# Properties

 Label 2151.1.d.e Level $2151$ Weight $1$ Character orbit 2151.d Self dual yes Analytic conductor $1.073$ Analytic rank $0$ Dimension $4$ Projective image $D_{15}$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{15})^+$$ Defining polynomial: $$x^{4} - x^{3} - 4 x^{2} + 4 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 239) Projective image $$D_{15}$$ Projective field Galois closure of 15.1.44543599279432079.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{1} - \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{3} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{1} - \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{3} ) q^{8} + ( -1 + \beta_{3} ) q^{10} + ( \beta_{2} + \beta_{3} ) q^{11} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{16} + q^{17} + ( \beta_{1} - \beta_{2} ) q^{20} + ( -1 - \beta_{2} - 2 \beta_{3} ) q^{22} + ( 1 - \beta_{2} - \beta_{3} ) q^{25} -\beta_{2} q^{29} + \beta_{1} q^{31} + ( 1 - \beta_{1} - \beta_{3} ) q^{32} -\beta_{1} q^{34} + ( -1 + \beta_{1} - \beta_{2} ) q^{40} + ( 2 + \beta_{2} + 2 \beta_{3} ) q^{44} + q^{49} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{50} + ( 1 - \beta_{1} + 2 \beta_{3} ) q^{55} + ( \beta_{1} + \beta_{3} ) q^{58} - q^{61} + ( -2 - \beta_{2} ) q^{62} + ( 1 - \beta_{1} + \beta_{2} ) q^{64} - q^{67} + ( 1 + \beta_{2} ) q^{68} -\beta_{3} q^{71} + ( -2 + \beta_{1} + \beta_{3} ) q^{80} -\beta_{2} q^{83} + ( -1 + \beta_{1} - \beta_{3} ) q^{85} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{88} -\beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - q^{2} + 5q^{4} - q^{5} + q^{8} + O(q^{10})$$ $$4q - q^{2} + 5q^{4} - q^{5} + q^{8} - 6q^{10} - q^{11} + 6q^{16} + 4q^{17} - q^{22} + 5q^{25} - q^{29} + q^{31} + 5q^{32} - q^{34} - 4q^{40} + 5q^{44} + 4q^{49} - q^{55} - q^{58} - 4q^{61} - 9q^{62} + 4q^{64} - 4q^{67} + 5q^{68} + 2q^{71} - 9q^{80} - q^{83} - q^{85} - 4q^{88} - q^{98} + O(q^{100})$$

## 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
 1.82709 1.33826 −0.209057 −1.95630
−1.82709 0 2.33826 0.209057 0 0 −2.44512 0 −0.381966
955.2 −1.33826 0 0.790943 1.95630 0 0 0.279773 0 −2.61803
955.3 0.209057 0 −0.956295 −1.82709 0 0 −0.408977 0 −0.381966
955.4 1.95630 0 2.82709 −1.33826 0 0 3.57433 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.e 4
3.b odd 2 1 239.1.b.c 4
12.b even 2 1 3824.1.h.c 4
239.b odd 2 1 CM 2151.1.d.e 4
717.b even 2 1 239.1.b.c 4
2868.e odd 2 1 3824.1.h.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
239.1.b.c 4 3.b odd 2 1
239.1.b.c 4 717.b even 2 1
2151.1.d.e 4 1.a even 1 1 trivial
2151.1.d.e 4 239.b odd 2 1 CM
3824.1.h.c 4 12.b even 2 1
3824.1.h.c 4 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 $$S_{1}^{\mathrm{new}}(2151, [\chi])$$:

 $$T_{2}^{4} + T_{2}^{3} - 4 T_{2}^{2} - 4 T_{2} + 1$$ $$T_{5}^{4} + T_{5}^{3} - 4 T_{5}^{2} - 4 T_{5} + 1$$

## Hecke characteristic polynomials

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