Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [819,2,Mod(370,819)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(819, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 6, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("819.370");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 819.fm (of order \(12\), degree \(4\), 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: | \(6.53974792554\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{12})\) |
Twist minimal: | no (minimal twist has level 273) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
370.1 | −0.687394 | − | 2.56539i | 0 | −4.37666 | + | 2.52687i | 1.17771 | + | 1.17771i | 0 | −1.53750 | − | 2.15316i | 5.73490 | + | 5.73490i | 0 | 2.21174 | − | 3.83085i | ||||||
370.2 | −0.385482 | − | 1.43864i | 0 | −0.189037 | + | 0.109141i | 1.07205 | + | 1.07205i | 0 | 1.81743 | − | 1.92274i | −1.87643 | − | 1.87643i | 0 | 1.12904 | − | 1.95556i | ||||||
370.3 | −0.0578232 | − | 0.215799i | 0 | 1.68883 | − | 0.975044i | −1.08760 | − | 1.08760i | 0 | −0.725943 | + | 2.54421i | −0.624019 | − | 0.624019i | 0 | −0.171815 | + | 0.297592i | ||||||
370.4 | −0.0473445 | − | 0.176692i | 0 | 1.70307 | − | 0.983269i | −2.80040 | − | 2.80040i | 0 | −0.467560 | − | 2.60411i | −0.513062 | − | 0.513062i | 0 | −0.362225 | + | 0.627392i | ||||||
370.5 | 0.189683 | + | 0.707908i | 0 | 1.26690 | − | 0.731443i | 1.23329 | + | 1.23329i | 0 | −2.64473 | − | 0.0736014i | 1.79455 | + | 1.79455i | 0 | −0.639122 | + | 1.10699i | ||||||
370.6 | 0.281068 | + | 1.04896i | 0 | 0.710736 | − | 0.410344i | 1.02734 | + | 1.02734i | 0 | 2.23270 | − | 1.41953i | 2.16598 | + | 2.16598i | 0 | −0.788887 | + | 1.36639i | ||||||
370.7 | 0.500642 | + | 1.86842i | 0 | −1.50830 | + | 0.870817i | −2.44068 | − | 2.44068i | 0 | 2.02526 | + | 1.70244i | 0.353388 | + | 0.353388i | 0 | 3.33831 | − | 5.78213i | ||||||
370.8 | 0.706652 | + | 2.63726i | 0 | −4.72373 | + | 2.72725i | 2.18431 | + | 2.18431i | 0 | 0.666364 | + | 2.56046i | −6.66928 | − | 6.66928i | 0 | −4.21705 | + | 7.30414i | ||||||
496.1 | −2.37607 | − | 0.636667i | 0 | 3.50833 | + | 2.02554i | 0.498430 | + | 0.498430i | 0 | −2.62820 | − | 0.304236i | −3.56765 | − | 3.56765i | 0 | −0.866973 | − | 1.50164i | ||||||
496.2 | −1.96303 | − | 0.525993i | 0 | 1.84478 | + | 1.06508i | 1.56698 | + | 1.56698i | 0 | 2.60496 | + | 0.462799i | −0.187059 | − | 0.187059i | 0 | −2.25182 | − | 3.90026i | ||||||
496.3 | −1.34112 | − | 0.359352i | 0 | −0.0625832 | − | 0.0361324i | −2.52867 | − | 2.52867i | 0 | −0.324044 | + | 2.62583i | 2.03448 | + | 2.03448i | 0 | 2.48257 | + | 4.29993i | ||||||
496.4 | −0.604240 | − | 0.161906i | 0 | −1.39316 | − | 0.804341i | −0.965431 | − | 0.965431i | 0 | 2.12303 | − | 1.57884i | 1.59624 | + | 1.59624i | 0 | 0.427043 | + | 0.739660i | ||||||
496.5 | 1.11595 | + | 0.299019i | 0 | −0.576113 | − | 0.332619i | −0.549341 | − | 0.549341i | 0 | 1.61214 | + | 2.09786i | −2.17732 | − | 2.17732i | 0 | −0.448776 | − | 0.777302i | ||||||
