Newspace parameters
| Level: | \( N \) | \(=\) | \( 33 = 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 33.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(10.3087058410\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-11}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-11})\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).
| \(n\) | \(13\) | \(23\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 32.1 |
|
0 | −41.5000 | − | 21.5581i | −128.000 | − | 235.480i | 0 | 0 | 0 | 1257.50 | + | 1789.32i | 0 | |||||||||||||||||||
| 32.2 | 0 | −41.5000 | + | 21.5581i | −128.000 | 235.480i | 0 | 0 | 0 | 1257.50 | − | 1789.32i | 0 | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-11}) \) |
| 3.b | odd | 2 | 1 | inner |
| 33.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 33.8.d.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | inner | 33.8.d.a | ✓ | 2 |
| 11.b | odd | 2 | 1 | CM | 33.8.d.a | ✓ | 2 |
| 33.d | even | 2 | 1 | inner | 33.8.d.a | ✓ | 2 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.8.d.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 33.8.d.a | ✓ | 2 | 3.b | odd | 2 | 1 | inner |
| 33.8.d.a | ✓ | 2 | 11.b | odd | 2 | 1 | CM |
| 33.8.d.a | ✓ | 2 | 33.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{8}^{\mathrm{new}}(33, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 83T + 2187 \)
$5$
\( T^{2} + 55451 \)
$7$
\( T^{2} \)
$11$
\( T^{2} + 19487171 \)
$13$
\( T^{2} \)
$17$
\( T^{2} \)
$19$
\( T^{2} \)
$23$
\( T^{2} + 5253089699 \)
$29$
\( T^{2} \)
$31$
\( (T - 39065)^{2} \)
$37$
\( (T + 562471)^{2} \)
$41$
\( T^{2} \)
$43$
\( T^{2} \)
$47$
\( T^{2} + 297467356076 \)
$53$
\( T^{2} + 4566800988464 \)
$59$
\( T^{2} + 9932119437251 \)
$61$
\( T^{2} \)
$67$
\( (T - 684671)^{2} \)
$71$
\( T^{2} + 3556553200475 \)
$73$
\( T^{2} \)
$79$
\( T^{2} \)
$83$
\( T^{2} \)
$89$
\( T^{2} + \cdots + 157352492959475 \)
$97$
\( (T + 15182479)^{2} \)
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