Newspace parameters
| Level: | \( N \) | \(=\) | \( 144 = 2^{4} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 144.q (of order \(6\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(58.6625198488\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Relative dimension: | \(7\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{14} - x^{13} + 427 x^{12} - 1362 x^{11} + 135762 x^{10} - 371244 x^{9} + 18261508 x^{8} + \cdots + 872385888256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{14}\cdot 3^{21} \) |
| Twist minimal: | no (minimal twist has level 9) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - x^{13} + 427 x^{12} - 1362 x^{11} + 135762 x^{10} - 371244 x^{9} + 18261508 x^{8} + \cdots + 872385888256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 41\!\cdots\!29 \nu^{13} + \cdots + 23\!\cdots\!20 ) / 15\!\cdots\!16 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 14\!\cdots\!61 \nu^{13} + \cdots - 10\!\cdots\!32 ) / 15\!\cdots\!16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 19\!\cdots\!77 \nu^{13} + \cdots - 11\!\cdots\!12 ) / 15\!\cdots\!16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 13\!\cdots\!63 \nu^{13} + \cdots - 10\!\cdots\!00 ) / 13\!\cdots\!72 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 28\!\cdots\!05 \nu^{13} + \cdots + 95\!\cdots\!88 ) / 31\!\cdots\!16 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 12\!\cdots\!41 \nu^{13} + \cdots + 60\!\cdots\!44 ) / 65\!\cdots\!36 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 67\!\cdots\!74 \nu^{13} + \cdots - 10\!\cdots\!12 ) / 16\!\cdots\!84 \)
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| \(\beta_{8}\) | \(=\) |
\( ( - 84\!\cdots\!35 \nu^{13} + \cdots - 33\!\cdots\!44 ) / 18\!\cdots\!96 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 73\!\cdots\!77 \nu^{13} + \cdots - 19\!\cdots\!36 ) / 13\!\cdots\!72 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 75\!\cdots\!33 \nu^{13} + \cdots - 77\!\cdots\!84 ) / 13\!\cdots\!72 \)
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| \(\beta_{11}\) | \(=\) |
\( ( 38\!\cdots\!31 \nu^{13} + \cdots + 69\!\cdots\!16 ) / 62\!\cdots\!32 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 14\!\cdots\!23 \nu^{13} + \cdots + 24\!\cdots\!28 ) / 21\!\cdots\!12 \)
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| \(\beta_{13}\) | \(=\) |
\( ( - 14\!\cdots\!83 \nu^{13} + \cdots - 33\!\cdots\!80 ) / 13\!\cdots\!72 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{3} + 3\beta_{2} + \beta_1 ) / 24 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -4\beta_{12} + 4\beta_{6} - 8\beta_{5} + \beta_{3} + 2923\beta_{2} - 6\beta _1 - 2923 ) / 24 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 24 \beta_{13} + 8 \beta_{11} - 8 \beta_{10} + 24 \beta_{9} - 24 \beta_{8} - 24 \beta_{7} + \cdots + 14667 ) / 72 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 496 \beta_{13} + 2988 \beta_{12} + 40 \beta_{11} + 80 \beta_{10} + 104 \beta_{9} + 208 \beta_{8} + \cdots - 120 ) / 72 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 6200 \beta_{13} - 14628 \beta_{12} - 7120 \beta_{11} - 3560 \beta_{10} - 18064 \beta_{9} + \cdots - 8413915 ) / 72 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 44760 \beta_{13} + 2824 \beta_{11} - 2824 \beta_{10} + 19592 \beta_{9} - 19592 \beta_{8} + \cdots + 144224747 ) / 24 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 323792 \beta_{13} + 1820460 \beta_{12} + 390328 \beta_{11} + 780656 \beta_{10} + 921080 \beta_{9} + \cdots - 525480 ) / 24 