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: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
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| Defining polynomial: |
\( x^{8} - 13x^{6} - 2x^{5} + 48x^{4} + 14x^{3} - 41x^{2} - 6x + 9 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 13x^{6} - 2x^{5} + 48x^{4} + 14x^{3} - 41x^{2} - 6x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{7} + 3\nu^{6} + 26\nu^{5} - 26\nu^{4} - 93\nu^{3} + 44\nu^{2} + 61\nu - 21 ) / 9 \)
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| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} + 3\nu^{6} - 13\nu^{5} - 41\nu^{4} + 42\nu^{3} + 149\nu^{2} + \nu - 66 ) / 9 \)
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| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + 13\nu^{5} + 2\nu^{4} - 45\nu^{3} - 14\nu^{2} + 20\nu + 3 ) / 3 \)
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| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - 13\nu^{5} - 2\nu^{4} + 48\nu^{3} + 14\nu^{2} - 38\nu - 6 ) / 3 \)
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| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - 12\nu^{5} - 2\nu^{4} + 38\nu^{3} + 14\nu^{2} - 17\nu - 5 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 6\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( -\beta_{5} - \beta_{4} + \beta_{3} + 7\beta_{2} + 17 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{7} + 7\beta_{6} + 10\beta_{5} + 39\beta _1 + 9 \)
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| \(\nu^{6}\) | \(=\) |
\( \beta_{6} - 11\beta_{5} - 10\beta_{4} + 13\beta_{3} + 46\beta_{2} - \beta _1 + 108 \)
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| \(\nu^{7}\) | \(=\) |
\( 13\beta_{7} + 46\beta_{6} + 80\beta_{5} - 2\beta_{4} + 2\beta_{3} + 257\beta _1 + 67 \)
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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 |
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−2.58988 | 0 | 4.70748 | 1.40075 | 0 | 4.44977 | −7.01203 | 0 | −3.62778 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00968 | 0 | 2.03880 | −3.45828 | 0 | 1.27221 | −0.0779704 | 0 | 6.95002 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.10130 | 0 | −0.787129 | −2.64935 | 0 | 1.85760 | 3.06948 | 0 | 2.91774 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.628929 | 0 | −1.60445 | 1.61629 | 0 | 0.675307 | 2.26694 | 0 | −1.01653 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.516018 | 0 | −1.73373 | −1.33904 | 0 | −3.45000 | −1.92667 | 0 | −0.690966 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.759065 | 0 | −1.42382 | 1.92387 | 0 | 4.69933 | −2.59890 | 0 | 1.46034 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.40940 | 0 | 3.80519 | 2.63365 | 0 | 2.31733 | 4.34941 | 0 | 6.34550 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.64531 | 0 | 4.99766 | −0.127899 | 0 | −1.82156 | 7.92974 | 0 | −0.338332 | |||||||||||||||||||||||||||||||||||||||||||
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.t | yes | 8 |
| 3.b | odd | 2 | 1 | 2277.2.a.s | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2277.2.a.s | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 2277.2.a.t | yes | 8 | 1.a | even | 1 | 1 | trivial |
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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\( T_{2}^{8} - 13T_{2}^{6} - 2T_{2}^{5} + 48T_{2}^{4} + 14T_{2}^{3} - 41T_{2}^{2} - 6T_{2} + 9 \)
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\( T_{5}^{8} - 18T_{5}^{6} + 10T_{5}^{5} + 95T_{5}^{4} - 88T_{5}^{3} - 128T_{5}^{2} + 126T_{5} + 18 \)
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\( T_{17}^{8} - 46T_{17}^{6} - 2T_{17}^{5} + 618T_{17}^{4} - 240T_{17}^{3} - 2480T_{17}^{2} + 3040T_{17} - 864 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 13 T^{6} + \cdots + 9 \)
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| $3$ |
\( T^{8} \)
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| $5$ |
\( T^{8} - 18 T^{6} + \cdots + 18 \)
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| $7$ |
\( T^{8} - 10 T^{7} + \cdots + 486 \)
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| $11$ |
\( (T - 1)^{8} \)
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| $13$ |
\( T^{8} - 10 T^{7} + \cdots - 17408 \)
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| $17$ |
\( T^{8} - 46 T^{6} + \cdots - 864 \)
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| $19$ |
\( T^{8} - 16 T^{7} + \cdots + 512 \)
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| $23$ |
\( (T + 1)^{8} \)
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| $29$ |
\( T^{8} + 6 T^{7} + \cdots + 21312 \)
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| $31$ |
\( T^{8} - 4 T^{7} + \cdots + 32 \)
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| $37$ |
\( T^{8} - 10 T^{7} + \cdots - 6144 \)
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| $41$ |
\( T^{8} - 6 T^{7} + \cdots + 60 \)
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| $43$ |
\( T^{8} - 22 T^{7} + \cdots + 446 \)
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| $47$ |
\( T^{8} + 12 T^{7} + \cdots + 3072 \)
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| $53$ |
\( T^{8} + 2 T^{7} + \cdots - 318390 \)
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| $59$ |
\( T^{8} + 24 T^{7} + \cdots - 248832 \)
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| $61$ |
\( T^{8} - 28 T^{7} + \cdots - 51456 \)
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| $67$ |
\( T^{8} + 2 T^{7} + \cdots + 61536 \)
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| $71$ |
\( T^{8} - 16 T^{7} + \cdots + 61440 \)
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| $73$ |
\( T^{8} - 24 T^{7} + \cdots - 3156992 \)
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| $79$ |
\( T^{8} - 12 T^{7} + \cdots - 19169690 \)
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| $83$ |
\( T^{8} + 10 T^{7} + \cdots + 236928 \)
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| $89$ |
\( T^{8} - 24 T^{7} + \cdots - 143670 \)
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| $97$ |
\( T^{8} - 34 T^{7} + \cdots - 4103296 \)
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