Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [39,3,Mod(31,39)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(39, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("39.31");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 39 = 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 39.g (of order \(4\), 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: | \(1.06267303101\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.1579585536.10 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} - 4x^{6} + 28x^{5} - 38x^{4} + 8x^{3} + 200x^{2} - 352x + 484 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} - 2x^{7} - 4x^{6} + 28x^{5} - 38x^{4} + 8x^{3} + 200x^{2} - 352x + 484 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{7} + 46\nu^{6} - 150\nu^{5} + 27\nu^{4} + 720\nu^{3} - 1768\nu^{2} + 2792\nu - 1881 ) / 935 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -118\nu^{7} - 607\nu^{6} + 490\nu^{5} - 384\nu^{4} - 8710\nu^{3} + 19856\nu^{2} - 35024\nu - 91398 ) / 68510 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 3448 \nu^{7} + 21163 \nu^{6} + 45670 \nu^{5} - 200494 \nu^{4} + 126360 \nu^{3} + 635936 \nu^{2} + \cdots + 2421122 ) / 753610 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 634\nu^{7} - 1964\nu^{6} + 4915\nu^{5} + 902\nu^{4} - 39130\nu^{3} + 104652\nu^{2} - 103278\nu + 63754 ) / 68510 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -86\nu^{7} + 227\nu^{6} - 162\nu^{5} - 758\nu^{4} + 1794\nu^{3} - 8124\nu^{2} + 8408\nu - 13662 ) / 8866 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 3777 \nu^{7} + 11052 \nu^{6} + 45875 \nu^{5} - 222466 \nu^{4} + 199615 \nu^{3} + 521764 \nu^{2} + \cdots + 3415148 ) / 376805 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 10422 \nu^{7} - 42767 \nu^{6} + 8395 \nu^{5} + 210416 \nu^{4} - 556790 \nu^{3} + 515236 \nu^{2} + \cdots - 1719828 ) / 753610 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} - \beta_{4} + \beta_{3} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{7} - \beta_{6} - 3\beta_{5} + \beta_{3} + 4\beta_{2} + 2 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -4\beta_{5} - 4\beta_{4} + \beta_{3} - \beta_{2} - \beta _1 - 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( -12\beta_{7} - 5\beta_{6} - 8\beta_{5} + 2\beta_{4} - 3\beta_{3} - \beta_{2} - 4\beta _1 + 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( -3\beta_{7} - 2\beta_{6} - 2\beta_{5} - 7\beta_{4} + \beta_{3} - 34\beta_{2} - 12\beta _1 - 58 \)
|
| \(\nu^{6}\) | \(=\) |
\( -70\beta_{7} - 22\beta_{6} + 4\beta_{5} + 70\beta_{4} - 22\beta_{3} - 22\beta_{2} + 22 \)
|
| \(\nu^{7}\) | \(=\) |
\( 70\beta_{7} + 37\beta_{6} + 40\beta_{5} + 48\beta_{4} - 11\beta_{3} - 195\beta_{2} + 37\beta _1 - 309 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).
