Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6003,2,Mod(1,6003)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6003, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("6003.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6003 = 3^{2} \cdot 23 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6003.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: | \(47.9341963334\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{10} - 3x^{9} - 10x^{8} + 32x^{7} + 32x^{6} - 118x^{5} - 29x^{4} + 182x^{3} - 28x^{2} - 101x + 43 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 667) |
| 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
\( x^{10} - 3x^{9} - 10x^{8} + 32x^{7} + 32x^{6} - 118x^{5} - 29x^{4} + 182x^{3} - 28x^{2} - 101x + 43 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{8} - 3\nu^{7} - 8\nu^{6} + 26\nu^{5} + 16\nu^{4} - 66\nu^{3} + 3\nu^{2} + 50\nu - 22 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{8} - 4\nu^{7} - 6\nu^{6} + 36\nu^{5} - \nu^{4} - 98\nu^{3} + 45\nu^{2} + 83\nu - 53 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{9} + 4\nu^{8} + 5\nu^{7} - 34\nu^{6} + 10\nu^{5} + 82\nu^{4} - 69\nu^{3} - 47\nu^{2} + 72\nu - 22 \)
|
| \(\beta_{6}\) | \(=\) |
\( 2\nu^{8} - 7\nu^{7} - 14\nu^{6} + 63\nu^{5} + 14\nu^{4} - 172\nu^{3} + 54\nu^{2} + 146\nu - 83 \)
|
| \(\beta_{7}\) | \(=\) |
\( 3\nu^{9} - 10\nu^{8} - 23\nu^{7} + 91\nu^{6} + 39\nu^{5} - 253\nu^{4} + 32\nu^{3} + 226\nu^{2} - 84\nu - 17 \)
|
| \(\beta_{8}\) | \(=\) |
\( \nu^{9} - \nu^{8} - 16\nu^{7} + 14\nu^{6} + 89\nu^{5} - 69\nu^{4} - 200\nu^{3} + 144\nu^{2} + 153\nu - 109 \)
|
| \(\beta_{9}\) | \(=\) |
\( 2\nu^{9} - 9\nu^{8} - 7\nu^{7} + 77\nu^{6} - 49\nu^{5} - 186\nu^{4} + 225\nu^{3} + 94\nu^{2} - 226\nu + 78 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} - \beta_{6} + \beta_{5} - \beta_{4} + 2\beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{9} + \beta_{7} + \beta_{5} - 2\beta_{4} - \beta_{3} + 8\beta_{2} + 2\beta _1 + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{9} + 8\beta_{8} + \beta_{7} - 7\beta_{6} + 9\beta_{5} - 11\beta_{4} - 2\beta_{3} + 18\beta_{2} + 21\beta _1 + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( -8\beta_{9} + 3\beta_{8} + 9\beta_{7} + 14\beta_{5} - 24\beta_{4} - 11\beta_{3} + 57\beta_{2} + 23\beta _1 + 69 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 9 \beta_{9} + 54 \beta_{8} + 11 \beta_{7} - 38 \beta_{6} + 69 \beta_{5} - 93 \beta_{4} - 24 \beta_{3} + \cdots + 90 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 49 \beta_{9} + 44 \beta_{8} + 63 \beta_{7} + 2 \beta_{6} + 135 \beta_{5} - 219 \beta_{4} - 91 \beta_{3} + \cdots + 407 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 61 \beta_{9} + 355 \beta_{8} + 93 \beta_{7} - 183 \beta_{6} + 511 \beta_{5} - 730 \beta_{4} + \cdots + 676 \)
|
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 |
|