496.6 | 1.38083 | + | 0.369991i | 0 | 0.0377371 | + | 0.0217876i | −0.512287 | − | 0.512287i | 0 | −2.41005 | − | 1.09162i | −1.97762 | − | 1.97762i | 0 | −0.517838 | − | 0.896921i | ||||||
496.7 | 2.11902 | + | 0.567791i | 0 | 2.43582 | + | 1.40632i | 3.00219 | + | 3.00219i | 0 | −2.60148 | + | 0.481982i | 1.26060 | + | 1.26060i | 0 | 4.65710 | + | 8.06633i | ||||||
496.8 | 2.16866 | + | 0.581092i | 0 | 2.63339 | + | 1.52039i | −1.87790 | − | 1.87790i | 0 | 1.25762 | − | 2.32774i | 1.65230 | + | 1.65230i | 0 | −2.98130 | − | 5.16377i | ||||||
622.1 | −0.687394 | + | 2.56539i | 0 | −4.37666 | − | 2.52687i | 1.17771 | − | 1.17771i | 0 | −1.53750 | + | 2.15316i | 5.73490 | − | 5.73490i | 0 | 2.21174 | + | 3.83085i | ||||||
622.2 | −0.385482 | + | 1.43864i | 0 | −0.189037 | − | 0.109141i | 1.07205 | − | 1.07205i | 0 | 1.81743 | + | 1.92274i | −1.87643 | + | 1.87643i | 0 | 1.12904 | + | 1.95556i | ||||||
622.3 | −0.0578232 | + | 0.215799i | 0 | 1.68883 | + | 0.975044i | −1.08760 | + | 1.08760i | 0 | −0.725943 | − | 2.54421i | −0.624019 | + | 0.624019i | 0 | −0.171815 | − | 0.297592i | ||||||
622.4 | −0.0473445 | + | 0.176692i | 0 | 1.70307 | + | 0.983269i | −2.80040 | + | 2.80040i | 0 | −0.467560 | + | 2.60411i | −0.513062 | + | 0.513062i | 0 | −0.362225 | − | 0.627392i | ||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
91.bc | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 819.2.fm.e | 32 | |
3.b | odd | 2 | 1 | 273.2.by.d | yes | 32 | |
7.b | odd | 2 | 1 | 819.2.fm.f | 32 | ||
13.f | odd | 12 | 1 | 819.2.fm.f | 32 | ||
21.c | even | 2 | 1 | 273.2.by.c | ✓ | 32 | |
39.k | even | 12 | 1 | 273.2.by.c | ✓ | 32 | |
91.bc | even | 12 | 1 | inner | 819.2.fm.e | 32 | |
273.ca | odd | 12 | 1 | 273.2.by.d | yes | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
273.2.by.c | ✓ | 32 | 21.c | even | 2 | 1 | |
273.2.by.c | ✓ | 32 | 39.k | even | 12 | 1 | |
273.2.by.d | yes | 32 | 3.b | odd | 2 | 1 | |
273.2.by.d | yes | 32 | 273.ca | odd | 12 | 1 | |
819.2.fm.e | 32 | 1.a | even | 1 | 1 | trivial | |
819.2.fm.e | 32 | 91.bc | even | 12 | 1 | inner | |
819.2.fm.f | 32 | 7.b | odd | 2 | 1 | ||
819.2.fm.f | 32 | 13.f | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\):
\( T_{2}^{32} - 2 T_{2}^{31} - T_{2}^{30} + 8 T_{2}^{29} - 67 T_{2}^{28} + 96 T_{2}^{27} + 112 T_{2}^{26} - 404 T_{2}^{25} + 3019 T_{2}^{24} - 4794 T_{2}^{23} - 8633 T_{2}^{22} + 26164 T_{2}^{21} - 40469 T_{2}^{20} + 3296 T_{2}^{19} + 182806 T_{2}^{18} + \cdots + 576 \) |
\( T_{5}^{32} + 2 T_{5}^{31} + 2 T_{5}^{30} - 18 T_{5}^{29} + 501 T_{5}^{28} + 916 T_{5}^{27} + 992 T_{5}^{26} - 9260 T_{5}^{25} + 71882 T_{5}^{24} + 85244 T_{5}^{23} + 115940 T_{5}^{22} - 1087128 T_{5}^{21} + 4494824 T_{5}^{20} + \cdots + 404653456 \) |
\( T_{19}^{32} - 2 T_{19}^{31} - 10 T_{19}^{30} - 116 T_{19}^{29} - 1607 T_{19}^{28} + 8838 T_{19}^{27} + 18222 T_{19}^{26} + 79068 T_{19}^{25} + 1914778 T_{19}^{24} - 12478412 T_{19}^{23} + 16914440 T_{19}^{22} + \cdots + 12\!\cdots\!36 \) |