# Properties

 Label 8034.2.a.j Level $8034$ Weight $2$ Character orbit 8034.a Self dual yes Analytic conductor $64.152$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8034 = 2 \cdot 3 \cdot 13 \cdot 103$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8034.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.1518129839$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + 2q^{5} + q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + 2q^{5} + q^{6} + q^{8} + q^{9} + 2q^{10} + q^{12} + q^{13} + 2q^{15} + q^{16} + 2q^{17} + q^{18} + 4q^{19} + 2q^{20} - 8q^{23} + q^{24} - q^{25} + q^{26} + q^{27} - 2q^{29} + 2q^{30} - 4q^{31} + q^{32} + 2q^{34} + q^{36} + 10q^{37} + 4q^{38} + q^{39} + 2q^{40} + 10q^{41} - 4q^{43} + 2q^{45} - 8q^{46} + 4q^{47} + q^{48} - 7q^{49} - q^{50} + 2q^{51} + q^{52} + 6q^{53} + q^{54} + 4q^{57} - 2q^{58} + 12q^{59} + 2q^{60} + 14q^{61} - 4q^{62} + q^{64} + 2q^{65} + 16q^{67} + 2q^{68} - 8q^{69} - 4q^{71} + q^{72} + 6q^{73} + 10q^{74} - q^{75} + 4q^{76} + q^{78} - 8q^{79} + 2q^{80} + q^{81} + 10q^{82} + 12q^{83} + 4q^{85} - 4q^{86} - 2q^{87} - 2q^{89} + 2q^{90} - 8q^{92} - 4q^{93} + 4q^{94} + 8q^{95} + q^{96} - 14q^{97} - 7q^{98} + O(q^{100})$$

## 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 1.00000 1.00000 2.00000 1.00000 0 1.00000 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$13$$ $$-1$$
$$103$$ $$1$$

## 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 8034.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.j 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8034))$$:

 $$T_{5} - 2$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-1 + T$$
$5$ $$-2 + T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$-1 + T$$
$17$ $$-2 + T$$
$19$ $$-4 + T$$
$23$ $$8 + T$$
$29$ $$2 + T$$
$31$ $$4 + T$$
$37$ $$-10 + T$$
$41$ $$-10 + T$$
$43$ $$4 + T$$
$47$ $$-4 + T$$
$53$ $$-6 + T$$
$59$ $$-12 + T$$
$61$ $$-14 + T$$
$67$ $$-16 + T$$
$71$ $$4 + T$$
$73$ $$-6 + T$$
$79$ $$8 + T$$
$83$ $$-12 + T$$
$89$ $$2 + T$$
$97$ $$14 + T$$