Newspace parameters
| Level: | \( N \) | \(=\) | \( 42 = 2 \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 42.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(32.2704135835\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{8} - 2 x^{7} + 131491 x^{6} + 44838722 x^{5} + 18152831051 x^{4} + 2926931386118 x^{3} + \cdots + 82\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{3}\cdot 7^{6} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{8} - 2 x^{7} + 131491 x^{6} + 44838722 x^{5} + 18152831051 x^{4} + 2926931386118 x^{3} + \cdots + 82\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 14\!\cdots\!09 \nu^{7} + \cdots + 29\!\cdots\!80 ) / 21\!\cdots\!30 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 21\!\cdots\!41 \nu^{7} + \cdots + 10\!\cdots\!40 ) / 77\!\cdots\!30 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 12\!\cdots\!87 \nu^{7} + \cdots + 26\!\cdots\!20 ) / 22\!\cdots\!58 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 93\!\cdots\!99 \nu^{7} + \cdots + 14\!\cdots\!40 ) / 77\!\cdots\!30 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 36\!\cdots\!81 \nu^{7} + \cdots + 16\!\cdots\!50 ) / 22\!\cdots\!58 \)
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| \(\beta_{6}\) | \(=\) |
\( ( - 64\!\cdots\!67 \nu^{7} + \cdots + 85\!\cdots\!30 ) / 38\!\cdots\!65 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 11\!\cdots\!02 \nu^{7} + \cdots + 23\!\cdots\!15 ) / 55\!\cdots\!95 \)
|
| \(\nu\) | \(=\) |
\( ( 15\beta_{7} + 4\beta_{6} - 20\beta_{5} - 14\beta_{4} - 3\beta_{3} - 21\beta_{2} - 586\beta _1 + 583 ) / 1176 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 130\beta_{7} - 26\beta_{6} - 19\beta_{5} + 33\beta_{4} - 169\beta_{3} - 66\beta_{2} - 1840859\beta _1 + 45 ) / 28 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 301471 \beta_{7} - 215538 \beta_{6} + 1682665 \beta_{5} + 1485519 \beta_{4} - 1574896 \beta_{3} + \cdots - 9944720524 ) / 588 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 45665593 \beta_{7} + 11008259 \beta_{6} + 62094934 \beta_{5} + 28353066 \beta_{4} + \cdots - 470313624937 ) / 49 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 96839960170 \beta_{7} + 19367992034 \beta_{6} - 3301296147 \beta_{5} - 42037280215 \beta_{4} + \cdots - 16066695887 ) / 294 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 201077819895 \beta_{7} - 10034029932 \beta_{6} - 1629215759604 \beta_{5} - 1221517964268 \beta_{4} + \cdots + 11\!\cdots\!97 ) / 7 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 11\!\cdots\!72 \beta_{7} + \cdots + 10\!\cdots\!27 ) / 147 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(31\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
16.0000 | + | 27.7128i | 121.500 | − | 210.444i | −512.000 | + | 886.810i | −4438.44 | − | 7687.60i | 7776.00 | −14427.2 | − | 42061.6i | −32768.0 | −29524.5 | − | 51137.9i | 142030. | − | 246003.i | ||||||||||||||||||||||||||||
| 25.2 | 16.0000 | + | 27.7128i | 121.500 | − | 210.444i | −512.000 | + | 886.810i | −1639.98 | − | 2840.53i | 7776.00 | −17084.7 | + | 41054.1i | −32768.0 | −29524.5 | − | 51137.9i | 52479.3 | − | 90896.9i | |||||||||||||||||||||||||||||
| 25.3 | 16.0000 | + | 27.7128i | 121.500 | − | 210.444i | −512.000 | + | 886.810i | −1017.54 | − | 1762.43i | 7776.00 | 42078.2 | + | 14378.9i | −32768.0 | −29524.5 | − | 51137.9i | 32561.3 | − | 56397.9i | |||||||||||||||||||||||||||||
| 25.4 | 16.0000 | + | 27.7128i | 121.500 | − | 210.444i | −512.000 | + | 886.810i | 6385.96 | + | 11060.8i | 7776.00 | −43747.2 | − | 7969.08i | −32768.0 | −29524.5 | − | 51137.9i | −204351. | + | 353946.i | |||||||||||||||||||||||||||||
| 37.1 | 16.0000 | − | 27.7128i | 121.500 | + | 210.444i | −512.000 | − | 886.810i | −4438.44 | + | 7687.60i | 7776.00 | −14427.2 | + | 42061.6i | −32768.0 | −29524.5 | + | 51137.9i | 142030. | + | 246003.i | |||||||||||||||||||||||||||||
