Newspace parameters
| Level: | \( N \) | \(=\) | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 150.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(15.5054944626\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-2}) \) |
|
|
|
| Defining polynomial: |
\( x^{2} + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 6) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 101.1 |
|
− | 2.82843i | 3.00000 | + | 8.48528i | −8.00000 | 0 | 24.0000 | − | 8.48528i | −26.0000 | 22.6274i | −63.0000 | + | 50.9117i | 0 | |||||||||||||||||
| 101.2 | 2.82843i | 3.00000 | − | 8.48528i | −8.00000 | 0 | 24.0000 | + | 8.48528i | −26.0000 | − | 22.6274i | −63.0000 | − | 50.9117i | 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 | 150.5.d.a | 2 | |
| 3.b | odd | 2 | 1 | inner | 150.5.d.a | 2 | |
| 5.b | even | 2 | 1 | 6.5.b.a | ✓ | 2 | |
| 5.c | odd | 4 | 2 | 150.5.b.a | 4 | ||
| 15.d | odd | 2 | 1 | 6.5.b.a | ✓ | 2 | |
| 15.e | even | 4 | 2 | 150.5.b.a | 4 | ||
| 20.d | odd | 2 | 1 | 48.5.e.b | 2 | ||
| 35.c | odd | 2 | 1 | 294.5.b.a | 2 | ||
| 40.e | odd | 2 | 1 | 192.5.e.c | 2 | ||
| 40.f | even | 2 | 1 | 192.5.e.d | 2 | ||
| 45.h | odd | 6 | 2 | 162.5.d.a | 4 | ||
| 45.j | even | 6 | 2 | 162.5.d.a | 4 | ||
| 60.h | even | 2 | 1 | 48.5.e.b | 2 | ||
| 105.g | even | 2 | 1 | 294.5.b.a | 2 | ||
| 120.i | odd | 2 | 1 | 192.5.e.d | 2 | ||
| 120.m | even | 2 | 1 | 192.5.e.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6.5.b.a | ✓ | 2 | 5.b | even | 2 | 1 | |
| 6.5.b.a | ✓ | 2 | 15.d | odd | 2 | 1 | |
| 48.5.e.b | 2 | 20.d | odd | 2 | 1 | ||
| 48.5.e.b | 2 | 60.h | even | 2 | 1 | ||
| 150.5.b.a | 4 | 5.c | odd | 4 | 2 | ||
| 150.5.b.a | 4 | 15.e | even | 4 | 2 | ||
| 150.5.d.a | 2 | 1.a | even | 1 | 1 | trivial | |
| 150.5.d.a | 2 | 3.b | odd | 2 | 1 | inner | |
| 162.5.d.a | 4 | 45.h | odd | 6 | 2 | ||
| 162.5.d.a | 4 | 45.j | even | 6 | 2 | ||
| 192.5.e.c | 2 | 40.e | odd | 2 | 1 | ||
| 192.5.e.c | 2 | 120.m | even | 2 | 1 | ||
| 192.5.e.d | 2 | 40.f | even | 2 | 1 | ||
| 192.5.e.d | 2 | 120.i | odd | 2 | 1 | ||
| 294.5.b.a | 2 | 35.c | odd | 2 | 1 | ||
| 294.5.b.a | 2 | 105.g | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7} + 26 \)
acting on \(S_{5}^{\mathrm{new}}(150, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 8 \)
|
| $3$ |
\( T^{2} - 6T + 81 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( (T + 26)^{2} \)
|
| $11$ |
\( T^{2} + 14112 \)
|
| $13$ |
\( (T + 50)^{2} \)
|
| $17$ |
\( T^{2} + 41472 \)
|
| $19$ |
\( (T + 358)^{2} \)
|
| $23$ |
\( T^{2} + 139392 \)
|
| $29$ |
\( T^{2} + 2080800 \)
|
| $31$ |
\( (T + 742)^{2} \)
|
| $37$ |
\( (T + 1874)^{2} \)
|
| $41$ |
\( T^{2} + 5807232 \)
|
| $43$ |
\( (T - 262)^{2} \)
|
| $47$ |
\( T^{2} + 2880000 \)
|
| $53$ |
\( T^{2} + 209952 \)
|
| $59$ |
\( T^{2} + 3297312 \)
|
| $61$ |
\( (T + 1486)^{2} \)
|
| $67$ |
\( (T - 4486)^{2} \)
|
| $71$ |
\( T^{2} + 12700800 \)
|
| $73$ |
\( (T + 290)^{2} \)
|
| $79$ |
\( (T - 9818)^{2} \)
|
| $83$ |
\( T^{2} + 50561568 \)
|
| $89$ |
\( T^{2} + 61471872 \)
|
| $97$ |
\( (T - 478)^{2} \)
|
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