Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [48,23,Mod(17,48)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(48, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 23, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("48.17");
S:= CuspForms(chi, 23);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 23 \) |
| Character orbit: | \([\chi]\) | \(=\) | 48.e (of order \(2\), degree \(1\), not 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: | \(147.219568724\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 126474x^{4} + 3861674040x^{2} + 9831214131200 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{33}\cdot 3^{22} \) |
| Twist minimal: | no (minimal twist has level 3) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 126474x^{4} + 3861674040x^{2} + 9831214131200 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 65\nu^{5} + 12256\nu^{4} + 8098250\nu^{3} + 909321664\nu^{2} + 602488341560\nu + 4539979294720 ) / 61954080 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 65\nu^{5} + 12256\nu^{4} + 8098250\nu^{3} + 909321664\nu^{2} + 221842474040\nu + 4539979294720 ) / 61954080 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 1369\nu^{5} + 6128\nu^{4} + 101935546\nu^{3} + 454660832\nu^{2} + 307543517560\nu + 2269989647360 ) / 5162840 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1391 \nu^{5} + 134816 \nu^{4} - 173302550 \nu^{3} + 10002538304 \nu^{2} + \cdots + 49939772241920 ) / 12390816 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1105 \nu^{5} + 208352 \nu^{4} + 137670250 \nu^{3} + 39248835008 \nu^{2} + 3898204014520 \nu + 10\!\cdots\!00 ) / 20651360 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 6144 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 50\beta_{2} - \beta _1 - 48566016 ) / 1152 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -792\beta_{4} - 520\beta_{3} + 119817\beta_{2} - 73137\beta_1 ) / 6912 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -111291\beta_{5} + 53920\beta_{4} + 11226150\beta_{2} + 219131\beta _1 + 4764858967296 ) / 1728 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 58006768\beta_{4} + 64786000\beta_{3} - 8770198683\beta_{2} + 5191110443\beta_1 ) / 6912 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).
| \(n\) | \(17\) | \(31\) | \(37\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
0 | −169408. | − | 51788.6i | 0 | − | 6.36810e7i | 0 | 2.60308e9 | 0 | 2.60170e10 | + | 1.75468e10i | 0 | |||||||||||||||||||||||||||||||
| 17.2 | 0 | −169408. | + | 51788.6i | 0 | 6.36810e7i | 0 | 2.60308e9 | 0 | 2.60170e10 | − | 1.75468e10i | 0 | |||||||||||||||||||||||||||||||||
| 17.3 | 0 | −48229.8 | − | 170455.i | 0 | 2.56634e6i | 0 | −8.97619e8 | 0 | −2.67288e10 | + | 1.64420e10i | 0 | |||||||||||||||||||||||||||||||||
| 17.4 | 0 | −48229.8 | + | 170455.i | 0 | − | 2.56634e6i | 0 | −8.97619e8 | 0 | −2.67288e10 | − | 1.64420e10i | 0 | ||||||||||||||||||||||||||||||||
| 17.5 | 0 | 174303. | − | 31617.4i | 0 | 4.89124e7i | 0 | 1.80726e7 | 0 | 2.93817e10 | − | 1.10220e10i | 0 | |||||||||||||||||||||||||||||||||
| 17.6 | 0 | 174303. | + | 31617.4i | 0 | − | 4.89124e7i | 0 | 1.80726e7 | 0 | 2.93817e10 | + | 1.10220e10i | 0 | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 48.23.e.b | 6 | |
| 3.b | odd | 2 | 1 | inner | 48.23.e.b | 6 | |
| 4.b | odd | 2 | 1 | 3.23.b.a | ✓ | 6 | |
| 12.b | even | 2 | 1 | 3.23.b.a | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.23.b.a | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 3.23.b.a | ✓ | 6 | 12.b | even | 2 | 1 | |
| 48.23.e.b | 6 | 1.a | even | 1 | 1 | trivial | |
| 48.23.e.b | 6 | 3.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + \cdots + 63\!\cdots\!00 \)
acting on \(S_{23}^{\mathrm{new}}(48, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots + 30\!\cdots\!29 \)
|
| $5$ |
\( T^{6} + \cdots + 63\!\cdots\!00 \)
|
| $7$ |
\( (T^{3} + \cdots + 42\!\cdots\!00)^{2} \)
|
| $11$ |
\( T^{6} + \cdots + 20\!\cdots\!00 \)
|
| $13$ |
\( (T^{3} + \cdots - 23\!\cdots\!80)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 63\!\cdots\!00 \)
|
| $19$ |
\( (T^{3} + \cdots - 73\!\cdots\!92)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 20\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots + 15\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots + 66\!\cdots\!68)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots + 17\!\cdots\!20)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 35\!\cdots\!00 \)
|
| $43$ |
\( (T^{3} + \cdots - 10\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 53\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots + 46\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots + 15\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} + \cdots + 56\!\cdots\!32)^{2} \)
|
| $67$ |
\( (T^{3} + \cdots + 38\!\cdots\!60)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 21\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots + 76\!\cdots\!80)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots - 16\!\cdots\!72)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $89$ |
\( T^{6} + \cdots + 21\!\cdots\!00 \)
|
| $97$ |
\( (T^{3} + \cdots - 29\!\cdots\!40)^{2} \)
|
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