Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,14,Mod(49,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.49");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.b (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: | \(80.4231967139\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{3121})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1561x^{2} + 608400 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | no (minimal twist has level 15) |
| 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
\( x^{4} + 1561x^{2} + 608400 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 781\nu ) / 780 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 3121\nu ) / 780 \)
|
| \(\beta_{3}\) | \(=\) |
\( 3\nu^{2} + 2342 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 2342 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -781\beta_{2} + 3121\beta_1 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
| \(n\) | \(26\) | \(52\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
− | 149.299i | 729.000i | −14098.2 | 0 | 108839. | − | 384225.i | 881782.i | −531441. | 0 | ||||||||||||||||||||||||||||
| 49.2 | − | 18.2989i | − | 729.000i | 7857.15 | 0 | −13339.9 | 112047.i | − | 293681.i | −531441. | 0 | ||||||||||||||||||||||||||||
| 49.3 | 18.2989i | 729.000i | 7857.15 | 0 | −13339.9 | − | 112047.i | 293681.i | −531441. | 0 | ||||||||||||||||||||||||||||||
| 49.4 | 149.299i | − | 729.000i | −14098.2 | 0 | 108839. | 384225.i | − | 881782.i | −531441. | 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 | 75.14.b.e | 4 | |
| 5.b | even | 2 | 1 | inner | 75.14.b.e | 4 | |
| 5.c | odd | 4 | 1 | 15.14.a.c | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 75.14.a.c | 2 | ||
| 15.e | even | 4 | 1 | 45.14.a.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.14.a.c | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 45.14.a.b | 2 | 15.e | even | 4 | 1 | ||
| 75.14.a.c | 2 | 5.c | odd | 4 | 1 | ||
| 75.14.b.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 75.14.b.e | 4 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 22625T_{2}^{2} + 7463824 \)
acting on \(S_{14}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 22625 T^{2} + 7463824 \)
|
| $3$ |
\( (T^{2} + 531441)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 18\!\cdots\!00 \)
|
| $11$ |
\( (T^{2} - 6245888 T + 877043529472)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 50\!\cdots\!84 \)
|
| $17$ |
\( T^{4} + \cdots + 41\!\cdots\!16 \)
|
| $19$ |
\( (T^{2} + \cdots - 12\!\cdots\!20)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 15\!\cdots\!00 \)
|
| $29$ |
\( (T^{2} + \cdots - 22\!\cdots\!40)^{2} \)
|
| $31$ |
\( (T^{2} + \cdots - 12\!\cdots\!00)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 27\!\cdots\!00 \)
|
| $41$ |
\( (T^{2} + \cdots + 17\!\cdots\!80)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 18\!\cdots\!76 \)
|
| $47$ |
\( T^{4} + \cdots + 14\!\cdots\!36 \)
|
| $53$ |
\( T^{4} + \cdots + 17\!\cdots\!00 \)
|
| $59$ |
\( (T^{2} + \cdots + 12\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots - 71\!\cdots\!24)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 34\!\cdots\!96 \)
|
| $71$ |
\( (T^{2} + \cdots - 45\!\cdots\!56)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 26\!\cdots\!00 \)
|
| $79$ |
\( (T^{2} + \cdots - 18\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 16\!\cdots\!24 \)
|
| $89$ |
\( (T^{2} + \cdots - 66\!\cdots\!20)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 12\!\cdots\!56 \)
|
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