Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4004,2,Mod(1,4004)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4004, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("4004.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4004.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: | \(31.9721009693\) |
| 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: |
\( x^{10} - x^{9} - 23x^{8} + 23x^{7} + 170x^{6} - 165x^{5} - 411x^{4} + 360x^{3} + 111x^{2} - 48x - 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - x^{9} - 23x^{8} + 23x^{7} + 170x^{6} - 165x^{5} - 411x^{4} + 360x^{3} + 111x^{2} - 48x - 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 194 \nu^{9} + 241 \nu^{8} + 4395 \nu^{7} - 5633 \nu^{6} - 31377 \nu^{5} + 41297 \nu^{4} + \cdots + 9704 ) / 557 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 905 \nu^{9} + 1219 \nu^{8} + 20557 \nu^{7} - 27785 \nu^{6} - 147644 \nu^{5} + 198405 \nu^{4} + \cdots + 43902 ) / 2228 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 945 \nu^{9} + 1039 \nu^{8} + 21601 \nu^{7} - 24077 \nu^{6} - 157444 \nu^{5} + 173385 \nu^{4} + \cdots + 31662 ) / 2228 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 721 \nu^{9} + 933 \nu^{8} + 16423 \nu^{7} - 21225 \nu^{6} - 118160 \nu^{5} + 150853 \nu^{4} + \cdots + 28910 ) / 1114 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 439 \nu^{9} + 531 \nu^{8} + 9954 \nu^{7} - 12164 \nu^{6} - 71350 \nu^{5} + 87177 \nu^{4} + \cdots + 17727 ) / 557 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 439 \nu^{9} - 531 \nu^{8} - 9954 \nu^{7} + 12164 \nu^{6} + 71350 \nu^{5} - 87177 \nu^{4} + \cdots - 17170 ) / 557 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 1955 \nu^{9} - 2621 \nu^{8} - 44063 \nu^{7} + 60231 \nu^{6} + 312432 \nu^{5} - 433387 \nu^{4} + \cdots - 99134 ) / 2228 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + \beta_{7} + 8\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} + 2\beta_{4} + \beta_{3} + 9\beta_{2} - \beta _1 + 41 \)
|
| \(\nu^{5}\) | \(=\) |
\( 13\beta_{8} + 13\beta_{7} + 2\beta_{6} - \beta_{5} - 3\beta_{4} + \beta_{3} - \beta_{2} + 68\beta _1 - 14 \)
|
| \(\nu^{6}\) | \(=\) |
\( 15 \beta_{9} - 3 \beta_{8} + 14 \beta_{7} - 14 \beta_{6} - 16 \beta_{5} + 29 \beta_{4} + 11 \beta_{3} + \cdots + 362 \)
|
| \(\nu^{7}\) | \(=\) |
\( 5 \beta_{9} + 148 \beta_{8} + 144 \beta_{7} + 38 \beta_{6} - 16 \beta_{5} - 45 \beta_{4} + 23 \beta_{3} + \cdots - 161 \)
|
| \(\nu^{8}\) | \(=\) |
\( 175 \beta_{9} - 58 \beta_{8} + 154 \beta_{7} - 157 \beta_{6} - 198 \beta_{5} + 349 \beta_{4} + \cdots + 3313 \)
|
| \(\nu^{9}\) | \(=\) |
\( 108 \beta_{9} + 1614 \beta_{8} + 1506 \beta_{7} + 536 \beta_{6} - 195 \beta_{5} - 517 \beta_{4} + \cdots - 1740 \)
|
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 | −3.18458 | 0 | −2.71592 | 0 | −1.00000 | 0 | 7.14153 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.90534 | 0 | 1.80243 | 0 | −1.00000 | 0 | 5.44100 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.02595 | 0 | 3.89239 | 0 | −1.00000 | 0 | 1.10449 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.290239 | 0 | 0.725041 | 0 | −1.00000 | 0 | −2.91576 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.257439 | 0 | −3.68825 | 0 | −1.00000 | 0 | −2.93373 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.418919 | 0 | 0.0566209 | 0 | −1.00000 | 0 | −2.82451 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.02188 | 0 | 3.35739 | 0 | −1.00000 | 0 | −1.95576 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.24381 | 0 | −2.78728 | 0 | −1.00000 | 0 | 2.03468 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 2.85872 | 0 | 3.60422 | 0 | −1.00000 | 0 | 5.17231 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 3.12022 | 0 | −0.246641 | 0 | −1.00000 | 0 | 6.73575 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(11\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 4004.2.a.k | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4004.2.a.k | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - T_{3}^{9} - 23T_{3}^{8} + 23T_{3}^{7} + 170T_{3}^{6} - 165T_{3}^{5} - 411T_{3}^{4} + 360T_{3}^{3} + 111T_{3}^{2} - 48T_{3} - 12 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( T^{10} - T^{9} + \cdots - 12 \)
|
| $5$ |
\( T^{10} - 4 T^{9} + \cdots + 24 \)
|
| $7$ |
\( (T + 1)^{10} \)
|
| $11$ |
\( (T - 1)^{10} \)
|
| $13$ |
\( (T - 1)^{10} \)
|
| $17$ |
\( T^{10} - 3 T^{9} + \cdots + 5100 \)
|
| $19$ |
\( T^{10} - 6 T^{9} + \cdots + 321996 \)
|
| $23$ |
\( T^{10} - 4 T^{9} + \cdots - 297600 \)
|
| $29$ |
\( T^{10} - 10 T^{9} + \cdots + 60000 \)
|
| $31$ |
\( T^{10} + T^{9} + \cdots - 64448 \)
|
| $37$ |
\( T^{10} - 20 T^{9} + \cdots - 57600 \)
|
| $41$ |
\( T^{10} + \cdots - 125445888 \)
|
| $43$ |
\( T^{10} - 237 T^{8} + \cdots + 89933566 \)
|
| $47$ |
\( T^{10} - 4 T^{9} + \cdots - 297600 \)
|
| $53$ |
\( T^{10} + 5 T^{9} + \cdots - 714146 \)
|
| $59$ |
\( T^{10} - 11 T^{9} + \cdots + 167936 \)
|
| $61$ |
\( T^{10} - 12 T^{9} + \cdots + 12373204 \)
|
| $67$ |
\( T^{10} + 2 T^{9} + \cdots - 3414728 \)
|
| $71$ |
\( T^{10} + \cdots + 290303744 \)
|
| $73$ |
\( T^{10} - 11 T^{9} + \cdots + 10568832 \)
|
| $79$ |
\( T^{10} + 10 T^{9} + \cdots + 80596382 \)
|
| $83$ |
\( T^{10} + \cdots - 645741612 \)
|
| $89$ |
\( T^{10} + \cdots - 1977816760 \)
|
| $97$ |
\( T^{10} + \cdots - 6882663168 \)
|
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