Newspace parameters
Level: | \( N \) | \(=\) | \( 99 = 3^{2} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 99.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(40.3304823961\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 11) |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).
\(n\) | \(46\) | \(56\) |
\(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
10.1 |
|
0 | 0 | 256.000 | −1151.00 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-11}) \) |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 99.9.c.a | 1 | |
3.b | odd | 2 | 1 | 11.9.b.a | ✓ | 1 | |
11.b | odd | 2 | 1 | CM | 99.9.c.a | 1 | |
12.b | even | 2 | 1 | 176.9.h.a | 1 | ||
33.d | even | 2 | 1 | 11.9.b.a | ✓ | 1 | |
132.d | odd | 2 | 1 | 176.9.h.a | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
11.9.b.a | ✓ | 1 | 3.b | odd | 2 | 1 | |
11.9.b.a | ✓ | 1 | 33.d | even | 2 | 1 | |
99.9.c.a | 1 | 1.a | even | 1 | 1 | trivial | |
99.9.c.a | 1 | 11.b | odd | 2 | 1 | CM | |
176.9.h.a | 1 | 12.b | even | 2 | 1 | ||
176.9.h.a | 1 | 132.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{9}^{\mathrm{new}}(99, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T \)
$5$
\( T + 1151 \)
$7$
\( T \)
$11$
\( T + 14641 \)
$13$
\( T \)
$17$
\( T \)
$19$
\( T \)
$23$
\( T - 531793 \)
$29$
\( T \)
$31$
\( T + 1541233 \)
$37$
\( T - 716447 \)
$41$
\( T \)
$43$
\( T \)
$47$
\( T - 6080638 \)
$53$
\( T - 15265438 \)
$59$
\( T - 4101553 \)
$61$
\( T \)
$67$
\( T - 19806767 \)
$71$
\( T + 7043087 \)
$73$
\( T \)
$79$
\( T \)
$83$
\( T \)
$89$
\( T - 84100993 \)
$97$
\( T + 81155713 \)
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