Newspace parameters
| Level: | \( N \) | \(=\) | \( 69 = 3 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 69.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(11.0664835671\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{29}) \) |
| Defining polynomial: |
\( x^{2} - x - 7 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(\beta = \sqrt{29}\). 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 |
|
−10.7703 | −9.00000 | 84.0000 | 41.6148 | 96.9330 | 0.236813 | −560.057 | 81.0000 | −448.205 | ||||||||||||||||||||||||
| 1.2 | 10.7703 | −9.00000 | 84.0000 | 52.3852 | −96.9330 | −118.237 | 560.057 | 81.0000 | 564.205 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(23\) | \(-1\) |
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 | 69.6.a.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 207.6.a.a | 2 | ||
| 4.b | odd | 2 | 1 | 1104.6.a.h | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.6.a.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 207.6.a.a | 2 | 3.b | odd | 2 | 1 | ||
| 1104.6.a.h | 2 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 116 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(69))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 116 \)
$3$
\( (T + 9)^{2} \)
$5$
\( T^{2} - 94T + 2180 \)
$7$
\( T^{2} + 118T - 28 \)
$11$
\( T^{2} - 320T - 312656 \)
$13$
\( T^{2} + 288T - 138068 \)
$17$
\( T^{2} + 1810 T + 814124 \)
$19$
\( T^{2} - 730 T - 2671684 \)
$23$
\( (T - 529)^{2} \)
$29$
\( T^{2} - 8208 T + 14790892 \)
$31$
\( T^{2} - 1772 T - 65688688 \)
$37$
\( T^{2} + 23112 T + 132310492 \)
$41$
\( T^{2} - 5516 T - 46982572 \)
$43$
\( T^{2} - 10322 T - 179637860 \)
$47$
\( T^{2} - 42952 T + 422732560 \)
$53$
\( T^{2} + 25350 T - 392901044 \)
$59$
\( T^{2} - 18344 T + 76641728 \)
$61$
\( T^{2} - 37224 T - 105150020 \)
$67$
\( T^{2} + 7482 T - 601513940 \)
$71$
\( T^{2} - 126848 T + 3896171200 \)
$73$
\( T^{2} - 137660 T + 4595673524 \)
$79$
\( T^{2} - 62286 T + 906450660 \)
$83$
\( T^{2} - 83120 T - 1861324304 \)
$89$
\( T^{2} - 69770 T + 1061277044 \)
$97$
\( T^{2} + 170104 T + 4172745500 \)
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