Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [882,2,Mod(37,882)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(882, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([0, 32]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("882.37");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 882.z (of order \(21\), degree \(12\), 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: | \(7.04280545828\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{21})\) |
Twist minimal: | no (minimal twist has level 98) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{21}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 | −0.826239 | + | 0.563320i | 0 | 0.365341 | − | 0.930874i | −3.17919 | − | 0.980650i | 0 | −0.0867132 | + | 2.64433i | 0.222521 | + | 0.974928i | 0 | 3.17919 | − | 0.980650i | ||||||
37.2 | −0.826239 | + | 0.563320i | 0 | 0.365341 | − | 0.930874i | 0.427005 | + | 0.131714i | 0 | 1.60505 | − | 2.10329i | 0.222521 | + | 0.974928i | 0 | −0.427005 | + | 0.131714i | ||||||
109.1 | 0.733052 | − | 0.680173i | 0 | 0.0747301 | − | 0.997204i | −0.904721 | + | 2.30519i | 0 | −1.30835 | − | 2.29961i | −0.623490 | − | 0.781831i | 0 | 0.904721 | + | 2.30519i | ||||||
109.2 | 0.733052 | − | 0.680173i | 0 | 0.0747301 | − | 0.997204i | −0.578784 | + | 1.47472i | 0 | 2.58946 | + | 0.542854i | −0.623490 | − | 0.781831i | 0 | 0.578784 | + | 1.47472i | ||||||
163.1 | −0.365341 | − | 0.930874i | 0 | −0.733052 | + | 0.680173i | −1.87725 | + | 1.27989i | 0 | −2.59910 | + | 0.494638i | 0.900969 | + | 0.433884i | 0 | 1.87725 | + | 1.27989i | ||||||
163.2 | −0.365341 | − | 0.930874i | 0 | −0.733052 | + | 0.680173i | 3.07015 | − | 2.09319i | 0 | 2.50800 | − | 0.842585i | 0.900969 | + | 0.433884i | 0 | −3.07015 | − | 2.09319i | ||||||
235.1 | 0.988831 | + | 0.149042i | 0 | 0.955573 | + | 0.294755i | −0.269665 | − | 3.59844i | 0 | 0.415648 | − | 2.61290i | 0.900969 | + | 0.433884i | 0 | 0.269665 | − | 3.59844i | ||||||
235.2 | 0.988831 | + | 0.149042i | 0 | 0.955573 | + | 0.294755i | −0.0772188 | − | 1.03041i | 0 | 1.36748 | + | 2.26495i | 0.900969 | + | 0.433884i | 0 | 0.0772188 | − | 1.03041i | ||||||
289.1 | 0.988831 | − | 0.149042i | 0 | 0.955573 | − | 0.294755i | −0.269665 | + | 3.59844i | 0 | 0.415648 | + | 2.61290i | 0.900969 | − | 0.433884i | 0 | 0.269665 | + | 3.59844i | ||||||
289.2 | 0.988831 | − | 0.149042i | 0 | 0.955573 | − | 0.294755i | −0.0772188 | + | 1.03041i | 0 | 1.36748 | − | 2.26495i | 0.900969 | − | 0.433884i | 0 | 0.0772188 | + | 1.03041i | ||||||
415.1 | −0.0747301 | + | 0.997204i | 0 | −0.988831 | − | 0.149042i | −0.958118 | − | 0.889004i | 0 | −2.43754 | + | 1.02877i | 0.222521 | − | 0.974928i | 0 | 0.958118 | − | 0.889004i | ||||||
415.2 | −0.0747301 | + | 0.997204i | 0 | −0.988831 | − | 0.149042i | 2.18585 | + | 2.02817i | 0 | −2.12971 | − | 1.56982i | 0.222521 | − | 0.974928i | 0 | −2.18585 | + | 2.02817i | ||||||
487.1 | −0.365341 | + | 0.930874i | 0 | −0.733052 | − | 0.680173i | −1.87725 | − | 1.27989i | 0 | −2.59910 | − | 0.494638i | 0.900969 | − | 0.433884i | 0 | 1.87725 | − | 1.27989i | ||||||
