Newspace parameters
| Level: | \( N \) | \(=\) | \( 2000 = 2^{4} \cdot 5^{3} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2000.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(118.003820011\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.497918125.1 |
|
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 29x^{4} - 6x^{3} + 216x^{2} + 280x + 80 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 125) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 29x^{4} - 6x^{3} + 216x^{2} + 280x + 80 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -3\nu^{5} - 19\nu^{4} + 161\nu^{3} + 260\nu^{2} - 1344\nu - 712 ) / 128 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 3\nu^{4} - 23\nu^{3} + 40\nu^{2} + 152\nu + 8 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 17\nu^{5} - 31\nu^{4} - 475\nu^{3} + 372\nu^{2} + 3200\nu + 1304 ) / 128 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7\nu^{5} - 9\nu^{4} - 237\nu^{3} + 44\nu^{2} + 2048\nu + 1640 ) / 64 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 25\nu^{5} - 55\nu^{4} - 595\nu^{3} + 500\nu^{2} + 3840\nu + 2264 ) / 128 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - 4\beta_{3} + 3\beta_{2} - 2\beta _1 + 1 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -4\beta_{5} - 2\beta_{4} - 2\beta_{3} + 9\beta_{2} - 6\beta _1 + 100 ) / 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 8\beta_{5} + 3\beta_{4} - 62\beta_{3} + 44\beta_{2} - 36\beta _1 + 169 ) / 10 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -28\beta_{5} - 49\beta_{4} - 104\beta_{3} + 163\beta_{2} - 182\beta _1 + 1635 ) / 10 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 52\beta_{5} - 30\beta_{4} - 210\beta_{3} + 153\beta_{2} - 166\beta _1 + 912 ) / 2 \)
|
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.
| 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 | −9.75552 | 0 | 0 | 0 | 14.1661 | 0 | 68.1701 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −8.30353 | 0 | 0 | 0 | −29.0783 | 0 | 41.9487 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −5.51148 | 0 | 0 | 0 | −2.28738 | 0 | 3.37638 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.11531 | 0 | 0 | 0 | −1.57558 | 0 | −25.7561 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 4.57894 | 0 | 0 | 0 | −18.9048 | 0 | −6.03327 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 6.10690 | 0 | 0 | 0 | −29.3201 | 0 | 10.2942 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( +1 \) |
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 | 2000.4.a.k | 6 | |
| 4.b | odd | 2 | 1 | 125.4.a.c | yes | 6 | |
| 5.b | even | 2 | 1 | 2000.4.a.n | 6 | ||
| 12.b | even | 2 | 1 | 1125.4.a.f | 6 | ||
| 20.d | odd | 2 | 1 | 125.4.a.b | ✓ | 6 | |
| 20.e | even | 4 | 2 | 125.4.b.c | 12 | ||
| 60.h | even | 2 | 1 | 1125.4.a.k | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 125.4.a.b | ✓ | 6 | 20.d | odd | 2 | 1 | |
| 125.4.a.c | yes | 6 | 4.b | odd | 2 | 1 | |
| 125.4.b.c | 12 | 20.e | even | 4 | 2 | ||
| 1125.4.a.f | 6 | 12.b | even | 2 | 1 | ||
| 1125.4.a.k | 6 | 60.h | even | 2 | 1 | ||
| 2000.4.a.k | 6 | 1.a | even | 1 | 1 | trivial | |
| 2000.4.a.n | 6 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 14T_{3}^{5} - 29T_{3}^{4} - 872T_{3}^{3} - 641T_{3}^{2} + 12794T_{3} + 13924 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2000))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 14 T^{5} + \cdots + 13924 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 67 T^{5} + \cdots - 822879 \)
|
| $11$ |
\( T^{6} + 27 T^{5} + \cdots - 47445831 \)
|
| $13$ |
\( T^{6} + \cdots - 7178567571 \)
|
| $17$ |
\( T^{6} + \cdots + 27354562736 \)
|
| $19$ |
\( T^{6} + \cdots + 29290968875 \)
|
| $23$ |
\( T^{6} + \cdots - 53831893136 \)
|
| $29$ |
\( T^{6} + \cdots - 369868315500 \)
|
| $31$ |
\( T^{6} + \cdots + 3206882909804 \)
|
| $37$ |
\( T^{6} + \cdots - 462416505264 \)
|
| $41$ |
\( T^{6} + \cdots + 18008321955939 \)
|
| $43$ |
\( T^{6} + \cdots + 395909856491184 \)
|
| $47$ |
\( T^{6} + \cdots - 9620559208829 \)
|
| $53$ |
\( T^{6} + \cdots + 10657074605569 \)
|
| $59$ |
\( T^{6} + \cdots + 10\!\cdots\!75 \)
|
| $61$ |
\( T^{6} + \cdots + 153598743998924 \)
|
| $67$ |
\( T^{6} + \cdots - 59\!\cdots\!64 \)
|
| $71$ |
\( T^{6} + \cdots + 39\!\cdots\!84 \)
|
| $73$ |
\( T^{6} + \cdots + 13\!\cdots\!04 \)
|
| $79$ |
\( T^{6} + \cdots + 14\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 42\!\cdots\!64 \)
|
| $89$ |
\( T^{6} + \cdots + 62\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 64\!\cdots\!64 \)
|
| show more | |
| show less |