Properties

 Label 129.2.a.a Level 129 Weight 2 Character orbit 129.a Self dual yes Analytic conductor 1.030 Analytic rank 1 Dimension 1 CM no Inner twists 1

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$129 = 3 \cdot 43$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 129.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$1.03007018607$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} - 2q^{4} - 2q^{5} - 2q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} - 2q^{4} - 2q^{5} - 2q^{7} + q^{9} - 5q^{11} + 2q^{12} + 3q^{13} + 2q^{15} + 4q^{16} - 3q^{17} + 2q^{19} + 4q^{20} + 2q^{21} - q^{23} - q^{25} - q^{27} + 4q^{28} - 5q^{31} + 5q^{33} + 4q^{35} - 2q^{36} + 8q^{37} - 3q^{39} - 7q^{41} - q^{43} + 10q^{44} - 2q^{45} - 8q^{47} - 4q^{48} - 3q^{49} + 3q^{51} - 6q^{52} + 3q^{53} + 10q^{55} - 2q^{57} + 12q^{59} - 4q^{60} - 8q^{61} - 2q^{63} - 8q^{64} - 6q^{65} - 15q^{67} + 6q^{68} + q^{69} - 14q^{71} + 12q^{73} + q^{75} - 4q^{76} + 10q^{77} - 16q^{79} - 8q^{80} + q^{81} + 15q^{83} - 4q^{84} + 6q^{85} + 10q^{89} - 6q^{91} + 2q^{92} + 5q^{93} - 4q^{95} + 11q^{97} - 5q^{99} + O(q^{100})$$

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
 0
0 −1.00000 −2.00000 −2.00000 0 −2.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

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 129.2.a.a 1
3.b odd 2 1 387.2.a.c 1
4.b odd 2 1 2064.2.a.k 1
5.b even 2 1 3225.2.a.g 1
7.b odd 2 1 6321.2.a.e 1
8.b even 2 1 8256.2.a.bm 1
8.d odd 2 1 8256.2.a.s 1
12.b even 2 1 6192.2.a.v 1
15.d odd 2 1 9675.2.a.m 1
43.b odd 2 1 5547.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
129.2.a.a 1 1.a even 1 1 trivial
387.2.a.c 1 3.b odd 2 1
2064.2.a.k 1 4.b odd 2 1
3225.2.a.g 1 5.b even 2 1
5547.2.a.c 1 43.b odd 2 1
6192.2.a.v 1 12.b even 2 1
6321.2.a.e 1 7.b odd 2 1
8256.2.a.s 1 8.d odd 2 1
8256.2.a.bm 1 8.b even 2 1
9675.2.a.m 1 15.d odd 2 1

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$43$$ $$1$$

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(129))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T^{2}$$
$3$ $$1 + T$$
$5$ $$1 + 2 T + 5 T^{2}$$
$7$ $$1 + 2 T + 7 T^{2}$$
$11$ $$1 + 5 T + 11 T^{2}$$
$13$ $$1 - 3 T + 13 T^{2}$$
$17$ $$1 + 3 T + 17 T^{2}$$
$19$ $$1 - 2 T + 19 T^{2}$$
$23$ $$1 + T + 23 T^{2}$$
$29$ $$1 + 29 T^{2}$$
$31$ $$1 + 5 T + 31 T^{2}$$
$37$ $$1 - 8 T + 37 T^{2}$$
$41$ $$1 + 7 T + 41 T^{2}$$
$43$ $$1 + T$$
$47$ $$1 + 8 T + 47 T^{2}$$
$53$ $$1 - 3 T + 53 T^{2}$$
$59$ $$1 - 12 T + 59 T^{2}$$
$61$ $$1 + 8 T + 61 T^{2}$$
$67$ $$1 + 15 T + 67 T^{2}$$
$71$ $$1 + 14 T + 71 T^{2}$$
$73$ $$1 - 12 T + 73 T^{2}$$
$79$ $$1 + 16 T + 79 T^{2}$$
$83$ $$1 - 15 T + 83 T^{2}$$
$89$ $$1 - 10 T + 89 T^{2}$$
$97$ $$1 - 11 T + 97 T^{2}$$