# Properties

 Label 900.2.a.a Level $900$ Weight $2$ Character orbit 900.a Self dual yes Analytic conductor $7.187$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 900.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.18653618192$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 60) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 4q^{7} + O(q^{10})$$ $$q - 4q^{7} + 4q^{11} + 4q^{17} + 4q^{23} + 6q^{29} + 4q^{31} + 8q^{37} + 10q^{41} - 4q^{43} - 4q^{47} + 9q^{49} - 12q^{53} - 4q^{59} + 2q^{61} + 4q^{67} + 8q^{73} - 16q^{77} - 12q^{79} + 4q^{83} + 10q^{89} - 8q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 0 0 −4.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.a.a 1
3.b odd 2 1 300.2.a.a 1
4.b odd 2 1 3600.2.a.bm 1
5.b even 2 1 900.2.a.h 1
5.c odd 4 2 180.2.d.a 2
12.b even 2 1 1200.2.a.s 1
15.d odd 2 1 300.2.a.d 1
15.e even 4 2 60.2.d.a 2
20.d odd 2 1 3600.2.a.d 1
20.e even 4 2 720.2.f.c 2
24.f even 2 1 4800.2.a.bf 1
24.h odd 2 1 4800.2.a.bn 1
40.i odd 4 2 2880.2.f.l 2
40.k even 4 2 2880.2.f.p 2
45.k odd 12 4 1620.2.r.d 4
45.l even 12 4 1620.2.r.c 4
60.h even 2 1 1200.2.a.a 1
60.l odd 4 2 240.2.f.b 2
105.k odd 4 2 2940.2.k.c 2
105.w odd 12 4 2940.2.bb.e 4
105.x even 12 4 2940.2.bb.d 4
120.i odd 2 1 4800.2.a.bj 1
120.m even 2 1 4800.2.a.bk 1
120.q odd 4 2 960.2.f.c 2
120.w even 4 2 960.2.f.f 2
240.z odd 4 2 3840.2.d.b 2
240.bb even 4 2 3840.2.d.r 2
240.bd odd 4 2 3840.2.d.be 2
240.bf even 4 2 3840.2.d.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.d.a 2 15.e even 4 2
180.2.d.a 2 5.c odd 4 2
240.2.f.b 2 60.l odd 4 2
300.2.a.a 1 3.b odd 2 1
300.2.a.d 1 15.d odd 2 1
720.2.f.c 2 20.e even 4 2
900.2.a.a 1 1.a even 1 1 trivial
900.2.a.h 1 5.b even 2 1
960.2.f.c 2 120.q odd 4 2
960.2.f.f 2 120.w even 4 2
1200.2.a.a 1 60.h even 2 1
1200.2.a.s 1 12.b even 2 1
1620.2.r.c 4 45.l even 12 4
1620.2.r.d 4 45.k odd 12 4
2880.2.f.l 2 40.i odd 4 2
2880.2.f.p 2 40.k even 4 2
2940.2.k.c 2 105.k odd 4 2
2940.2.bb.d 4 105.x even 12 4
2940.2.bb.e 4 105.w odd 12 4
3600.2.a.d 1 20.d odd 2 1
3600.2.a.bm 1 4.b odd 2 1
3840.2.d.b 2 240.z odd 4 2
3840.2.d.o 2 240.bf even 4 2
3840.2.d.r 2 240.bb even 4 2
3840.2.d.be 2 240.bd odd 4 2
4800.2.a.bf 1 24.f even 2 1
4800.2.a.bj 1 120.i odd 2 1
4800.2.a.bk 1 120.m even 2 1
4800.2.a.bn 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(900))$$:

 $$T_{7} + 4$$ $$T_{11} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$4 + T$$
$11$ $$-4 + T$$
$13$ $$T$$
$17$ $$-4 + T$$
$19$ $$T$$
$23$ $$-4 + T$$
$29$ $$-6 + T$$
$31$ $$-4 + T$$
$37$ $$-8 + T$$
$41$ $$-10 + T$$
$43$ $$4 + T$$
$47$ $$4 + T$$
$53$ $$12 + T$$
$59$ $$4 + T$$
$61$ $$-2 + T$$
$67$ $$-4 + T$$
$71$ $$T$$
$73$ $$-8 + T$$
$79$ $$12 + T$$
$83$ $$-4 + T$$
$89$ $$-10 + T$$
$97$ $$8 + T$$