| 37.2 | 16.0000 | − | 27.7128i | 121.500 | + | 210.444i | −512.000 | − | 886.810i | −1639.98 | + | 2840.53i | 7776.00 | −17084.7 | − | 41054.1i | −32768.0 | −29524.5 | + | 51137.9i | 52479.3 | + | 90896.9i | |||||||||||||||||||||||||||||
| 37.3 | 16.0000 | − | 27.7128i | 121.500 | + | 210.444i | −512.000 | − | 886.810i | −1017.54 | + | 1762.43i | 7776.00 | 42078.2 | − | 14378.9i | −32768.0 | −29524.5 | + | 51137.9i | 32561.3 | + | 56397.9i | |||||||||||||||||||||||||||||
| 37.4 | 16.0000 | − | 27.7128i | 121.500 | + | 210.444i | −512.000 | − | 886.810i | 6385.96 | − | 11060.8i | 7776.00 | −43747.2 | + | 7969.08i | −32768.0 | −29524.5 | + | 51137.9i | −204351. | − | 353946.i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 42.12.e.d | ✓ | 8 |
| 3.b | odd | 2 | 1 | 126.12.g.d | 8 | ||
| 7.c | even | 3 | 1 | inner | 42.12.e.d | ✓ | 8 |
| 21.h | odd | 6 | 1 | 126.12.g.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.12.e.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 42.12.e.d | ✓ | 8 | 7.c | even | 3 | 1 | inner |
| 126.12.g.d | 8 | 3.b | odd | 2 | 1 | ||
| 126.12.g.d | 8 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 1420 T_{5}^{7} + 129418525 T_{5}^{6} + 1076271246300 T_{5}^{5} + \cdots + 57\!\cdots\!00 \)
acting on \(S_{12}^{\mathrm{new}}(42, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 32 T + 1024)^{4} \)
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| $3$ |
\( (T^{2} - 243 T + 59049)^{4} \)
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| $5$ |
\( T^{8} + \cdots + 57\!\cdots\!00 \)
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| $7$ |
\( T^{8} + \cdots + 15\!\cdots\!01 \)
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| $11$ |
\( T^{8} + \cdots + 26\!\cdots\!56 \)
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| $13$ |
\( (T^{4} + \cdots + 93\!\cdots\!04)^{2} \)
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| $17$ |
\( T^{8} + \cdots + 15\!\cdots\!00 \)
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| $19$ |
\( T^{8} + \cdots + 38\!\cdots\!04 \)
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| $23$ |
\( T^{8} + \cdots + 12\!\cdots\!56 \)
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| $29$ |
\( (T^{4} + \cdots + 20\!\cdots\!08)^{2} \)
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| $31$ |
\( T^{8} + \cdots + 67\!\cdots\!25 \)
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| $37$ |
\( T^{8} + \cdots + 55\!\cdots\!84 \)
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| $41$ |
\( (T^{4} + \cdots + 41\!\cdots\!92)^{2} \)
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| $43$ |
\( (T^{4} + \cdots + 52\!\cdots\!24)^{2} \)
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| $47$ |
\( T^{8} + \cdots + 50\!\cdots\!00 \)
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| $53$ |
\( T^{8} + \cdots + 29\!\cdots\!44 \)
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| $59$ |
\( T^{8} + \cdots + 56\!\cdots\!56 \)
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| $61$ |
\( T^{8} + \cdots + 13\!\cdots\!00 \)
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| $67$ |
\( T^{8} + \cdots + 21\!\cdots\!00 \)
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| $71$ |
\( (T^{4} + \cdots - 19\!\cdots\!56)^{2} \)
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| $73$ |
\( T^{8} + \cdots + 16\!\cdots\!96 \)
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| $79$ |
\( T^{8} + \cdots + 10\!\cdots\!01 \)
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| $83$ |
\( (T^{4} + \cdots - 53\!\cdots\!52)^{2} \)
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| $89$ |
\( T^{8} + \cdots + 18\!\cdots\!00 \)
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| $97$ |
\( (T^{4} + \cdots + 33\!\cdots\!48)^{2} \)
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