Newspace parameters
| Level: | \( N \) | \(=\) | \( 1127 = 7^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1127.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(66.4951525765\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
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| Defining polynomial: |
\( x^{8} - 44x^{6} - 23x^{5} + 587x^{4} + 594x^{3} - 2430x^{2} - 3403x + 110 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 161) |
| 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
\( x^{8} - 44x^{6} - 23x^{5} + 587x^{4} + 594x^{3} - 2430x^{2} - 3403x + 110 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( -10\nu^{7} + 18\nu^{6} + 433\nu^{5} - 524\nu^{4} - 5765\nu^{3} + 3040\nu^{2} + 24035\nu + 4483 ) / 381 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 55\nu^{7} - 99\nu^{6} - 2191\nu^{5} + 2882\nu^{4} + 25421\nu^{3} - 18625\nu^{2} - 88949\nu - 82 ) / 1524 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 55\nu^{7} - 99\nu^{6} - 2191\nu^{5} + 2882\nu^{4} + 25421\nu^{3} - 20149\nu^{2} - 87425\nu + 16682 ) / 1524 \)
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| \(\beta_{5}\) | \(=\) |
\( ( -57\nu^{7} + 128\nu^{6} + 2303\nu^{5} - 3825\nu^{4} - 27590\nu^{3} + 25583\nu^{2} + 100614\nu - 8140 ) / 762 \)
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| \(\beta_{6}\) | \(=\) |
\( ( -176\nu^{7} + 393\nu^{6} + 6935\nu^{5} - 11356\nu^{4} - 80509\nu^{3} + 71411\nu^{2} + 289666\nu - 12844 ) / 762 \)
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| \(\beta_{7}\) | \(=\) |
\( ( - 721 \nu^{7} + 1501 \nu^{6} + 28565 \nu^{5} - 42962 \nu^{4} - 332535 \nu^{3} + 266555 \nu^{2} + \cdots - 32998 ) / 1524 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta _1 + 11 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - 2\beta_{6} + 2\beta_{5} + 2\beta_{4} + \beta_{3} - 2\beta_{2} + 17\beta _1 + 11 \)
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| \(\nu^{4}\) | \(=\) |
\( 3\beta_{7} - 4\beta_{6} - 17\beta_{4} + 30\beta_{3} - \beta_{2} + 24\beta _1 + 197 \)
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| \(\nu^{5}\) | \(=\) |
\( 33\beta_{7} - 66\beta_{6} + 66\beta_{5} + 56\beta_{4} + 51\beta_{3} - 55\beta_{2} + 344\beta _1 + 344 \)
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| \(\nu^{6}\) | \(=\) |
\( 106\beta_{7} - 146\beta_{6} + 44\beta_{5} - 287\beta_{4} + 789\beta_{3} - 61\beta_{2} + 608\beta _1 + 4206 \)
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| \(\nu^{7}\) | \(=\) |
\( 886\beta_{7} - 1758\beta_{6} + 1784\beta_{5} + 1342\beta_{4} + 1784\beta_{3} - 1324\beta_{2} + 7639\beta _1 + 9594 \)
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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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−5.10754 | −4.52103 | 18.0870 | −1.28161 | 23.0913 | 0 | −51.5197 | −6.56032 | 6.54585 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.78325 | 9.05000 | 6.31297 | 4.56958 | −34.2384 | 0 | 6.38246 | 54.9026 | −17.2879 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.91250 | −3.10924 | 0.482647 | −2.69280 | 9.05565 | 0 | 21.8943 | −17.3326 | 7.84277 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.0316163 | −9.11982 | −7.99900 | 18.5636 | 0.288335 | 0 | 0.505830 | 56.1711 | −0.586913 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.07589 | −3.18430 | −3.69068 | 8.32072 | −6.61027 | 0 | −24.2686 | −16.8602 | 17.2729 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.14518 | 0.0184068 | −3.39821 | −1.27994 | 0.0394858 | 0 | −24.4512 | −26.9997 | −2.74571 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.02588 | 9.11775 | 1.15594 | −18.5973 | 27.5892 | 0 | −20.7093 | 56.1334 | −56.2732 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.58796 | 4.74823 | 13.0493 | 16.3977 | 21.7847 | 0 | 23.1662 | −4.45434 | 75.2321 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( -1 \) |
| \(23\) | \( -1 \) |
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 | 1127.4.a.e | 8 | |
| 7.b | odd | 2 | 1 | 161.4.a.b | ✓ | 8 | |
| 21.c | even | 2 | 1 | 1449.4.a.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 161.4.a.b | ✓ | 8 | 7.b | odd | 2 | 1 | |
| 1127.4.a.e | 8 | 1.a | even | 1 | 1 | trivial | |
| 1449.4.a.i | 8 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1127))\):
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\( T_{2}^{8} - 44T_{2}^{6} + 23T_{2}^{5} + 587T_{2}^{4} - 594T_{2}^{3} - 2430T_{2}^{2} + 3403T_{2} + 110 \)
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\( T_{3}^{8} - 3T_{3}^{7} - 151T_{3}^{6} + 231T_{3}^{5} + 6672T_{3}^{4} + 3444T_{3}^{3} - 85750T_{3}^{2} - 158364T_{3} + 2944 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 44 T^{6} + \cdots + 110 \)
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| $3$ |
\( T^{8} - 3 T^{7} + \cdots + 2944 \)
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| $5$ |
\( T^{8} - 24 T^{7} + \cdots + 950784 \)
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| $7$ |
\( T^{8} \)
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| $11$ |
\( T^{8} + 98 T^{7} + \cdots + 228169216 \)
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| $13$ |
\( T^{8} + \cdots - 45149692647552 \)
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| $17$ |
\( T^{8} + \cdots - 999803228258304 \)
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| $19$ |
\( T^{8} + \cdots + 59039918237696 \)
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| $23$ |
\( (T - 23)^{8} \)
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| $29$ |
\( T^{8} + \cdots - 77\!\cdots\!32 \)
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| $31$ |
\( T^{8} + \cdots - 98\!\cdots\!24 \)
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| $37$ |
\( T^{8} + \cdots + 283373860839168 \)
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| $41$ |
\( T^{8} + \cdots + 63\!\cdots\!56 \)
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| $43$ |
\( T^{8} + \cdots - 43\!\cdots\!84 \)
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| $47$ |
\( T^{8} + \cdots - 98\!\cdots\!36 \)
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| $53$ |
\( T^{8} + \cdots + 29\!\cdots\!36 \)
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| $59$ |
\( T^{8} + \cdots - 91\!\cdots\!24 \)
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| $61$ |
\( T^{8} + \cdots + 69\!\cdots\!20 \)
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| $67$ |
\( T^{8} + \cdots - 56\!\cdots\!92 \)
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| $71$ |
\( T^{8} + \cdots + 49\!\cdots\!60 \)
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| $73$ |
\( T^{8} + \cdots - 26\!\cdots\!24 \)
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| $79$ |
\( T^{8} + \cdots - 48\!\cdots\!20 \)
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| $83$ |
\( T^{8} + \cdots - 36\!\cdots\!64 \)
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| $89$ |
\( T^{8} + \cdots + 43\!\cdots\!36 \)
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| $97$ |
\( T^{8} + \cdots - 22\!\cdots\!52 \)
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