Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [48,22,Mod(47,48)]
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([1, 0, 1]))
N = Newforms(chi, 22, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("48.47");
S:= CuspForms(chi, 22);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 48.c (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: | \(134.149125258\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 6 x^{11} + \cdots + 19\!\cdots\!25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{68}\cdot 3^{56}\cdot 7^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 6 x^{11} + \cdots + 19\!\cdots\!25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 61\!\cdots\!80 \nu^{11} + \cdots - 36\!\cdots\!05 ) / 48\!\cdots\!75 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 13\!\cdots\!60 \nu^{11} + \cdots - 12\!\cdots\!70 ) / 48\!\cdots\!75 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 14\!\cdots\!00 \nu^{11} + \cdots + 76\!\cdots\!45 ) / 48\!\cdots\!75 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 90\!\cdots\!64 \nu^{11} + \cdots - 18\!\cdots\!78 ) / 19\!\cdots\!39 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 47\!\cdots\!04 \nu^{11} + \cdots + 15\!\cdots\!00 ) / 88\!\cdots\!85 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 76\!\cdots\!00 \nu^{11} + \cdots + 15\!\cdots\!05 ) / 48\!\cdots\!75 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 22\!\cdots\!44 \nu^{11} + \cdots + 64\!\cdots\!10 ) / 96\!\cdots\!95 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 47\!\cdots\!32 \nu^{11} + \cdots - 25\!\cdots\!80 ) / 87\!\cdots\!25 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 47\!\cdots\!32 \nu^{11} + \cdots - 45\!\cdots\!20 ) / 87\!\cdots\!25 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 94\!\cdots\!44 \nu^{11} + \cdots - 80\!\cdots\!60 ) / 87\!\cdots\!25 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 21\!\cdots\!28 \nu^{11} + \cdots + 39\!\cdots\!80 ) / 79\!\cdots\!75 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{6} + \beta_{5} + 2\beta_{3} + 3727\beta _1 + 18 ) / 36 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 11262058 \beta_{10} - 32937612 \beta_{9} - 10413496 \beta_{8} + 1962 \beta_{7} + \cdots - 34\!\cdots\!64 ) / 648 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1488287001125 \beta_{11} + 232913784090159 \beta_{10} - 458407819047049 \beta_{9} + \cdots - 92\!\cdots\!44 ) / 11664 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 26789166020250 \beta_{11} + \cdots + 73\!\cdots\!84 ) / 104976 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 22\!\cdots\!65 \beta_{11} + \cdots + 12\!\cdots\!12 ) / 69984 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 67\!\cdots\!45 \beta_{11} + \cdots - 71\!\cdots\!64 ) / 69984 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 91\!\cdots\!80 \beta_{11} + \cdots - 49\!\cdots\!12 ) / 139968 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 54\!\cdots\!10 \beta_{11} + \cdots + 31\!\cdots\!04 ) / 209952 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 11\!\cdots\!35 \beta_{11} + \cdots + 62\!\cdots\!64 ) / 93312 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 50\!\cdots\!55 \beta_{11} + \cdots - 18\!\cdots\!24 ) / 839808 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 12\!\cdots\!35 \beta_{11} + \cdots - 66\!\cdots\!88 ) / 559872 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 47.1 |
|
