# Properties

 Label 900.1.c.a Level $900$ Weight $1$ Character orbit 900.c Self dual yes Analytic conductor $0.449$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -4, -15, 60 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 900.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.449158511370$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 180) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(i, \sqrt{15})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.13500.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} - q^{8} + q^{16} + 2 q^{17} - q^{32} - 2 q^{34} + q^{49} + 2 q^{53} - 2 q^{61} + q^{64} + 2 q^{68} - q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$451$$ $$577$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 0
−1.00000 0 1.00000 0 0 0 −1.00000 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
60.h even 2 1 RM by $$\Q(\sqrt{15})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.1.c.a 1
3.b odd 2 1 900.1.c.b 1
4.b odd 2 1 CM 900.1.c.a 1
5.b even 2 1 900.1.c.b 1
5.c odd 4 2 180.1.f.a 2
12.b even 2 1 900.1.c.b 1
15.d odd 2 1 CM 900.1.c.a 1
15.e even 4 2 180.1.f.a 2
20.d odd 2 1 900.1.c.b 1
20.e even 4 2 180.1.f.a 2
40.i odd 4 2 2880.1.j.b 2
40.k even 4 2 2880.1.j.b 2
45.k odd 12 4 1620.1.p.b 4
45.l even 12 4 1620.1.p.b 4
60.h even 2 1 RM 900.1.c.a 1
60.l odd 4 2 180.1.f.a 2
120.q odd 4 2 2880.1.j.b 2
120.w even 4 2 2880.1.j.b 2
180.v odd 12 4 1620.1.p.b 4
180.x even 12 4 1620.1.p.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.1.f.a 2 5.c odd 4 2
180.1.f.a 2 15.e even 4 2
180.1.f.a 2 20.e even 4 2
180.1.f.a 2 60.l odd 4 2
900.1.c.a 1 1.a even 1 1 trivial
900.1.c.a 1 4.b odd 2 1 CM
900.1.c.a 1 15.d odd 2 1 CM
900.1.c.a 1 60.h even 2 1 RM
900.1.c.b 1 3.b odd 2 1
900.1.c.b 1 5.b even 2 1
900.1.c.b 1 12.b even 2 1
900.1.c.b 1 20.d odd 2 1
1620.1.p.b 4 45.k odd 12 4
1620.1.p.b 4 45.l even 12 4
1620.1.p.b 4 180.v odd 12 4
1620.1.p.b 4 180.x even 12 4
2880.1.j.b 2 40.i odd 4 2
2880.1.j.b 2 40.k even 4 2
2880.1.j.b 2 120.q odd 4 2
2880.1.j.b 2 120.w even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{17} - 2$$ acting on $$S_{1}^{\mathrm{new}}(900, [\chi])$$.

## Hecke characteristic polynomials

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