Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1388,3,Mod(693,1388)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1388, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1388.693");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1388 = 2^{2} \cdot 347 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1388.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: | yes |
| Analytic conductor: | \(37.8202606932\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 30x^{8} + 315x^{6} - 25x^{5} - 1350x^{4} + 375x^{3} + 2025x^{2} - 1125x - 104 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 5^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 30x^{8} + 315x^{6} - 25x^{5} - 1350x^{4} + 375x^{3} + 2025x^{2} - 1125x - 104 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{8} - 24\nu^{7} - 24\nu^{6} + 504\nu^{5} + 180\nu^{4} - 3024\nu^{3} - 241\nu^{2} + 4536\nu - 984 ) / 191 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 15\nu^{3} + 45\nu - 13 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{9} - 81\nu^{7} - 25\nu^{6} + 729\nu^{5} + 450\nu^{4} - 2430\nu^{3} - 2025\nu^{2} + 2378\nu + 1350 ) / 191 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{9} + 135\nu^{7} - 22\nu^{6} - 1215\nu^{5} + 396\nu^{4} + 4050\nu^{3} - 1782\nu^{2} - 3454\nu + 1188 ) / 191 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -6\nu^{9} + 162\nu^{7} + 50\nu^{6} - 1458\nu^{5} - 709\nu^{4} + 4860\nu^{3} + 1758\nu^{2} - 4756\nu + 738 ) / 191 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -11\nu^{9} + 297\nu^{7} + 28\nu^{6} - 2673\nu^{5} - 313\nu^{4} + 8910\nu^{3} - 24\nu^{2} - 8974\nu + 1926 ) / 191 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -17\nu^{8} + 26\nu^{7} + 408\nu^{6} - 546\nu^{5} - 3060\nu^{4} + 3658\nu^{3} + 6771\nu^{2} - 8352\nu + 684 ) / 191 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 25\nu^{8} - 27\nu^{7} - 600\nu^{6} + 567\nu^{5} + 4500\nu^{4} - 3593\nu^{3} - 10418\nu^{2} + 6822\nu + 1758 ) / 191 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} + \beta_{6} + \beta_{5} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta _1 + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{9} + 3\beta_{8} - 9\beta_{7} + 9\beta_{6} + 9\beta_{5} + \beta_{2} + 4\beta_1 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 2\beta_{4} + 12\beta _1 + 54 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 30\beta_{9} + 45\beta_{8} - 90\beta_{7} + 90\beta_{6} + 90\beta_{5} + 4\beta_{3} + 15\beta_{2} + 60\beta _1 + 52 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( -2\beta_{7} + 20\beta_{6} - \beta_{5} + 31\beta_{4} + 135\beta _1 + 540 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 380 \beta_{9} + 568 \beta_{8} - 945 \beta_{7} + 945 \beta_{6} + 945 \beta_{5} + 84 \beta_{3} + \cdots + 1092 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( 12\beta_{9} + 6\beta_{8} - 48\beta_{7} + 300\beta_{6} - 24\beta_{5} + 384\beta_{4} - 7\beta_{2} + 1513\beta _1 + 5670 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 4590 \beta_{9} + 6831 \beta_{8} - 10209 \beta_{7} + 10209 \beta_{6} + 10109 \beta_{5} + 88 \beta_{4} + \cdots + 16848 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1388\mathbb{Z}\right)^\times\).
| \(n\) | \(349\) | \(695\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 693.1 |
|
0 | −5.99346 | 0 | 0 | 0 | 0 | 0 | 26.9215 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 693.2 | 0 | −5.36135 | 0 | 0 | 0 | 0 | 0 | 19.7441 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 693.3 | 0 | −4.21859 | 0 | 0 | 0 | 0 | 0 | 8.79654 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 693.4 | 0 | −2.11851 | 0 | 0 | 0 | 0 | 0 | −4.51189 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 693.5 | 0 | −1.58564 | 0 | 0 | 0 | 0 | 0 | −6.48573 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 693.6 | 0 | 0.905099 | 0 | 0 | 0 | 0 | 0 | −8.18080 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 693.7 | 0 | 2.75411 | 0 | 0 | 0 | 0 | 0 | −1.41486 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 693.8 | 0 | 4.68414 | 0 | 0 | 0 | 0 | 0 | 12.9412 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 693.9 | 0 | 5.01347 | 0 | 0 | 0 | 0 | 0 | 16.1349 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 693.10 | 0 | 5.92073 | 0 | 0 | 0 | 0 | 0 | 26.0551 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 347.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-347}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1388.3.d.a | ✓ | 10 |
| 347.b | odd | 2 | 1 | CM | 1388.3.d.a | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1388.3.d.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 1388.3.d.a | ✓ | 10 | 347.b | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - 90T_{3}^{8} + 2835T_{3}^{6} + 139T_{3}^{5} - 36450T_{3}^{4} - 6255T_{3}^{3} + 164025T_{3}^{2} + 56295T_{3} - 157826 \)
acting on \(S_{3}^{\mathrm{new}}(1388, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( T^{10} - 90 T^{8} + \cdots - 157826 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} \)
|
| $11$ |
\( T^{10} + \cdots + 24268615198 \)
|
| $13$ |
\( (T^{5} - 845 T^{3} + \cdots - 692618)^{2} \)
|
| $17$ |
\( T^{10} \)
|
| $19$ |
\( T^{10} \)
|
| $23$ |
\( T^{10} \)
|
| $29$ |
\( T^{10} + \cdots - 11\!\cdots\!02 \)
|
| $31$ |
\( (T^{5} - 4805 T^{3} + \cdots + 13009198)^{2} \)
|
| $37$ |
\( T^{10} \)
|
| $41$ |
\( T^{10} \)
|
| $43$ |
\( T^{10} + \cdots - 46\!\cdots\!98 \)
|
| $47$ |
\( T^{10} \)
|
| $53$ |
\( (T^{5} - 14045 T^{3} + \cdots + 824817382)^{2} \)
|
| $59$ |
\( (T^{5} - 17405 T^{3} + \cdots - 1403797226)^{2} \)
|
| $61$ |
\( T^{10} + \cdots - 21\!\cdots\!02 \)
|
| $67$ |
\( T^{10} + \cdots + 14\!\cdots\!02 \)
|
| $71$ |
\( (T^{5} - 25205 T^{3} + \cdots - 3603627074)^{2} \)
|
| $73$ |
\( T^{10} + \cdots - 11\!\cdots\!98 \)
|
| $79$ |
\( T^{10} \)
|
| $83$ |
\( T^{10} + \cdots - 63\!\cdots\!26 \)
|
| $89$ |
\( (T^{2} - 169 T + 4798)^{5} \)
|
| $97$ |
\( T^{10} \)
|
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