Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5,34,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, 34, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5.1");
S:= CuspForms(chi, 34);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5 \) |
| Weight: | \( k \) | \(=\) | \( 34 \) |
| 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: | \(34.4914144405\) |
| 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: |
\( x^{5} - 2x^{4} - 1372039866x^{3} - 648067657640x^{2} + 285631173782445856x - 33409741805340964224 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{21}\cdot 3^{5}\cdot 5^{4}\cdot 11 \) |
| 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
\( x^{5} - 2x^{4} - 1372039866x^{3} - 648067657640x^{2} + 285631173782445856x - 33409741805340964224 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 77\nu^{4} - 192270\nu^{3} - 92052810618\nu^{2} + 119506052126816\nu + 10209085202875426224 ) / 84151650384 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 539 \nu^{4} + 1345890 \nu^{3} + 1317582877398 \nu^{2} + \cdots - 44\!\cdots\!72 ) / 42075825192 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 11597 \nu^{4} - 699627858 \nu^{3} + 15491770165050 \nu^{2} + \cdots - 21\!\cdots\!68 ) / 28050550128 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 14\beta_{2} + 2838\beta _1 + 8781056288 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -308\beta_{4} + 1117\beta_{3} - 123526\beta_{2} + 1911420946\beta _1 + 3124660933052 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -769080\beta_{4} + 600534687\beta_{3} + 16803019914\beta_{2} + 3365191419490\beta _1 + 4195949021597623688 ) / 8 \)
|
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 |
|
−129081. | −6.82881e7 | 8.07187e9 | −1.52588e11 | 8.81467e12 | −1.39338e14 | 6.68716e13 | −8.95800e14 | 1.96961e16 | |||||||||||||||||||||||||||||||||
| 1.2 | −56118.7 | 5.48591e7 | −5.44063e9 | −1.52588e11 | −3.07862e12 | 7.38557e13 | 7.87377e14 | −2.54954e15 | 8.56303e15 | ||||||||||||||||||||||||||||||||||
| 1.3 | 5627.97 | −1.24605e8 | −8.55826e9 | −1.52588e11 | −7.01272e11 | 7.01560e13 | −9.65096e13 | 9.96727e15 | −8.58760e14 | ||||||||||||||||||||||||||||||||||
| 1.4 | 71937.8 | 1.37573e8 | −3.41489e9 | −1.52588e11 | 9.89671e12 | −1.07684e14 | −8.63600e14 | 1.33673e16 | −1.09768e16 | ||||||||||||||||||||||||||||||||||
| 1.5 | 138106. | −1.45282e7 | 1.04832e10 | −1.52588e11 | −2.00642e12 | 3.75584e13 | 2.61472e14 | −5.34799e15 | −2.10732e16 | ||||||||||||||||||||||||||||||||||
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 | 5.34.a.a | ✓ | 5 |
| 5.b | even | 2 | 1 | 25.34.a.b | 5 | ||
| 5.c | odd | 4 | 2 | 25.34.b.b | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.34.a.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 25.34.a.b | 5 | 5.b | even | 2 | 1 | ||
| 25.34.b.b | 10 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 30472 T_{2}^{4} - 21581220768 T_{2}^{3} + 440682607210496 T_{2}^{2} + \cdots - 40\!\cdots\!32 \)
acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(5))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + \cdots - 40\!\cdots\!32 \)
|
| $3$ |
\( T^{5} + \cdots + 93\!\cdots\!24 \)
|
| $5$ |
\( (T + 152587890625)^{5} \)
|
| $7$ |
\( T^{5} + \cdots - 29\!\cdots\!32 \)
|
| $11$ |
\( T^{5} + \cdots + 62\!\cdots\!68 \)
|
| $13$ |
\( T^{5} + \cdots - 12\!\cdots\!76 \)
|
| $17$ |
\( T^{5} + \cdots + 24\!\cdots\!68 \)
|
| $19$ |
\( T^{5} + \cdots - 97\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots - 72\!\cdots\!76 \)
|
| $29$ |
\( T^{5} + \cdots - 58\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots - 18\!\cdots\!32 \)
|
| $37$ |
\( T^{5} + \cdots - 14\!\cdots\!32 \)
|
| $41$ |
\( T^{5} + \cdots + 14\!\cdots\!68 \)
|
| $43$ |
\( T^{5} + \cdots + 73\!\cdots\!24 \)
|
| $47$ |
\( T^{5} + \cdots + 52\!\cdots\!68 \)
|
| $53$ |
\( T^{5} + \cdots - 16\!\cdots\!76 \)
|
| $59$ |
\( T^{5} + \cdots - 50\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots - 88\!\cdots\!32 \)
|
| $67$ |
\( T^{5} + \cdots - 10\!\cdots\!32 \)
|
| $71$ |
\( T^{5} + \cdots - 43\!\cdots\!32 \)
|
| $73$ |
\( T^{5} + \cdots - 47\!\cdots\!76 \)
|
| $79$ |
\( T^{5} + \cdots + 25\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots - 10\!\cdots\!76 \)
|
| $89$ |
\( T^{5} + \cdots - 20\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots - 42\!\cdots\!32 \)
|
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