Newspace parameters
| Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 126.s (of order \(6\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(28.9868145361\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{16} - 8580 x^{14} - 15624 x^{13} + 31160874 x^{12} + 100477944 x^{11} - 62268032912 x^{10} + \cdots + 62\!\cdots\!04 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{6}\cdot 7^{6} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 8580 x^{14} - 15624 x^{13} + 31160874 x^{12} + 100477944 x^{11} - 62268032912 x^{10} + \cdots + 62\!\cdots\!04 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 10\!\cdots\!56 \nu^{15} + \cdots + 22\!\cdots\!28 ) / 38\!\cdots\!68 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 29\!\cdots\!55 \nu^{15} + \cdots - 52\!\cdots\!68 ) / 75\!\cdots\!08 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 17\!\cdots\!28 \nu^{15} + \cdots + 27\!\cdots\!60 ) / 42\!\cdots\!92 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 14\!\cdots\!42 \nu^{15} + \cdots + 26\!\cdots\!60 ) / 16\!\cdots\!36 \)
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| \(\beta_{5}\) | \(=\) |
\( ( - 41\!\cdots\!01 \nu^{15} + \cdots - 79\!\cdots\!92 ) / 32\!\cdots\!72 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 35\!\cdots\!41 \nu^{15} + \cdots + 89\!\cdots\!52 ) / 54\!\cdots\!12 \)
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| \(\beta_{7}\) | \(=\) |
\( ( 27\!\cdots\!04 \nu^{15} + \cdots + 64\!\cdots\!88 ) / 32\!\cdots\!72 \)
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| \(\beta_{8}\) | \(=\) |
\( ( 14\!\cdots\!25 \nu^{15} + \cdots + 11\!\cdots\!60 ) / 16\!\cdots\!36 \)
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| \(\beta_{9}\) | \(=\) |
\( ( - 45\!\cdots\!12 \nu^{15} + \cdots + 58\!\cdots\!24 ) / 32\!\cdots\!72 \)
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| \(\beta_{10}\) | \(=\) |
\( ( - 46\!\cdots\!01 \nu^{15} + \cdots - 85\!\cdots\!56 ) / 32\!\cdots\!72 \)
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| \(\beta_{11}\) | \(=\) |
\( ( 12\!\cdots\!13 \nu^{15} + \cdots + 22\!\cdots\!64 ) / 82\!\cdots\!68 \)
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| \(\beta_{12}\) | \(=\) |
\( ( - 16\!\cdots\!25 \nu^{15} + \cdots - 33\!\cdots\!20 ) / 32\!\cdots\!72 \)
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| \(\beta_{13}\) | \(=\) |
\( ( 86\!\cdots\!19 \nu^{15} + \cdots + 36\!\cdots\!24 ) / 16\!\cdots\!36 \)
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| \(\beta_{14}\) | \(=\) |
\( ( 83\!\cdots\!32 \nu^{15} + \cdots + 22\!\cdots\!32 ) / 82\!\cdots\!68 \)
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| \(\beta_{15}\) | \(=\) |
\( ( 37\!\cdots\!84 \nu^{15} + \cdots + 89\!\cdots\!40 ) / 16\!\cdots\!36 \)
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| \(\nu\) | \(=\) |
\( ( -\beta_{11} + \beta_{8} + 2\beta_{6} + 6\beta_{5} - 42\beta_{2} - \beta _1 + 1 ) / 42 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 2 \beta_{14} + 3 \beta_{13} - 11 \beta_{12} - 31 \beta_{11} - \beta_{9} + 9 \beta_{8} - \beta_{7} + \cdots + 22516 ) / 21 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 33 \beta_{15} + 45 \beta_{14} + 48 \beta_{13} - 318 \beta_{12} - 1328 \beta_{11} + 18 \beta_{10} + \cdots + 62378 ) / 21 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 1264 \beta_{15} - 1864 \beta_{14} + 7192 \beta_{13} - 23681 \beta_{12} - 68566 \beta_{11} + \cdots + 29546461 ) / 21 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 118045 \beta_{15} + 140875 \beta_{14} + 201355 \beta_{13} - 732101 \beta_{12} - 2525148 \beta_{11} + \cdots + 221176769 ) / 21 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 4336238 \beta_{15} - 328686 \beta_{14} + 15516758 \beta_{13} - 42585389 \beta_{12} + \cdots + 43502870551 ) / 21 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 295124795 \beta_{15} + 314604899 \beta_{14} + 556395161 \beta_{13} - 1396353343 \beta_{12} + \cdots + 533812235135 ) / 21 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 10924433016 \beta_{15} + 3159314200 \beta_{14} + 32915989512 \beta_{13} - 73953808373 \beta_{12} + \cdots + 67764529603957 ) / 21 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 652065076989 \beta_{15} + 633396324639 \beta_{14} + 1309431813783 \beta_{13} - 2548128202731 \beta_{12} + \cdots + 11\!