Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2205,2,Mod(1324,2205)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2205, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2205.1324");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2205.d (of order \(2\), degree \(1\), 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: | \(17.6070136457\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.2058981376.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} + 2x^{6} + 18x^{4} - 34x^{3} + 32x^{2} - 8x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 105) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{8} - 2x^{7} + 2x^{6} + 18x^{4} - 34x^{3} + 32x^{2} - 8x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -97\nu^{7} + 102\nu^{6} - 30\nu^{5} - 432\nu^{4} - 1769\nu^{3} + 1637\nu^{2} - 408\nu - 4773 ) / 1631 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 145\nu^{7} + 167\nu^{6} - 241\nu^{5} + 444\nu^{4} + 3132\nu^{3} + 2984\nu^{2} - 3930\nu + 4226 ) / 1631 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 485\nu^{7} - 510\nu^{6} + 150\nu^{5} + 529\nu^{4} + 8845\nu^{3} - 8185\nu^{2} + 2040\nu + 4293 ) / 1631 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -552\nu^{7} + 984\nu^{6} - 961\nu^{5} - 138\nu^{4} - 9966\nu^{3} + 16176\nu^{2} - 15353\nu + 2213 ) / 1631 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -137\nu^{7} + 305\nu^{6} - 309\nu^{5} + 24\nu^{4} - 2400\nu^{3} + 5281\nu^{2} - 4948\nu + 1236 ) / 233 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -1084\nu^{7} + 2216\nu^{6} - 1899\nu^{5} - 271\nu^{4} - 19500\nu^{3} + 38219\nu^{2} - 30067\nu + 4334 ) / 1631 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -1661\nu^{7} + 2890\nu^{6} - 2481\nu^{5} - 823\nu^{4} - 30006\nu^{3} + 49100\nu^{2} - 37656\nu + 1769 ) / 1631 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 - 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 2\beta_{4} - \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{6} + 5\beta_{5} - 3\beta_{4} - 5\beta_{3} + 3\beta_{2} + 3\beta _1 + 5 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{3} - 5\beta _1 - 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -34\beta_{7} + 23\beta_{6} + 11\beta_{5} + 13\beta_{4} - 23\beta_{3} - 11\beta_{2} + 11\beta _1 + 21 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -\beta_{7} + 7\beta_{6} - 23\beta_{5} + 35\beta_{4} + 22\beta_{2} \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 103\beta_{6} - 105\beta_{5} + 61\beta_{4} + 103\beta_{3} - 43\beta_{2} - 43\beta _1 - 85 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times\).
| \(n\) | \(442\) | \(1081\) | \(1226\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1324.1 |
|
− | 2.51658i | 0 | −4.33317 | 1.11922 | + | 1.93581i | 0 | 0 | 5.87162i | 0 | 4.87162 | − | 2.81659i | |||||||||||||||||||||||||||||||||||||
| 1324.2 | − | 1.78165i | 0 | −1.17429 | −2.22038 | + | 0.264435i | 0 | 0 | − | 1.47113i | 0 | 0.471131 | + | 3.95594i | |||||||||||||||||||||||||||||||||||||
| 1324.3 | − | 1.55241i | 0 | −0.409975 | −0.0917505 | − | 2.23418i | 0 | 0 | − | 2.46837i | 0 | −3.46837 | + | 0.142434i | |||||||||||||||||||||||||||||||||||||
| 1324.4 | − | 0.287336i | 0 | 1.91744 | 2.19291 | + | 0.437190i | 0 | 0 | − | 1.12562i | 0 | 0.125620 | − | 0.630102i | |||||||||||||||||||||||||||||||||||||
| 1324.5 | 0.287336i | 0 | 1.91744 | 2.19291 | − | 0.437190i | 0 | 0 | 1.12562i | 0 | 0.125620 | + | 0.630102i | |||||||||||||||||||||||||||||||||||||||
| 1324.6 | 1.55241i | 0 | −0.409975 | −0.0917505 | + | 2.23418i | 0 | 0 | 2.46837i | 0 | −3.46837 | − | 0.142434i | |||||||||||||||||||||||||||||||||||||||
| 1324.7 | 1.78165i | 0 | −1.17429 | −2.22038 | − | 0.264435i | 0 | 0 | 1.47113i | 0 | 0.471131 | − | 3.95594i | |||||||||||||||||||||||||||||||||||||||
