# Properties

 Label 58.2.a.b Level 58 Weight 2 Character orbit 58.a Self dual yes Analytic conductor 0.463 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$58 = 2 \cdot 29$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 58.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.463132331723$$ Analytic rank: $$0$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - 2q^{7} + q^{8} - 2q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - 2q^{7} + q^{8} - 2q^{9} + q^{10} - 3q^{11} - q^{12} - q^{13} - 2q^{14} - q^{15} + q^{16} + 8q^{17} - 2q^{18} + q^{20} + 2q^{21} - 3q^{22} + 4q^{23} - q^{24} - 4q^{25} - q^{26} + 5q^{27} - 2q^{28} - q^{29} - q^{30} - 3q^{31} + q^{32} + 3q^{33} + 8q^{34} - 2q^{35} - 2q^{36} + 8q^{37} + q^{39} + q^{40} + 2q^{41} + 2q^{42} - 11q^{43} - 3q^{44} - 2q^{45} + 4q^{46} + 13q^{47} - q^{48} - 3q^{49} - 4q^{50} - 8q^{51} - q^{52} - 11q^{53} + 5q^{54} - 3q^{55} - 2q^{56} - q^{58} - q^{60} - 8q^{61} - 3q^{62} + 4q^{63} + q^{64} - q^{65} + 3q^{66} - 12q^{67} + 8q^{68} - 4q^{69} - 2q^{70} + 2q^{71} - 2q^{72} + 4q^{73} + 8q^{74} + 4q^{75} + 6q^{77} + q^{78} + 15q^{79} + q^{80} + q^{81} + 2q^{82} + 4q^{83} + 2q^{84} + 8q^{85} - 11q^{86} + q^{87} - 3q^{88} - 10q^{89} - 2q^{90} + 2q^{91} + 4q^{92} + 3q^{93} + 13q^{94} - q^{96} - 2q^{97} - 3q^{98} + 6q^{99} + O(q^{100})$$

## 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
1.00000 −1.00000 1.00000 1.00000 −1.00000 −2.00000 1.00000 −2.00000 1.00000
 $$n$$: 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 58.2.a.b 1
3.b odd 2 1 522.2.a.b 1
4.b odd 2 1 464.2.a.e 1
5.b even 2 1 1450.2.a.c 1
5.c odd 4 2 1450.2.b.b 2
7.b odd 2 1 2842.2.a.e 1
8.b even 2 1 1856.2.a.k 1
8.d odd 2 1 1856.2.a.f 1
11.b odd 2 1 7018.2.a.a 1
12.b even 2 1 4176.2.a.n 1
13.b even 2 1 9802.2.a.a 1
29.b even 2 1 1682.2.a.d 1
29.c odd 4 2 1682.2.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.2.a.b 1 1.a even 1 1 trivial
464.2.a.e 1 4.b odd 2 1
522.2.a.b 1 3.b odd 2 1
1450.2.a.c 1 5.b even 2 1
1450.2.b.b 2 5.c odd 4 2
1682.2.a.d 1 29.b even 2 1
1682.2.b.a 2 29.c odd 4 2
1856.2.a.f 1 8.d odd 2 1
1856.2.a.k 1 8.b even 2 1
2842.2.a.e 1 7.b odd 2 1
4176.2.a.n 1 12.b even 2 1
7018.2.a.a 1 11.b odd 2 1
9802.2.a.a 1 13.b even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$29$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(58))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T$$
$3$ $$1 + T + 3 T^{2}$$
$5$ $$1 - T + 5 T^{2}$$
$7$ $$1 + 2 T + 7 T^{2}$$
$11$ $$1 + 3 T + 11 T^{2}$$
$13$ $$1 + T + 13 T^{2}$$
$17$ $$1 - 8 T + 17 T^{2}$$
$19$ $$1 + 19 T^{2}$$
$23$ $$1 - 4 T + 23 T^{2}$$
$29$ $$1 + T$$
$31$ $$1 + 3 T + 31 T^{2}$$
$37$ $$1 - 8 T + 37 T^{2}$$
$41$ $$1 - 2 T + 41 T^{2}$$
$43$ $$1 + 11 T + 43 T^{2}$$
$47$ $$1 - 13 T + 47 T^{2}$$
$53$ $$1 + 11 T + 53 T^{2}$$
$59$ $$1 + 59 T^{2}$$
$61$ $$1 + 8 T + 61 T^{2}$$
$67$ $$1 + 12 T + 67 T^{2}$$
$71$ $$1 - 2 T + 71 T^{2}$$
$73$ $$1 - 4 T + 73 T^{2}$$
$79$ $$1 - 15 T + 79 T^{2}$$
$83$ $$1 - 4 T + 83 T^{2}$$
$89$ $$1 + 10 T + 89 T^{2}$$
$97$ $$1 + 2 T + 97 T^{2}$$