Newspace parameters
Level: | \( N \) | \(=\) | \( 1210 = 2 \cdot 5 \cdot 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 1210.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(32.9701119876\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{5})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 6x^{2} + 4 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2^{2} \) |
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 \( x^{4} + 6x^{2} + 4 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{3} + 4\nu ) / 2 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{3} + 8\nu ) / 2 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{2} + 3 \) |
\(\nu\) | \(=\) | \( ( \beta_{2} - \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{3} - 3 \) |
\(\nu^{3}\) | \(=\) | \( -2\beta_{2} + 4\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1210\mathbb{Z}\right)^\times\).
\(n\) | \(727\) | \(1091\) |
\(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
241.1 |
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− | 1.41421i | −1.23607 | −2.00000 | −2.23607 | 1.74806i | 2.49458i | 2.82843i | −7.47214 | 3.16228i | |||||||||||||||||||||||||||||
241.2 | − | 1.41421i | 3.23607 | −2.00000 | 2.23607 | − | 4.57649i | 8.81913i | 2.82843i | 1.47214 | − | 3.16228i | ||||||||||||||||||||||||||||
241.3 | 1.41421i | −1.23607 | −2.00000 | −2.23607 | − | 1.74806i | − | 2.49458i | − | 2.82843i | −7.47214 | − | 3.16228i | |||||||||||||||||||||||||||
241.4 | 1.41421i | 3.23607 | −2.00000 | 2.23607 | 4.57649i | − | 8.81913i | − | 2.82843i | 1.47214 | 3.16228i | |||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1210.3.d.b | ✓ | 4 |
11.b | odd | 2 | 1 | inner | 1210.3.d.b | ✓ | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1210.3.d.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
1210.3.d.b | ✓ | 4 | 11.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 2T_{3} - 4 \)
acting on \(S_{3}^{\mathrm{new}}(1210, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 2)^{2} \)
$3$
\( (T^{2} - 2 T - 4)^{2} \)
$5$
\( (T^{2} - 5)^{2} \)
$7$
\( T^{4} + 84T^{2} + 484 \)
$11$
\( T^{4} \)
$13$
\( T^{4} + 56T^{2} + 64 \)
$17$
\( T^{4} + 184T^{2} + 7744 \)
$19$
\( T^{4} + 1104 T^{2} + 53824 \)
$23$
\( (T^{2} + 6 T - 836)^{2} \)
$29$
\( T^{4} + 84T^{2} + 1444 \)
$31$
\( (T^{2} + 24 T - 576)^{2} \)
$37$
\( (T^{2} + 100 T + 2180)^{2} \)
$41$
\( T^{4} + 324T^{2} + 324 \)
$43$
\( T^{4} + 5444 T^{2} + \cdots + 5750404 \)
$47$
\( (T^{2} - 90 T + 1420)^{2} \)
$53$
\( (T^{2} + 36 T - 396)^{2} \)
$59$
\( (T^{2} - 4 T - 16)^{2} \)
$61$
\( T^{4} + 4644 T^{2} + \cdots + 4743684 \)
$67$
\( (T^{2} - 102 T - 3524)^{2} \)
$71$
\( (T^{2} + 80 T + 1520)^{2} \)
$73$
\( T^{4} + 6456 T^{2} + \cdots + 5053504 \)
$79$
\( T^{4} + 11664 T^{2} + \cdots + 32353344 \)
$83$
\( T^{4} + 20484 T^{2} + \cdots + 104816644 \)
$89$
\( (T^{2} - 48 T - 7424)^{2} \)
$97$
\( (T^{2} - 48 T - 21204)^{2} \)
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