Newspace parameters
| Level: | \( N \) | \(=\) | \( 15 = 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 15.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(2.40575729719\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{89})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 45x^{2} + 484 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3^{4} \) |
| Twist minimal: | yes |
| 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
\( x^{4} + 45x^{2} + 484 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{3} - 35\nu ) / 11 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{3} - 137\nu ) / 22 \)
|
| \(\beta_{3}\) | \(=\) |
\( 9\nu^{2} + 203 \)
|
| \(\nu\) | \(=\) |
\( ( -4\beta_{2} + 5\beta_1 ) / 9 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 203 ) / 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 70\beta_{2} - 137\beta_1 ) / 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) | \(11\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4.1 |
|
− | 9.21699i | − | 9.00000i | −52.9529 | 30.0000 | + | 47.1699i | −82.9529 | − | 167.208i | 193.123i | −81.0000 | 434.765 | − | 276.510i | |||||||||||||||||||||||
| 4.2 | − | 0.216991i | 9.00000i | 31.9529 | 30.0000 | + | 47.1699i | 1.95292 | − | 59.2078i | − | 13.8772i | −81.0000 | 10.2354 | − | 6.50972i | ||||||||||||||||||||||||
| 4.3 | 0.216991i | − | 9.00000i | 31.9529 | 30.0000 | − | 47.1699i | 1.95292 | 59.2078i | 13.8772i | −81.0000 | 10.2354 | + | 6.50972i | ||||||||||||||||||||||||||
| 4.4 | 9.21699i | 9.00000i | −52.9529 | 30.0000 | − | 47.1699i | −82.9529 | 167.208i | − | 193.123i | −81.0000 | 434.765 | + | 276.510i | ||||||||||||||||||||||||||
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 | 15.6.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 45.6.b.c | 4 | ||
| 4.b | odd | 2 | 1 | 240.6.f.c | 4 | ||
| 5.b | even | 2 | 1 | inner | 15.6.b.a | ✓ | 4 |
| 5.c | odd | 4 | 1 | 75.6.a.f | 2 | ||
| 5.c | odd | 4 | 1 | 75.6.a.j | 2 | ||
| 12.b | even | 2 | 1 | 720.6.f.h | 4 | ||
| 15.d | odd | 2 | 1 | 45.6.b.c | 4 | ||
| 15.e | even | 4 | 1 | 225.6.a.i | 2 | ||
| 15.e | even | 4 | 1 | 225.6.a.u | 2 | ||
| 20.d | odd | 2 | 1 | 240.6.f.c | 4 | ||
| 60.h | even | 2 | 1 | 720.6.f.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.6.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 15.6.b.a | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 45.6.b.c | 4 | 3.b | odd | 2 | 1 | ||
| 45.6.b.c | 4 | 15.d | odd | 2 | 1 | ||
| 75.6.a.f | 2 | 5.c | odd | 4 | 1 | ||
| 75.6.a.j | 2 | 5.c | odd | 4 | 1 | ||
| 225.6.a.i | 2 | 15.e | even | 4 | 1 | ||
| 225.6.a.u | 2 | 15.e | even | 4 | 1 | ||
| 240.6.f.c | 4 | 4.b | odd | 2 | 1 | ||
| 240.6.f.c | 4 | 20.d | odd | 2 | 1 | ||
| 720.6.f.h | 4 | 12.b | even | 2 | 1 | ||
| 720.6.f.h | 4 | 60.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(15, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 85T^{2} + 4 \)
|
| $3$ |
\( (T^{2} + 81)^{2} \)
|
| $5$ |
\( (T^{2} - 60 T + 3125)^{2} \)
|
| $7$ |
\( T^{4} + 31464 T^{2} + 98010000 \)
|
| $11$ |
\( (T^{2} - 168 T - 252468)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 69120616464 \)
|
| $17$ |
\( T^{4} + \cdots + 124719160336 \)
|
| $19$ |
\( (T^{2} - 1336 T + 330880)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 72965764000000 \)
|
| $29$ |
\( (T^{2} - 3552 T - 13455360)^{2} \)
|
| $31$ |
\( (T^{2} + 11648 T + 33457600)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 25\!\cdots\!00 \)
|
| $41$ |
\( (T^{2} - 1812 T - 202645980)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 670943443435776 \)
|
| $47$ |
\( T^{4} + \cdots + 11\!\cdots\!16 \)
|
| $53$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $59$ |
\( (T^{2} + 57336 T + 572451660)^{2} \)
|
| $61$ |
\( (T^{2} + 30140 T + 3798916)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 20\!\cdots\!96 \)
|
| $71$ |
\( (T^{2} - 17424 T - 157672656)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 120354942009600 \)
|
| $79$ |
\( (T^{2} - 57520 T - 1122176000)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 11\!\cdots\!24 \)
|
| $89$ |
\( (T^{2} + 136764 T + 4173659460)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 12\!\cdots\!76 \)
|
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