Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [13,12,Mod(1,13)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 13 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 13.a (trivial) |
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: | \(9.98846134727\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 6448x^{3} - 12116x^{2} + 9682560x + 66650112 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 6448x^{3} - 12116x^{2} + 9682560x + 66650112 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{4} + 133\nu^{3} - 17726\nu^{2} - 457048\nu - 169824 ) / 24096 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 127\nu^{3} - 6256\nu^{2} - 427348\nu + 5904896 ) / 16064 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 124\nu^{3} + 11527\nu^{2} - 448642\nu - 22129536 ) / 24096 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{4} - 2\beta_{3} + \beta_{2} + 3\beta _1 + 2579 \)
|
| \(\nu^{3}\) | \(=\) |
\( 54\beta_{4} + 106\beta_{3} - 21\beta_{2} + 3427\beta _1 + 10481 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5654\beta_{4} - 9910\beta_{3} + 8923\beta_{2} + 10887\beta _1 + 8898241 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−71.7031 | −710.121 | 3093.33 | 10598.3 | 50917.9 | −48491.8 | −74953.5 | 327125. | −759930. | |||||||||||||||||||||||||||||||||
| 1.2 | −60.4714 | −54.5312 | 1608.79 | −233.583 | 3297.58 | 81481.1 | 26559.7 | −174173. | 14125.1 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.947150 | 680.292 | −2047.10 | −10968.3 | −644.339 | −18559.5 | 3878.68 | 285650. | 10388.7 | ||||||||||||||||||||||||||||||||||
| 1.4 | 36.2939 | −248.328 | −730.754 | 8212.56 | −9012.78 | −51189.3 | −100852. | −115480. | 298066. | ||||||||||||||||||||||||||||||||||
| 1.5 | 55.8277 | −163.312 | 1068.74 | −10150.9 | −9117.33 | 463.524 | −54670.1 | −150476. | −566704. | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(13\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 13.12.a.a | ✓ | 5 |
| 3.b | odd | 2 | 1 | 117.12.a.b | 5 | ||
| 4.b | odd | 2 | 1 | 208.12.a.f | 5 | ||
| 13.b | even | 2 | 1 | 169.12.a.b | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.12.a.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 117.12.a.b | 5 | 3.b | odd | 2 | 1 | ||
| 169.12.a.b | 5 | 13.b | even | 2 | 1 | ||
| 208.12.a.f | 5 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 41T_{2}^{4} - 5776T_{2}^{3} - 137132T_{2}^{2} + 8660928T_{2} + 8321280 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(13))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 41 T^{4} + \cdots + 8321280 \)
|
| $3$ |
\( T^{5} + \cdots - 1068353853588 \)
|
| $5$ |
\( T^{5} + \cdots + 22\!\cdots\!50 \)
|
| $7$ |
\( T^{5} + \cdots + 17\!\cdots\!68 \)
|
| $11$ |
\( T^{5} + \cdots + 66\!\cdots\!48 \)
|
| $13$ |
\( (T - 371293)^{5} \)
|
| $17$ |
\( T^{5} + \cdots + 16\!\cdots\!82 \)
|
| $19$ |
\( T^{5} + \cdots + 15\!\cdots\!48 \)
|
| $23$ |
\( T^{5} + \cdots + 29\!\cdots\!48 \)
|
| $29$ |
\( T^{5} + \cdots + 67\!\cdots\!08 \)
|
| $31$ |
\( T^{5} + \cdots - 84\!\cdots\!48 \)
|
| $37$ |
\( T^{5} + \cdots + 47\!\cdots\!06 \)
|
| $41$ |
\( T^{5} + \cdots - 16\!\cdots\!24 \)
|
| $43$ |
\( T^{5} + \cdots - 47\!\cdots\!92 \)
|
| $47$ |
\( T^{5} + \cdots - 51\!\cdots\!60 \)
|
| $53$ |
\( T^{5} + \cdots - 27\!\cdots\!64 \)
|
| $59$ |
\( T^{5} + \cdots + 88\!\cdots\!04 \)
|
| $61$ |
\( T^{5} + \cdots - 40\!\cdots\!64 \)
|
| $67$ |
\( T^{5} + \cdots - 33\!\cdots\!84 \)
|
| $71$ |
\( T^{5} + \cdots - 15\!\cdots\!64 \)
|
| $73$ |
\( T^{5} + \cdots - 49\!\cdots\!48 \)
|
| $79$ |
\( T^{5} + \cdots - 50\!\cdots\!24 \)
|
| $83$ |
\( T^{5} + \cdots + 37\!\cdots\!48 \)
|
| $89$ |
\( T^{5} + \cdots + 11\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots + 17\!\cdots\!00 \)
|
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