0 | −94418.3 | − | 39313.4i | 0 | − | 2.52523e7i | 0 | − | 1.86298e8i | 0 | 7.36926e9 | + | 7.42381e9i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 47.2 | 0 | −94418.3 | + | 39313.4i | 0 | 2.52523e7i | 0 | 1.86298e8i | 0 | 7.36926e9 | − | 7.42381e9i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.3 | 0 | −81435.3 | − | 61876.0i | 0 | 1.06005e7i | 0 | 3.89377e8i | 0 | 2.80307e9 | + | 1.00778e10i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.4 | 0 | −81435.3 | + | 61876.0i | 0 | − | 1.06005e7i | 0 | − | 3.89377e8i | 0 | 2.80307e9 | − | 1.00778e10i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 47.5 | 0 | −30363.0 | − | 97665.0i | 0 | 3.55520e7i | 0 | − | 1.36883e9i | 0 | −8.61653e9 | + | 5.93080e9i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.6 | 0 | −30363.0 | + | 97665.0i | 0 | − | 3.55520e7i | 0 | 1.36883e9i | 0 | −8.61653e9 | − | 5.93080e9i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.7 | 0 | 30363.0 | − | 97665.0i | 0 | − | 3.55520e7i | 0 | − | 1.36883e9i | 0 | −8.61653e9 | − | 5.93080e9i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 47.8 | 0 | 30363.0 | + | 97665.0i | 0 | 3.55520e7i | 0 | 1.36883e9i | 0 | −8.61653e9 | + | 5.93080e9i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.9 | 0 | 81435.3 | − | 61876.0i | 0 | − | 1.06005e7i | 0 | 3.89377e8i | 0 | 2.80307e9 | − | 1.00778e10i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.10 | 0 | 81435.3 | + | 61876.0i | 0 | 1.06005e7i | 0 | − | 3.89377e8i | 0 | 2.80307e9 | + | 1.00778e10i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.11 | 0 | 94418.3 | − | 39313.4i | 0 | 2.52523e7i | 0 | − | 1.86298e8i | 0 | 7.36926e9 | − | 7.42381e9i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.12 | 0 | 94418.3 | + | 39313.4i | 0 | − | 2.52523e7i | 0 | 1.86298e8i | 0 | 7.36926e9 | + | 7.42381e9i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 4.b | odd | 2 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 48.22.c.b | ✓ | 12 |
| 3.b | odd | 2 | 1 | inner | 48.22.c.b | ✓ | 12 |
| 4.b | odd | 2 | 1 | inner | 48.22.c.b | ✓ | 12 |
| 12.b | even | 2 | 1 | inner | 48.22.c.b | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 48.22.c.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 48.22.c.b | ✓ | 12 | 3.b | odd | 2 | 1 | inner |
| 48.22.c.b | ✓ | 12 | 4.b | odd | 2 | 1 | inner |
| 48.22.c.b | ✓ | 12 | 12.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + \cdots + 90\!\cdots\!00 \)
acting on \(S_{22}^{\mathrm{new}}(48, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + \cdots + 13\!\cdots\!29 \)
|
| $5$ |
\( (T^{6} + \cdots + 90\!\cdots\!00)^{2} \)
|
| $7$ |
\( (T^{6} + \cdots + 98\!\cdots\!68)^{2} \)
|
| $11$ |
\( (T^{6} + \cdots - 53\!\cdots\!00)^{2} \)
|
| $13$ |
\( (T^{3} + \cdots - 32\!\cdots\!40)^{4} \)
|
| $17$ |
\( (T^{6} + \cdots + 75\!\cdots\!00)^{2} \)
|
| $19$ |
\( (T^{6} + \cdots + 41\!\cdots\!00)^{2} \)
|
| $23$ |
\( (T^{6} + \cdots - 34\!\cdots\!80)^{2} \)
|
| $29$ |
\( (T^{6} + \cdots + 78\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots + 68\!\cdots\!20)^{4} \)
|
| $41$ |
\( (T^{6} + \cdots + 36\!\cdots\!40)^{2} \)
|
| $43$ |
\( (T^{6} + \cdots + 36\!\cdots\!92)^{2} \)
|
| $47$ |
\( (T^{6} + \cdots - 34\!\cdots\!00)^{2} \)
|
| $53$ |
\( (T^{6} + \cdots + 21\!\cdots\!00)^{2} \)
|
| $59$ |
\( (T^{6} + \cdots - 11\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots - 67\!\cdots\!68)^{4} \)
|
| $67$ |
\( (T^{6} + \cdots + 53\!\cdots\!28)^{2} \)
|
| $71$ |
\( (T^{6} + \cdots - 84\!\cdots\!00)^{2} \)
|
| $73$ |
\( (T^{3} + \cdots - 44\!\cdots\!80)^{4} \)
|
| $79$ |
\( (T^{6} + \cdots + 15\!\cdots\!00)^{2} \)
|
| $83$ |
\( (T^{6} + \cdots - 96\!\cdots\!80)^{2} \)
|
| $89$ |
\( (T^{6} + \cdots + 34\!\cdots\!40)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots - 19\!\cdots\!00)^{4} \)
|
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