487.2 | −0.365341 | + | 0.930874i | 0 | −0.733052 | − | 0.680173i | 3.07015 | + | 2.09319i | 0 | 2.50800 | + | 0.842585i | 0.900969 | − | 0.433884i | 0 | −3.07015 | + | 2.09319i | ||||||
541.1 | −0.955573 | − | 0.294755i | 0 | 0.826239 | + | 0.563320i | 0.0821245 | − | 0.0123783i | 0 | −2.45519 | − | 0.985929i | −0.623490 | − | 0.781831i | 0 | −0.0821245 | − | 0.0123783i | ||||||
541.2 | −0.955573 | − | 0.294755i | 0 | 0.826239 | + | 0.563320i | 2.07983 | − | 0.313484i | 0 | 2.53097 | − | 0.770831i | −0.623490 | − | 0.781831i | 0 | −2.07983 | − | 0.313484i | ||||||
613.1 | −0.955573 | + | 0.294755i | 0 | 0.826239 | − | 0.563320i | 0.0821245 | + | 0.0123783i | 0 | −2.45519 | + | 0.985929i | −0.623490 | + | 0.781831i | 0 | −0.0821245 | + | 0.0123783i | ||||||
613.2 | −0.955573 | + | 0.294755i | 0 | 0.826239 | − | 0.563320i | 2.07983 | + | 0.313484i | 0 | 2.53097 | + | 0.770831i | −0.623490 | + | 0.781831i | 0 | −2.07983 | + | 0.313484i | ||||||
739.1 | −0.826239 | − | 0.563320i | 0 | 0.365341 | + | 0.930874i | −3.17919 | + | 0.980650i | 0 | −0.0867132 | − | 2.64433i | 0.222521 | − | 0.974928i | 0 | 3.17919 | + | 0.980650i | ||||||
739.2 | −0.826239 | − | 0.563320i | 0 | 0.365341 | + | 0.930874i | 0.427005 | − | 0.131714i | 0 | 1.60505 | + | 2.10329i | 0.222521 | − | 0.974928i | 0 | −0.427005 | − | 0.131714i | ||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
49.g | even | 21 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 882.2.z.b | 24 | |
3.b | odd | 2 | 1 | 98.2.g.b | ✓ | 24 | |
12.b | even | 2 | 1 | 784.2.bg.b | 24 | ||
21.c | even | 2 | 1 | 686.2.g.e | 24 | ||
21.g | even | 6 | 1 | 686.2.e.d | 24 | ||
21.g | even | 6 | 1 | 686.2.g.d | 24 | ||
21.h | odd | 6 | 1 | 686.2.e.c | 24 | ||
21.h | odd | 6 | 1 | 686.2.g.f | 24 | ||
49.g | even | 21 | 1 | inner | 882.2.z.b | 24 | |
147.k | even | 14 | 1 | 686.2.g.d | 24 | ||
147.l | odd | 14 | 1 | 686.2.g.f | 24 | ||
147.n | odd | 42 | 1 | 98.2.g.b | ✓ | 24 | |
147.n | odd | 42 | 1 | 686.2.e.c | 24 | ||
147.n | odd | 42 | 1 | 4802.2.a.o | 12 | ||
147.o | even | 42 | 1 | 686.2.e.d | 24 | ||
147.o | even | 42 | 1 | 686.2.g.e | 24 | ||
147.o | even | 42 | 1 | 4802.2.a.l | 12 | ||
588.bb | even | 42 | 1 | 784.2.bg.b | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
98.2.g.b | ✓ | 24 | 3.b | odd | 2 | 1 | |
98.2.g.b | ✓ | 24 | 147.n | odd | 42 | 1 | |
686.2.e.c | 24 | 21.h | odd | 6 | 1 | ||
686.2.e.c | 24 | 147.n | odd | 42 | 1 | ||
686.2.e.d | 24 | 21.g | even | 6 | 1 | ||
686.2.e.d | 24 | 147.o | even | 42 | 1 | ||
686.2.g.d | 24 | 21.g | even | 6 | 1 | ||
686.2.g.d | 24 | 147.k | even | 14 | 1 | ||
686.2.g.e | 24 | 21.c | even | 2 | 1 | ||
686.2.g.e | 24 | 147.o | even | 42 | 1 | ||
686.2.g.f | 24 | 21.h | odd | 6 | 1 | ||
686.2.g.f | 24 | 147.l | odd | 14 | 1 | ||
784.2.bg.b | 24 | 12.b | even | 2 | 1 | ||
784.2.bg.b | 24 | 588.bb | even | 42 | 1 | ||
882.2.z.b | 24 | 1.a | even | 1 | 1 | trivial | |
882.2.z.b | 24 | 49.g | even | 21 | 1 | inner | |
4802.2.a.l | 12 | 147.o | even | 42 | 1 | ||
4802.2.a.o | 12 | 147.n | odd | 42 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - T_{5}^{22} + 7 T_{5}^{21} + 7 T_{5}^{20} + 455 T_{5}^{19} + 1597 T_{5}^{18} - 1253 T_{5}^{17} + \cdots + 15625 \) acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\).