Newspace parameters
| Level: | \( N \) | \(=\) | \( 97 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 97.e (of order \(6\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(5.72318527056\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-65})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 65x^{2} + 4225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 65x^{2} + 4225 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 65 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 65 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 65\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 65\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/97\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) |
| \(\chi(n)\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 36.1 |
|
−0.500000 | + | 0.866025i | 2.00000 | − | 3.46410i | 3.50000 | + | 6.06218i | −12.9821 | − | 7.49523i | 2.00000 | + | 3.46410i | −31.4464 | − | 18.1556i | −15.0000 | 5.50000 | + | 9.52628i | 12.9821 | − | 7.49523i | ||||||||||||||
| 36.2 | −0.500000 | + | 0.866025i | 2.00000 | − | 3.46410i | 3.50000 | + | 6.06218i | 0.982120 | + | 0.567027i | 2.00000 | + | 3.46410i | 10.4464 | + | 6.03121i | −15.0000 | 5.50000 | + | 9.52628i | −0.982120 | + | 0.567027i | |||||||||||||||
| 62.1 | −0.500000 | − | 0.866025i | 2.00000 | + | 3.46410i | 3.50000 | − | 6.06218i | −12.9821 | + | 7.49523i | 2.00000 | − | 3.46410i | −31.4464 | + | 18.1556i | −15.0000 | 5.50000 | − | 9.52628i | 12.9821 | + | 7.49523i | |||||||||||||||
| 62.2 | −0.500000 | − | 0.866025i | 2.00000 | + | 3.46410i | 3.50000 | − | 6.06218i | 0.982120 | − | 0.567027i | 2.00000 | − | 3.46410i | 10.4464 | − | 6.03121i | −15.0000 | 5.50000 | − | 9.52628i | −0.982120 | − | 0.567027i | |||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 97.e | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 97.4.e.a | ✓ | 4 |
| 97.e | even | 6 | 1 | inner | 97.4.e.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 97.4.e.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 97.4.e.a | ✓ | 4 | 97.e | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + T_{2} + 1 \)
acting on \(S_{4}^{\mathrm{new}}(97, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{2} \)
|
| $3$ |
\( (T^{2} - 4 T + 16)^{2} \)
|
| $5$ |
\( T^{4} + 24 T^{3} + \cdots + 289 \)
|
| $7$ |
\( T^{4} + 42 T^{3} + \cdots + 191844 \)
|
| $11$ |
\( T^{4} + 22 T^{3} + \cdots + 5476 \)
|
| $13$ |
\( T^{4} - 60 T^{3} + \cdots + 1755625 \)
|
| $17$ |
\( T^{4} - 162 T^{3} + \cdots + 3892729 \)
|
| $19$ |
\( T^{4} + 30264 T^{2} + 199317924 \)
|
| $23$ |
\( T^{4} + 264 T^{3} + \cdots + 22733824 \)
|
| $29$ |
\( T^{4} - 276 T^{3} + \cdots + 37063744 \)
|
| $31$ |
\( T^{4} + 4 T^{3} + \cdots + 49224256 \)
|
| $37$ |
\( T^{4} + \cdots + 16670683225 \)
|
| $41$ |
\( T^{4} + \cdots + 11077983504 \)
|
| $43$ |
\( T^{4} + \cdots + 1948692736 \)
|
| $47$ |
\( (T^{2} + 462 T + 43806)^{2} \)
|
| $53$ |
\( T^{4} - 96 T^{3} + \cdots + 6610041 \)
|
| $59$ |
\( T^{4} + \cdots + 12225282624 \)
|
| $61$ |
\( T^{4} + \cdots + 9167105025 \)
|
| $67$ |
\( T^{4} + \cdots + 307102172224 \)
|
| $71$ |
\( T^{4} + 474 T^{3} + \cdots + 22486564 \)
|
| $73$ |
\( T^{4} + \cdots + 11762703936 \)
|
| $79$ |
\( (T^{2} - 256 T - 11696)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 1275164909824 \)
|
| $89$ |
\( (T^{2} - 714 T - 1183731)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 832972004929 \)
|
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