Newspace parameters
| Level: | \( N \) | \(=\) | \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 390.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(200.863976104\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{4} - x^{3} - 103194x^{2} + 6753414x - 65794500 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3\cdot 5 \) |
| 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
\( x^{4} - x^{3} - 103194x^{2} + 6753414x - 65794500 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 49\nu^{3} - 2413\nu^{2} - 2613312\nu + 369456608 ) / 192914 \)
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| \(\beta_{2}\) | \(=\) |
\( ( 221\nu^{3} + 12739\nu^{2} - 21345656\nu + 450312956 ) / 192914 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 605\nu^{3} + 1703\nu^{2} - 65589344\nu + 2946160850 ) / 192914 \)
|
| \(\nu\) | \(=\) |
\( ( -3\beta_{3} + 4\beta_{2} + 19\beta _1 + 91 ) / 320 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -607\beta_{3} + 2116\beta_{2} - 2049\beta _1 + 8254839 ) / 160 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -109891\beta_{3} + 210868\beta_{2} + 1035683\beta _1 - 797453573 ) / 160 \)
|
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 | 625.000 | −1296.00 | −11809.6 | −4096.00 | 6561.00 | −10000.0 | ||||||||||||||||||||||||||||||
| 1.2 | −16.0000 | 81.0000 | 256.000 | 625.000 | −1296.00 | −1281.24 | −4096.00 | 6561.00 | −10000.0 | |||||||||||||||||||||||||||||||
| 1.3 | −16.0000 | 81.0000 | 256.000 | 625.000 | −1296.00 | 2491.76 | −4096.00 | 6561.00 | −10000.0 | |||||||||||||||||||||||||||||||
| 1.4 | −16.0000 | 81.0000 | 256.000 | 625.000 | −1296.00 | 8281.12 | −4096.00 | 6561.00 | −10000.0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(13\) | \( +1 \) |
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 | 390.10.a.e | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 390.10.a.e | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 2318T_{7}^{3} - 105260859T_{7}^{2} + 107120277040T_{7} + 312219392439500 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(390))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 16)^{4} \)
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| $3$ |
\( (T - 81)^{4} \)
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| $5$ |
\( (T - 625)^{4} \)
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| $7$ |
\( T^{4} + \cdots + 312219392439500 \)
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| $11$ |
\( T^{4} + \cdots + 29\!\cdots\!24 \)
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| $13$ |
\( (T + 28561)^{4} \)
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| $17$ |
\( T^{4} + \cdots + 17\!\cdots\!60 \)
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| $19$ |
\( T^{4} + \cdots + 59\!\cdots\!16 \)
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| $23$ |
\( T^{4} + \cdots + 10\!\cdots\!48 \)
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| $29$ |
\( T^{4} + \cdots - 45\!\cdots\!92 \)
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| $31$ |
\( T^{4} + \cdots - 72\!\cdots\!32 \)
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| $37$ |
\( T^{4} + \cdots + 32\!\cdots\!00 \)
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| $41$ |
\( T^{4} + \cdots - 64\!\cdots\!44 \)
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| $43$ |
\( T^{4} + \cdots + 17\!\cdots\!48 \)
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| $47$ |
\( T^{4} + \cdots + 74\!\cdots\!20 \)
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| $53$ |
\( T^{4} + \cdots - 13\!\cdots\!00 \)
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| $59$ |
\( T^{4} + \cdots - 12\!\cdots\!40 \)
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| $61$ |
\( T^{4} + \cdots + 19\!\cdots\!00 \)
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| $67$ |
\( T^{4} + \cdots + 16\!\cdots\!52 \)
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| $71$ |
\( T^{4} + \cdots - 24\!\cdots\!72 \)
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| $73$ |
\( T^{4} + \cdots - 27\!\cdots\!00 \)
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| $79$ |
\( T^{4} + \cdots - 11\!\cdots\!64 \)
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| $83$ |
\( T^{4} + \cdots + 85\!\cdots\!08 \)
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| $89$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
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| $97$ |
\( T^{4} + \cdots - 40\!\cdots\!92 \)
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