Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4527,2,Mod(1,4527)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4527, 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("4527.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4527 = 3^{2} \cdot 503 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4527.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: | \(36.1482769950\) |
| Analytic rank: | \(1\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 2x^{9} - 9x^{8} + 14x^{7} + 27x^{6} - 27x^{5} - 34x^{4} + 14x^{3} + 17x^{2} + x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 503) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2x^{9} - 9x^{8} + 14x^{7} + 27x^{6} - 27x^{5} - 34x^{4} + 14x^{3} + 17x^{2} + x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 3\nu^{9} - 8\nu^{8} - 22\nu^{7} + 57\nu^{6} + 46\nu^{5} - 113\nu^{4} - 34\nu^{3} + 65\nu^{2} + 12\nu - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( 5\nu^{9} - 13\nu^{8} - 37\nu^{7} + 92\nu^{6} + 78\nu^{5} - 181\nu^{4} - 57\nu^{3} + 104\nu^{2} + 20\nu - 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( -6\nu^{9} + 15\nu^{8} + 47\nu^{7} - 108\nu^{6} - 113\nu^{5} + 221\nu^{4} + 108\nu^{3} - 139\nu^{2} - 44\nu + 13 \)
|
| \(\beta_{5}\) | \(=\) |
\( 10\nu^{9} - 25\nu^{8} - 78\nu^{7} + 180\nu^{6} + 184\nu^{5} - 368\nu^{4} - 165\nu^{3} + 230\nu^{2} + 60\nu - 21 \)
|
| \(\beta_{6}\) | \(=\) |
\( -11\nu^{9} + 28\nu^{8} + 84\nu^{7} - 200\nu^{6} - 191\nu^{5} + 402\nu^{4} + 165\nu^{3} - 244\nu^{2} - 63\nu + 22 \)
|
| \(\beta_{7}\) | \(=\) |
\( -18\nu^{9} + 45\nu^{8} + 139\nu^{7} - 321\nu^{6} - 321\nu^{5} + 645\nu^{4} + 278\nu^{3} - 394\nu^{2} - 102\nu + 38 \)
|
| \(\beta_{8}\) | \(=\) |
\( -21\nu^{9} + 53\nu^{8} + 162\nu^{7} - 380\nu^{6} - 375\nu^{5} + 770\nu^{4} + 331\nu^{3} - 475\nu^{2} - 127\nu + 45 \)
|
| \(\beta_{9}\) | \(=\) |
\( -22\nu^{9} + 55\nu^{8} + 170\nu^{7} - 392\nu^{6} - 394\nu^{5} + 785\nu^{4} + 346\nu^{3} - 473\nu^{2} - 130\nu + 42 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{6} + \beta_{4} - \beta_{3} + \beta _1 + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} - 2\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 5\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{9} + \beta_{8} + 2\beta_{7} - 9\beta_{6} + 2\beta_{5} + 8\beta_{4} - 6\beta_{3} - 2\beta_{2} + 10\beta _1 - 1 \)
|
| \(\nu^{5}\) | \(=\) |
\( - \beta_{9} + 8 \beta_{8} + 3 \beta_{7} - 22 \beta_{6} + 9 \beta_{5} + 12 \beta_{4} - 9 \beta_{3} + \cdots - 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 8 \beta_{9} + 12 \beta_{8} + 19 \beta_{7} - 73 \beta_{6} + 22 \beta_{5} + 58 \beta_{4} - 37 \beta_{3} + \cdots - 28 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 12 \beta_{9} + 58 \beta_{8} + 37 \beta_{7} - 192 \beta_{6} + 73 \beta_{5} + 113 \beta_{4} + \cdots - 107 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 58 \beta_{9} + 113 \beta_{8} + 152 \beta_{7} - 578 \beta_{6} + 192 \beta_{5} + 425 \beta_{4} + \cdots - 278 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 113 \beta_{9} + 425 \beta_{8} + 345 \beta_{7} - 1565 \beta_{6} + 578 \beta_{5} + 967 \beta_{4} + \cdots - 879 \)
|
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 |
|
