Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4900,2,Mod(2549,4900)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4900, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("4900.2549");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4900.e (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: | \(39.1266969904\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.4227136.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 9x^{4} + 22x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | no (minimal twist has level 700) |
| 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
\( x^{6} + 9x^{4} + 22x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} + 5\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 6\nu^{3} + 7\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 9\nu^{3} + 13\nu ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( -2\nu^{4} - 9\nu^{2} - 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{4} + 6\nu^{2} + 6 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{5} + \beta_{4} - 10 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{3} - 5\beta_{2} - 2\beta_1 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -3\beta_{5} - 2\beta_{4} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -23\beta_{3} + 32\beta_{2} + 5\beta_1 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(1177\) | \(2451\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2549.1 |
|
0 | − | 3.20440i | 0 | 0 | 0 | 0 | 0 | −7.26819 | 0 | |||||||||||||||||||||||||||||||||||
| 2549.2 | 0 | − | 2.56885i | 0 | 0 | 0 | 0 | 0 | −3.59899 | 0 | ||||||||||||||||||||||||||||||||||||
| 2549.3 | 0 | − | 0.364448i | 0 | 0 | 0 | 0 | 0 | 2.86718 | 0 | ||||||||||||||||||||||||||||||||||||
| 2549.4 | 0 | 0.364448i | 0 | 0 | 0 | 0 | 0 | 2.86718 | 0 | |||||||||||||||||||||||||||||||||||||
| 2549.5 | 0 | 2.56885i | 0 | 0 | 0 | 0 | 0 | −3.59899 | 0 | |||||||||||||||||||||||||||||||||||||
| 2549.6 | 0 | 3.20440i | 0 | 0 | 0 | 0 | 0 | −7.26819 | 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 | 4900.2.e.s | 6 | |
| 5.b | even | 2 | 1 | inner | 4900.2.e.s | 6 | |
| 5.c | odd | 4 | 1 | 4900.2.a.bb | 3 | ||
| 5.c | odd | 4 | 1 | 4900.2.a.bd | 3 | ||
| 7.b | odd | 2 | 1 | 4900.2.e.t | 6 | ||
| 7.d | odd | 6 | 2 | 700.2.r.d | 12 | ||
| 35.c | odd | 2 | 1 | 4900.2.e.t | 6 | ||
| 35.f | even | 4 | 1 | 4900.2.a.ba | 3 | ||
| 35.f | even | 4 | 1 | 4900.2.a.bc | 3 | ||
| 35.i | odd | 6 | 2 | 700.2.r.d | 12 | ||
| 35.k | even | 12 | 2 | 700.2.i.d | ✓ | 6 | |
| 35.k | even | 12 | 2 | 700.2.i.e | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 700.2.i.d | ✓ | 6 | 35.k | even | 12 | 2 | |
| 700.2.i.e | yes | 6 | 35.k | even | 12 | 2 | |
| 700.2.r.d | 12 | 7.d | odd | 6 | 2 | ||
| 700.2.r.d | 12 | 35.i | odd | 6 | 2 | ||
| 4900.2.a.ba | 3 | 35.f | even | 4 | 1 | ||
| 4900.2.a.bb | 3 | 5.c | odd | 4 | 1 | ||
| 4900.2.a.bc | 3 | 35.f | even | 4 | 1 | ||
| 4900.2.a.bd | 3 | 5.c | odd | 4 | 1 | ||
| 4900.2.e.s | 6 | 1.a | even | 1 | 1 | trivial | |
| 4900.2.e.s | 6 | 5.b | even | 2 | 1 | inner | |
| 4900.2.e.t | 6 | 7.b | odd | 2 | 1 | ||
| 4900.2.e.t | 6 | 35.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4900, [\chi])\):
|
\( T_{3}^{6} + 17T_{3}^{4} + 70T_{3}^{2} + 9 \)
|
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\( T_{11}^{3} - 4T_{11}^{2} - 3T_{11} + 9 \)
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\( T_{19}^{3} + 2T_{19}^{2} - 23T_{19} + 21 \)
|
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\( T_{31}^{3} - 3T_{31}^{2} - 42T_{31} - 37 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 17 T^{4} + \cdots + 9 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( (T^{3} - 4 T^{2} - 3 T + 9)^{2} \)
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| $13$ |
\( T^{6} + 38 T^{4} + \cdots + 9 \)
|
| $17$ |
\( T^{6} + 82 T^{4} + \cdots + 6561 \)
|
| $19$ |
\( (T^{3} + 2 T^{2} - 23 T + 21)^{2} \)
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| $23$ |
\( T^{6} + 81 T^{4} + \cdots + 6561 \)
|
| $29$ |
\( (T^{3} - 45 T - 81)^{2} \)
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| $31$ |
\( (T^{3} - 3 T^{2} - 42 T - 37)^{2} \)
|
| $37$ |
\( T^{6} + 162 T^{4} + \cdots + 22201 \)
|
| $41$ |
\( (T^{3} + 11 T^{2} + \cdots - 873)^{2} \)
|
| $43$ |
\( T^{6} + 93 T^{4} + \cdots + 5041 \)
|
| $47$ |
\( T^{6} + 58 T^{4} + \cdots + 81 \)
|
| $53$ |
\( T^{6} + 226 T^{4} + \cdots + 301401 \)
|
| $59$ |
\( (T^{3} + 5 T^{2} - 9)^{2} \)
|
| $61$ |
\( (T^{3} + 17 T^{2} + 26 T + 1)^{2} \)
|
| $67$ |
\( T^{6} + 200 T^{4} + \cdots + 5184 \)
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| $71$ |
\( (T^{3} + 19 T^{2} + \cdots + 45)^{2} \)
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| $73$ |
\( (T^{2} + 16)^{3} \)
|
| $79$ |
\( (T^{3} - T^{2} - 118 T - 449)^{2} \)
|
| $83$ |
\( T^{6} + 526 T^{4} + \cdots + 962361 \)
|
| $89$ |
\( (T^{3} + 14 T^{2} + \cdots - 45)^{2} \)
|
| $97$ |
\( T^{6} + 261 T^{4} + \cdots + 25 \)
|
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