Newspace parameters
Level: | \( N \) | \(=\) | \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 2352.m (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(64.0873581775\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 2x^{2} + 4 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
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 \( x^{4} + 2x^{2} + 4 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{3} ) / 2 \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} + 1 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} + 4\nu ) / 2 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} - \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} - 1 \) |
\(\nu^{3}\) | \(=\) | \( 2\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).
\(n\) | \(785\) | \(1471\) | \(1765\) | \(2257\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1471.1 |
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0 | − | 1.73205i | 0 | 4.58579 | 0 | 0 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||
1471.2 | 0 | − | 1.73205i | 0 | 7.41421 | 0 | 0 | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||
1471.3 | 0 | 1.73205i | 0 | 4.58579 | 0 | 0 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||
1471.4 | 0 | 1.73205i | 0 | 7.41421 | 0 | 0 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2352.3.m.l | yes | 4 |
4.b | odd | 2 | 1 | inner | 2352.3.m.l | yes | 4 |
7.b | odd | 2 | 1 | 2352.3.m.d | ✓ | 4 | |
28.d | even | 2 | 1 | 2352.3.m.d | ✓ | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2352.3.m.d | ✓ | 4 | 7.b | odd | 2 | 1 | |
2352.3.m.d | ✓ | 4 | 28.d | even | 2 | 1 | |
2352.3.m.l | yes | 4 | 1.a | even | 1 | 1 | trivial |
2352.3.m.l | yes | 4 | 4.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 12T_{5} + 34 \)
acting on \(S_{3}^{\mathrm{new}}(2352, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( (T^{2} + 3)^{2} \)
$5$
\( (T^{2} - 12 T + 34)^{2} \)
$7$
\( T^{4} \)
$11$
\( T^{4} + 408T^{2} + 144 \)
$13$
\( (T^{2} - 24 T + 46)^{2} \)
$17$
\( (T^{2} - 12 T - 14)^{2} \)
$19$
\( (T^{2} + 216)^{2} \)
$23$
\( T^{4} + 2136 T^{2} + 767376 \)
$29$
\( (T^{2} - 48 T + 504)^{2} \)
$31$
\( T^{4} + 3984 T^{2} + \cdots + 3594816 \)
$37$
\( (T^{2} - 1152)^{2} \)
$41$
\( (T^{2} + 36 T + 226)^{2} \)
$43$
\( T^{4} + 5664 T^{2} + \cdots + 3873024 \)
$47$
\( T^{4} + 816T^{2} + 576 \)
$53$
\( (T^{2} + 44 T - 2108)^{2} \)
$59$
\( T^{4} + 8592 T^{2} + 166464 \)
$61$
\( (T^{2} - 48 T - 2786)^{2} \)
$67$
\( T^{4} + 1056 T^{2} + 112896 \)
$71$
\( T^{4} + 8856 T^{2} + \cdots + 6170256 \)
$73$
\( (T^{2} - 24 T - 2306)^{2} \)
$79$
\( T^{4} + 8256 T^{2} + 451584 \)
$83$
\( T^{4} + 4128 T^{2} + 112896 \)
$89$
\( (T^{2} + 60 T + 178)^{2} \)
$97$
\( (T^{2} - 216 T + 11422)^{2} \)
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