# Properties

 Label 900.3.b.a Level $900$ Weight $3$ Character orbit 900.b Analytic conductor $24.523$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 900.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{8}^{2} q^{7} +O(q^{10})$$ $$q + \zeta_{8}^{2} q^{7} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{11} -7 \zeta_{8}^{2} q^{13} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{17} + 7 q^{19} + ( -21 \zeta_{8} + 21 \zeta_{8}^{3} ) q^{23} + ( -21 \zeta_{8} - 21 \zeta_{8}^{3} ) q^{29} + 17 q^{31} + 16 \zeta_{8}^{2} q^{37} + ( -36 \zeta_{8} - 36 \zeta_{8}^{3} ) q^{41} -55 \zeta_{8}^{2} q^{43} + ( 33 \zeta_{8} - 33 \zeta_{8}^{3} ) q^{47} + 48 q^{49} + ( -60 \zeta_{8} + 60 \zeta_{8}^{3} ) q^{53} + ( -39 \zeta_{8} - 39 \zeta_{8}^{3} ) q^{59} + 65 q^{61} + 49 \zeta_{8}^{2} q^{67} + ( -36 \zeta_{8} - 36 \zeta_{8}^{3} ) q^{71} -88 \zeta_{8}^{2} q^{73} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{77} + 40 q^{79} + ( -111 \zeta_{8} + 111 \zeta_{8}^{3} ) q^{83} + ( -72 \zeta_{8} - 72 \zeta_{8}^{3} ) q^{89} + 7 q^{91} -41 \zeta_{8}^{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 28q^{19} + 68q^{31} + 192q^{49} + 260q^{61} + 160q^{79} + 28q^{91} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
0 0 0 0 0 1.00000i 0 0 0
449.2 0 0 0 0 0 1.00000i 0 0 0
449.3 0 0 0 0 0 1.00000i 0 0 0
449.4 0 0 0 0 0 1.00000i 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
5.b even 2 1 inner
15.d 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.b.a 4
3.b odd 2 1 inner 900.3.b.a 4
4.b odd 2 1 3600.3.c.d 4
5.b even 2 1 inner 900.3.b.a 4
5.c odd 4 1 900.3.g.a 2
5.c odd 4 1 900.3.g.b yes 2
12.b even 2 1 3600.3.c.d 4
15.d odd 2 1 inner 900.3.b.a 4
15.e even 4 1 900.3.g.a 2
15.e even 4 1 900.3.g.b yes 2
20.d odd 2 1 3600.3.c.d 4
20.e even 4 1 3600.3.l.f 2
20.e even 4 1 3600.3.l.g 2
60.h even 2 1 3600.3.c.d 4
60.l odd 4 1 3600.3.l.f 2
60.l odd 4 1 3600.3.l.g 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 1$$ acting on $$S_{3}^{\mathrm{new}}(900, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( 18 + T^{2} )^{2}$$
$13$ $$( 49 + T^{2} )^{2}$$
$17$ $$( -18 + T^{2} )^{2}$$
$19$ $$( -7 + T )^{4}$$
$23$ $$( -882 + T^{2} )^{2}$$
$29$ $$( 882 + T^{2} )^{2}$$
$31$ $$( -17 + T )^{4}$$
$37$ $$( 256 + T^{2} )^{2}$$
$41$ $$( 2592 + T^{2} )^{2}$$
$43$ $$( 3025 + T^{2} )^{2}$$
$47$ $$( -2178 + T^{2} )^{2}$$
$53$ $$( -7200 + T^{2} )^{2}$$
$59$ $$( 3042 + T^{2} )^{2}$$
$61$ $$( -65 + T )^{4}$$
$67$ $$( 2401 + T^{2} )^{2}$$
$71$ $$( 2592 + T^{2} )^{2}$$
$73$ $$( 7744 + T^{2} )^{2}$$
$79$ $$( -40 + T )^{4}$$
$83$ $$( -24642 + T^{2} )^{2}$$
$89$ $$( 10368 + T^{2} )^{2}$$
$97$ $$( 1681 + T^{2} )^{2}$$