Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5,30,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, 30, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5.1");
S:= CuspForms(chi, 30);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5 \) |
| Weight: | \( k \) | \(=\) | \( 30 \) |
| 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: | \(26.6390211915\) |
| 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: |
\( x^{4} - 2x^{3} - 38120625x^{2} - 35295444214x + 207047907801890 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{3}\cdot 5^{2}\cdot 7 \) |
| 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
\( x^{4} - 2x^{3} - 38120625x^{2} - 35295444214x + 207047907801890 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 188\nu^{3} - 1135172\nu^{2} - 3221059076\nu + 16650937075160 ) / 192942225 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -10904\nu^{3} + 65839976\nu^{2} + 464658230408\nu - 965893268761280 ) / 192942225 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -21832\nu^{3} + 66142408\nu^{2} + 488735626264\nu - 681763422106840 ) / 38588445 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 58\beta _1 + 720 ) / 1440 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -47\beta_{3} + 97\beta_{2} - 21664\beta _1 + 1524825080 ) / 80 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5108274\beta_{3} + 27675901\beta_{2} + 116998078\beta _1 + 38201420304000 ) / 1440 \)
|
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 |
|
−42557.3 | 2.86347e6 | 1.27425e9 | −6.10352e9 | −1.21862e11 | −2.58797e11 | −3.13809e13 | −6.04309e13 | 2.59749e14 | ||||||||||||||||||||||||||||||
| 1.2 | −15043.9 | −9.45262e6 | −3.10551e8 | −6.10352e9 | 1.42205e11 | 6.01698e11 | 1.27486e13 | 2.07217e13 | 9.18209e13 | |||||||||||||||||||||||||||||||
| 1.3 | 8721.19 | 7.48706e6 | −4.60812e8 | −6.10352e9 | 6.52961e10 | 2.73294e12 | −8.70098e12 | −1.25744e13 | −5.32299e13 | |||||||||||||||||||||||||||||||
| 1.4 | 33280.0 | 1.81470e6 | 5.70689e8 | −6.10352e9 | 6.03931e10 | −1.91372e12 | 1.12547e12 | −6.53373e13 | −2.03125e14 | |||||||||||||||||||||||||||||||
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 | 5.30.a.a | ✓ | 4 |
| 5.b | even | 2 | 1 | 25.30.a.b | 4 | ||
| 5.c | odd | 4 | 2 | 25.30.b.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.30.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 25.30.a.b | 4 | 5.b | even | 2 | 1 | ||
| 25.30.b.b | 8 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 15600T_{2}^{3} - 1488850608T_{2}^{2} - 10172141670400T_{2} + 185821106961186816 \)
acting on \(S_{30}^{\mathrm{new}}(\Gamma_0(5))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + \cdots + 18\!\cdots\!16 \)
|
| $3$ |
\( T^{4} + \cdots - 36\!\cdots\!24 \)
|
| $5$ |
\( (T + 6103515625)^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 81\!\cdots\!96 \)
|
| $11$ |
\( T^{4} + \cdots - 17\!\cdots\!04 \)
|
| $13$ |
\( T^{4} + \cdots + 78\!\cdots\!56 \)
|
| $17$ |
\( T^{4} + \cdots + 96\!\cdots\!56 \)
|
| $19$ |
\( T^{4} + \cdots + 18\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 31\!\cdots\!36 \)
|
| $29$ |
\( T^{4} + \cdots - 12\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots + 36\!\cdots\!56 \)
|
| $37$ |
\( T^{4} + \cdots + 26\!\cdots\!76 \)
|
| $41$ |
\( T^{4} + \cdots + 15\!\cdots\!36 \)
|
| $43$ |
\( T^{4} + \cdots - 10\!\cdots\!04 \)
|
| $47$ |
\( T^{4} + \cdots - 11\!\cdots\!64 \)
|
| $53$ |
\( T^{4} + \cdots - 25\!\cdots\!24 \)
|
| $59$ |
\( T^{4} + \cdots + 95\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 32\!\cdots\!96 \)
|
| $67$ |
\( T^{4} + \cdots + 15\!\cdots\!56 \)
|
| $71$ |
\( T^{4} + \cdots + 66\!\cdots\!76 \)
|
| $73$ |
\( T^{4} + \cdots - 10\!\cdots\!64 \)
|
| $79$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 30\!\cdots\!16 \)
|
| $89$ |
\( T^{4} + \cdots + 92\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 83\!\cdots\!36 \)
|
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