Newspace parameters
Level: | \( N \) | \(=\) | \( 387 = 3^{2} \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 387.h (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.09021055822\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{5})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{3} + 2x^{2} + x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 43) |
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 \( x^{4} - x^{3} + 2x^{2} + x + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{3} + 1 ) / 2 \) |
\(\beta_{3}\) | \(=\) | \( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{3} + \beta_{2} + \beta_1 \) |
\(\nu^{3}\) | \(=\) | \( 2\beta_{2} - 1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/387\mathbb{Z}\right)^\times\).
\(n\) | \(46\) | \(173\) |
\(\chi(n)\) | \(\beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
208.1 |
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0.381966 | 0 | −1.85410 | 0.618034 | + | 1.07047i | 0 | 2.11803 | − | 3.66854i | −1.47214 | 0 | 0.236068 | + | 0.408882i | ||||||||||||||||||||||||
208.2 | 2.61803 | 0 | 4.85410 | −1.61803 | − | 2.80252i | 0 | −0.118034 | + | 0.204441i | 7.47214 | 0 | −4.23607 | − | 7.33708i | |||||||||||||||||||||||||
307.1 | 0.381966 | 0 | −1.85410 | 0.618034 | − | 1.07047i | 0 | 2.11803 | + | 3.66854i | −1.47214 | 0 | 0.236068 | − | 0.408882i | |||||||||||||||||||||||||
307.2 | 2.61803 | 0 | 4.85410 | −1.61803 | + | 2.80252i | 0 | −0.118034 | − | 0.204441i | 7.47214 | 0 | −4.23607 | + | 7.33708i | |||||||||||||||||||||||||
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 | 387.2.h.d | 4 | |
3.b | odd | 2 | 1 | 43.2.c.b | ✓ | 4 | |
12.b | even | 2 | 1 | 688.2.i.e | 4 | ||
43.c | even | 3 | 1 | inner | 387.2.h.d | 4 | |
129.f | odd | 6 | 1 | 43.2.c.b | ✓ | 4 | |
129.f | odd | 6 | 1 | 1849.2.a.e | 2 | ||
129.h | even | 6 | 1 | 1849.2.a.h | 2 | ||
516.p | even | 6 | 1 | 688.2.i.e | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
43.2.c.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
43.2.c.b | ✓ | 4 | 129.f | odd | 6 | 1 | |
387.2.h.d | 4 | 1.a | even | 1 | 1 | trivial | |
387.2.h.d | 4 | 43.c | even | 3 | 1 | inner | |
688.2.i.e | 4 | 12.b | even | 2 | 1 | ||
688.2.i.e | 4 | 516.p | even | 6 | 1 | ||
1849.2.a.e | 2 | 129.f | odd | 6 | 1 | ||
1849.2.a.h | 2 | 129.h | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 3T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(387, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - 3 T + 1)^{2} \)
$3$
\( T^{4} \)
$5$
\( T^{4} + 2 T^{3} + 8 T^{2} - 8 T + 16 \)
$7$
\( T^{4} - 4 T^{3} + 17 T^{2} + 4 T + 1 \)
$11$
\( (T^{2} - 5 T + 5)^{2} \)
$13$
\( T^{4} + 5 T^{3} + 20 T^{2} + 25 T + 25 \)
$17$
\( T^{4} - T^{3} + 32 T^{2} + 31 T + 961 \)
$19$
\( T^{4} + 2 T^{3} + 8 T^{2} - 8 T + 16 \)
$23$
\( T^{4} - 11 T^{3} + 92 T^{2} + \cdots + 841 \)
$29$
\( (T^{2} - 3 T + 9)^{2} \)
$31$
\( T^{4} \)
$37$
\( T^{4} - 3 T^{3} + 18 T^{2} + 27 T + 81 \)
$41$
\( (T^{2} + 10 T + 5)^{2} \)
$43$
\( (T^{2} + 13 T + 43)^{2} \)
$47$
\( (T^{2} + 9 T + 9)^{2} \)
$53$
\( T^{4} + 5 T^{3} + 20 T^{2} + 25 T + 25 \)
$59$
\( (T^{2} - T - 31)^{2} \)
$61$
\( T^{4} - T^{3} + 12 T^{2} + 11 T + 121 \)
$67$
\( T^{4} - T^{3} + 12 T^{2} + 11 T + 121 \)
$71$
\( T^{4} - 3 T^{3} + 158 T^{2} + \cdots + 22201 \)
$73$
\( T^{4} - 3 T^{3} + 18 T^{2} + 27 T + 81 \)
$79$
\( T^{4} - 5 T^{3} + 20 T^{2} - 25 T + 25 \)
$83$
\( T^{4} - 3 T^{3} + 218 T^{2} + \cdots + 43681 \)
$89$
\( T^{4} - 3 T^{3} + 18 T^{2} + 27 T + 81 \)
$97$
\( (T^{2} + 14 T + 44)^{2} \)
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