# Properties

 Label 900.3.c.a Level $900$ Weight $3$ Character orbit 900.c Self dual yes Analytic conductor $24.523$ Analytic rank $0$ Dimension $1$ CM discriminant -4 Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{2} + 4q^{4} - 8q^{8} + O(q^{10})$$ $$q - 2q^{2} + 4q^{4} - 8q^{8} - 24q^{13} + 16q^{16} + 16q^{17} + 48q^{26} + 42q^{29} - 32q^{32} - 32q^{34} + 24q^{37} + 18q^{41} + 49q^{49} - 96q^{52} - 56q^{53} - 84q^{58} + 22q^{61} + 64q^{64} + 64q^{68} + 96q^{73} - 48q^{74} - 36q^{82} - 78q^{89} + 144q^{97} - 98q^{98} + O(q^{100})$$

## 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
−2.00000 0 4.00000 0 0 0 −8.00000 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 $$\Q(\sqrt{-1})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.c.a 1
3.b odd 2 1 100.3.b.b 1
4.b odd 2 1 CM 900.3.c.a 1
5.b even 2 1 900.3.c.d 1
5.c odd 4 2 180.3.f.c 2
12.b even 2 1 100.3.b.b 1
15.d odd 2 1 100.3.b.a 1
15.e even 4 2 20.3.d.c 2
20.d odd 2 1 900.3.c.d 1
20.e even 4 2 180.3.f.c 2
24.f even 2 1 1600.3.b.c 1
24.h odd 2 1 1600.3.b.c 1
60.h even 2 1 100.3.b.a 1
60.l odd 4 2 20.3.d.c 2
120.i odd 2 1 1600.3.b.a 1
120.m even 2 1 1600.3.b.a 1
120.q odd 4 2 320.3.h.d 2
120.w even 4 2 320.3.h.d 2
240.z odd 4 2 1280.3.e.d 2
240.bb even 4 2 1280.3.e.d 2
240.bd odd 4 2 1280.3.e.a 2
240.bf even 4 2 1280.3.e.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.d.c 2 15.e even 4 2
20.3.d.c 2 60.l odd 4 2
100.3.b.a 1 15.d odd 2 1
100.3.b.a 1 60.h even 2 1
100.3.b.b 1 3.b odd 2 1
100.3.b.b 1 12.b even 2 1
180.3.f.c 2 5.c odd 4 2
180.3.f.c 2 20.e even 4 2
320.3.h.d 2 120.q odd 4 2
320.3.h.d 2 120.w even 4 2
900.3.c.a 1 1.a even 1 1 trivial
900.3.c.a 1 4.b odd 2 1 CM
900.3.c.d 1 5.b even 2 1
900.3.c.d 1 20.d odd 2 1
1280.3.e.a 2 240.bd odd 4 2
1280.3.e.a 2 240.bf even 4 2
1280.3.e.d 2 240.z odd 4 2
1280.3.e.d 2 240.bb even 4 2
1600.3.b.a 1 120.i odd 2 1
1600.3.b.a 1 120.m even 2 1
1600.3.b.c 1 24.f even 2 1
1600.3.b.c 1 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(900, [\chi])$$:

 $$T_{7}$$ $$T_{13} + 24$$ $$T_{17} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$24 + T$$
$17$ $$-16 + T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$-42 + T$$
$31$ $$T$$
$37$ $$-24 + T$$
$41$ $$-18 + T$$
$43$ $$T$$
$47$ $$T$$
$53$ $$56 + T$$
$59$ $$T$$
$61$ $$-22 + T$$
$67$ $$T$$
$71$ $$T$$
$73$ $$-96 + T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$78 + T$$
$97$ $$-144 + T$$