Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,2,Mod(64,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 10]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.64");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 441.u (of order \(7\), degree \(6\), 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: | \(3.52140272914\) |
Analytic rank: | \(0\) |
Dimension: | \(60\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{7})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{7}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
64.1 | −0.605649 | + | 2.65352i | 0 | −4.87244 | − | 2.34644i | 1.34310 | − | 1.68419i | 0 | −1.61866 | + | 2.09283i | 5.78335 | − | 7.25209i | 0 | 3.65560 | + | 4.58397i | ||||||
64.2 | −0.458811 | + | 2.01018i | 0 | −2.02838 | − | 0.976818i | −2.04151 | + | 2.55997i | 0 | 2.15910 | + | 1.52915i | 0.323108 | − | 0.405164i | 0 | −4.20934 | − | 5.27834i | ||||||
64.3 | −0.418387 | + | 1.83307i | 0 | −1.38317 | − | 0.666101i | 0.364972 | − | 0.457660i | 0 | −1.75476 | − | 1.98010i | −0.544876 | + | 0.683253i | 0 | 0.686225 | + | 0.860499i | ||||||
64.4 | −0.150352 | + | 0.658735i | 0 | 1.39061 | + | 0.669683i | 1.64892 | − | 2.06768i | 0 | −1.22090 | + | 2.34721i | −1.49278 | + | 1.87188i | 0 | 1.11413 | + | 1.39708i | ||||||
64.5 | −0.131442 | + | 0.575886i | 0 | 1.48757 | + | 0.716376i | 1.01472 | − | 1.27242i | 0 | 2.21270 | − | 1.45050i | −1.34467 | + | 1.68616i | 0 | 0.599392 | + | 0.751614i | ||||||
64.6 | 0.131442 | − | 0.575886i | 0 | 1.48757 | + | 0.716376i | −1.01472 | + | 1.27242i | 0 | 2.21270 | − | 1.45050i | 1.34467 | − | 1.68616i | 0 | 0.599392 | + | 0.751614i | ||||||
64.7 | 0.150352 | − | 0.658735i | 0 | 1.39061 | + | 0.669683i | −1.64892 | + | 2.06768i | 0 | −1.22090 | + | 2.34721i | 1.49278 | − | 1.87188i | 0 | 1.11413 | + | 1.39708i | ||||||
64.8 | 0.418387 | − | 1.83307i | 0 | −1.38317 | − | 0.666101i | −0.364972 | + | 0.457660i | 0 | −1.75476 | − | 1.98010i | 0.544876 | − | 0.683253i | 0 | 0.686225 | + | 0.860499i | ||||||
64.9 | 0.458811 | − | 2.01018i | 0 | −2.02838 | − | 0.976818i | 2.04151 | − | 2.55997i | 0 | 2.15910 | + | 1.52915i | −0.323108 | + | 0.405164i | 0 | −4.20934 | − | 5.27834i | ||||||
64.10 | 0.605649 | − | 2.65352i | 0 | −4.87244 | − | 2.34644i | −1.34310 | + | 1.68419i | 0 | −1.61866 | + | 2.09283i | −5.78335 | + | 7.25209i | 0 | 3.65560 | + | 4.58397i | ||||||
127.1 | −2.47259 | − | 1.19074i | 0 | 3.44888 | + | 4.32476i | 0.875748 | + | 3.83690i | 0 | 2.35340 | + | 1.20894i | −2.15666 | − | 9.44894i | 0 | 2.40338 | − | 10.5299i | ||||||
127.2 | −2.10278 | − | 1.01265i | 0 | 2.14925 | + | 2.69508i | −0.460720 | − | 2.01854i | 0 | −2.61455 | − | 0.405131i | −0.751559 | − | 3.29280i | 0 | −1.07528 | + | 4.71110i | ||||||
127.3 | −1.23234 | − | 0.593462i | 0 | −0.0805240 | − | 0.100974i | 0.303726 | + | 1.33071i | 0 | −0.112387 | − | 2.64336i | 0.648032 | + | 2.83922i | 0 | 0.415434 | − | 1.82013i | ||||||
127.4 | −0.964230 | − | 0.464348i | 0 | −0.532861 | − | 0.668186i | −0.00423780 | − | 0.0185670i | 0 | 1.74036 | + | 1.99277i | 0.679819 | + | 2.97848i | 0 | −0.00453535 | + | 0.0198707i | ||||||
127.5 | −0.0642773 | − | 0.0309543i | 0 | −1.24381 | − | 1.55968i | −0.848708 | − | 3.71843i | 0 | −2.26779 | + | 1.36276i | 0.0634200 | + | 0.277861i | 0 | −0.0605489 | + | 0.265282i | ||||||
127.6 | 0.0642773 | + | 0.0309543i | 0 | −1.24381 | − | 1.55968i | 0.848708 | + | 3.71843i | 0 | −2.26779 | + | 1.36276i | −0.0634200 | − | 0.277861i | 0 | −0.0605489 | + | 0.265282i | ||||||
127.7 | 0.964230 | + | 0.464348i | 0 | −0.532861 | − | 0.668186i | 0.00423780 | + | 0.0185670i | 0 | 1.74036 | + | 1.99277i | −0.679819 | − | 2.97848i | 0 | −0.00453535 | + | 0.0198707i | ||||||
127.8 | 1.23234 | + | 0.593462i | 0 | −0.0805240 | − | 0.100974i | −0.303726 | − | 1.33071i | 0 | −0.112387 | − | 2.64336i | −0.648032 | − | 2.83922i | 0 | 0.415434 | − | 1.82013i | ||||||
127.9 | 2.10278 | + | 1.01265i | 0 | 2.14925 | + | 2.69508i | 0.460720 | + | 2.01854i | 0 | −2.61455 | − | 0.405131i | 0.751559 | + | 3.29280i | 0 | −1.07528 | + | 4.71110i | ||||||
127.10 | 2.47259 | + | 1.19074i | 0 | 3.44888 | + | 4.32476i | −0.875748 | − | 3.83690i | 0 | 2.35340 | + | 1.20894i | 2.15666 | + | 9.44894i | 0 | 2.40338 | − | 10.5299i | ||||||
See all 60 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
49.e | even | 7 | 1 | inner |
147.l | odd | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 441.2.u.e | ✓ | 60 |
3.b | odd | 2 | 1 | inner | 441.2.u.e | ✓ | 60 |
49.e | even | 7 | 1 | inner | 441.2.u.e | ✓ | 60 |
147.l | odd | 14 | 1 | inner | 441.2.u.e | ✓ | 60 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
441.2.u.e | ✓ | 60 | 1.a | even | 1 | 1 | trivial |
441.2.u.e | ✓ | 60 | 3.b | odd | 2 | 1 | inner |
441.2.u.e | ✓ | 60 | 49.e | even | 7 | 1 | inner |
441.2.u.e | ✓ | 60 | 147.l | odd | 14 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} + 16 T_{2}^{58} + 166 T_{2}^{56} + 1544 T_{2}^{54} + 13794 T_{2}^{52} + 113435 T_{2}^{50} + \cdots + 35721 \) acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\).