Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2001,1,Mod(68,2001)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2001, base_ring=CyclotomicField(28))
chi = DirichletCharacter(H, H._module([14, 14, 23]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2001.68");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2001 = 3 \cdot 23 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2001.bf (of order \(28\), 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: | \(0.998629090279\) |
| Analytic rank: | \(0\) |
| Dimension: | \(24\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{28})\) |
| Coefficient field: | \(\Q(\zeta_{84})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{84}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{84} - \cdots)\) |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2001\mathbb{Z}\right)^\times\).
| \(n\) | \(553\) | \(668\) | \(1132\) |
| \(\chi(n)\) | \(-\zeta_{84}^{27}\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 68.1 |
|
−0.0397866 | − | 0.0633201i | −0.680173 | − | 0.733052i | 0.431457 | − | 0.895930i | 0 | −0.0193551 | + | 0.0722342i | 0 | −0.148209 | + | 0.0166991i | −0.0747301 | + | 0.997204i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 68.2 | 0.940755 | + | 1.49720i | −0.294755 | + | 0.955573i | −0.922715 | + | 1.91604i | 0 | −1.70798 | + | 0.457652i | 0 | −1.97963 | + | 0.223051i | −0.826239 | − | 0.563320i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 137.1 | −1.85486 | − | 0.649042i | −0.563320 | − | 0.826239i | 2.23740 | + | 1.78427i | 0 | 0.508614 | + | 1.89817i | 0 | −1.94648 | − | 3.09781i | −0.365341 | + | 0.930874i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 137.2 | 1.23137 | + | 0.430874i | 0.997204 | − | 0.0747301i | 0.548780 | + | 0.437637i | 0 | 1.26012 | + | 0.337649i | 0 | −0.206893 | − | 0.329269i | 0.988831 | − | 0.149042i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 206.1 | −0.0397866 | + | 0.0633201i | −0.680173 | + | 0.733052i | 0.431457 | + | 0.895930i | 0 | −0.0193551 | − | 0.0722342i | 0 | −0.148209 | − | 0.0166991i | −0.0747301 | − | 0.997204i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 206.2 | 0.940755 | − | 1.49720i | −0.294755 | − | 0.955573i | −0.922715 | − | 1.91604i | 0 | −1.70798 | − | 0.457652i | 0 | −1.97963 | − | 0.223051i | −0.826239 | + | 0.563320i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 275.1 | −1.59908 | + | 0.180173i | 0.149042 | + | 0.988831i | 1.54966 | − | 0.353699i | 0 | −0.416490 | − | 1.55436i | 0 | −0.895403 | + | 0.313315i | −0.955573 | + | 0.294755i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 275.2 | 1.82160 | − | 0.205245i | −0.930874 | − | 0.365341i | 2.30117 | − | 0.525226i | 0 | −1.77066 | − | 0.474448i | 0 | 2.35375 | − | 0.823611i | 0.733052 | + | 0.680173i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 482.1 | −1.85486 | + | 0.649042i | −0.563320 | + | 0.826239i | 2.23740 | − | 1.78427i | 0 | 0.508614 | − | 1.89817i | 0 | −1.94648 | + | 3.09781i | −0.365341 | − | 0.930874i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 482.2 | 1.23137 | − | 0.430874i | 0.997204 | + | 0.0747301i | 0.548780 | − | 0.437637i | 0 | 1.26012 | − | 0.337649i | 0 | −0.206893 | + | 0.329269i | 0.988831 | + | 0.149042i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 620.1 | −0.500684 | − | 1.43087i | −0.997204 | − | 0.0747301i | −1.01488 | + | 0.809342i | 0 | 0.392355 | + | 1.46429i | 0 | 0.382617 | + | 0.240414i | 0.988831 | + | 0.149042i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 620.2 | −0.122805 | − | 0.350958i | 0.563320 | − | 0.826239i | 0.673741 | − | 0.537291i | 0 | −0.359154 | − | 0.0962349i | 0 | −0.586137 | − | 0.368294i | −0.365341 | − | 0.930874i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 827.1 | 0.0895474 | + | 0.794755i | 0.930874 | + | 0.365341i | 0.351311 | − | 0.0801844i | 0 | −0.206999 | + | 0.772532i | 0 | 0.359338 | + | 1.02693i | 