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: | 6.6.4222000.1 |
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| Defining polynomial: |
\( x^{6} - 2x^{5} - 6x^{4} + 10x^{3} + 8x^{2} - 8x + 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 759) |
| 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
\( x^{6} - 2x^{5} - 6x^{4} + 10x^{3} + 8x^{2} - 8x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 5\nu^{2} + 11\nu - 4 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 6\nu^{3} + 9\nu^{2} + 9\nu - 4 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + \beta_{3} + 5\beta_{2} + 7\beta _1 + 8 \)
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| \(\nu^{5}\) | \(=\) |
\( -\beta_{5} + 2\beta_{4} + 8\beta_{3} + 7\beta_{2} + 26\beta _1 + 2 \)
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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 |
|
−2.44912 | 0 | 3.99818 | −0.147021 | 0 | 0.804741 | −4.89379 | 0 | 0.360072 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.06427 | 0 | 2.26120 | −4.04066 | 0 | 4.81590 | −0.539176 | 0 | 8.34099 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.537960 | 0 | −1.71060 | −3.67524 | 0 | −2.55698 | 1.99615 | 0 | 1.97713 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.152120 | 0 | −1.97686 | 2.81666 | 0 | −3.18555 | 0.604961 | 0 | −0.428471 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.21639 | 0 | −0.520404 | 0.460065 | 0 | 2.84179 | −3.06579 | 0 | 0.559617 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.98708 | 0 | 1.94848 | −1.41380 | 0 | −2.71990 | −0.102367 | 0 | −2.80934 | |||||||||||||||||||||||||||||||||||||
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.n | 6 | |
| 3.b | odd | 2 | 1 | 759.2.a.h | ✓ | 6 | |
| 33.d | even | 2 | 1 | 8349.2.a.n | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 759.2.a.h | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 2277.2.a.n | 6 | 1.a | even | 1 | 1 | trivial | |
| 8349.2.a.n | 6 | 33.d | even | 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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\( T_{2}^{6} + 2T_{2}^{5} - 6T_{2}^{4} - 10T_{2}^{3} + 8T_{2}^{2} + 8T_{2} + 1 \)
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\( T_{5}^{6} + 6T_{5}^{5} - 2T_{5}^{4} - 52T_{5}^{3} - 43T_{5}^{2} + 22T_{5} + 4 \)
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\( T_{17}^{6} + 10T_{17}^{5} + 4T_{17}^{4} - 208T_{17}^{3} - 576T_{17}^{2} - 256T_{17} + 256 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 2 T^{5} + \cdots + 1 \)
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| $3$ |
\( T^{6} \)
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| $5$ |
\( T^{6} + 6 T^{5} + \cdots + 4 \)
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| $7$ |
\( T^{6} - 28 T^{4} + \cdots - 244 \)
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| $11$ |
\( (T - 1)^{6} \)
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| $13$ |
\( (T^{2} - 20)^{3} \)
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| $17$ |
\( T^{6} + 10 T^{5} + \cdots + 256 \)
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| $19$ |
\( T^{6} + 2 T^{5} + \cdots - 64 \)
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| $23$ |
\( (T + 1)^{6} \)
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| $29$ |
\( T^{6} + 16 T^{5} + \cdots + 2304 \)
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| $31$ |
\( T^{6} + 12 T^{5} + \cdots - 3824 \)
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| $37$ |
\( T^{6} - 12 T^{5} + \cdots - 21824 \)
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| $41$ |
\( T^{6} + 22 T^{5} + \cdots - 36884 \)
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| $43$ |
\( T^{6} - 132 T^{4} + \cdots + 1844 \)
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| $47$ |
\( T^{6} + 8 T^{5} + \cdots + 38656 \)
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| $53$ |
\( T^{6} + 10 T^{5} + \cdots + 250924 \)
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| $59$ |
\( T^{6} + 12 T^{5} + \cdots - 1024 \)
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| $61$ |
\( T^{6} - 24 T^{5} + \cdots + 84736 \)
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| $67$ |
\( T^{6} + 18 T^{5} + \cdots - 22336 \)
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| $71$ |
\( T^{6} + 24 T^{5} + \cdots + 277504 \)
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| $73$ |
\( T^{6} + 4 T^{5} + \cdots + 704 \)
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| $79$ |
\( T^{6} - 16 T^{5} + \cdots + 10924 \)
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| $83$ |
\( T^{6} + 16 T^{5} + \cdots + 446464 \)
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| $89$ |
\( T^{6} + 2 T^{5} + \cdots + 31844 \)
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| $97$ |
\( T^{6} + 12 T^{5} + \cdots + 282944 \)
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