Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [13,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 13 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(4.06100533129\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{337}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 84 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{337})\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−18.6788 | 50.0363 | 220.897 | −277.467 | −934.618 | −921.891 | −1735.20 | 316.635 | 5182.74 | ||||||||||||||||||||||||
| 1.2 | −0.321220 | −5.03634 | −127.897 | −75.5334 | 1.61777 | −1087.11 | 82.1992 | −2161.64 | 24.2629 | |||||||||||||||||||||||||
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 | 13.8.a.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 117.8.a.c | 2 | ||
| 4.b | odd | 2 | 1 | 208.8.a.g | 2 | ||
| 5.b | even | 2 | 1 | 325.8.a.b | 2 | ||
| 13.b | even | 2 | 1 | 169.8.a.b | 2 | ||
| 13.d | odd | 4 | 2 | 169.8.b.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.8.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 117.8.a.c | 2 | 3.b | odd | 2 | 1 | ||
| 169.8.a.b | 2 | 13.b | even | 2 | 1 | ||
| 169.8.b.b | 4 | 13.d | odd | 4 | 2 | ||
| 208.8.a.g | 2 | 4.b | odd | 2 | 1 | ||
| 325.8.a.b | 2 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 19T_{2} + 6 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(13))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 19T + 6 \)
|
| $3$ |
\( T^{2} - 45T - 252 \)
|
| $5$ |
\( T^{2} + 353T + 20958 \)
|
| $7$ |
\( T^{2} + 2009 T + 1002196 \)
|
| $11$ |
\( T^{2} + 1810 T - 31775952 \)
|
| $13$ |
\( (T - 2197)^{2} \)
|
| $17$ |
\( T^{2} + 25361 T - 177216678 \)
|
| $19$ |
\( T^{2} + \cdots - 1646636688 \)
|
| $23$ |
\( T^{2} + \cdots - 3712002048 \)
|
| $29$ |
\( T^{2} + \cdots - 1049753004 \)
|
| $31$ |
\( T^{2} - 39744 T - 634808464 \)
|
| $37$ |
\( T^{2} + \cdots - 7868862106 \)
|
| $41$ |
\( T^{2} + \cdots - 115487893152 \)
|
| $43$ |
\( T^{2} + \cdots + 11546069484 \)
|
| $47$ |
\( T^{2} + \cdots + 669689975052 \)
|
| $53$ |
\( T^{2} + \cdots - 648240166944 \)
|
| $59$ |
\( T^{2} + \cdots - 140057008272 \)
|
| $61$ |
\( T^{2} + \cdots + 4947441275696 \)
|
| $67$ |
\( T^{2} + \cdots - 5405517426256 \)
|
| $71$ |
\( T^{2} + \cdots - 24787522246284 \)
|
| $73$ |
\( T^{2} + \cdots + 10156859198164 \)
|
| $79$ |
\( T^{2} + \cdots - 22472459585984 \)
|
| $83$ |
\( T^{2} + \cdots + 34361915476704 \)
|
| $89$ |
\( T^{2} + \cdots - 10393898367132 \)
|
| $97$ |
\( T^{2} + \cdots + 45398949212204 \)
|
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