Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5,26,Mod(1,5)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 26, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5.1");
S:= CuspForms(chi, 26);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5 \) |
| Weight: | \( k \) | \(=\) | \( 26 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5.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: | \(19.7998389976\) |
| 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: |
\( x^{4} - x^{3} - 1769856x^{2} + 106836475x + 628040620025 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{14}\cdot 3^{2}\cdot 5^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 1769856x^{2} + 106836475x + 628040620025 \)
:
| \(\beta_{1}\) | \(=\) |
\( 8\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8\nu^{3} + 3344\nu^{2} - 8182184\nu - 2326754810 ) / 1863 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 32\nu^{3} + 132608\nu^{2} - 22087280\nu - 114821444708 ) / 1863 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 4\beta_{2} - 714\beta _1 + 56635408 ) / 64 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -209\beta_{3} + 8288\beta_{2} + 4240318\beta _1 - 2521598848 ) / 32 \)
|
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 |
|
−8221.63 | 987550. | 3.40407e7 | −2.44141e8 | −8.11927e9 | −1.93681e10 | −3.99805e9 | 1.27967e11 | 2.00723e12 | ||||||||||||||||||||||||||||||
| 1.2 | −6133.72 | −1.60154e6 | 4.06809e6 | −2.44141e8 | 9.82339e9 | −2.17553e9 | 1.80861e11 | 1.71764e12 | 1.49749e12 | |||||||||||||||||||||||||||||||
| 1.3 | 5288.75 | 887362. | −5.58351e6 | −2.44141e8 | 4.69304e9 | −4.61897e10 | −2.06991e11 | −5.98771e10 | −1.29120e12 | |||||||||||||||||||||||||||||||
| 1.4 | 9666.59 | −1.07197e6 | 5.98886e7 | −2.44141e8 | −1.03623e10 | 1.87952e10 | 2.54562e11 | 3.01839e11 | −2.36001e12 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
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 | 5.26.a.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 45.26.a.c | 4 | ||
| 5.b | even | 2 | 1 | 25.26.a.b | 4 | ||
| 5.c | odd | 4 | 2 | 25.26.b.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.26.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 25.26.a.b | 4 | 5.b | even | 2 | 1 | ||
| 25.26.b.b | 8 | 5.c | odd | 4 | 2 | ||
| 45.26.a.c | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 600T_{2}^{3} - 113135808T_{2}^{2} - 20279449600T_{2} + 2578152318959616 \)
acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(5))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + \cdots + 25\!\cdots\!16 \)
|
| $3$ |
\( T^{4} + \cdots + 15\!\cdots\!56 \)
|
| $5$ |
\( (T + 244140625)^{4} \)
|
| $7$ |
\( T^{4} + \cdots - 36\!\cdots\!04 \)
|
| $11$ |
\( T^{4} + \cdots - 12\!\cdots\!84 \)
|
| $13$ |
\( T^{4} + \cdots - 27\!\cdots\!84 \)
|
| $17$ |
\( T^{4} + \cdots + 10\!\cdots\!56 \)
|
| $19$ |
\( T^{4} + \cdots + 33\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots - 75\!\cdots\!24 \)
|
| $29$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 11\!\cdots\!84 \)
|
| $37$ |
\( T^{4} + \cdots - 10\!\cdots\!24 \)
|
| $41$ |
\( T^{4} + \cdots - 11\!\cdots\!84 \)
|
| $43$ |
\( T^{4} + \cdots + 55\!\cdots\!96 \)
|
| $47$ |
\( T^{4} + \cdots + 38\!\cdots\!36 \)
|
| $53$ |
\( T^{4} + \cdots - 26\!\cdots\!44 \)
|
| $59$ |
\( T^{4} + \cdots + 24\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots - 88\!\cdots\!84 \)
|
| $67$ |
\( T^{4} + \cdots + 19\!\cdots\!56 \)
|
| $71$ |
\( T^{4} + \cdots - 14\!\cdots\!84 \)
|
| $73$ |
\( T^{4} + \cdots - 89\!\cdots\!24 \)
|
| $79$ |
\( T^{4} + \cdots - 89\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 33\!\cdots\!36 \)
|
| $89$ |
\( T^{4} + \cdots + 71\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 18\!\cdots\!64 \)
|
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