# Properties

 Label 3520.1.i.a Level $3520$ Weight $1$ Character orbit 3520.i Self dual yes Analytic conductor $1.757$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -11, -55, 5 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3520 = 2^{6} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3520.i (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.75670884447$$ 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 4.2.17600.1

## $q$-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 $$\left(\mathbb{Z}/3520\mathbb{Z}\right)^\times$$.

 $$n$$ $$321$$ $$1541$$ $$2751$$ $$2817$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$

## 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 3520.1.i.a 1
4.b odd 2 1 3520.1.i.b 1
5.b even 2 1 RM 3520.1.i.a 1
8.b even 2 1 880.1.i.a 1
8.d odd 2 1 55.1.d.a 1
11.b odd 2 1 CM 3520.1.i.a 1
20.d odd 2 1 3520.1.i.b 1
24.f even 2 1 495.1.h.a 1
40.e odd 2 1 55.1.d.a 1
40.f even 2 1 880.1.i.a 1
40.k even 4 2 275.1.c.a 1
44.c even 2 1 3520.1.i.b 1
55.d odd 2 1 CM 3520.1.i.a 1
56.e even 2 1 2695.1.g.c 1
56.k odd 6 2 2695.1.q.c 2
56.m even 6 2 2695.1.q.b 2
88.b odd 2 1 880.1.i.a 1
88.g even 2 1 55.1.d.a 1
88.k even 10 4 605.1.h.a 4
88.l odd 10 4 605.1.h.a 4
120.m even 2 1 495.1.h.a 1
120.q odd 4 2 2475.1.b.a 1
220.g even 2 1 3520.1.i.b 1
264.p odd 2 1 495.1.h.a 1
280.n even 2 1 2695.1.g.c 1
280.ba even 6 2 2695.1.q.b 2
280.bi odd 6 2 2695.1.q.c 2
440.c even 2 1 55.1.d.a 1
440.o odd 2 1 880.1.i.a 1
440.w odd 4 2 275.1.c.a 1
440.bh odd 10 4 605.1.h.a 4
440.bm even 10 4 605.1.h.a 4
440.br odd 20 8 3025.1.x.a 4
440.bs even 20 8 3025.1.x.a 4
616.g odd 2 1 2695.1.g.c 1
616.y even 6 2 2695.1.q.c 2
616.z odd 6 2 2695.1.q.b 2
1320.b odd 2 1 495.1.h.a 1
1320.bt even 4 2 2475.1.b.a 1
3080.bc odd 2 1 2695.1.g.c 1
3080.ck odd 6 2 2695.1.q.b 2
3080.cq even 6 2 2695.1.q.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.1.d.a 1 8.d odd 2 1
55.1.d.a 1 40.e odd 2 1
55.1.d.a 1 88.g even 2 1
55.1.d.a 1 440.c even 2 1
275.1.c.a 1 40.k even 4 2
275.1.c.a 1 440.w odd 4 2
495.1.h.a 1 24.f even 2 1
495.1.h.a 1 120.m even 2 1
495.1.h.a 1 264.p odd 2 1
495.1.h.a 1 1320.b odd 2 1
605.1.h.a 4 88.k even 10 4
605.1.h.a 4 88.l odd 10 4
605.1.h.a 4 440.bh odd 10 4
605.1.h.a 4 440.bm even 10 4
880.1.i.a 1 8.b even 2 1
880.1.i.a 1 40.f even 2 1
880.1.i.a 1 88.b odd 2 1
880.1.i.a 1 440.o odd 2 1
2475.1.b.a 1 120.q odd 4 2
2475.1.b.a 1 1320.bt even 4 2
2695.1.g.c 1 56.e even 2 1
2695.1.g.c 1 280.n even 2 1
2695.1.g.c 1 616.g odd 2 1
2695.1.g.c 1 3080.bc odd 2 1
2695.1.q.b 2 56.m even 6 2
2695.1.q.b 2 280.ba even 6 2
2695.1.q.b 2 616.z odd 6 2
2695.1.q.b 2 3080.ck odd 6 2
2695.1.q.c 2 56.k odd 6 2
2695.1.q.c 2 280.bi odd 6 2
2695.1.q.c 2 616.y even 6 2
2695.1.q.c 2 3080.cq even 6 2
3025.1.x.a 4 440.br odd 20 8
3025.1.x.a 4 440.bs even 20 8
3520.1.i.a 1 1.a even 1 1 trivial
3520.1.i.a 1 5.b even 2 1 RM
3520.1.i.a 1 11.b odd 2 1 CM
3520.1.i.a 1 55.d odd 2 1 CM
3520.1.i.b 1 4.b odd 2 1
3520.1.i.b 1 20.d odd 2 1
3520.1.i.b 1 44.c even 2 1
3520.1.i.b 1 220.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3520, [\chi])$$:

 $$T_{3}$$ T3 $$T_{31} - 2$$ T31 - 2

## Hecke characteristic polynomials

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