Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1450,2,Mod(349,1450)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1450, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1450.349");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1450 = 2 \cdot 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1450.b (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: | \(11.5783082931\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.399424.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} - \nu^{2} + 2\nu - 2 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} - 3\nu^{3} + 4\nu^{2} - 2\nu + 8 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} - 3\nu^{3} + 6\nu^{2} - 2\nu + 4 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} + 3\nu^{3} - 6\nu^{2} + 10\nu - 8 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{5} - 2\nu^{4} + 5\nu^{3} - 6\nu^{2} + 2\nu - 12 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 2\beta_{3} + \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{4} + \beta_{3} - 2\beta_{2} + 2\beta _1 + 3 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{5} + 2\beta_{3} - 5\beta_{2} - \beta _1 + 4 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4\beta_{5} + \beta_{4} + 3\beta_{3} + 2\beta_{2} - 2\beta _1 + 5 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times\).
| \(n\) | \(901\) | \(1277\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 349.1 |
|
− | 1.00000i | − | 2.34292i | −1.00000 | 0 | −2.34292 | 3.83221i | 1.00000i | −2.48929 | 0 | ||||||||||||||||||||||||||||||||||
| 349.2 | − | 1.00000i | − | 0.470683i | −1.00000 | 0 | −0.470683 | − | 3.30777i | 1.00000i | 2.77846 | 0 | ||||||||||||||||||||||||||||||||||
| 349.3 | − | 1.00000i | 1.81361i | −1.00000 | 0 | 1.81361 | − | 2.52444i | 1.00000i | −0.289169 | 0 | |||||||||||||||||||||||||||||||||||
| 349.4 | 1.00000i | − | 1.81361i | −1.00000 | 0 | 1.81361 | 2.52444i | − | 1.00000i | −0.289169 | 0 | |||||||||||||||||||||||||||||||||||
| 349.5 | 1.00000i | 0.470683i | −1.00000 | 0 | −0.470683 | 3.30777i | − | 1.00000i | 2.77846 | 0 | ||||||||||||||||||||||||||||||||||||
| 349.6 | 1.00000i | 2.34292i | −1.00000 | 0 | −2.34292 | − | 3.83221i | − | 1.00000i | −2.48929 | 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 | 1450.2.b.k | 6 | |
| 5.b | even | 2 | 1 | inner | 1450.2.b.k | 6 | |
| 5.c | odd | 4 | 1 | 1450.2.a.q | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 1450.2.a.s | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1450.2.a.q | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 1450.2.a.s | yes | 3 | 5.c | odd | 4 | 1 | |
| 1450.2.b.k | 6 | 1.a | even | 1 | 1 | trivial | |
| 1450.2.b.k | 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 \(S_{2}^{\mathrm{new}}(1450, [\chi])\):
|
\( T_{3}^{6} + 9T_{3}^{4} + 20T_{3}^{2} + 4 \)
|
|
\( T_{7}^{6} + 32T_{7}^{4} + 324T_{7}^{2} + 1024 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{3} \)
|
| $3$ |
\( T^{6} + 9 T^{4} + 20 T^{2} + 4 \)
|
| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} + 32 T^{4} + 324 T^{2} + \cdots + 1024 \)
|
| $11$ |
\( (T^{3} - 5 T^{2} - 8 T + 44)^{2} \)
|
| $13$ |
\( T^{6} + 16 T^{4} + 68 T^{2} + 64 \)
|
| $17$ |
\( T^{6} + 49 T^{4} + 56 T^{2} + 16 \)
|
| $19$ |
\( (T^{3} - T^{2} - 4 T + 2)^{2} \)
|
| $23$ |
\( T^{6} + 132 T^{4} + 5444 T^{2} + \cdots + 71824 \)
|
| $29$ |
\( (T + 1)^{6} \)
|
| $31$ |
\( (T^{3} - 4 T^{2} - 79 T + 158)^{2} \)
|
| $37$ |
\( T^{6} + 114 T^{4} + 3305 T^{2} + \cdots + 3364 \)
|
| $41$ |
\( (T^{3} - 17 T^{2} + 72 T + 2)^{2} \)
|
| $43$ |
\( T^{6} + 128 T^{4} + 5184 T^{2} + \cdots + 65536 \)
|
| $47$ |
\( T^{6} + 174 T^{4} + 6033 T^{2} + \cdots + 58564 \)
|
| $53$ |
\( T^{6} + 144 T^{4} + 3300 T^{2} + \cdots + 1024 \)
|
| $59$ |
\( (T^{3} + 16 T^{2} + 61 T + 68)^{2} \)
|
| $61$ |
\( (T^{3} - 2 T^{2} - 107 T + 146)^{2} \)
|
| $67$ |
\( T^{6} + 111 T^{4} + 2783 T^{2} + \cdots + 14641 \)
|
| $71$ |
\( (T^{3} - 8 T^{2} - 34 T + 268)^{2} \)
|
| $73$ |
\( T^{6} + 273 T^{4} + 10868 T^{2} + \cdots + 98596 \)
|
| $79$ |
\( (T^{3} - 252 T + 432)^{2} \)
|
| $83$ |
\( T^{6} + 209 T^{4} + 6600 T^{2} + \cdots + 44944 \)
|
| $89$ |
\( (T^{3} + 11 T^{2} - 158)^{2} \)
|
| $97$ |
\( T^{6} + 260 T^{4} + 17092 T^{2} + \cdots + 26896 \)
|
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