# Properties

 Label 927.1.d.c Level $927$ Weight $1$ Character orbit 927.d Self dual yes Analytic conductor $0.463$ Analytic rank $0$ Dimension $4$ Projective image $D_{10}$ CM discriminant -103 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$927 = 3^{2} \cdot 103$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 927.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.462633266711$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{20})^+$$ Defining polynomial: $$x^{4} - 5 x^{2} + 5$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{10}$$ Projective field: Galois closure of 10.0.27349864083.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} + ( -1 - \beta_{2} ) q^{7} + ( -\beta_{1} - \beta_{3} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} + ( -1 - \beta_{2} ) q^{7} + ( -\beta_{1} - \beta_{3} ) q^{8} -\beta_{2} q^{13} + ( \beta_{1} + \beta_{3} ) q^{14} + ( 2 + 2 \beta_{2} ) q^{16} + \beta_{3} q^{17} + \beta_{2} q^{19} + \beta_{1} q^{23} + q^{25} + \beta_{3} q^{26} + ( -3 - 2 \beta_{2} ) q^{28} -\beta_{3} q^{29} + ( -\beta_{1} - \beta_{3} ) q^{32} + ( -1 - 2 \beta_{2} ) q^{34} -\beta_{3} q^{38} + \beta_{1} q^{41} + ( -3 - \beta_{2} ) q^{46} + ( 1 + \beta_{2} ) q^{49} -\beta_{1} q^{50} + ( -1 - \beta_{2} ) q^{52} + ( 2 \beta_{1} + \beta_{3} ) q^{56} + ( 1 + 2 \beta_{2} ) q^{58} -\beta_{3} q^{59} + ( 1 + \beta_{2} ) q^{61} + ( 2 + \beta_{2} ) q^{64} + ( \beta_{1} + \beta_{3} ) q^{68} + ( 1 + \beta_{2} ) q^{76} -\beta_{2} q^{79} + ( -3 - \beta_{2} ) q^{82} + \beta_{3} q^{83} + q^{91} + ( 2 \beta_{1} + \beta_{3} ) q^{92} + ( 1 + \beta_{2} ) q^{97} + ( -\beta_{1} - \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{4} - 2q^{7} + O(q^{10})$$ $$4q + 6q^{4} - 2q^{7} + 2q^{13} + 4q^{16} - 2q^{19} + 4q^{25} - 8q^{28} - 10q^{46} + 2q^{49} - 2q^{52} + 2q^{61} + 6q^{64} + 2q^{76} + 2q^{79} - 10q^{82} + 4q^{91} + 2q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/927\mathbb{Z}\right)^\times$$.

 $$n$$ $$722$$ $$829$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
514.1
 1.90211 1.17557 −1.17557 −1.90211
−1.90211 0 2.61803 0 0 −1.61803 −3.07768 0 0
514.2 −1.17557 0 0.381966 0 0 0.618034 0.726543 0 0
514.3 1.17557 0 0.381966 0 0 0.618034 −0.726543 0 0
514.4 1.90211 0 2.61803 0 0 −1.61803 3.07768 0 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
103.b odd 2 1 CM by $$\Q(\sqrt{-103})$$
3.b odd 2 1 inner
309.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 927.1.d.c 4
3.b odd 2 1 inner 927.1.d.c 4
103.b odd 2 1 CM 927.1.d.c 4
309.c even 2 1 inner 927.1.d.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
927.1.d.c 4 1.a even 1 1 trivial
927.1.d.c 4 3.b odd 2 1 inner
927.1.d.c 4 103.b odd 2 1 CM
927.1.d.c 4 309.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 5 T_{2}^{2} + 5$$ acting on $$S_{1}^{\mathrm{new}}(927, [\chi])$$.

## Hecke characteristic polynomials

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