Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [425,4,Mod(324,425)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(425, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("425.324");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 425 = 5^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 425.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: | \(25.0758117524\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.27793984.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{3} + 49x^{2} - 14x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 17) |
| 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
\( x^{6} - 2x^{3} + 49x^{2} - 14x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 49\nu^{4} + 7\nu^{3} - \nu^{2} + 1696 ) / 342 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -7\nu^{5} - \nu^{4} - 49\nu^{3} + 7\nu^{2} + 98 ) / 342 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{5} + \nu^{4} + 49\nu^{3} - 7\nu^{2} + 684\nu - 98 ) / 342 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -49\nu^{5} - 7\nu^{4} - \nu^{3} + 49\nu^{2} - 2394\nu + 344 ) / 342 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -245\nu^{5} - 35\nu^{4} - 5\nu^{3} + 587\nu^{2} - 11970\nu + 1720 ) / 342 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 5\beta_{4} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{4} + 7\beta_{3} - 7\beta_{2} + 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{2} + 7\beta _1 - 35 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{5} - 24\beta_{4} - 49\beta_{3} - 49\beta_{2} - 2\beta _1 + 24 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).
| \(n\) | \(52\) | \(326\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 324.1 |
|
− | 5.03251i | − | 8.47535i | −17.3261 | 0 | −42.6523 | 3.81828i | 46.9339i | −44.8316 | 0 | ||||||||||||||||||||||||||||||||||
| 324.2 | − | 4.67129i | − | 7.62999i | −13.8209 | 0 | −35.6419 | − | 26.1222i | 27.1912i | −31.2167 | 0 | ||||||||||||||||||||||||||||||||||
| 324.3 | − | 1.36122i | 3.15463i | 6.14708 | 0 | 4.29415 | 7.94049i | − | 19.2573i | 17.0483 | 0 | |||||||||||||||||||||||||||||||||||
| 324.4 | 1.36122i | − | 3.15463i | 6.14708 | 0 | 4.29415 | − | 7.94049i | 19.2573i | 17.0483 | 0 | |||||||||||||||||||||||||||||||||||
| 324.5 | 4.67129i | 7.62999i | −13.8209 | 0 | −35.6419 | 26.1222i | − | 27.1912i | −31.2167 | 0 | ||||||||||||||||||||||||||||||||||||
| 324.6 | 5.03251i | 8.47535i | −17.3261 | 0 | −42.6523 | − | 3.81828i | − | 46.9339i | −44.8316 | 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 | 425.4.b.f | 6 | |
| 5.b | even | 2 | 1 | inner | 425.4.b.f | 6 | |
| 5.c | odd | 4 | 1 | 17.4.a.b | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 425.4.a.g | 3 | ||
| 15.e | even | 4 | 1 | 153.4.a.g | 3 | ||
| 20.e | even | 4 | 1 | 272.4.a.h | 3 | ||
| 35.f | even | 4 | 1 | 833.4.a.d | 3 | ||
| 40.i | odd | 4 | 1 | 1088.4.a.v | 3 | ||
| 40.k | even | 4 | 1 | 1088.4.a.x | 3 | ||
| 55.e | even | 4 | 1 | 2057.4.a.e | 3 | ||
| 60.l | odd | 4 | 1 | 2448.4.a.bi | 3 | ||
| 85.f | odd | 4 | 1 | 289.4.b.b | 6 | ||
| 85.g | odd | 4 | 1 | 289.4.a.b | 3 | ||
| 85.i | odd | 4 | 1 | 289.4.b.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 17.4.a.b | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 153.4.a.g | 3 | 15.e | even | 4 | 1 | ||
| 272.4.a.h | 3 | 20.e | even | 4 | 1 | ||
| 289.4.a.b | 3 | 85.g | odd | 4 | 1 | ||
| 289.4.b.b | 6 | 85.f | odd | 4 | 1 | ||
| 289.4.b.b | 6 | 85.i | odd | 4 | 1 | ||
| 425.4.a.g | 3 | 5.c | odd | 4 | 1 | ||
| 425.4.b.f | 6 | 1.a | even | 1 | 1 | trivial | |
| 425.4.b.f | 6 | 5.b | even | 2 | 1 | inner | |
| 833.4.a.d | 3 | 35.f | even | 4 | 1 | ||
| 1088.4.a.v | 3 | 40.i | odd | 4 | 1 | ||
| 1088.4.a.x | 3 | 40.k | even | 4 | 1 | ||
| 2057.4.a.e | 3 | 55.e | even | 4 | 1 | ||
| 2448.4.a.bi | 3 | 60.l | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(425, [\chi])\):
|
\( T_{2}^{6} + 49T_{2}^{4} + 640T_{2}^{2} + 1024 \)
|
|
\( T_{3}^{6} + 140T_{3}^{4} + 5476T_{3}^{2} + 41616 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 49 T^{4} + \cdots + 1024 \)
|
| $3$ |
\( T^{6} + 140 T^{4} + \cdots + 41616 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 760 T^{4} + \cdots + 627264 \)
|
| $11$ |
\( (T^{3} + 28 T^{2} + \cdots - 4692)^{2} \)
|
| $13$ |
\( T^{6} + 3844 T^{4} + \cdots + 88209664 \)
|
| $17$ |
\( (T^{2} + 289)^{3} \)
|
| $19$ |
\( (T^{3} + 80 T^{2} + \cdots - 340128)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 2561741095936 \)
|
| $29$ |
\( (T^{3} - 456 T^{2} + \cdots - 1518624)^{2} \)
|
| $31$ |
\( (T^{3} - 230 T^{2} + \cdots - 81608)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 38152265269504 \)
|
| $41$ |
\( (T^{3} + 294 T^{2} + \cdots - 1638744)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 52856854953984 \)
|
| $47$ |
\( T^{6} + \cdots + 2792802484224 \)
|
| $53$ |
\( T^{6} + \cdots + 329860859333184 \)
|
| $59$ |
\( (T^{3} + 636 T^{2} + \cdots - 49419072)^{2} \)
|
| $61$ |
\( (T^{3} + 84 T^{2} + \cdots - 6792784)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 586682466304 \)
|
| $71$ |
\( (T^{3} + 402 T^{2} + \cdots - 274866016)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 398302285230144 \)
|
| $79$ |
\( (T^{3} - 594 T^{2} + \cdots + 742135824)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 20\!\cdots\!16 \)
|
| $89$ |
\( (T^{3} - 170 T^{2} + \cdots + 446571376)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 42\!\cdots\!00 \)
|
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