Newspace parameters
Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 48.e (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.96175822802\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{13})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - 2x^{3} - x^{2} + 2x + 27 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{8}\cdot 3 \) |
Twist minimal: | no (minimal twist has level 24) |
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 \( x^{4} - 2x^{3} - x^{2} + 2x + 27 \) :
\(\beta_{1}\) | \(=\) | \( ( 8\nu^{3} + 72\nu^{2} - 224\nu - 12 ) / 21 \) |
\(\beta_{2}\) | \(=\) | \( ( -10\nu^{3} + 36\nu^{2} + 28\nu - 48 ) / 21 \) |
\(\beta_{3}\) | \(=\) | \( ( 32\nu^{3} - 48\nu^{2} + 112\nu - 48 ) / 21 \) |
\(\nu\) | \(=\) | \( ( 3\beta_{3} + 8\beta_{2} - 2\beta _1 + 24 ) / 48 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{3} + 4\beta_{2} + \beta _1 + 12 ) / 8 \) |
\(\nu^{3}\) | \(=\) | \( ( 15\beta_{3} + 4\beta_{2} + 8\beta _1 + 48 ) / 24 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).
\(n\) | \(17\) | \(31\) | \(37\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 |
|
0 | −6.21110 | − | 6.51323i | 0 | 16.4520i | 0 | −49.2666 | 0 | −3.84441 | + | 80.9087i | 0 | ||||||||||||||||||||||||||
17.2 | 0 | −6.21110 | + | 6.51323i | 0 | − | 16.4520i | 0 | −49.2666 | 0 | −3.84441 | − | 80.9087i | 0 | ||||||||||||||||||||||||||
17.3 | 0 | 8.21110 | − | 3.68481i | 0 | 44.7363i | 0 | 37.2666 | 0 | 53.8444 | − | 60.5126i | 0 | |||||||||||||||||||||||||||
17.4 | 0 | 8.21110 | + | 3.68481i | 0 | − | 44.7363i | 0 | 37.2666 | 0 | 53.8444 | + | 60.5126i | 0 | ||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 48.5.e.c | 4 | |
3.b | odd | 2 | 1 | inner | 48.5.e.c | 4 | |
4.b | odd | 2 | 1 | 24.5.e.a | ✓ | 4 | |
8.b | even | 2 | 1 | 192.5.e.e | 4 | ||
8.d | odd | 2 | 1 | 192.5.e.f | 4 | ||
12.b | even | 2 | 1 | 24.5.e.a | ✓ | 4 | |
20.d | odd | 2 | 1 | 600.5.l.a | 4 | ||
20.e | even | 4 | 2 | 600.5.c.a | 8 | ||
24.f | even | 2 | 1 | 192.5.e.f | 4 | ||
24.h | odd | 2 | 1 | 192.5.e.e | 4 | ||
36.f | odd | 6 | 2 | 648.5.m.e | 8 | ||
36.h | even | 6 | 2 | 648.5.m.e | 8 | ||
60.h | even | 2 | 1 | 600.5.l.a | 4 | ||
60.l | odd | 4 | 2 | 600.5.c.a | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
24.5.e.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
24.5.e.a | ✓ | 4 | 12.b | even | 2 | 1 | |
48.5.e.c | 4 | 1.a | even | 1 | 1 | trivial | |
48.5.e.c | 4 | 3.b | odd | 2 | 1 | inner | |
192.5.e.e | 4 | 8.b | even | 2 | 1 | ||
192.5.e.e | 4 | 24.h | odd | 2 | 1 | ||
192.5.e.f | 4 | 8.d | odd | 2 | 1 | ||
192.5.e.f | 4 | 24.f | even | 2 | 1 | ||
600.5.c.a | 8 | 20.e | even | 4 | 2 | ||
600.5.c.a | 8 | 60.l | odd | 4 | 2 | ||
600.5.l.a | 4 | 20.d | odd | 2 | 1 | ||
600.5.l.a | 4 | 60.h | even | 2 | 1 | ||
648.5.m.e | 8 | 36.f | odd | 6 | 2 | ||
648.5.m.e | 8 | 36.h | even | 6 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 2272T_{5}^{2} + 541696 \)
acting on \(S_{5}^{\mathrm{new}}(48, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} - 4 T^{3} - 42 T^{2} + \cdots + 6561 \)
$5$
\( T^{4} + 2272 T^{2} + 541696 \)
$7$
\( (T^{2} + 12 T - 1836)^{2} \)
$11$
\( T^{4} + 43744 T^{2} + \cdots + 25240576 \)
$13$
\( (T^{2} + 124 T - 26108)^{2} \)
$17$
\( T^{4} + 122368 T^{2} + \cdots + 975437824 \)
$19$
\( (T^{2} + 412 T + 40564)^{2} \)
$23$
\( T^{4} + 485248 T^{2} + \cdots + 15447506944 \)
$29$
\( T^{4} + 227808 T^{2} + \cdots + 12680561664 \)
$31$
\( (T^{2} + 1900 T + 675988)^{2} \)
$37$
\( (T^{2} + 1404 T - 974844)^{2} \)
$41$
\( T^{4} + 6966144 T^{2} + \cdots + 1432636637184 \)
$43$
\( (T^{2} - 484 T - 33164)^{2} \)
$47$
\( T^{4} + 1027584 T^{2} + \cdots + 55228760064 \)
$53$
\( T^{4} + 28706272 T^{2} + \cdots + 22497339114496 \)
$59$
\( T^{4} + 14492128 T^{2} + \cdots + 1354747012096 \)
$61$
\( (T^{2} - 4292 T + 981124)^{2} \)
$67$
\( (T^{2} - 7556 T + 14226484)^{2} \)
$71$
\( T^{4} + 29260672 T^{2} + \cdots + 23483250786304 \)
$73$
\( (T^{2} - 2756 T - 2414204)^{2} \)
$79$
\( (T^{2} - 532 T - 36098156)^{2} \)
$83$
\( T^{4} + 90476512 T^{2} + \cdots + 16\!\cdots\!64 \)
$89$
\( T^{4} + \cdots + 919970046296064 \)
$97$
\( (T^{2} - 16228 T + 47116996)^{2} \)
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