Newspace parameters
Level: | \( N \) | \(=\) | \( 33 = 3 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 33.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.899184872389\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-11}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{2} - x + 3 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-11}\). 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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 |
|
− | 3.31662i | 3.00000 | −7.00000 | 6.63325i | − | 9.94987i | −8.00000 | 9.94987i | 9.00000 | 22.0000 | ||||||||||||||||||||||
23.2 | 3.31662i | 3.00000 | −7.00000 | − | 6.63325i | 9.94987i | −8.00000 | − | 9.94987i | 9.00000 | 22.0000 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 33.3.b.a | ✓ | 2 |
3.b | odd | 2 | 1 | inner | 33.3.b.a | ✓ | 2 |
4.b | odd | 2 | 1 | 528.3.i.a | 2 | ||
11.b | odd | 2 | 1 | 363.3.b.d | 2 | ||
11.c | even | 5 | 4 | 363.3.h.e | 8 | ||
11.d | odd | 10 | 4 | 363.3.h.d | 8 | ||
12.b | even | 2 | 1 | 528.3.i.a | 2 | ||
33.d | even | 2 | 1 | 363.3.b.d | 2 | ||
33.f | even | 10 | 4 | 363.3.h.d | 8 | ||
33.h | odd | 10 | 4 | 363.3.h.e | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
33.3.b.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
33.3.b.a | ✓ | 2 | 3.b | odd | 2 | 1 | inner |
363.3.b.d | 2 | 11.b | odd | 2 | 1 | ||
363.3.b.d | 2 | 33.d | even | 2 | 1 | ||
363.3.h.d | 8 | 11.d | odd | 10 | 4 | ||
363.3.h.d | 8 | 33.f | even | 10 | 4 | ||
363.3.h.e | 8 | 11.c | even | 5 | 4 | ||
363.3.h.e | 8 | 33.h | odd | 10 | 4 | ||
528.3.i.a | 2 | 4.b | odd | 2 | 1 | ||
528.3.i.a | 2 | 12.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 11 \)
acting on \(S_{3}^{\mathrm{new}}(33, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 11 \)
$3$
\( (T - 3)^{2} \)
$5$
\( T^{2} + 44 \)
$7$
\( (T + 8)^{2} \)
$11$
\( T^{2} + 11 \)
$13$
\( (T - 4)^{2} \)
$17$
\( T^{2} + 176 \)
$19$
\( (T + 6)^{2} \)
$23$
\( T^{2} + 44 \)
$29$
\( T^{2} + 1584 \)
$31$
\( (T + 26)^{2} \)
$37$
\( (T - 30)^{2} \)
$41$
\( T^{2} + 176 \)
$43$
\( (T - 42)^{2} \)
$47$
\( T^{2} + 7436 \)
$53$
\( T^{2} + 3564 \)
$59$
\( T^{2} + 4400 \)
$61$
\( (T - 12)^{2} \)
$67$
\( (T - 2)^{2} \)
$71$
\( T^{2} + 3564 \)
$73$
\( (T + 74)^{2} \)
$79$
\( (T + 40)^{2} \)
$83$
\( T^{2} + 1584 \)
$89$
\( T^{2} + 14256 \)
$97$
\( (T - 62)^{2} \)
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