# Properties

 Label 2151.1.d.b Level $2151$ Weight $1$ Character orbit 2151.d Analytic conductor $1.073$ Analytic rank $0$ Dimension $2$ Projective image $S_{4}$ CM/RM no 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: no Analytic conductor: $$1.07348884217$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$S_{4}$$ Projective field Galois closure of 4.2.2151.1 Artin image $\GL(2,3)$ Artin field Galois closure of 8.2.9952248951.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{5} -\beta q^{7} + q^{8} +O(q^{10})$$ $$q - q^{2} + q^{5} -\beta q^{7} + q^{8} - q^{10} + q^{11} + \beta q^{13} + \beta q^{14} - q^{16} + q^{17} + \beta q^{19} - q^{22} + \beta q^{23} -\beta q^{26} + q^{29} - q^{31} - q^{34} -\beta q^{35} -\beta q^{37} -\beta q^{38} + q^{40} -\beta q^{41} -\beta q^{46} + \beta q^{47} - q^{49} + q^{55} -\beta q^{56} - q^{58} + q^{61} + q^{62} + q^{64} + \beta q^{65} - q^{67} + \beta q^{70} + \beta q^{74} -\beta q^{77} - q^{80} + \beta q^{82} - q^{83} + q^{85} + q^{88} + 2 q^{91} -\beta q^{94} + \beta q^{95} + q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{5} + 2q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{5} + 2q^{8} - 2q^{10} + 2q^{11} - 2q^{16} + 2q^{17} - 2q^{22} + 2q^{29} - 2q^{31} - 2q^{34} + 2q^{40} - 2q^{49} + 2q^{55} - 2q^{58} + 2q^{61} + 2q^{62} + 2q^{64} - 2q^{67} - 2q^{80} - 2q^{83} + 2q^{85} + 2q^{88} + 4q^{91} + 2q^{98} + O(q^{100})$$

## 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
 1.41421i − 1.41421i
−1.00000 0 0 1.00000 0 1.41421i 1.00000 0 −1.00000
955.2 −1.00000 0 0 1.00000 0 1.41421i 1.00000 0 −1.00000
 $$n$$: 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2151.1.d.b 2
3.b odd 2 1 2151.1.d.d yes 2
239.b odd 2 1 inner 2151.1.d.b 2
717.b even 2 1 2151.1.d.d yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2151.1.d.b 2 1.a even 1 1 trivial
2151.1.d.b 2 239.b odd 2 1 inner
2151.1.d.d yes 2 3.b odd 2 1
2151.1.d.d yes 2 717.b even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

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