# Properties

 Label 6014.2.a.b Level $6014$ Weight $2$ Character orbit 6014.a Self dual yes Analytic conductor $48.022$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6014 = 2 \cdot 31 \cdot 97$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6014.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.0220317756$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + 2q^{5} - 4q^{7} + q^{8} - 3q^{9} + O(q^{10})$$ $$q + q^{2} + q^{4} + 2q^{5} - 4q^{7} + q^{8} - 3q^{9} + 2q^{10} + 2q^{13} - 4q^{14} + q^{16} - 2q^{17} - 3q^{18} + 4q^{19} + 2q^{20} - q^{25} + 2q^{26} - 4q^{28} + 2q^{29} - q^{31} + q^{32} - 2q^{34} - 8q^{35} - 3q^{36} - 6q^{37} + 4q^{38} + 2q^{40} + 2q^{41} - 8q^{43} - 6q^{45} + 9q^{49} - q^{50} + 2q^{52} - 14q^{53} - 4q^{56} + 2q^{58} - 4q^{59} + 2q^{61} - q^{62} + 12q^{63} + q^{64} + 4q^{65} + 12q^{67} - 2q^{68} - 8q^{70} - 12q^{71} - 3q^{72} + 2q^{73} - 6q^{74} + 4q^{76} - 16q^{79} + 2q^{80} + 9q^{81} + 2q^{82} + 4q^{83} - 4q^{85} - 8q^{86} - 6q^{89} - 6q^{90} - 8q^{91} + 8q^{95} + q^{97} + 9q^{98} + O(q^{100})$$

## 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 0 1.00000 2.00000 0 −4.00000 1.00000 −3.00000 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$31$$ $$1$$
$$97$$ $$-1$$

## 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 6014.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6014.2.a.b 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6014))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T$$
$3$ $$1 + 3 T^{2}$$
$5$ $$1 - 2 T + 5 T^{2}$$
$7$ $$1 + 4 T + 7 T^{2}$$
$11$ $$1 + 11 T^{2}$$
$13$ $$1 - 2 T + 13 T^{2}$$
$17$ $$1 + 2 T + 17 T^{2}$$
$19$ $$1 - 4 T + 19 T^{2}$$
$23$ $$1 + 23 T^{2}$$
$29$ $$1 - 2 T + 29 T^{2}$$
$31$ $$1 + T$$
$37$ $$1 + 6 T + 37 T^{2}$$
$41$ $$1 - 2 T + 41 T^{2}$$
$43$ $$1 + 8 T + 43 T^{2}$$
$47$ $$1 + 47 T^{2}$$
$53$ $$1 + 14 T + 53 T^{2}$$
$59$ $$1 + 4 T + 59 T^{2}$$
$61$ $$1 - 2 T + 61 T^{2}$$
$67$ $$1 - 12 T + 67 T^{2}$$
$71$ $$1 + 12 T + 71 T^{2}$$
$73$ $$1 - 2 T + 73 T^{2}$$
$79$ $$1 + 16 T + 79 T^{2}$$
$83$ $$1 - 4 T + 83 T^{2}$$
$89$ $$1 + 6 T + 89 T^{2}$$
$97$ $$1 - T$$