Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3648,2,Mod(607,3648)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3648, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3648.607");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3648 = 2^{6} \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3648.e (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: | \(29.1294266574\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 5x^{10} + 11x^{8} - 26x^{6} + 99x^{4} - 405x^{2} + 729 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 5x^{10} + 11x^{8} - 26x^{6} + 99x^{4} - 405x^{2} + 729 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{8} + 5\nu^{6} - 2\nu^{4} - 19\nu^{2} - 9 ) / 72 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{10} - \nu^{8} - 8\nu^{6} + 41\nu^{4} - 51\nu^{2} + 162 ) / 108 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -7\nu^{10} - \nu^{8} + 22\nu^{6} - 133\nu^{4} + 405\nu^{2} + 324 ) / 648 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4\nu^{10} - 11\nu^{8} - \nu^{6} - 86\nu^{4} + 243\nu^{2} - 891 ) / 324 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -7\nu^{11} + 8\nu^{9} - 23\nu^{7} + 533\nu^{5} - 1368\nu^{3} + 2997\nu ) / 1944 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5\nu^{11} - 10\nu^{9} + 7\nu^{7} - 19\nu^{5} + 402\nu^{3} - 1161\nu ) / 1296 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{11} + 5\nu^{9} - 11\nu^{7} + 26\nu^{5} - 99\nu^{3} + 162\nu ) / 243 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -\nu^{11} + 5\nu^{9} - 11\nu^{7} + 26\nu^{5} - 99\nu^{3} + 648\nu ) / 243 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 5\nu^{10} - 13\nu^{8} + 22\nu^{6} - 25\nu^{4} + 345\nu^{2} - 1188 ) / 216 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 13\nu^{11} - 2\nu^{9} + 71\nu^{7} - 131\nu^{5} + 378\nu^{3} - 2673\nu ) / 1944 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( -31\nu^{11} + 56\nu^{9} + 73\nu^{7} + 365\nu^{5} + 72\nu^{3} + 1701\nu ) / 3888 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{8} - \beta_{7} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{9} - \beta_{4} + \beta_{3} - 2\beta _1 + 2 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{11} - \beta_{10} + 2\beta_{8} - \beta_{7} + 6\beta_{6} - \beta_{5} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 3\beta_{9} - 5\beta_{4} - \beta_{3} + 2\beta_{2} - 2\beta_1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4\beta_{11} + \beta_{8} + \beta_{7} + 16\beta_{6} + 6\beta_{5} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3\beta_{9} - 7\beta_{4} + \beta_{3} - 3\beta_{2} + 6\beta _1 + 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 16\beta_{11} + 26\beta_{10} + 3\beta_{8} - 25\beta_{7} - 28\beta_{6} + 8\beta_{5} ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 5\beta_{9} - 41\beta_{4} - 7\beta_{3} - 34\beta_{2} - 42\beta _1 - 36 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 34\beta_{11} + 77\beta_{10} + 38\beta_{8} + 45\beta_{7} + 2\beta_{6} - 25\beta_{5} ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 19\beta_{9} - \beta_{4} - 101\beta_{3} - 52\beta_{2} - 34\beta _1 + 226 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( -100\beta_{11} + 198\beta_{10} + 147\beta_{8} - 23\beta_{7} + 140\beta_{6} + 42\beta_{5} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3648\mathbb{Z}\right)^\times\).
