Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [169,10,Mod(1,169)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(169, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("169.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 169 = 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 169.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: | \(87.0410563117\) |
| 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: |
\( x^{4} - x^{3} - 1602x^{2} + 1544x + 342272 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 13) |
| 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
\( x^{4} - x^{3} - 1602x^{2} + 1544x + 342272 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{3} + 17\nu^{2} - 3586\nu - 12856 ) / 332 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 105\nu^{2} + 1306\nu - 84248 ) / 664 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 6\beta_{3} + \beta_{2} - \beta _1 + 800 \)
|
| \(\nu^{3}\) | \(=\) |
\( -34\beta_{3} + 105\beta_{2} + 1201\beta _1 - 248 \)
|
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 |
|
−28.8028 | −204.594 | 317.603 | 258.914 | 5892.88 | 8862.28 | 5599.18 | 22175.6 | −7457.46 | ||||||||||||||||||||||||||||||
| 1.2 | −7.35673 | 42.6243 | −457.879 | 1236.25 | −313.575 | −892.010 | 7135.13 | −17866.2 | −9094.74 | |||||||||||||||||||||||||||||||
| 1.3 | 24.5360 | 49.9972 | 90.0171 | −1814.98 | 1226.73 | 8707.31 | −10353.8 | −17183.3 | −44532.3 | |||||||||||||||||||||||||||||||
| 1.4 | 44.6235 | −51.0278 | 1479.26 | −151.187 | −2277.04 | −5436.58 | 43162.5 | −17079.2 | −6746.48 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(13\) | \( +1 \) |
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 | 169.10.a.a | 4 | |
| 13.b | even | 2 | 1 | 13.10.a.a | ✓ | 4 | |
| 39.d | odd | 2 | 1 | 117.10.a.c | 4 | ||
| 52.b | odd | 2 | 1 | 208.10.a.g | 4 | ||
| 65.d | even | 2 | 1 | 325.10.a.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.10.a.a | ✓ | 4 | 13.b | even | 2 | 1 | |
| 117.10.a.c | 4 | 39.d | odd | 2 | 1 | ||
| 169.10.a.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 208.10.a.g | 4 | 52.b | odd | 2 | 1 | ||
| 325.10.a.a | 4 | 65.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 33T_{2}^{3} - 1194T_{2}^{2} + 24936T_{2} + 232000 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(169))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 33 T^{3} + \cdots + 232000 \)
|
| $3$ |
\( T^{4} + 163 T^{3} + \cdots + 22248576 \)
|
| $5$ |
\( T^{4} + \cdots + 87830562190 \)
|
| $7$ |
\( T^{4} + \cdots + 374218195104754 \)
|
| $11$ |
\( T^{4} + \cdots + 27\!\cdots\!36 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} + \cdots + 25\!\cdots\!18 \)
|
| $19$ |
\( T^{4} + \cdots + 50\!\cdots\!08 \)
|
| $23$ |
\( T^{4} + \cdots - 17\!\cdots\!36 \)
|
| $29$ |
\( T^{4} + \cdots + 37\!\cdots\!32 \)
|
| $31$ |
\( T^{4} + \cdots - 58\!\cdots\!60 \)
|
| $37$ |
\( T^{4} + \cdots + 61\!\cdots\!62 \)
|
| $41$ |
\( T^{4} + \cdots + 15\!\cdots\!52 \)
|
| $43$ |
\( T^{4} + \cdots + 26\!\cdots\!40 \)
|
| $47$ |
\( T^{4} + \cdots + 10\!\cdots\!50 \)
|
| $53$ |
\( T^{4} + \cdots - 27\!\cdots\!48 \)
|
| $59$ |
\( T^{4} + \cdots - 26\!\cdots\!68 \)
|
| $61$ |
\( T^{4} + \cdots + 60\!\cdots\!72 \)
|
| $67$ |
\( T^{4} + \cdots + 19\!\cdots\!64 \)
|
| $71$ |
\( T^{4} + \cdots + 45\!\cdots\!54 \)
|
| $73$ |
\( T^{4} + \cdots - 47\!\cdots\!72 \)
|
| $79$ |
\( T^{4} + \cdots + 25\!\cdots\!64 \)
|
| $83$ |
\( T^{4} + \cdots - 12\!\cdots\!88 \)
|
| $89$ |
\( T^{4} + \cdots - 34\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 21\!\cdots\!60 \)
|
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