# Properties

 Label 880.1.i.a Level $880$ Weight $1$ Character orbit 880.i Self dual yes Analytic conductor $0.439$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -11, -55, 5 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$880 = 2^{4} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 880.i (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.439177211117$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{5}, \sqrt{-11})$$ Artin image: $D_4$ Artin field: Galois closure of 4.2.4400.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{5} + q^{9}+O(q^{10})$$ q - q^5 + q^9 $$q - q^{5} + q^{9} + q^{11} + q^{25} + 2 q^{31} - q^{45} - q^{49} - q^{55} - 2 q^{59} - 2 q^{71} + q^{81} - 2 q^{89} + q^{99}+O(q^{100})$$ q - q^5 + q^9 + q^11 + q^25 + 2 * q^31 - q^45 - q^49 - q^55 - 2 * q^59 - 2 * q^71 + q^81 - 2 * q^89 + q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$$.

 $$n$$ $$111$$ $$177$$ $$321$$ $$661$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
769.1
 0
0 0 0 −1.00000 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 $$\Q(\sqrt{5})$$
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
55.d odd 2 1 CM by $$\Q(\sqrt{-55})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.1.i.a 1
4.b odd 2 1 55.1.d.a 1
5.b even 2 1 RM 880.1.i.a 1
8.b even 2 1 3520.1.i.a 1
8.d odd 2 1 3520.1.i.b 1
11.b odd 2 1 CM 880.1.i.a 1
12.b even 2 1 495.1.h.a 1
20.d odd 2 1 55.1.d.a 1
20.e even 4 2 275.1.c.a 1
28.d even 2 1 2695.1.g.c 1
28.f even 6 2 2695.1.q.b 2
28.g odd 6 2 2695.1.q.c 2
40.e odd 2 1 3520.1.i.b 1
40.f even 2 1 3520.1.i.a 1
44.c even 2 1 55.1.d.a 1
44.g even 10 4 605.1.h.a 4
44.h odd 10 4 605.1.h.a 4
55.d odd 2 1 CM 880.1.i.a 1
60.h even 2 1 495.1.h.a 1
60.l odd 4 2 2475.1.b.a 1
88.b odd 2 1 3520.1.i.a 1
88.g even 2 1 3520.1.i.b 1
132.d odd 2 1 495.1.h.a 1
140.c even 2 1 2695.1.g.c 1
140.p odd 6 2 2695.1.q.c 2
140.s even 6 2 2695.1.q.b 2
220.g even 2 1 55.1.d.a 1
220.i odd 4 2 275.1.c.a 1
220.n odd 10 4 605.1.h.a 4
220.o even 10 4 605.1.h.a 4
220.v even 20 8 3025.1.x.a 4
220.w odd 20 8 3025.1.x.a 4
308.g odd 2 1 2695.1.g.c 1
308.m odd 6 2 2695.1.q.b 2
308.n even 6 2 2695.1.q.c 2
440.c even 2 1 3520.1.i.b 1
440.o odd 2 1 3520.1.i.a 1
660.g odd 2 1 495.1.h.a 1
660.q even 4 2 2475.1.b.a 1
1540.b odd 2 1 2695.1.g.c 1
1540.be even 6 2 2695.1.q.c 2
1540.bj odd 6 2 2695.1.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.1.d.a 1 4.b odd 2 1
55.1.d.a 1 20.d odd 2 1
55.1.d.a 1 44.c even 2 1
55.1.d.a 1 220.g even 2 1
275.1.c.a 1 20.e even 4 2
275.1.c.a 1 220.i odd 4 2
495.1.h.a 1 12.b even 2 1
495.1.h.a 1 60.h even 2 1
495.1.h.a 1 132.d odd 2 1
495.1.h.a 1 660.g odd 2 1
605.1.h.a 4 44.g even 10 4
605.1.h.a 4 44.h odd 10 4
605.1.h.a 4 220.n odd 10 4
605.1.h.a 4 220.o even 10 4
880.1.i.a 1 1.a even 1 1 trivial
880.1.i.a 1 5.b even 2 1 RM
880.1.i.a 1 11.b odd 2 1 CM
880.1.i.a 1 55.d odd 2 1 CM
2475.1.b.a 1 60.l odd 4 2
2475.1.b.a 1 660.q even 4 2
2695.1.g.c 1 28.d even 2 1
2695.1.g.c 1 140.c even 2 1
2695.1.g.c 1 308.g odd 2 1
2695.1.g.c 1 1540.b odd 2 1
2695.1.q.b 2 28.f even 6 2
2695.1.q.b 2 140.s even 6 2
2695.1.q.b 2 308.m odd 6 2
2695.1.q.b 2 1540.bj odd 6 2
2695.1.q.c 2 28.g odd 6 2
2695.1.q.c 2 140.p odd 6 2
2695.1.q.c 2 308.n even 6 2
2695.1.q.c 2 1540.be even 6 2
3025.1.x.a 4 220.v even 20 8
3025.1.x.a 4 220.w odd 20 8
3520.1.i.a 1 8.b even 2 1
3520.1.i.a 1 40.f even 2 1
3520.1.i.a 1 88.b odd 2 1
3520.1.i.a 1 440.o odd 2 1
3520.1.i.b 1 8.d odd 2 1
3520.1.i.b 1 40.e odd 2 1
3520.1.i.b 1 88.g even 2 1
3520.1.i.b 1 440.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{1}^{\mathrm{new}}(880, [\chi])$$.

## Hecke characteristic polynomials

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