Newspace parameters
| Level: | \( N \) | \(=\) | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 150.f (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(4.08720396540\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{6})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 30) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−1.00000 | − | 1.00000i | −1.22474 | + | 1.22474i | 2.00000i | 0 | 2.44949 | −0.898979 | − | 0.898979i | 2.00000 | − | 2.00000i | − | 3.00000i | 0 | |||||||||||||||||||||
| 7.2 | −1.00000 | − | 1.00000i | 1.22474 | − | 1.22474i | 2.00000i | 0 | −2.44949 | 8.89898 | + | 8.89898i | 2.00000 | − | 2.00000i | − | 3.00000i | 0 | ||||||||||||||||||||||
| 43.1 | −1.00000 | + | 1.00000i | −1.22474 | − | 1.22474i | − | 2.00000i | 0 | 2.44949 | −0.898979 | + | 0.898979i | 2.00000 | + | 2.00000i | 3.00000i | 0 | ||||||||||||||||||||||
| 43.2 | −1.00000 | + | 1.00000i | 1.22474 | + | 1.22474i | − | 2.00000i | 0 | −2.44949 | 8.89898 | − | 8.89898i | 2.00000 | + | 2.00000i | 3.00000i | 0 | ||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 150.3.f.b | 4 | |
| 3.b | odd | 2 | 1 | 450.3.g.j | 4 | ||
| 4.b | odd | 2 | 1 | 1200.3.bg.d | 4 | ||
| 5.b | even | 2 | 1 | 30.3.f.a | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 30.3.f.a | ✓ | 4 | |
| 5.c | odd | 4 | 1 | inner | 150.3.f.b | 4 | |
| 15.d | odd | 2 | 1 | 90.3.g.d | 4 | ||
| 15.e | even | 4 | 1 | 90.3.g.d | 4 | ||
| 15.e | even | 4 | 1 | 450.3.g.j | 4 | ||
| 20.d | odd | 2 | 1 | 240.3.bg.b | 4 | ||
| 20.e | even | 4 | 1 | 240.3.bg.b | 4 | ||
| 20.e | even | 4 | 1 | 1200.3.bg.d | 4 | ||
| 40.e | odd | 2 | 1 | 960.3.bg.g | 4 | ||
| 40.f | even | 2 | 1 | 960.3.bg.e | 4 | ||
| 40.i | odd | 4 | 1 | 960.3.bg.e | 4 | ||
| 40.k | even | 4 | 1 | 960.3.bg.g | 4 | ||
| 60.h | even | 2 | 1 | 720.3.bh.i | 4 | ||
| 60.l | odd | 4 | 1 | 720.3.bh.i | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 30.3.f.a | ✓ | 4 | 5.b | even | 2 | 1 | |
| 30.3.f.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 90.3.g.d | 4 | 15.d | odd | 2 | 1 | ||
| 90.3.g.d | 4 | 15.e | even | 4 | 1 | ||
| 150.3.f.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 150.3.f.b | 4 | 5.c | odd | 4 | 1 | inner | |
| 240.3.bg.b | 4 | 20.d | odd | 2 | 1 | ||
| 240.3.bg.b | 4 | 20.e | even | 4 | 1 | ||
| 450.3.g.j | 4 | 3.b | odd | 2 | 1 | ||
| 450.3.g.j | 4 | 15.e | even | 4 | 1 | ||
| 720.3.bh.i | 4 | 60.h | even | 2 | 1 | ||
| 720.3.bh.i | 4 | 60.l | odd | 4 | 1 | ||
| 960.3.bg.e | 4 | 40.f | even | 2 | 1 | ||
| 960.3.bg.e | 4 | 40.i | odd | 4 | 1 | ||
| 960.3.bg.g | 4 | 40.e | odd | 2 | 1 | ||
| 960.3.bg.g | 4 | 40.k | even | 4 | 1 | ||
| 1200.3.bg.d | 4 | 4.b | odd | 2 | 1 | ||
| 1200.3.bg.d | 4 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} - 16T_{7}^{3} + 128T_{7}^{2} + 256T_{7} + 256 \)
acting on \(S_{3}^{\mathrm{new}}(150, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 2 T + 2)^{2} \)
$3$
\( T^{4} + 9 \)
$5$
\( T^{4} \)
$7$
\( T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 256 \)
$11$
\( (T^{2} + 8 T - 80)^{2} \)
$13$
\( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 30276 \)
$17$
\( T^{4} + 44 T^{3} + 968 T^{2} + \cdots + 37636 \)
$19$
\( T^{4} + 704 T^{2} + 25600 \)
$23$
\( T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 160000 \)
$29$
\( T^{4} + 704 T^{2} + 25600 \)
$31$
\( (T^{2} + 40 T + 16)^{2} \)
$37$
\( (T^{2} - 54 T + 1458)^{2} \)
$41$
\( (T^{2} - 16 T - 32)^{2} \)
$43$
\( T^{4} + 48 T^{3} + 1152 T^{2} + \cdots + 831744 \)
$47$
\( T^{4} + 96 T^{3} + 4608 T^{2} + \cdots + 518400 \)
$53$
\( T^{4} + 100 T^{3} + 5000 T^{2} + \cdots + 6959044 \)
$59$
\( (T^{2} + 400)^{2} \)
$61$
\( (T^{2} + 48 T - 960)^{2} \)
$67$
\( T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 14868736 \)
$71$
\( (T^{2} - 32 T + 160)^{2} \)
$73$
\( T^{4} - 188 T^{3} + \cdots + 17859076 \)
$79$
\( T^{4} + 32736 T^{2} + \cdots + 258566400 \)
$83$
\( T^{4} + 192 T^{3} + 18432 T^{2} + \cdots + 5089536 \)
$89$
\( T^{4} + 17216 T^{2} + \cdots + 47334400 \)
$97$
\( T^{4} + 132 T^{3} + 8712 T^{2} + \cdots + 6874884 \)
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