# Properties

 Label 2151.1.d.a Level $2151$ Weight $1$ Character orbit 2151.d Self dual yes Analytic conductor $1.073$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 239) Projective image $$D_{3}$$ Projective field Galois closure of 3.1.239.1 Artin image $D_6$ Artin field Galois closure of 6.0.1542267.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{5} - q^{8} + O(q^{10})$$ $$q + q^{2} + q^{5} - q^{8} + q^{10} + q^{11} - q^{16} + q^{17} + q^{22} + q^{29} - q^{31} + q^{34} - q^{40} + q^{49} + q^{55} + q^{58} - q^{61} - q^{62} + q^{64} - q^{67} - 2q^{71} - q^{80} + q^{83} + q^{85} - q^{88} + q^{98} + O(q^{100})$$

## 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
 0
1.00000 0 0 1.00000 0 0 −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 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.a 1
3.b odd 2 1 239.1.b.a 1
12.b even 2 1 3824.1.h.a 1
239.b odd 2 1 CM 2151.1.d.a 1
717.b even 2 1 239.1.b.a 1
2868.e odd 2 1 3824.1.h.a 1

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

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

## Hecke characteristic polynomials

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