Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3648,2,Mod(1825,3648)]
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([0, 1, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3648.1825");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3648 = 2^{6} \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3648.g (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: |
\( x^{12} - 12x^{10} + 108x^{8} - 430x^{6} + 1284x^{4} - 36x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{12} \) |
| 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
\( x^{12} - 12x^{10} + 108x^{8} - 430x^{6} + 1284x^{4} - 36x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{10} - 9\nu^{8} + 41\nu^{6} - 107\nu^{4} + 3\nu^{2} - 1987 ) / 480 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{10} - 9\nu^{8} + 61\nu^{6} - 107\nu^{4} + 3\nu^{2} + 913 ) / 240 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -29\nu^{10} + 341\nu^{8} - 3089\nu^{6} + 12223\nu^{4} - 36967\nu^{2} + 523 ) / 720 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 169\nu^{11} - 2041\nu^{9} + 18409\nu^{7} - 73763\nu^{5} + 220787\nu^{3} - 12323\nu ) / 1440 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11\nu^{10} - 131\nu^{8} + 1181\nu^{6} - 4681\nu^{4} + 14065\nu^{2} - 199 ) / 72 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -27\nu^{11} + 323\nu^{9} - 2907\nu^{7} + 11529\nu^{5} - 34401\nu^{3} + 9\nu ) / 160 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -251\nu^{10} + 3059\nu^{8} - 27371\nu^{6} + 109417\nu^{4} - 320593\nu^{2} + 4537 ) / 1440 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -649\nu^{11} + 7801\nu^{9} - 70249\nu^{7} + 280643\nu^{5} - 839987\nu^{3} + 46883\nu ) / 1440 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -737\nu^{11} + 8873\nu^{9} - 79937\nu^{7} + 319819\nu^{5} - 957091\nu^{3} + 53419\nu ) / 1440 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( -309\nu^{11} + 3701\nu^{9} - 33269\nu^{7} + 131943\nu^{5} - 392647\nu^{3} + 103\nu ) / 480 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( -42\nu^{11} + 503\nu^{9} - 4522\nu^{7} + 17934\nu^{5} - 53401\nu^{3} + 14\nu ) / 60 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{4} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} + 3\beta_{5} + 7\beta_{3} - \beta_{2} + \beta _1 + 8 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{11} + 3\beta_{10} + \beta_{6} \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 5\beta_{7} + 17\beta_{5} + 43\beta_{3} + 5\beta_{2} - 7\beta _1 - 48 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -37\beta_{11} + 35\beta_{10} + 35\beta_{9} - 27\beta_{8} + 20\beta_{6} + 49\beta_{4} ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( 12\beta_{2} - 24\beta _1 - 145 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 229\beta_{11} - 205\beta_{10} + 205\beta_{9} - 145\beta_{8} - 168\beta_{6} + 337\beta_{4} ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( -109\beta_{7} - 555\beta_{5} - 1639\beta_{3} + 109\beta_{2} - 325\beta _1 - 1760 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( 711\beta_{11} - 603\beta_{10} - 649\beta_{6} \)
|
| \(\nu^{10}\) | \(=\) |
\( ( -449\beta_{7} - 3185\beta_{5} - 10171\beta_{3} - 449\beta_{2} + 2179\beta _1 + 10728 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 8857\beta_{11} - 7127\beta_{10} - 7127\beta_{9} + 4083\beta_{8} - 9548\beta_{6} - 15361\beta_{4} ) / 4 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1825.1 |
