Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1900,4,Mod(1749,1900)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1900, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1900.1749");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1900.c (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: | \(112.103629011\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{33})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 17x^{2} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 5^{2} \) |
| Twist minimal: | no (minimal twist has level 76) |
| 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
\( x^{4} + 17x^{2} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} - 5\nu ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{3} - 35\nu ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( 5\nu^{2} + 43 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{2} + 3\beta_1 ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 43 ) / 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} - 7\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).
| \(n\) | \(77\) | \(401\) | \(951\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1749.1 |
|
0 | − | 5.37228i | 0 | 0 | 0 | 26.4891i | 0 | −1.86141 | 0 | |||||||||||||||||||||||||||||
| 1749.2 | 0 | − | 0.372281i | 0 | 0 | 0 | − | 3.51087i | 0 | 26.8614 | 0 | |||||||||||||||||||||||||||||
| 1749.3 | 0 | 0.372281i | 0 | 0 | 0 | 3.51087i | 0 | 26.8614 | 0 | |||||||||||||||||||||||||||||||
| 1749.4 | 0 | 5.37228i | 0 | 0 | 0 | − | 26.4891i | 0 | −1.86141 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1900.4.c.b | 4 | |
| 5.b | even | 2 | 1 | inner | 1900.4.c.b | 4 | |
| 5.c | odd | 4 | 1 | 76.4.a.a | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 1900.4.a.b | 2 | ||
| 15.e | even | 4 | 1 | 684.4.a.g | 2 | ||
| 20.e | even | 4 | 1 | 304.4.a.f | 2 | ||
| 40.i | odd | 4 | 1 | 1216.4.a.o | 2 | ||
| 40.k | even | 4 | 1 | 1216.4.a.h | 2 | ||
| 95.g | even | 4 | 1 | 1444.4.a.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 76.4.a.a | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 304.4.a.f | 2 | 20.e | even | 4 | 1 | ||
| 684.4.a.g | 2 | 15.e | even | 4 | 1 | ||
| 1216.4.a.h | 2 | 40.k | even | 4 | 1 | ||
| 1216.4.a.o | 2 | 40.i | odd | 4 | 1 | ||
| 1444.4.a.d | 2 | 95.g | even | 4 | 1 | ||
| 1900.4.a.b | 2 | 5.c | odd | 4 | 1 | ||
| 1900.4.c.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 1900.4.c.b | 4 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 29T_{3}^{2} + 4 \)
acting on \(S_{4}^{\mathrm{new}}(1900, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 29T^{2} + 4 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 714T^{2} + 8649 \)
|
| $11$ |
\( (T^{2} + 71 T + 1054)^{2} \)
|
| $13$ |
\( T^{4} + 2609 T^{2} + 478864 \)
|
| $17$ |
\( (T^{2} + 297)^{2} \)
|
| $19$ |
\( (T - 19)^{4} \)
|
| $23$ |
\( T^{4} + 57449 T^{2} + 824378944 \)
|
| $29$ |
\( (T^{2} + 155 T - 28850)^{2} \)
|
| $31$ |
\( (T^{2} + 88 T - 50864)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 1265367184 \)
|
| $41$ |
\( (T^{2} + 142 T - 2384)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 11433597184 \)
|
| $47$ |
\( T^{4} + \cdots + 2246001664 \)
|
| $53$ |
\( T^{4} + \cdots + 11216504464 \)
|
| $59$ |
\( (T^{2} - 873 T + 180426)^{2} \)
|
| $61$ |
\( (T^{2} - 445 T - 203150)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 5374062864 \)
|
| $71$ |
\( (T^{2} + 1712 T + 729436)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 44619380289 \)
|
| $79$ |
\( (T^{2} - 1274 T + 404944)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 11065356864 \)
|
| $89$ |
\( (T^{2} - 888 T + 144336)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 2574510102784 \)
|
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