Newspace parameters
| Level: | \( N \) | \(=\) | \( 11 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 11.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(3.43623528033\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{15}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 15 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(\beta = 2\sqrt{15}\). 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 |
|
−11.7460 | 43.4758 | 9.96773 | −389.919 | −510.665 | −1249.17 | 1386.40 | −296.855 | 4579.98 | ||||||||||||||||||||||||
| 1.2 | 3.74597 | −49.4758 | −113.968 | −80.0807 | −185.335 | 21.1693 | −906.403 | 260.855 | −299.980 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(-1\) |
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 | 11.8.a.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 99.8.a.c | 2 | ||
| 4.b | odd | 2 | 1 | 176.8.a.d | 2 | ||
| 5.b | even | 2 | 1 | 275.8.a.a | 2 | ||
| 7.b | odd | 2 | 1 | 539.8.a.a | 2 | ||
| 11.b | odd | 2 | 1 | 121.8.a.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.8.a.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 99.8.a.c | 2 | 3.b | odd | 2 | 1 | ||
| 121.8.a.b | 2 | 11.b | odd | 2 | 1 | ||
| 176.8.a.d | 2 | 4.b | odd | 2 | 1 | ||
| 275.8.a.a | 2 | 5.b | even | 2 | 1 | ||
| 539.8.a.a | 2 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 8T_{2} - 44 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(11))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 8T - 44 \)
$3$
\( T^{2} + 6T - 2151 \)
$5$
\( T^{2} + 470T + 31225 \)
$7$
\( T^{2} + 1228T - 26444 \)
$11$
\( (T - 1331)^{2} \)
$13$
\( T^{2} - 344 T - 16069856 \)
$17$
\( T^{2} + 8468 T - 788446604 \)
$19$
\( T^{2} + 35280 T - 222369840 \)
$23$
\( T^{2} + 61486 T - 159113951 \)
$29$
\( T^{2} - 179040 T + 122928960 \)
$31$
\( T^{2} + 57166 T - 23689658111 \)
$37$
\( T^{2} + 877698 T + 191745594561 \)
$41$
\( T^{2} + 283616 T - 264475105136 \)
$43$
\( T^{2} - 275484 T + 9203802564 \)
$47$
\( T^{2} - 1662512 T + 677029127296 \)
$53$
\( T^{2} - 1616484 T + 388813137924 \)
$59$
\( T^{2} + 2454130 T + 207772229185 \)
$61$
\( T^{2} + 6019176 T + 9007507329744 \)
$67$
\( T^{2} + 174698 T - 9239984638199 \)
$71$
\( T^{2} + 1151466 T - 11975092638711 \)
$73$
\( T^{2} - 885944 T - 2090754692576 \)
$79$
\( T^{2} - 3801460 T - 2681742596060 \)
$83$
\( T^{2} + 2282916 T - 536001911676 \)
$89$
\( T^{2} + 13481970 T + 42998937114225 \)
$97$
\( T^{2} + 68078 T - 1834997592239 \)
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