# Properties

 Label 450.2.a.g Level $450$ Weight $2$ Character orbit 450.a Self dual yes Analytic conductor $3.593$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 50) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + 2q^{7} + q^{8} + O(q^{10})$$ $$q + q^{2} + q^{4} + 2q^{7} + q^{8} + 3q^{11} - 4q^{13} + 2q^{14} + q^{16} + 3q^{17} + 5q^{19} + 3q^{22} - 6q^{23} - 4q^{26} + 2q^{28} + 2q^{31} + q^{32} + 3q^{34} + 2q^{37} + 5q^{38} + 3q^{41} - 4q^{43} + 3q^{44} - 6q^{46} - 12q^{47} - 3q^{49} - 4q^{52} - 6q^{53} + 2q^{56} + 2q^{61} + 2q^{62} + q^{64} - 13q^{67} + 3q^{68} - 12q^{71} + 11q^{73} + 2q^{74} + 5q^{76} + 6q^{77} - 10q^{79} + 3q^{82} + 9q^{83} - 4q^{86} + 3q^{88} - 15q^{89} - 8q^{91} - 6q^{92} - 12q^{94} + 2q^{97} - 3q^{98} + O(q^{100})$$

## 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.g 1
3.b odd 2 1 50.2.a.a 1
4.b odd 2 1 3600.2.a.l 1
5.b even 2 1 450.2.a.c 1
5.c odd 4 2 450.2.c.c 2
12.b even 2 1 400.2.a.d 1
15.d odd 2 1 50.2.a.b yes 1
15.e even 4 2 50.2.b.a 2
20.d odd 2 1 3600.2.a.bc 1
20.e even 4 2 3600.2.f.f 2
21.c even 2 1 2450.2.a.g 1
24.f even 2 1 1600.2.a.p 1
24.h odd 2 1 1600.2.a.j 1
33.d even 2 1 6050.2.a.bi 1
39.d odd 2 1 8450.2.a.v 1
60.h even 2 1 400.2.a.f 1
60.l odd 4 2 400.2.c.c 2
105.g even 2 1 2450.2.a.bd 1
105.k odd 4 2 2450.2.c.m 2
120.i odd 2 1 1600.2.a.q 1
120.m even 2 1 1600.2.a.i 1
120.q odd 4 2 1600.2.c.h 2
120.w even 4 2 1600.2.c.i 2
165.d even 2 1 6050.2.a.h 1
195.e odd 2 1 8450.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.a.a 1 3.b odd 2 1
50.2.a.b yes 1 15.d odd 2 1
50.2.b.a 2 15.e even 4 2
400.2.a.d 1 12.b even 2 1
400.2.a.f 1 60.h even 2 1
400.2.c.c 2 60.l odd 4 2
450.2.a.c 1 5.b even 2 1
450.2.a.g 1 1.a even 1 1 trivial
450.2.c.c 2 5.c odd 4 2
1600.2.a.i 1 120.m even 2 1
1600.2.a.j 1 24.h odd 2 1
1600.2.a.p 1 24.f even 2 1
1600.2.a.q 1 120.i odd 2 1
1600.2.c.h 2 120.q odd 4 2
1600.2.c.i 2 120.w even 4 2
2450.2.a.g 1 21.c even 2 1
2450.2.a.bd 1 105.g even 2 1
2450.2.c.m 2 105.k odd 4 2
3600.2.a.l 1 4.b odd 2 1
3600.2.a.bc 1 20.d odd 2 1
3600.2.f.f 2 20.e even 4 2
6050.2.a.h 1 165.d even 2 1
6050.2.a.bi 1 33.d even 2 1
8450.2.a.d 1 195.e odd 2 1
8450.2.a.v 1 39.d odd 2 1

## Hecke kernels

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

 $$T_{7} - 2$$ $$T_{11} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-2 + T$$
$11$ $$-3 + T$$
$13$ $$4 + T$$
$17$ $$-3 + T$$
$19$ $$-5 + T$$
$23$ $$6 + T$$
$29$ $$T$$
$31$ $$-2 + T$$
$37$ $$-2 + T$$
$41$ $$-3 + T$$
$43$ $$4 + T$$
$47$ $$12 + T$$
$53$ $$6 + T$$
$59$ $$T$$
$61$ $$-2 + T$$
$67$ $$13 + T$$
$71$ $$12 + T$$
$73$ $$-11 + T$$
$79$ $$10 + T$$
$83$ $$-9 + T$$
$89$ $$15 + T$$
$97$ $$-2 + T$$