Newspace parameters
| Level: | \( N \) | \(=\) | \( 10 = 2 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 10.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(7.68343180560\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 198x^{3} + 3568321x^{2} - 6762620x + 6408200 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{19}\cdot 5^{5} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 2x^{4} + 198x^{3} + 3568321x^{2} - 6762620x + 6408200 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 807955 \nu^{5} + 842118 \nu^{4} - 3200275588 \nu^{3} + 4808106038 \nu^{2} + \cdots + 2990639720130 ) / 604523721605 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 5715200 \nu^{5} + 6014576 \nu^{4} - 315216 \nu^{3} - 11372932784 \nu^{2} + \cdots + 19324355382560 ) / 604523721605 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 4247878015 \nu^{5} - 14980310918 \nu^{4} + 509621899488 \nu^{3} + 3127575282762 \nu^{2} + \cdots - 31\!\cdots\!45 ) / 5440713494445 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 4735302175 \nu^{5} - 13979944358 \nu^{4} - 412097187072 \nu^{3} + 3079320290922 \nu^{2} + \cdots - 29\!\cdots\!30 ) / 5440713494445 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 2690537304 \nu^{5} + 4096032176 \nu^{4} + 26667276564 \nu^{3} + 1873448147736 \nu^{2} + \cdots + 12\!\cdots\!63 ) / 1088142698889 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{4} - \beta_{3} - 14\beta_{2} - 32\beta _1 + 213 ) / 640 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 54\beta_{5} + 18\beta_{4} - 15781\beta_{2} - 6\beta _1 - 18 ) / 400 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 576\beta_{5} - 9523\beta_{4} + 9235\beta_{3} - 79334\beta_{2} - 302144\beta _1 - 316047 ) / 3200 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 34038\beta_{5} - 52599\beta_{4} - 49515\beta_{3} + 34038\beta_{2} + 34038\beta _1 - 951686251 ) / 400 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -114624\beta_{5} - 3714187\beta_{4} + 3484939\beta_{3} + 32565482\beta_{2} + 114269024\beta _1 - 198490599 ) / 640 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9.1 |
|
− | 32.0000i | − | 532.146i | −1024.00 | 1594.62 | − | 6803.33i | −17028.7 | 24477.0i | 32768.0i | −106033. | −217707. | − | 51027.9i | ||||||||||||||||||||||||||||||
| 9.2 | − | 32.0000i | 100.951i | −1024.00 | 2827.02 | + | 6390.31i | 3230.43 | 15908.8i | 32768.0i | 166956. | 204490. | − | 90464.5i | ||||||||||||||||||||||||||||||||
| 9.3 | − | 32.0000i | 697.195i | −1024.00 | −4156.64 | − | 5616.98i | 22310.3 | − | 73603.8i | 32768.0i | −308934. | −179743. | + | 133012.i | |||||||||||||||||||||||||||||||
| 9.4 | 32.0000i | − | 697.195i | −1024.00 | −4156.64 | + | 5616.98i | 22310.3 | 73603.8i | − | 32768.0i | −308934. | −179743. | − | 133012.i | |||||||||||||||||||||||||||||||
| 9.5 | 32.0000i | − | 100.951i | −1024.00 | 2827.02 | − | 6390.31i | 3230.43 | − | 15908.8i | − | 32768.0i | 166956. | 204490. | + | 90464.5i | ||||||||||||||||||||||||||||||
| 9.6 | 32.0000i | 532.146i | −1024.00 | 1594.62 | + | 6803.33i | −17028.7 | − | 24477.0i | − | 32768.0i | −106033. | −217707. | + | 51027.9i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 10.12.b.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 90.12.c.b | 6 | ||
| 4.b | odd | 2 | 1 | 80.12.c.c | 6 | ||
| 5.b | even | 2 | 1 | inner | 10.12.b.a | ✓ | 6 |
| 5.c | odd | 4 | 1 | 50.12.a.i | 3 | ||
| 5.c | odd | 4 | 1 | 50.12.a.j | 3 | ||
| 15.d | odd | 2 | 1 | 90.12.c.b | 6 | ||
| 20.d | odd | 2 | 1 | 80.12.c.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.12.b.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 10.12.b.a | ✓ | 6 | 5.b | even | 2 | 1 | inner |
| 50.12.a.i | 3 | 5.c | odd | 4 | 1 | ||
| 50.12.a.j | 3 | 5.c | odd | 4 | 1 | ||
| 80.12.c.c | 6 | 4.b | odd | 2 | 1 | ||
| 80.12.c.c | 6 | 20.d | odd | 2 | 1 | ||
| 90.12.c.b | 6 | 3.b | odd | 2 | 1 | ||
| 90.12.c.b | 6 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(10, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1024)^{3} \)
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| $3$ |
\( T^{6} + \cdots + 14\!\cdots\!04 \)
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| $5$ |
\( T^{6} + \cdots + 11\!\cdots\!25 \)
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| $7$ |
\( T^{6} + \cdots + 82\!\cdots\!56 \)
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| $11$ |
\( (T^{3} + \cdots - 11\!\cdots\!28)^{2} \)
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| $13$ |
\( T^{6} + \cdots + 42\!\cdots\!64 \)
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| $17$ |
\( T^{6} + \cdots + 79\!\cdots\!76 \)
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| $19$ |
\( (T^{3} + \cdots - 41\!\cdots\!00)^{2} \)
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| $23$ |
\( T^{6} + \cdots + 14\!\cdots\!24 \)
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| $29$ |
\( (T^{3} + \cdots + 26\!\cdots\!00)^{2} \)
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| $31$ |
\( (T^{3} + \cdots + 25\!\cdots\!32)^{2} \)
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| $37$ |
\( T^{6} + \cdots + 30\!\cdots\!16 \)
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| $41$ |
\( (T^{3} + \cdots + 54\!\cdots\!12)^{2} \)
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| $43$ |
\( T^{6} + \cdots + 18\!\cdots\!44 \)
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| $47$ |
\( T^{6} + \cdots + 23\!\cdots\!36 \)
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| $53$ |
\( T^{6} + \cdots + 10\!\cdots\!04 \)
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| $59$ |
\( (T^{3} + \cdots + 23\!\cdots\!00)^{2} \)
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| $61$ |
\( (T^{3} + \cdots + 54\!\cdots\!72)^{2} \)
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| $67$ |
\( T^{6} + \cdots + 57\!\cdots\!76 \)
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| $71$ |
\( (T^{3} + \cdots - 73\!\cdots\!48)^{2} \)
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| $73$ |
\( T^{6} + \cdots + 49\!\cdots\!24 \)
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| $79$ |
\( (T^{3} + \cdots + 26\!\cdots\!00)^{2} \)
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| $83$ |
\( T^{6} + \cdots + 87\!\cdots\!84 \)
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| $89$ |
\( (T^{3} + \cdots + 19\!\cdots\!00)^{2} \)
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| $97$ |
\( T^{6} + \cdots + 12\!\cdots\!36 \)
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