Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,13,Mod(74,75)]
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([1, 1]))
N = Newforms(chi, 13, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.74");
S:= CuspForms(chi, 13);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 13 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.d (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: | \(68.5495362957\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{26})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 169 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{4}\cdot 5^{4} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 169 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 25\nu^{2} ) / 13 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -18\nu^{3} + 234\nu ) / 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 450\nu^{3} + 5850\nu ) / 13 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 25\beta_{2} ) / 900 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 13\beta_1 ) / 25 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 13\beta_{3} - 325\beta_{2} ) / 900 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 74.1 |
|
−91.7824 | −275.347 | − | 675.000i | 4328.00 | 0 | 25272.0 | + | 61953.1i | − | 40250.0i | −21293.5 | −379809. | + | 371719.i | 0 | |||||||||||||||||||||||
| 74.2 | −91.7824 | −275.347 | + | 675.000i | 4328.00 | 0 | 25272.0 | − | 61953.1i | 40250.0i | −21293.5 | −379809. | − | 371719.i | 0 | |||||||||||||||||||||||||
| 74.3 | 91.7824 | 275.347 | − | 675.000i | 4328.00 | 0 | 25272.0 | − | 61953.1i | − | 40250.0i | 21293.5 | −379809. | − | 371719.i | 0 | ||||||||||||||||||||||||
| 74.4 | 91.7824 | 275.347 | + | 675.000i | 4328.00 | 0 | 25272.0 | + | 61953.1i | 40250.0i | 21293.5 | −379809. | + | 371719.i | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 15.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 75.13.d.b | 4 | |
| 3.b | odd | 2 | 1 | inner | 75.13.d.b | 4 | |
| 5.b | even | 2 | 1 | inner | 75.13.d.b | 4 | |
| 5.c | odd | 4 | 1 | 3.13.b.b | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 75.13.c.c | 2 | ||
| 15.d | odd | 2 | 1 | inner | 75.13.d.b | 4 | |
| 15.e | even | 4 | 1 | 3.13.b.b | ✓ | 2 | |
| 15.e | even | 4 | 1 | 75.13.c.c | 2 | ||
| 20.e | even | 4 | 1 | 48.13.e.b | 2 | ||
| 40.i | odd | 4 | 1 | 192.13.e.d | 2 | ||
| 40.k | even | 4 | 1 | 192.13.e.c | 2 | ||
| 45.k | odd | 12 | 2 | 81.13.d.c | 4 | ||
| 45.l | even | 12 | 2 | 81.13.d.c | 4 | ||
| 60.l | odd | 4 | 1 | 48.13.e.b | 2 | ||
| 120.q | odd | 4 | 1 | 192.13.e.c | 2 | ||
| 120.w | even | 4 | 1 | 192.13.e.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.13.b.b | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 3.13.b.b | ✓ | 2 | 15.e | even | 4 | 1 | |
| 48.13.e.b | 2 | 20.e | even | 4 | 1 | ||
| 48.13.e.b | 2 | 60.l | odd | 4 | 1 | ||
| 75.13.c.c | 2 | 5.c | odd | 4 | 1 | ||
| 75.13.c.c | 2 | 15.e | even | 4 | 1 | ||
| 75.13.d.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 75.13.d.b | 4 | 3.b | odd | 2 | 1 | inner | |
| 75.13.d.b | 4 | 5.b | even | 2 | 1 | inner | |
| 75.13.d.b | 4 | 15.d | odd | 2 | 1 | inner | |
| 81.13.d.c | 4 | 45.k | odd | 12 | 2 | ||
| 81.13.d.c | 4 | 45.l | even | 12 | 2 | ||
| 192.13.e.c | 2 | 40.k | even | 4 | 1 | ||
| 192.13.e.c | 2 | 120.q | odd | 4 | 1 | ||
| 192.13.e.d | 2 | 40.i | odd | 4 | 1 | ||
| 192.13.e.d | 2 | 120.w | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 8424 \)
acting on \(S_{13}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 8424)^{2} \)
|
| $3$ |
\( T^{4} + \cdots + 282429536481 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 1620062500)^{2} \)
|
| $11$ |
\( (T^{2} + 1348029540000)^{2} \)
|
| $13$ |
\( (T^{2} + 1648784402500)^{2} \)
|
| $17$ |
\( (T^{2} - 220359487855104)^{2} \)
|
| $19$ |
\( (T + 53343578)^{4} \)
|
| $23$ |
\( (T^{2} - 11\!\cdots\!44)^{2} \)
|
| $29$ |
\( (T^{2} + 14\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T - 66526202)^{4} \)
|
| $37$ |
\( (T^{2} + 49\!\cdots\!00)^{2} \)
|
| $41$ |
\( (T^{2} + 67\!\cdots\!00)^{2} \)
|
| $43$ |
\( (T^{2} + 80\!\cdots\!00)^{2} \)
|
| $47$ |
\( (T^{2} - 11\!\cdots\!04)^{2} \)
|
| $53$ |
\( (T^{2} - 16\!\cdots\!04)^{2} \)
|
| $59$ |
\( (T^{2} + 21\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T + 40679935918)^{4} \)
|
| $67$ |
\( (T^{2} + 14\!\cdots\!00)^{2} \)
|
| $71$ |
\( (T^{2} + 20\!\cdots\!00)^{2} \)
|
| $73$ |
\( (T^{2} + 37\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T - 252324997702)^{4} \)
|
| $83$ |
\( (T^{2} - 16\!\cdots\!84)^{2} \)
|
| $89$ |
\( (T^{2} + 12\!\cdots\!00)^{2} \)
|
| $97$ |
\( (T^{2} + 42\!\cdots\!00)^{2} \)
|
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