Newspace parameters
Level: | \( N \) | \(=\) | \( 264 = 2^{3} \cdot 3 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 264.p (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(0.131753163335\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{2}\) |
Projective field: | Galois closure of \(\Q(\sqrt{-2}, \sqrt{33})\) |
Artin image: | $D_4$ |
Artin field: | Galois closure of 4.0.2112.2 |
$q$-expansion
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/264\mathbb{Z}\right)^\times\).
\(n\) | \(89\) | \(133\) | \(145\) | \(199\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
131.1 |
|
−1.00000 | −1.00000 | 1.00000 | 0 | 1.00000 | 0 | −1.00000 | 1.00000 | 0 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2}) \) |
33.d | even | 2 | 1 | RM by \(\Q(\sqrt{33}) \) |
264.p | odd | 2 | 1 | CM by \(\Q(\sqrt{-66}) \) |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 264.1.p.a | ✓ | 1 |
3.b | odd | 2 | 1 | 264.1.p.b | yes | 1 | |
4.b | odd | 2 | 1 | 1056.1.p.a | 1 | ||
8.b | even | 2 | 1 | 1056.1.p.a | 1 | ||
8.d | odd | 2 | 1 | CM | 264.1.p.a | ✓ | 1 |
11.b | odd | 2 | 1 | 264.1.p.b | yes | 1 | |
11.c | even | 5 | 4 | 2904.1.r.f | 4 | ||
11.d | odd | 10 | 4 | 2904.1.r.b | 4 | ||
12.b | even | 2 | 1 | 1056.1.p.b | 1 | ||
24.f | even | 2 | 1 | 264.1.p.b | yes | 1 | |
24.h | odd | 2 | 1 | 1056.1.p.b | 1 | ||
33.d | even | 2 | 1 | RM | 264.1.p.a | ✓ | 1 |
33.f | even | 10 | 4 | 2904.1.r.f | 4 | ||
33.h | odd | 10 | 4 | 2904.1.r.b | 4 | ||
44.c | even | 2 | 1 | 1056.1.p.b | 1 | ||
88.b | odd | 2 | 1 | 1056.1.p.b | 1 | ||
88.g | even | 2 | 1 | 264.1.p.b | yes | 1 | |
88.k | even | 10 | 4 | 2904.1.r.b | 4 | ||
88.l | odd | 10 | 4 | 2904.1.r.f | 4 | ||
132.d | odd | 2 | 1 | 1056.1.p.a | 1 | ||
264.m | even | 2 | 1 | 1056.1.p.a | 1 | ||
264.p | odd | 2 | 1 | CM | 264.1.p.a | ✓ | 1 |
264.r | odd | 10 | 4 | 2904.1.r.f | 4 | ||
264.w | even | 10 | 4 | 2904.1.r.b | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
264.1.p.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
264.1.p.a | ✓ | 1 | 8.d | odd | 2 | 1 | CM |
264.1.p.a | ✓ | 1 | 33.d | even | 2 | 1 | RM |
264.1.p.a | ✓ | 1 | 264.p | odd | 2 | 1 | CM |
264.1.p.b | yes | 1 | 3.b | odd | 2 | 1 | |
264.1.p.b | yes | 1 | 11.b | odd | 2 | 1 | |
264.1.p.b | yes | 1 | 24.f | even | 2 | 1 | |
264.1.p.b | yes | 1 | 88.g | even | 2 | 1 | |
1056.1.p.a | 1 | 4.b | odd | 2 | 1 | ||
1056.1.p.a | 1 | 8.b | even | 2 | 1 | ||
1056.1.p.a | 1 | 132.d | odd | 2 | 1 | ||
1056.1.p.a | 1 | 264.m | even | 2 | 1 | ||
1056.1.p.b | 1 | 12.b | even | 2 | 1 | ||
1056.1.p.b | 1 | 24.h | odd | 2 | 1 | ||
1056.1.p.b | 1 | 44.c | even | 2 | 1 | ||
1056.1.p.b | 1 | 88.b | odd | 2 | 1 | ||
2904.1.r.b | 4 | 11.d | odd | 10 | 4 | ||
2904.1.r.b | 4 | 33.h | odd | 10 | 4 | ||
2904.1.r.b | 4 | 88.k | even | 10 | 4 | ||
2904.1.r.b | 4 | 264.w | even | 10 | 4 | ||
2904.1.r.f | 4 | 11.c | even | 5 | 4 | ||
2904.1.r.f | 4 | 33.f | even | 10 | 4 | ||
2904.1.r.f | 4 | 88.l | odd | 10 | 4 | ||
2904.1.r.f | 4 | 264.r | odd | 10 | 4 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{17} - 2 \)
acting on \(S_{1}^{\mathrm{new}}(264, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 1 \)
$3$
\( T + 1 \)
$5$
\( T \)
$7$
\( T \)
$11$
\( T - 1 \)
$13$
\( T \)
$17$
\( T - 2 \)
$19$
\( T \)
$23$
\( T \)
$29$
\( T \)
$31$
\( T \)
$37$
\( T \)
$41$
\( T + 2 \)
$43$
\( T \)
$47$
\( T \)
$53$
\( T \)
$59$
\( T \)
$61$
\( T \)
$67$
\( T - 2 \)
$71$
\( T \)
$73$
\( T \)
$79$
\( T \)
$83$
\( T + 2 \)
$89$
\( T \)
$97$
\( T + 2 \)
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