# Properties

 Label 450.6.a.o Level $450$ Weight $6$ Character orbit 450.a Self dual yes Analytic conductor $72.173$ 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: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.1727189158$$ Analytic rank: $$0$$ 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

 $$f(q)$$ $$=$$ $$q + 4q^{2} + 16q^{4} - 98q^{7} + 64q^{8} + O(q^{10})$$ $$q + 4q^{2} + 16q^{4} - 98q^{7} + 64q^{8} - 354q^{11} - 404q^{13} - 392q^{14} + 256q^{16} + 654q^{17} + 1796q^{19} - 1416q^{22} - 1080q^{23} - 1616q^{26} - 1568q^{28} + 5754q^{29} + 10196q^{31} + 1024q^{32} + 2616q^{34} - 5552q^{37} + 7184q^{38} + 12960q^{41} + 8968q^{43} - 5664q^{44} - 4320q^{46} - 5400q^{47} - 7203q^{49} - 6464q^{52} + 8214q^{53} - 6272q^{56} + 23016q^{58} - 3954q^{59} + 962q^{61} + 40784q^{62} + 4096q^{64} + 17956q^{67} + 10464q^{68} + 56148q^{71} + 85690q^{73} - 22208q^{74} + 28736q^{76} + 34692q^{77} - 26044q^{79} + 51840q^{82} - 93468q^{83} + 35872q^{86} - 22656q^{88} - 73428q^{89} + 39592q^{91} - 17280q^{92} - 21600q^{94} - 128978q^{97} - 28812q^{98} + O(q^{100})$$

## 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
4.00000 0 16.0000 0 0 −98.0000 64.0000 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 450.6.a.o 1
3.b odd 2 1 450.6.a.d 1
5.b even 2 1 90.6.a.c 1
5.c odd 4 2 450.6.c.d 2
15.d odd 2 1 90.6.a.e yes 1
15.e even 4 2 450.6.c.l 2
20.d odd 2 1 720.6.a.o 1
60.h even 2 1 720.6.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.6.a.c 1 5.b even 2 1
90.6.a.e yes 1 15.d odd 2 1
450.6.a.d 1 3.b odd 2 1
450.6.a.o 1 1.a even 1 1 trivial
450.6.c.d 2 5.c odd 4 2
450.6.c.l 2 15.e even 4 2
720.6.a.c 1 60.h even 2 1
720.6.a.o 1 20.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 $$S_{6}^{\mathrm{new}}(\Gamma_0(450))$$:

 $$T_{7} + 98$$ $$T_{11} + 354$$ $$T_{17} - 654$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$98 + T$$
$11$ $$354 + T$$
$13$ $$404 + T$$
$17$ $$-654 + T$$
$19$ $$-1796 + T$$
$23$ $$1080 + T$$
$29$ $$-5754 + T$$
$31$ $$-10196 + T$$
$37$ $$5552 + T$$
$41$ $$-12960 + T$$
$43$ $$-8968 + T$$
$47$ $$5400 + T$$
$53$ $$-8214 + T$$
$59$ $$3954 + T$$
$61$ $$-962 + T$$
$67$ $$-17956 + T$$
$71$ $$-56148 + T$$
$73$ $$-85690 + T$$
$79$ $$26044 + T$$
$83$ $$93468 + T$$
$89$ $$73428 + T$$
$97$ $$128978 + T$$