Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [350,8,Mod(99,350)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(350, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("350.99");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 350.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: | \(109.334758919\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{8761})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 4381x^{2} + 4796100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 70) |
| 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
\( x^{4} + 4381x^{2} + 4796100 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 2191\nu ) / 2190 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 2191 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 2191 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2190\beta_{2} - 2191\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 99.1 |
|
− | 8.00000i | − | 49.3001i | −64.0000 | 0 | −394.401 | − | 343.000i | 512.000i | −243.501 | 0 | |||||||||||||||||||||||||||
| 99.2 | − | 8.00000i | 44.3001i | −64.0000 | 0 | 354.401 | − | 343.000i | 512.000i | 224.501 | 0 | |||||||||||||||||||||||||||||
| 99.3 | 8.00000i | − | 44.3001i | −64.0000 | 0 | 354.401 | 343.000i | − | 512.000i | 224.501 | 0 | |||||||||||||||||||||||||||||
| 99.4 | 8.00000i | 49.3001i | −64.0000 | 0 | −394.401 | 343.000i | − | 512.000i | −243.501 | 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 | 350.8.c.i | 4 | |
| 5.b | even | 2 | 1 | inner | 350.8.c.i | 4 | |
| 5.c | odd | 4 | 1 | 70.8.a.g | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 350.8.a.l | 2 | ||
| 20.e | even | 4 | 1 | 560.8.a.h | 2 | ||
| 35.f | even | 4 | 1 | 490.8.a.k | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.8.a.g | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 350.8.a.l | 2 | 5.c | odd | 4 | 1 | ||
| 350.8.c.i | 4 | 1.a | even | 1 | 1 | trivial | |
| 350.8.c.i | 4 | 5.b | even | 2 | 1 | inner | |
| 490.8.a.k | 2 | 35.f | even | 4 | 1 | ||
| 560.8.a.h | 2 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 4393T_{3}^{2} + 4769856 \)
acting on \(S_{8}^{\mathrm{new}}(350, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 64)^{2} \)
|
| $3$ |
\( T^{4} + 4393 T^{2} + 4769856 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 117649)^{2} \)
|
| $11$ |
\( (T^{2} - 4515 T - 31351644)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 204929589191044 \)
|
| $17$ |
\( T^{4} + \cdots + 11\!\cdots\!56 \)
|
| $19$ |
\( (T^{2} + 43702 T + 476756560)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 22\!\cdots\!00 \)
|
| $29$ |
\( (T^{2} + 9261 T - 5453221950)^{2} \)
|
| $31$ |
\( (T^{2} - 207028 T + 8327915872)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 18\!\cdots\!16 \)
|
| $41$ |
\( (T^{2} - 264798 T - 46196345448)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 47\!\cdots\!84 \)
|
| $47$ |
\( T^{4} + \cdots + 40\!\cdots\!64 \)
|
| $53$ |
\( T^{4} + \cdots + 11\!\cdots\!04 \)
|
| $59$ |
\( (T^{2} + 1905408 T + 907564170240)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots - 6065095799840)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 31\!\cdots\!36 \)
|
| $71$ |
\( (T^{2} + \cdots - 29167155883008)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 82\!\cdots\!00 \)
|
| $79$ |
\( (T^{2} + \cdots + 38120740022200)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 21\!\cdots\!96 \)
|
| $89$ |
\( (T^{2} + \cdots + 26223919716600)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 23\!\cdots\!00 \)
|
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