# Properties

 Label 900.3.c.c 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 36) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{2} + 4q^{4} + 8q^{8} + O(q^{10})$$ $$q + 2q^{2} + 4q^{4} + 8q^{8} + 10q^{13} + 16q^{16} + 16q^{17} + 20q^{26} - 40q^{29} + 32q^{32} + 32q^{34} + 70q^{37} + 80q^{41} + 49q^{49} + 40q^{52} - 56q^{53} - 80q^{58} - 22q^{61} + 64q^{64} + 64q^{68} - 110q^{73} + 140q^{74} + 160q^{82} - 160q^{89} + 130q^{97} + 98q^{98} + O(q^{100})$$

## 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.c 1
3.b odd 2 1 900.3.c.b 1
4.b odd 2 1 CM 900.3.c.c 1
5.b even 2 1 36.3.d.a 1
5.c odd 4 2 900.3.f.b 2
12.b even 2 1 900.3.c.b 1
15.d odd 2 1 36.3.d.b yes 1
15.e even 4 2 900.3.f.a 2
20.d odd 2 1 36.3.d.a 1
20.e even 4 2 900.3.f.b 2
40.e odd 2 1 576.3.g.a 1
40.f even 2 1 576.3.g.a 1
45.h odd 6 2 324.3.f.e 2
45.j even 6 2 324.3.f.f 2
60.h even 2 1 36.3.d.b yes 1
60.l odd 4 2 900.3.f.a 2
80.k odd 4 2 2304.3.b.d 2
80.q even 4 2 2304.3.b.d 2
120.i odd 2 1 576.3.g.c 1
120.m even 2 1 576.3.g.c 1
180.n even 6 2 324.3.f.e 2
180.p odd 6 2 324.3.f.f 2
240.t even 4 2 2304.3.b.e 2
240.bm odd 4 2 2304.3.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.d.a 1 5.b even 2 1
36.3.d.a 1 20.d odd 2 1
36.3.d.b yes 1 15.d odd 2 1
36.3.d.b yes 1 60.h even 2 1
324.3.f.e 2 45.h odd 6 2
324.3.f.e 2 180.n even 6 2
324.3.f.f 2 45.j even 6 2
324.3.f.f 2 180.p odd 6 2
576.3.g.a 1 40.e odd 2 1
576.3.g.a 1 40.f even 2 1
576.3.g.c 1 120.i odd 2 1
576.3.g.c 1 120.m even 2 1
900.3.c.b 1 3.b odd 2 1
900.3.c.b 1 12.b even 2 1
900.3.c.c 1 1.a even 1 1 trivial
900.3.c.c 1 4.b odd 2 1 CM
900.3.f.a 2 15.e even 4 2
900.3.f.a 2 60.l odd 4 2
900.3.f.b 2 5.c odd 4 2
900.3.f.b 2 20.e even 4 2
2304.3.b.d 2 80.k odd 4 2
2304.3.b.d 2 80.q even 4 2
2304.3.b.e 2 240.t even 4 2
2304.3.b.e 2 240.bm odd 4 2

## Hecke kernels

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

 $$T_{7}$$ $$T_{13} - 10$$ $$T_{17} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$-10 + T$$
$17$ $$-16 + T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$40 + T$$
$31$ $$T$$
$37$ $$-70 + T$$
$41$ $$-80 + T$$
$43$ $$T$$
$47$ $$T$$
$53$ $$56 + T$$
$59$ $$T$$
$61$ $$22 + T$$
$67$ $$T$$
$71$ $$T$$
$73$ $$110 + T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$160 + T$$
$97$ $$-130 + T$$