0.733052 | + | 0.680173i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 827.2 | 0.132974 | + | 1.18017i | −0.149042 | − | 0.988831i | −0.400198 | + | 0.0913425i | 0 | 1.14717 | − | 0.307384i | 0 | 0.231237 | + | 0.660838i | −0.955573 | + | 0.294755i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 896.1 | −0.791295 | − | 0.497204i | 0.294755 | + | 0.955573i | −0.0549471 | − | 0.114099i | 0 | 0.241876 | − | 0.902694i | 0 | −0.117886 | + | 1.04627i | −0.826239 | + | 0.563320i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 896.2 | 1.69226 | + | 1.06332i | 0.680173 | − | 0.733052i | 1.29922 | + | 2.69787i | 0 | 1.93050 | − | 0.517276i | 0 | −0.446293 | + | 3.96096i | −0.0747301 | − | 0.997204i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 965.1 | −0.500684 | + | 1.43087i | −0.997204 | + | 0.0747301i | −1.01488 | − | 0.809342i | 0 | 0.392355 | − | 1.46429i | 0 | 0.382617 | − | 0.240414i | 0.988831 | − | 0.149042i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 965.2 | −0.122805 | + | 0.350958i | 0.563320 | + | 0.826239i | 0.673741 | + | 0.537291i | 0 | −0.359154 | + | 0.0962349i | 0 | −0.586137 | + | 0.368294i | −0.365341 | + | 0.930874i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1034.1 | −0.791295 | + | 0.497204i | 0.294755 | − | 0.955573i | −0.0549471 | + | 0.114099i | 0 | 0.241876 | + | 0.902694i | 0 | −0.117886 | − | 1.04627i | −0.826239 | − | 0.563320i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1034.2 | 1.69226 | − | 1.06332i | 0.680173 | + | 0.733052i | 1.29922 | − | 2.69787i | 0 | 1.93050 | + | 0.517276i | 0 | −0.446293 | − | 3.96096i | −0.0747301 | + | 0.997204i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| See all 24 embeddings | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-23}) \) |
| 87.k | even | 28 | 1 | inner |
| 2001.bf | odd | 28 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2001.1.bf.d | yes | 24 |
| 3.b | odd | 2 | 1 | 2001.1.bf.c | ✓ | 24 | |
| 23.b | odd | 2 | 1 | CM | 2001.1.bf.d | yes | 24 |
| 29.f | odd | 28 | 1 | 2001.1.bf.c | ✓ | 24 | |
| 69.c | even | 2 | 1 | 2001.1.bf.c | ✓ | 24 | |
| 87.k | even | 28 | 1 | inner | 2001.1.bf.d | yes | 24 |
| 667.o | even | 28 | 1 | 2001.1.bf.c | ✓ | 24 | |
| 2001.bf | odd | 28 | 1 | inner | 2001.1.bf.d | yes | 24 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2001.1.bf.c | ✓ | 24 | 3.b | odd | 2 | 1 | |
| 2001.1.bf.c | ✓ | 24 | 29.f | odd | 28 | 1 | |
| 2001.1.bf.c | ✓ | 24 | 69.c | even | 2 | 1 | |
| 2001.1.bf.c | ✓ | 24 | 667.o | even | 28 | 1 | |
| 2001.1.bf.d | yes | 24 | 1.a | even | 1 | 1 | trivial |
| 2001.1.bf.d | yes | 24 | 23.b | odd | 2 | 1 | CM |
| 2001.1.bf.d | yes | 24 | 87.k | even | 28 | 1 | inner |
| 2001.1.bf.d | yes | 24 | 2001.bf | odd | 28 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{24} - 2 T_{2}^{23} - 5 T_{2}^{22} + 16 T_{2}^{21} + T_{2}^{20} - 42 T_{2}^{19} + 42 T_{2}^{18} + \cdots + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2001, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{24} - 2 T^{23} + \cdots + 1 \)
|
| $3$ |
\( T^{24} + T^{22} + \cdots + 1 \)
|
| $5$ |
\( T^{24} \)
|
| $7$ |
\( T^{24} \)
|
| $11$ |
\( T^{24} \)
|
| $13$ |
\( T^{24} - 2 T^{22} + \cdots + 1 \)
|
| $17$ |
\( T^{24} \)
|
| $19$ |
\( T^{24} \)
|
| $23$ |
\( (T^{12} - T^{10} + T^{8} + \cdots + 1)^{2} \)
|
| $29$ |
\( T^{24} + T^{22} + \cdots + 1 \)
|
| $31$ |
\( T^{24} + 2 T^{23} + \cdots + 1 \)
|
| $37$ |
\( T^{24} \)
|
| $41$ |
\( T^{24} + 2 T^{23} + \cdots + 1 \)
|
| $43$ |
\( T^{24} \)
|
| $47$ |
\( T^{24} - 2 T^{23} + \cdots + 1 \)
|
| $53$ |
\( T^{24} \)
|
| $59$ |
\( (T^{6} + 5 T^{4} + 6 T^{2} + 1)^{4} \)
|
| $61$ |
\( T^{24} \)
|
| $67$ |
\( T^{24} \)
|
| $71$ |
\( T^{24} + 6 T^{22} + \cdots + 1 \)
|
| $73$ |
\( T^{24} + 2 T^{23} + \cdots + 1 \)
|
| $79$ |
\( T^{24} \)
|
| $83$ |
\( T^{24} \)
|
| $89$ |
\( T^{24} \)
|
| $97$ |
\( T^{24} \)
|
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