Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.49");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(12.0287864860\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{409})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 205x^{2} + 10404 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} + 205x^{2} + 10404 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 103\nu ) / 102 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 103 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 103 \)
|
| \(\nu^{3}\) | \(=\) |
\( 102\beta_{2} - 103\beta_1 \)
|
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 |
|
− | 10.6119i | 9.00000i | −80.6119 | 0 | 95.5069 | 105.790i | 515.863i | −81.0000 | 0 | |||||||||||||||||||||||||||||
| 49.2 | − | 9.61187i | − | 9.00000i | −60.3881 | 0 | −86.5069 | 217.790i | 272.863i | −81.0000 | 0 | |||||||||||||||||||||||||||||
| 49.3 | 9.61187i | 9.00000i | −60.3881 | 0 | −86.5069 | − | 217.790i | − | 272.863i | −81.0000 | 0 | |||||||||||||||||||||||||||||
| 49.4 | 10.6119i | − | 9.00000i | −80.6119 | 0 | 95.5069 | − | 105.790i | − | 515.863i | −81.0000 | 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 | 75.6.b.e | 4 | |
| 3.b | odd | 2 | 1 | 225.6.b.g | 4 | ||
| 5.b | even | 2 | 1 | inner | 75.6.b.e | 4 | |
| 5.c | odd | 4 | 1 | 15.6.a.c | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 75.6.a.h | 2 | ||
| 15.d | odd | 2 | 1 | 225.6.b.g | 4 | ||
| 15.e | even | 4 | 1 | 45.6.a.e | 2 | ||
| 15.e | even | 4 | 1 | 225.6.a.m | 2 | ||
| 20.e | even | 4 | 1 | 240.6.a.q | 2 | ||
| 35.f | even | 4 | 1 | 735.6.a.g | 2 | ||
| 40.i | odd | 4 | 1 | 960.6.a.bj | 2 | ||
| 40.k | even | 4 | 1 | 960.6.a.bf | 2 | ||
| 60.l | odd | 4 | 1 | 720.6.a.bd | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.6.a.c | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 45.6.a.e | 2 | 15.e | even | 4 | 1 | ||
| 75.6.a.h | 2 | 5.c | odd | 4 | 1 | ||
| 75.6.b.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 75.6.b.e | 4 | 5.b | even | 2 | 1 | inner | |
| 225.6.a.m | 2 | 15.e | even | 4 | 1 | ||
| 225.6.b.g | 4 | 3.b | odd | 2 | 1 | ||
| 225.6.b.g | 4 | 15.d | odd | 2 | 1 | ||
| 240.6.a.q | 2 | 20.e | even | 4 | 1 | ||
| 720.6.a.bd | 2 | 60.l | odd | 4 | 1 | ||
| 735.6.a.g | 2 | 35.f | even | 4 | 1 | ||
| 960.6.a.bf | 2 | 40.k | even | 4 | 1 | ||
| 960.6.a.bj | 2 | 40.i | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 205T_{2}^{2} + 10404 \)
acting on \(S_{6}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 205 T^{2} + 10404 \)
|
| $3$ |
\( (T^{2} + 81)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 58624 T^{2} + 530841600 \)
|
| $11$ |
\( (T^{2} - 248 T - 89328)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 27445886224 \)
|
| $17$ |
\( T^{4} + \cdots + 145865941776 \)
|
| $19$ |
\( (T^{2} + 1464 T - 3887920)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 5522876006400 \)
|
| $29$ |
\( (T^{2} + 1948 T + 529860)^{2} \)
|
| $31$ |
\( (T^{2} - 2672 T - 1382400)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 24\!\cdots\!00 \)
|
| $41$ |
\( (T^{2} + 7628 T - 15712860)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 45\!\cdots\!56 \)
|
| $47$ |
\( T^{4} + \cdots + 37\!\cdots\!16 \)
|
| $53$ |
\( T^{4} + \cdots + 22\!\cdots\!00 \)
|
| $59$ |
\( (T^{2} - 904 T - 1067881200)^{2} \)
|
| $61$ |
\( (T^{2} - 20220 T - 149182204)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 15\!\cdots\!16 \)
|
| $71$ |
\( (T^{2} + 40976 T - 627281856)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 69\!\cdots\!00 \)
|
| $79$ |
\( (T^{2} + 107600 T + 1448870400)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 42\!\cdots\!24 \)
|
| $89$ |
\( (T^{2} + 103764 T - 2895368220)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 24\!\cdots\!76 \)
|
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