Newspace parameters
Level: | \( N \) | \(=\) | \( 105 = 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 105.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(0.838429221223\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{5}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x - 1 \) |
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 \(\beta = \sqrt{5}\). 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 |
|
−2.23607 | −1.00000 | 3.00000 | −1.00000 | 2.23607 | 1.00000 | −2.23607 | 1.00000 | 2.23607 | ||||||||||||||||||||||||
1.2 | 2.23607 | −1.00000 | 3.00000 | −1.00000 | −2.23607 | 1.00000 | 2.23607 | 1.00000 | −2.23607 | |||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(5\) | \(1\) |
\(7\) | \(-1\) |
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 | 105.2.a.b | ✓ | 2 |
3.b | odd | 2 | 1 | 315.2.a.d | 2 | ||
4.b | odd | 2 | 1 | 1680.2.a.v | 2 | ||
5.b | even | 2 | 1 | 525.2.a.g | 2 | ||
5.c | odd | 4 | 2 | 525.2.d.c | 4 | ||
7.b | odd | 2 | 1 | 735.2.a.k | 2 | ||
7.c | even | 3 | 2 | 735.2.i.k | 4 | ||
7.d | odd | 6 | 2 | 735.2.i.i | 4 | ||
8.b | even | 2 | 1 | 6720.2.a.cx | 2 | ||
8.d | odd | 2 | 1 | 6720.2.a.cs | 2 | ||
12.b | even | 2 | 1 | 5040.2.a.bw | 2 | ||
15.d | odd | 2 | 1 | 1575.2.a.r | 2 | ||
15.e | even | 4 | 2 | 1575.2.d.d | 4 | ||
20.d | odd | 2 | 1 | 8400.2.a.cx | 2 | ||
21.c | even | 2 | 1 | 2205.2.a.w | 2 | ||
35.c | odd | 2 | 1 | 3675.2.a.y | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
105.2.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
315.2.a.d | 2 | 3.b | odd | 2 | 1 | ||
525.2.a.g | 2 | 5.b | even | 2 | 1 | ||
525.2.d.c | 4 | 5.c | odd | 4 | 2 | ||
735.2.a.k | 2 | 7.b | odd | 2 | 1 | ||
735.2.i.i | 4 | 7.d | odd | 6 | 2 | ||
735.2.i.k | 4 | 7.c | even | 3 | 2 | ||
1575.2.a.r | 2 | 15.d | odd | 2 | 1 | ||
1575.2.d.d | 4 | 15.e | even | 4 | 2 | ||
1680.2.a.v | 2 | 4.b | odd | 2 | 1 | ||
2205.2.a.w | 2 | 21.c | even | 2 | 1 | ||
3675.2.a.y | 2 | 35.c | odd | 2 | 1 | ||
5040.2.a.bw | 2 | 12.b | even | 2 | 1 | ||
6720.2.a.cs | 2 | 8.d | odd | 2 | 1 | ||
6720.2.a.cx | 2 | 8.b | even | 2 | 1 | ||
8400.2.a.cx | 2 | 20.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 5 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(105))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 5 \)
$3$
\( (T + 1)^{2} \)
$5$
\( (T + 1)^{2} \)
$7$
\( (T - 1)^{2} \)
$11$
\( T^{2} - 4T - 16 \)
$13$
\( T^{2} - 20 \)
$17$
\( (T + 2)^{2} \)
$19$
\( T^{2} - 4T - 16 \)
$23$
\( (T - 4)^{2} \)
$29$
\( (T + 2)^{2} \)
$31$
\( T^{2} - 12T + 16 \)
$37$
\( T^{2} - 4T - 76 \)
$41$
\( (T + 2)^{2} \)
$43$
\( T^{2} - 80 \)
$47$
\( T^{2} - 8T - 64 \)
$53$
\( T^{2} + 16T + 44 \)
$59$
\( T^{2} - 80 \)
$61$
\( (T + 2)^{2} \)
$67$
\( (T + 4)^{2} \)
$71$
\( T^{2} - 20T + 80 \)
$73$
\( T^{2} + 16T + 44 \)
$79$
\( T^{2} - 8T - 64 \)
$83$
\( T^{2} + 16T - 16 \)
$89$
\( (T + 2)^{2} \)
$97$
\( T^{2} - 8T - 4 \)
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