Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1400,4,Mod(449,1400)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1400, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1400.449");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1400.g (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: | \(82.6026740080\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.7807489600.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 63x^{4} + 1041x^{2} + 1600 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 280) |
| 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
\( x^{6} + 63x^{4} + 1041x^{2} + 1600 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} - 23\nu^{3} + 239\nu ) / 360 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{4} - 58\nu^{2} + 43 ) / 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 43\nu^{3} + 321\nu ) / 60 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 35\nu^{2} + 106 ) / 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 13\nu^{5} + 539\nu^{3} + 5053\nu ) / 360 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - 2\beta_{3} + \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 6\beta_{4} + 3\beta_{2} - 85 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -7\beta_{5} + 17\beta_{3} + 11\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -174\beta_{4} - 105\beta_{2} + 2551 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 883\beta_{5} - 2042\beta_{3} - 2213\beta_1 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).
| \(n\) | \(351\) | \(701\) | \(801\) | \(1177\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 449.1 |
|
0 | − | 9.69516i | 0 | 0 | 0 | − | 7.00000i | 0 | −66.9961 | 0 | ||||||||||||||||||||||||||||||||||
| 449.2 | 0 | − | 3.45643i | 0 | 0 | 0 | 7.00000i | 0 | 15.0531 | 0 | ||||||||||||||||||||||||||||||||||||
| 449.3 | 0 | − | 0.238730i | 0 | 0 | 0 | 7.00000i | 0 | 26.9430 | 0 | ||||||||||||||||||||||||||||||||||||
| 449.4 | 0 | 0.238730i | 0 | 0 | 0 | − | 7.00000i | 0 | 26.9430 | 0 | ||||||||||||||||||||||||||||||||||||
| 449.5 | 0 | 3.45643i | 0 | 0 | 0 | − | 7.00000i | 0 | 15.0531 | 0 | ||||||||||||||||||||||||||||||||||||
| 449.6 | 0 | 9.69516i | 0 | 0 | 0 | 7.00000i | 0 | −66.9961 | 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 | 1400.4.g.l | 6 | |
| 5.b | even | 2 | 1 | inner | 1400.4.g.l | 6 | |
| 5.c | odd | 4 | 1 | 280.4.a.f | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 1400.4.a.m | 3 | ||
| 20.e | even | 4 | 1 | 560.4.a.w | 3 | ||
| 35.f | even | 4 | 1 | 1960.4.a.r | 3 | ||
| 40.i | odd | 4 | 1 | 2240.4.a.bz | 3 | ||
| 40.k | even | 4 | 1 | 2240.4.a.br | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 280.4.a.f | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 560.4.a.w | 3 | 20.e | even | 4 | 1 | ||
| 1400.4.a.m | 3 | 5.c | odd | 4 | 1 | ||
| 1400.4.g.l | 6 | 1.a | even | 1 | 1 | trivial | |
| 1400.4.g.l | 6 | 5.b | even | 2 | 1 | inner | |
| 1960.4.a.r | 3 | 35.f | even | 4 | 1 | ||
| 2240.4.a.br | 3 | 40.k | even | 4 | 1 | ||
| 2240.4.a.bz | 3 | 40.i | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1400, [\chi])\):
|
\( T_{3}^{6} + 106T_{3}^{4} + 1129T_{3}^{2} + 64 \)
|
|
\( T_{11}^{3} + 6T_{11}^{2} - 2855T_{11} - 24620 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 106 T^{4} + \cdots + 64 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( (T^{2} + 49)^{3} \)
|
| $11$ |
\( (T^{3} + 6 T^{2} + \cdots - 24620)^{2} \)
|
| $13$ |
\( T^{6} + 7134 T^{4} + \cdots + 416568100 \)
|
| $17$ |
\( T^{6} + 13278 T^{4} + \cdots + 102697956 \)
|
| $19$ |
\( (T^{3} - 152 T^{2} + \cdots + 2087776)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 2160382590976 \)
|
| $29$ |
\( (T^{3} - 468 T^{2} + \cdots + 3237134)^{2} \)
|
| $31$ |
\( (T^{3} + 268 T^{2} + \cdots - 921984)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 122116113979456 \)
|
| $41$ |
\( (T^{3} + 74 T^{2} + \cdots - 3221920)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 12675308857600 \)
|
| $47$ |
\( T^{6} + \cdots + 72428133662784 \)
|
| $53$ |
\( T^{6} + \cdots + 33641856025600 \)
|
| $59$ |
\( (T^{3} - 1088 T^{2} + \cdots - 19501056)^{2} \)
|
| $61$ |
\( (T^{3} - 174 T^{2} + \cdots + 187440544)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 17\!\cdots\!00 \)
|
| $71$ |
\( (T^{3} + 1056 T^{2} + \cdots - 270950400)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 64\!\cdots\!00 \)
|
| $79$ |
\( (T^{3} + 362 T^{2} + \cdots - 362644136)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 81\!\cdots\!84 \)
|
| $89$ |
\( (T^{3} + 158 T^{2} + \cdots - 860463840)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 56\!\cdots\!64 \)
|
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