Properties

 Label 450.6.a.h 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 10) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - 4q^{2} + 16q^{4} + 22q^{7} - 64q^{8} + O(q^{10})$$ $$q - 4q^{2} + 16q^{4} + 22q^{7} - 64q^{8} + 768q^{11} + 46q^{13} - 88q^{14} + 256q^{16} + 378q^{17} + 1100q^{19} - 3072q^{22} - 1986q^{23} - 184q^{26} + 352q^{28} + 5610q^{29} - 3988q^{31} - 1024q^{32} - 1512q^{34} + 142q^{37} - 4400q^{38} - 1542q^{41} + 5026q^{43} + 12288q^{44} + 7944q^{46} + 24738q^{47} - 16323q^{49} + 736q^{52} - 14166q^{53} - 1408q^{56} - 22440q^{58} - 28380q^{59} + 5522q^{61} + 15952q^{62} + 4096q^{64} + 24742q^{67} + 6048q^{68} - 42372q^{71} + 52126q^{73} - 568q^{74} + 17600q^{76} + 16896q^{77} - 39640q^{79} + 6168q^{82} - 59826q^{83} - 20104q^{86} - 49152q^{88} - 57690q^{89} + 1012q^{91} - 31776q^{92} - 98952q^{94} + 144382q^{97} + 65292q^{98} + O(q^{100})$$

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 22.0000 −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.h 1
3.b odd 2 1 50.6.a.g 1
5.b even 2 1 90.6.a.f 1
5.c odd 4 2 450.6.c.o 2
12.b even 2 1 400.6.a.a 1
15.d odd 2 1 10.6.a.a 1
15.e even 4 2 50.6.b.d 2
20.d odd 2 1 720.6.a.r 1
60.h even 2 1 80.6.a.h 1
60.l odd 4 2 400.6.c.a 2
105.g even 2 1 490.6.a.j 1
120.i odd 2 1 320.6.a.p 1
120.m even 2 1 320.6.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.a 1 15.d odd 2 1
50.6.a.g 1 3.b odd 2 1
50.6.b.d 2 15.e even 4 2
80.6.a.h 1 60.h even 2 1
90.6.a.f 1 5.b even 2 1
320.6.a.a 1 120.m even 2 1
320.6.a.p 1 120.i odd 2 1
400.6.a.a 1 12.b even 2 1
400.6.c.a 2 60.l odd 4 2
450.6.a.h 1 1.a even 1 1 trivial
450.6.c.o 2 5.c odd 4 2
490.6.a.j 1 105.g even 2 1
720.6.a.r 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(450))$$:

 $$T_{7} - 22$$ $$T_{11} - 768$$ $$T_{17} - 378$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-22 + T$$
$11$ $$-768 + T$$
$13$ $$-46 + T$$
$17$ $$-378 + T$$
$19$ $$-1100 + T$$
$23$ $$1986 + T$$
$29$ $$-5610 + T$$
$31$ $$3988 + T$$
$37$ $$-142 + T$$
$41$ $$1542 + T$$
$43$ $$-5026 + T$$
$47$ $$-24738 + T$$
$53$ $$14166 + T$$
$59$ $$28380 + T$$
$61$ $$-5522 + T$$
$67$ $$-24742 + T$$
$71$ $$42372 + T$$
$73$ $$-52126 + T$$
$79$ $$39640 + T$$
$83$ $$59826 + T$$
$89$ $$57690 + T$$
$97$ $$-144382 + T$$