Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [148,4,Mod(73,148)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(148, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("148.73");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 148 = 2^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 148.d (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: | \(8.73228268085\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 114x^{6} + 3352x^{4} + 27006x^{2} + 343 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{2}\cdot 7 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 114x^{6} + 3352x^{4} + 27006x^{2} + 343 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 93\nu^{4} + 1399\nu^{2} - 2373 ) / 896 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + 93\nu^{4} + 1847\nu^{2} + 10619 ) / 896 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{6} - 521\nu^{4} - 11923\nu^{2} - 39151 ) / 2688 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -11\nu^{7} - 1247\nu^{5} - 36445\nu^{3} - 300265\nu ) / 2688 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5\nu^{7} + 605\nu^{5} + 19987\nu^{3} + 186235\nu ) / 1344 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 19\nu^{7} + 2159\nu^{5} + 62869\nu^{3} + 495705\nu ) / 896 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{3} - 2\beta_{2} - 29 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -10\beta_{7} + 2\beta_{6} - 50\beta_{5} - 55\beta_1 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( -48\beta_{4} - 176\beta_{3} + 96\beta_{2} + 1641 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 896\beta_{7} + 32\beta_{6} + 4672\beta_{5} + 3625\beta_1 ) / 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( 4464\beta_{4} + 13570\beta_{3} - 5234\beta_{2} - 109669 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -68442\beta_{7} - 10254\beta_{6} - 365442\beta_{5} - 256015\beta_1 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/148\mathbb{Z}\right)^\times\).
| \(n\) | \(75\) | \(113\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 73.1 |
|
0 | −9.11907 | 0 | − | 18.9371i | 0 | −0.695173 | 0 | 56.1574 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 73.2 | 0 | −9.11907 | 0 | 18.9371i | 0 | −0.695173 | 0 | 56.1574 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 73.3 | 0 | −1.07092 | 0 | − | 9.73153i | 0 | 23.8403 | 0 | −25.8531 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 73.4 | 0 | −1.07092 | 0 | 9.73153i | 0 | 23.8403 | 0 | −25.8531 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 73.5 | 0 | 2.66828 | 0 | − | 12.5795i | 0 | −29.0023 | 0 | −19.8803 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 73.6 | 0 | 2.66828 | 0 | 12.5795i | 0 | −29.0023 | 0 | −19.8803 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 73.7 | 0 | 7.52170 | 0 | − | 10.9289i | 0 | 6.85725 | 0 | 29.5760 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 73.8 | 0 | 7.52170 | 0 | 10.9289i | 0 | 6.85725 | 0 | 29.5760 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 37.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 148.4.d.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1332.4.e.c | 8 | ||
| 4.b | odd | 2 | 1 | 592.4.g.b | 8 | ||
| 37.b | even | 2 | 1 | inner | 148.4.d.b | ✓ | 8 |
| 111.d | odd | 2 | 1 | 1332.4.e.c | 8 | ||
| 148.b | odd | 2 | 1 | 592.4.g.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 148.4.d.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 148.4.d.b | ✓ | 8 | 37.b | even | 2 | 1 | inner |
| 592.4.g.b | 8 | 4.b | odd | 2 | 1 | ||
| 592.4.g.b | 8 | 148.b | odd | 2 | 1 | ||
| 1332.4.e.c | 8 | 3.b | odd | 2 | 1 | ||
| 1332.4.e.c | 8 | 111.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 74T_{3}^{2} + 105T_{3} + 196 \)
acting on \(S_{4}^{\mathrm{new}}(148, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} - 74 T^{2} + \cdots + 196)^{2} \)
|
| $5$ |
\( T^{8} + 731 T^{6} + \cdots + 641898432 \)
|
| $7$ |
\( (T^{4} - T^{3} - 728 T^{2} + \cdots + 3296)^{2} \)
|
| $11$ |
\( (T^{4} + 82 T^{3} + \cdots - 481188)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 1221991468992 \)
|
| $17$ |
\( T^{8} + \cdots + 132244192346112 \)
|
| $19$ |
\( T^{8} + \cdots + 43\!\cdots\!32 \)
|
| $23$ |
\( T^{8} + \cdots + 2258431616448 \)
|
| $29$ |
\( T^{8} + \cdots + 35\!\cdots\!68 \)
|
| $31$ |
\( T^{8} + \cdots + 274815431589888 \)
|
| $37$ |
\( T^{8} + \cdots + 65\!\cdots\!81 \)
|
| $41$ |
\( (T^{4} - 238 T^{3} + \cdots - 549833802)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 19\!\cdots\!32 \)
|
| $47$ |
\( (T^{4} - 119 T^{3} + \cdots + 2276613696)^{2} \)
|
| $53$ |
\( (T^{4} - 625 T^{3} + \cdots - 262619400)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 49\!\cdots\!32 \)
|
| $61$ |
\( T^{8} + \cdots + 30\!\cdots\!72 \)
|
| $67$ |
\( (T^{4} - 303 T^{3} + \cdots + 1950735952)^{2} \)
|
| $71$ |
\( (T^{4} + 1509 T^{3} + \cdots - 75685992864)^{2} \)
|
| $73$ |
\( (T^{4} + 728 T^{3} + \cdots - 33987851674)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 13\!\cdots\!48 \)
|
| $83$ |
\( (T^{4} + 1197 T^{3} + \cdots - 3332116032)^{2} \)
|
| $89$ |
\( T^{8} + \cdots + 11\!\cdots\!32 \)
|
| $97$ |
\( T^{8} + \cdots + 22\!\cdots\!72 \)
|
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