Newspace parameters
| Level: | \( N \) | \(=\) | \( 1668 = 2^{2} \cdot 3 \cdot 139 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1668.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(13.3190470571\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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| Defining polynomial: |
\( x^{7} - 28x^{5} - x^{4} + 217x^{3} + 51x^{2} - 456x - 300 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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
\( x^{7} - 28x^{5} - x^{4} + 217x^{3} + 51x^{2} - 456x - 300 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{6} + 123\nu^{5} + 105\nu^{4} - 2636\nu^{3} - 4321\nu^{2} + 10752\nu + 19942 ) / 1738 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{6} + 21\nu^{5} + 95\nu^{4} - 504\nu^{3} - 757\nu^{2} + 2618\nu + 1528 ) / 316 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -43\nu^{6} - 15\nu^{5} + 835\nu^{4} - 272\nu^{3} - 3161\nu^{2} + 3818\nu + 4732 ) / 3476 \)
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| \(\beta_{5}\) | \(=\) |
\( ( -89\nu^{6} + 70\nu^{5} + 2476\nu^{4} - 1917\nu^{3} - 17981\nu^{2} + 11439\nu + 27740 ) / 1738 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 259\nu^{6} - 233\nu^{5} - 6727\nu^{4} + 5276\nu^{3} + 43129\nu^{2} - 20410\nu - 55340 ) / 3476 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} - \beta_{2} - \beta _1 + 8 \)
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| \(\nu^{3}\) | \(=\) |
\( -2\beta_{6} - 2\beta_{5} - 2\beta_{4} - 2\beta_{3} + \beta_{2} + 14\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( 3\beta_{6} + 6\beta_{5} - 23\beta_{4} + 16\beta_{3} - 17\beta_{2} - 24\beta _1 + 101 \)
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| \(\nu^{5}\) | \(=\) |
\( -49\beta_{6} - 54\beta_{5} - 41\beta_{4} - 32\beta_{3} + 26\beta_{2} + 217\beta _1 - 6 \)
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| \(\nu^{6}\) | \(=\) |
\( 88\beta_{6} + 148\beta_{5} - 427\beta_{4} + 261\beta_{3} - 272\beta_{2} - 468\beta _1 + 1479 \)
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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 |
|
0 | 1.00000 | 0 | −3.98745 | 0 | −4.32675 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 1.00000 | 0 | −2.91888 | 0 | 3.61860 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.00000 | 0 | −2.11277 | 0 | 1.19918 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.00000 | 0 | 0.831175 | 0 | 4.60071 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.00000 | 0 | 1.28035 | 0 | −4.41213 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.00000 | 0 | 2.77217 | 0 | −1.71894 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.00000 | 0 | 4.13540 | 0 | 2.03932 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(139\) | \( -1 \) |
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 | 1668.2.a.h | ✓ | 7 |
| 3.b | odd | 2 | 1 | 5004.2.a.m | 7 | ||
| 4.b | odd | 2 | 1 | 6672.2.a.bn | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1668.2.a.h | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 5004.2.a.m | 7 | 3.b | odd | 2 | 1 | ||
| 6672.2.a.bn | 7 | 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1668))\):
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\( T_{5}^{7} - 28T_{5}^{5} + T_{5}^{4} + 217T_{5}^{3} - 51T_{5}^{2} - 456T_{5} + 300 \)
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\( T_{7}^{7} - T_{7}^{6} - 40T_{7}^{5} + 46T_{7}^{4} + 450T_{7}^{3} - 599T_{7}^{2} - 1040T_{7} + 1336 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
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| $3$ |
\( (T - 1)^{7} \)
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| $5$ |
\( T^{7} - 28 T^{5} + \cdots + 300 \)
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| $7$ |
\( T^{7} - T^{6} + \cdots + 1336 \)
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| $11$ |
\( T^{7} - 31 T^{5} + \cdots + 45 \)
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| $13$ |
\( T^{7} - 8 T^{6} + \cdots + 3964 \)
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| $17$ |
\( T^{7} - 27 T^{6} + \cdots - 9608 \)
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| $19$ |
\( T^{7} - 72 T^{5} + \cdots + 6632 \)
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| $23$ |
\( T^{7} + 8 T^{6} + \cdots - 6048 \)
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| $29$ |
\( T^{7} - 6 T^{6} + \cdots + 7680 \)
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| $31$ |
\( T^{7} - 7 T^{6} + \cdots + 17724 \)
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| $37$ |
\( T^{7} - 14 T^{6} + \cdots + 367971 \)
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| $41$ |
\( T^{7} - T^{6} + \cdots - 78372 \)
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| $43$ |
\( T^{7} - 27 T^{6} + \cdots + 26944 \)
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| $47$ |
\( T^{7} + 6 T^{6} + \cdots + 6885 \)
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| $53$ |
\( T^{7} - 18 T^{6} + \cdots - 854784 \)
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| $59$ |
\( T^{7} - 17 T^{6} + \cdots + 5278880 \)
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| $61$ |
\( T^{7} - 13 T^{6} + \cdots + 31744 \)
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| $67$ |
\( T^{7} - 31 T^{6} + \cdots - 29637792 \)
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| $71$ |
\( T^{7} - 367 T^{5} + \cdots + 1752240 \)
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| $73$ |
\( T^{7} - 6 T^{6} + \cdots + 38080 \)
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| $79$ |
\( T^{7} - 2 T^{6} + \cdots - 23696 \)
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| $83$ |
\( T^{7} + 2 T^{6} + \cdots - 375108 \)
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| $89$ |
\( T^{7} - 13 T^{6} + \cdots - 32328 \)
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| $97$ |
\( T^{7} - 7 T^{6} + \cdots - 106848 \)
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