Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,22,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, 22, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.49");
S:= CuspForms(chi, 22);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| 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: | \(209.608008215\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 12791593x^{4} + 40900620507408x^{2} + 7800754619229474816 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2}\cdot 5^{2}\cdot 7^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 12791593x^{4} + 40900620507408x^{2} + 7800754619229474816 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 2805774697\nu^{3} + 17904253578121296\nu ) / 7818617969394105600 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 6395797\nu^{2} - 2792983104 ) / 2799378900 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 164\nu^{4} + 1748755433\nu^{2} + 2983432113138294 ) / 699844725 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -55944935\nu^{5} - 596523660427583\nu^{3} - 1424011700176015769136\nu ) / 156372359387882112 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - 656\beta_{3} - 4263646 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2797246750\beta_{2} - 6396452\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -6395797\beta_{4} + 6995021732\beta_{3} + 27272207278966 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2805774697\beta_{5} - 29826183021379150\beta_{2} + 42749594053748\beta_1 \)
|
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 |
|
− | 2540.56i | 59049.0i | −4.35728e6 | 0 | 1.50017e8 | − | 4.55015e8i | 5.74199e9i | −3.48678e9 | 0 | ||||||||||||||||||||||||||||||||||
| 49.2 | − | 2456.63i | − | 59049.0i | −3.93788e6 | 0 | −1.45062e8 | − | 5.82386e8i | 4.52198e9i | −3.48678e9 | 0 | ||||||||||||||||||||||||||||||||||
| 49.3 | − | 719.072i | 59049.0i | 1.58009e6 | 0 | 4.24605e7 | 1.45023e9i | − | 2.64420e9i | −3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||
| 49.4 | 719.072i | − | 59049.0i | 1.58009e6 | 0 | 4.24605e7 | − | 1.45023e9i | 2.64420e9i | −3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||
| 49.5 | 2456.63i | 59049.0i | −3.93788e6 | 0 | −1.45062e8 | 5.82386e8i | − | 4.52198e9i | −3.48678e9 | 0 | ||||||||||||||||||||||||||||||||||||
| 49.6 | 2540.56i | − | 59049.0i | −4.35728e6 | 0 | 1.50017e8 | 4.55015e8i | − | 5.74199e9i | −3.48678e9 | 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 | 75.22.b.e | 6 | |
| 5.b | even | 2 | 1 | inner | 75.22.b.e | 6 | |
| 5.c | odd | 4 | 1 | 15.22.a.d | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 75.22.a.e | 3 | ||
| 15.e | even | 4 | 1 | 45.22.a.b | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.22.a.d | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 45.22.a.b | 3 | 15.e | even | 4 | 1 | ||
| 75.22.a.e | 3 | 5.c | odd | 4 | 1 | ||
| 75.22.b.e | 6 | 1.a | even | 1 | 1 | trivial | |
| 75.22.b.e | 6 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 13006529T_{2}^{4} + 45410564694544T_{2}^{2} + 20141061724362571776 \)
acting on \(S_{22}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots + 20\!\cdots\!76 \)
|
| $3$ |
\( (T^{2} + 3486784401)^{3} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots + 14\!\cdots\!00 \)
|
| $11$ |
\( (T^{3} + \cdots + 25\!\cdots\!36)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 10\!\cdots\!84 \)
|
| $17$ |
\( T^{6} + \cdots + 27\!\cdots\!64 \)
|
| $19$ |
\( (T^{3} + \cdots - 39\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 15\!\cdots\!00 \)
|
| $29$ |
\( (T^{3} + \cdots + 19\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{3} + \cdots + 10\!\cdots\!00)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 34\!\cdots\!00 \)
|
| $41$ |
\( (T^{3} + \cdots - 86\!\cdots\!00)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 39\!\cdots\!16 \)
|
| $47$ |
\( T^{6} + \cdots + 38\!\cdots\!24 \)
|
| $53$ |
\( T^{6} + \cdots + 92\!\cdots\!00 \)
|
| $59$ |
\( (T^{3} + \cdots + 67\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots - 41\!\cdots\!32)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 89\!\cdots\!04 \)
|
| $71$ |
\( (T^{3} + \cdots + 14\!\cdots\!52)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $79$ |
\( (T^{3} + \cdots + 61\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 10\!\cdots\!64 \)
|
| $89$ |
\( (T^{3} + \cdots - 26\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 26\!\cdots\!04 \)
|
| show more | |
| show less |