−2.17460 | 0 | 2.72887 | 1.26068 | 0 | 1.97284 | −1.58501 | 0 | −2.74147 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.57663 | 0 | 0.485749 | 1.88241 | 0 | −0.868755 | 2.38741 | 0 | −2.96785 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.35371 | 0 | −0.167476 | −1.38626 | 0 | −1.97610 | 2.93413 | 0 | 1.87659 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.31926 | 0 | −0.259562 | 4.11119 | 0 | −2.74867 | 2.98094 | 0 | −5.42371 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.685481 | 0 | −1.53012 | 0.380940 | 0 | −0.405330 | −2.41983 | 0 | 0.261127 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.788514 | 0 | −1.37825 | 4.42591 | 0 | 4.56405 | −2.66379 | 0 | 3.48989 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.21378 | 0 | −0.526738 | −2.40259 | 0 | −0.534912 | −3.06690 | 0 | −2.91621 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.68137 | 0 | 0.827018 | 1.60084 | 0 | −4.81497 | −1.97222 | 0 | 2.69161 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.37954 | 0 | 3.66223 | 2.05928 | 0 | 5.08987 | 3.95536 | 0 | 4.90014 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.67549 | 0 | 5.15827 | −1.93239 | 0 | 0.721979 | 8.44992 | 0 | −5.17011 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(23\) | \(1\) |
| \(29\) | \(1\) |
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 | 6003.2.a.l | 10 | |
| 3.b | odd | 2 | 1 | 667.2.a.a | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 667.2.a.a | ✓ | 10 | 3.b | odd | 2 | 1 | |
| 6003.2.a.l | 10 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):
|
\( T_{2}^{10} - 3T_{2}^{9} - 10T_{2}^{8} + 32T_{2}^{7} + 32T_{2}^{6} - 118T_{2}^{5} - 29T_{2}^{4} + 182T_{2}^{3} - 28T_{2}^{2} - 101T_{2} + 43 \)
|
|
\( T_{5}^{10} - 10 T_{5}^{9} + 20 T_{5}^{8} + 82 T_{5}^{7} - 310 T_{5}^{6} - 27 T_{5}^{5} + 1109 T_{5}^{4} + \cdots - 349 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 3 T^{9} + \cdots + 43 \)
|
| $3$ |
\( T^{10} \)
|
| $5$ |
\( T^{10} - 10 T^{9} + \cdots - 349 \)
|
| $7$ |
\( T^{10} - T^{9} + \cdots + 163 \)
|
| $11$ |
\( T^{10} - 64 T^{8} + \cdots + 9599 \)
|
| $13$ |
\( T^{10} + 13 T^{9} + \cdots + 18783 \)
|
| $17$ |
\( T^{10} - 22 T^{9} + \cdots + 7129 \)
|
| $19$ |
\( T^{10} + 2 T^{9} + \cdots + 5125 \)
|
| $23$ |
\( (T + 1)^{10} \)
|
| $29$ |
\( (T + 1)^{10} \)
|
| $31$ |
\( T^{10} + 22 T^{9} + \cdots - 416511 \)
|
| $37$ |
\( T^{10} + 9 T^{9} + \cdots + 113329 \)
|
| $41$ |
\( T^{10} - 25 T^{9} + \cdots + 4666817 \)
|
| $43$ |
\( T^{10} - 3 T^{9} + \cdots - 66033 \)
|
| $47$ |
\( T^{10} + \cdots - 176953097 \)
|
| $53$ |
\( T^{10} - 43 T^{9} + \cdots + 17317103 \)
|
| $59$ |
\( T^{10} - 7 T^{9} + \cdots - 603431 \)
|
| $61$ |
\( T^{10} + 6 T^{9} + \cdots + 315433 \)
|
| $67$ |
\( T^{10} - 11 T^{9} + \cdots - 5078341 \)
|
| $71$ |
\( T^{10} + \cdots - 249705089 \)
|
| $73$ |
\( T^{10} + 44 T^{9} + \cdots + 874591 \)
|
| $79$ |
\( T^{10} - 5 T^{9} + \cdots - 6631087 \)
|
| $83$ |
\( T^{10} - 32 T^{9} + \cdots + 3705241 \)
|
| $89$ |
\( T^{10} - 10 T^{9} + \cdots + 12231211 \)
|
| $97$ |
\( T^{10} - 6 T^{9} + \cdots + 86444003 \)
|
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