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 | −1.15218 | − | 0.355400i | 0 | 2.64164 | + | 0.147471i | −0.222521 | − | 0.974928i | 0 | −1.15218 | + | 0.355400i | ||||||
37.2 | 0.826239 | − | 0.563320i | 0 | 0.365341 | − | 0.930874i | 0.596780 | + | 0.184082i | 0 | −2.13962 | − | 1.55628i | −0.222521 | − | 0.974928i | 0 | 0.596780 | − | 0.184082i | ||||||
109.1 | −0.733052 | + | 0.680173i | 0 | 0.0747301 | − | 0.997204i | −0.459870 | + | 1.17173i | 0 | −1.68832 | + | 2.03705i | 0.623490 | + | 0.781831i | 0 | −0.459870 | − | 1.17173i | ||||||
109.2 | −0.733052 | + | 0.680173i | 0 | 0.0747301 | − | 0.997204i | 1.10349 | − | 2.81164i | 0 | 1.42353 | − | 2.23015i | 0.623490 | + | 0.781831i | 0 | 1.10349 | + | 2.81164i | ||||||
163.1 | 0.365341 | + | 0.930874i | 0 | −0.733052 | + | 0.680173i | −1.52046 | + | 1.03663i | 0 | 2.03934 | − | 1.68555i | −0.900969 | − | 0.433884i | 0 | −1.52046 | − | 1.03663i | ||||||
163.2 | 0.365341 | + | 0.930874i | 0 | −0.733052 | + | 0.680173i | 1.09780 | − | 0.748464i | 0 | 2.60370 | + | 0.469838i | −0.900969 | − | 0.433884i | 0 | 1.09780 | + | 0.748464i | ||||||
235.1 | −0.988831 | − | 0.149042i | 0 | 0.955573 | + | 0.294755i | −0.169008 | − | 2.25526i | 0 | 2.40601 | − | 1.10050i | −0.900969 | − | 0.433884i | 0 | −0.169008 | + | 2.25526i | ||||||
235.2 | −0.988831 | − | 0.149042i | 0 | 0.955573 | + | 0.294755i | 0.0807922 | + | 1.07810i | 0 | −2.64324 | − | 0.115218i | −0.900969 | − | 0.433884i | 0 | 0.0807922 | − | 1.07810i | ||||||
289.1 | −0.988831 | + | 0.149042i | 0 | 0.955573 | − | 0.294755i | −0.169008 | + | 2.25526i | 0 | 2.40601 | + | 1.10050i | −0.900969 | + | 0.433884i | 0 | −0.169008 | − | 2.25526i | ||||||
289.2 | −0.988831 | + | 0.149042i | 0 | 0.955573 | − | 0.294755i | 0.0807922 | − | 1.07810i | 0 | −2.64324 | + | 0.115218i | −0.900969 | + | 0.433884i | 0 | 0.0807922 | + | 1.07810i | ||||||
415.1 | 0.0747301 | − | 0.997204i | 0 | −0.988831 | − | 0.149042i | −2.80532 | − | 2.60296i | 0 | −2.61464 | + | 0.404557i | −0.222521 | + | 0.974928i | 0 | −2.80532 | + | 2.60296i | ||||||
415.2 | 0.0747301 | − | 0.997204i | 0 | −0.988831 | − | 0.149042i | 0.144243 | + | 0.133838i | 0 | 2.44775 | + | 1.00425i | −0.222521 | + | 0.974928i | 0 | 0.144243 | − | 0.133838i | ||||||
487.1 | 0.365341 | − | 0.930874i | 0 | −0.733052 | − | 0.680173i | −1.52046 | − | 1.03663i | 0 | 2.03934 | + | 1.68555i | −0.900969 | + | 0.433884i | 0 | −1.52046 | + | 1.03663i | ||||||
487.2 | 0.365341 | − | 0.930874i | 0 | −0.733052 | − | 0.680173i | 1.09780 | + | 0.748464i | 0 | 2.60370 | − | 0.469838i | −0.900969 | + | 0.433884i | 0 | 1.09780 | − | 0.748464i | ||||||
541.1 | 0.955573 | + | 0.294755i | 0 | 0.826239 | + | 0.563320i | −0.926902 | + | 0.139708i | 0 | −2.29875 | + | 1.30987i | 0.623490 | + | 0.781831i | 0 | −0.926902 | − | 0.139708i | ||||||
