Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6018,2,Mod(1,6018)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6018, 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("6018.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6018.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: | \(48.0539719364\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} - 17x^{6} + 37x^{5} + 105x^{4} - 117x^{3} - 238x^{2} + 42x + 90 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} - 17x^{6} + 37x^{5} + 105x^{4} - 117x^{3} - 238x^{2} + 42x + 90 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 23\nu^{7} - 126\nu^{6} - 82\nu^{5} + 1079\nu^{4} - 276\nu^{3} - 2202\nu^{2} + 184\nu + 810 ) / 30 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 34\nu^{7} - 183\nu^{6} - 146\nu^{5} + 1612\nu^{4} - 213\nu^{3} - 3456\nu^{2} - 58\nu + 1200 ) / 30 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 34\nu^{7} - 183\nu^{6} - 146\nu^{5} + 1612\nu^{4} - 213\nu^{3} - 3486\nu^{2} - 28\nu + 1350 ) / 30 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 43\nu^{7} - 231\nu^{6} - 182\nu^{5} + 2029\nu^{4} - 321\nu^{3} - 4362\nu^{2} + 164\nu + 1710 ) / 30 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -77\nu^{7} + 414\nu^{6} + 328\nu^{5} - 3641\nu^{4} + 549\nu^{3} + 7803\nu^{2} - 196\nu - 2925 ) / 15 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 169\nu^{7} - 903\nu^{6} - 746\nu^{5} + 7957\nu^{4} - 993\nu^{3} - 17106\nu^{2} + 92\nu + 6420 ) / 30 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta _1 + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + 2\beta_{5} - \beta_{4} + 3\beta_{3} + 7\beta _1 + 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{7} + 2\beta_{6} + 9\beta_{5} - 12\beta_{4} + 16\beta_{3} - 2\beta_{2} + 12\beta _1 + 45 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2\beta_{7} + 17\beta_{6} + 49\beta_{5} - 31\beta_{4} + 58\beta_{3} - 3\beta_{2} + 62\beta _1 + 106 \)
|
| \(\nu^{6}\) | \(=\) |
\( -10\beta_{7} + 59\beta_{6} + 206\beta_{5} - 175\beta_{4} + 249\beta_{3} - 26\beta_{2} + 153\beta _1 + 520 \)
|
| \(\nu^{7}\) | \(=\) |
\( -15\beta_{7} + 302\beta_{6} + 905\beta_{5} - 614\beta_{4} + 952\beta_{3} - 58\beta_{2} + 668\beta _1 + 1631 \)
|
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 |
|
−1.00000 | −1.00000 | 1.00000 | −3.85158 | 1.00000 | 4.10228 | −1.00000 | 1.00000 | 3.85158 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.00000 | 1.00000 | −2.25098 | 1.00000 | 0.744783 | −1.00000 | 1.00000 | 2.25098 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.00000 | 1.00000 | −1.89711 | 1.00000 | 1.64158 | −1.00000 | 1.00000 | 1.89711 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.00000 | 1.00000 | −1.03120 | 1.00000 | −2.85327 | −1.00000 | 1.00000 | 1.03120 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | −1.00000 | 1.00000 | −0.462452 | 1.00000 | −3.09978 | −1.00000 | 1.00000 | 0.462452 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | −1.00000 | 1.00000 | 2.51738 | 1.00000 | 4.42797 | −1.00000 | 1.00000 | −2.51738 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | −1.00000 | 1.00000 | 2.86769 | 1.00000 | 0.636539 | −1.00000 | 1.00000 | −2.86769 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | −1.00000 | 1.00000 | 3.10826 | 1.00000 | 0.399898 | −1.00000 | 1.00000 | −3.10826 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(17\) | \(1\) |
| \(59\) | \(-1\) |
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 | 6018.2.a.s | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6018.2.a.s | ✓ | 8 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):
|
\( T_{5}^{8} + T_{5}^{7} - 24T_{5}^{6} - 22T_{5}^{5} + 175T_{5}^{4} + 194T_{5}^{3} - 380T_{5}^{2} - 580T_{5} - 176 \)
|
|
\( T_{7}^{8} - 6T_{7}^{7} - 11T_{7}^{6} + 102T_{7}^{5} - 40T_{7}^{4} - 377T_{7}^{3} + 566T_{7}^{2} - 291T_{7} + 50 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{8} \)
|
| $3$ |
\( (T + 1)^{8} \)
|
| $5$ |
\( T^{8} + T^{7} + \cdots - 176 \)
|
| $7$ |
\( T^{8} - 6 T^{7} + \cdots + 50 \)
|
| $11$ |
\( T^{8} - 48 T^{6} + \cdots + 4416 \)
|
| $13$ |
\( T^{8} - 6 T^{7} + \cdots - 24 \)
|
| $17$ |
\( (T + 1)^{8} \)
|
| $19$ |
\( T^{8} + 7 T^{7} + \cdots - 3492 \)
|
| $23$ |
\( T^{8} + 5 T^{7} + \cdots + 4338 \)
|
| $29$ |
\( T^{8} + 15 T^{7} + \cdots - 45904 \)
|
| $31$ |
\( T^{8} - 21 T^{7} + \cdots + 3312 \)
|
| $37$ |
\( T^{8} - 7 T^{7} + \cdots - 304 \)
|
| $41$ |
\( T^{8} + T^{7} + \cdots + 2858 \)
|
| $43$ |
\( T^{8} - 14 T^{7} + \cdots - 3680 \)
|
| $47$ |
\( T^{8} + 8 T^{7} + \cdots - 473072 \)
|
| $53$ |
\( T^{8} - 8 T^{7} + \cdots - 12958 \)
|
| $59$ |
\( (T - 1)^{8} \)
|
| $61$ |
\( T^{8} - 242 T^{6} + \cdots + 1362080 \)
|
| $67$ |
\( T^{8} - 15 T^{7} + \cdots - 24048 \)
|
| $71$ |
\( T^{8} + 22 T^{7} + \cdots - 3215304 \)
|
| $73$ |
\( T^{8} - 13 T^{7} + \cdots + 1929118 \)
|
| $79$ |
\( T^{8} - 26 T^{7} + \cdots + 1328328 \)
|
| $83$ |
\( T^{8} - 30 T^{7} + \cdots - 29588796 \)
|
| $89$ |
\( T^{8} + 6 T^{7} + \cdots + 504146 \)
|
| $97$ |
\( T^{8} - 23 T^{7} + \cdots - 3151542 \)
|
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