Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [525,4,Mod(274,525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(525, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("525.274");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 525.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: | \(30.9760027530\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{57})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 29x^{2} + 196 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 21) |
| 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
\( x^{4} + 29x^{2} + 196 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 15\nu ) / 14 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 15 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 15 \)
|
| \(\nu^{3}\) | \(=\) |
\( 14\beta_{2} - 15\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(176\) | \(451\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 |
|
− | 5.27492i | − | 3.00000i | −19.8248 | 0 | −15.8248 | 7.00000i | 62.3746i | −9.00000 | 0 | ||||||||||||||||||||||||||||
| 274.2 | − | 2.27492i | 3.00000i | 2.82475 | 0 | 6.82475 | − | 7.00000i | − | 24.6254i | −9.00000 | 0 | ||||||||||||||||||||||||||||
| 274.3 | 2.27492i | − | 3.00000i | 2.82475 | 0 | 6.82475 | 7.00000i | 24.6254i | −9.00000 | 0 | ||||||||||||||||||||||||||||||
| 274.4 | 5.27492i | 3.00000i | −19.8248 | 0 | −15.8248 | − | 7.00000i | − | 62.3746i | −9.00000 | 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 | 525.4.d.g | 4 | |
| 5.b | even | 2 | 1 | inner | 525.4.d.g | 4 | |
| 5.c | odd | 4 | 1 | 21.4.a.c | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 525.4.a.n | 2 | ||
| 15.e | even | 4 | 1 | 63.4.a.e | 2 | ||
| 15.e | even | 4 | 1 | 1575.4.a.p | 2 | ||
| 20.e | even | 4 | 1 | 336.4.a.m | 2 | ||
| 35.f | even | 4 | 1 | 147.4.a.i | 2 | ||
| 35.k | even | 12 | 2 | 147.4.e.m | 4 | ||
| 35.l | odd | 12 | 2 | 147.4.e.l | 4 | ||
| 40.i | odd | 4 | 1 | 1344.4.a.bg | 2 | ||
| 40.k | even | 4 | 1 | 1344.4.a.bo | 2 | ||
| 60.l | odd | 4 | 1 | 1008.4.a.ba | 2 | ||
| 105.k | odd | 4 | 1 | 441.4.a.r | 2 | ||
| 105.w | odd | 12 | 2 | 441.4.e.p | 4 | ||
| 105.x | even | 12 | 2 | 441.4.e.q | 4 | ||
| 140.j | odd | 4 | 1 | 2352.4.a.bz | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.4.a.c | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 63.4.a.e | 2 | 15.e | even | 4 | 1 | ||
| 147.4.a.i | 2 | 35.f | even | 4 | 1 | ||
| 147.4.e.l | 4 | 35.l | odd | 12 | 2 | ||
| 147.4.e.m | 4 | 35.k | even | 12 | 2 | ||
| 336.4.a.m | 2 | 20.e | even | 4 | 1 | ||
| 441.4.a.r | 2 | 105.k | odd | 4 | 1 | ||
| 441.4.e.p | 4 | 105.w | odd | 12 | 2 | ||
| 441.4.e.q | 4 | 105.x | even | 12 | 2 | ||
| 525.4.a.n | 2 | 5.c | odd | 4 | 1 | ||
| 525.4.d.g | 4 | 1.a | even | 1 | 1 | trivial | |
| 525.4.d.g | 4 | 5.b | even | 2 | 1 | inner | |
| 1008.4.a.ba | 2 | 60.l | odd | 4 | 1 | ||
| 1344.4.a.bg | 2 | 40.i | odd | 4 | 1 | ||
| 1344.4.a.bo | 2 | 40.k | even | 4 | 1 | ||
| 1575.4.a.p | 2 | 15.e | even | 4 | 1 | ||
| 2352.4.a.bz | 2 | 140.j | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(525, [\chi])\):
|
\( T_{2}^{4} + 33T_{2}^{2} + 144 \)
|
|
\( T_{11}^{2} + 6T_{11} - 1416 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 33T^{2} + 144 \)
|
| $3$ |
\( (T^{2} + 9)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 49)^{2} \)
|
| $11$ |
\( (T^{2} + 6 T - 1416)^{2} \)
|
| $13$ |
\( T^{4} + 4232 T^{2} + 3952144 \)
|
| $17$ |
\( T^{4} + 132T^{2} + 2304 \)
|
| $19$ |
\( (T^{2} + 64 T - 7184)^{2} \)
|
| $23$ |
\( T^{4} + 32964 T^{2} + 271063296 \)
|
| $29$ |
\( (T^{2} - 252 T + 7668)^{2} \)
|
| $31$ |
\( (T^{2} - 40 T - 73472)^{2} \)
|
| $37$ |
\( T^{4} + 67688 T^{2} + 9560464 \)
|
| $41$ |
\( (T^{2} + 450 T + 37800)^{2} \)
|
| $43$ |
\( T^{4} + 136352 T^{2} + 6310144 \)
|
| $47$ |
\( T^{4} + \cdots + 4337012736 \)
|
| $53$ |
\( T^{4} + \cdots + 92705634576 \)
|
| $59$ |
\( (T^{2} + 804 T - 30144)^{2} \)
|
| $61$ |
\( (T^{2} + 428 T - 28076)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 25836061696 \)
|
| $71$ |
\( (T^{2} - 954 T + 214704)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 81364139536 \)
|
| $79$ |
\( (T^{2} - 572 T - 84416)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 661710663936 \)
|
| $89$ |
\( (T^{2} + 366 T - 253848)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 850622533264 \)
|
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