Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [23,5,Mod(22,23)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(23, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("23.22");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 23 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 23.b (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(2.37750915093\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{4} + 426x^{2} + 35283 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 426x^{2} + 35283 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 213\nu ) / 41 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 459\nu ) / 41 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{2} + 213 ) / 41 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( 41\beta_{3} - 213 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -71\beta_{2} + 153\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 22.1 |
|
−4.44949 | 4.34847 | 3.79796 | − | 25.9890i | −19.3485 | − | 89.6486i | 54.2929 | −62.0908 | 115.638i | ||||||||||||||||||||||||||||
| 22.2 | −4.44949 | 4.34847 | 3.79796 | 25.9890i | −19.3485 | 89.6486i | 54.2929 | −62.0908 | − | 115.638i | ||||||||||||||||||||||||||||||
| 22.3 | 0.449490 | −10.3485 | −15.7980 | − | 43.3656i | −4.65153 | 62.8580i | −14.2929 | 26.0908 | − | 19.4924i | |||||||||||||||||||||||||||||
| 22.4 | 0.449490 | −10.3485 | −15.7980 | 43.3656i | −4.65153 | − | 62.8580i | −14.2929 | 26.0908 | 19.4924i | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 23.5.b.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 207.5.d.b | 4 | ||
| 4.b | odd | 2 | 1 | 368.5.f.b | 4 | ||
| 23.b | odd | 2 | 1 | inner | 23.5.b.b | ✓ | 4 |
| 69.c | even | 2 | 1 | 207.5.d.b | 4 | ||
| 92.b | even | 2 | 1 | 368.5.f.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 23.5.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 23.5.b.b | ✓ | 4 | 23.b | odd | 2 | 1 | inner |
| 207.5.d.b | 4 | 3.b | odd | 2 | 1 | ||
| 207.5.d.b | 4 | 69.c | even | 2 | 1 | ||
| 368.5.f.b | 4 | 4.b | odd | 2 | 1 | ||
| 368.5.f.b | 4 | 92.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 4T_{2} - 2 \)
acting on \(S_{5}^{\mathrm{new}}(23, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4 T - 2)^{2} \)
|
| $3$ |
\( (T^{2} + 6 T - 45)^{2} \)
|
| $5$ |
\( T^{4} + 2556 T^{2} + 1270188 \)
|
| $7$ |
\( T^{4} + 11988 T^{2} + 31754700 \)
|
| $11$ |
\( T^{4} + 11988 T^{2} + 31754700 \)
|
| $13$ |
\( (T^{2} + 2 T - 9599)^{2} \)
|
| $17$ |
\( T^{4} + 119772 T^{2} + 285792300 \)
|
| $19$ |
\( T^{4} + \cdots + 18290707200 \)
|
| $23$ |
\( T^{4} + \cdots + 78310985281 \)
|
| $29$ |
\( (T^{2} - 530 T + 25849)^{2} \)
|
| $31$ |
\( (T^{2} - 454 T - 922925)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 4755869664300 \)
|
| $41$ |
\( (T^{2} - 386 T - 1804247)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 10682281080000 \)
|
| $47$ |
\( (T^{2} - 6302 T + 9896827)^{2} \)
|
| $53$ |
\( T^{4} + \cdots + 114316920000 \)
|
| $59$ |
\( (T^{2} + 1108 T - 3287540)^{2} \)
|
| $61$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots + 263203308362700 \)
|
| $71$ |
\( (T^{2} + 10294 T + 16021363)^{2} \)
|
| $73$ |
\( (T^{2} - 3958 T - 3343559)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 19\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 179116354687500 \)
|
| $89$ |
\( T^{4} + \cdots + 60473650680000 \)
|
| $97$ |
\( T^{4} + \cdots + 38\!\cdots\!00 \)
|
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