# Properties

 Label 2151.2.a.c Level $2151$ Weight $2$ Character orbit 2151.a Self dual yes Analytic conductor $17.176$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2151 = 3^{2} \cdot 239$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2151.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$17.1758214748$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 717) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.61803 −0.618034
−1.61803 0 0.618034 1.00000 0 0 2.23607 0 −1.61803
1.2 0.618034 0 −1.61803 1.00000 0 0 −2.23607 0 0.618034
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$239$$ $$-1$$

## 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 2151.2.a.c 2
3.b odd 2 1 717.2.a.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
717.2.a.a 2 3.b odd 2 1
2151.2.a.c 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2151))$$:

 $$T_{2}^{2} + T_{2} - 1$$ $$T_{5} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$-1 + 4 T + T^{2}$$
$13$ $$4 + 6 T + T^{2}$$
$17$ $$( -5 + T )^{2}$$
$19$ $$4 - 6 T + T^{2}$$
$23$ $$-16 - 4 T + T^{2}$$
$29$ $$-11 + 6 T + T^{2}$$
$31$ $$-41 + 4 T + T^{2}$$
$37$ $$( 6 + T )^{2}$$
$41$ $$-20 + 10 T + T^{2}$$
$43$ $$( 10 + T )^{2}$$
$47$ $$-44 + 2 T + T^{2}$$
$53$ $$-16 - 4 T + T^{2}$$
$59$ $$-4 + 2 T + T^{2}$$
$61$ $$-55 - 10 T + T^{2}$$
$67$ $$11 + 8 T + T^{2}$$
$71$ $$-64 + 8 T + T^{2}$$
$73$ $$-44 - 12 T + T^{2}$$
$79$ $$-44 + 12 T + T^{2}$$
$83$ $$-1 - 4 T + T^{2}$$
$89$ $$-64 + 8 T + T^{2}$$
$97$ $$-116 - 6 T + T^{2}$$