# Properties

 Label 8001.2.a.h Level $8001$ Weight $2$ Character orbit 8001.a Self dual yes Analytic conductor $63.888$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8001 = 3^{2} \cdot 7 \cdot 127$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8001.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$63.8883066572$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2667) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + ( 2 + \beta ) q^{4} + ( -1 - \beta ) q^{5} - q^{7} + ( -4 - \beta ) q^{8} +O(q^{10})$$ $$q -\beta q^{2} + ( 2 + \beta ) q^{4} + ( -1 - \beta ) q^{5} - q^{7} + ( -4 - \beta ) q^{8} + ( 4 + 2 \beta ) q^{10} + ( 2 - 2 \beta ) q^{11} + ( -1 - \beta ) q^{13} + \beta q^{14} + 3 \beta q^{16} -4 q^{17} -4 q^{19} + ( -6 - 4 \beta ) q^{20} + 8 q^{22} + ( 3 - \beta ) q^{23} + 3 \beta q^{25} + ( 4 + 2 \beta ) q^{26} + ( -2 - \beta ) q^{28} + ( -5 + \beta ) q^{29} + ( 5 - \beta ) q^{31} + ( -4 - \beta ) q^{32} + 4 \beta q^{34} + ( 1 + \beta ) q^{35} + ( -3 + \beta ) q^{37} + 4 \beta q^{38} + ( 8 + 6 \beta ) q^{40} + ( -2 - 2 \beta ) q^{41} + ( 2 + 2 \beta ) q^{43} + ( -4 - 4 \beta ) q^{44} + ( 4 - 2 \beta ) q^{46} + 10 q^{47} + q^{49} + ( -12 - 3 \beta ) q^{50} + ( -6 - 4 \beta ) q^{52} + ( 1 + 3 \beta ) q^{53} + ( 6 + 2 \beta ) q^{55} + ( 4 + \beta ) q^{56} + ( -4 + 4 \beta ) q^{58} + ( 1 + 3 \beta ) q^{59} + ( -7 + \beta ) q^{61} + ( 4 - 4 \beta ) q^{62} + ( 4 - \beta ) q^{64} + ( 5 + 3 \beta ) q^{65} + ( 2 + 2 \beta ) q^{67} + ( -8 - 4 \beta ) q^{68} + ( -4 - 2 \beta ) q^{70} + 12 q^{71} + ( -7 + \beta ) q^{73} + ( -4 + 2 \beta ) q^{74} + ( -8 - 4 \beta ) q^{76} + ( -2 + 2 \beta ) q^{77} + ( -4 + 4 \beta ) q^{79} + ( -12 - 6 \beta ) q^{80} + ( 8 + 4 \beta ) q^{82} + ( -3 + 3 \beta ) q^{83} + ( 4 + 4 \beta ) q^{85} + ( -8 - 4 \beta ) q^{86} + 8 \beta q^{88} + ( -13 + 3 \beta ) q^{89} + ( 1 + \beta ) q^{91} + 2 q^{92} -10 \beta q^{94} + ( 4 + 4 \beta ) q^{95} + ( -12 - 2 \beta ) q^{97} -\beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + 5q^{4} - 3q^{5} - 2q^{7} - 9q^{8} + O(q^{10})$$ $$2q - q^{2} + 5q^{4} - 3q^{5} - 2q^{7} - 9q^{8} + 10q^{10} + 2q^{11} - 3q^{13} + q^{14} + 3q^{16} - 8q^{17} - 8q^{19} - 16q^{20} + 16q^{22} + 5q^{23} + 3q^{25} + 10q^{26} - 5q^{28} - 9q^{29} + 9q^{31} - 9q^{32} + 4q^{34} + 3q^{35} - 5q^{37} + 4q^{38} + 22q^{40} - 6q^{41} + 6q^{43} - 12q^{44} + 6q^{46} + 20q^{47} + 2q^{49} - 27q^{50} - 16q^{52} + 5q^{53} + 14q^{55} + 9q^{56} - 4q^{58} + 5q^{59} - 13q^{61} + 4q^{62} + 7q^{64} + 13q^{65} + 6q^{67} - 20q^{68} - 10q^{70} + 24q^{71} - 13q^{73} - 6q^{74} - 20q^{76} - 2q^{77} - 4q^{79} - 30q^{80} + 20q^{82} - 3q^{83} + 12q^{85} - 20q^{86} + 8q^{88} - 23q^{89} + 3q^{91} + 4q^{92} - 10q^{94} + 12q^{95} - 26q^{97} - q^{98} + O(q^{100})$$

## 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
 2.56155 −1.56155
−2.56155 0 4.56155 −3.56155 0 −1.00000 −6.56155 0 9.12311
1.2 1.56155 0 0.438447 0.561553 0 −1.00000 −2.43845 0 0.876894
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$1$$
$$127$$ $$-1$$

## 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 8001.2.a.h 2
3.b odd 2 1 2667.2.a.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.i 2 3.b odd 2 1
8001.2.a.h 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8001))$$:

 $$T_{2}^{2} + T_{2} - 4$$ $$T_{5}^{2} + 3 T_{5} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$-2 + 3 T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-16 - 2 T + T^{2}$$
$13$ $$-2 + 3 T + T^{2}$$
$17$ $$( 4 + T )^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$2 - 5 T + T^{2}$$
$29$ $$16 + 9 T + T^{2}$$
$31$ $$16 - 9 T + T^{2}$$
$37$ $$2 + 5 T + T^{2}$$
$41$ $$-8 + 6 T + T^{2}$$
$43$ $$-8 - 6 T + T^{2}$$
$47$ $$( -10 + T )^{2}$$
$53$ $$-32 - 5 T + T^{2}$$
$59$ $$-32 - 5 T + T^{2}$$
$61$ $$38 + 13 T + T^{2}$$
$67$ $$-8 - 6 T + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$38 + 13 T + T^{2}$$
$79$ $$-64 + 4 T + T^{2}$$
$83$ $$-36 + 3 T + T^{2}$$
$89$ $$94 + 23 T + T^{2}$$
$97$ $$152 + 26 T + T^{2}$$