# Properties

 Label 450.2.a.f 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 30) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + 2 q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + 2 * q^7 + q^8 $$q + q^{2} + q^{4} + 2 q^{7} + q^{8} - 2 q^{11} + 6 q^{13} + 2 q^{14} + q^{16} - 2 q^{17} - 2 q^{22} + 4 q^{23} + 6 q^{26} + 2 q^{28} - 8 q^{31} + q^{32} - 2 q^{34} + 2 q^{37} - 2 q^{41} - 4 q^{43} - 2 q^{44} + 4 q^{46} + 8 q^{47} - 3 q^{49} + 6 q^{52} - 6 q^{53} + 2 q^{56} - 10 q^{59} + 2 q^{61} - 8 q^{62} + q^{64} - 8 q^{67} - 2 q^{68} - 12 q^{71} - 4 q^{73} + 2 q^{74} - 4 q^{77} - 2 q^{82} + 4 q^{83} - 4 q^{86} - 2 q^{88} + 10 q^{89} + 12 q^{91} + 4 q^{92} + 8 q^{94} - 8 q^{97} - 3 q^{98}+O(q^{100})$$ q + q^2 + q^4 + 2 * q^7 + q^8 - 2 * q^11 + 6 * q^13 + 2 * q^14 + q^16 - 2 * q^17 - 2 * q^22 + 4 * q^23 + 6 * q^26 + 2 * q^28 - 8 * q^31 + q^32 - 2 * q^34 + 2 * q^37 - 2 * q^41 - 4 * q^43 - 2 * q^44 + 4 * q^46 + 8 * q^47 - 3 * q^49 + 6 * q^52 - 6 * q^53 + 2 * q^56 - 10 * q^59 + 2 * q^61 - 8 * q^62 + q^64 - 8 * q^67 - 2 * q^68 - 12 * q^71 - 4 * q^73 + 2 * q^74 - 4 * q^77 - 2 * q^82 + 4 * q^83 - 4 * q^86 - 2 * q^88 + 10 * q^89 + 12 * q^91 + 4 * q^92 + 8 * q^94 - 8 * q^97 - 3 * q^98

## 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.f 1
3.b odd 2 1 150.2.a.a 1
4.b odd 2 1 3600.2.a.o 1
5.b even 2 1 450.2.a.b 1
5.c odd 4 2 90.2.c.a 2
12.b even 2 1 1200.2.a.m 1
15.d odd 2 1 150.2.a.c 1
15.e even 4 2 30.2.c.a 2
20.d odd 2 1 3600.2.a.bg 1
20.e even 4 2 720.2.f.f 2
21.c even 2 1 7350.2.a.bg 1
24.f even 2 1 4800.2.a.m 1
24.h odd 2 1 4800.2.a.cg 1
40.i odd 4 2 2880.2.f.e 2
40.k even 4 2 2880.2.f.c 2
45.k odd 12 4 810.2.i.b 4
45.l even 12 4 810.2.i.e 4
60.h even 2 1 1200.2.a.g 1
60.l odd 4 2 240.2.f.a 2
105.g even 2 1 7350.2.a.cc 1
105.k odd 4 2 1470.2.g.g 2
105.w odd 12 4 1470.2.n.a 4
105.x even 12 4 1470.2.n.h 4
120.i odd 2 1 4800.2.a.l 1
120.m even 2 1 4800.2.a.cj 1
120.q odd 4 2 960.2.f.i 2
120.w even 4 2 960.2.f.h 2
240.z odd 4 2 3840.2.d.j 2
240.bb even 4 2 3840.2.d.y 2
240.bd odd 4 2 3840.2.d.x 2
240.bf even 4 2 3840.2.d.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.c.a 2 15.e even 4 2
90.2.c.a 2 5.c odd 4 2
150.2.a.a 1 3.b odd 2 1
150.2.a.c 1 15.d odd 2 1
240.2.f.a 2 60.l odd 4 2
450.2.a.b 1 5.b even 2 1
450.2.a.f 1 1.a even 1 1 trivial
720.2.f.f 2 20.e even 4 2
810.2.i.b 4 45.k odd 12 4
810.2.i.e 4 45.l even 12 4
960.2.f.h 2 120.w even 4 2
960.2.f.i 2 120.q odd 4 2
1200.2.a.g 1 60.h even 2 1
1200.2.a.m 1 12.b even 2 1
1470.2.g.g 2 105.k odd 4 2
1470.2.n.a 4 105.w odd 12 4
1470.2.n.h 4 105.x even 12 4
2880.2.f.c 2 40.k even 4 2
2880.2.f.e 2 40.i odd 4 2
3600.2.a.o 1 4.b odd 2 1
3600.2.a.bg 1 20.d odd 2 1
3840.2.d.g 2 240.bf even 4 2
3840.2.d.j 2 240.z odd 4 2
3840.2.d.x 2 240.bd odd 4 2
3840.2.d.y 2 240.bb even 4 2
4800.2.a.l 1 120.i odd 2 1
4800.2.a.m 1 24.f even 2 1
4800.2.a.cg 1 24.h odd 2 1
4800.2.a.cj 1 120.m even 2 1
7350.2.a.bg 1 21.c even 2 1
7350.2.a.cc 1 105.g even 2 1

## Hecke kernels

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

 $$T_{7} - 2$$ T7 - 2 $$T_{11} + 2$$ T11 + 2

## Hecke characteristic polynomials

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