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 6378488 \beta_{13} - 67671844 \beta_{12} - 2058320 \beta_{11} - 1029160 \beta_{10} - 15663952 \beta_{9} + \cdots - 37935512347 ) / 24 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 24441480 \beta_{13} + 343293032 \beta_{11} - 343293032 \beta_{10} + 804239976 \beta_{9} + \cdots + 1129904003811 ) / 72 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 16592549776 \beta_{13} + 56229724044 \beta_{12} + 1365007480 \beta_{11} + 2730014960 \beta_{10} + \cdots - 6238735848 ) / 72 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 122689125368 \beta_{13} - 631438680900 \beta_{12} - 192850574416 \beta_{11} - 96425287208 \beta_{10} + \cdots - 371220192029107 ) / 72 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 930383856888 \beta_{13} + 185537979752 \beta_{11} - 185537979752 \beta_{10} + 915630048232 \beta_{9} + \cdots + 29\!\cdots\!35 ) / 24 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 16744043516560 \beta_{13} + 67758080937612 \beta_{12} + 8942189679608 \beta_{11} + \cdots - 12023834435688 ) / 24 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(65\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 65.1 |
|
0 | −80.5605 | − | 8.42672i | 0 | −46.9888 | + | 27.1290i | 0 | 921.012 | − | 1595.24i | 0 | 6418.98 | + | 1357.72i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 65.2 | 0 | −54.3339 | − | 60.0735i | 0 | 676.216 | − | 390.413i | 0 | −2168.61 | + | 3756.14i | 0 | −656.650 | + | 6528.06i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 65.3 | 0 | −51.0315 | + | 62.9030i | 0 | −896.557 | + | 517.627i | 0 | 21.9132 | − | 37.9547i | 0 | −1352.58 | − | 6420.07i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 65.4 | 0 | 29.7082 | + | 75.3553i | 0 | 331.396 | − | 191.331i | 0 | −467.516 | + | 809.762i | 0 | −4795.84 | + | 4477.35i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 65.5 | 0 | 44.1317 | − | 67.9220i | 0 | −604.549 | + | 349.037i | 0 | 1124.45 | − | 1947.61i | 0 | −2665.79 | − | 5995.02i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 65.6 | 0 | 77.6435 | − | 23.0757i | 0 | 189.131 | − | 109.195i | 0 | −1404.67 | + | 2432.96i | 0 | 5496.03 | − | 3583.35i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 65.7 | 0 | 80.9424 | + | 3.05302i | 0 | 570.352 | − | 329.293i | 0 | 1512.41 | − | 2619.58i | 0 | 6542.36 | + | 494.238i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.1 | 0 | −80.5605 | + | 8.42672i | 0 | −46.9888 | − | 27.1290i | 0 | 921.012 | + | 1595.24i | 0 | 6418.98 | − | 1357.72i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.2 | 0 | −54.3339 | + | 60.0735i | 0 | 676.216 | + | 390.413i | 0 | −2168.61 | − | 3756.14i | 0 | −656.650 | − | 6528.06i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.3 | 0 | −51.0315 | − | 62.9030i | 0 | −896.557 | − | 517.627i | 0 | 21.9132 | + | 37.9547i | 0 | −1352.58 | + | 6420.07i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.4 | 0 | 29.7082 | − | 75.3553i | 0 | 331.396 | + | 191.331i | 0 | −467.516 | − | 809.762i | 0 | −4795.84 | − | 4477.35i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.5 | 0 | 44.1317 | + | 67.9220i | 0 | −604.549 | − | 349.037i | 0 | 1124.45 | + | 1947.61i | 0 | −2665.79 | + | 5995.02i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.6 | 0 | 77.6435 | + | 23.0757i | 0 | 189.131 | + | 109.195i | 0 | −1404.67 | − | 2432.96i | 0 | 5496.03 | + | 3583.35i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.7 | 0 | 80.9424 | − | 3.05302i | 0 | 570.352 | + | 329.293i | 0 | 1512.41 | + | 2619.58i | 0 | 6542.36 | − | 494.238i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 144.9.q.a | 14 | |
