Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [350,10,Mod(1,350)]
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([0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("350.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 350.a (trivial) |
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: | yes |
| Analytic conductor: | \(180.262542657\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.2997373.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 373x - 2632 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{3}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 70) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 373x - 2632 \)
:
| \(\beta_{1}\) | \(=\) |
\( -9\nu^{2} + 225\nu + 2166 \)
|
| \(\beta_{2}\) | \(=\) |
\( 9\nu^{2} - 105\nu - 2206 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 40 ) / 120 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 15\beta_{2} + 7\beta _1 + 17928 ) / 72 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−16.0000 | −222.983 | 256.000 | 0 | 3567.73 | 2401.00 | −4096.00 | 30038.4 | 0 | |||||||||||||||||||||||||||
| 1.2 | −16.0000 | 47.7937 | 256.000 | 0 | −764.699 | 2401.00 | −4096.00 | −17398.8 | 0 | ||||||||||||||||||||||||||||
| 1.3 | −16.0000 | 241.189 | 256.000 | 0 | −3859.03 | 2401.00 | −4096.00 | 38489.3 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(1\) |
| \(7\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 350.10.a.l | 3 | |
| 5.b | even | 2 | 1 | 70.10.a.h | ✓ | 3 | |
| 5.c | odd | 4 | 2 | 350.10.c.k | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.10.a.h | ✓ | 3 | 5.b | even | 2 | 1 | |
| 350.10.a.l | 3 | 1.a | even | 1 | 1 | trivial | |
| 350.10.c.k | 6 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 66T_{3}^{2} - 52911T_{3} + 2570400 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(350))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 16)^{3} \)
|
| $3$ |
\( T^{3} - 66 T^{2} + \cdots + 2570400 \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( (T - 2401)^{3} \)
|
| $11$ |
\( T^{3} + \cdots - 130003844724100 \)
|
| $13$ |
\( T^{3} + \cdots - 10\!\cdots\!06 \)
|
| $17$ |
\( T^{3} + \cdots - 27\!\cdots\!38 \)
|
| $19$ |
\( T^{3} + \cdots - 10\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots - 28\!\cdots\!00 \)
|
| $29$ |
\( T^{3} + \cdots + 42\!\cdots\!94 \)
|
| $31$ |
\( T^{3} + \cdots - 10\!\cdots\!08 \)
|
| $37$ |
\( T^{3} + \cdots + 39\!\cdots\!60 \)
|
| $41$ |
\( T^{3} + \cdots - 11\!\cdots\!12 \)
|
| $43$ |
\( T^{3} + \cdots - 10\!\cdots\!44 \)
|
| $47$ |
\( T^{3} + \cdots - 76\!\cdots\!48 \)
|
| $53$ |
\( T^{3} + \cdots + 20\!\cdots\!08 \)
|
| $59$ |
\( T^{3} + \cdots - 33\!\cdots\!12 \)
|
| $61$ |
\( T^{3} + \cdots - 88\!\cdots\!44 \)
|
| $67$ |
\( T^{3} + \cdots + 30\!\cdots\!68 \)
|
| $71$ |
\( T^{3} + \cdots + 23\!\cdots\!00 \)
|
| $73$ |
\( T^{3} + \cdots - 67\!\cdots\!72 \)
|
| $79$ |
\( T^{3} + \cdots + 75\!\cdots\!72 \)
|
| $83$ |
\( T^{3} + \cdots + 21\!\cdots\!88 \)
|
| $89$ |
\( T^{3} + \cdots - 63\!\cdots\!60 \)
|
| $97$ |
\( T^{3} + \cdots + 44\!\cdots\!50 \)
|
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