Newspace parameters
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 20 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(6.86450089669\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{87481}) \) |
| Defining polynomial: |
\( x^{2} - x - 21870 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{87481}\). 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 |
|
−536.316 | −19683.0 | −236654. | −683163. | 1.05563e7 | 1.13677e8 | 4.08105e8 | 3.87420e8 | 3.66391e8 | ||||||||||||||||||||||||
| 1.2 | 1238.32 | −19683.0 | 1.00914e6 | 6.69930e6 | −2.43738e7 | 214621. | 6.00397e8 | 3.87420e8 | 8.29585e9 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
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 | 3.20.a.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 9.20.a.c | 2 | ||
| 4.b | odd | 2 | 1 | 48.20.a.j | 2 | ||
| 5.b | even | 2 | 1 | 75.20.a.b | 2 | ||
| 5.c | odd | 4 | 2 | 75.20.b.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.20.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 9.20.a.c | 2 | 3.b | odd | 2 | 1 | ||
| 48.20.a.j | 2 | 4.b | odd | 2 | 1 | ||
| 75.20.a.b | 2 | 5.b | even | 2 | 1 | ||
| 75.20.b.b | 4 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 702T_{2} - 664128 \)
acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 702T - 664128 \)
$3$
\( (T + 19683)^{2} \)
$5$
\( T^{2} - 6016140 T - 4576715617500 \)
$7$
\( T^{2} - 113892064 T + 24397550813440 \)
$11$
\( T^{2} + 6650071272 T - 84\!\cdots\!28 \)
$13$
\( T^{2} + 44072356148 T - 77\!\cdots\!08 \)
$17$
\( T^{2} - 336281471748 T - 23\!\cdots\!24 \)
$19$
\( T^{2} - 602118925096 T - 75\!\cdots\!20 \)
$23$
\( T^{2} - 2368252165968 T - 13\!\cdots\!80 \)
$29$
\( T^{2} - 280977251970492 T + 19\!\cdots\!80 \)
$31$
\( T^{2} - 41610149253712 T + 91\!\cdots\!00 \)
$37$
\( T^{2} - 637994163989884 T - 97\!\cdots\!80 \)
$41$
\( T^{2} + \cdots - 99\!\cdots\!60 \)
$43$
\( T^{2} + \cdots + 83\!\cdots\!76 \)
$47$
\( T^{2} + \cdots - 97\!\cdots\!76 \)
$53$
\( T^{2} + \cdots + 17\!\cdots\!80 \)
$59$
\( T^{2} + \cdots + 76\!\cdots\!40 \)
$61$
\( T^{2} + \cdots + 18\!\cdots\!16 \)
$67$
\( T^{2} + \cdots + 60\!\cdots\!76 \)
$71$
\( T^{2} + \cdots - 75\!\cdots\!16 \)
$73$
\( T^{2} + \cdots - 23\!\cdots\!00 \)
$79$
\( T^{2} + \cdots - 12\!\cdots\!00 \)
$83$
\( T^{2} + \cdots - 53\!\cdots\!12 \)
$89$
\( T^{2} + \cdots - 82\!\cdots\!00 \)
$97$
\( T^{2} + \cdots + 10\!\cdots\!16 \)
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