Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2008,2,Mod(1,2008)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2008, 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("2008.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2008 = 2^{3} \cdot 251 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2008.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: | \(16.0339607259\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 3 x^{11} - 17 x^{10} + 49 x^{9} + 106 x^{8} - 277 x^{7} - 317 x^{6} + 644 x^{5} + 537 x^{4} + \cdots + 104 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 3 x^{11} - 17 x^{10} + 49 x^{9} + 106 x^{8} - 277 x^{7} - 317 x^{6} + 644 x^{5} + 537 x^{4} + \cdots + 104 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5 \nu^{11} - 13 \nu^{10} - 72 \nu^{9} + 159 \nu^{8} + 331 \nu^{7} - 431 \nu^{6} - 551 \nu^{5} + \cdots - 728 ) / 26 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 15 \nu^{11} - 13 \nu^{10} + 385 \nu^{9} + 251 \nu^{8} - 3424 \nu^{7} - 1801 \nu^{6} + 12521 \nu^{5} + \cdots + 2132 ) / 52 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 32 \nu^{11} - 117 \nu^{10} - 427 \nu^{9} + 1743 \nu^{8} + 1601 \nu^{7} - 8390 \nu^{6} - 1033 \nu^{5} + \cdots + 1612 ) / 26 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 29 \nu^{11} - 13 \nu^{10} + 701 \nu^{9} + 331 \nu^{8} - 5994 \nu^{7} - 3007 \nu^{6} + 21401 \nu^{5} + \cdots + 4680 ) / 52 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 113 \nu^{11} + 91 \nu^{10} + 2363 \nu^{9} - 1009 \nu^{8} - 18008 \nu^{7} + 1249 \nu^{6} + \cdots + 10556 ) / 52 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 74 \nu^{11} + 91 \nu^{10} + 1466 \nu^{9} - 1165 \nu^{8} - 10624 \nu^{7} + 3628 \nu^{6} + 33549 \nu^{5} + \cdots + 5304 ) / 26 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 45 \nu^{11} - 52 \nu^{10} - 895 \nu^{9} + 638 \nu^{8} + 6515 \nu^{7} - 1669 \nu^{6} - 20676 \nu^{5} + \cdots - 3757 ) / 13 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 124 \nu^{11} - 208 \nu^{10} - 2303 \nu^{9} + 2820 \nu^{8} + 15585 \nu^{7} - 10694 \nu^{6} - 46274 \nu^{5} + \cdots - 6396 ) / 26 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 127 \nu^{11} - 39 \nu^{10} + 3017 \nu^{9} + 1177 \nu^{8} - 25492 \nu^{7} - 11813 \nu^{6} + \cdots + 20696 ) / 52 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} - \beta_{9} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} + 2\beta_{10} + \beta_{8} + \beta_{6} - 3\beta_{5} + 10\beta_{2} + 10\beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2 \beta_{11} + 11 \beta_{10} - 8 \beta_{9} + 12 \beta_{6} - 14 \beta_{5} - 8 \beta_{4} + 9 \beta_{3} + \cdots + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( 12 \beta_{11} + 27 \beta_{10} - 2 \beta_{9} + 11 \beta_{8} + 20 \beta_{6} - 41 \beta_{5} + 2 \beta_{3} + \cdots + 106 \)
|
| \(\nu^{7}\) | \(=\) |
\( 29 \beta_{11} + 106 \beta_{10} - 55 \beta_{9} + 5 \beta_{8} + 2 \beta_{7} + 125 \beta_{6} - 151 \beta_{5} + \cdots + 111 \)
|
| \(\nu^{8}\) | \(=\) |
\( 120 \beta_{11} + 285 \beta_{10} - 25 \beta_{9} + 106 \beta_{8} - 3 \beta_{7} + 268 \beta_{6} + \cdots + 772 \)
|
| \(\nu^{9}\) | \(=\) |
\( 324 \beta_{11} + 986 \beta_{10} - 357 \beta_{9} + 110 \beta_{8} + 18 \beta_{7} + 1256 \beta_{6} + \cdots + 1089 \)
|
| \(\nu^{10}\) | \(=\) |
\( 1156 \beta_{11} + 2790 \beta_{10} - 201 \beta_{9} + 1028 \beta_{8} - 77 \beta_{7} + 3092 \beta_{6} + \cdots + 5946 \)
