Newspace parameters
| Level: | \( N \) | \(=\) | \( 74 = 2 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 74.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(11.8684026662\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.324233.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - 77x - 140 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 77x - 140 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 52 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{2} + \beta _1 + 104 ) / 2 \)
|
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 |
|
−4.00000 | −25.5252 | 16.0000 | −14.0000 | 102.101 | 203.072 | −64.0000 | 408.537 | 56.0000 | |||||||||||||||||||||||||||
| 1.2 | −4.00000 | 1.11745 | 16.0000 | −14.0000 | −4.46979 | 32.2315 | −64.0000 | −241.751 | 56.0000 | ||||||||||||||||||||||||||||
| 1.3 | −4.00000 | 16.4078 | 16.0000 | −14.0000 | −65.6311 | −151.304 | −64.0000 | 26.2148 | 56.0000 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(37\) | \( +1 \) |
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 | 74.6.a.c | ✓ | 3 |
| 3.b | odd | 2 | 1 | 666.6.a.i | 3 | ||
| 4.b | odd | 2 | 1 | 592.6.a.c | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 74.6.a.c | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 592.6.a.c | 3 | 4.b | odd | 2 | 1 | ||
| 666.6.a.i | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 8T_{3}^{2} - 429T_{3} + 468 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(74))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{3} \)
|
| $3$ |
\( T^{3} + 8 T^{2} + \cdots + 468 \)
|
| $5$ |
\( (T + 14)^{3} \)
|
| $7$ |
\( T^{3} - 84 T^{2} + \cdots + 990332 \)
|
| $11$ |
\( T^{3} - 304 T^{2} + \cdots + 11483660 \)
|
| $13$ |
\( T^{3} + 806 T^{2} + \cdots - 922776 \)
|
| $17$ |
\( T^{3} + \cdots + 2114271552 \)
|
| $19$ |
\( T^{3} + \cdots + 2579057312 \)
|
| $23$ |
\( T^{3} + \cdots - 1430959880 \)
|
| $29$ |
\( T^{3} + \cdots + 23321731784 \)
|
| $31$ |
\( T^{3} + \cdots - 41137100320 \)
|
| $37$ |
\( (T + 1369)^{3} \)
|
| $41$ |
\( T^{3} + \cdots - 237018444942 \)
|
| $43$ |
\( T^{3} + \cdots - 531931362712 \)
|
| $47$ |
\( T^{3} + \cdots + 2444798953316 \)
|
| $53$ |
\( T^{3} + \cdots + 246191549806 \)
|
| $59$ |
\( T^{3} + \cdots - 3503222325504 \)
|
| $61$ |
\( T^{3} + \cdots - 1303290680000 \)
|
| $67$ |
\( T^{3} + \cdots - 103615970816 \)
|
| $71$ |
\( T^{3} + \cdots - 29653094181780 \)
|
| $73$ |
\( T^{3} + \cdots - 38779212138142 \)
|
| $79$ |
\( T^{3} + \cdots - 696052558500480 \)
|
| $83$ |
\( T^{3} + \cdots - 11\!\cdots\!48 \)
|
| $89$ |
\( T^{3} + \cdots - 5026391398400 \)
|
| $97$ |
\( T^{3} + \cdots + 730628639852224 \)
|
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