Newspace parameters
Level: | \( N \) | \(=\) | \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 252.z (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(26.0492306971\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{14})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 14x^{2} + 196 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{2}\cdot 3^{3} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \( x^{4} + 14x^{2} + 196 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{2} ) / 14 \) |
\(\beta_{2}\) | \(=\) | \( ( 3\nu^{3} + 84\nu ) / 7 \) |
\(\beta_{3}\) | \(=\) | \( ( -3\nu^{3} + 42\nu ) / 7 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} + \beta_{2} ) / 18 \) |
\(\nu^{2}\) | \(=\) | \( 14\beta_1 \) |
\(\nu^{3}\) | \(=\) | \( ( -14\beta_{3} + 7\beta_{2} ) / 9 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).
\(n\) | \(29\) | \(73\) | \(127\) |
\(\chi(n)\) | \(1\) | \(-\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
73.1 |
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0 | 0 | 0 | −33.6749 | + | 19.4422i | 0 | −38.5000 | + | 30.3109i | 0 | 0 | 0 | ||||||||||||||||||||||||||
73.2 | 0 | 0 | 0 | 33.6749 | − | 19.4422i | 0 | −38.5000 | + | 30.3109i | 0 | 0 | 0 | |||||||||||||||||||||||||||
145.1 | 0 | 0 | 0 | −33.6749 | − | 19.4422i | 0 | −38.5000 | − | 30.3109i | 0 | 0 | 0 | |||||||||||||||||||||||||||
145.2 | 0 | 0 | 0 | 33.6749 | + | 19.4422i | 0 | −38.5000 | − | 30.3109i | 0 | 0 | 0 | |||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
21.g | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 252.5.z.c | ✓ | 4 |
3.b | odd | 2 | 1 | inner | 252.5.z.c | ✓ | 4 |
7.d | odd | 6 | 1 | inner | 252.5.z.c | ✓ | 4 |
21.g | even | 6 | 1 | inner | 252.5.z.c | ✓ | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
252.5.z.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
252.5.z.c | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
252.5.z.c | ✓ | 4 | 7.d | odd | 6 | 1 | inner |
252.5.z.c | ✓ | 4 | 21.g | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 1512T_{5}^{2} + 2286144 \)
acting on \(S_{5}^{\mathrm{new}}(252, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} \)
$5$
\( T^{4} - 1512 T^{2} + \cdots + 2286144 \)
$7$
\( (T^{2} + 77 T + 2401)^{2} \)
$11$
\( T^{4} + 18144 T^{2} + \cdots + 329204736 \)
$13$
\( (T^{2} + 9747)^{2} \)
$17$
\( T^{4} - 1512 T^{2} + \cdots + 2286144 \)
$19$
\( (T^{2} - 471 T + 73947)^{2} \)
$23$
\( T^{4} + 113400 T^{2} + \cdots + 12859560000 \)
$29$
\( (T^{2} - 653184)^{2} \)
$31$
\( (T^{2} + 489 T + 79707)^{2} \)
$37$
\( (T^{2} + 2041 T + 4165681)^{2} \)
$41$
\( (T^{2} + 1747872)^{2} \)
$43$
\( (T - 1723)^{4} \)
$47$
\( T^{4} - 545832 T^{2} + \cdots + 297932572224 \)
$53$
\( T^{4} + \cdots + 584302214454336 \)
$59$
\( T^{4} - 40172328 T^{2} + \cdots + 16\!\cdots\!84 \)
$61$
\( (T^{2} - 7224 T + 17395392)^{2} \)
$67$
\( (T^{2} - 5035 T + 25351225)^{2} \)
$71$
\( (T^{2} - 70875000)^{2} \)
$73$
\( (T^{2} + 6861 T + 15691107)^{2} \)
$79$
\( (T^{2} - 235 T + 55225)^{2} \)
$83$
\( (T^{2} + 5263272)^{2} \)
$89$
\( T^{4} - 209236608 T^{2} + \cdots + 43\!\cdots\!64 \)
$97$
\( (T^{2} + 42232512)^{2} \)
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