Newspace parameters
Level: | \( N \) | \(=\) | \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2310.p (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(18.4454428669\) |
Analytic rank: | \(1\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(i, \sqrt{13})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 7x^{2} + 9 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
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 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 \( x^{4} + 7x^{2} + 9 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{3} + 4\nu ) / 3 \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} + \nu + 4 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} - 3\nu^{2} + 7\nu - 12 ) / 3 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{3} + \beta_{2} + \beta _1 - 8 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( -2\beta_{3} - 2\beta_{2} + 5\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2310\mathbb{Z}\right)^\times\).
\(n\) | \(211\) | \(661\) | \(1387\) | \(1541\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1231.1 |
|
− | 1.00000i | 1.00000i | −1.00000 | 1.00000i | 1.00000 | −2.30278 | + | 1.30278i | 1.00000i | −1.00000 | 1.00000 | |||||||||||||||||||||||||||
1231.2 | − | 1.00000i | 1.00000i | −1.00000 | 1.00000i | 1.00000 | 1.30278 | − | 2.30278i | 1.00000i | −1.00000 | 1.00000 | ||||||||||||||||||||||||||||
1231.3 | 1.00000i | − | 1.00000i | −1.00000 | − | 1.00000i | 1.00000 | −2.30278 | − | 1.30278i | − | 1.00000i | −1.00000 | 1.00000 | ||||||||||||||||||||||||||
1231.4 | 1.00000i | − | 1.00000i | −1.00000 | − | 1.00000i | 1.00000 | 1.30278 | + | 2.30278i | − | 1.00000i | −1.00000 | 1.00000 | ||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
77.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2310.2.p.b | yes | 4 |
7.b | odd | 2 | 1 | 2310.2.p.a | ✓ | 4 | |
11.b | odd | 2 | 1 | 2310.2.p.a | ✓ | 4 | |
77.b | even | 2 | 1 | inner | 2310.2.p.b | yes | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2310.2.p.a | ✓ | 4 | 7.b | odd | 2 | 1 | |
2310.2.p.a | ✓ | 4 | 11.b | odd | 2 | 1 | |
2310.2.p.b | yes | 4 | 1.a | even | 1 | 1 | trivial |
2310.2.p.b | yes | 4 | 77.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{13} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(2310, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 1)^{2} \)
$3$
\( (T^{2} + 1)^{2} \)
$5$
\( (T^{2} + 1)^{2} \)
$7$
\( T^{4} + 2 T^{3} + 2 T^{2} + 14 T + 49 \)
$11$
\( T^{4} - 6 T^{3} + 18 T^{2} - 66 T + 121 \)
$13$
\( (T + 4)^{4} \)
$17$
\( (T^{2} - 2 T - 12)^{2} \)
$19$
\( (T^{2} + 10 T + 12)^{2} \)
$23$
\( T^{4} \)
$29$
\( T^{4} + 28T^{2} + 144 \)
$31$
\( (T^{2} + 52)^{2} \)
$37$
\( (T - 4)^{4} \)
$41$
\( (T + 10)^{4} \)
$43$
\( (T^{2} + 4)^{2} \)
$47$
\( T^{4} + 28T^{2} + 144 \)
$53$
\( (T^{2} + 2 T - 12)^{2} \)
$59$
\( (T^{2} + 64)^{2} \)
$61$
\( (T^{2} + 2 T - 12)^{2} \)
$67$
\( (T^{2} + 16 T + 12)^{2} \)
$71$
\( (T^{2} + 4 T - 48)^{2} \)
$73$
\( (T^{2} + 10 T - 92)^{2} \)
$79$
\( T^{4} + 136T^{2} + 1296 \)
$83$
\( (T^{2} - 10 T + 12)^{2} \)
$89$
\( T^{4} + 188T^{2} + 4624 \)
$97$
\( T^{4} + 232T^{2} + 144 \)
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