# Properties

 Label 900.3.g.a Level $900$ Weight $3$ Character orbit 900.g Analytic conductor $24.523$ Analytic rank $0$ Dimension $2$ CM no 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.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 3\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{7} +O(q^{10})$$ $$q - q^{7} + \beta q^{11} -7 q^{13} -\beta q^{17} -7 q^{19} -7 \beta q^{23} -7 \beta q^{29} + 17 q^{31} -16 q^{37} + 12 \beta q^{41} -55 q^{43} -11 \beta q^{47} -48 q^{49} -20 \beta q^{53} -13 \beta q^{59} + 65 q^{61} -49 q^{67} + 12 \beta q^{71} -88 q^{73} -\beta q^{77} -40 q^{79} -37 \beta q^{83} -24 \beta q^{89} + 7 q^{91} + 41 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{7} + O(q^{10})$$ $$2q - 2q^{7} - 14q^{13} - 14q^{19} + 34q^{31} - 32q^{37} - 110q^{43} - 96q^{49} + 130q^{61} - 98q^{67} - 176q^{73} - 80q^{79} + 14q^{91} + 82q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
701.1
 − 1.41421i 1.41421i
0 0 0 0 0 −1.00000 0 0 0
701.2 0 0 0 0 0 −1.00000 0 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.g.a 2
3.b odd 2 1 inner 900.3.g.a 2
4.b odd 2 1 3600.3.l.g 2
5.b even 2 1 900.3.g.b yes 2
5.c odd 4 2 900.3.b.a 4
12.b even 2 1 3600.3.l.g 2
15.d odd 2 1 900.3.g.b yes 2
15.e even 4 2 900.3.b.a 4
20.d odd 2 1 3600.3.l.f 2
20.e even 4 2 3600.3.c.d 4
60.h even 2 1 3600.3.l.f 2
60.l odd 4 2 3600.3.c.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.3.b.a 4 5.c odd 4 2
900.3.b.a 4 15.e even 4 2
900.3.g.a 2 1.a even 1 1 trivial
900.3.g.a 2 3.b odd 2 1 inner
900.3.g.b yes 2 5.b even 2 1
900.3.g.b yes 2 15.d odd 2 1
3600.3.c.d 4 20.e even 4 2
3600.3.c.d 4 60.l odd 4 2
3600.3.l.f 2 20.d odd 2 1
3600.3.l.f 2 60.h even 2 1
3600.3.l.g 2 4.b odd 2 1
3600.3.l.g 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} + 1$$ acting on $$S_{3}^{\mathrm{new}}(900, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$18 + T^{2}$$
$13$ $$( 7 + T )^{2}$$
$17$ $$18 + T^{2}$$
$19$ $$( 7 + T )^{2}$$
$23$ $$882 + T^{2}$$
$29$ $$882 + T^{2}$$
$31$ $$( -17 + T )^{2}$$
$37$ $$( 16 + T )^{2}$$
$41$ $$2592 + T^{2}$$
$43$ $$( 55 + T )^{2}$$
$47$ $$2178 + T^{2}$$
$53$ $$7200 + T^{2}$$
$59$ $$3042 + T^{2}$$
$61$ $$( -65 + T )^{2}$$
$67$ $$( 49 + T )^{2}$$
$71$ $$2592 + T^{2}$$
$73$ $$( 88 + T )^{2}$$
$79$ $$( 40 + T )^{2}$$
$83$ $$24642 + T^{2}$$
$89$ $$10368 + T^{2}$$
$97$ $$( -41 + T )^{2}$$