Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2151,1,Mod(238,2151)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2151, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 3]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2151.238");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2151 = 3^{2} \cdot 239 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2151.f (of order \(6\), degree \(2\), 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: | \(1.07348884217\) |
| Analytic rank: | \(0\) |
| Dimension: | \(24\) |
| Relative dimension: | \(12\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\zeta_{45})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{45}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{45} - \cdots)\) |
$q$-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 \(\left(\mathbb{Z}/2151\mathbb{Z}\right)^\times\).
| \(n\) | \(479\) | \(1441\) |
| \(\chi(n)\) | \(\zeta_{90}^{30}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 238.1 |
|
−0.990268 | − | 1.71519i | 0.669131 | + | 0.743145i | −1.46126 | + | 2.53098i | −0.848048 | + | 1.46886i | 0.612019 | − | 1.88360i | 0 | 3.80763 | −0.104528 | + | 0.994522i | 3.35918 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 238.2 | −0.961262 | − | 1.66495i | −0.104528 | − | 0.994522i | −1.34805 | + | 2.33489i | −0.438371 | + | 0.759281i | −1.55535 | + | 1.13003i | 0 | 3.26078 | −0.978148 | + | 0.207912i | 1.68556 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 238.3 | −0.848048 | − | 1.46886i | −0.978148 | − | 0.207912i | −0.938371 | + | 1.62531i | 0.615661 | − | 1.06636i | 0.524123 | + | 1.61308i | 0 | 1.48704 | 0.913545 | + | 0.406737i | −2.08844 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 238.4 | −0.559193 | − | 0.968551i | 0.913545 | − | 0.406737i | −0.125393 | + | 0.217188i | 0.719340 | − | 1.24593i | −0.904793 | − | 0.657371i | 0 | −0.837909 | 0.669131 | − | 0.743145i | −1.60900 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 238.5 | −0.438371 | − | 0.759281i | 0.913545 | − | 0.406737i | 0.115661 | − | 0.200332i | 0.241922 | − | 0.419021i | −0.709299 | − | 0.515336i | 0 | −1.07955 | 0.669131 | − | 0.743145i | −0.424206 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 238.6 | −0.0348995 | − | 0.0604477i | −0.978148 | − | 0.207912i | 0.497564 | − | 0.861806i | −0.990268 | + | 1.71519i | 0.0215691 | + | 0.0663828i | 0 | −0.139258 | 0.913545 | + | 0.406737i | 0.138239 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 238.7 | 0.241922 | + | 0.419021i | −0.104528 | − | 0.994522i | 0.382948 | − | 0.663285i | −0.559193 | + | 0.968551i | 0.391438 | − | 0.284396i | 0 | 0.854417 | −0.978148 | + | 0.207912i | −0.541124 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 238.8 | 0.374607 | + | 0.648838i | 0.669131 | + | 0.743145i | 0.219340 | − | 0.379908i | −0.0348995 | + | 0.0604477i | −0.231520 | + | 0.712544i | 0 | 1.07788 | −0.104528 | + | 0.994522i | −0.0522943 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 238.9 | 0.615661 | + | 1.06636i | 0.669131 | + | 0.743145i | −0.258078 | + | 0.447004i | 0.882948 | − | 1.52931i | −0.380500 | + | 1.17106i | 0 | 0.595768 | −0.104528 | + | 0.994522i | 2.17439 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 238.10 | 0.719340 | + | 1.24593i | −0.104528 | − | 0.994522i | −0.534899 | + | 0.926473i | 0.997564 | − | 1.72783i | 1.16392 | − | 0.845635i | 0 | −0.100418 | −0.978148 | + | 0.207912i | 2.87035 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 238.11 | 0.882948 | + | 1.52931i | −0.978148 | − | 0.207912i | −1.05919 | + | 1.83458i | 0.374607 | − | 0.648838i | −0.545692 | − | 1.67947i | 0 | −1.97495 | 0.913545 | + | 0.406737i | 1.32303 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 238.12 | 0.997564 | + | 1.72783i | 0.913545 | − | 0.406737i | −1.49027 | + | 2.58122i | −0.961262 | + | 1.66495i | 1.61409 | + | 1.17271i | 0 | −3.95142 | 0.669131 | − | 0.743145i | −3.83568 