Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [102,14,Mod(1,102)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(102, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("102.1");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 102 = 2 \cdot 3 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 102.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: | \(109.375547531\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 13078169x^{2} - 5525405910x + 22928880943080 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 13078169x^{2} - 5525405910x + 22928880943080 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -324\nu^{3} + 451039\nu^{2} + 3228342050\nu - 1602037001250 ) / 139744605 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} + 38293\nu^{2} - 1607610\nu - 258737722380 ) / 19963515 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 62\nu^{3} - 143818\nu^{2} - 308030704\nu + 682444569135 ) / 27948921 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + \beta _1 + 7 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -223\beta_{3} + 2693\beta_{2} - 97\beta _1 + 39235811 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4671572\beta_{3} + 8730923\beta_{2} + 2259113\beta _1 + 24987525146 ) / 6 \)
|
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 |
|
64.0000 | −729.000 | 4096.00 | −54016.2 | −46656.0 | −335501. | 262144. | 531441. | −3.45704e6 | ||||||||||||||||||||||||||||||
| 1.2 | 64.0000 | −729.000 | 4096.00 | −36242.2 | −46656.0 | 157321. | 262144. | 531441. | −2.31950e6 | |||||||||||||||||||||||||||||||
| 1.3 | 64.0000 | −729.000 | 4096.00 | 24273.0 | −46656.0 | 561533. | 262144. | 531441. | 1.55347e6 | |||||||||||||||||||||||||||||||
| 1.4 | 64.0000 | −729.000 | 4096.00 | 29235.4 | −46656.0 | −207141. | 262144. | 531441. | 1.87107e6 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(17\) | \( -1 \) |
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 | 102.14.a.c | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 102.14.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 36750T_{5}^{3} - 2162286135T_{5}^{2} - 40701412457500T_{5} + 1389219666096442500 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(102))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 64)^{4} \)
|
| $3$ |
\( (T + 729)^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $7$ |
\( T^{4} + \cdots + 61\!\cdots\!44 \)
|
| $11$ |
\( T^{4} + \cdots - 77\!\cdots\!00 \)
|
| $13$ |
\( T^{4} + \cdots + 97\!\cdots\!56 \)
|
| $17$ |
\( (T - 24137569)^{4} \)
|
| $19$ |
\( T^{4} + \cdots + 63\!\cdots\!64 \)
|
| $23$ |
\( T^{4} + \cdots + 13\!\cdots\!96 \)
|
| $29$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 14\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots - 37\!\cdots\!64 \)
|
| $41$ |
\( T^{4} + \cdots - 21\!\cdots\!04 \)
|
| $43$ |
\( T^{4} + \cdots - 32\!\cdots\!84 \)
|
| $47$ |
\( T^{4} + \cdots - 79\!\cdots\!36 \)
|
| $53$ |
\( T^{4} + \cdots - 12\!\cdots\!76 \)
|
| $59$ |
\( T^{4} + \cdots + 96\!\cdots\!80 \)
|
| $61$ |
\( T^{4} + \cdots + 87\!\cdots\!80 \)
|
| $67$ |
\( T^{4} + \cdots - 97\!\cdots\!80 \)
|
| $71$ |
\( T^{4} + \cdots + 32\!\cdots\!36 \)
|
| $73$ |
\( T^{4} + \cdots + 59\!\cdots\!44 \)
|
| $79$ |
\( T^{4} + \cdots + 17\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 62\!\cdots\!24 \)
|
| $89$ |
\( T^{4} + \cdots - 67\!\cdots\!84 \)
|
| $97$ |
\( T^{4} + \cdots - 13\!\cdots\!80 \)
|
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