Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [400,8,Mod(49,400)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(400, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("400.49");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 400.c (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: | \(124.954010194\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{19})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 9x^{2} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 5) |
| 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
\( x^{4} - 9x^{2} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu^{3} - 8\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -32\nu^{3} + 448\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 32\nu^{2} - 144 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 16\beta_1 ) / 320 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 144 ) / 32 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{2} + 56\beta_1 ) / 80 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(177\) | \(351\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | − | 79.7424i | 0 | 0 | 0 | − | 538.197i | 0 | −4171.85 | 0 | ||||||||||||||||||||||||||||
| 49.2 | 0 | − | 59.7424i | 0 | 0 | 0 | − | 438.197i | 0 | −1382.15 | 0 | |||||||||||||||||||||||||||||
| 49.3 | 0 | 59.7424i | 0 | 0 | 0 | 438.197i | 0 | −1382.15 | 0 | |||||||||||||||||||||||||||||||
| 49.4 | 0 | 79.7424i | 0 | 0 | 0 | 538.197i | 0 | −4171.85 | 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 | 400.8.c.m | 4 | |
| 4.b | odd | 2 | 1 | 25.8.b.c | 4 | ||
| 5.b | even | 2 | 1 | inner | 400.8.c.m | 4 | |
| 5.c | odd | 4 | 1 | 80.8.a.g | 2 | ||
| 5.c | odd | 4 | 1 | 400.8.a.bb | 2 | ||
| 12.b | even | 2 | 1 | 225.8.b.m | 4 | ||
| 20.d | odd | 2 | 1 | 25.8.b.c | 4 | ||
| 20.e | even | 4 | 1 | 5.8.a.b | ✓ | 2 | |
| 20.e | even | 4 | 1 | 25.8.a.b | 2 | ||
| 40.i | odd | 4 | 1 | 320.8.a.u | 2 | ||
| 40.k | even | 4 | 1 | 320.8.a.l | 2 | ||
| 60.h | even | 2 | 1 | 225.8.b.m | 4 | ||
| 60.l | odd | 4 | 1 | 45.8.a.h | 2 | ||
| 60.l | odd | 4 | 1 | 225.8.a.w | 2 | ||
| 140.j | odd | 4 | 1 | 245.8.a.c | 2 | ||
| 220.i | odd | 4 | 1 | 605.8.a.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.8.a.b | ✓ | 2 | 20.e | even | 4 | 1 | |
| 25.8.a.b | 2 | 20.e | even | 4 | 1 | ||
| 25.8.b.c | 4 | 4.b | odd | 2 | 1 | ||
| 25.8.b.c | 4 | 20.d | odd | 2 | 1 | ||
| 45.8.a.h | 2 | 60.l | odd | 4 | 1 | ||
| 80.8.a.g | 2 | 5.c | odd | 4 | 1 | ||
| 225.8.a.w | 2 | 60.l | odd | 4 | 1 | ||
| 225.8.b.m | 4 | 12.b | even | 2 | 1 | ||
| 225.8.b.m | 4 | 60.h | even | 2 | 1 | ||
| 245.8.a.c | 2 | 140.j | odd | 4 | 1 | ||
| 320.8.a.l | 2 | 40.k | even | 4 | 1 | ||
| 320.8.a.u | 2 | 40.i | odd | 4 | 1 | ||
| 400.8.a.bb | 2 | 5.c | odd | 4 | 1 | ||
| 400.8.c.m | 4 | 1.a | even | 1 | 1 | trivial | |
| 400.8.c.m | 4 | 5.b | even | 2 | 1 | inner | |
| 605.8.a.d | 2 | 220.i | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 9928T_{3}^{2} + 22695696 \)
acting on \(S_{8}^{\mathrm{new}}(400, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 9928 T^{2} + 22695696 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 55618618896 \)
|
| $11$ |
\( (T^{2} + 4544 T - 6998016)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 623079677326096 \)
|
| $17$ |
\( T^{4} + \cdots + 64\!\cdots\!36 \)
|
| $19$ |
\( (T^{2} - 38760 T + 367802000)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 14\!\cdots\!96 \)
|
| $29$ |
\( (T^{2} - 72260 T - 27652933500)^{2} \)
|
| $31$ |
\( (T^{2} + 306824 T + 22939401744)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 20\!\cdots\!16 \)
|
| $41$ |
\( (T^{2} - 264364 T - 227722158876)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 94\!\cdots\!96 \)
|
| $47$ |
\( T^{4} + \cdots + 23\!\cdots\!56 \)
|
| $53$ |
\( T^{4} + \cdots + 14\!\cdots\!96 \)
|
| $59$ |
\( (T^{2} + \cdots - 3614968086000)^{2} \)
|
| $61$ |
\( (T^{2} - 2257044 T - 672038095516)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 21\!\cdots\!36 \)
|
| $71$ |
\( (T^{2} + 621784 T - 275746164336)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 17\!\cdots\!96 \)
|
| $79$ |
\( (T^{2} + \cdots - 12272229720000)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 32\!\cdots\!96 \)
|
| $89$ |
\( (T^{2} + \cdots + 1403196358500)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 34\!\cdots\!56 \)
|
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