Newspace parameters
| Level: | \( N \) | \(=\) | \( 4641 = 3 \cdot 7 \cdot 13 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4641.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(37.0585715781\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | 7.7.97824733.1 |
|
|
|
| Defining polynomial: |
\( x^{7} - 2x^{6} - 7x^{5} + 15x^{4} + 6x^{3} - 22x^{2} + 9x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 7x^{5} + 15x^{4} + 6x^{3} - 22x^{2} + 9x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 8\nu^{4} + 7\nu^{3} + 13\nu^{2} - 10\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 7\nu^{4} + 15\nu^{3} + 7\nu^{2} - 22\nu + 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( -2\nu^{6} + 3\nu^{5} + 15\nu^{4} - 22\nu^{3} - 19\nu^{2} + 32\nu - 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( -3\nu^{6} + 6\nu^{5} + 22\nu^{4} - 45\nu^{3} - 26\nu^{2} + 66\nu - 14 \)
|
| \(\beta_{6}\) | \(=\) |
\( 6\nu^{6} - 10\nu^{5} - 45\nu^{4} + 75\nu^{3} + 58\nu^{2} - 112\nu + 22 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 5\beta_{4} + 8\beta_{3} + 5\beta_{2} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{6} + 9\beta_{5} + 7\beta_{4} + \beta_{3} - 8\beta_{2} + 20\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 10\beta_{5} + 27\beta_{4} + 52\beta_{3} + 27\beta_{2} + 2\beta _1 + 74 \)
|
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 |
|
−2.48067 | −1.00000 | 4.15370 | 2.06438 | 2.48067 | 1.00000 | −5.34261 | 1.00000 | −5.12103 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.57760 | −1.00000 | 0.488818 | −3.13008 | 1.57760 | 1.00000 | 2.38404 | 1.00000 | 4.93801 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.25216 | −1.00000 | −0.432105 | 1.65020 | 1.25216 | 1.00000 | 3.04538 | 1.00000 | −2.06631 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.286902 | −1.00000 | −1.91769 | 1.54274 | 0.286902 | 1.00000 | 1.12399 | 1.00000 | −0.442616 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.208559 | −1.00000 | −1.95650 | 0.639086 | 0.208559 | 1.00000 | 0.825164 | 1.00000 | −0.133287 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.44391 | −1.00000 | 0.0848831 | −2.96406 | −1.44391 | 1.00000 | −2.76526 | 1.00000 | −4.27984 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.36197 | −1.00000 | 3.57889 | −0.802266 | −2.36197 | 1.00000 | 3.72930 | 1.00000 | −1.89493 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(7\) | \( -1 \) |
| \(13\) | \( -1 \) |
| \(17\) | \( +1 \) |
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 | 4641.2.a.l | ✓ | 7 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4641.2.a.l | ✓ | 7 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4641))\):
|
\( T_{2}^{7} + 2T_{2}^{6} - 7T_{2}^{5} - 15T_{2}^{4} + 6T_{2}^{3} + 22T_{2}^{2} + 9T_{2} + 1 \)
|
|
\( T_{5}^{7} + T_{5}^{6} - 14T_{5}^{5} - T_{5}^{4} + 60T_{5}^{3} - 41T_{5}^{2} - 35T_{5} + 25 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 2 T^{6} + \cdots + 1 \)
|
| $3$ |
\( (T + 1)^{7} \)
|
| $5$ |
\( T^{7} + T^{6} + \cdots + 25 \)
|
| $7$ |
\( (T - 1)^{7} \)
|
| $11$ |
\( T^{7} - 2 T^{6} + \cdots - 304 \)
|
| $13$ |
\( (T - 1)^{7} \)
|
| $17$ |
\( (T + 1)^{7} \)
|
| $19$ |
\( T^{7} + 12 T^{6} + \cdots - 2069 \)
|
| $23$ |
\( T^{7} - 5 T^{6} + \cdots + 727 \)
|
| $29$ |
\( T^{7} - 12 T^{6} + \cdots + 107623 \)
|
| $31$ |
\( T^{7} + 9 T^{6} + \cdots - 4432 \)
|
| $37$ |
\( T^{7} + 12 T^{6} + \cdots - 3092 \)
|
| $41$ |
\( T^{7} + 16 T^{6} + \cdots - 21053 \)
|
| $43$ |
\( T^{7} + 22 T^{6} + \cdots + 4139 \)
|
| $47$ |
\( T^{7} - 13 T^{6} + \cdots + 6128 \)
|
| $53$ |
\( T^{7} + 3 T^{6} + \cdots - 332032 \)
|
| $59$ |
\( T^{7} + 33 T^{6} + \cdots + 387100 \)
|
| $61$ |
\( T^{7} + 11 T^{6} + \cdots - 10181 \)
|
| $67$ |
\( T^{7} + 5 T^{6} + \cdots + 5392 \)
|
| $71$ |
\( T^{7} + 14 T^{6} + \cdots + 869744 \)
|
| $73$ |
\( T^{7} + 16 T^{6} + \cdots - 113600 \)
|
| $79$ |
\( T^{7} - 121 T^{5} + \cdots + 29948 \)
|
| $83$ |
\( T^{7} + 17 T^{6} + \cdots + 293732 \)
|
| $89$ |
\( T^{7} - 17 T^{6} + \cdots + 2330128 \)
|
| $97$ |
\( T^{7} + 21 T^{6} + \cdots - 78800 \)
|
| show more | |
| show less |