Newspace parameters
Level: | \( N \) | \(=\) | \( 105 = 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 105.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(6.19520055060\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{65}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x - 16 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{65})\). 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 |
|
−3.53113 | 3.00000 | 4.46887 | 5.00000 | −10.5934 | −7.00000 | 12.4689 | 9.00000 | −17.6556 | ||||||||||||||||||||||||
1.2 | 4.53113 | 3.00000 | 12.5311 | 5.00000 | 13.5934 | −7.00000 | 20.5311 | 9.00000 | 22.6556 | |||||||||||||||||||||||||
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 | 105.4.a.f | ✓ | 2 |
3.b | odd | 2 | 1 | 315.4.a.i | 2 | ||
4.b | odd | 2 | 1 | 1680.4.a.bg | 2 | ||
5.b | even | 2 | 1 | 525.4.a.k | 2 | ||
5.c | odd | 4 | 2 | 525.4.d.h | 4 | ||
7.b | odd | 2 | 1 | 735.4.a.p | 2 | ||
15.d | odd | 2 | 1 | 1575.4.a.w | 2 | ||
21.c | even | 2 | 1 | 2205.4.a.z | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
105.4.a.f | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
315.4.a.i | 2 | 3.b | odd | 2 | 1 | ||
525.4.a.k | 2 | 5.b | even | 2 | 1 | ||
525.4.d.h | 4 | 5.c | odd | 4 | 2 | ||
735.4.a.p | 2 | 7.b | odd | 2 | 1 | ||
1575.4.a.w | 2 | 15.d | odd | 2 | 1 | ||
1680.4.a.bg | 2 | 4.b | odd | 2 | 1 | ||
2205.4.a.z | 2 | 21.c | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - T_{2} - 16 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(105))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - T - 16 \)
$3$
\( (T - 3)^{2} \)
$5$
\( (T - 5)^{2} \)
$7$
\( (T + 7)^{2} \)
$11$
\( T^{2} + 22T + 56 \)
$13$
\( T^{2} + 22T + 56 \)
$17$
\( T^{2} - 116T - 796 \)
$19$
\( T^{2} - 102T - 584 \)
$23$
\( T^{2} - 260T + 10400 \)
$29$
\( T^{2} + 196T - 27836 \)
$31$
\( T^{2} - 150T - 23040 \)
$37$
\( T^{2} + 96T - 18756 \)
$41$
\( T^{2} + 176T - 154756 \)
$43$
\( T^{2} + 344T + 12944 \)
$47$
\( T^{2} - 560T + 40960 \)
$53$
\( T^{2} - 326T - 93616 \)
$59$
\( T^{2} + 844T + 63424 \)
$61$
\( T^{2} + 204T + 1044 \)
$67$
\( T^{2} + 104T - 63856 \)
$71$
\( T^{2} - 1670 T + 668560 \)
$73$
\( T^{2} + 386T - 625816 \)
$79$
\( T^{2} + 888T + 21376 \)
$83$
\( T^{2} - 928T - 542864 \)
$89$
\( T^{2} - 588T + 85396 \)
$97$
\( T^{2} - 522 T - 1534064 \)
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