541.2 | 0.955573 | + | 0.294755i | 0 | 0.826239 | + | 0.563320i | 4.01065 | − | 0.604508i | 0 | −2.17740 | − | 1.50297i | 0.623490 | + | 0.781831i | 0 | 4.01065 | + | 0.604508i | ||||||
613.1 | 0.955573 | − | 0.294755i | 0 | 0.826239 | − | 0.563320i | −0.926902 | − | 0.139708i | 0 | −2.29875 | − | 1.30987i | 0.623490 | − | 0.781831i | 0 | −0.926902 | + | 0.139708i | ||||||
613.2 | 0.955573 | − | 0.294755i | 0 | 0.826239 | − | 0.563320i | 4.01065 | + | 0.604508i | 0 | −2.17740 | + | 1.50297i | 0.623490 | − | 0.781831i | 0 | 4.01065 | − | 0.604508i | ||||||
739.1 | 0.826239 | + | 0.563320i | 0 | 0.365341 | + | 0.930874i | −1.15218 | + | 0.355400i | 0 | 2.64164 | − | 0.147471i | −0.222521 | + | 0.974928i | 0 | −1.15218 | − | 0.355400i | ||||||
739.2 | 0.826239 | + | 0.563320i | 0 | 0.365341 | + | 0.930874i | 0.596780 | − | 0.184082i | 0 | −2.13962 | + | 1.55628i | −0.222521 | + | 0.974928i | 0 | 0.596780 | + | 0.184082i | ||||||
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.d | 24 | |
3.b | odd | 2 | 1 | 98.2.g.a | ✓ | 24 | |
12.b | even | 2 | 1 | 784.2.bg.a | 24 | ||
21.c | even | 2 | 1 | 686.2.g.b | 24 | ||
21.g | even | 6 | 1 | 686.2.e.e | 24 | ||
21.g | even | 6 | 1 | 686.2.g.c | 24 | ||
21.h | odd | 6 | 1 | 686.2.e.f | 24 | ||
21.h | odd | 6 | 1 | 686.2.g.a | 24 | ||
49.g | even | 21 | 1 | inner | 882.2.z.d | 24 | |
147.k | even | 14 | 1 | 686.2.g.c | 24 | ||
147.l | odd | 14 | 1 | 686.2.g.a | 24 | ||
147.n | odd | 42 | 1 | 98.2.g.a | ✓ | 24 | |
147.n | odd | 42 | 1 | 686.2.e.f | 24 | ||
147.n | odd | 42 | 1 | 4802.2.a.i | 12 | ||
147.o | even | 42 | 1 | 686.2.e.e | 24 | ||
147.o | even | 42 | 1 | 686.2.g.b | 24 | ||
147.o | even | 42 | 1 | 4802.2.a.k | 12 | ||
588.bb | even | 42 | 1 | 784.2.bg.a | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
98.2.g.a | ✓ | 24 | 3.b | odd | 2 | 1 | |
98.2.g.a | ✓ | 24 | 147.n | odd | 42 | 1 | |
686.2.e.e | 24 | 21.g | even | 6 | 1 | ||
686.2.e.e | 24 | 147.o | even | 42 | 1 | ||
686.2.e.f | 24 | 21.h | odd | 6 | 1 | ||
686.2.e.f | 24 | 147.n | odd | 42 | 1 | ||
686.2.g.a | 24 | 21.h | odd | 6 | 1 | ||
686.2.g.a | 24 | 147.l | odd | 14 | 1 | ||
686.2.g.b | 24 | 21.c | even | 2 | 1 | ||
686.2.g.b | 24 | 147.o | even | 42 | 1 | ||
686.2.g.c | 24 | 21.g | even | 6 | 1 | ||
686.2.g.c | 24 | 147.k | even | 14 | 1 | ||
784.2.bg.a | 24 | 12.b | even | 2 | 1 | ||
784.2.bg.a | 24 | 588.bb | even | 42 | 1 | ||
882.2.z.d | 24 | 1.a | even | 1 | 1 | trivial | |
882.2.z.d | 24 | 49.g | even | 21 | 1 | inner | |
4802.2.a.i | 12 | 147.n | odd | 42 | 1 | ||
4802.2.a.k | 12 | 147.o | even | 42 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 7 T_{5}^{22} - 49 T_{5}^{21} + 7 T_{5}^{20} - 343 T_{5}^{19} + 469 T_{5}^{18} + 3871 T_{5}^{17} + \cdots + 2401 \) acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\).