Newspace parameters
Level: | \( N \) | \(=\) | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 150.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(8.85028650086\) |
Analytic rank: | \(1\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 30) |
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 |
|
2.00000 | −3.00000 | 4.00000 | 0 | −6.00000 | −32.0000 | 8.00000 | 9.00000 | 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 | 150.4.a.e | 1 | |
3.b | odd | 2 | 1 | 450.4.a.b | 1 | ||
4.b | odd | 2 | 1 | 1200.4.a.bk | 1 | ||
5.b | even | 2 | 1 | 30.4.a.a | ✓ | 1 | |
5.c | odd | 4 | 2 | 150.4.c.a | 2 | ||
15.d | odd | 2 | 1 | 90.4.a.d | 1 | ||
15.e | even | 4 | 2 | 450.4.c.k | 2 | ||
20.d | odd | 2 | 1 | 240.4.a.c | 1 | ||
20.e | even | 4 | 2 | 1200.4.f.u | 2 | ||
35.c | odd | 2 | 1 | 1470.4.a.a | 1 | ||
40.e | odd | 2 | 1 | 960.4.a.s | 1 | ||
40.f | even | 2 | 1 | 960.4.a.j | 1 | ||
45.h | odd | 6 | 2 | 810.4.e.e | 2 | ||
45.j | even | 6 | 2 | 810.4.e.m | 2 | ||
60.h | even | 2 | 1 | 720.4.a.b | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
30.4.a.a | ✓ | 1 | 5.b | even | 2 | 1 | |
90.4.a.d | 1 | 15.d | odd | 2 | 1 | ||
150.4.a.e | 1 | 1.a | even | 1 | 1 | trivial | |
150.4.c.a | 2 | 5.c | odd | 4 | 2 | ||
240.4.a.c | 1 | 20.d | odd | 2 | 1 | ||
450.4.a.b | 1 | 3.b | odd | 2 | 1 | ||
450.4.c.k | 2 | 15.e | even | 4 | 2 | ||
720.4.a.b | 1 | 60.h | even | 2 | 1 | ||
810.4.e.e | 2 | 45.h | odd | 6 | 2 | ||
810.4.e.m | 2 | 45.j | even | 6 | 2 | ||
960.4.a.j | 1 | 40.f | even | 2 | 1 | ||
960.4.a.s | 1 | 40.e | odd | 2 | 1 | ||
1200.4.a.bk | 1 | 4.b | odd | 2 | 1 | ||
1200.4.f.u | 2 | 20.e | even | 4 | 2 | ||
1470.4.a.a | 1 | 35.c | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7} + 32 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(150))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T - 2 \)
$3$
\( T + 3 \)
$5$
\( T \)
$7$
\( T + 32 \)
$11$
\( T + 60 \)
$13$
\( T - 34 \)
$17$
\( T + 42 \)
$19$
\( T + 76 \)
$23$
\( T \)
$29$
\( T - 6 \)
$31$
\( T + 232 \)
$37$
\( T + 134 \)
$41$
\( T - 234 \)
$43$
\( T - 412 \)
$47$
\( T - 360 \)
$53$
\( T + 222 \)
$59$
\( T - 660 \)
$61$
\( T + 490 \)
$67$
\( T + 812 \)
$71$
\( T - 120 \)
$73$
\( T + 746 \)
$79$
\( T - 152 \)
$83$
\( T - 804 \)
$89$
\( T + 678 \)
$97$
\( T + 194 \)
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