# Properties

 Label 900.2.a.e Level $900$ Weight $2$ Character orbit 900.a Self dual yes Analytic conductor $7.187$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 300) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{7} + O(q^{10})$$ $$q + q^{7} - 6q^{11} - 5q^{13} - 6q^{17} + 5q^{19} - 6q^{23} + 6q^{29} - q^{31} - 2q^{37} + q^{43} + 6q^{47} - 6q^{49} - 12q^{53} + 6q^{59} - 13q^{61} - 11q^{67} - 2q^{73} - 6q^{77} + 8q^{79} - 6q^{83} - 5q^{91} + 7q^{97} + O(q^{100})$$

## 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 1.00000 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.e 1
3.b odd 2 1 300.2.a.b 1
4.b odd 2 1 3600.2.a.s 1
5.b even 2 1 900.2.a.c 1
5.c odd 4 2 900.2.d.a 2
12.b even 2 1 1200.2.a.n 1
15.d odd 2 1 300.2.a.c yes 1
15.e even 4 2 300.2.d.a 2
20.d odd 2 1 3600.2.a.z 1
20.e even 4 2 3600.2.f.v 2
24.f even 2 1 4800.2.a.p 1
24.h odd 2 1 4800.2.a.ce 1
60.h even 2 1 1200.2.a.f 1
60.l odd 4 2 1200.2.f.a 2
120.i odd 2 1 4800.2.a.o 1
120.m even 2 1 4800.2.a.cf 1
120.q odd 4 2 4800.2.f.bi 2
120.w even 4 2 4800.2.f.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.a.b 1 3.b odd 2 1
300.2.a.c yes 1 15.d odd 2 1
300.2.d.a 2 15.e even 4 2
900.2.a.c 1 5.b even 2 1
900.2.a.e 1 1.a even 1 1 trivial
900.2.d.a 2 5.c odd 4 2
1200.2.a.f 1 60.h even 2 1
1200.2.a.n 1 12.b even 2 1
1200.2.f.a 2 60.l odd 4 2
3600.2.a.s 1 4.b odd 2 1
3600.2.a.z 1 20.d odd 2 1
3600.2.f.v 2 20.e even 4 2
4800.2.a.o 1 120.i odd 2 1
4800.2.a.p 1 24.f even 2 1
4800.2.a.ce 1 24.h odd 2 1
4800.2.a.cf 1 120.m even 2 1
4800.2.f.b 2 120.w even 4 2
4800.2.f.bi 2 120.q odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(900))$$:

 $$T_{7} - 1$$ $$T_{11} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-1 + T$$
$11$ $$6 + T$$
$13$ $$5 + T$$
$17$ $$6 + T$$
$19$ $$-5 + T$$
$23$ $$6 + T$$
$29$ $$-6 + T$$
$31$ $$1 + T$$
$37$ $$2 + T$$
$41$ $$T$$
$43$ $$-1 + T$$
$47$ $$-6 + T$$
$53$ $$12 + T$$
$59$ $$-6 + T$$
$61$ $$13 + T$$
$67$ $$11 + T$$
$71$ $$T$$
$73$ $$2 + T$$
$79$ $$-8 + T$$
$83$ $$6 + T$$
$89$ $$T$$
$97$ $$-7 + T$$