# Properties

 Label 450.2.a.a Level $450$ Weight $2$ Character orbit 450.a Self dual yes Analytic conductor $3.593$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.59326809096$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 90) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - 2q^{7} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} - 2q^{7} - q^{8} - 6q^{11} + 4q^{13} + 2q^{14} + q^{16} - 6q^{17} - 4q^{19} + 6q^{22} - 4q^{26} - 2q^{28} + 6q^{29} - 4q^{31} - q^{32} + 6q^{34} - 8q^{37} + 4q^{38} - 8q^{43} - 6q^{44} - 3q^{49} + 4q^{52} - 6q^{53} + 2q^{56} - 6q^{58} - 6q^{59} + 2q^{61} + 4q^{62} + q^{64} + 4q^{67} - 6q^{68} + 12q^{71} + 10q^{73} + 8q^{74} - 4q^{76} + 12q^{77} - 4q^{79} + 12q^{83} + 8q^{86} + 6q^{88} - 12q^{89} - 8q^{91} - 2q^{97} + 3q^{98} + O(q^{100})$$

## 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
−1.00000 0 1.00000 0 0 −2.00000 −1.00000 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 450.2.a.a 1
3.b odd 2 1 450.2.a.e 1
4.b odd 2 1 3600.2.a.bj 1
5.b even 2 1 90.2.a.b yes 1
5.c odd 4 2 450.2.c.a 2
12.b even 2 1 3600.2.a.ba 1
15.d odd 2 1 90.2.a.a 1
15.e even 4 2 450.2.c.d 2
20.d odd 2 1 720.2.a.b 1
20.e even 4 2 3600.2.f.u 2
35.c odd 2 1 4410.2.a.bf 1
40.e odd 2 1 2880.2.a.u 1
40.f even 2 1 2880.2.a.bf 1
45.h odd 6 2 810.2.e.h 2
45.j even 6 2 810.2.e.e 2
60.h even 2 1 720.2.a.g 1
60.l odd 4 2 3600.2.f.a 2
105.g even 2 1 4410.2.a.k 1
120.i odd 2 1 2880.2.a.k 1
120.m even 2 1 2880.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.a.a 1 15.d odd 2 1
90.2.a.b yes 1 5.b even 2 1
450.2.a.a 1 1.a even 1 1 trivial
450.2.a.e 1 3.b odd 2 1
450.2.c.a 2 5.c odd 4 2
450.2.c.d 2 15.e even 4 2
720.2.a.b 1 20.d odd 2 1
720.2.a.g 1 60.h even 2 1
810.2.e.e 2 45.j even 6 2
810.2.e.h 2 45.h odd 6 2
2880.2.a.h 1 120.m even 2 1
2880.2.a.k 1 120.i odd 2 1
2880.2.a.u 1 40.e odd 2 1
2880.2.a.bf 1 40.f even 2 1
3600.2.a.ba 1 12.b even 2 1
3600.2.a.bj 1 4.b odd 2 1
3600.2.f.a 2 60.l odd 4 2
3600.2.f.u 2 20.e even 4 2
4410.2.a.k 1 105.g even 2 1
4410.2.a.bf 1 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(450))$$:

 $$T_{7} + 2$$ $$T_{11} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$2 + T$$
$11$ $$6 + T$$
$13$ $$-4 + T$$
$17$ $$6 + T$$
$19$ $$4 + T$$
$23$ $$T$$
$29$ $$-6 + T$$
$31$ $$4 + T$$
$37$ $$8 + T$$
$41$ $$T$$
$43$ $$8 + T$$
$47$ $$T$$
$53$ $$6 + T$$
$59$ $$6 + T$$
$61$ $$-2 + T$$
$67$ $$-4 + T$$
$71$ $$-12 + T$$
$73$ $$-10 + T$$
$79$ $$4 + T$$
$83$ $$-12 + T$$
$89$ $$12 + T$$
$97$ $$2 + T$$