−2.15783 | 0 | 2.65622 | 2.23445 | 0 | −3.60329 | −1.41602 | 0 | −4.82156 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.36113 | 0 | −0.147314 | −0.590303 | 0 | 1.95900 | 2.92278 | 0 | 0.803481 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.17266 | 0 | −0.624870 | −0.178789 | 0 | −0.0809018 | 3.07808 | 0 | 0.209658 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.0830530 | 0 | −1.99310 | −2.25024 | 0 | −3.20647 | 0.331639 | 0 | 0.186890 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.392284 | 0 | −1.84611 | 2.28693 | 0 | 2.71022 | −1.50877 | 0 | 0.897127 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.756417 | 0 | −1.42783 | −0.386144 | 0 | 0.194914 | −2.59287 | 0 | −0.292086 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.37178 | 0 | −0.118218 | −1.17276 | 0 | 0.469303 | −2.90573 | 0 | −1.60876 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.62786 | 0 | 0.649933 | 1.79865 | 0 | 0.552233 | −2.19772 | 0 | 2.92795 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.03947 | 0 | 2.15945 | 0.701114 | 0 | −2.02991 | 0.325186 | 0 | 1.42990 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.58686 | 0 | 4.69185 | −1.44291 | 0 | −1.96509 | 6.96343 | 0 | −3.73261 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(503\) | \(-1\) |
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 | 4527.2.a.k | 10 | |
| 3.b | odd | 2 | 1 | 503.2.a.e | ✓ | 10 | |
| 12.b | even | 2 | 1 | 8048.2.a.p | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 503.2.a.e | ✓ | 10 | 3.b | odd | 2 | 1 | |
| 4527.2.a.k | 10 | 1.a | even | 1 | 1 | trivial | |
| 8048.2.a.p | 10 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4527))\):
|
\( T_{2}^{10} - 4T_{2}^{9} - 4T_{2}^{8} + 31T_{2}^{7} - 13T_{2}^{6} - 66T_{2}^{5} + 56T_{2}^{4} + 37T_{2}^{3} - 46T_{2}^{2} + 8T_{2} + 1 \)
|
|
\( T_{5}^{10} - T_{5}^{9} - 11T_{5}^{8} + 7T_{5}^{7} + 41T_{5}^{6} - 7T_{5}^{5} - 59T_{5}^{4} - 16T_{5}^{3} + 18T_{5}^{2} + 9T_{5} + 1 \)
|
|
\( T_{7}^{10} + 5T_{7}^{9} - 9T_{7}^{8} - 64T_{7}^{7} + 8T_{7}^{6} + 228T_{7}^{5} + 23T_{7}^{4} - 214T_{7}^{3} + 86T_{7}^{2} - 4T_{7} - 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 4 T^{9} + \cdots + 1 \)
|
| $3$ |
\( T^{10} \)
|
| $5$ |
\( T^{10} - T^{9} + \cdots + 1 \)
|
| $7$ |
\( T^{10} + 5 T^{9} + \cdots - 1 \)
|
| $11$ |
\( T^{10} - 3 T^{9} + \cdots + 311 \)
|
| $13$ |
\( T^{10} + 18 T^{9} + \cdots - 19 \)
|
| $17$ |
\( T^{10} - 11 T^{9} + \cdots + 30151 \)
|
| $19$ |
\( T^{10} - 73 T^{8} + \cdots + 8863 \)
|
| $23$ |
\( T^{10} - 2 T^{9} + \cdots - 2281 \)
|
| $29$ |
\( T^{10} - 9 T^{9} + \cdots - 1397 \)
|
| $31$ |
\( T^{10} + 22 T^{9} + \cdots + 8207 \)
|
| $37$ |
\( T^{10} + 35 T^{9} + \cdots - 3774629 \)
|
| $41$ |
\( T^{10} - 4 T^{9} + \cdots - 17357 \)
|
| $43$ |
\( T^{10} + 20 T^{9} + \cdots - 147629 \)
|
| $47$ |
\( T^{10} + 7 T^{9} + \cdots + 34183 \)
|
| $53$ |
\( T^{10} - 24 T^{9} + \cdots + 30585517 \)
|
| $59$ |
\( T^{10} + 17 T^{9} + \cdots - 3373 \)
|
| $61$ |
\( T^{10} + \cdots + 160395869 \)
|
| $67$ |
\( T^{10} + 6 T^{9} + \cdots - 52161527 \)
|
| $71$ |
\( T^{10} - T^{9} + \cdots + 14183807 \)
|
| $73$ |
\( T^{10} + 31 T^{9} + \cdots - 3955559 \)
|
| $79$ |
\( T^{10} + 10 T^{9} + \cdots + 8912581 \)
|
| $83$ |
\( T^{10} + 22 T^{9} + \cdots + 40035623 \)
|
| $89$ |
\( T^{10} + T^{9} + \cdots - 789547 \)
|
| $97$ |
\( T^{10} + \cdots + 3229338523 \)
|
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