Newspace parameters
| Level: | \( N \) | \(=\) | \( 93 = 3 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 93.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.742608738798\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/93\mathbb{Z}\right)^\times\).
| \(n\) | \(32\) | \(34\) |
| \(\chi(n)\) | \(1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
−1.41421 | 0.500000 | + | 0.866025i | 0 | 1.70711 | − | 2.95680i | −0.707107 | − | 1.22474i | 2.41421 | + | 4.18154i | 2.82843 | −0.500000 | + | 0.866025i | −2.41421 | + | 4.18154i | ||||||||||||||||||
| 25.2 | 1.41421 | 0.500000 | + | 0.866025i | 0 | 0.292893 | − | 0.507306i | 0.707107 | + | 1.22474i | −0.414214 | − | 0.717439i | −2.82843 | −0.500000 | + | 0.866025i | 0.414214 | − | 0.717439i | |||||||||||||||||||
| 67.1 | −1.41421 | 0.500000 | − | 0.866025i | 0 | 1.70711 | + | 2.95680i | −0.707107 | + | 1.22474i | 2.41421 | − | 4.18154i | 2.82843 | −0.500000 | − | 0.866025i | −2.41421 | − | 4.18154i | |||||||||||||||||||
| 67.2 | 1.41421 | 0.500000 | − | 0.866025i | 0 | 0.292893 | + | 0.507306i | 0.707107 | − | 1.22474i | −0.414214 | + | 0.717439i | −2.82843 | −0.500000 | − | 0.866025i | 0.414214 | + | 0.717439i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 93.2.e.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 279.2.h.b | 4 | ||
| 4.b | odd | 2 | 1 | 1488.2.q.g | 4 | ||
| 31.c | even | 3 | 1 | inner | 93.2.e.a | ✓ | 4 |
| 31.c | even | 3 | 1 | 2883.2.a.b | 2 | ||
| 31.e | odd | 6 | 1 | 2883.2.a.c | 2 | ||
| 93.g | even | 6 | 1 | 8649.2.a.i | 2 | ||
| 93.h | odd | 6 | 1 | 279.2.h.b | 4 | ||
| 93.h | odd | 6 | 1 | 8649.2.a.h | 2 | ||
| 124.i | odd | 6 | 1 | 1488.2.q.g | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 93.2.e.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 93.2.e.a | ✓ | 4 | 31.c | even | 3 | 1 | inner |
| 279.2.h.b | 4 | 3.b | odd | 2 | 1 | ||
| 279.2.h.b | 4 | 93.h | odd | 6 | 1 | ||
| 1488.2.q.g | 4 | 4.b | odd | 2 | 1 | ||
| 1488.2.q.g | 4 | 124.i | odd | 6 | 1 | ||
| 2883.2.a.b | 2 | 31.c | even | 3 | 1 | ||
| 2883.2.a.c | 2 | 31.e | odd | 6 | 1 | ||
| 8649.2.a.h | 2 | 93.h | odd | 6 | 1 | ||
| 8649.2.a.i | 2 | 93.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 2 \)
acting on \(S_{2}^{\mathrm{new}}(93, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2)^{2} \)
|
| $3$ |
\( (T^{2} - T + 1)^{2} \)
|
| $5$ |
\( T^{4} - 4 T^{3} + \cdots + 4 \)
|
| $7$ |
\( T^{4} - 4 T^{3} + \cdots + 16 \)
|
| $11$ |
\( T^{4} + 2T^{2} + 4 \)
|
| $13$ |
\( T^{4} - 2 T^{3} + \cdots + 49 \)
|
| $17$ |
\( T^{4} + 8 T^{3} + \cdots + 64 \)
|
| $19$ |
\( T^{4} - 2 T^{3} + \cdots + 49 \)
|
| $23$ |
\( (T^{2} + 4 T - 28)^{2} \)
|
| $29$ |
\( (T^{2} + 4 T + 2)^{2} \)
|
| $31$ |
\( (T^{2} - 4 T + 31)^{2} \)
|
| $37$ |
\( (T^{2} + T + 1)^{2} \)
|
| $41$ |
\( T^{4} + 18T^{2} + 324 \)
|
| $43$ |
\( T^{4} + 6 T^{3} + \cdots + 3969 \)
|
| $47$ |
\( (T^{2} - 4 T + 2)^{2} \)
|
| $53$ |
\( T^{4} + 2T^{2} + 4 \)
|
| $59$ |
\( T^{4} + 72T^{2} + 5184 \)
|
| $61$ |
\( (T^{2} + 8 T - 56)^{2} \)
|
| $67$ |
\( T^{4} - 12 T^{3} + \cdots + 1296 \)
|
| $71$ |
\( (T^{2} + 6 T + 36)^{2} \)
|
| $73$ |
\( T^{4} + 2 T^{3} + \cdots + 49 \)
|
| $79$ |
\( T^{4} + 20 T^{3} + \cdots + 4624 \)
|
| $83$ |
\( T^{4} - 12 T^{3} + \cdots + 1156 \)
|
| $89$ |
\( (T^{2} + 16 T - 34)^{2} \)
|
| $97$ |
\( (T^{2} + 6 T - 63)^{2} \)
|
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