Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [784,2,Mod(165,784)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(784, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 3, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("784.165");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 784 = 2^{4} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 784.x (of order \(12\), degree \(4\), not 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: | \(6.26027151847\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
165.1 | −1.21510 | − | 0.723563i | −0.608516 | − | 2.27101i | 0.952912 | + | 1.75840i | −0.400455 | + | 1.49452i | −0.903818 | + | 3.19980i | 0 | 0.114433 | − | 2.82611i | −2.18914 | + | 1.26390i | 1.56797 | − | 1.52623i | ||
165.2 | −1.21510 | − | 0.723563i | 0.608516 | + | 2.27101i | 0.952912 | + | 1.75840i | 0.400455 | − | 1.49452i | 0.903818 | − | 3.19980i | 0 | 0.114433 | − | 2.82611i | −2.18914 | + | 1.26390i | −1.56797 | + | 1.52623i | ||
165.3 | −0.953167 | + | 1.04474i | −0.554848 | − | 2.07072i | −0.182947 | − | 1.99162i | −0.265812 | + | 0.992025i | 2.69222 | + | 1.39407i | 0 | 2.25509 | + | 1.70721i | −1.38196 | + | 0.797875i | −0.783041 | − | 1.22327i | ||
165.4 | −0.953167 | + | 1.04474i | 0.554848 | + | 2.07072i | −0.182947 | − | 1.99162i | 0.265812 | − | 0.992025i | −2.69222 | − | 1.39407i | 0 | 2.25509 | + | 1.70721i | −1.38196 | + | 0.797875i | 0.783041 | + | 1.22327i | ||
165.5 | 0.164410 | + | 1.40462i | −0.338847 | − | 1.26459i | −1.94594 | + | 0.461868i | 0.958008 | − | 3.57533i | 1.72057 | − | 0.683864i | 0 | −0.968683 | − | 2.65738i | 1.11370 | − | 0.642992i | 5.17951 | + | 0.757821i | ||
165.6 | 0.164410 | + | 1.40462i | 0.338847 | + | 1.26459i | −1.94594 | + | 0.461868i | −0.958008 | + | 3.57533i | −1.72057 | + | 0.683864i | 0 | −0.968683 | − | 2.65738i | 1.11370 | − | 0.642992i | −5.17951 | − | 0.757821i | ||
165.7 | 0.674626 | − | 1.24293i | −0.839505 | − | 3.13308i | −1.08976 | − | 1.67703i | 0.734404 | − | 2.74083i | −4.46055 | − | 1.07021i | 0 | −2.81961 | + | 0.223131i | −6.51332 | + | 3.76047i | −2.91122 | − | 2.76185i | ||
165.8 | 0.674626 | − | 1.24293i | 0.839505 | + | 3.13308i | −1.08976 | − | 1.67703i | −0.734404 | + | 2.74083i | 4.46055 | + | 1.07021i | 0 | −2.81961 | + | 0.223131i | −6.51332 | + | 3.76047i | 2.91122 | + | 2.76185i | ||
165.9 | 1.32923 | − | 0.482865i | −0.331601 | − | 1.23755i | 1.53368 | − | 1.28367i | −0.850527 | + | 3.17421i | −1.03834 | − | 1.48487i | 0 | 1.41877 | − | 2.44685i | 1.17650 | − | 0.679252i | 0.402172 | + | 4.62993i | ||
165.10 | 1.32923 | − | 0.482865i | 0.331601 | + | 1.23755i | 1.53368 | − | 1.28367i | 0.850527 | − | 3.17421i | 1.03834 | + | 1.48487i | 0 | 1.41877 | − | 2.44685i | 1.17650 | − | 0.679252i | −0.402172 | − | 4.62993i | ||
373.1 | −1.41372 | + | 0.0372230i | −3.13308 | − | 0.839505i | 1.99723 | − | 0.105246i | 2.74083 | − | 0.734404i | 4.46055 | + | 1.07021i | 0 | −2.81961 | + | 0.223131i | 6.51332 | + | 3.76047i | −3.84744 | + | 1.14027i | ||
373.2 | −1.41372 | + | 0.0372230i | 3.13308 | + | 0.839505i | 1.99723 | − | 0.105246i | −2.74083 | + | 0.734404i | −4.46055 | − | 1.07021i | 0 | −2.81961 | + | 0.223131i | 6.51332 | + | 3.76047i | 3.84744 | − | 1.14027i | ||
373.3 | −1.08279 | − | 0.909711i | −1.23755 | − | 0.331601i | 0.344852 | + | 1.97004i | −3.17421 | + | 0.850527i | 1.03834 | + | 1.48487i | 0 | 1.41877 | − | 2.44685i | −1.17650 | − | 0.679252i | 4.21072 | + | 1.96667i | ||
