# Properties

 Label 327.1.d.a Level $327$ Weight $1$ Character orbit 327.d Self dual yes Analytic conductor $0.163$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -327 Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$327 = 3 \cdot 109$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 327.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.163194259131$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.327.1 Artin image: $S_3$ Artin field: Galois closure of 3.1.327.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} - q^{6} - q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + q^{3} - q^{6} - q^{7} + q^{8} + q^{9} + 2q^{11} + q^{14} - q^{16} - q^{17} - q^{18} - q^{21} - 2q^{22} - q^{23} + q^{24} + q^{25} + q^{27} - q^{31} + 2q^{33} + q^{34} - q^{41} + q^{42} - q^{43} + q^{46} - q^{47} - q^{48} - q^{50} - q^{51} + 2q^{53} - q^{54} - q^{56} - q^{59} - q^{61} + q^{62} - q^{63} + q^{64} - 2q^{66} - q^{69} + q^{72} - q^{73} + q^{75} - 2q^{77} + q^{81} + q^{82} + q^{86} + 2q^{88} - q^{93} + q^{94} - q^{97} + 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/327\mathbb{Z}\right)^\times$$.

 $$n$$ $$110$$ $$115$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
326.1
 0
−1.00000 1.00000 0 0 −1.00000 −1.00000 1.00000 1.00000 0
 $$n$$: 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
327.d odd 2 1 CM by $$\Q(\sqrt{-327})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 327.1.d.a 1
3.b odd 2 1 327.1.d.c yes 1
109.b even 2 1 327.1.d.c yes 1
327.d odd 2 1 CM 327.1.d.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
327.1.d.a 1 1.a even 1 1 trivial
327.1.d.a 1 327.d odd 2 1 CM
327.1.d.c yes 1 3.b odd 2 1
327.1.d.c yes 1 109.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(327, [\chi])$$.

## Hecke characteristic polynomials

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