Newspace parameters
| Level: | \( N \) | \(=\) | \( 3366 = 2 \cdot 3^{2} \cdot 11 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3366.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(26.8776453204\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.257.1 |
|
|
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| Defining polynomial: |
\( x^{3} - x^{2} - 4x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 374) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 4x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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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 |
|
−1.00000 | 0 | 1.00000 | −1.77733 | 0 | 2.71354 | −1.00000 | 0 | 1.77733 | |||||||||||||||||||||||||||
| 1.2 | −1.00000 | 0 | 1.00000 | −1.25561 | 0 | 0.0877708 | −1.00000 | 0 | 1.25561 | ||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0 | 1.00000 | 4.03293 | 0 | 4.19869 | −1.00000 | 0 | −4.03293 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(11\) | \( -1 \) |
| \(17\) | \( -1 \) |
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 | 3366.2.a.ba | 3 | |
| 3.b | odd | 2 | 1 | 374.2.a.c | ✓ | 3 | |
| 12.b | even | 2 | 1 | 2992.2.a.o | 3 | ||
| 15.d | odd | 2 | 1 | 9350.2.a.bw | 3 | ||
| 33.d | even | 2 | 1 | 4114.2.a.p | 3 | ||
| 51.c | odd | 2 | 1 | 6358.2.a.r | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 374.2.a.c | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 2992.2.a.o | 3 | 12.b | even | 2 | 1 | ||
| 3366.2.a.ba | 3 | 1.a | even | 1 | 1 | trivial | |
| 4114.2.a.p | 3 | 33.d | even | 2 | 1 | ||
| 6358.2.a.r | 3 | 51.c | odd | 2 | 1 | ||
| 9350.2.a.bw | 3 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3366))\):
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\( T_{5}^{3} - T_{5}^{2} - 10T_{5} - 9 \)
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\( T_{7}^{3} - 7T_{7}^{2} + 12T_{7} - 1 \)
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\( T_{13}^{3} + 3T_{13}^{2} - 2T_{13} - 7 \)
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\( T_{19}^{3} - 5T_{19}^{2} - 18T_{19} + 63 \)
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\( T_{23}^{3} + 4T_{23}^{2} - 76T_{23} - 360 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{3} \)
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| $3$ |
\( T^{3} \)
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| $5$ |
\( T^{3} - T^{2} - 10T - 9 \)
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| $7$ |
\( T^{3} - 7 T^{2} + \cdots - 1 \)
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| $11$ |
\( (T - 1)^{3} \)
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| $13$ |
\( T^{3} + 3 T^{2} + \cdots - 7 \)
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| $17$ |
\( (T - 1)^{3} \)
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| $19$ |
\( T^{3} - 5 T^{2} + \cdots + 63 \)
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| $23$ |
\( T^{3} + 4 T^{2} + \cdots - 360 \)
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| $29$ |
\( T^{3} - 4 T^{2} + \cdots + 360 \)
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| $31$ |
\( T^{3} - 2 T^{2} + \cdots + 360 \)
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| $37$ |
\( T^{3} + 9 T^{2} + \cdots - 47 \)
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| $41$ |
\( T^{3} - T^{2} + \cdots - 105 \)
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| $43$ |
\( T^{3} - 5 T^{2} + \cdots + 193 \)
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| $47$ |
\( T^{3} - 21 T^{2} + \cdots + 45 \)
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| $53$ |
\( T^{3} - 80T + 192 \)
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| $59$ |
\( T^{3} - 20 T^{2} + \cdots - 72 \)
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| $61$ |
\( T^{3} - 6 T^{2} + \cdots + 56 \)
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| $67$ |
\( T^{3} - 2 T^{2} + \cdots + 904 \)
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| $71$ |
\( T^{3} + 2 T^{2} + \cdots + 216 \)
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| $73$ |
\( T^{3} + T^{2} + \cdots - 45 \)
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| $79$ |
\( T^{3} - 19 T^{2} + \cdots + 587 \)
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| $83$ |
\( T^{3} - 17 T^{2} + \cdots - 9 \)
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| $89$ |
\( T^{3} - 3 T^{2} + \cdots + 147 \)
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| $97$ |
\( T^{3} - 24 T^{2} + \cdots + 320 \)
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