Newspace parameters
Level: | \( N \) | \(=\) | \( 215 = 5 \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 215.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.71678364346\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{13})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) |
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 \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} + 7 ) / 4 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{3} + 3\beta_{2} + \beta _1 - 1 \) |
\(\nu^{3}\) | \(=\) | \( 4\beta_{3} - 7 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/215\mathbb{Z}\right)^\times\).
\(n\) | \(46\) | \(87\) |
\(\chi(n)\) | \(-1 - \beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
6.1 |
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−1.30278 | 0.151388 | − | 0.262211i | −0.302776 | −0.500000 | + | 0.866025i | −0.197224 | + | 0.341603i | −0.500000 | − | 0.866025i | 3.00000 | 1.45416 | + | 2.51868i | 0.651388 | − | 1.12824i | ||||||||||||||||||
6.2 | 2.30278 | −1.65139 | + | 2.86029i | 3.30278 | −0.500000 | + | 0.866025i | −3.80278 | + | 6.58660i | −0.500000 | − | 0.866025i | 3.00000 | −3.95416 | − | 6.84881i | −1.15139 | + | 1.99426i | |||||||||||||||||||
36.1 | −1.30278 | 0.151388 | + | 0.262211i | −0.302776 | −0.500000 | − | 0.866025i | −0.197224 | − | 0.341603i | −0.500000 | + | 0.866025i | 3.00000 | 1.45416 | − | 2.51868i | 0.651388 | + | 1.12824i | |||||||||||||||||||
36.2 | 2.30278 | −1.65139 | − | 2.86029i | 3.30278 | −0.500000 | − | 0.866025i | −3.80278 | − | 6.58660i | −0.500000 | + | 0.866025i | 3.00000 | −3.95416 | + | 6.84881i | −1.15139 | − | 1.99426i | |||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
43.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 215.2.e.b | ✓ | 4 |
43.c | even | 3 | 1 | inner | 215.2.e.b | ✓ | 4 |
43.c | even | 3 | 1 | 9245.2.a.e | 2 | ||
43.d | odd | 6 | 1 | 9245.2.a.d | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
215.2.e.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
215.2.e.b | ✓ | 4 | 43.c | even | 3 | 1 | inner |
9245.2.a.d | 2 | 43.d | odd | 6 | 1 | ||
9245.2.a.e | 2 | 43.c | even | 3 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - T_{2} - 3 \)
acting on \(S_{2}^{\mathrm{new}}(215, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - T - 3)^{2} \)
$3$
\( T^{4} + 3 T^{3} + 10 T^{2} - 3 T + 1 \)
$5$
\( (T^{2} + T + 1)^{2} \)
$7$
\( (T^{2} + T + 1)^{2} \)
$11$
\( (T^{2} + T - 29)^{2} \)
$13$
\( T^{4} - 9 T^{3} + 64 T^{2} - 153 T + 289 \)
$17$
\( T^{4} - 7 T^{3} + 40 T^{2} - 63 T + 81 \)
$19$
\( T^{4} \)
$23$
\( T^{4} - 3 T^{3} + 36 T^{2} + 81 T + 729 \)
$29$
\( T^{4} + 13T^{2} + 169 \)
$31$
\( T^{4} - 6 T^{3} + 40 T^{2} + 24 T + 16 \)
$37$
\( T^{4} + 7 T^{3} + 66 T^{2} - 119 T + 289 \)
$41$
\( (T + 7)^{4} \)
$43$
\( (T^{2} - 13 T + 43)^{2} \)
$47$
\( (T^{2} + 17 T + 69)^{2} \)
$53$
\( T^{4} - 5 T^{3} + 100 T^{2} + \cdots + 5625 \)
$59$
\( (T^{2} + 13 T + 39)^{2} \)
$61$
\( T^{4} + 23 T^{3} + 400 T^{2} + \cdots + 16641 \)
$67$
\( T^{4} - 3 T^{3} + 36 T^{2} + 81 T + 729 \)
$71$
\( T^{4} - 19 T^{3} + 300 T^{2} + \cdots + 3721 \)
$73$
\( T^{4} - 7 T^{3} + 66 T^{2} + 119 T + 289 \)
$79$
\( T^{4} - 9 T^{3} + 64 T^{2} - 153 T + 289 \)
$83$
\( T^{4} - 9 T^{3} + 90 T^{2} + 81 T + 81 \)
$89$
\( T^{4} + 17 T^{3} + 220 T^{2} + \cdots + 4761 \)
$97$
\( (T - 6)^{4} \)
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