# Properties

 Label 775.1.d.a Level $775$ Weight $1$ Character orbit 775.d Self dual yes Analytic conductor $0.387$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -31, -155, 5 Inner twists $4$

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## 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 155) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{5}, \sqrt{-31})$$ Artin image: $D_4$ Artin field: Galois closure of 4.2.3875.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{4} + q^{9} + O(q^{10})$$ $$q - q^{4} + q^{9} + q^{16} + 2 q^{19} - q^{31} - q^{36} + 2 q^{41} - q^{49} - 2 q^{59} - q^{64} - 2 q^{71} - 2 q^{76} + q^{81} + O(q^{100})$$

## 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
 0
0 0 −1.00000 0 0 0 0 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
5.b even 2 1 RM by $$\Q(\sqrt{5})$$
31.b odd 2 1 CM by $$\Q(\sqrt{-31})$$
155.c odd 2 1 CM by $$\Q(\sqrt{-155})$$

## Twists

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

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

## Hecke kernels

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

## Hecke characteristic polynomials

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