Newspace parameters
| Level: | \( N \) | \(=\) | \( 2299 = 11^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2299.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(18.3576074247\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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| Defining polynomial: |
\( x^{7} - 2x^{6} - 8x^{5} + 14x^{4} + 17x^{3} - 18x^{2} - 16x - 1 \)
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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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 8x^{5} + 14x^{4} + 17x^{3} - 18x^{2} - 16x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 5\nu^{2} + 8\nu \)
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| \(\beta_{3}\) | \(=\) |
\( -\nu^{5} + \nu^{4} + 8\nu^{3} - 5\nu^{2} - 13\nu - 1 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 8\nu^{4} - 3\nu^{3} + 13\nu^{2} + 13\nu + 2 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 9\nu^{4} - 2\nu^{3} + 19\nu^{2} + 8\nu - 2 \)
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| \(\beta_{6}\) | \(=\) |
\( -\nu^{6} + 8\nu^{4} + 3\nu^{3} - 12\nu^{2} - 14\nu - 5 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{4} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( 6\beta_{6} - \beta_{5} + 7\beta_{4} + \beta_{3} + \beta_{2} + 6\beta _1 + 15 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{6} - \beta_{5} + 2\beta_{4} + 8\beta_{3} + 9\beta_{2} + 28\beta _1 + 7 \)
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| \(\nu^{6}\) | \(=\) |
\( 35\beta_{6} - 8\beta_{5} + 44\beta_{4} + 11\beta_{3} + 11\beta_{2} + 37\beta _1 + 82 \)
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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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−2.31092 | −1.96348 | 3.34035 | −1.91976 | 4.53745 | −0.391807 | −3.09745 | 0.855254 | 4.43642 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.09476 | −0.248246 | −0.801502 | 1.63186 | 0.271770 | −4.48973 | 3.06697 | −2.93837 | −1.78650 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.692399 | 3.33751 | −1.52058 | −0.880738 | −2.31089 | 3.64547 | 2.43765 | 8.13895 | 0.609822 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.0680200 | −0.141369 | −1.99537 | 3.58517 | 0.00961589 | 3.20234 | 0.271765 | −2.98001 | −0.243863 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.79190 | −2.48473 | 1.21090 | 0.260416 | −4.45238 | 3.76880 | −1.41399 | 3.17387 | 0.466638 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.87135 | −1.09616 | 1.50197 | −4.30775 | −2.05131 | −3.98475 | −0.931995 | −1.79843 | −8.06132 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.50285 | 1.59648 | 4.26424 | 0.630803 | 3.99574 | 0.249675 | 5.66706 | −0.451261 | 1.57880 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \( -1 \) |
| \(19\) | \( +1 \) |
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 | 2299.2.a.r | yes | 7 |
| 11.b | odd | 2 | 1 | 2299.2.a.p | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2299.2.a.p | ✓ | 7 | 11.b | odd | 2 | 1 | |
| 2299.2.a.r | yes | 7 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2299))\):
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\( T_{2}^{7} - 2T_{2}^{6} - 8T_{2}^{5} + 14T_{2}^{4} + 17T_{2}^{3} - 18T_{2}^{2} - 16T_{2} - 1 \)
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\( T_{3}^{7} + T_{3}^{6} - 12T_{3}^{5} - 18T_{3}^{4} + 20T_{3}^{3} + 38T_{3}^{2} + 12T_{3} + 1 \)
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\( T_{7}^{7} - 2T_{7}^{6} - 35T_{7}^{5} + 79T_{7}^{4} + 313T_{7}^{3} - 753T_{7}^{2} - 141T_{7} + 77 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 2 T^{6} + \cdots - 1 \)
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| $3$ |
\( T^{7} + T^{6} - 12 T^{5} + \cdots + 1 \)
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| $5$ |
\( T^{7} + T^{6} - 19 T^{5} + \cdots + 7 \)
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| $7$ |
\( T^{7} - 2 T^{6} + \cdots + 77 \)
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| $11$ |
\( T^{7} \)
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| $13$ |
\( T^{7} - 12 T^{6} + \cdots - 7777 \)
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| $17$ |
\( T^{7} - 12 T^{6} + \cdots + 7 \)
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| $19$ |
\( (T + 1)^{7} \)
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| $23$ |
\( T^{7} + 9 T^{6} + \cdots - 3789 \)
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| $29$ |
\( T^{7} - 18 T^{6} + \cdots + 1363 \)
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| $31$ |
\( T^{7} + 5 T^{6} + \cdots - 368903 \)
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| $37$ |
\( T^{7} - 4 T^{6} + \cdots + 10109 \)
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| $41$ |
\( T^{7} - 25 T^{6} + \cdots + 61253 \)
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| $43$ |
\( T^{7} + 10 T^{6} + \cdots + 1963 \)
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| $47$ |
\( T^{7} + 14 T^{6} + \cdots + 98963 \)
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| $53$ |
\( T^{7} + 15 T^{6} + \cdots + 16651 \)
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| $59$ |
\( T^{7} + 9 T^{6} + \cdots + 50353 \)
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| $61$ |
\( T^{7} - 3 T^{6} + \cdots - 1310837 \)
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| $67$ |
\( T^{7} + 15 T^{6} + \cdots + 484987 \)
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| $71$ |
\( T^{7} - 13 T^{6} + \cdots - 3979 \)
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| $73$ |
\( T^{7} + 28 T^{6} + \cdots - 304013 \)
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| $79$ |
\( T^{7} - 18 T^{6} + \cdots + 108143 \)
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| $83$ |
\( T^{7} + 10 T^{6} + \cdots - 158647 \)
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| $89$ |
\( T^{7} - 2 T^{6} + \cdots + 2211325 \)
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| $97$ |
\( T^{7} - 29 T^{6} + \cdots + 2132767 \)
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