Newspace parameters
| Level: | \( N \) | \(=\) | \( 153 = 3^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 153.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(9.02729223088\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-37 +3 \sqrt{33}})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 74x^{2} + 1072 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 17) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 74x^{2} + 1072 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + 40 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 46\nu ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 6\beta_{2} - 40 \)
|
| \(\nu^{3}\) | \(=\) |
\( 6\beta_{3} - 46\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/153\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(137\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 118.1 |
|
−2.37228 | 0 | −2.37228 | − | 19.4389i | 0 | − | 14.9929i | 24.6060 | 0 | 46.1145i | ||||||||||||||||||||||||||||
| 118.2 | −2.37228 | 0 | −2.37228 | 19.4389i | 0 | 14.9929i | 24.6060 | 0 | − | 46.1145i | ||||||||||||||||||||||||||||||
| 118.3 | 3.37228 | 0 | 3.37228 | − | 10.1060i | 0 | − | 17.4703i | −15.6060 | 0 | − | 34.0802i | ||||||||||||||||||||||||||||
| 118.4 | 3.37228 | 0 | 3.37228 | 10.1060i | 0 | 17.4703i | −15.6060 | 0 | 34.0802i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 153.4.d.b | 4 | |
| 3.b | odd | 2 | 1 | 17.4.b.a | ✓ | 4 | |
| 12.b | even | 2 | 1 | 272.4.b.d | 4 | ||
| 15.d | odd | 2 | 1 | 425.4.d.c | 4 | ||
| 15.e | even | 4 | 2 | 425.4.c.c | 8 | ||
| 17.b | even | 2 | 1 | inner | 153.4.d.b | 4 | |
| 51.c | odd | 2 | 1 | 17.4.b.a | ✓ | 4 | |
| 51.f | odd | 4 | 2 | 289.4.a.e | 4 | ||
| 204.h | even | 2 | 1 | 272.4.b.d | 4 | ||
| 255.h | odd | 2 | 1 | 425.4.d.c | 4 | ||
| 255.o | even | 4 | 2 | 425.4.c.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 17.4.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 17.4.b.a | ✓ | 4 | 51.c | odd | 2 | 1 | |
| 153.4.d.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 153.4.d.b | 4 | 17.b | even | 2 | 1 | inner | |
| 272.4.b.d | 4 | 12.b | even | 2 | 1 | ||
| 272.4.b.d | 4 | 204.h | even | 2 | 1 | ||
| 289.4.a.e | 4 | 51.f | odd | 4 | 2 | ||
| 425.4.c.c | 8 | 15.e | even | 4 | 2 | ||
| 425.4.c.c | 8 | 255.o | even | 4 | 2 | ||
| 425.4.d.c | 4 | 15.d | odd | 2 | 1 | ||
| 425.4.d.c | 4 | 255.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - T_{2} - 8 \)
acting on \(S_{4}^{\mathrm{new}}(153, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T - 8)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 480 T^{2} + 38592 \)
|
| $7$ |
\( T^{4} + 530 T^{2} + 68608 \)
|
| $11$ |
\( T^{4} + 3626 T^{2} + 2573872 \)
|
| $13$ |
\( (T^{2} - 70 T - 392)^{2} \)
|
| $17$ |
\( T^{4} + 112 T^{3} + \cdots + 24137569 \)
|
| $19$ |
\( (T + 28)^{4} \)
|
| $23$ |
\( T^{4} + 28322 T^{2} + 10295488 \)
|
| $29$ |
\( T^{4} + 23520 T^{2} + 92659392 \)
|
| $31$ |
\( T^{4} + 4274 T^{2} + 4390912 \)
|
| $37$ |
\( T^{4} + \cdots + 1245754048 \)
|
| $41$ |
\( T^{4} + \cdots + 2799480832 \)
|
| $43$ |
\( (T^{2} + 260 T - 30752)^{2} \)
|
| $47$ |
\( (T^{2} + 476 T + 50176)^{2} \)
|
| $53$ |
\( (T^{2} - 288 T - 49092)^{2} \)
|
| $59$ |
\( (T^{2} - 588 T - 75264)^{2} \)
|
| $61$ |
\( T^{4} + \cdots + 7146728128 \)
|
| $67$ |
\( (T^{2} + 120 T - 30192)^{2} \)
|
| $71$ |
\( T^{4} + 52626 T^{2} + 92659392 \)
|
| $73$ |
\( T^{4} + \cdots + 647466319872 \)
|
| $79$ |
\( T^{4} + \cdots + 70925616832 \)
|
| $83$ |
\( (T^{2} - 1428 T + 451584)^{2} \)
|
| $89$ |
\( (T^{2} + 994 T + 232456)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 16878665728 \)
|
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