|
| \(\nu^{11}\) | \(=\) |
\( 3324 \beta_{11} + 9088 \beta_{10} - 2190 \beta_{9} + 1637 \beta_{8} + 22 \beta_{7} + 12417 \beta_{6} + \cdots + 10436 \)
|
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 |
|
0 | −2.54171 | 0 | 1.59809 | 0 | 2.13408 | 0 | 3.46027 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.41194 | 0 | −1.89365 | 0 | 0.462635 | 0 | 2.81747 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.20604 | 0 | −1.79940 | 0 | −2.78727 | 0 | −1.54547 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.18772 | 0 | 0.261605 | 0 | 3.40145 | 0 | −1.58932 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.770730 | 0 | 3.57480 | 0 | 0.727506 | 0 | −2.40598 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −0.589492 | 0 | −1.64401 | 0 | −1.73430 | 0 | −2.65250 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0.571724 | 0 | 0.953980 | 0 | −4.33542 | 0 | −2.67313 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.57790 | 0 | −2.90606 | 0 | 2.41450 | 0 | −0.510222 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 1.58102 | 0 | 3.46315 | 0 | 1.91212 | 0 | −0.500388 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.20058 | 0 | 2.09881 | 0 | 3.80569 | 0 | 1.84254 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 2.69680 | 0 | −1.00937 | 0 | −0.288703 | 0 | 4.27276 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 3.07960 | 0 | 2.30205 | 0 | −0.712294 | 0 | 6.48396 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(251\) | \(-1\) |
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 | 2008.2.a.b | ✓ | 12 |
| 4.b | odd | 2 | 1 | 4016.2.a.i | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2008.2.a.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 4016.2.a.i | 12 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} - 3 T_{3}^{11} - 17 T_{3}^{10} + 49 T_{3}^{9} + 106 T_{3}^{8} - 277 T_{3}^{7} - 317 T_{3}^{6} + \cdots + 104 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2008))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} - 3 T^{11} + \cdots + 104 \)
|
| $5$ |
\( T^{12} - 5 T^{11} + \cdots - 392 \)
|
| $7$ |
\( T^{12} - 5 T^{11} + \cdots - 185 \)
|
| $11$ |
\( T^{12} - 10 T^{11} + \cdots + 88 \)
|
| $13$ |
\( T^{12} - 3 T^{11} + \cdots + 5912 \)
|
| $17$ |
\( T^{12} - 2 T^{11} + \cdots + 17 \)
|
| $19$ |
\( T^{12} - 15 T^{11} + \cdots + 2560 \)
|
| $23$ |
\( T^{12} - 20 T^{11} + \cdots + 91363 \)
|
| $29$ |
\( T^{12} - 6 T^{11} + \cdots - 177464 \)
|
| $31$ |
\( T^{12} - 14 T^{11} + \cdots + 194260 \)
|
| $37$ |
\( T^{12} - 5 T^{11} + \cdots + 20949032 \)
|
| $41$ |
\( T^{12} - 234 T^{10} + \cdots - 299045 \)
|
| $43$ |
\( T^{12} - 21 T^{11} + \cdots - 10896776 \)
|
| $47$ |
\( T^{12} - 27 T^{11} + \cdots - 2083520 \)
|
| $53$ |
\( T^{12} + \cdots - 8160829600 \)
|
| $59$ |
\( T^{12} + \cdots + 18510238208 \)
|
| $61$ |
\( T^{12} - 4 T^{11} + \cdots + 147224 \)
|
| $67$ |
\( T^{12} - 26 T^{11} + \cdots - 20820056 \)
|
| $71$ |
\( T^{12} - 23 T^{11} + \cdots - 68742656 \)
|
| $73$ |
\( T^{12} + 8 T^{11} + \cdots + 13278400 \)
|
| $79$ |
\( T^{12} + \cdots + 201852769 \)
|
| $83$ |
\( T^{12} - 30 T^{11} + \cdots + 17740288 \)
|
| $89$ |
\( T^{12} - 3 T^{11} + \cdots - 31637125 \)
|
| $97$ |
\( T^{12} + 4 T^{11} + \cdots + 7482944 \)
|
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