Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9801,2,Mod(1,9801)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9801, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("9801.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9801 = 3^{4} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9801.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: | \(78.2613790211\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{5}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 99) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−2.61803 | 0 | 4.85410 | −3.23607 | 0 | 1.00000 | −7.47214 | 0 | 8.47214 | ||||||||||||||||||||||||
| 1.2 | −0.381966 | 0 | −1.85410 | 1.23607 | 0 | 1.00000 | 1.47214 | 0 | −0.472136 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(11\) | \( -1 \) |
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 | 9801.2.a.m | 2 | |
| 3.b | odd | 2 | 1 | 9801.2.a.bc | 2 | ||
| 9.d | odd | 6 | 2 | 1089.2.e.d | 4 | ||
| 11.b | odd | 2 | 1 | 9801.2.a.bb | 2 | ||
| 11.d | odd | 10 | 2 | 891.2.f.a | 4 | ||
| 33.d | even | 2 | 1 | 9801.2.a.n | 2 | ||
| 33.f | even | 10 | 2 | 891.2.f.b | 4 | ||
| 99.g | even | 6 | 2 | 1089.2.e.g | 4 | ||
| 99.o | odd | 30 | 4 | 297.2.n.a | 8 | ||
| 99.p | even | 30 | 4 | 99.2.m.a | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 99.2.m.a | ✓ | 8 | 99.p | even | 30 | 4 | |
| 297.2.n.a | 8 | 99.o | odd | 30 | 4 | ||
| 891.2.f.a | 4 | 11.d | odd | 10 | 2 | ||
| 891.2.f.b | 4 | 33.f | even | 10 | 2 | ||
| 1089.2.e.d | 4 | 9.d | odd | 6 | 2 | ||
| 1089.2.e.g | 4 | 99.g | even | 6 | 2 | ||
| 9801.2.a.m | 2 | 1.a | even | 1 | 1 | trivial | |
| 9801.2.a.n | 2 | 33.d | even | 2 | 1 | ||
| 9801.2.a.bb | 2 | 11.b | odd | 2 | 1 | ||
| 9801.2.a.bc | 2 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9801))\):
|
\( T_{2}^{2} + 3T_{2} + 1 \)
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\( T_{5}^{2} + 2T_{5} - 4 \)
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\( T_{7} - 1 \)
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\( T_{17}^{2} - 3T_{17} - 9 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 3T + 1 \)
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| $3$ |
\( T^{2} \)
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| $5$ |
\( T^{2} + 2T - 4 \)
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| $7$ |
\( (T - 1)^{2} \)
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| $11$ |
\( T^{2} \)
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| $13$ |
\( T^{2} - 4T - 16 \)
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| $17$ |
\( T^{2} - 3T - 9 \)
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| $19$ |
\( (T + 1)^{2} \)
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| $23$ |
\( T^{2} + 7T + 11 \)
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| $29$ |
\( T^{2} + 3T - 9 \)
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| $31$ |
\( T^{2} + T - 1 \)
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| $37$ |
\( T^{2} - T - 31 \)
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| $41$ |
\( T^{2} + 14T + 29 \)
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| $43$ |
\( T^{2} - 3T - 9 \)
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| $47$ |
\( T^{2} + 6T - 71 \)
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| $53$ |
\( T^{2} - 3T - 29 \)
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| $59$ |
\( T^{2} + T - 1 \)
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| $61$ |
\( T^{2} + 15T + 45 \)
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| $67$ |
\( (T - 6)^{2} \)
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| $71$ |
\( T^{2} - 17T + 41 \)
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| $73$ |
\( T^{2} - 7T + 1 \)
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| $79$ |
\( T^{2} + T - 101 \)
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| $83$ |
\( T^{2} + 24T + 139 \)
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| $89$ |
\( T^{2} - 18T + 76 \)
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| $97$ |
\( (T - 6)^{2} \)
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