Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [350,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("350.99");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(180.262542657\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 14) |
| 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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
− | 16.0000i | 6.00000i | −256.000 | 0 | 96.0000 | − | 2401.00i | 4096.00i | 19647.0 | 0 | ||||||||||||||||||||||
| 99.2 | 16.0000i | − | 6.00000i | −256.000 | 0 | 96.0000 | 2401.00i | − | 4096.00i | 19647.0 | 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 | 350.10.c.b | 2 | |
| 5.b | even | 2 | 1 | inner | 350.10.c.b | 2 | |
| 5.c | odd | 4 | 1 | 14.10.a.a | ✓ | 1 | |
| 5.c | odd | 4 | 1 | 350.10.a.c | 1 | ||
| 15.e | even | 4 | 1 | 126.10.a.e | 1 | ||
| 20.e | even | 4 | 1 | 112.10.a.b | 1 | ||
| 35.f | even | 4 | 1 | 98.10.a.a | 1 | ||
| 35.k | even | 12 | 2 | 98.10.c.e | 2 | ||
| 35.l | odd | 12 | 2 | 98.10.c.f | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.10.a.a | ✓ | 1 | 5.c | odd | 4 | 1 | |
| 98.10.a.a | 1 | 35.f | even | 4 | 1 | ||
| 98.10.c.e | 2 | 35.k | even | 12 | 2 | ||
| 98.10.c.f | 2 | 35.l | odd | 12 | 2 | ||
| 112.10.a.b | 1 | 20.e | even | 4 | 1 | ||
| 126.10.a.e | 1 | 15.e | even | 4 | 1 | ||
| 350.10.a.c | 1 | 5.c | odd | 4 | 1 | ||
| 350.10.c.b | 2 | 1.a | even | 1 | 1 | trivial | |
| 350.10.c.b | 2 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 36 \)
acting on \(S_{10}^{\mathrm{new}}(350, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 256 \)
|
| $3$ |
\( T^{2} + 36 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} + 5764801 \)
|
| $11$ |
\( (T + 54152)^{2} \)
|
| $13$ |
\( T^{2} + 12807901584 \)
|
| $17$ |
\( T^{2} + 39212644 \)
|
| $19$ |
\( (T + 257078)^{2} \)
|
| $23$ |
\( T^{2} + 70756000000 \)
|
| $29$ |
\( (T + 1574714)^{2} \)
|
| $31$ |
\( (T + 4637484)^{2} \)
|
| $37$ |
\( T^{2} + 142712602352644 \)
|
| $41$ |
\( (T - 21909126)^{2} \)
|
| $43$ |
\( T^{2} + 757382984030464 \)
|
| $47$ |
\( T^{2} + 28\!\cdots\!96 \)
|
| $53$ |
\( T^{2} + 263128043173284 \)
|
| $59$ |
\( (T - 140509618)^{2} \)
|
| $61$ |
\( (T + 202963560)^{2} \)
|
| $67$ |
\( T^{2} + 23\!\cdots\!84 \)
|
| $71$ |
\( (T - 279655936)^{2} \)
|
| $73$ |
\( T^{2} + 16\!\cdots\!00 \)
|
| $79$ |
\( (T - 130689816)^{2} \)
|
| $83$ |
\( T^{2} + 17\!\cdots\!96 \)
|
| $89$ |
\( (T - 469542390)^{2} \)
|
| $97$ |
\( T^{2} + 76\!\cdots\!00 \)
|
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