Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6800,2,Mod(1,6800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6800, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6800.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6800 = 2^{4} \cdot 5^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6800.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: | \(54.2982733745\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.6224.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 6x^{2} - 2x + 5 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 85) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 6x^{2} - 2x + 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −1.44579 | 0 | 0 | 0 | 2.86537 | 0 | −0.909700 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0.203185 | 0 | 0 | 0 | 0.683735 | 0 | −2.95872 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 2.37033 | 0 | 0 | 0 | 5.03032 | 0 | 2.61845 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.87228 | 0 | 0 | 0 | 1.42058 | 0 | 5.24997 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(17\) | \( -1 \) |
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 | 6800.2.a.bw | 4 | |
| 4.b | odd | 2 | 1 | 425.2.a.g | 4 | ||
| 5.b | even | 2 | 1 | 6800.2.a.bt | 4 | ||
| 5.c | odd | 4 | 2 | 1360.2.e.d | 8 | ||
| 12.b | even | 2 | 1 | 3825.2.a.bj | 4 | ||
| 20.d | odd | 2 | 1 | 425.2.a.h | 4 | ||
| 20.e | even | 4 | 2 | 85.2.b.a | ✓ | 8 | |
| 60.h | even | 2 | 1 | 3825.2.a.bh | 4 | ||
| 60.l | odd | 4 | 2 | 765.2.b.c | 8 | ||
| 68.d | odd | 2 | 1 | 7225.2.a.v | 4 | ||
| 340.d | odd | 2 | 1 | 7225.2.a.w | 4 | ||
| 340.r | even | 4 | 2 | 1445.2.b.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 85.2.b.a | ✓ | 8 | 20.e | even | 4 | 2 | |
| 425.2.a.g | 4 | 4.b | odd | 2 | 1 | ||
| 425.2.a.h | 4 | 20.d | odd | 2 | 1 | ||
| 765.2.b.c | 8 | 60.l | odd | 4 | 2 | ||
| 1360.2.e.d | 8 | 5.c | odd | 4 | 2 | ||
| 1445.2.b.e | 8 | 340.r | even | 4 | 2 | ||
| 3825.2.a.bh | 4 | 60.h | even | 2 | 1 | ||
| 3825.2.a.bj | 4 | 12.b | even | 2 | 1 | ||
| 6800.2.a.bt | 4 | 5.b | even | 2 | 1 | ||
| 6800.2.a.bw | 4 | 1.a | even | 1 | 1 | trivial | |
| 7225.2.a.v | 4 | 68.d | odd | 2 | 1 | ||
| 7225.2.a.w | 4 | 340.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6800))\):
|
\( T_{3}^{4} - 4T_{3}^{3} + 10T_{3} - 2 \)
|
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\( T_{7}^{4} - 10T_{7}^{3} + 32T_{7}^{2} - 38T_{7} + 14 \)
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\( T_{11}^{4} - 2T_{11}^{3} - 14T_{11}^{2} + 26T_{11} + 2 \)
|
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\( T_{13}^{4} + 6T_{13}^{3} - 28T_{13}^{2} - 192T_{13} - 164 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 4 T^{3} + \cdots - 2 \)
|
| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} - 10 T^{3} + \cdots + 14 \)
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| $11$ |
\( T^{4} - 2 T^{3} + \cdots + 2 \)
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| $13$ |
\( T^{4} + 6 T^{3} + \cdots - 164 \)
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| $17$ |
\( (T - 1)^{4} \)
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| $19$ |
\( T^{4} + 4 T^{3} + \cdots + 112 \)
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| $23$ |
\( T^{4} - 4 T^{3} + \cdots - 10 \)
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| $29$ |
\( T^{4} + 4 T^{3} + \cdots - 80 \)
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| $31$ |
\( T^{4} - 12 T^{3} + \cdots + 74 \)
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| $37$ |
\( T^{4} + 12 T^{3} + \cdots - 16 \)
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| $41$ |
\( T^{4} + 6 T^{3} + \cdots + 392 \)
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| $43$ |
\( T^{4} - 18 T^{3} + \cdots - 316 \)
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| $47$ |
\( T^{4} - 6 T^{3} + \cdots - 164 \)
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| $53$ |
\( T^{4} + 8 T^{3} + \cdots + 16 \)
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| $59$ |
\( (T - 2)^{4} \)
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| $61$ |
\( T^{4} - 6 T^{3} + \cdots + 1880 \)
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| $67$ |
\( T^{4} - 6 T^{3} + \cdots - 52 \)
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| $71$ |
\( T^{4} - 10 T^{3} + \cdots + 242 \)
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| $73$ |
\( T^{4} + 2 T^{3} + \cdots + 1256 \)
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| $79$ |
\( T^{4} - 12 T^{3} + \cdots - 526 \)
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| $83$ |
\( T^{4} + 14 T^{3} + \cdots - 956 \)
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| $89$ |
\( T^{4} - 24 T^{3} + \cdots - 484 \)
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| $97$ |
\( T^{4} - 4 T^{3} + \cdots + 2000 \)
|
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