# Properties

 Label 775.1.d.c Level $775$ Weight $1$ Character orbit 775.d Self dual yes Analytic conductor $0.387$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -31 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$775 = 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 775.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.386775384791$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 155) Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.120125.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + 2 q^{4} + \beta q^{7} -\beta q^{8} + q^{9} +O(q^{10})$$ $$q -\beta q^{2} + 2 q^{4} + \beta q^{7} -\beta q^{8} + q^{9} -3 q^{14} + q^{16} -\beta q^{18} - q^{19} + 2 \beta q^{28} - q^{31} + 2 q^{36} + \beta q^{38} - q^{41} + 2 q^{49} -3 q^{56} + q^{59} + \beta q^{62} + \beta q^{63} - q^{64} + q^{71} -\beta q^{72} -2 q^{76} + q^{81} + \beta q^{82} -\beta q^{97} -2 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4} + 2 q^{9} + O(q^{10})$$ $$2 q + 4 q^{4} + 2 q^{9} - 6 q^{14} + 2 q^{16} - 2 q^{19} - 2 q^{31} + 4 q^{36} - 2 q^{41} + 4 q^{49} - 6 q^{56} + 2 q^{59} - 2 q^{64} + 2 q^{71} - 4 q^{76} + 2 q^{81} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/775\mathbb{Z}\right)^\times$$.

 $$n$$ $$251$$ $$652$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
526.1
 1.73205 −1.73205
−1.73205 0 2.00000 0 0 1.73205 −1.73205 1.00000 0
526.2 1.73205 0 2.00000 0 0 −1.73205 1.73205 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
31.b odd 2 1 CM by $$\Q(\sqrt{-31})$$
5.b even 2 1 inner
155.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.1.d.c 2
5.b even 2 1 inner 775.1.d.c 2
5.c odd 4 2 155.1.c.b 2
15.e even 4 2 1395.1.b.c 2
20.e even 4 2 2480.1.k.e 2
31.b odd 2 1 CM 775.1.d.c 2
155.c odd 2 1 inner 775.1.d.c 2
155.f even 4 2 155.1.c.b 2
465.m odd 4 2 1395.1.b.c 2
620.m odd 4 2 2480.1.k.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.1.c.b 2 5.c odd 4 2
155.1.c.b 2 155.f even 4 2
775.1.d.c 2 1.a even 1 1 trivial
775.1.d.c 2 5.b even 2 1 inner
775.1.d.c 2 31.b odd 2 1 CM
775.1.d.c 2 155.c odd 2 1 inner
1395.1.b.c 2 15.e even 4 2
1395.1.b.c 2 465.m odd 4 2
2480.1.k.e 2 20.e even 4 2
2480.1.k.e 2 620.m odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 3$$ acting on $$S_{1}^{\mathrm{new}}(775, [\chi])$$.

## Hecke characteristic polynomials

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