Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [325,6,Mod(274,325)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(325, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("325.274");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 325 = 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 325.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: | \(52.1247414392\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{17})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 9x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 13) |
| 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
\( x^{4} + 9x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 5\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{2} - 5\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(301\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 274.1 |
|
− | 4.56155i | 1.63068i | 11.1922 | 0 | 7.43845 | 126.309i | − | 197.024i | 240.341 | 0 | ||||||||||||||||||||||||||||
| 274.2 | − | 0.438447i | 26.3693i | 31.8078 | 0 | 11.5616 | − | 162.309i | − | 27.9763i | −452.341 | 0 | ||||||||||||||||||||||||||||
| 274.3 | 0.438447i | − | 26.3693i | 31.8078 | 0 | 11.5616 | 162.309i | 27.9763i | −452.341 | 0 | ||||||||||||||||||||||||||||||
| 274.4 | 4.56155i | − | 1.63068i | 11.1922 | 0 | 7.43845 | − | 126.309i | 197.024i | 240.341 | 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 | 325.6.b.b | 4 | |
| 5.b | even | 2 | 1 | inner | 325.6.b.b | 4 | |
| 5.c | odd | 4 | 1 | 13.6.a.a | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 325.6.a.b | 2 | ||
| 15.e | even | 4 | 1 | 117.6.a.c | 2 | ||
| 20.e | even | 4 | 1 | 208.6.a.h | 2 | ||
| 35.f | even | 4 | 1 | 637.6.a.a | 2 | ||
| 40.i | odd | 4 | 1 | 832.6.a.p | 2 | ||
| 40.k | even | 4 | 1 | 832.6.a.i | 2 | ||
| 65.f | even | 4 | 1 | 169.6.b.a | 4 | ||
| 65.h | odd | 4 | 1 | 169.6.a.a | 2 | ||
| 65.k | even | 4 | 1 | 169.6.b.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.6.a.a | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 117.6.a.c | 2 | 15.e | even | 4 | 1 | ||
| 169.6.a.a | 2 | 65.h | odd | 4 | 1 | ||
| 169.6.b.a | 4 | 65.f | even | 4 | 1 | ||
| 169.6.b.a | 4 | 65.k | even | 4 | 1 | ||
| 208.6.a.h | 2 | 20.e | even | 4 | 1 | ||
| 325.6.a.b | 2 | 5.c | odd | 4 | 1 | ||
| 325.6.b.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 325.6.b.b | 4 | 5.b | even | 2 | 1 | inner | |
| 637.6.a.a | 2 | 35.f | even | 4 | 1 | ||
| 832.6.a.i | 2 | 40.k | even | 4 | 1 | ||
| 832.6.a.p | 2 | 40.i | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 21T_{2}^{2} + 4 \)
acting on \(S_{6}^{\mathrm{new}}(325, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 21T^{2} + 4 \)
|
| $3$ |
\( T^{4} + 698T^{2} + 1849 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 42298 T^{2} + 420291001 \)
|
| $11$ |
\( (T^{2} + 376 T + 5356)^{2} \)
|
| $13$ |
\( (T^{2} + 28561)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 2754248925649 \)
|
| $19$ |
\( (T^{2} - 312 T + 21004)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 46241958420736 \)
|
| $29$ |
\( (T^{2} - 812 T + 155044)^{2} \)
|
| $31$ |
\( (T^{2} - 7720 T + 14046608)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 32\!\cdots\!81 \)
|
| $41$ |
\( (T^{2} - 7840 T - 827392)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 86\!\cdots\!29 \)
|
| $47$ |
\( T^{4} + \cdots + 13\!\cdots\!41 \)
|
| $53$ |
\( T^{4} + \cdots + 78\!\cdots\!44 \)
|
| $59$ |
\( (T^{2} - 38936 T + 59682572)^{2} \)
|
| $61$ |
\( (T^{2} - 1984 T - 366282304)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 14\!\cdots\!84 \)
|
| $71$ |
\( (T^{2} - 67396 T + 1078437091)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 63\!\cdots\!96 \)
|
| $79$ |
\( (T^{2} - 55296 T + 361555696)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 46\!\cdots\!96 \)
|
| $89$ |
\( (T^{2} + 116508 T + 1045666948)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 20\!\cdots\!56 \)
|
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