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: | \(6\) |
| Coefficient field: | 6.6.8639957.1 |
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| Defining polynomial: |
\( x^{6} - 3x^{5} - 4x^{4} + 10x^{3} + 6x^{2} - 7x - 2 \)
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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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 4x^{4} + 10x^{3} + 6x^{2} - 7x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 2\nu^{2} + 5\nu + 1 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 4\nu^{4} + 9\nu^{2} - \nu - 3 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 4\nu^{4} + \nu^{3} + 7\nu^{2} - 5\nu - 1 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 4\nu^{4} + 10\nu^{2} - 3\nu - 5 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{3} + 2\beta _1 + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{5} + \beta_{4} - 3\beta_{3} + 8\beta _1 + 2 \)
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| \(\nu^{4}\) | \(=\) |
\( 8\beta_{5} + 3\beta_{4} - 11\beta_{3} + \beta_{2} + 23\beta _1 + 9 \)
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| \(\nu^{5}\) | \(=\) |
\( 23\beta_{5} + 12\beta_{4} - 34\beta_{3} + 4\beta_{2} + 75\beta _1 + 21 \)
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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 |
|
−2.34987 | 0 | 3.52188 | 2.57623 | 0 | −4.37344 | −3.57623 | 0 | −6.05380 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.17801 | 0 | 2.74373 | 0.619849 | 0 | 2.65532 | −1.61985 | 0 | −1.35004 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.25551 | 0 | −0.423685 | −4.04297 | 0 | 0.798084 | 3.04297 | 0 | 5.07601 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.0985770 | 0 | −1.99028 | −1.39335 | 0 | −2.57601 | 0.393350 | 0 | 0.137352 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0.735479 | 0 | −1.45907 | 1.54407 | 0 | 0.781955 | −2.54407 | 0 | 1.13563 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.14649 | 0 | 2.60742 | −2.30383 | 0 | 1.71408 | 1.30383 | 0 | −4.94515 | |||||||||||||||||||||||||||||||||||||
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.m | 6 | |
| 3.b | odd | 2 | 1 | 253.2.a.d | ✓ | 6 | |
| 12.b | even | 2 | 1 | 4048.2.a.bc | 6 | ||
| 15.d | odd | 2 | 1 | 6325.2.a.m | 6 | ||
| 33.d | even | 2 | 1 | 2783.2.a.h | 6 | ||
| 69.c | even | 2 | 1 | 5819.2.a.e | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 253.2.a.d | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 2277.2.a.m | 6 | 1.a | even | 1 | 1 | trivial | |
| 2783.2.a.h | 6 | 33.d | even | 2 | 1 | ||
| 4048.2.a.bc | 6 | 12.b | even | 2 | 1 | ||
| 5819.2.a.e | 6 | 69.c | even | 2 | 1 | ||
| 6325.2.a.m | 6 | 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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\( T_{2}^{6} + 3T_{2}^{5} - 4T_{2}^{4} - 16T_{2}^{3} - 3T_{2}^{2} + 10T_{2} + 1 \)
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\( T_{5}^{6} + 3T_{5}^{5} - 12T_{5}^{4} - 25T_{5}^{3} + 38T_{5}^{2} + 40T_{5} - 32 \)
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\( T_{17}^{6} + 5T_{17}^{5} - 48T_{17}^{4} - 253T_{17}^{3} - 98T_{17}^{2} + 596T_{17} + 296 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 3 T^{5} + \cdots + 1 \)
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| $3$ |
\( T^{6} \)
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| $5$ |
\( T^{6} + 3 T^{5} + \cdots - 32 \)
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| $7$ |
\( T^{6} + T^{5} + \cdots + 32 \)
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| $11$ |
\( (T + 1)^{6} \)
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| $13$ |
\( T^{6} + 3 T^{5} + \cdots + 502 \)
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| $17$ |
\( T^{6} + 5 T^{5} + \cdots + 296 \)
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| $19$ |
\( T^{6} - T^{5} + \cdots - 13616 \)
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| $23$ |
\( (T - 1)^{6} \)
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| $29$ |
\( T^{6} + 6 T^{5} + \cdots - 6302 \)
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| $31$ |
\( T^{6} + 8 T^{5} + \cdots + 9616 \)
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| $37$ |
\( T^{6} - 2 T^{5} + \cdots - 248 \)
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| $41$ |
\( T^{6} - 2 T^{5} + \cdots + 206 \)
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| $43$ |
\( T^{6} - 10 T^{5} + \cdots - 127184 \)
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| $47$ |
\( T^{6} + 14 T^{5} + \cdots - 17536 \)
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| $53$ |
\( T^{6} - 4 T^{5} + \cdots + 808 \)
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| $59$ |
\( T^{6} + 39 T^{5} + \cdots + 16064 \)
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| $61$ |
\( T^{6} + 22 T^{5} + \cdots + 27656 \)
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| $67$ |
\( T^{6} - 8 T^{5} + \cdots - 8576 \)
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| $71$ |
\( T^{6} + 18 T^{5} + \cdots + 192376 \)
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| $73$ |
\( T^{6} + 9 T^{5} + \cdots - 2878 \)
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| $79$ |
\( T^{6} + 15 T^{5} + \cdots + 321344 \)
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| $83$ |
\( T^{6} + 43 T^{5} + \cdots - 59216 \)
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| $89$ |
\( T^{6} + 17 T^{5} + \cdots - 115096 \)
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| $97$ |
\( T^{6} - 19 T^{5} + \cdots - 33464 \)
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