| \(n\) | \(1217\) | \(1921\) | \(2053\) | \(2623\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 607.1 |
|
0 | − | 1.00000i | 0 | − | 3.44500i | 0 | − | 3.14134i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 607.2 | 0 | − | 1.00000i | 0 | − | 2.98277i | 0 | 2.62620i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 607.3 | 0 | − | 1.00000i | 0 | − | 1.49502i | 0 | − | 0.484862i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 607.4 | 0 | − | 1.00000i | 0 | 1.49502i | 0 | − | 0.484862i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 607.5 | 0 | − | 1.00000i | 0 | 2.98277i | 0 | 2.62620i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 607.6 | 0 | − | 1.00000i | 0 | 3.44500i | 0 | − | 3.14134i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 607.7 | 0 | 1.00000i | 0 | − | 3.44500i | 0 | 3.14134i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 607.8 | 0 | 1.00000i | 0 | − | 2.98277i | 0 | − | 2.62620i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 607.9 | 0 | 1.00000i | 0 | − | 1.49502i | 0 | 0.484862i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 607.10 | 0 | 1.00000i | 0 | 1.49502i | 0 | 0.484862i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 607.11 | 0 | 1.00000i | 0 | 2.98277i | 0 | − | 2.62620i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 607.12 | 0 | 1.00000i | 0 | 3.44500i | 0 | 3.14134i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 152.b | even | 2 | 1 | inner |
| 152.g | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3648.2.e.b | yes | 12 |
| 4.b | odd | 2 | 1 | inner | 3648.2.e.b | yes | 12 |
| 8.b | even | 2 | 1 | 3648.2.e.a | ✓ | 12 | |
| 8.d | odd | 2 | 1 | 3648.2.e.a | ✓ | 12 | |
| 19.b | odd | 2 | 1 | 3648.2.e.a | ✓ | 12 | |
| 76.d | even | 2 | 1 | 3648.2.e.a | ✓ | 12 | |
| 152.b | even | 2 | 1 | inner | 3648.2.e.b | yes | 12 |
| 152.g | odd | 2 | 1 | inner | 3648.2.e.b | yes | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3648.2.e.a | ✓ | 12 | 8.b | even | 2 | 1 | |
| 3648.2.e.a | ✓ | 12 | 8.d | odd | 2 | 1 | |
| 3648.2.e.a | ✓ | 12 | 19.b | odd | 2 | 1 | |
| 3648.2.e.a | ✓ | 12 | 76.d | even | 2 | 1 | |
| 3648.2.e.b | yes | 12 | 1.a | even | 1 | 1 | trivial |
| 3648.2.e.b | yes | 12 | 4.b | odd | 2 | 1 | inner |
| 3648.2.e.b | yes | 12 | 152.b | even | 2 | 1 | inner |
| 3648.2.e.b | yes | 12 | 152.g | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3648, [\chi])\):
|
\( T_{5}^{6} + 23T_{5}^{4} + 152T_{5}^{2} + 236 \)
|
|
\( T_{13}^{3} - 4T_{13}^{2} - 14T_{13} - 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( (T^{2} + 1)^{6} \)
|
| $5$ |
\( (T^{6} + 23 T^{4} + \cdots + 236)^{2} \)
|
| $7$ |
\( (T^{6} + 17 T^{4} + \cdots + 16)^{2} \)
|
| $11$ |
\( (T^{6} - 51 T^{4} + \cdots - 944)^{2} \)
|
| $13$ |
\( (T^{3} - 4 T^{2} - 14 T - 8)^{4} \)
|
| $17$ |
\( (T^{3} + T^{2} - 8 T - 4)^{4} \)
|
| $19$ |
\( T^{12} - 34 T^{10} + \cdots + 47045881 \)
|
| $23$ |
\( (T^{6} + 44 T^{4} + \cdots + 64)^{2} \)
|
| $29$ |
\( (T^{3} - 4 T^{2} - 28 T + 32)^{4} \)
|
| $31$ |
\( (T^{6} - 172 T^{4} + \cdots - 3776)^{2} \)
|
| $37$ |
\( (T^{3} + 4 T^{2} + \cdots - 200)^{4} \)
|
| $41$ |
\( (T^{6} + 168 T^{4} + \cdots + 15104)^{2} \)
|
| $43$ |
\( (T^{6} - 51 T^{4} + \cdots - 944)^{2} \)
|
| $47$ |
\( (T^{6} + 125 T^{4} + \cdots + 4)^{2} \)
|
| $53$ |
\( (T^{3} - 8 T^{2} - 4 T + 16)^{4} \)
|
| $59$ |
\( (T^{6} + 280 T^{4} + \cdots + 350464)^{2} \)
|
| $61$ |
\( (T^{6} + 179 T^{4} + \cdots + 3776)^{2} \)
|
| $67$ |
\( (T^{6} + 176 T^{4} + \cdots + 4096)^{2} \)
|
| $71$ |
\( (T^{6} - 192 T^{4} + \cdots - 60416)^{2} \)
|
| $73$ |
\( (T^{3} - 5 T^{2} + \cdots + 352)^{4} \)
|
| $79$ |
\( (T^{6} - 220 T^{4} + \cdots - 60416)^{2} \)
|
| $83$ |
\( (T^{6} - 148 T^{4} + \cdots - 94400)^{2} \)
|
| $89$ |
\( (T^{6} + 192 T^{4} + \cdots + 60416)^{2} \)
|
| $97$ |
\( (T^{6} + 368 T^{4} + \cdots + 966656)^{2} \)
|
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