# Properties

 Label 450.6.a.l Level $450$ Weight $6$ Character orbit 450.a Self dual yes Analytic conductor $72.173$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more about

## 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

 $$f(q)$$ $$=$$ $$q - 4q^{2} + 16q^{4} + 172q^{7} - 64q^{8} + O(q^{10})$$ $$q - 4q^{2} + 16q^{4} + 172q^{7} - 64q^{8} - 132q^{11} + 946q^{13} - 688q^{14} + 256q^{16} - 222q^{17} + 500q^{19} + 528q^{22} + 3564q^{23} - 3784q^{26} + 2752q^{28} - 2190q^{29} + 2312q^{31} - 1024q^{32} + 888q^{34} + 11242q^{37} - 2000q^{38} - 1242q^{41} - 20624q^{43} - 2112q^{44} - 14256q^{46} + 6588q^{47} + 12777q^{49} + 15136q^{52} - 21066q^{53} - 11008q^{56} + 8760q^{58} - 7980q^{59} + 16622q^{61} - 9248q^{62} + 4096q^{64} - 1808q^{67} - 3552q^{68} + 24528q^{71} - 20474q^{73} - 44968q^{74} + 8000q^{76} - 22704q^{77} - 46240q^{79} + 4968q^{82} - 51576q^{83} + 82496q^{86} + 8448q^{88} + 110310q^{89} + 162712q^{91} + 57024q^{92} - 26352q^{94} + 78382q^{97} - 51108q^{98} + O(q^{100})$$

## 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 0 172.000 −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.l 1
3.b odd 2 1 50.6.a.d 1
5.b even 2 1 90.6.a.d 1
5.c odd 4 2 450.6.c.h 2
12.b even 2 1 400.6.a.n 1
15.d odd 2 1 10.6.a.b 1
15.e even 4 2 50.6.b.a 2
20.d odd 2 1 720.6.a.j 1
60.h even 2 1 80.6.a.a 1
60.l odd 4 2 400.6.c.b 2
105.g even 2 1 490.6.a.a 1
120.i odd 2 1 320.6.a.b 1
120.m even 2 1 320.6.a.o 1

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

 $$T_{7} - 172$$ $$T_{11} + 132$$ $$T_{17} + 222$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-172 + T$$
$11$ $$132 + T$$
$13$ $$-946 + T$$
$17$ $$222 + T$$
$19$ $$-500 + T$$
$23$ $$-3564 + T$$
$29$ $$2190 + T$$
$31$ $$-2312 + T$$
$37$ $$-11242 + T$$
$41$ $$1242 + T$$
$43$ $$20624 + T$$
$47$ $$-6588 + T$$
$53$ $$21066 + T$$
$59$ $$7980 + T$$
$61$ $$-16622 + T$$
$67$ $$1808 + T$$
$71$ $$-24528 + T$$
$73$ $$20474 + T$$
$79$ $$46240 + T$$
$83$ $$51576 + T$$
$89$ $$-110310 + T$$
$97$ $$-78382 + T$$
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