Newspace parameters
| Level: | \( N \) | \(=\) | \( 345 = 3 \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 345.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(20.3556589520\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 34x^{4} + 38x^{3} + 293x^{2} - 12x - 144 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 34x^{4} + 38x^{3} + 293x^{2} - 12x - 144 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 18\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} - 31\nu^{3} - 37\nu^{2} + 200\nu + 174 ) / 18 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{5} + 7\nu^{4} + 62\nu^{3} - 142\nu^{2} - 481\nu + 174 ) / 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 19\beta _1 + 10 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 4\beta_{4} + 24\beta_{2} + 33\beta _1 + 230 \)
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| \(\nu^{5}\) | \(=\) |
\( -\beta_{5} + 14\beta_{4} + 31\beta_{3} + 44\beta_{2} + 393\beta _1 + 350 \)
|
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 |
|
−5.33023 | −3.00000 | 20.4114 | 5.00000 | 15.9907 | 18.3327 | −66.1555 | 9.00000 | −26.6512 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −3.94479 | −3.00000 | 7.56134 | 5.00000 | 11.8344 | −23.8531 | 1.73043 | 9.00000 | −19.7239 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.73897 | −3.00000 | −4.97597 | 5.00000 | 5.21692 | −19.6129 | 22.5649 | 9.00000 | −8.69487 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.289779 | −3.00000 | −7.91603 | 5.00000 | 0.869338 | 4.21992 | 4.61214 | 9.00000 | −1.44890 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 3.30064 | −3.00000 | 2.89421 | 5.00000 | −9.90191 | 4.06396 | −16.8524 | 9.00000 | 16.5032 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.00313 | −3.00000 | 8.02508 | 5.00000 | −12.0094 | −23.1506 | 0.100409 | 9.00000 | 20.0157 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(23\) | \( +1 \) |
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 | 345.4.a.i | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1035.4.a.q | 6 | ||
| 5.b | even | 2 | 1 | 1725.4.a.x | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 345.4.a.i | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1035.4.a.q | 6 | 3.b | odd | 2 | 1 | ||
| 1725.4.a.x | 6 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 4T_{2}^{5} - 29T_{2}^{4} - 98T_{2}^{3} + 198T_{2}^{2} + 548T_{2} + 140 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(345))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 4 T^{5} + \cdots + 140 \)
|
| $3$ |
\( (T + 3)^{6} \)
|
| $5$ |
\( (T - 5)^{6} \)
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| $7$ |
\( T^{6} + 40 T^{5} + \cdots - 3405104 \)
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| $11$ |
\( T^{6} + 48 T^{5} + \cdots + 79041536 \)
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| $13$ |
\( T^{6} + \cdots - 17569769904 \)
|
| $17$ |
\( T^{6} + \cdots + 4644582060 \)
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| $19$ |
\( T^{6} + \cdots + 240989265664 \)
|
| $23$ |
\( (T + 23)^{6} \)
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| $29$ |
\( T^{6} + \cdots - 981064322660 \)
|
| $31$ |
\( T^{6} + \cdots + 14465095567616 \)
|
| $37$ |
\( T^{6} + \cdots + 667948862332 \)
|
| $41$ |
\( T^{6} + \cdots + 38665089298772 \)
|
| $43$ |
\( T^{6} + \cdots + 393309913017600 \)
|
| $47$ |
\( T^{6} + \cdots + 5540106113280 \)
|
| $53$ |
\( T^{6} + \cdots + 694355179374996 \)
|
| $59$ |
\( T^{6} + \cdots - 383618353311168 \)
|
| $61$ |
\( T^{6} + \cdots - 15\!\cdots\!72 \)
|
| $67$ |
\( T^{6} + \cdots + 30\!\cdots\!56 \)
|
| $71$ |
\( T^{6} + \cdots - 722494790804464 \)
|
| $73$ |
\( T^{6} + \cdots - 510701189362704 \)
|
| $79$ |
\( T^{6} + \cdots - 20\!\cdots\!40 \)
|
| $83$ |
\( T^{6} + \cdots + 133188280336656 \)
|
| $89$ |
\( T^{6} + \cdots + 86\!\cdots\!48 \)
|
| $97$ |
\( T^{6} + \cdots - 16\!\cdots\!44 \)
|
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