Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3465,2,Mod(1,3465)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3465, 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("3465.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3465.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: | \(27.6681643004\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.352076.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 8x^{3} + 3x^{2} + 8x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 1155) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 8x^{3} + 3x^{2} + 8x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 8\nu^{2} + 3\nu + 6 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 9\nu^{2} - 5\nu + 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} + \nu^{3} - 10\nu^{2} - 9\nu + 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( 2\nu^{4} - \nu^{3} - 15\nu^{2} - 4\nu + 8 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 2\beta_{2} + 2\beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{4} + \beta_{3} - 4\beta_{2} - 2\beta _1 + 13 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{4} + 9\beta_{3} - 16\beta_{2} + 6\beta _1 + 13 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 9\beta_{4} + 7\beta_{3} - 21\beta_{2} - 4\beta _1 + 45 ) / 2 \)
|
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 |
|
−2.66516 | 0 | 5.10307 | 1.00000 | 0 | 1.00000 | −8.27017 | 0 | −2.66516 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.88068 | 0 | 1.53695 | 1.00000 | 0 | 1.00000 | 0.870842 | 0 | −1.88068 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.346466 | 0 | −1.87996 | 1.00000 | 0 | 1.00000 | 1.34427 | 0 | −0.346466 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.36955 | 0 | −0.124319 | 1.00000 | 0 | 1.00000 | −2.90937 | 0 | 1.36955 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.52275 | 0 | 4.36426 | 1.00000 | 0 | 1.00000 | 5.96443 | 0 | 2.52275 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
| \(11\) | \(1\) |
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 | 3465.2.a.bm | 5 | |
| 3.b | odd | 2 | 1 | 1155.2.a.w | ✓ | 5 | |
| 15.d | odd | 2 | 1 | 5775.2.a.cg | 5 | ||
| 21.c | even | 2 | 1 | 8085.2.a.bv | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1155.2.a.w | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 3465.2.a.bm | 5 | 1.a | even | 1 | 1 | trivial | |
| 5775.2.a.cg | 5 | 15.d | odd | 2 | 1 | ||
| 8085.2.a.bv | 5 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3465))\):
|
\( T_{2}^{5} + T_{2}^{4} - 9T_{2}^{3} - 7T_{2}^{2} + 16T_{2} + 6 \)
|
|
\( T_{13}^{5} - 8T_{13}^{4} - 10T_{13}^{3} + 164T_{13}^{2} - 40T_{13} - 784 \)
|
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\( T_{17}^{5} - 63T_{17}^{3} - 70T_{17}^{2} + 380T_{17} + 504 \)
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\( T_{19}^{5} - 6T_{19}^{4} - 3T_{19}^{3} + 44T_{19}^{2} - 64 \)
|
|
\( T_{23}^{5} - 2T_{23}^{4} - 47T_{23}^{3} - 76T_{23}^{2} + 128T_{23} + 192 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + T^{4} - 9 T^{3} + \cdots + 6 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( (T - 1)^{5} \)
|
| $7$ |
\( (T - 1)^{5} \)
|
| $11$ |
\( (T + 1)^{5} \)
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| $13$ |
\( T^{5} - 8 T^{4} + \cdots - 784 \)
|
| $17$ |
\( T^{5} - 63 T^{3} + \cdots + 504 \)
|
| $19$ |
\( T^{5} - 6 T^{4} + \cdots - 64 \)
|
| $23$ |
\( T^{5} - 2 T^{4} + \cdots + 192 \)
|
| $29$ |
\( T^{5} + 6 T^{4} + \cdots + 4872 \)
|
| $31$ |
\( T^{5} - 10 T^{4} + \cdots - 128 \)
|
| $37$ |
\( T^{5} - 4 T^{4} + \cdots - 784 \)
|
| $41$ |
\( T^{5} - 122 T^{3} + \cdots - 10128 \)
|
| $43$ |
\( T^{5} - 167 T^{3} + \cdots + 11072 \)
|
| $47$ |
\( T^{5} - 2 T^{4} + \cdots + 9216 \)
|
| $53$ |
\( T^{5} - 8 T^{4} + \cdots - 3576 \)
|
| $59$ |
\( T^{5} - 4 T^{4} + \cdots - 16704 \)
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| $61$ |
\( T^{5} - 16 T^{4} + \cdots - 17144 \)
|
| $67$ |
\( T^{5} - 16 T^{4} + \cdots + 256 \)
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| $71$ |
\( T^{5} - 6 T^{4} + \cdots - 24576 \)
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| $73$ |
\( T^{5} - 2 T^{4} + \cdots - 67264 \)
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| $79$ |
\( T^{5} - 42 T^{4} + \cdots - 25216 \)
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| $83$ |
\( T^{5} - 22 T^{4} + \cdots - 17088 \)
|
| $89$ |
\( T^{5} - 10 T^{4} + \cdots - 78696 \)
|
| $97$ |
\( T^{5} + 2 T^{4} + \cdots - 59192 \)
|
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