Newspace parameters
| Level: | \( N \) | \(=\) | \( 158 = 2 \cdot 79 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 158.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(25.3406435305\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{7} - x^{6} - 424x^{5} - 1337x^{4} + 29651x^{3} + 148738x^{2} - 123584x - 916096 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 424x^{5} - 1337x^{4} + 29651x^{3} + 148738x^{2} - 123584x - 916096 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 1142627 \nu^{6} - 6275971 \nu^{5} - 452741336 \nu^{4} + 490167493 \nu^{3} + 30254411881 \nu^{2} + \cdots - 218115501536 ) / 3043454000 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 2454079 \nu^{6} + 20672167 \nu^{5} + 913246272 \nu^{4} - 3471927161 \nu^{3} + \cdots + 462272418672 ) / 6086908000 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7024587 \nu^{6} - 45776051 \nu^{5} - 2724211616 \nu^{4} + 5432597133 \nu^{3} + \cdots - 1346908240816 ) / 6086908000 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1778481 \nu^{6} - 11828713 \nu^{5} - 674900608 \nu^{4} + 1382827479 \nu^{3} + \cdots - 230853083408 ) / 1217381600 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 596759 \nu^{6} + 4911707 \nu^{5} + 221330412 \nu^{4} - 827969081 \nu^{3} - 13177518177 \nu^{2} + \cdots + 94619921712 ) / 304345400 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 20204979 \nu^{6} - 152921867 \nu^{5} - 7584347472 \nu^{4} + 22979672261 \nu^{3} + \cdots - 2623925485872 ) / 6086908000 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -9\beta_{6} - 9\beta_{5} + 6\beta_{4} + 13\beta_{3} + 10\beta_{2} - 20\beta _1 + 740 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -21\beta_{6} - 40\beta_{5} + 4\beta_{4} + 111\beta_{3} + 176\beta_{2} - 191\beta _1 + 1649 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -3300\beta_{6} - 3624\beta_{5} + 2598\beta_{4} + 5525\beta_{3} + 6467\beta_{2} - 9898\beta _1 + 218866 ) / 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 38145 \beta_{6} - 59979 \beta_{5} + 18612 \beta_{4} + 130469 \beta_{3} + 212264 \beta_{2} + \cdots + 2607346 ) / 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 417247 \beta_{6} - 491864 \beta_{5} + 322538 \beta_{4} + 796643 \beta_{3} + 1069408 \beta_{2} + \cdots + 26804639 ) / 2 \)
|
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 |
|
−4.00000 | −26.2667 | 16.0000 | −80.1028 | 105.067 | −64.9267 | −64.0000 | 446.941 | 320.411 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.00000 | −18.7272 | 16.0000 | 2.23983 | 74.9088 | −36.2874 | −64.0000 | 107.708 | −8.95932 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.00000 | −11.0374 | 16.0000 | 98.5110 | 44.1494 | −177.207 | −64.0000 | −121.177 | −394.044 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −4.00000 | −3.80292 | 16.0000 | 19.7521 | 15.2117 | 78.5412 | −64.0000 | −228.538 | −79.0086 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | −4.00000 | 8.80993 | 16.0000 | 81.5613 | −35.2397 | −2.47382 | −64.0000 | −165.385 | −326.245 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | −4.00000 | 14.7900 | 16.0000 | −31.3434 | −59.1599 | 55.4706 | −64.0000 | −24.2567 | 125.374 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | −4.00000 | 27.2343 | 16.0000 | −59.6181 | −108.937 | −88.1168 | −64.0000 | 498.707 | 238.472 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(79\) | \( +1 \) |
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 | 158.6.a.c | ✓ | 7 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 158.6.a.c | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} + 9T_{3}^{6} - 1067T_{3}^{5} - 9189T_{3}^{4} + 270094T_{3}^{3} + 1789416T_{3}^{2} - 16645608T_{3} - 73268496 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(158))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{7} \)
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| $3$ |
\( T^{7} + 9 T^{6} + \cdots - 73268496 \)
|
| $5$ |
\( T^{7} + \cdots + 53207103217 \)
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| $7$ |
\( T^{7} + \cdots + 396503830080 \)
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| $11$ |
\( T^{7} + \cdots - 628877735535592 \)
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| $13$ |
\( T^{7} + \cdots - 87\!\cdots\!49 \)
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| $17$ |
\( T^{7} + \cdots + 15\!\cdots\!36 \)
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| $19$ |
\( T^{7} + \cdots + 15\!\cdots\!00 \)
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| $23$ |
\( T^{7} + \cdots + 14\!\cdots\!00 \)
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| $29$ |
\( T^{7} + \cdots + 59\!\cdots\!80 \)
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| $31$ |
\( T^{7} + \cdots - 10\!\cdots\!24 \)
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| $37$ |
\( T^{7} + \cdots + 63\!\cdots\!48 \)
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| $41$ |
\( T^{7} + \cdots - 16\!\cdots\!80 \)
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| $43$ |
\( T^{7} + \cdots - 39\!\cdots\!00 \)
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| $47$ |
\( T^{7} + \cdots + 30\!\cdots\!00 \)
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| $53$ |
\( T^{7} + \cdots - 44\!\cdots\!24 \)
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| $59$ |
\( T^{7} + \cdots - 39\!\cdots\!80 \)
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| $61$ |
\( T^{7} + \cdots - 20\!\cdots\!00 \)
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| $67$ |
\( T^{7} + \cdots - 18\!\cdots\!00 \)
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| $71$ |
\( T^{7} + \cdots + 73\!\cdots\!20 \)
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| $73$ |
\( T^{7} + \cdots + 17\!\cdots\!48 \)
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| $79$ |
\( (T + 6241)^{7} \)
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| $83$ |
\( T^{7} + \cdots + 65\!\cdots\!20 \)
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| $89$ |
\( T^{7} + \cdots + 20\!\cdots\!55 \)
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| $97$ |
\( T^{7} + \cdots + 37\!\cdots\!95 \)
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