Newspace parameters
| Level: | \( N \) | \(=\) | \( 90 = 2 \cdot 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 90.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(69.1508862504\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{1129})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 565x^{2} + 79524 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 30) |
| 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
\( x^{4} + 565x^{2} + 79524 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{3} - 566\nu ) / 141 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 1410\nu^{2} + 1693\nu + 398466 ) / 282 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 1410\nu^{2} - 1693\nu + 398466 ) / 282 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{3} + 2\beta_{2} + \beta_1 ) / 20 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 2826 ) / 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 566\beta_{3} - 566\beta_{2} - 1693\beta_1 ) / 20 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) | \(37\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
− | 32.0000i | 0 | −1024.00 | −6697.60 | − | 1992.57i | 0 | − | 10004.9i | 32768.0i | 0 | −63762.1 | + | 214323.i | ||||||||||||||||||||||||
| 19.2 | − | 32.0000i | 0 | −1024.00 | 4222.60 | + | 5567.57i | 0 | − | 17733.1i | 32768.0i | 0 | 178162. | − | 135123.i | |||||||||||||||||||||||||
| 19.3 | 32.0000i | 0 | −1024.00 | −6697.60 | + | 1992.57i | 0 | 10004.9i | − | 32768.0i | 0 | −63762.1 | − | 214323.i | ||||||||||||||||||||||||||
| 19.4 | 32.0000i | 0 | −1024.00 | 4222.60 | − | 5567.57i | 0 | 17733.1i | − | 32768.0i | 0 | 178162. | + | 135123.i | ||||||||||||||||||||||||||
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 | 90.12.c.a | 4 | |
| 3.b | odd | 2 | 1 | 30.12.c.a | ✓ | 4 | |
| 5.b | even | 2 | 1 | inner | 90.12.c.a | 4 | |
| 12.b | even | 2 | 1 | 240.12.f.a | 4 | ||
| 15.d | odd | 2 | 1 | 30.12.c.a | ✓ | 4 | |
| 15.e | even | 4 | 1 | 150.12.a.k | 2 | ||
| 15.e | even | 4 | 1 | 150.12.a.r | 2 | ||
| 60.h | even | 2 | 1 | 240.12.f.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 30.12.c.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 30.12.c.a | ✓ | 4 | 15.d | odd | 2 | 1 | |
| 90.12.c.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 90.12.c.a | 4 | 5.b | even | 2 | 1 | inner | |
| 150.12.a.k | 2 | 15.e | even | 4 | 1 | ||
| 150.12.a.r | 2 | 15.e | even | 4 | 1 | ||
| 240.12.f.a | 4 | 12.b | even | 2 | 1 | ||
| 240.12.f.a | 4 | 60.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 414560372T_{7}^{2} + 31477194981714496 \)
acting on \(S_{12}^{\mathrm{new}}(90, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1024)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 23\!\cdots\!25 \)
|
| $7$ |
\( T^{4} + \cdots + 31\!\cdots\!96 \)
|
| $11$ |
\( (T^{2} - 226026 T + 12358695944)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 24\!\cdots\!16 \)
|
| $17$ |
\( T^{4} + \cdots + 11\!\cdots\!76 \)
|
| $19$ |
\( (T^{2} + \cdots + 81964096704000)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 18\!\cdots\!76 \)
|
| $29$ |
\( (T^{2} + \cdots - 12\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{2} + \cdots - 43\!\cdots\!76)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 30\!\cdots\!36 \)
|
| $41$ |
\( (T^{2} + \cdots + 38\!\cdots\!64)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 13\!\cdots\!96 \)
|
| $47$ |
\( T^{4} + \cdots + 61\!\cdots\!16 \)
|
| $53$ |
\( T^{4} + \cdots + 86\!\cdots\!56 \)
|
| $59$ |
\( (T^{2} + \cdots - 54\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots - 17\!\cdots\!56)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 67\!\cdots\!76 \)
|
| $71$ |
\( (T^{2} + \cdots - 21\!\cdots\!16)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 30\!\cdots\!76 \)
|
| $79$ |
\( (T^{2} + \cdots - 33\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 11\!\cdots\!36 \)
|
| $89$ |
\( (T^{2} + \cdots - 51\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 46\!\cdots\!16 \)
|
| show more | |
| show less |