Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7225,2,Mod(1,7225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7225, 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("7225.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7225 = 5^{2} \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7225.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: | \(57.6919154604\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.199789929.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 9x^{4} + 5x^{3} + 21x^{2} - 3x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 9x^{4} + 5x^{3} + 21x^{2} - 3x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 7\nu \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 6\nu^{3} + 9\nu^{2} + 8\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 6\beta_{2} + 7\beta _1 + 14 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + 10\beta_{3} + 9\beta_{2} + 27\beta _1 + 11 \)
|
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 |
|
−2.63233 | 2.78135 | 4.92916 | 0 | −7.32143 | 3.24325 | −7.71051 | 4.73590 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.08568 | −1.61993 | 2.35007 | 0 | 3.37867 | −1.02312 | −0.730139 | −0.375818 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.451353 | 0.0828196 | −1.79628 | 0 | −0.0373808 | 3.20219 | 1.71346 | −2.99314 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.330308 | 3.17609 | −1.89090 | 0 | 1.04909 | −1.66459 | −1.28519 | 7.08755 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.78030 | 0.309133 | 1.16947 | 0 | 0.550349 | −2.80925 | −1.47860 | −2.90444 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.05876 | −2.72946 | 2.23848 | 0 | −5.61929 | 5.05151 | 0.490977 | 4.44994 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(17\) | \( +1 \) |
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 | 7225.2.a.bd | yes | 6 |
| 5.b | even | 2 | 1 | 7225.2.a.be | yes | 6 | |
| 17.b | even | 2 | 1 | 7225.2.a.bc | ✓ | 6 | |
| 85.c | even | 2 | 1 | 7225.2.a.bf | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7225.2.a.bc | ✓ | 6 | 17.b | even | 2 | 1 | |
| 7225.2.a.bd | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 7225.2.a.be | yes | 6 | 5.b | even | 2 | 1 | |
| 7225.2.a.bf | yes | 6 | 85.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7225))\):
|
\( T_{2}^{6} + T_{2}^{5} - 9T_{2}^{4} - 5T_{2}^{3} + 21T_{2}^{2} + 3T_{2} - 3 \)
|
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\( T_{3}^{6} - 2T_{3}^{5} - 12T_{3}^{4} + 17T_{3}^{3} + 34T_{3}^{2} - 15T_{3} + 1 \)
|
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\( T_{7}^{6} - 6T_{7}^{5} - 11T_{7}^{4} + 82T_{7}^{3} + 54T_{7}^{2} - 280T_{7} - 251 \)
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\( T_{11}^{6} + 6T_{11}^{5} - 32T_{11}^{4} - 161T_{11}^{3} + 348T_{11}^{2} + 747T_{11} - 1317 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} - 9 T^{4} + \cdots - 3 \)
|
| $3$ |
\( T^{6} - 2 T^{5} + \cdots + 1 \)
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| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 6 T^{5} + \cdots - 251 \)
|
| $11$ |
\( T^{6} + 6 T^{5} + \cdots - 1317 \)
|
| $13$ |
\( T^{6} + T^{5} + \cdots + 529 \)
|
| $17$ |
\( T^{6} \)
|
| $19$ |
\( T^{6} + 6 T^{5} + \cdots + 27 \)
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| $23$ |
\( T^{6} - 16 T^{5} + \cdots + 81 \)
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| $29$ |
\( T^{6} - 19 T^{5} + \cdots - 216 \)
|
| $31$ |
\( T^{6} - 7 T^{5} + \cdots - 8523 \)
|
| $37$ |
\( T^{6} - 5 T^{5} + \cdots - 293 \)
|
| $41$ |
\( T^{6} - 13 T^{5} + \cdots + 53691 \)
|
| $43$ |
\( T^{6} + T^{5} + \cdots + 1657 \)
|
| $47$ |
\( T^{6} - 14 T^{5} + \cdots + 35829 \)
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| $53$ |
\( T^{6} - 5 T^{5} + \cdots + 170109 \)
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| $59$ |
\( T^{6} - 11 T^{5} + \cdots - 115377 \)
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| $61$ |
\( T^{6} - 19 T^{5} + \cdots - 2699 \)
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| $67$ |
\( T^{6} - 23 T^{5} + \cdots + 338167 \)
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| $71$ |
\( T^{6} - 8 T^{5} + \cdots + 11691 \)
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| $73$ |
\( T^{6} - 33 T^{5} + \cdots - 34184 \)
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| $79$ |
\( T^{6} - 12 T^{5} + \cdots - 2393 \)
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| $83$ |
\( T^{6} - 3 T^{5} + \cdots + 14961 \)
|
| $89$ |
\( T^{6} - 23 T^{5} + \cdots + 843021 \)
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| $97$ |
\( T^{6} + 10 T^{5} + \cdots + 137939 \)
|
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