Newspace parameters
| Level: | \( N \) | \(=\) | \( 342 = 2 \cdot 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 342.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(20.1786532220\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{73}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 38) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{73})\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−2.00000 | 0 | 4.00000 | −8.31601 | 0 | 8.08801 | −8.00000 | 0 | 16.6320 | ||||||||||||||||||||||||
| 1.2 | −2.00000 | 0 | 4.00000 | 17.3160 | 0 | −26.0880 | −8.00000 | 0 | −34.6320 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 342.4.a.h | 2 | |
| 3.b | odd | 2 | 1 | 38.4.a.c | ✓ | 2 | |
| 12.b | even | 2 | 1 | 304.4.a.c | 2 | ||
| 15.d | odd | 2 | 1 | 950.4.a.e | 2 | ||
| 15.e | even | 4 | 2 | 950.4.b.i | 4 | ||
| 21.c | even | 2 | 1 | 1862.4.a.e | 2 | ||
| 24.f | even | 2 | 1 | 1216.4.a.p | 2 | ||
| 24.h | odd | 2 | 1 | 1216.4.a.g | 2 | ||
| 57.d | even | 2 | 1 | 722.4.a.f | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.4.a.c | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 304.4.a.c | 2 | 12.b | even | 2 | 1 | ||
| 342.4.a.h | 2 | 1.a | even | 1 | 1 | trivial | |
| 722.4.a.f | 2 | 57.d | even | 2 | 1 | ||
| 950.4.a.e | 2 | 15.d | odd | 2 | 1 | ||
| 950.4.b.i | 4 | 15.e | even | 4 | 2 | ||
| 1216.4.a.g | 2 | 24.h | odd | 2 | 1 | ||
| 1216.4.a.p | 2 | 24.f | even | 2 | 1 | ||
| 1862.4.a.e | 2 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 9T_{5} - 144 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(342))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T + 2)^{2} \)
$3$
\( T^{2} \)
$5$
\( T^{2} - 9T - 144 \)
$7$
\( T^{2} + 18T - 211 \)
$11$
\( T^{2} - 17T + 54 \)
$13$
\( T^{2} - 17T - 3012 \)
$17$
\( T^{2} - 80T + 1527 \)
$19$
\( (T - 19)^{2} \)
$23$
\( T^{2} + 73T - 1752 \)
$29$
\( T^{2} + 3T - 8046 \)
$31$
\( T^{2} - 212T - 24096 \)
$37$
\( T^{2} - 192T - 5092 \)
$41$
\( T^{2} - 50T - 1200 \)
$43$
\( T^{2} - 677T + 113688 \)
$47$
\( T^{2} - 389T - 54168 \)
$53$
\( T^{2} - 1219 T + 366216 \)
$59$
\( T^{2} - 287T + 9186 \)
$61$
\( T^{2} - 313T - 200366 \)
$67$
\( T^{2} - 1223 T + 265728 \)
$71$
\( T^{2} + 200T - 235572 \)
$73$
\( T^{2} - 378T - 582151 \)
$79$
\( T^{2} - 1350 T + 394232 \)
$83$
\( T^{2} - 670T - 574632 \)
$89$
\( T^{2} - 236T - 631104 \)
$97$
\( T^{2} - 1294 T + 228736 \)
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