Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [46,14,Mod(1,46)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(46, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("46.1");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 46 = 2 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 46.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: | \(49.3262273179\) |
| 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: |
\( x^{5} - x^{4} - 3257825x^{3} + 1372618617x^{2} + 279354108456x - 130815431589168 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 59 \) |
| 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
\( x^{5} - x^{4} - 3257825x^{3} + 1372618617x^{2} + 279354108456x - 130815431589168 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -12481\nu^{4} - 12829265\nu^{3} + 32956482977\nu^{2} + 17271508397517\nu - 3307289103907668 ) / 4598290128456 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 259893 \nu^{4} - 582275233 \nu^{3} - 659872651633 \nu^{2} + \cdots - 66\!\cdots\!20 ) / 16860397137672 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 3998666 \nu^{4} - 1423823965 \nu^{3} + 12709676152996 \nu^{2} + \cdots - 16\!\cdots\!06 ) / 12645297853254 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 23195195 \nu^{4} - 8660318164 \nu^{3} + 72151790122993 \nu^{2} + \cdots - 78\!\cdots\!20 ) / 25290595706508 \)
|
| \(\nu\) | \(=\) |
\( ( 11\beta_{4} - 28\beta_{3} + 13\beta_{2} - 381\beta _1 + 385 ) / 1888 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -5815\beta_{4} + 20144\beta_{3} + 15655\beta_{2} - 293031\beta _1 + 1230145423 ) / 944 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 38119673\beta_{4} - 96700036\beta_{3} + 8974447\beta_{2} - 1563865983\beta _1 - 1551179111813 ) / 1888 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 6833834341 \beta_{4} + 20879125655 \beta_{3} + 11429988907 \beta_{2} - 145354014783 \beta _1 + 948897413820376 ) / 236 \)
|
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 | −1710.21 | 4096.00 | −66208.2 | 109454. | −138269. | −262144. | 1.33050e6 | 4.23733e6 | |||||||||||||||||||||||||||||||||
| 1.2 | −64.0000 | −1136.86 | 4096.00 | 57154.9 | 72759.2 | −17496.6 | −262144. | −301866. | −3.65791e6 | ||||||||||||||||||||||||||||||||||
| 1.3 | −64.0000 | −115.417 | 4096.00 | 8488.20 | 7386.68 | 418547. | −262144. | −1.58100e6 | −543245. | ||||||||||||||||||||||||||||||||||
| 1.4 | −64.0000 | 1281.56 | 4096.00 | −32225.1 | −82019.7 | 32545.6 | −262144. | 48065.7 | 2.06240e6 | ||||||||||||||||||||||||||||||||||
| 1.5 | −64.0000 | 2370.93 | 4096.00 | −39007.8 | −151740. | −263961. | −262144. | 4.02700e6 | 2.49650e6 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(23\) | \( +1 \) |
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 | 46.14.a.a | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 46.14.a.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 690T_{3}^{4} - 5509108T_{3}^{3} + 924234162T_{3}^{2} + 6086476932219T_{3} + 681843043869048 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(46))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 64)^{5} \)
|
| $3$ |
\( T^{5} + \cdots + 681843043869048 \)
|
| $5$ |
\( T^{5} + \cdots + 40\!\cdots\!00 \)
|
| $7$ |
\( T^{5} + \cdots + 86\!\cdots\!24 \)
|
| $11$ |
\( T^{5} + \cdots - 25\!\cdots\!40 \)
|
| $13$ |
\( T^{5} + \cdots - 14\!\cdots\!74 \)
|
| $17$ |
\( T^{5} + \cdots + 60\!\cdots\!20 \)
|
| $19$ |
\( T^{5} + \cdots - 34\!\cdots\!12 \)
|
| $23$ |
\( (T + 148035889)^{5} \)
|
| $29$ |
\( T^{5} + \cdots + 10\!\cdots\!30 \)
|
| $31$ |
\( T^{5} + \cdots - 15\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots - 58\!\cdots\!28 \)
|
| $41$ |
\( T^{5} + \cdots - 52\!\cdots\!18 \)
|
| $43$ |
\( T^{5} + \cdots + 16\!\cdots\!00 \)
|
| $47$ |
\( T^{5} + \cdots + 13\!\cdots\!72 \)
|
| $53$ |
\( T^{5} + \cdots - 27\!\cdots\!08 \)
|
| $59$ |
\( T^{5} + \cdots - 15\!\cdots\!40 \)
|
| $61$ |
\( T^{5} + \cdots - 20\!\cdots\!52 \)
|
| $67$ |
\( T^{5} + \cdots - 32\!\cdots\!80 \)
|
| $71$ |
\( T^{5} + \cdots - 92\!\cdots\!36 \)
|
| $73$ |
\( T^{5} + \cdots + 63\!\cdots\!46 \)
|
| $79$ |
\( T^{5} + \cdots + 51\!\cdots\!16 \)
|
| $83$ |
\( T^{5} + \cdots + 86\!\cdots\!76 \)
|
| $89$ |
\( T^{5} + \cdots + 94\!\cdots\!04 \)
|
| $97$ |
\( T^{5} + \cdots + 32\!\cdots\!88 \)
|
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