Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2006,2,Mod(1,2006)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2006, 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("2006.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2006 = 2 \cdot 17 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2006.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: | \(16.0179906455\) |
| 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: |
\( x^{8} - 3x^{7} - 13x^{6} + 24x^{5} + 50x^{4} - 50x^{3} - 62x^{2} + 28x + 24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{8} - 3x^{7} - 13x^{6} + 24x^{5} + 50x^{4} - 50x^{3} - 62x^{2} + 28x + 24 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 4\nu^{7} - 3\nu^{6} - 91\nu^{5} + 31\nu^{4} + 474\nu^{3} - 58\nu^{2} - 486\nu - 100 ) / 86 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -13\nu^{7} - \nu^{6} + 285\nu^{5} + 168\nu^{4} - 1390\nu^{3} - 822\nu^{2} + 1386\nu + 540 ) / 172 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -9\nu^{7} - 4\nu^{6} + 194\nu^{5} + 199\nu^{4} - 916\nu^{3} - 966\nu^{2} + 986\nu + 784 ) / 86 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 49\nu^{7} - 155\nu^{6} - 545\nu^{5} + 1100\nu^{4} + 1442\nu^{3} - 1850\nu^{2} - 342\nu + 796 ) / 172 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 29\nu^{7} - 54\nu^{6} - 434\nu^{5} + 171\nu^{4} + 1652\nu^{3} + 676\nu^{2} - 1266\nu - 940 ) / 86 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -57\nu^{7} + 161\nu^{6} + 727\nu^{5} - 1162\nu^{4} - 2390\nu^{3} + 1966\nu^{2} + 1658\nu - 768 ) / 172 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 75\nu^{7} - 239\nu^{6} - 857\nu^{5} + 1710\nu^{4} + 2674\nu^{3} - 2614\nu^{2} - 1910\nu + 748 ) / 172 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + \beta_{4} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} + \beta_{4} - 2\beta_{3} + 4\beta_{2} + 3\beta _1 + 9 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 5\beta_{6} - 2\beta_{5} + 5\beta_{4} - 5\beta_{3} + 5\beta_{2} + 7\beta _1 + 11 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{7} + 25\beta_{6} - 12\beta_{5} + 31\beta_{4} - 46\beta_{3} + 64\beta_{2} + 57\beta _1 + 113 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 24\beta_{7} + 143\beta_{6} - 74\beta_{5} + 163\beta_{4} - 208\beta_{3} + 232\beta_{2} + 241\beta _1 + 419 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 27\beta_{7} + 255\beta_{6} - 141\beta_{5} + 320\beta_{4} - 447\beta_{3} + 550\beta_{2} + 514\beta _1 + 945 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 334 \beta_{7} + 2393 \beta_{6} - 1328 \beta_{5} + 2899 \beta_{4} - 3890 \beta_{3} + 4480 \beta_{2} + \cdots + 7769 ) / 2 \)
|
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 |
|
1.00000 | −1.94000 | 1.00000 | −3.29166 | −1.94000 | −4.13061 | 1.00000 | 0.763600 | −3.29166 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.82114 | 1.00000 | 3.81371 | −1.82114 | −0.872031 | 1.00000 | 0.316568 | 3.81371 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.60537 | 1.00000 | −3.23933 | −1.60537 | 5.21303 | 1.00000 | −0.422801 | −3.23933 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −0.530249 | 1.00000 | −4.02337 | −0.530249 | −3.82299 | 1.00000 | −2.71884 | −4.02337 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.80762 | 1.00000 | 0.972911 | 1.80762 | 0.354079 | 1.00000 | 0.267502 | 0.972911 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.62733 | 1.00000 | 2.88492 | 2.62733 | 3.10512 | 1.00000 | 3.90288 | 2.88492 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 3.14832 | 1.00000 | 0.589028 | 3.14832 | −2.62587 | 1.00000 | 6.91190 | 0.589028 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 3.31349 | 1.00000 | −2.70620 | 3.31349 | 2.77927 | 1.00000 | 7.97920 | −2.70620 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(17\) | \(1\) |
| \(59\) | \(1\) |
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 | 2006.2.a.t | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2006.2.a.t | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2006))\):
|
\( T_{3}^{8} - 5T_{3}^{7} - 8T_{3}^{6} + 57T_{3}^{5} + 26T_{3}^{4} - 225T_{3}^{3} - 80T_{3}^{2} + 301T_{3} + 149 \)
|
|
\( T_{5}^{8} + 5T_{5}^{7} - 22T_{5}^{6} - 127T_{5}^{5} + 107T_{5}^{4} + 896T_{5}^{3} - 32T_{5}^{2} - 1540T_{5} + 732 \)
|
|
\( T_{11}^{8} + 5T_{11}^{7} - 35T_{11}^{6} - 144T_{11}^{5} + 456T_{11}^{4} + 1168T_{11}^{3} - 2288T_{11}^{2} - 2176T_{11} + 2304 \)
|
|
\( T_{31}^{8} - 18 T_{31}^{7} + 9 T_{31}^{6} + 1434 T_{31}^{5} - 6327 T_{31}^{4} - 24238 T_{31}^{3} + \cdots - 492244 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( T^{8} - 5 T^{7} + \cdots + 149 \)
|
| $5$ |
\( T^{8} + 5 T^{7} + \cdots + 732 \)
|
| $7$ |
\( T^{8} - 42 T^{6} + \cdots + 576 \)
|
| $11$ |
\( T^{8} + 5 T^{7} + \cdots + 2304 \)
|
| $13$ |
\( T^{8} - T^{7} + \cdots + 79036 \)
|
| $17$ |
\( (T + 1)^{8} \)
|
| $19$ |
\( T^{8} - 8 T^{7} + \cdots + 151328 \)
|
| $23$ |
\( T^{8} + 8 T^{7} + \cdots - 10896 \)
|
| $29$ |
\( T^{8} - 20 T^{7} + \cdots + 28848 \)
|
| $31$ |
\( T^{8} - 18 T^{7} + \cdots - 492244 \)
|
| $37$ |
\( T^{8} + 7 T^{7} + \cdots - 50944 \)
|
| $41$ |
\( T^{8} - 12 T^{7} + \cdots + 972864 \)
|
| $43$ |
\( T^{8} + 8 T^{7} + \cdots + 72640 \)
|
| $47$ |
\( T^{8} - 30 T^{7} + \cdots - 17280 \)
|
| $53$ |
\( T^{8} + 5 T^{7} + \cdots + 845760 \)
|
| $59$ |
\( (T + 1)^{8} \)
|
| $61$ |
\( T^{8} - 12 T^{7} + \cdots + 12256256 \)
|
| $67$ |
\( T^{8} + 4 T^{7} + \cdots - 255600 \)
|
| $71$ |
\( T^{8} - 4 T^{7} + \cdots - 10420992 \)
|
| $73$ |
\( T^{8} + 18 T^{7} + \cdots + 336368 \)
|
| $79$ |
\( T^{8} + 17 T^{7} + \cdots - 16 \)
|
| $83$ |
\( T^{8} - 9 T^{7} + \cdots - 1323984 \)
|
| $89$ |
\( T^{8} + 12 T^{7} + \cdots - 1007232 \)
|
| $97$ |
\( T^{8} + 11 T^{7} + \cdots + 2257868 \)
|
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