Newspace parameters
| Level: | \( N \) | \(=\) | \( 231 = 3 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 231.i (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.84454428669\) |
| 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 \(\left(\mathbb{Z}/231\mathbb{Z}\right)^\times\).
| \(n\) | \(155\) | \(199\) | \(211\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 67.1 |
|
−0.207107 | − | 0.358719i | −0.500000 | + | 0.866025i | 0.914214 | − | 1.58346i | 1.00000 | + | 1.73205i | 0.414214 | 1.00000 | + | 2.44949i | −1.58579 | −0.500000 | − | 0.866025i | 0.414214 | − | 0.717439i | ||||||||||||||||
| 67.2 | 1.20711 | + | 2.09077i | −0.500000 | + | 0.866025i | −1.91421 | + | 3.31552i | 1.00000 | + | 1.73205i | −2.41421 | 1.00000 | − | 2.44949i | −4.41421 | −0.500000 | − | 0.866025i | −2.41421 | + | 4.18154i | |||||||||||||||||
| 100.1 | −0.207107 | + | 0.358719i | −0.500000 | − | 0.866025i | 0.914214 | + | 1.58346i | 1.00000 | − | 1.73205i | 0.414214 | 1.00000 | − | 2.44949i | −1.58579 | −0.500000 | + | 0.866025i | 0.414214 | + | 0.717439i | |||||||||||||||||
| 100.2 | 1.20711 | − | 2.09077i | −0.500000 | − | 0.866025i | −1.91421 | − | 3.31552i | 1.00000 | − | 1.73205i | −2.41421 | 1.00000 | + | 2.44949i | −4.41421 | −0.500000 | + | 0.866025i | −2.41421 | − | 4.18154i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 231.2.i.d | ✓ | 4 |
| 3.b | odd | 2 | 1 | 693.2.i.f | 4 | ||
| 7.c | even | 3 | 1 | inner | 231.2.i.d | ✓ | 4 |
| 7.c | even | 3 | 1 | 1617.2.a.n | 2 | ||
| 7.d | odd | 6 | 1 | 1617.2.a.m | 2 | ||
| 21.g | even | 6 | 1 | 4851.2.a.bd | 2 | ||
| 21.h | odd | 6 | 1 | 693.2.i.f | 4 | ||
| 21.h | odd | 6 | 1 | 4851.2.a.be | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 231.2.i.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 231.2.i.d | ✓ | 4 | 7.c | even | 3 | 1 | inner |
| 693.2.i.f | 4 | 3.b | odd | 2 | 1 | ||
| 693.2.i.f | 4 | 21.h | odd | 6 | 1 | ||
| 1617.2.a.m | 2 | 7.d | odd | 6 | 1 | ||
| 1617.2.a.n | 2 | 7.c | even | 3 | 1 | ||
| 4851.2.a.bd | 2 | 21.g | even | 6 | 1 | ||
| 4851.2.a.be | 2 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(231, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1 \)
$3$
\( (T^{2} + T + 1)^{2} \)
$5$
\( (T^{2} - 2 T + 4)^{2} \)
$7$
\( (T^{2} - 2 T + 7)^{2} \)
$11$
\( (T^{2} + T + 1)^{2} \)
$13$
\( (T^{2} - 4 T - 4)^{2} \)
$17$
\( T^{4} - 6 T^{3} + 29 T^{2} - 42 T + 49 \)
$19$
\( T^{4} + 6 T^{3} + 45 T^{2} - 54 T + 81 \)
$23$
\( (T^{2} - 7 T + 49)^{2} \)
$29$
\( (T^{2} + 2 T - 17)^{2} \)
$31$
\( T^{4} + 32T^{2} + 1024 \)
$37$
\( T^{4} - 2 T^{3} + 75 T^{2} + \cdots + 5041 \)
$41$
\( (T^{2} + 8 T + 8)^{2} \)
$43$
\( (T^{2} + 14 T + 31)^{2} \)
$47$
\( T^{4} + 14 T^{3} + 155 T^{2} + \cdots + 1681 \)
$53$
\( T^{4} - 20 T^{3} + 308 T^{2} + \cdots + 8464 \)
$59$
\( T^{4} - 6 T^{3} + 59 T^{2} + 138 T + 529 \)
$61$
\( (T^{2} - 4 T + 16)^{2} \)
$67$
\( T^{4} - 12 T^{3} + 116 T^{2} + \cdots + 784 \)
$71$
\( (T^{2} + 14 T + 41)^{2} \)
$73$
\( T^{4} + 12 T^{3} + 140 T^{2} + \cdots + 16 \)
$79$
\( T^{4} + 4 T^{3} + 140 T^{2} + \cdots + 15376 \)
$83$
\( (T^{2} - 8)^{2} \)
$89$
\( T^{4} + 200 T^{2} + 40000 \)
$97$
\( (T^{2} + 6 T - 63)^{2} \)
show more
show less