Newspace parameters
| Level: | \( N \) | \(=\) | \( 2277 = 3^{2} \cdot 11 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2277.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(18.1819365402\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\Q(\zeta_{18})^+\) |
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|
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| Defining polynomial: |
\( x^{3} - 3x - 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 253) |
| 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 \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
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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 |
|
−2.53209 | 0 | 4.41147 | −1.34730 | 0 | −0.879385 | −6.10607 | 0 | 3.41147 | |||||||||||||||||||||||||||
| 1.2 | −1.34730 | 0 | −0.184793 | 0.879385 | 0 | 2.53209 | 2.94356 | 0 | −1.18479 | ||||||||||||||||||||||||||||
| 1.3 | 0.879385 | 0 | −1.22668 | −2.53209 | 0 | 1.34730 | −2.83750 | 0 | −2.22668 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(11\) | \( -1 \) |
| \(23\) | \( +1 \) |
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 | 2277.2.a.g | 3 | |
| 3.b | odd | 2 | 1 | 253.2.a.b | ✓ | 3 | |
| 12.b | even | 2 | 1 | 4048.2.a.o | 3 | ||
| 15.d | odd | 2 | 1 | 6325.2.a.i | 3 | ||
| 33.d | even | 2 | 1 | 2783.2.a.e | 3 | ||
| 69.c | even | 2 | 1 | 5819.2.a.c | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 253.2.a.b | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 2277.2.a.g | 3 | 1.a | even | 1 | 1 | trivial | |
| 2783.2.a.e | 3 | 33.d | even | 2 | 1 | ||
| 4048.2.a.o | 3 | 12.b | even | 2 | 1 | ||
| 5819.2.a.c | 3 | 69.c | even | 2 | 1 | ||
| 6325.2.a.i | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2277))\):
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\( T_{2}^{3} + 3T_{2}^{2} - 3 \)
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\( T_{5}^{3} + 3T_{5}^{2} - 3 \)
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\( T_{17}^{3} + 3T_{17}^{2} - 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 3T^{2} - 3 \)
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| $3$ |
\( T^{3} \)
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| $5$ |
\( T^{3} + 3T^{2} - 3 \)
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| $7$ |
\( T^{3} - 3T^{2} + 3 \)
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| $11$ |
\( (T - 1)^{3} \)
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| $13$ |
\( T^{3} + 3 T^{2} + \cdots - 17 \)
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| $17$ |
\( T^{3} + 3T^{2} - 1 \)
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| $19$ |
\( T^{3} - 9 T^{2} + \cdots + 17 \)
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| $23$ |
\( (T + 1)^{3} \)
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| $29$ |
\( T^{3} - 63T - 9 \)
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| $31$ |
\( T^{3} - 21T + 17 \)
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| $37$ |
\( T^{3} + 18 T^{2} + \cdots + 27 \)
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| $41$ |
\( T^{3} + 12 T^{2} + \cdots - 17 \)
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| $43$ |
\( T^{3} - 39T - 19 \)
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| $47$ |
\( T^{3} + 6 T^{2} + \cdots + 8 \)
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| $53$ |
\( T^{3} + 18 T^{2} + \cdots + 73 \)
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| $59$ |
\( T^{3} - 9 T^{2} + \cdots + 153 \)
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| $61$ |
\( T^{3} + 6 T^{2} + \cdots - 159 \)
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| $67$ |
\( T^{3} - 6 T^{2} + \cdots + 456 \)
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| $71$ |
\( T^{3} - 18 T^{2} + \cdots + 1007 \)
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| $73$ |
\( T^{3} + 9 T^{2} + \cdots - 73 \)
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| $79$ |
\( T^{3} - 9 T^{2} + \cdots - 19 \)
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| $83$ |
\( T^{3} + 21 T^{2} + \cdots - 541 \)
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| $89$ |
\( T^{3} + 21 T^{2} + \cdots - 2071 \)
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| $97$ |
\( T^{3} + 9 T^{2} + \cdots - 19 \)
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