Newspace parameters
Level: | \( N \) | \(=\) | \( 2112 = 2^{6} \cdot 3 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2112.p (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.05402530668\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(i)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{2}\) |
Projective field: | Galois closure of \(\Q(\sqrt{-6}, \sqrt{-11})\) |
Artin image: | $D_4:C_2$ |
Artin field: | Galois closure of 8.0.8635613184.2 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times\).
\(n\) | \(133\) | \(1409\) | \(1729\) | \(2047\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1055.1 |
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0 | − | 1.00000i | 0 | −2.00000 | 0 | 0 | 0 | −1.00000 | 0 | |||||||||||||||||||||||
1055.2 | 0 | 1.00000i | 0 | −2.00000 | 0 | 0 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-11}) \) |
24.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-6}) \) |
264.m | even | 2 | 1 | RM by \(\Q(\sqrt{66}) \) |
4.b | odd | 2 | 1 | inner |
24.f | even | 2 | 1 | inner |
44.c | even | 2 | 1 | inner |
264.p | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2112.1.p.a | ✓ | 2 |
3.b | odd | 2 | 1 | 2112.1.p.b | yes | 2 | |
4.b | odd | 2 | 1 | inner | 2112.1.p.a | ✓ | 2 |
8.b | even | 2 | 1 | 2112.1.p.b | yes | 2 | |
8.d | odd | 2 | 1 | 2112.1.p.b | yes | 2 | |
11.b | odd | 2 | 1 | CM | 2112.1.p.a | ✓ | 2 |
12.b | even | 2 | 1 | 2112.1.p.b | yes | 2 | |
24.f | even | 2 | 1 | inner | 2112.1.p.a | ✓ | 2 |
24.h | odd | 2 | 1 | CM | 2112.1.p.a | ✓ | 2 |
33.d | even | 2 | 1 | 2112.1.p.b | yes | 2 | |
44.c | even | 2 | 1 | inner | 2112.1.p.a | ✓ | 2 |
88.b | odd | 2 | 1 | 2112.1.p.b | yes | 2 | |
88.g | even | 2 | 1 | 2112.1.p.b | yes | 2 | |
132.d | odd | 2 | 1 | 2112.1.p.b | yes | 2 | |
264.m | even | 2 | 1 | RM | 2112.1.p.a | ✓ | 2 |
264.p | odd | 2 | 1 | inner | 2112.1.p.a | ✓ | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2112.1.p.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
2112.1.p.a | ✓ | 2 | 4.b | odd | 2 | 1 | inner |
2112.1.p.a | ✓ | 2 | 11.b | odd | 2 | 1 | CM |
2112.1.p.a | ✓ | 2 | 24.f | even | 2 | 1 | inner |
2112.1.p.a | ✓ | 2 | 24.h | odd | 2 | 1 | CM |
2112.1.p.a | ✓ | 2 | 44.c | even | 2 | 1 | inner |
2112.1.p.a | ✓ | 2 | 264.m | even | 2 | 1 | RM |
2112.1.p.a | ✓ | 2 | 264.p | odd | 2 | 1 | inner |
2112.1.p.b | yes | 2 | 3.b | odd | 2 | 1 | |
2112.1.p.b | yes | 2 | 8.b | even | 2 | 1 | |
2112.1.p.b | yes | 2 | 8.d | odd | 2 | 1 | |
2112.1.p.b | yes | 2 | 12.b | even | 2 | 1 | |
2112.1.p.b | yes | 2 | 33.d | even | 2 | 1 | |
2112.1.p.b | yes | 2 | 88.b | odd | 2 | 1 | |
2112.1.p.b | yes | 2 | 88.g | even | 2 | 1 | |
2112.1.p.b | yes | 2 | 132.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} + 2 \)
acting on \(S_{1}^{\mathrm{new}}(2112, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 1 \)
$5$
\( (T + 2)^{2} \)
$7$
\( T^{2} \)
$11$
\( T^{2} + 1 \)
$13$
\( T^{2} \)
$17$
\( T^{2} \)
$19$
\( T^{2} \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( T^{2} + 4 \)
$37$
\( T^{2} \)
$41$
\( T^{2} \)
$43$
\( T^{2} \)
$47$
\( T^{2} \)
$53$
\( (T - 2)^{2} \)
$59$
\( T^{2} + 4 \)
$61$
\( T^{2} \)
$67$
\( T^{2} \)
$71$
\( T^{2} \)
$73$
\( T^{2} \)
$79$
\( T^{2} \)
$83$
\( T^{2} \)
$89$
\( T^{2} \)
$97$
\( (T + 2)^{2} \)
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