\cdots\!53 ) / 21 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 24611078858350 \beta_{15} + 10407967819946 \beta_{14} + 68615396770978 \beta_{13} + \cdots + 10\!\cdots\!23 ) / 21 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 13\!\cdots\!75 \beta_{15} + \cdots + 21\!\cdots\!23 ) / 21 \)
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| \(\nu^{12}\) | \(=\) |
\( ( - 52\!\cdots\!36 \beta_{15} + \cdots + 17\!\cdots\!41 ) / 21 \)
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| \(\nu^{13}\) | \(=\) |
\( ( - 27\!\cdots\!45 \beta_{15} + \cdots + 41\!\cdots\!85 ) / 21 \)
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| \(\nu^{14}\) | \(=\) |
\( ( - 10\!\cdots\!22 \beta_{15} + \cdots + 28\!\cdots\!03 ) / 21 \)
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| \(\nu^{15}\) | \(=\) |
\( ( - 54\!\cdots\!59 \beta_{15} + \cdots + 75\!\cdots\!03 ) / 21 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(73\) |
| \(\chi(n)\) | \(-1\) | \(-1 - \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 53.1 |
|
−4.89898 | + | 2.82843i | 0 | 16.0000 | − | 27.7128i | −168.052 | + | 97.0249i | 0 | 157.769 | − | 304.562i | 181.019i | 0 | 548.856 | − | 950.646i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 53.2 | −4.89898 | + | 2.82843i | 0 | 16.0000 | − | 27.7128i | 20.9715 | − | 12.1079i | 0 | 94.7935 | − | 329.641i | 181.019i | 0 | −68.4925 | + | 118.633i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 53.3 | −4.89898 | + | 2.82843i | 0 | 16.0000 | − | 27.7128i | 61.9031 | − | 35.7398i | 0 | −336.264 | + | 67.6424i | 181.019i | 0 | −202.175 | + | 350.177i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 53.4 | −4.89898 | + | 2.82843i | 0 | 16.0000 | − | 27.7128i | 68.0310 | − | 39.2777i | 0 | 221.702 | + | 261.720i | 181.019i | 0 | −222.188 | + | 384.842i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 53.5 | 4.89898 | − | 2.82843i | 0 | 16.0000 | − | 27.7128i | −68.0310 | + | 39.2777i | 0 | 221.702 | + | 261.720i | − | 181.019i | 0 | −222.188 | + | 384.842i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 53.6 | 4.89898 | − | 2.82843i | 0 | 16.0000 | − | 27.7128i | −61.9031 | + | 35.7398i | 0 | −336.264 | + | 67.6424i | − | 181.019i | 0 | −202.175 | + | 350.177i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 53.7 | 4.89898 | − | 2.82843i | 0 | 16.0000 | − | 27.7128i | −20.9715 | + | 12.1079i | 0 | 94.7935 | − | 329.641i | − | 181.019i | 0 | −68.4925 | + | 118.633i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 53.8 | 4.89898 | − | 2.82843i | 0 | 16.0000 | − | 27.7128i | 168.052 | − | 97.0249i | 0 | 157.769 | − | 304.562i | − | 181.019i | 0 | 548.856 | − | 950.646i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.1 | −4.89898 | − | 2.82843i | 0 | 16.0000 | + | 27.7128i | −168.052 | − | 97.0249i | 0 | 157.769 | + | 304.562i | − | 181.019i | 0 | 548.856 | + | 950.646i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.2 | −4.89898 | − | 2.82843i | 0 | 16.0000 | + | 27.7128i | 20.9715 | + | 12.1079i | 0 | 94.7935 | + | 329.641i | − | 181.019i | 0 | −68.4925 | − | 118.633i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.3 | −4.89898 | − | 2.82843i | 0 | 16.0000 | + | 27.7128i | 61.9031 | + | 35.7398i | 0 | −336.264 | − | 67.6424i | − | 181.019i | 0 | −202.175 | − | 350.177i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.4 | −4.89898 | − | 2.82843i | 0 | 16.0000 | + | 27.7128i | 68.0310 | + | 39.2777i | 0 | 221.702 | − | 261.720i | − | 181.019i | 0 | −222.188 | − | 384.842i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.5 | 4.89898 | + | 2.82843i | 0 | 16.0000 | + | 27.7128i | −68.0310 | − | 39.2777i | 0 | 221.702 | − | 261.720i | 181.019i | 0 | −222.188 | − | 384.842i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.6 | 4.89898 | + | 2.82843i | 0 | 16.0000 | + | 27.7128i | −61.9031 | − | 35.7398i | 0 | −336.264 | − | 67.6424i | 181.019i | 0 | −202.175 | − | 350.177i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.7 | 4.89898 | + | 2.82843i | 0 | 16.0000 | + | 27.7128i | −20.9715 | − | 12.1079i | 0 | 94.7935 | + | 329.641i | 181.019i | 0 | −68.4925 | − | 118.633i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 107.8 | 4.89898 | + | 2.82843i | 0 | 16.0000 | + | 27.7128i | 168.052 | + | 97.0249i | 0 | 157.769 | + | 304.562i | 181.019i | 0 | 548.856 | + | 950.646i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 21.h | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 126.7.s.a | ✓ | 16 |
| 3.b | odd | 2 | 1 | inner | 126.7.s.a | ✓ | 16 |
| 7.c | even | 3 | 1 | inner | 126.7.s.a | ✓ | 16 |
| 21.h | odd | 6 | 1 | inner | 126.7.s.a | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 126.7.s.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 126.7.s.a | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
| 126.7.s.a | ✓ | 16 | 7.c | even | 3 | 1 | inner |
| 126.7.s.a | ✓ | 16 | 21.h | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{16} - 49522 T_{5}^{14} + 1967440359 T_{5}^{12} - 21107937231250 T_{5}^{10} + \cdots + 48\!\cdots\!00 \)
acting on \(S_{7}^{\mathrm{new}}(126, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 32 T^{2} + 1024)^{4} \)
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| $3$ |
\( T^{16} \)
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| $5$ |
\( T^{16} + \cdots + 48\!\cdots\!00 \)
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| $7$ |
\( (T^{8} + \cdots + 19\!\cdots\!01)^{2} \)
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| $11$ |
\( T^{16} + \cdots + 12\!\cdots\!00 \)
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| $13$ |
\( (T^{4} + \cdots - 1535891822730)^{4} \)
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| $17$ |
\( T^{16} + \cdots + 10\!\cdots\!56 \)
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| $19$ |
\( (T^{8} + \cdots + 78\!\cdots\!16)^{2} \)
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| $23$ |
\( T^{16} + \cdots + 27\!\cdots\!16 \)
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| $29$ |
\( (T^{8} + \cdots + 36\!\cdots\!96)^{2} \)
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| $31$ |
\( (T^{8} + \cdots + 19\!\cdots\!09)^{2} \)
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| $37$ |
\( (T^{8} + \cdots + 38\!\cdots\!64)^{2} \)
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| $41$ |
\( (T^{8} + \cdots + 52\!\cdots\!24)^{2} \)
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| $43$ |
\( (T^{4} + \cdots - 48\!\cdots\!70)^{4} \)
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| $47$ |
\( T^{16} + \cdots + 47\!\cdots\!00 \)
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| $53$ |
\( T^{16} + \cdots + 12\!\cdots\!16 \)
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| $59$ |
\( T^{16} + \cdots + 85\!\cdots\!36 \)
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| $61$ |
\( (T^{8} + \cdots + 24\!\cdots\!24)^{2} \)
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| $67$ |
\( (T^{8} + \cdots + 45\!\cdots\!16)^{2} \)
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| $71$ |
\( (T^{8} + \cdots + 13\!\cdots\!64)^{2} \)
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| $73$ |
\( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \)
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| $79$ |
\( (T^{8} + \cdots + 25\!\cdots\!61)^{2} \)
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| $83$ |
\( (T^{8} + \cdots + 99\!\cdots\!44)^{2} \)
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| $89$ |
\( T^{16} + \cdots + 52\!\cdots\!16 \)
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| $97$ |
\( (T^{4} + \cdots - 10\!\cdots\!16)^{4} \)
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