373.4 | −1.08279 | − | 0.909711i | 1.23755 | + | 0.331601i | 0.344852 | + | 1.97004i | 3.17421 | − | 0.850527i | −1.03834 | − | 1.48487i | 0 | 1.41877 | − | 2.44685i | −1.17650 | − | 0.679252i | −4.21072 | − | 1.96667i | ||
373.5 | −0.0190769 | + | 1.41408i | −2.27101 | − | 0.608516i | −1.99927 | − | 0.0539526i | −1.49452 | + | 0.400455i | 0.903818 | − | 3.19980i | 0 | 0.114433 | − | 2.82611i | 2.18914 | + | 1.26390i | −0.537766 | − | 2.12101i | ||
373.6 | −0.0190769 | + | 1.41408i | 2.27101 | + | 0.608516i | −1.99927 | − | 0.0539526i | 1.49452 | − | 0.400455i | −0.903818 | + | 3.19980i | 0 | 0.114433 | − | 2.82611i | 2.18914 | + | 1.26390i | 0.537766 | + | 2.12101i | ||
373.7 | 1.13424 | − | 0.844695i | −1.26459 | − | 0.338847i | 0.572980 | − | 1.91617i | 3.57533 | − | 0.958008i | −1.72057 | + | 0.683864i | 0 | −0.968683 | − | 2.65738i | −1.11370 | − | 0.642992i | 3.24604 | − | 4.10667i | ||
373.8 | 1.13424 | − | 0.844695i | 1.26459 | + | 0.338847i | 0.572980 | − | 1.91617i | −3.57533 | + | 0.958008i | 1.72057 | − | 0.683864i | 0 | −0.968683 | − | 2.65738i | −1.11370 | − | 0.642992i | −3.24604 | + | 4.10667i | ||
373.9 | 1.38135 | + | 0.303098i | −2.07072 | − | 0.554848i | 1.81626 | + | 0.837371i | −0.992025 | + | 0.265812i | −2.69222 | − | 1.39407i | 0 | 2.25509 | + | 1.70721i | 1.38196 | + | 0.797875i | −1.45090 | + | 0.0664989i | ||
373.10 | 1.38135 | + | 0.303098i | 2.07072 | + | 0.554848i | 1.81626 | + | 0.837371i | 0.992025 | − | 0.265812i | 2.69222 | + | 1.39407i | 0 | 2.25509 | + | 1.70721i | 1.38196 | + | 0.797875i | 1.45090 | − | 0.0664989i | ||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
7.d | odd | 6 | 1 | inner |
16.e | even | 4 | 1 | inner |
112.l | odd | 4 | 1 | inner |
112.w | even | 12 | 1 | inner |
112.x | odd | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 784.2.x.n | 40 | |
7.b | odd | 2 | 1 | inner | 784.2.x.n | 40 | |
7.c | even | 3 | 1 | 784.2.m.i | ✓ | 20 | |
7.c | even | 3 | 1 | inner | 784.2.x.n | 40 | |
7.d | odd | 6 | 1 | 784.2.m.i | ✓ | 20 | |
7.d | odd | 6 | 1 | inner | 784.2.x.n | 40 | |
16.e | even | 4 | 1 | inner | 784.2.x.n | 40 | |
112.l | odd | 4 | 1 | inner | 784.2.x.n | 40 | |
112.w | even | 12 | 1 | 784.2.m.i | ✓ | 20 | |
112.w | even | 12 | 1 | inner | 784.2.x.n | 40 | |
112.x | odd | 12 | 1 | 784.2.m.i | ✓ | 20 | |
112.x | odd | 12 | 1 | inner | 784.2.x.n | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
784.2.m.i | ✓ | 20 | 7.c | even | 3 | 1 | |
784.2.m.i | ✓ | 20 | 7.d | odd | 6 | 1 | |
784.2.m.i | ✓ | 20 | 112.w | even | 12 | 1 | |
784.2.m.i | ✓ | 20 | 112.x | odd | 12 | 1 | |
784.2.x.n | 40 | 1.a | even | 1 | 1 | trivial | |
784.2.x.n | 40 | 7.b | odd | 2 | 1 | inner | |
784.2.x.n | 40 | 7.c | even | 3 | 1 | inner | |
784.2.x.n | 40 | 7.d | odd | 6 | 1 | inner | |
784.2.x.n | 40 | 16.e | even | 4 | 1 | inner | |
784.2.x.n | 40 | 112.l | odd | 4 | 1 | inner | |
784.2.x.n | 40 | 112.w | even | 12 | 1 | inner | |
784.2.x.n | 40 | 112.x | odd | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\):
\( T_{3}^{40} - 168 T_{3}^{36} + 20936 T_{3}^{32} - 1007232 T_{3}^{28} + 34421424 T_{3}^{24} + \cdots + 319794774016 \) |
\( T_{5}^{40} - 376 T_{5}^{36} + 97224 T_{5}^{32} - 13188608 T_{5}^{28} + 1297863856 T_{5}^{24} + \cdots + 81867462148096 \) |