# Properties

 Label 2151.1.d.c Level $2151$ Weight $1$ Character orbit 2151.d Self dual yes Analytic conductor $1.073$ Analytic rank $0$ Dimension $2$ Projective image $D_{5}$ CM discriminant -239 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2151 = 3^{2} \cdot 239$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2151.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.07348884217$$ Analytic rank: $$0$$ 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 239) Projective image $$D_{5}$$ Projective field Galois closure of 5.1.57121.1 Artin image $D_{10}$ Artin field Galois closure of 10.0.792862499763.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta ) q^{2} + ( 1 - \beta ) q^{4} + ( 1 - \beta ) q^{5} + q^{8} +O(q^{10})$$ $$q + ( 1 - \beta ) q^{2} + ( 1 - \beta ) q^{4} + ( 1 - \beta ) q^{5} + q^{8} + ( 2 - \beta ) q^{10} + \beta q^{11} -2 q^{17} + ( 2 - \beta ) q^{20} - q^{22} + ( 1 - \beta ) q^{25} + \beta q^{29} + ( -1 + \beta ) q^{31} - q^{32} + ( -2 + 2 \beta ) q^{34} + ( 1 - \beta ) q^{40} - q^{44} + q^{49} + ( 2 - \beta ) q^{50} - q^{55} - q^{58} + 2 q^{61} + ( -2 + \beta ) q^{62} + ( -1 + \beta ) q^{64} + 2 q^{67} + ( -2 + 2 \beta ) q^{68} + \beta q^{71} + \beta q^{83} + ( -2 + 2 \beta ) q^{85} + \beta q^{88} + ( 1 - \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + q^{4} + q^{5} + 2q^{8} + O(q^{10})$$ $$2q + q^{2} + q^{4} + q^{5} + 2q^{8} + 3q^{10} + q^{11} - 4q^{17} + 3q^{20} - 2q^{22} + q^{25} + q^{29} - q^{31} - 2q^{32} - 2q^{34} + q^{40} - 2q^{44} + 2q^{49} + 3q^{50} - 2q^{55} - 2q^{58} + 4q^{61} - 3q^{62} - q^{64} + 4q^{67} - 2q^{68} + q^{71} + q^{83} - 2q^{85} + q^{88} + q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2151\mathbb{Z}\right)^\times$$.

 $$n$$ $$479$$ $$1441$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
955.1
 1.61803 −0.618034
−0.618034 0 −0.618034 −0.618034 0 0 1.00000 0 0.381966
955.2 1.61803 0 1.61803 1.61803 0 0 1.00000 0 2.61803
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
239.b odd 2 1 CM by $$\Q(\sqrt{-239})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2151.1.d.c 2
3.b odd 2 1 239.1.b.b 2
12.b even 2 1 3824.1.h.b 2
239.b odd 2 1 CM 2151.1.d.c 2
717.b even 2 1 239.1.b.b 2
2868.e odd 2 1 3824.1.h.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
239.1.b.b 2 3.b odd 2 1
239.1.b.b 2 717.b even 2 1
2151.1.d.c 2 1.a even 1 1 trivial
2151.1.d.c 2 239.b odd 2 1 CM
3824.1.h.b 2 12.b even 2 1
3824.1.h.b 2 2868.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2151, [\chi])$$:

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

## Hecke characteristic polynomials

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