Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [350,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("350.99");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(56.1343369345\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 1286x^{4} + 413449x^{2} + 2377764 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 5^{2} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 1286x^{4} + 413449x^{2} + 2377764 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 643\nu ) / 1542 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 643\nu^{2} ) / 1542 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{4} + 2057\nu^{2} + 330759 ) / 771 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 1072\nu^{3} + 269679\nu ) / 1542 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - 4\beta_{3} - 429 \)
|
| \(\nu^{3}\) | \(=\) |
\( 1542\beta_{2} - 643\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -643\beta_{4} + 4114\beta_{3} + 275847 \)
|
| \(\nu^{5}\) | \(=\) |
\( 1542\beta_{5} - 1653024\beta_{2} + 419617\beta_1 \)
|
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 |
|
− | 4.00000i | − | 23.4808i | −16.0000 | 0 | −93.9231 | − | 49.0000i | 64.0000i | −308.346 | 0 | |||||||||||||||||||||||||||||||||
| 99.2 | − | 4.00000i | 5.42018i | −16.0000 | 0 | 21.6807 | − | 49.0000i | 64.0000i | 213.622 | 0 | |||||||||||||||||||||||||||||||||||
| 99.3 | − | 4.00000i | 27.0606i | −16.0000 | 0 | 108.242 | − | 49.0000i | 64.0000i | −489.275 | 0 | |||||||||||||||||||||||||||||||||||
| 99.4 | 4.00000i | − | 27.0606i | −16.0000 | 0 | 108.242 | 49.0000i | − | 64.0000i | −489.275 | 0 | |||||||||||||||||||||||||||||||||||
| 99.5 | 4.00000i | − | 5.42018i | −16.0000 | 0 | 21.6807 | 49.0000i | − | 64.0000i | 213.622 | 0 | |||||||||||||||||||||||||||||||||||
| 99.6 | 4.00000i | 23.4808i | −16.0000 | 0 | −93.9231 | 49.0000i | − | 64.0000i | −308.346 | 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 | 350.6.c.n | 6 | |
| 5.b | even | 2 | 1 | inner | 350.6.c.n | 6 | |
| 5.c | odd | 4 | 1 | 350.6.a.u | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 350.6.a.x | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 350.6.a.u | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 350.6.a.x | yes | 3 | 5.c | odd | 4 | 1 | |
| 350.6.c.n | 6 | 1.a | even | 1 | 1 | trivial | |
| 350.6.c.n | 6 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(350, [\chi])\):
|
\( T_{3}^{6} + 1313T_{3}^{4} + 441448T_{3}^{2} + 11861136 \)
|
|
\( T_{11}^{3} + 107T_{11}^{2} - 139269T_{11} - 21103983 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16)^{3} \)
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| $3$ |
\( T^{6} + 1313 T^{4} + \cdots + 11861136 \)
|
| $5$ |
\( T^{6} \)
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| $7$ |
\( (T^{2} + 2401)^{3} \)
|
| $11$ |
\( (T^{3} + 107 T^{2} + \cdots - 21103983)^{2} \)
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| $13$ |
\( T^{6} + \cdots + 909945082767616 \)
|
| $17$ |
\( T^{6} + \cdots + 13\!\cdots\!44 \)
|
| $19$ |
\( (T^{3} + 441 T^{2} + \cdots + 289884980)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 18\!\cdots\!00 \)
|
| $29$ |
\( (T^{3} + 1308 T^{2} + \cdots - 32353852950)^{2} \)
|
| $31$ |
\( (T^{3} - 4724 T^{2} + \cdots - 9395311072)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 39\!\cdots\!84 \)
|
| $41$ |
\( (T^{3} + \cdots - 1366682605092)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 90\!\cdots\!76 \)
|
| $47$ |
\( T^{6} + \cdots + 20\!\cdots\!04 \)
|
| $53$ |
\( T^{6} + \cdots + 66\!\cdots\!36 \)
|
| $59$ |
\( (T^{3} - 16076 T^{2} + \cdots + 533329584480)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots - 28835515765000)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 12\!\cdots\!09 \)
|
| $71$ |
\( (T^{3} + \cdots - 128336306971662)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 70\!\cdots\!00 \)
|
| $79$ |
\( (T^{3} + \cdots + 10\!\cdots\!50)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 32\!\cdots\!76 \)
|
| $89$ |
\( (T^{3} + \cdots - 65722788999000)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 13\!\cdots\!00 \)
|
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