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: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
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| Defining polynomial: |
\( x^{8} - 11x^{6} - 2x^{5} + 36x^{4} + 10x^{3} - 35x^{2} - 14x + 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{8} - 11x^{6} - 2x^{5} + 36x^{4} + 10x^{3} - 35x^{2} - 14x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 3\nu^{2} + 10\nu + 2 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 7\nu^{4} + 4\nu^{3} + 10\nu^{2} - 2\nu - 1 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 6\nu^{4} + 10\nu^{3} + 7\nu^{2} - 7\nu - 2 \)
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| \(\beta_{6}\) | \(=\) |
\( -\nu^{7} + 2\nu^{6} + 7\nu^{5} - 11\nu^{4} - 14\nu^{3} + 10\nu^{2} + 12\nu + 3 \)
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| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - 2\nu^{6} - 7\nu^{5} + 12\nu^{4} + 13\nu^{3} - 16\nu^{2} - 9\nu + 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} - \beta_{3} + 5\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 6\beta_{2} + 2\beta _1 + 14 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{7} + \beta_{6} - 8\beta_{5} + 8\beta_{4} - 7\beta_{3} + 3\beta_{2} + 27\beta _1 + 10 \)
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| \(\nu^{6}\) | \(=\) |
\( 8\beta_{7} + 8\beta_{6} - 11\beta_{5} + 12\beta_{4} - 10\beta_{3} + 35\beta_{2} + 23\beta _1 + 75 \)
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| \(\nu^{7}\) | \(=\) |
\( 12\beta_{7} + 11\beta_{6} - 53\beta_{5} + 55\beta_{4} - 44\beta_{3} + 35\beta_{2} + 155\beta _1 + 85 \)
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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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−2.59781 | 0 | 4.74863 | 4.21910 | 0 | −1.76707 | −7.14042 | 0 | −10.9604 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.74429 | 0 | 1.04255 | 0.210629 | 0 | −0.492181 | 1.67007 | 0 | −0.367398 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.40881 | 0 | −0.0152604 | 0.124472 | 0 | −1.04695 | 2.83911 | 0 | −0.175357 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.0620205 | 0 | −1.99615 | 2.44556 | 0 | 3.80059 | 0.247843 | 0 | −0.151675 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.512135 | 0 | −1.73772 | 2.18114 | 0 | −4.04698 | −1.91422 | 0 | 1.11704 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.13892 | 0 | −0.702863 | −1.09402 | 0 | 1.67173 | −3.07834 | 0 | −1.24600 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.07090 | 0 | 2.28862 | 1.71731 | 0 | −4.65170 | 0.597693 | 0 | 3.55637 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.09098 | 0 | 2.37220 | −1.80420 | 0 | 0.532553 | 0.778259 | 0 | −3.77254 | |||||||||||||||||||||||||||||||||||||||||||
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.r | yes | 8 |
| 3.b | odd | 2 | 1 | 2277.2.a.q | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2277.2.a.q | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 2277.2.a.r | 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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\( T_{2}^{8} - 11T_{2}^{6} + 2T_{2}^{5} + 36T_{2}^{4} - 10T_{2}^{3} - 35T_{2}^{2} + 14T_{2} + 1 \)
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\( T_{5}^{8} - 8T_{5}^{7} + 14T_{5}^{6} + 26T_{5}^{5} - 81T_{5}^{4} + 8T_{5}^{3} + 80T_{5}^{2} - 26T_{5} + 2 \)
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\( T_{17}^{8} - 74T_{17}^{6} - 58T_{17}^{5} + 1282T_{17}^{4} + 560T_{17}^{3} - 5552T_{17}^{2} + 3296T_{17} + 544 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 11 T^{6} + \cdots + 1 \)
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| $3$ |
\( T^{8} \)
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| $5$ |
\( T^{8} - 8 T^{7} + \cdots + 2 \)
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| $7$ |
\( T^{8} + 6 T^{7} + \cdots - 58 \)
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| $11$ |
\( (T + 1)^{8} \)
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| $13$ |
\( T^{8} + 10 T^{7} + \cdots + 4352 \)
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| $17$ |
\( T^{8} - 74 T^{6} + \cdots + 544 \)
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| $19$ |
\( T^{8} + 16 T^{7} + \cdots - 17312 \)
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| $23$ |
\( (T + 1)^{8} \)
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| $29$ |
\( T^{8} + 2 T^{7} + \cdots - 339968 \)
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| $31$ |
\( T^{8} + 20 T^{7} + \cdots + 10784 \)
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| $37$ |
\( T^{8} + 6 T^{7} + \cdots + 1306112 \)
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| $41$ |
\( T^{8} + 6 T^{7} + \cdots - 326644 \)
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| $43$ |
\( T^{8} + 26 T^{7} + \cdots - 441234 \)
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| $47$ |
\( T^{8} - 12 T^{7} + \cdots - 947456 \)
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| $53$ |
\( T^{8} - 14 T^{7} + \cdots - 3923542 \)
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| $59$ |
\( T^{8} - 280 T^{6} + \cdots - 6036736 \)
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| $61$ |
\( T^{8} + 12 T^{7} + \cdots - 27594752 \)
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| $67$ |
\( T^{8} + 50 T^{7} + \cdots - 3165728 \)
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| $71$ |
\( T^{8} + 32 T^{7} + \cdots + 7168 \)
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| $73$ |
\( T^{8} + 24 T^{7} + \cdots + 1024 \)
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| $79$ |
\( T^{8} + 4 T^{7} + \cdots - 47741146 \)
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| $83$ |
\( T^{8} + 10 T^{7} + \cdots + 326528 \)
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| $89$ |
\( T^{8} - 16 T^{7} + \cdots - 4465622 \)
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| $97$ |
\( T^{8} + 22 T^{7} + \cdots + 335488 \)
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