Newspace parameters
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(3.21692786856\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{1969}) \) |
| Defining polynomial: |
\( x^{2} - x - 492 \)
|
| 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 \(\beta = 3\sqrt{1969}\). 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 |
|
−160.120 | 729.000 | 17446.5 | 3318.61 | −116728. | 449560. | −1.48183e6 | 531441. | −531376. | ||||||||||||||||||||||||
| 1.2 | 106.120 | 729.000 | 3069.51 | 37397.4 | 77361.7 | −470568. | −543600. | 531441. | 3.96862e6 | |||||||||||||||||||||||||
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 | 3.14.a.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 9.14.a.c | 2 | ||
| 4.b | odd | 2 | 1 | 48.14.a.g | 2 | ||
| 5.b | even | 2 | 1 | 75.14.a.e | 2 | ||
| 5.c | odd | 4 | 2 | 75.14.b.c | 4 | ||
| 7.b | odd | 2 | 1 | 147.14.a.b | 2 | ||
| 8.b | even | 2 | 1 | 192.14.a.k | 2 | ||
| 8.d | odd | 2 | 1 | 192.14.a.o | 2 | ||
| 12.b | even | 2 | 1 | 144.14.a.m | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.14.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 9.14.a.c | 2 | 3.b | odd | 2 | 1 | ||
| 48.14.a.g | 2 | 4.b | odd | 2 | 1 | ||
| 75.14.a.e | 2 | 5.b | even | 2 | 1 | ||
| 75.14.b.c | 4 | 5.c | odd | 4 | 2 | ||
| 144.14.a.m | 2 | 12.b | even | 2 | 1 | ||
| 147.14.a.b | 2 | 7.b | odd | 2 | 1 | ||
| 192.14.a.k | 2 | 8.b | even | 2 | 1 | ||
| 192.14.a.o | 2 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 54T_{2} - 16992 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 54T - 16992 \)
$3$
\( (T - 729)^{2} \)
$5$
\( T^{2} - 40716 T + 124107300 \)
$7$
\( T^{2} + 21008 T - 211548155840 \)
$11$
\( T^{2} - 672408 T - 23635688182128 \)
$13$
\( T^{2} - 17532604 T - 66233088387452 \)
$17$
\( T^{2} - 83838564 T + 16\!\cdots\!24 \)
$19$
\( T^{2} - 256293544 T + 59\!\cdots\!80 \)
$23$
\( T^{2} - 859581936 T - 57\!\cdots\!20 \)
$29$
\( T^{2} + 4728475332 T + 49\!\cdots\!00 \)
$31$
\( T^{2} + 5982551648 T - 12\!\cdots\!00 \)
$37$
\( T^{2} - 27411194092 T + 18\!\cdots\!80 \)
$41$
\( T^{2} - 15258974292 T - 11\!\cdots\!60 \)
$43$
\( T^{2} + 11314499240 T - 16\!\cdots\!36 \)
$47$
\( T^{2} + 69035142240 T - 72\!\cdots\!44 \)
$53$
\( T^{2} + 226336894164 T - 17\!\cdots\!80 \)
$59$
\( T^{2} + 927820824264 T + 21\!\cdots\!80 \)
$61$
\( T^{2} - 179395461340 T - 34\!\cdots\!64 \)
$67$
\( T^{2} + 698315061176 T - 53\!\cdots\!56 \)
$71$
\( T^{2} + 784458549936 T - 27\!\cdots\!76 \)
$73$
\( T^{2} - 1857400245076 T - 31\!\cdots\!00 \)
$79$
\( T^{2} + 714025470080 T + 10\!\cdots\!00 \)
$83$
\( T^{2} + 4574293917912 T + 51\!\cdots\!92 \)
$89$
\( T^{2} - 3270178701684 T - 61\!\cdots\!80 \)
$97$
\( T^{2} + 9874926156476 T + 30\!\cdots\!44 \)
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