Newspace parameters
| Level: | \( N \) | \(=\) | \( 18 = 2 \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 46 \) |
| Character orbit: | \([\chi]\) | \(=\) | 18.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(230.860306934\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + \cdots - 31\!\cdots\!80 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{25}\cdot 3^{26}\cdot 5\cdot 7^{2}\cdot 11 \) |
| 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
\( x^{4} - x^{3} + \cdots - 31\!\cdots\!80 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 46050393747888 \nu^{3} + \cdots - 77\!\cdots\!60 ) / 67\!\cdots\!45 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 366920244333792 \nu^{3} + \cdots - 10\!\cdots\!00 ) / 67\!\cdots\!45 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 44\!\cdots\!92 \nu^{3} + \cdots + 64\!\cdots\!40 ) / 96\!\cdots\!35 \)
|
| \(\nu\) | \(=\) |
\( ( 29\beta_{3} + 27931\beta_{2} - 19296698\beta _1 + 1257091799040 ) / 5028367196160 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 1661845552603 \beta_{3} + \cdots + 47\!\cdots\!40 ) / 1676122398720 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 15\!\cdots\!19 \beta_{3} + \cdots - 14\!\cdots\!60 ) / 457124290560 \)
|
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 |
|
−4.19430e6 | 0 | 1.75922e13 | −9.60262e15 | 0 | −1.99306e19 | −7.37870e19 | 0 | 4.02763e22 | ||||||||||||||||||||||||||||||
| 1.2 | −4.19430e6 | 0 | 1.75922e13 | −4.39541e15 | 0 | 1.69267e19 | −7.37870e19 | 0 | 1.84357e22 | |||||||||||||||||||||||||||||||
| 1.3 | −4.19430e6 | 0 | 1.75922e13 | 1.65959e15 | 0 | −1.80250e18 | −7.37870e19 | 0 | −6.96082e21 | |||||||||||||||||||||||||||||||
| 1.4 | −4.19430e6 | 0 | 1.75922e13 | 8.51555e15 | 0 | −2.76759e18 | −7.37870e19 | 0 | −3.57168e22 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
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 | 18.46.a.g | ✓ | 4 |
| 3.b | odd | 2 | 1 | 18.46.a.h | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.46.a.g | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 18.46.a.h | yes | 4 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + \cdots + 59\!\cdots\!00 \)
acting on \(S_{46}^{\mathrm{new}}(\Gamma_0(18))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4194304)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 59\!\cdots\!00 \)
|
| $7$ |
\( T^{4} + \cdots - 16\!\cdots\!84 \)
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| $11$ |
\( T^{4} + \cdots - 10\!\cdots\!24 \)
|
| $13$ |
\( T^{4} + \cdots - 15\!\cdots\!24 \)
|
| $17$ |
\( T^{4} + \cdots + 30\!\cdots\!36 \)
|
| $19$ |
\( T^{4} + \cdots + 34\!\cdots\!96 \)
|
| $23$ |
\( T^{4} + \cdots + 22\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots - 83\!\cdots\!44 \)
|
| $31$ |
\( T^{4} + \cdots - 56\!\cdots\!04 \)
|
| $37$ |
\( T^{4} + \cdots - 79\!\cdots\!84 \)
|
| $41$ |
\( T^{4} + \cdots + 71\!\cdots\!00 \)
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| $43$ |
\( T^{4} + \cdots - 40\!\cdots\!04 \)
|
| $47$ |
\( T^{4} + \cdots + 20\!\cdots\!00 \)
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| $53$ |
\( T^{4} + \cdots - 45\!\cdots\!44 \)
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| $59$ |
\( T^{4} + \cdots + 34\!\cdots\!16 \)
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| $61$ |
\( T^{4} + \cdots + 61\!\cdots\!96 \)
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| $67$ |
\( T^{4} + \cdots + 61\!\cdots\!96 \)
|
| $71$ |
\( T^{4} + \cdots - 49\!\cdots\!04 \)
|
| $73$ |
\( T^{4} + \cdots + 73\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 38\!\cdots\!16 \)
|
| $83$ |
\( T^{4} + \cdots - 95\!\cdots\!24 \)
|
| $89$ |
\( T^{4} + \cdots - 52\!\cdots\!84 \)
|
| $97$ |
\( T^{4} + \cdots + 57\!\cdots\!76 \)
|
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