Newspace parameters
| Level: | \( N \) | \(=\) | \( 465 = 3 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 465.n (of order \(5\), degree \(4\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(3.71304369399\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | 8.0.13140625.1 |
|
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-expansion
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
\( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} + 2\nu^{6} - 3\nu^{5} - 4\nu^{3} - 7\nu^{2} - 12\nu - 7 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 7\nu^{5} + 20\nu^{4} - 16\nu^{3} + 19\nu^{2} + 6\nu + 9 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 4\nu^{6} - 9\nu^{5} + 12\nu^{4} - 16\nu^{3} + 13\nu^{2} - 10\nu - 1 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{7} + 10\nu^{6} - 17\nu^{5} + 8\nu^{4} - 4\nu^{3} - 13\nu^{2} - 8\nu - 5 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{7} - 12\nu^{6} + 23\nu^{5} - 20\nu^{4} + 16\nu^{3} + \nu^{2} + 6\nu - 1 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -5\nu^{7} + 18\nu^{6} - 35\nu^{5} + 32\nu^{4} - 28\nu^{3} - 11\nu^{2} - 12\nu - 7 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{4} - 3\beta_{2} - 2\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{7} + 4\beta_{6} + \beta_{4} - \beta_{3} - 5\beta_{2} - 4\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{7} - 6\beta_{5} - 4\beta_{3} - 6\beta_{2} - 6\beta _1 - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -16\beta_{6} - 16\beta_{5} - 6\beta_{4} - 6\beta_{3} - 7\beta _1 - 6 \)
|
| \(\nu^{7}\) | \(=\) |
\( -16\beta_{7} - 51\beta_{6} - 29\beta_{5} - 23\beta_{4} + 29\beta_{2} - 16 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).
| \(n\) | \(187\) | \(311\) | \(406\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1 + \beta_{3} - \beta_{4} - \beta_{7}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 16.1 |
|
−1.43376 | + | 1.04169i | −0.809017 | − | 0.587785i | 0.352519 | − | 1.08494i | 1.00000 | 1.77222 | −0.238630 | + | 0.734428i | −0.470553 | − | 1.44821i | 0.309017 | + | 0.951057i | −1.43376 | + | 1.04169i | ||||||||||||||||||||||||||||
| 16.2 | 1.93376 | − | 1.40496i | −0.809017 | − | 0.587785i | 1.14748 | − | 3.53158i | 1.00000 | −2.39026 | 1.04765 | − | 3.22433i | −1.26552 | − | 3.89486i | 0.309017 | + | 0.951057i | 1.93376 | − | 1.40496i | |||||||||||||||||||||||||||||
| 256.1 | 0.0501062 | − | 0.154211i | 0.309017 | + | 0.951057i | 1.59676 | + | 1.16012i | 1.00000 | 0.162147 | −0.677837 | − | 0.492478i | 0.521270 | − | 0.378725i | −0.809017 | + | 0.587785i | 0.0501062 | − | 0.154211i | |||||||||||||||||||||||||||||
| 256.2 | 0.449894 | − | 1.38463i | 0.309017 | + | 0.951057i | −0.0967635 | − | 0.0703028i | 1.00000 | 1.45589 | 0.368820 | + | 0.267964i | 2.21480 | − | 1.60914i | −0.809017 | + | 0.587785i | 0.449894 | − | 1.38463i | |||||||||||||||||||||||||||||
| 376.1 | 0.0501062 | + | 0.154211i | 0.309017 | − | 0.951057i | 1.59676 | − | 1.16012i | 1.00000 | 0.162147 | −0.677837 | + | 0.492478i | 0.521270 | + | 0.378725i | −0.809017 | − | 0.587785i | 0.0501062 | + | 0.154211i | |||||||||||||||||||||||||||||
| 376.2 | 0.449894 | + | 1.38463i | 0.309017 | − | 0.951057i | −0.0967635 | + | 0.0703028i | 1.00000 | 1.45589 | 0.368820 | − | 0.267964i | 2.21480 | + | 1.60914i | −0.809017 | − | 0.587785i | 0.449894 | + | 1.38463i | |||||||||||||||||||||||||||||
| 436.1 | −1.43376 | − | 1.04169i | −0.809017 | + | 0.587785i | 0.352519 | + | 1.08494i | 1.00000 | 1.77222 | −0.238630 | − | 0.734428i | −0.470553 | + | 1.44821i | 0.309017 | − | 0.951057i | −1.43376 | − | 1.04169i | |||||||||||||||||||||||||||||
| 436.2 | 1.93376 | + | 1.40496i | −0.809017 | + | 0.587785i | 1.14748 | + | 3.53158i | 1.00000 | −2.39026 | 1.04765 | + | 3.22433i | −1.26552 | + | 3.89486i | 0.309017 | − | 0.951057i | 1.93376 | + | 1.40496i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.d | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 465.2.n.c | ✓ | 8 |
| 31.d | even | 5 | 1 | inner | 465.2.n.c | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 465.2.n.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 465.2.n.c | ✓ | 8 | 31.d | even | 5 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 2T_{2}^{7} + T_{2}^{6} + 4T_{2}^{5} + 9T_{2}^{4} - 8T_{2}^{3} + 39T_{2}^{2} - 4T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(465, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 2 T^{7} + \cdots + 1 \)
|
| $3$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $5$ |
\( (T - 1)^{8} \)
|
| $7$ |
\( T^{8} - T^{7} + 10 T^{6} + \cdots + 1 \)
|
| $11$ |
\( T^{8} - 6 T^{7} + \cdots + 1 \)
|
| $13$ |
\( T^{8} + 4 T^{7} + \cdots + 121 \)
|
| $17$ |
\( (T^{4} + T^{3} + 6 T^{2} + \cdots + 1)^{2} \)
|
| $19$ |
\( T^{8} - 5 T^{7} + \cdots + 625 \)
|
| $23$ |
\( T^{8} + 8 T^{7} + \cdots + 271441 \)
|
| $29$ |
\( T^{8} + 3 T^{7} + \cdots + 3721 \)
|
| $31$ |
\( T^{8} - 3 T^{7} + \cdots + 923521 \)
|
| $37$ |
\( (T^{4} - 4 T^{3} + \cdots + 121)^{2} \)
|
| $41$ |
\( T^{8} - 14 T^{7} + \cdots + 2070721 \)
|
| $43$ |
\( T^{8} - 4 T^{7} + \cdots + 2401 \)
|
| $47$ |
\( T^{8} - 24 T^{7} + \cdots + 3041536 \)
|
| $53$ |
\( T^{8} - 4 T^{7} + \cdots + 546121 \)
|
| $59$ |
\( T^{8} - 6 T^{7} + \cdots + 361 \)
|
| $61$ |
\( (T^{4} - 17 T^{3} + \cdots - 8531)^{2} \)
|
| $67$ |
\( (T^{4} + 43 T^{3} + \cdots + 10811)^{2} \)
|
| $71$ |
\( T^{8} - 2 T^{7} + \cdots + 361 \)
|
| $73$ |
\( T^{8} + 6 T^{7} + \cdots + 256 \)
|
| $79$ |
\( T^{8} + 13 T^{7} + \cdots + 687241 \)
|
| $83$ |
\( T^{8} - 22 T^{7} + \cdots + 121 \)
|
| $89$ |
\( T^{8} + 44 T^{7} + \cdots + 34586161 \)
|
| $97$ |
\( T^{8} + 11 T^{7} + \cdots + 361201 \)
|
| show more | |
| show less |