Error: table mf_hecke_newspace_traces does not exist
Newspace parameters
| Level: | \( N \) | \(=\) | \( 152 = 2^{3} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 152.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(24.3783406116\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 1012x^{4} + 2938x^{3} + 184643x^{2} - 1214504x + 729856 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 1012x^{4} + 2938x^{3} + 184643x^{2} - 1214504x + 729856 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 44\nu^{5} - 137\nu^{4} - 38767\nu^{3} + 218113\nu^{2} + 4844027\nu - 60205124 ) / 423036 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -169\nu^{5} - 275\nu^{4} + 162521\nu^{3} + 220639\nu^{2} - 24714652\nu + 55807456 ) / 846072 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -91\nu^{5} - 1052\nu^{4} + 79376\nu^{3} + 936856\nu^{2} - 7485721\nu - 55818908 ) / 423036 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 171\nu^{5} + 1159\nu^{4} - 163749\nu^{3} - 854003\nu^{2} + 24374526\nu - 33927488 ) / 282024 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 3\beta_{3} + 2\beta_{2} - 4\beta _1 + 339 \)
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| \(\nu^{3}\) | \(=\) |
\( -16\beta_{5} - 34\beta_{4} + 12\beta_{3} + 46\beta_{2} + 605\beta _1 - 656 \)
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| \(\nu^{4}\) | \(=\) |
\( 1049\beta_{5} + 743\beta_{4} + 3111\beta_{3} + 1396\beta_{2} - 2624\beta _1 + 217703 \)
|
| \(\nu^{5}\) | \(=\) |
\( -15788\beta_{5} - 32600\beta_{4} + 5388\beta_{3} + 44576\beta_{2} + 434613\beta _1 - 212296 \)
|
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 |
|
0 | −28.6020 | 0 | −47.0403 | 0 | 92.6779 | 0 | 575.077 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −18.4790 | 0 | 70.0177 | 0 | −152.088 | 0 | 98.4744 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.330307 | 0 | −50.5728 | 0 | 153.008 | 0 | −242.891 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 6.12757 | 0 | 59.9535 | 0 | −44.7227 | 0 | −205.453 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 10.3428 | 0 | −9.61696 | 0 | 11.2548 | 0 | −136.027 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 26.9410 | 0 | −47.7412 | 0 | −221.130 | 0 | 482.820 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(19\) | \( +1 \) |
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 | 152.6.a.c | ✓ | 6 |
| 4.b | odd | 2 | 1 | 304.6.a.n | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 152.6.a.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 304.6.a.n | 6 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 4T_{3}^{5} - 1007T_{3}^{4} - 1110T_{3}^{3} + 187380T_{3}^{2} - 840456T_{3} - 298080 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(152))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 4 T^{5} + \cdots - 298080 \)
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| $5$ |
\( T^{6} + \cdots + 4585004992 \)
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| $7$ |
\( T^{6} + \cdots - 240049768320 \)
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| $11$ |
\( T^{6} + \cdots - 795454171823952 \)
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| $13$ |
\( T^{6} + \cdots - 20\!\cdots\!24 \)
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| $17$ |
\( T^{6} + \cdots + 96\!\cdots\!02 \)
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| $19$ |
\( (T + 361)^{6} \)
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| $23$ |
\( T^{6} + \cdots + 42\!\cdots\!08 \)
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| $29$ |
\( T^{6} + \cdots + 12\!\cdots\!32 \)
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| $31$ |
\( T^{6} + \cdots + 83\!\cdots\!00 \)
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| $37$ |
\( T^{6} + \cdots + 26\!\cdots\!52 \)
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| $41$ |
\( T^{6} + \cdots + 21\!\cdots\!24 \)
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| $43$ |
\( T^{6} + \cdots + 29\!\cdots\!76 \)
|
| $47$ |
\( T^{6} + \cdots + 13\!\cdots\!16 \)
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| $53$ |
\( T^{6} + \cdots - 10\!\cdots\!84 \)
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| $59$ |
\( T^{6} + \cdots + 11\!\cdots\!68 \)
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| $61$ |
\( T^{6} + \cdots + 54\!\cdots\!36 \)
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| $67$ |
\( T^{6} + \cdots + 62\!\cdots\!92 \)
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| $71$ |
\( T^{6} + \cdots - 35\!\cdots\!48 \)
|
| $73$ |
\( T^{6} + \cdots - 24\!\cdots\!78 \)
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| $79$ |
\( T^{6} + \cdots + 17\!\cdots\!72 \)
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| $83$ |
\( T^{6} + \cdots + 21\!\cdots\!76 \)
|
| $89$ |
\( T^{6} + \cdots + 59\!\cdots\!08 \)
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| $97$ |
\( T^{6} + \cdots + 81\!\cdots\!92 \)
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