Newspace parameters
| Level: | \( N \) | \(=\) | \( 102 = 2 \cdot 3 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 102.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(52.5336552887\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 520688x^{2} - 146431260x - 953767152 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 520688x^{2} - 146431260x - 953767152 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 553\nu^{2} + 184628\nu - 33817968 ) / 30804 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 553\nu^{2} + 307844\nu - 33838504 ) / 10268 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -127\nu^{3} + 49695\nu^{2} + 45729316\nu + 1048010256 ) / 154020 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 3\beta _1 + 2 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -90\beta_{3} + 1085\beta_{2} - 969\beta _1 + 3124234 ) / 12 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -49770\beta_{3} + 784633\beta_{2} - 1459389\beta _1 + 1322255042 ) / 12 \)
|
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 |
|
−16.0000 | −81.0000 | 256.000 | −2454.93 | 1296.00 | −11429.5 | −4096.00 | 6561.00 | 39278.8 | ||||||||||||||||||||||||||||||
| 1.2 | −16.0000 | −81.0000 | 256.000 | −1645.64 | 1296.00 | 3480.01 | −4096.00 | 6561.00 | 26330.3 | |||||||||||||||||||||||||||||||
| 1.3 | −16.0000 | −81.0000 | 256.000 | 901.022 | 1296.00 | −1949.06 | −4096.00 | 6561.00 | −14416.4 | |||||||||||||||||||||||||||||||
| 1.4 | −16.0000 | −81.0000 | 256.000 | 2257.54 | 1296.00 | 7814.54 | −4096.00 | 6561.00 | −36120.7 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(17\) | \( +1 \) |
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 | 102.10.a.h | ✓ | 4 |
| 3.b | odd | 2 | 1 | 306.10.a.m | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 102.10.a.h | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 306.10.a.m | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 942T_{5}^{3} - 6877887T_{5}^{2} - 4419421676T_{5} + 8217602109120 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(102))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 16)^{4} \)
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| $3$ |
\( (T + 81)^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 8217602109120 \)
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| $7$ |
\( T^{4} + \cdots + 605810903968704 \)
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| $11$ |
\( T^{4} + \cdots + 97\!\cdots\!00 \)
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| $13$ |
\( T^{4} + \cdots + 72\!\cdots\!60 \)
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| $17$ |
\( (T + 83521)^{4} \)
|
| $19$ |
\( T^{4} + \cdots + 39\!\cdots\!76 \)
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| $23$ |
\( T^{4} + \cdots - 20\!\cdots\!64 \)
|
| $29$ |
\( T^{4} + \cdots + 66\!\cdots\!76 \)
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| $31$ |
\( T^{4} + \cdots + 59\!\cdots\!56 \)
|
| $37$ |
\( T^{4} + \cdots + 22\!\cdots\!76 \)
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| $41$ |
\( T^{4} + \cdots - 31\!\cdots\!00 \)
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| $43$ |
\( T^{4} + \cdots + 16\!\cdots\!92 \)
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| $47$ |
\( T^{4} + \cdots + 42\!\cdots\!24 \)
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| $53$ |
\( T^{4} + \cdots + 89\!\cdots\!52 \)
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| $59$ |
\( T^{4} + \cdots - 74\!\cdots\!36 \)
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| $61$ |
\( T^{4} + \cdots + 11\!\cdots\!48 \)
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| $67$ |
\( T^{4} + \cdots + 76\!\cdots\!08 \)
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| $71$ |
\( T^{4} + \cdots - 48\!\cdots\!88 \)
|
| $73$ |
\( T^{4} + \cdots - 20\!\cdots\!32 \)
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| $79$ |
\( T^{4} + \cdots + 46\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 12\!\cdots\!40 \)
|
| $89$ |
\( T^{4} + \cdots + 27\!\cdots\!32 \)
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| $97$ |
\( T^{4} + \cdots - 32\!\cdots\!52 \)
|
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