| 3.b | odd | 2 | 1 | 432.9.q.a | 14 | ||
| 4.b | odd | 2 | 1 | 9.9.d.a | ✓ | 14 | |
| 9.c | even | 3 | 1 | 432.9.q.a | 14 | ||
| 9.d | odd | 6 | 1 | inner | 144.9.q.a | 14 | |
| 12.b | even | 2 | 1 | 27.9.d.a | 14 | ||
| 36.f | odd | 6 | 1 | 27.9.d.a | 14 | ||
| 36.f | odd | 6 | 1 | 81.9.b.a | 14 | ||
| 36.h | even | 6 | 1 | 9.9.d.a | ✓ | 14 | |
| 36.h | even | 6 | 1 | 81.9.b.a | 14 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9.9.d.a | ✓ | 14 | 4.b | odd | 2 | 1 | |
| 9.9.d.a | ✓ | 14 | 36.h | even | 6 | 1 | |
| 27.9.d.a | 14 | 12.b | even | 2 | 1 | ||
| 27.9.d.a | 14 | 36.f | odd | 6 | 1 | ||
| 81.9.b.a | 14 | 36.f | odd | 6 | 1 | ||
| 81.9.b.a | 14 | 36.h | even | 6 | 1 | ||
| 144.9.q.a | 14 | 1.a | even | 1 | 1 | trivial | |
| 144.9.q.a | 14 | 9.d | odd | 6 | 1 | inner | |
| 432.9.q.a | 14 | 3.b | odd | 2 | 1 | ||
| 432.9.q.a | 14 | 9.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{14} - 438 T_{5}^{13} - 1303854 T_{5}^{12} + 599097276 T_{5}^{11} + 1339373715651 T_{5}^{10} + \cdots + 28\!\cdots\!00 \)
acting on \(S_{9}^{\mathrm{new}}(144, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} \)
|
| $3$ |
\( T^{14} + \cdots + 52\!\cdots\!21 \)
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| $5$ |
\( T^{14} + \cdots + 28\!\cdots\!00 \)
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| $7$ |
\( T^{14} + \cdots + 39\!\cdots\!36 \)
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| $11$ |
\( T^{14} + \cdots + 15\!\cdots\!43 \)
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| $13$ |
\( T^{14} + \cdots + 41\!\cdots\!96 \)
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| $17$ |
\( T^{14} + \cdots + 19\!\cdots\!68 \)
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| $19$ |
\( (T^{7} + \cdots - 18\!\cdots\!88)^{2} \)
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| $23$ |
\( T^{14} + \cdots + 83\!\cdots\!52 \)
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| $29$ |
\( T^{14} + \cdots + 26\!\cdots\!88 \)
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| $31$ |
\( T^{14} + \cdots + 25\!\cdots\!84 \)
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| $37$ |
\( (T^{7} + \cdots + 33\!\cdots\!00)^{2} \)
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| $41$ |
\( T^{14} + \cdots + 20\!\cdots\!23 \)
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| $43$ |
\( T^{14} + \cdots + 11\!\cdots\!69 \)
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| $47$ |
\( T^{14} + \cdots + 35\!\cdots\!72 \)
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| $53$ |
\( T^{14} + \cdots + 35\!\cdots\!00 \)
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| $59$ |
\( T^{14} + \cdots + 10\!\cdots\!67 \)
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| $61$ |
\( T^{14} + \cdots + 26\!\cdots\!04 \)
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| $67$ |
\( T^{14} + \cdots + 48\!\cdots\!49 \)
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| $71$ |
\( T^{14} + \cdots + 37\!\cdots\!00 \)
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| $73$ |
\( (T^{7} + \cdots + 40\!\cdots\!00)^{2} \)
|
| $79$ |
\( T^{14} + \cdots + 49\!\cdots\!84 \)
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| $83$ |
\( T^{14} + \cdots + 55\!\cdots\!52 \)
|
| $89$ |
\( T^{14} + \cdots + 78\!\cdots\!00 \)
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| $97$ |
\( T^{14} + \cdots + 18\!\cdots\!25 \)
|
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