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: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 116x + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 116x + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 19\nu - 8 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 23\nu^{2} + 30\nu + 56 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 19\beta _1 + 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 4\beta_{3} + 23\beta_{2} + 31\beta _1 + 213 \)
|
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 |
|
−4.06714 | −3.00000 | 8.54164 | 5.00000 | 12.2014 | 12.7154 | −2.20294 | 9.00000 | −20.3357 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.64782 | −3.00000 | −5.28470 | 5.00000 | 4.94345 | 11.3892 | 21.8907 | 9.00000 | −8.23908 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.63251 | −3.00000 | −5.33490 | 5.00000 | −4.89754 | 32.5794 | −21.7694 | 9.00000 | 8.16257 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.73451 | −3.00000 | −0.522481 | 5.00000 | −8.20352 | −9.34911 | −23.3048 | 9.00000 | 13.6725 | ||||||||||||||||||||||||||||||||||
| 1.5 | 5.34794 | −3.00000 | 20.6004 | 5.00000 | −16.0438 | −1.33485 | 67.3864 | 9.00000 | 26.7397 | ||||||||||||||||||||||||||||||||||
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.h | ✓ | 5 |
| 3.b | odd | 2 | 1 | 1035.4.a.l | 5 | ||
| 5.b | even | 2 | 1 | 1725.4.a.t | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 345.4.a.h | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1035.4.a.l | 5 | 3.b | odd | 2 | 1 | ||
| 1725.4.a.t | 5 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 4T_{2}^{4} - 21T_{2}^{3} + 70T_{2}^{2} + 50T_{2} - 160 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(345))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 4 T^{4} + \cdots - 160 \)
|
| $3$ |
\( (T + 3)^{5} \)
|
| $5$ |
\( (T - 5)^{5} \)
|
| $7$ |
\( T^{5} - 46 T^{4} + \cdots - 58880 \)
|
| $11$ |
\( T^{5} - 42 T^{4} + \cdots - 4736000 \)
|
| $13$ |
\( T^{5} + 16 T^{4} + \cdots - 15301120 \)
|
| $17$ |
\( T^{5} + \cdots + 1797586600 \)
|
| $19$ |
\( T^{5} - 94 T^{4} + \cdots - 62264320 \)
|
| $23$ |
\( (T - 23)^{5} \)
|
| $29$ |
\( T^{5} + \cdots - 8170679360 \)
|
| $31$ |
\( T^{5} + \cdots + 3767990272 \)
|
| $37$ |
\( T^{5} + \cdots + 184774204800 \)
|
| $41$ |
\( T^{5} + \cdots + 1532412152 \)
|
| $43$ |
\( T^{5} + \cdots + 968409833472 \)
|
| $47$ |
\( T^{5} + \cdots + 278902558720 \)
|
| $53$ |
\( T^{5} + \cdots - 2953885614840 \)
|
| $59$ |
\( T^{5} + \cdots - 70032517376 \)
|
| $61$ |
\( T^{5} + \cdots + 4949359730240 \)
|
| $67$ |
\( T^{5} + \cdots - 1975638939264 \)
|
| $71$ |
\( T^{5} + \cdots + 67193285589632 \)
|
| $73$ |
\( T^{5} + \cdots + 27257990767776 \)
|
| $79$ |
\( T^{5} + \cdots + 2634349985792 \)
|
| $83$ |
\( T^{5} + \cdots + 180582360197120 \)
|
| $89$ |
\( T^{5} + \cdots - 99556146755584 \)
|
| $97$ |
\( T^{5} + \cdots + 160070991245312 \)
|
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