# Properties

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

## Newform invariants

 Self dual: yes Analytic conductor: $$72.1727189158$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 18) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 4q^{2} + 16q^{4} + 148q^{7} - 64q^{8} + O(q^{10})$$ $$q - 4q^{2} + 16q^{4} + 148q^{7} - 64q^{8} - 384q^{11} + 334q^{13} - 592q^{14} + 256q^{16} + 576q^{17} - 664q^{19} + 1536q^{22} - 3840q^{23} - 1336q^{26} + 2368q^{28} - 96q^{29} - 4564q^{31} - 1024q^{32} - 2304q^{34} - 5798q^{37} + 2656q^{38} + 6720q^{41} + 14872q^{43} - 6144q^{44} + 15360q^{46} - 19200q^{47} + 5097q^{49} + 5344q^{52} + 7776q^{53} - 9472q^{56} + 384q^{58} + 13056q^{59} + 42782q^{61} + 18256q^{62} + 4096q^{64} - 36656q^{67} + 9216q^{68} - 64512q^{71} + 16810q^{73} + 23192q^{74} - 10624q^{76} - 56832q^{77} + 28076q^{79} - 26880q^{82} - 66432q^{83} - 59488q^{86} + 24576q^{88} + 81792q^{89} + 49432q^{91} - 61440q^{92} + 76800q^{94} + 29938q^{97} - 20388q^{98} + O(q^{100})$$

## 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 0 148.000 −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.k 1
3.b odd 2 1 450.6.a.v 1
5.b even 2 1 18.6.a.c yes 1
5.c odd 4 2 450.6.c.c 2
15.d odd 2 1 18.6.a.a 1
15.e even 4 2 450.6.c.m 2
20.d odd 2 1 144.6.a.l 1
35.c odd 2 1 882.6.a.l 1
40.e odd 2 1 576.6.a.b 1
40.f even 2 1 576.6.a.a 1
45.h odd 6 2 162.6.c.l 2
45.j even 6 2 162.6.c.a 2
60.h even 2 1 144.6.a.a 1
105.g even 2 1 882.6.a.k 1
120.i odd 2 1 576.6.a.bh 1
120.m even 2 1 576.6.a.bi 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.6.a.a 1 15.d odd 2 1
18.6.a.c yes 1 5.b even 2 1
144.6.a.a 1 60.h even 2 1
144.6.a.l 1 20.d odd 2 1
162.6.c.a 2 45.j even 6 2
162.6.c.l 2 45.h odd 6 2
450.6.a.k 1 1.a even 1 1 trivial
450.6.a.v 1 3.b odd 2 1
450.6.c.c 2 5.c odd 4 2
450.6.c.m 2 15.e even 4 2
576.6.a.a 1 40.f even 2 1
576.6.a.b 1 40.e odd 2 1
576.6.a.bh 1 120.i odd 2 1
576.6.a.bi 1 120.m even 2 1
882.6.a.k 1 105.g even 2 1
882.6.a.l 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(450))$$:

 $$T_{7} - 148$$ $$T_{11} + 384$$ $$T_{17} - 576$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-148 + T$$
$11$ $$384 + T$$
$13$ $$-334 + T$$
$17$ $$-576 + T$$
$19$ $$664 + T$$
$23$ $$3840 + T$$
$29$ $$96 + T$$
$31$ $$4564 + T$$
$37$ $$5798 + T$$
$41$ $$-6720 + T$$
$43$ $$-14872 + T$$
$47$ $$19200 + T$$
$53$ $$-7776 + T$$
$59$ $$-13056 + T$$
$61$ $$-42782 + T$$
$67$ $$36656 + T$$
$71$ $$64512 + T$$
$73$ $$-16810 + T$$
$79$ $$-28076 + T$$
$83$ $$66432 + T$$
$89$ $$-81792 + T$$
$97$ $$-29938 + T$$