Newspace parameters
| Level: | \( N \) | \(=\) | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 150.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(24.0575729719\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
|
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| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(i = \sqrt{-1}\). 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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
− | 4.00000i | − | 9.00000i | −16.0000 | 0 | −36.0000 | − | 176.000i | 64.0000i | −81.0000 | 0 | |||||||||||||||||||||
| 49.2 | 4.00000i | 9.00000i | −16.0000 | 0 | −36.0000 | 176.000i | − | 64.0000i | −81.0000 | 0 | ||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 150.6.c.b | 2 | |
| 3.b | odd | 2 | 1 | 450.6.c.j | 2 | ||
| 5.b | even | 2 | 1 | inner | 150.6.c.b | 2 | |
| 5.c | odd | 4 | 1 | 6.6.a.a | ✓ | 1 | |
| 5.c | odd | 4 | 1 | 150.6.a.d | 1 | ||
| 15.d | odd | 2 | 1 | 450.6.c.j | 2 | ||
| 15.e | even | 4 | 1 | 18.6.a.b | 1 | ||
| 15.e | even | 4 | 1 | 450.6.a.m | 1 | ||
| 20.e | even | 4 | 1 | 48.6.a.c | 1 | ||
| 35.f | even | 4 | 1 | 294.6.a.m | 1 | ||
| 35.k | even | 12 | 2 | 294.6.e.a | 2 | ||
| 35.l | odd | 12 | 2 | 294.6.e.g | 2 | ||
| 40.i | odd | 4 | 1 | 192.6.a.o | 1 | ||
| 40.k | even | 4 | 1 | 192.6.a.g | 1 | ||
| 45.k | odd | 12 | 2 | 162.6.c.e | 2 | ||
| 45.l | even | 12 | 2 | 162.6.c.h | 2 | ||
| 55.e | even | 4 | 1 | 726.6.a.a | 1 | ||
| 60.l | odd | 4 | 1 | 144.6.a.j | 1 | ||
| 65.h | odd | 4 | 1 | 1014.6.a.c | 1 | ||
| 80.i | odd | 4 | 1 | 768.6.d.c | 2 | ||
| 80.j | even | 4 | 1 | 768.6.d.p | 2 | ||
| 80.s | even | 4 | 1 | 768.6.d.p | 2 | ||
| 80.t | odd | 4 | 1 | 768.6.d.c | 2 | ||
| 105.k | odd | 4 | 1 | 882.6.a.a | 1 | ||
| 120.q | odd | 4 | 1 | 576.6.a.i | 1 | ||
| 120.w | even | 4 | 1 | 576.6.a.j | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6.6.a.a | ✓ | 1 | 5.c | odd | 4 | 1 | |
| 18.6.a.b | 1 | 15.e | even | 4 | 1 | ||
| 48.6.a.c | 1 | 20.e | even | 4 | 1 | ||
| 144.6.a.j | 1 | 60.l | odd | 4 | 1 | ||
| 150.6.a.d | 1 | 5.c | odd | 4 | 1 | ||
| 150.6.c.b | 2 | 1.a | even | 1 | 1 | trivial | |
| 150.6.c.b | 2 | 5.b | even | 2 | 1 | inner | |
| 162.6.c.e | 2 | 45.k | odd | 12 | 2 | ||
| 162.6.c.h | 2 | 45.l | even | 12 | 2 | ||
| 192.6.a.g | 1 | 40.k | even | 4 | 1 | ||
| 192.6.a.o | 1 | 40.i | odd | 4 | 1 | ||
| 294.6.a.m | 1 | 35.f | even | 4 | 1 | ||
| 294.6.e.a | 2 | 35.k | even | 12 | 2 | ||
| 294.6.e.g | 2 | 35.l | odd | 12 | 2 | ||
| 450.6.a.m | 1 | 15.e | even | 4 | 1 | ||
| 450.6.c.j | 2 | 3.b | odd | 2 | 1 | ||
| 450.6.c.j | 2 | 15.d | odd | 2 | 1 | ||
| 576.6.a.i | 1 | 120.q | odd | 4 | 1 | ||
| 576.6.a.j | 1 | 120.w | even | 4 | 1 | ||
| 726.6.a.a | 1 | 55.e | even | 4 | 1 | ||
| 768.6.d.c | 2 | 80.i | odd | 4 | 1 | ||
| 768.6.d.c | 2 | 80.t | odd | 4 | 1 | ||
| 768.6.d.p | 2 | 80.j | even | 4 | 1 | ||
| 768.6.d.p | 2 | 80.s | even | 4 | 1 | ||
| 882.6.a.a | 1 | 105.k | odd | 4 | 1 | ||
| 1014.6.a.c | 1 | 65.h | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} + 30976 \)
acting on \(S_{6}^{\mathrm{new}}(150, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 16 \)
|
| $3$ |
\( T^{2} + 81 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} + 30976 \)
|
| $11$ |
\( (T + 60)^{2} \)
|
| $13$ |
\( T^{2} + 432964 \)
|
| $17$ |
\( T^{2} + 171396 \)
|
| $19$ |
\( (T + 956)^{2} \)
|
| $23$ |
\( T^{2} + 360000 \)
|
| $29$ |
\( (T + 5574)^{2} \)
|
| $31$ |
\( (T + 3592)^{2} \)
|
| $37$ |
\( T^{2} + 71537764 \)
|
| $41$ |
\( (T - 19194)^{2} \)
|
| $43$ |
\( T^{2} + 177315856 \)
|
| $47$ |
\( T^{2} + 387302400 \)
|
| $53$ |
\( T^{2} + 977562756 \)
|
| $59$ |
\( (T + 26340)^{2} \)
|
| $61$ |
\( (T + 31090)^{2} \)
|
| $67$ |
\( T^{2} + 282374416 \)
|
| $71$ |
\( (T - 6120)^{2} \)
|
| $73$ |
\( T^{2} + 653211364 \)
|
| $79$ |
\( (T + 74408)^{2} \)
|
| $83$ |
\( T^{2} + 41835024 \)
|
| $89$ |
\( (T - 32742)^{2} \)
|
| $97$ |
\( T^{2} + 27583230724 \)
|
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