| \(n\) | \(14\) | \(28\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
−1.26523 | + | 1.26523i | −1.73205 | 0.798403i | −4.65826 | + | 4.65826i | 2.19144 | − | 2.19144i | 1.87963 | + | 1.87963i | −6.07107 | − | 6.07107i | 3.00000 | − | 11.7875i | |||||||||||||||||||||||||||||||
| 31.2 | −0.676424 | + | 0.676424i | 1.73205 | 3.08490i | 1.58008 | − | 1.58008i | −1.17160 | + | 1.17160i | 3.96400 | + | 3.96400i | −4.79240 | − | 4.79240i | 3.00000 | 2.13761i | |||||||||||||||||||||||||||||||||
| 31.3 | 1.67642 | − | 1.67642i | 1.73205 | − | 1.62080i | −4.84803 | + | 4.84803i | 2.90365 | − | 2.90365i | −8.89220 | − | 8.89220i | 3.98855 | + | 3.98855i | 3.00000 | 16.2547i | ||||||||||||||||||||||||||||||||
| 31.4 | 2.26523 | − | 2.26523i | −1.73205 | − | 6.26250i | −2.07379 | + | 2.07379i | −3.92349 | + | 3.92349i | 7.04857 | + | 7.04857i | −5.12509 | − | 5.12509i | 3.00000 | 9.39521i | ||||||||||||||||||||||||||||||||
| 34.1 | −1.26523 | − | 1.26523i | −1.73205 | − | 0.798403i | −4.65826 | − | 4.65826i | 2.19144 | + | 2.19144i | 1.87963 | − | 1.87963i | −6.07107 | + | 6.07107i | 3.00000 | 11.7875i | ||||||||||||||||||||||||||||||||
| 34.2 | −0.676424 | − | 0.676424i | 1.73205 | − | 3.08490i | 1.58008 | + | 1.58008i | −1.17160 | − | 1.17160i | 3.96400 | − | 3.96400i | −4.79240 | + | 4.79240i | 3.00000 | − | 2.13761i | |||||||||||||||||||||||||||||||
| 34.3 | 1.67642 | + | 1.67642i | 1.73205 | 1.62080i | −4.84803 | − | 4.84803i | 2.90365 | + | 2.90365i | −8.89220 | + | 8.89220i | 3.98855 | − | 3.98855i | 3.00000 | − | 16.2547i | ||||||||||||||||||||||||||||||||
| 34.4 | 2.26523 | + | 2.26523i | −1.73205 | 6.26250i | −2.07379 | − | 2.07379i | −3.92349 | − | 3.92349i | 7.04857 | − | 7.04857i | −5.12509 | + | 5.12509i | 3.00000 | − | 9.39521i | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.d | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 39.3.g.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 117.3.j.b | 8 | ||
| 4.b | odd | 2 | 1 | 624.3.ba.b | 8 | ||
| 13.d | odd | 4 | 1 | inner | 39.3.g.a | ✓ | 8 |
| 39.f | even | 4 | 1 | 117.3.j.b | 8 | ||
| 52.f | even | 4 | 1 | 624.3.ba.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 39.3.g.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 39.3.g.a | ✓ | 8 | 13.d | odd | 4 | 1 | inner |
| 117.3.j.b | 8 | 3.b | odd | 2 | 1 | ||
| 117.3.j.b | 8 | 39.f | even | 4 | 1 | ||
| 624.3.ba.b | 8 | 4.b | odd | 2 | 1 | ||
| 624.3.ba.b | 8 | 52.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(39, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 4 T^{7} + \cdots + 169 \)
|
| $3$ |
\( (T^{2} - 3)^{4} \)
|
| $5$ |
\( T^{8} + 20 T^{7} + \cdots + 87616 \)
|
| $7$ |
\( T^{8} - 8 T^{7} + \cdots + 3489424 \)
|
| $11$ |
\( T^{8} + 20 T^{7} + \cdots + 602176 \)
|
| $13$ |
\( T^{8} - 8 T^{7} + \cdots + 815730721 \)
|
| $17$ |
\( T^{8} + 336 T^{6} + \cdots + 3504384 \)
|
| $19$ |
\( T^{8} + 40 T^{7} + \cdots + 3671056 \)
|
| $23$ |
\( T^{8} + \cdots + 8422834176 \)
|
| $29$ |
\( (T^{4} - 28 T^{3} + \cdots - 3824)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 246909610000 \)
|
| $37$ |
\( T^{8} + \cdots + 12456636124816 \)
|
| $41$ |
\( T^{8} + \cdots + 26504951703616 \)
|
| $43$ |
\( T^{8} + 4704 T^{6} + \cdots + 632623104 \)
|
| $47$ |
\( T^{8} - 100 T^{7} + \cdots + 326019136 \)
|
| $53$ |
\( (T^{4} + 116 T^{3} + \cdots + 304336)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 62781028208704 \)
|
| $61$ |
\( (T^{4} - 80 T^{3} + \cdots + 2335552)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 780982247824 \)
|
| $71$ |
\( T^{8} + \cdots + 34773288084544 \)
|
| $73$ |
\( T^{8} + \cdots + 450106810000 \)
|
| $79$ |
\( (T^{4} - 104 T^{3} + \cdots - 5173376)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 31365107407936 \)
|
| $89$ |
\( T^{8} + \cdots + 14\!\cdots\!16 \)
|
| $97$ |
\( T^{8} + \cdots + 190804383734416 \)
|
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