Newspace parameters
| Level: | \( N \) | \(=\) | \( 5194 = 2 \cdot 7^{2} \cdot 53 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5194.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(41.4742988099\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{7} - x^{6} - 14x^{5} + x^{4} + 57x^{3} + 34x^{2} - 36x - 21 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 742) |
| 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
\( x^{7} - x^{6} - 14x^{5} + x^{4} + 57x^{3} + 34x^{2} - 36x - 21 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{6} - 8\nu^{5} - 31\nu^{4} + 57\nu^{3} + 97\nu^{2} - 62\nu - 42 ) / 7 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{6} - 8\nu^{5} - 31\nu^{4} + 57\nu^{3} + 104\nu^{2} - 69\nu - 70 ) / 7 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 8\nu^{6} - 19\nu^{5} - 85\nu^{4} + 124\nu^{3} + 275\nu^{2} - 100\nu - 126 ) / 7 \)
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| \(\beta_{5}\) | \(=\) |
\( ( -9\nu^{6} + 24\nu^{5} + 86\nu^{4} - 150\nu^{3} - 263\nu^{2} + 116\nu + 112 ) / 7 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 9\nu^{6} - 24\nu^{5} - 86\nu^{4} + 157\nu^{3} + 249\nu^{2} - 151\nu - 91 ) / 7 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} + \beta _1 + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 2\beta_{3} - 2\beta_{2} + 7\beta _1 + 5 \)
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| \(\nu^{4}\) | \(=\) |
\( 3\beta_{6} + 2\beta_{5} + 10\beta_{3} - 13\beta_{2} + 15\beta _1 + 29 \)
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| \(\nu^{5}\) | \(=\) |
\( 15\beta_{6} + 14\beta_{5} + 3\beta_{4} + 27\beta_{3} - 38\beta_{2} + 64\beta _1 + 67 \)
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| \(\nu^{6}\) | \(=\) |
\( 52\beta_{6} + 39\beta_{5} + 8\beta_{4} + 105\beta_{3} - 163\beta_{2} + 181\beta _1 + 268 \)
|
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 |
|
−1.00000 | −3.38824 | 1.00000 | 2.16134 | 3.38824 | 0 | −1.00000 | 8.48014 | −2.16134 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.74428 | 1.00000 | −3.42627 | 2.74428 | 0 | −1.00000 | 4.53109 | 3.42627 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.56798 | 1.00000 | −0.829638 | 1.56798 | 0 | −1.00000 | −0.541448 | 0.829638 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −0.160973 | 1.00000 | 1.63435 | 0.160973 | 0 | −1.00000 | −2.97409 | −1.63435 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 1.19365 | 1.00000 | 0.537640 | −1.19365 | 0 | −1.00000 | −1.57521 | −0.537640 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 2.02957 | 1.00000 | 1.50585 | −2.02957 | 0 | −1.00000 | 1.11917 | −1.50585 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 2.63825 | 1.00000 | −2.58327 | −2.63825 | 0 | −1.00000 | 3.96034 | 2.58327 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(7\) | \( +1 \) |
| \(53\) | \( +1 \) |
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 | 5194.2.a.bs | 7 | |
| 7.b | odd | 2 | 1 | 5194.2.a.bt | 7 | ||
| 7.c | even | 3 | 2 | 742.2.e.g | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 742.2.e.g | ✓ | 14 | 7.c | even | 3 | 2 | |
| 5194.2.a.bs | 7 | 1.a | even | 1 | 1 | trivial | |
| 5194.2.a.bt | 7 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5194))\):
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\( T_{3}^{7} + 2T_{3}^{6} - 15T_{3}^{5} - 21T_{3}^{4} + 69T_{3}^{3} + 50T_{3}^{2} - 87T_{3} - 15 \)
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\( T_{5}^{7} + T_{5}^{6} - 14T_{5}^{5} - T_{5}^{4} + 57T_{5}^{3} - 34T_{5}^{2} - 36T_{5} + 21 \)
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\( T_{11}^{7} - 7T_{11}^{6} - 21T_{11}^{5} + 252T_{11}^{4} - 498T_{11}^{3} - 160T_{11}^{2} + 855T_{11} - 51 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{7} \)
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| $3$ |
\( T^{7} + 2 T^{6} + \cdots - 15 \)
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| $5$ |
\( T^{7} + T^{6} + \cdots + 21 \)
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| $7$ |
\( T^{7} \)
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| $11$ |
\( T^{7} - 7 T^{6} + \cdots - 51 \)
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| $13$ |
\( T^{7} - 9 T^{6} + \cdots + 71 \)
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| $17$ |
\( T^{7} - 2 T^{6} + \cdots + 22309 \)
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| $19$ |
\( T^{7} + 20 T^{6} + \cdots + 71367 \)
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| $23$ |
\( T^{7} + 5 T^{6} + \cdots + 49 \)
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| $29$ |
\( T^{7} - 11 T^{6} + \cdots - 153 \)
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| $31$ |
\( T^{7} + 23 T^{6} + \cdots + 36323 \)
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| $37$ |
\( T^{7} + 14 T^{6} + \cdots + 951 \)
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| $41$ |
\( T^{7} + 21 T^{6} + \cdots - 63 \)
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| $43$ |
\( T^{7} + 8 T^{6} + \cdots - 28943 \)
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| $47$ |
\( T^{7} - 6 T^{6} + \cdots + 46653 \)
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| $53$ |
\( (T + 1)^{7} \)
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| $59$ |
\( T^{7} + 31 T^{6} + \cdots + 34803 \)
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| $61$ |
\( T^{7} - 21 T^{6} + \cdots - 307071 \)
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| $67$ |
\( T^{7} + 27 T^{6} + \cdots + 166995 \)
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| $71$ |
\( T^{7} - 10 T^{6} + \cdots - 5019 \)
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| $73$ |
\( T^{7} - 18 T^{6} + \cdots + 95861 \)
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| $79$ |
\( T^{7} + 27 T^{6} + \cdots + 208751 \)
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| $83$ |
\( T^{7} + 21 T^{6} + \cdots - 4511781 \)
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| $89$ |
\( T^{7} + 9 T^{6} + \cdots - 19704027 \)
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| $97$ |
\( T^{7} + 2 T^{6} + \cdots - 3668623 \)
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