Newspace parameters
Level: | \( N \) | \(=\) | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 150.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(46.8577538226\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 6) |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 |
|
−8.00000 | −27.0000 | 64.0000 | 0 | 216.000 | 1576.00 | −512.000 | 729.000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(3\) | \(1\) |
\(5\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 150.8.a.e | 1 | |
3.b | odd | 2 | 1 | 450.8.a.ba | 1 | ||
5.b | even | 2 | 1 | 6.8.a.a | ✓ | 1 | |
5.c | odd | 4 | 2 | 150.8.c.k | 2 | ||
15.d | odd | 2 | 1 | 18.8.a.a | 1 | ||
15.e | even | 4 | 2 | 450.8.c.a | 2 | ||
20.d | odd | 2 | 1 | 48.8.a.b | 1 | ||
35.c | odd | 2 | 1 | 294.8.a.l | 1 | ||
35.i | odd | 6 | 2 | 294.8.e.d | 2 | ||
35.j | even | 6 | 2 | 294.8.e.c | 2 | ||
40.e | odd | 2 | 1 | 192.8.a.n | 1 | ||
40.f | even | 2 | 1 | 192.8.a.f | 1 | ||
45.h | odd | 6 | 2 | 162.8.c.i | 2 | ||
45.j | even | 6 | 2 | 162.8.c.d | 2 | ||
60.h | even | 2 | 1 | 144.8.a.h | 1 | ||
120.i | odd | 2 | 1 | 576.8.a.h | 1 | ||
120.m | even | 2 | 1 | 576.8.a.i | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6.8.a.a | ✓ | 1 | 5.b | even | 2 | 1 | |
18.8.a.a | 1 | 15.d | odd | 2 | 1 | ||
48.8.a.b | 1 | 20.d | odd | 2 | 1 | ||
144.8.a.h | 1 | 60.h | even | 2 | 1 | ||
150.8.a.e | 1 | 1.a | even | 1 | 1 | trivial | |
150.8.c.k | 2 | 5.c | odd | 4 | 2 | ||
162.8.c.d | 2 | 45.j | even | 6 | 2 | ||
162.8.c.i | 2 | 45.h | odd | 6 | 2 | ||
192.8.a.f | 1 | 40.f | even | 2 | 1 | ||
192.8.a.n | 1 | 40.e | odd | 2 | 1 | ||
294.8.a.l | 1 | 35.c | odd | 2 | 1 | ||
294.8.e.c | 2 | 35.j | even | 6 | 2 | ||
294.8.e.d | 2 | 35.i | odd | 6 | 2 | ||
450.8.a.ba | 1 | 3.b | odd | 2 | 1 | ||
450.8.c.a | 2 | 15.e | even | 4 | 2 | ||
576.8.a.h | 1 | 120.i | odd | 2 | 1 | ||
576.8.a.i | 1 | 120.m | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7} - 1576 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(150))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 8 \)
$3$
\( T + 27 \)
$5$
\( T \)
$7$
\( T - 1576 \)
$11$
\( T - 7332 \)
$13$
\( T - 3802 \)
$17$
\( T - 6606 \)
$19$
\( T - 24860 \)
$23$
\( T + 41448 \)
$29$
\( T + 41610 \)
$31$
\( T - 33152 \)
$37$
\( T - 36466 \)
$41$
\( T + 639078 \)
$43$
\( T - 156412 \)
$47$
\( T - 433776 \)
$53$
\( T + 786078 \)
$59$
\( T - 745140 \)
$61$
\( T + 1660618 \)
$67$
\( T - 3290836 \)
$71$
\( T - 5716152 \)
$73$
\( T + 2659898 \)
$79$
\( T - 3807440 \)
$83$
\( T + 2229468 \)
$89$
\( T - 5991210 \)
$97$
\( T - 4060126 \)
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