|
0 | − | 1.00000i | 0 | − | 3.99777i | 0 | 3.99777 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1825.2 | 0 | − | 1.00000i | 0 | − | 1.23519i | 0 | 1.23519 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1825.3 | 0 | − | 1.00000i | 0 | − | 0.701519i | 0 | 0.701519 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1825.4 | 0 | − | 1.00000i | 0 | 0.701519i | 0 | −0.701519 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1825.5 | 0 | − | 1.00000i | 0 | 1.23519i | 0 | −1.23519 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1825.6 | 0 | − | 1.00000i | 0 | 3.99777i | 0 | −3.99777 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1825.7 | 0 | 1.00000i | 0 | − | 3.99777i | 0 | −3.99777 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1825.8 | 0 | 1.00000i | 0 | − | 1.23519i | 0 | −1.23519 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1825.9 | 0 | 1.00000i | 0 | − | 0.701519i | 0 | −0.701519 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1825.10 | 0 | 1.00000i | 0 | 0.701519i | 0 | 0.701519 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1825.11 | 0 | 1.00000i | 0 | 1.23519i | 0 | 1.23519 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1825.12 | 0 | 1.00000i | 0 | 3.99777i | 0 | 3.99777 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 8.d | 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.g.i | ✓ | 12 |
| 4.b | odd | 2 | 1 | inner | 3648.2.g.i | ✓ | 12 |
| 8.b | even | 2 | 1 | inner | 3648.2.g.i | ✓ | 12 |
| 8.d | odd | 2 | 1 | inner | 3648.2.g.i | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3648.2.g.i | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 3648.2.g.i | ✓ | 12 | 4.b | odd | 2 | 1 | inner |
| 3648.2.g.i | ✓ | 12 | 8.b | even | 2 | 1 | inner |
| 3648.2.g.i | ✓ | 12 | 8.d | 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 \(S_{2}^{\mathrm{new}}(3648, [\chi])\):
|
\( T_{5}^{6} + 18T_{5}^{4} + 33T_{5}^{2} + 12 \)
|
|
\( T_{11}^{6} + 42T_{11}^{4} + 297T_{11}^{2} + 576 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( (T^{2} + 1)^{6} \)
|
| $5$ |
\( (T^{6} + 18 T^{4} + \cdots + 12)^{2} \)
|
| $7$ |
\( (T^{6} - 18 T^{4} + \cdots - 12)^{2} \)
|
| $11$ |
\( (T^{6} + 42 T^{4} + \cdots + 576)^{2} \)
|
| $13$ |
\( (T^{6} + 84 T^{4} + \cdots + 19200)^{2} \)
|
| $17$ |
\( (T^{3} + 6 T^{2} - 9 T - 18)^{4} \)
|
| $19$ |
\( (T^{2} + 1)^{6} \)
|
| $23$ |
\( (T^{6} - 120 T^{4} + \cdots - 3072)^{2} \)
|
| $29$ |
\( (T^{6} + 144 T^{4} + \cdots + 1728)^{2} \)
|
| $31$ |
\( (T^{2} - 12)^{6} \)
|
| $37$ |
\( T^{12} \)
|
| $41$ |
\( (T^{3} + 6 T^{2} - 48 T - 96)^{4} \)
|
| $43$ |
\( (T^{6} + 210 T^{4} + \cdots + 150544)^{2} \)
|
| $47$ |
\( (T^{6} - 294 T^{4} + \cdots - 539328)^{2} \)
|
| $53$ |
\( (T^{6} + 228 T^{4} + \cdots + 4800)^{2} \)
|
| $59$ |
\( (T^{6} + 60 T^{4} + \cdots + 2304)^{2} \)
|
| $61$ |
\( (T^{6} + 198 T^{4} + \cdots + 30000)^{2} \)
|
| $67$ |
\( (T^{6} + 348 T^{4} + \cdots + 565504)^{2} \)
|
| $71$ |
\( (T^{6} - 228 T^{4} + \cdots - 768)^{2} \)
|
| $73$ |
\( (T^{3} - 147 T + 430)^{4} \)
|
| $79$ |
\( (T^{6} - 384 T^{4} + \cdots - 1806528)^{2} \)
|
| $83$ |
\( (T^{6} + 240 T^{4} + \cdots + 147456)^{2} \)
|
| $89$ |
\( (T^{3} - 6 T^{2} + \cdots + 120)^{4} \)
|
| $97$ |
\( (T^{3} - 192 T + 776)^{4} \)
|
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