# Properties

 Label 8050.2.a.n Level $8050$ Weight $2$ Character orbit 8050.a Self dual yes Analytic conductor $64.280$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8050 = 2 \cdot 5^{2} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8050.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.2795736271$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 322) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - 2q^{3} + q^{4} - 2q^{6} - q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - 2q^{3} + q^{4} - 2q^{6} - q^{7} + q^{8} + q^{9} + 4q^{11} - 2q^{12} - q^{14} + q^{16} - 6q^{17} + q^{18} - 6q^{19} + 2q^{21} + 4q^{22} + q^{23} - 2q^{24} + 4q^{27} - q^{28} + 10q^{29} + 4q^{31} + q^{32} - 8q^{33} - 6q^{34} + q^{36} + 2q^{37} - 6q^{38} - 10q^{41} + 2q^{42} + 4q^{43} + 4q^{44} + q^{46} - 12q^{47} - 2q^{48} + q^{49} + 12q^{51} + 6q^{53} + 4q^{54} - q^{56} + 12q^{57} + 10q^{58} - 2q^{59} + 4q^{62} - q^{63} + q^{64} - 8q^{66} - 6q^{68} - 2q^{69} - 8q^{71} + q^{72} + 6q^{73} + 2q^{74} - 6q^{76} - 4q^{77} - 8q^{79} - 11q^{81} - 10q^{82} + 14q^{83} + 2q^{84} + 4q^{86} - 20q^{87} + 4q^{88} - 14q^{89} + q^{92} - 8q^{93} - 12q^{94} - 2q^{96} + 2q^{97} + q^{98} + 4q^{99} + O(q^{100})$$

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

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$1$$
$$23$$ $$-1$$

## 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 8050.2.a.n 1
5.b even 2 1 322.2.a.b 1
15.d odd 2 1 2898.2.a.p 1
20.d odd 2 1 2576.2.a.c 1
35.c odd 2 1 2254.2.a.a 1
115.c odd 2 1 7406.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.a.b 1 5.b even 2 1
2254.2.a.a 1 35.c odd 2 1
2576.2.a.c 1 20.d odd 2 1
2898.2.a.p 1 15.d odd 2 1
7406.2.a.e 1 115.c odd 2 1
8050.2.a.n 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8050))$$:

 $$T_{3} + 2$$ $$T_{11} - 4$$ $$T_{13}$$ $$T_{17} + 6$$

## Hecke characteristic polynomials

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