Newspace parameters
Level: | \( N \) | \(=\) | \( 381 = 3 \cdot 127 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 381.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(10.3814980721\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/381\mathbb{Z}\right)^\times\).
\(n\) | \(128\) | \(130\) |
\(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
253.1 |
|
−2.00000 | − | 1.73205i | 0 | − | 8.66025i | 3.46410i | 6.92820i | 8.00000 | −3.00000 | 17.3205i | ||||||||||||||||||||||
253.2 | −2.00000 | 1.73205i | 0 | 8.66025i | − | 3.46410i | − | 6.92820i | 8.00000 | −3.00000 | − | 17.3205i | ||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
127.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 381.3.d.a | ✓ | 2 |
127.b | odd | 2 | 1 | inner | 381.3.d.a | ✓ | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
381.3.d.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
381.3.d.a | ✓ | 2 | 127.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 2 \)
acting on \(S_{3}^{\mathrm{new}}(381, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T + 2)^{2} \)
$3$
\( T^{2} + 3 \)
$5$
\( T^{2} + 75 \)
$7$
\( T^{2} + 48 \)
$11$
\( (T - 10)^{2} \)
$13$
\( (T - 23)^{2} \)
$17$
\( (T - 10)^{2} \)
$19$
\( (T - 26)^{2} \)
$23$
\( T^{2} + 75 \)
$29$
\( T^{2} + 147 \)
$31$
\( (T + 25)^{2} \)
$37$
\( (T + 37)^{2} \)
$41$
\( (T - 16)^{2} \)
$43$
\( T^{2} + 588 \)
$47$
\( (T - 82)^{2} \)
$53$
\( T^{2} + 6075 \)
$59$
\( T^{2} + 8427 \)
$61$
\( (T + 79)^{2} \)
$67$
\( T^{2} + 6348 \)
$71$
\( (T + 26)^{2} \)
$73$
\( (T - 17)^{2} \)
$79$
\( (T - 26)^{2} \)
$83$
\( T^{2} + 3675 \)
$89$
\( T^{2} + 3267 \)
$97$
\( T^{2} + 14700 \)
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