| 1324.8 | 2.51658i | 0 | −4.33317 | 1.11922 | − | 1.93581i | 0 | 0 | − | 5.87162i | 0 | 4.87162 | + | 2.81659i | ||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2205.2.d.s | 8 | |
| 3.b | odd | 2 | 1 | 735.2.d.d | 8 | ||
| 5.b | even | 2 | 1 | inner | 2205.2.d.s | 8 | |
| 7.b | odd | 2 | 1 | 2205.2.d.o | 8 | ||
| 7.c | even | 3 | 2 | 315.2.bf.b | 16 | ||
| 15.d | odd | 2 | 1 | 735.2.d.d | 8 | ||
| 15.e | even | 4 | 1 | 3675.2.a.bp | 4 | ||
| 15.e | even | 4 | 1 | 3675.2.a.bz | 4 | ||
| 21.c | even | 2 | 1 | 735.2.d.e | 8 | ||
| 21.g | even | 6 | 2 | 735.2.q.g | 16 | ||
| 21.h | odd | 6 | 2 | 105.2.q.a | ✓ | 16 | |
| 35.c | odd | 2 | 1 | 2205.2.d.o | 8 | ||
| 35.j | even | 6 | 2 | 315.2.bf.b | 16 | ||
| 84.n | even | 6 | 2 | 1680.2.di.d | 16 | ||
| 105.g | even | 2 | 1 | 735.2.d.e | 8 | ||
| 105.k | odd | 4 | 1 | 3675.2.a.bn | 4 | ||
| 105.k | odd | 4 | 1 | 3675.2.a.cb | 4 | ||
| 105.o | odd | 6 | 2 | 105.2.q.a | ✓ | 16 | |
| 105.p | even | 6 | 2 | 735.2.q.g | 16 | ||
| 105.x | even | 12 | 2 | 525.2.i.h | 8 | ||
| 105.x | even | 12 | 2 | 525.2.i.k | 8 | ||
| 420.ba | even | 6 | 2 | 1680.2.di.d | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.2.q.a | ✓ | 16 | 21.h | odd | 6 | 2 | |
| 105.2.q.a | ✓ | 16 | 105.o | odd | 6 | 2 | |
| 315.2.bf.b | 16 | 7.c | even | 3 | 2 | ||
| 315.2.bf.b | 16 | 35.j | even | 6 | 2 | ||
| 525.2.i.h | 8 | 105.x | even | 12 | 2 | ||
| 525.2.i.k | 8 | 105.x | even | 12 | 2 | ||
| 735.2.d.d | 8 | 3.b | odd | 2 | 1 | ||
| 735.2.d.d | 8 | 15.d | odd | 2 | 1 | ||
| 735.2.d.e | 8 | 21.c | even | 2 | 1 | ||
| 735.2.d.e | 8 | 105.g | even | 2 | 1 | ||
| 735.2.q.g | 16 | 21.g | even | 6 | 2 | ||
| 735.2.q.g | 16 | 105.p | even | 6 | 2 | ||
| 1680.2.di.d | 16 | 84.n | even | 6 | 2 | ||
| 1680.2.di.d | 16 | 420.ba | even | 6 | 2 | ||
| 2205.2.d.o | 8 | 7.b | odd | 2 | 1 | ||
| 2205.2.d.o | 8 | 35.c | odd | 2 | 1 | ||
| 2205.2.d.s | 8 | 1.a | even | 1 | 1 | trivial | |
| 2205.2.d.s | 8 | 5.b | even | 2 | 1 | inner | |
| 3675.2.a.bn | 4 | 105.k | odd | 4 | 1 | ||
| 3675.2.a.bp | 4 | 15.e | even | 4 | 1 | ||
| 3675.2.a.bz | 4 | 15.e | even | 4 | 1 | ||
| 3675.2.a.cb | 4 | 105.k | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2205, [\chi])\):
|
\( T_{2}^{8} + 12T_{2}^{6} + 44T_{2}^{4} + 52T_{2}^{2} + 4 \)
|
|
\( T_{11}^{4} - 18T_{11}^{2} + 14T_{11} + 30 \)
|
|
\( T_{13}^{8} + 60T_{13}^{6} + 1182T_{13}^{4} + 8408T_{13}^{2} + 16129 \)
|
|
\( T_{19}^{4} - 12T_{19}^{3} + 38T_{19}^{2} - 40T_{19} + 9 \)
|
|
\( T_{29}^{4} + 6T_{29}^{3} - 38T_{29}^{2} - 190T_{29} - 22 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 12 T^{6} + \cdots + 4 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} - 2 T^{7} + \cdots + 625 \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( (T^{4} - 18 T^{2} + \cdots + 30)^{2} \)
|
| $13$ |
\( T^{8} + 60 T^{6} + \cdots + 16129 \)
|
| $17$ |
\( T^{8} + 60 T^{6} + \cdots + 17956 \)
|
| $19$ |
\( (T^{4} - 12 T^{3} + 38 T^{2} + \cdots + 9)^{2} \)
|
| $23$ |
\( T^{8} + 164 T^{6} + \cdots + 2268036 \)
|
| $29$ |
\( (T^{4} + 6 T^{3} - 38 T^{2} + \cdots - 22)^{2} \)
|
| $31$ |
\( (T^{4} + 8 T^{3} - 6 T^{2} + \cdots + 61)^{2} \)
|
| $37$ |
\( T^{8} + 168 T^{6} + \cdots + 822649 \)
|
| $41$ |
\( (T^{4} + 4 T^{3} - 50 T^{2} + \cdots - 10)^{2} \)
|
| $43$ |
\( T^{8} + 128 T^{6} + \cdots + 2401 \)
|
| $47$ |
\( T^{8} + 68 T^{6} + \cdots + 14400 \)
|
| $53$ |
\( T^{8} + 160 T^{6} + \cdots + 9216 \)
|
| $59$ |
\( (T^{4} - 2 T^{3} - 6 T^{2} + \cdots + 10)^{2} \)
|
| $61$ |
\( (T^{4} + 8 T^{3} + \cdots + 500)^{2} \)
|
| $67$ |
\( T^{8} + 180 T^{6} + \cdots + 819025 \)
|
| $71$ |
\( (T^{4} - 14 T^{3} + \cdots - 3202)^{2} \)
|
| $73$ |
\( T^{8} + 272 T^{6} + \cdots + 1929321 \)
|
| $79$ |
\( (T^{4} - 8 T^{3} + \cdots + 7081)^{2} \)
|
| $83$ |
\( T^{8} + 148 T^{6} + \cdots + 131044 \)
|
| $89$ |
\( (T^{4} - 8 T^{3} + \cdots - 534)^{2} \)
|
| $97$ |
\( T^{8} + 20 T^{6} + \cdots + 16 \)
|
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