Newspace parameters
| Level: | \( N \) | \(=\) | \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 840.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(49.5616044048\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.821313.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 117x - 340 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
\( x^{3} - 117x - 340 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + \nu - 78 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 7\nu - 78 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 7\beta _1 + 624 ) / 8 \)
|
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 |
|
0 | −3.00000 | 0 | −5.00000 | 0 | −7.00000 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −3.00000 | 0 | −5.00000 | 0 | −7.00000 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | −3.00000 | 0 | −5.00000 | 0 | −7.00000 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(7\) | \( +1 \) |
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 | 840.4.a.p | ✓ | 3 |
| 4.b | odd | 2 | 1 | 1680.4.a.bs | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 840.4.a.p | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 1680.4.a.bs | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{3} + 6T_{11}^{2} - 3144T_{11} + 42752 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(840))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T + 3)^{3} \)
|
| $5$ |
\( (T + 5)^{3} \)
|
| $7$ |
\( (T + 7)^{3} \)
|
| $11$ |
\( T^{3} + 6 T^{2} + \cdots + 42752 \)
|
| $13$ |
\( T^{3} - 36 T^{2} + \cdots - 12912 \)
|
| $17$ |
\( T^{3} - 90 T^{2} + \cdots + 675112 \)
|
| $19$ |
\( T^{3} - 78 T^{2} + \cdots + 1928192 \)
|
| $23$ |
\( T^{3} - 7488 T - 174080 \)
|
| $29$ |
\( T^{3} - 162 T^{2} + \cdots + 916680 \)
|
| $31$ |
\( T^{3} - 114 T^{2} + \cdots + 115520 \)
|
| $37$ |
\( T^{3} - 246 T^{2} + \cdots + 9674936 \)
|
| $41$ |
\( T^{3} + 42 T^{2} + \cdots - 689768 \)
|
| $43$ |
\( T^{3} + 348 T^{2} + \cdots - 98481408 \)
|
| $47$ |
\( T^{3} + 480 T^{2} + \cdots - 2392704 \)
|
| $53$ |
\( T^{3} + 252 T^{2} + \cdots - 49294384 \)
|
| $59$ |
\( T^{3} + 408 T^{2} + \cdots - 171481216 \)
|
| $61$ |
\( T^{3} + 534 T^{2} + \cdots + 3668520 \)
|
| $67$ |
\( T^{3} + 372 T^{2} + \cdots - 133798400 \)
|
| $71$ |
\( T^{3} + 366 T^{2} + \cdots - 23564800 \)
|
| $73$ |
\( T^{3} + 60 T^{2} + \cdots + 35106768 \)
|
| $79$ |
\( T^{3} + 624 T^{2} + \cdots - 244063232 \)
|
| $83$ |
\( T^{3} + \cdots + 1064186432 \)
|
| $89$ |
\( T^{3} + 510 T^{2} + \cdots + 226497000 \)
|
| $97$ |
\( T^{3} + 2304 T^{2} + \cdots + 827040 \)
|
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