||||
89.18 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | 7.29745 | − | 8.47037i | 0 | 16.1725 | + | 9.02502i | −8.00000 | 0 | −7.37367 | − | 21.1099i | ||||||||
89.19 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | 8.81157 | + | 6.88159i | 0 | 9.44916 | + | 15.9284i | −8.00000 | 0 | 20.7308 | − | 8.38050i | ||||||||
89.20 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | 10.3654 | + | 4.19025i | 0 | −9.44916 | − | 15.9284i | −8.00000 | 0 | 17.6231 | − | 13.7632i | ||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
15.d | odd | 2 | 1 | inner |
105.p | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 630.4.bo.b | yes | 48 |
3.b | odd | 2 | 1 | 630.4.bo.a | ✓ | 48 | |
5.b | even | 2 | 1 | 630.4.bo.a | ✓ | 48 | |
7.d | odd | 6 | 1 | inner | 630.4.bo.b | yes | 48 |
15.d | odd | 2 | 1 | inner | 630.4.bo.b | yes | 48 |
21.g | even | 6 | 1 | 630.4.bo.a | ✓ | 48 | |
35.i | odd | 6 | 1 | 630.4.bo.a | ✓ | 48 | |
105.p | even | 6 | 1 | inner | 630.4.bo.b | yes | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
630.4.bo.a | ✓ | 48 | 3.b | odd | 2 | 1 | |
630.4.bo.a | ✓ | 48 | 5.b | even | 2 | 1 | |
630.4.bo.a | ✓ | 48 | 21.g | even | 6 | 1 | |
630.4.bo.a | ✓ | 48 | 35.i | odd | 6 | 1 | |
630.4.bo.b | yes | 48 | 1.a | even | 1 | 1 | trivial |
630.4.bo.b | yes | 48 | 7.d | odd | 6 | 1 | inner |
630.4.bo.b | yes | 48 | 15.d | odd | 2 | 1 | inner |
630.4.bo.b | yes | 48 | 105.p | even | 6 | 1 | inner |