Newspace parameters
| Level: | \( N \) | \(=\) | \( 33 = 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 33.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(1.94706303019\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{33}) \) |
| Defining polynomial: |
\( x^{2} - x - 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.37228 | 3.00000 | −2.37228 | 19.4891 | −7.11684 | 6.74456 | 24.6060 | 9.00000 | −46.2337 | ||||||||||||||||||||||||
| 1.2 | 3.37228 | 3.00000 | 3.37228 | −3.48913 | 10.1168 | −4.74456 | −15.6060 | 9.00000 | −11.7663 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(11\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 33.4.a.d | ✓ | 2 |
| 3.b | odd | 2 | 1 | 99.4.a.e | 2 | ||
| 4.b | odd | 2 | 1 | 528.4.a.o | 2 | ||
| 5.b | even | 2 | 1 | 825.4.a.k | 2 | ||
| 5.c | odd | 4 | 2 | 825.4.c.i | 4 | ||
| 7.b | odd | 2 | 1 | 1617.4.a.j | 2 | ||
| 8.b | even | 2 | 1 | 2112.4.a.ba | 2 | ||
| 8.d | odd | 2 | 1 | 2112.4.a.bh | 2 | ||
| 11.b | odd | 2 | 1 | 363.4.a.j | 2 | ||
| 12.b | even | 2 | 1 | 1584.4.a.x | 2 | ||
| 15.d | odd | 2 | 1 | 2475.4.a.o | 2 | ||
| 33.d | even | 2 | 1 | 1089.4.a.t | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.4.a.d | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 99.4.a.e | 2 | 3.b | odd | 2 | 1 | ||
| 363.4.a.j | 2 | 11.b | odd | 2 | 1 | ||
| 528.4.a.o | 2 | 4.b | odd | 2 | 1 | ||
| 825.4.a.k | 2 | 5.b | even | 2 | 1 | ||
| 825.4.c.i | 4 | 5.c | odd | 4 | 2 | ||
| 1089.4.a.t | 2 | 33.d | even | 2 | 1 | ||
| 1584.4.a.x | 2 | 12.b | even | 2 | 1 | ||
| 1617.4.a.j | 2 | 7.b | odd | 2 | 1 | ||
| 2112.4.a.ba | 2 | 8.b | even | 2 | 1 | ||
| 2112.4.a.bh | 2 | 8.d | odd | 2 | 1 | ||
| 2475.4.a.o | 2 | 15.d | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(33))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - T - 8 \)
$3$
\( (T - 3)^{2} \)
$5$
\( T^{2} - 16T - 68 \)
$7$
\( T^{2} - 2T - 32 \)
$11$
\( (T - 11)^{2} \)
$13$
\( T^{2} + 76T + 916 \)
$17$
\( T^{2} + 26T - 7256 \)
$19$
\( T^{2} + 54T - 1944 \)
$23$
\( (T - 112)^{2} \)
$29$
\( T^{2} - 222T - 5136 \)
$31$
\( T^{2} + 40T - 88832 \)
$37$
\( T^{2} + 48T - 15396 \)
$41$
\( T^{2} + 494T + 60976 \)
$43$
\( T^{2} + 66T - 59928 \)
$47$
\( T^{2} + 64T - 17984 \)
$53$
\( T^{2} + 84T - 133404 \)
$59$
\( (T - 196)^{2} \)
$61$
\( T^{2} + 1104 T + 282396 \)
$67$
\( T^{2} - 928T + 24688 \)
$71$
\( T^{2} - 456T - 227328 \)
$73$
\( T^{2} + 592T - 436292 \)
$79$
\( T^{2} + 230T - 31952 \)
$83$
\( T^{2} - 348T - 835776 \)
$89$
\( T^{2} - 972T + 235668 \)
$97$
\( T^{2} + 1184 T - 1104836 \)
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