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1672.1 | −0.990268 | + | 1.71519i | 0.669131 | − | 0.743145i | −1.46126 | − | 2.53098i | −0.848048 | − | 1.46886i | 0.612019 | + | 1.88360i | 0 | 3.80763 | −0.104528 | − | 0.994522i | 3.35918 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1672.2 | −0.961262 | + | 1.66495i | −0.104528 | + | 0.994522i | −1.34805 | − | 2.33489i | −0.438371 | − | 0.759281i | −1.55535 | − | 1.13003i | 0 | 3.26078 | −0.978148 | − | 0.207912i | 1.68556 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1672.3 | −0.848048 | + | 1.46886i | −0.978148 | + | 0.207912i | −0.938371 | − | 1.62531i | 0.615661 | + | 1.06636i | 0.524123 | − | 1.61308i | 0 | 1.48704 | 0.913545 | − | 0.406737i | −2.08844 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1672.4 | −0.559193 | + | 0.968551i | 0.913545 | + | 0.406737i | −0.125393 | − | 0.217188i | 0.719340 | + | 1.24593i | −0.904793 | + | 0.657371i | 0 | −0.837909 | 0.669131 | + | 0.743145i | −1.60900 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1672.5 | −0.438371 | + | 0.759281i | 0.913545 | + | 0.406737i | 0.115661 | + | 0.200332i | 0.241922 | + | 0.419021i | −0.709299 | + | 0.515336i | 0 | −1.07955 | 0.669131 | + | 0.743145i | −0.424206 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1672.6 | −0.0348995 | + | 0.0604477i | −0.978148 | + | 0.207912i | 0.497564 | + | 0.861806i | −0.990268 | − | 1.71519i | 0.0215691 | − | 0.0663828i | 0 | −0.139258 | 0.913545 | − | 0.406737i | 0.138239 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1672.7 | 0.241922 | − | 0.419021i | −0.104528 | + | 0.994522i | 0.382948 | + | 0.663285i | −0.559193 | − | 0.968551i | 0.391438 | + | 0.284396i | 0 | 0.854417 | −0.978148 | − | 0.207912i | −0.541124 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1672.8 | 0.374607 | − | 0.648838i | 0.669131 | − | 0.743145i | 0.219340 | + | 0.379908i | −0.0348995 | − | 0.0604477i | −0.231520 | − | 0.712544i | 0 | 1.07788 | −0.104528 | − | 0.994522i | −0.0522943 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| See all 24 embeddings | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 239.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-239}) \) |
| 9.c | even | 3 | 1 | inner |
| 2151.f | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2151.1.f.b | ✓ | 24 |
| 9.c | even | 3 | 1 | inner | 2151.1.f.b | ✓ | 24 |
| 239.b | odd | 2 | 1 | CM | 2151.1.f.b | ✓ | 24 |
| 2151.f | odd | 6 | 1 | inner | 2151.1.f.b | ✓ | 24 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2151.1.f.b | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
| 2151.1.f.b | ✓ | 24 | 9.c | even | 3 | 1 | inner |
| 2151.1.f.b | ✓ | 24 | 239.b | odd | 2 | 1 | CM |
| 2151.1.f.b | ✓ | 24 | 2151.f | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{24} + 12 T_{2}^{22} - 2 T_{2}^{21} + 90 T_{2}^{20} - 21 T_{2}^{19} + 425 T_{2}^{18} - 135 T_{2}^{17} + \cdots + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2151, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{24} + 12 T^{22} + \cdots + 1 \)
|
| $3$ |
\( (T^{8} - T^{7} + T^{5} + \cdots + 1)^{3} \)
|
| $5$ |
\( T^{24} + 12 T^{22} + \cdots + 1 \)
|
| $7$ |
\( T^{24} \)
|
| $11$ |
\( T^{24} + 12 T^{22} + \cdots + 1 \)
|
| $13$ |
\( T^{24} \)
|
| $17$ |
\( (T^{3} - 3 T + 1)^{8} \)
|
| $19$ |
\( T^{24} \)
|
| $23$ |
\( T^{24} \)
|
| $29$ |
\( T^{24} + 12 T^{22} + \cdots + 1 \)
|
| $31$ |
\( T^{24} + 12 T^{22} + \cdots + 1 \)
|
| $37$ |
\( T^{24} \)
|
| $41$ |
\( T^{24} \)
|
| $43$ |
\( T^{24} \)
|
| $47$ |
\( T^{24} \)
|
| $53$ |
\( T^{24} \)
|
| $59$ |
\( T^{24} \)
|
| $61$ |
\( (T^{6} + 3 T^{4} + 2 T^{3} + \cdots + 1)^{4} \)
|
| $67$ |
\( (T^{6} + 3 T^{4} + 2 T^{3} + \cdots + 1)^{4} \)
|
| $71$ |
\( (T^{2} + T - 1)^{12} \)
|
| $73$ |
\( T^{24} \)
|
| $79$ |
\( T^{24} \)
|
| $83$ |
\( T^{24} + 12 T^{22} + \cdots + 1 \)
|
| $89$ |
\( T^{24} \)
|
| $97$ |
\( T^{24} \)
|
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