# Properties

 Label 450.6.a.w 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

 $$f(q)$$ $$=$$ $$q + 4q^{2} + 16q^{4} + 158q^{7} + 64q^{8} + O(q^{10})$$ $$q + 4q^{2} + 16q^{4} + 158q^{7} + 64q^{8} + 148q^{11} + 684q^{13} + 632q^{14} + 256q^{16} - 2048q^{17} + 2220q^{19} + 592q^{22} + 1246q^{23} + 2736q^{26} + 2528q^{28} + 270q^{29} - 2048q^{31} + 1024q^{32} - 8192q^{34} - 4372q^{37} + 8880q^{38} + 2398q^{41} + 2294q^{43} + 2368q^{44} + 4984q^{46} + 10682q^{47} + 8157q^{49} + 10944q^{52} - 2964q^{53} + 10112q^{56} + 1080q^{58} + 39740q^{59} - 42298q^{61} - 8192q^{62} + 4096q^{64} + 32098q^{67} - 32768q^{68} + 4248q^{71} + 30104q^{73} - 17488q^{74} + 35520q^{76} + 23384q^{77} + 35280q^{79} + 9592q^{82} + 27826q^{83} + 9176q^{86} + 9472q^{88} + 85210q^{89} + 108072q^{91} + 19936q^{92} + 42728q^{94} - 97232q^{97} + 32628q^{98} + O(q^{100})$$

## 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 0 158.000 64.0000 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.w 1
3.b odd 2 1 50.6.a.c 1
5.b even 2 1 450.6.a.c 1
5.c odd 4 2 90.6.c.a 2
12.b even 2 1 400.6.a.c 1
15.d odd 2 1 50.6.a.e 1
15.e even 4 2 10.6.b.a 2
20.e even 4 2 720.6.f.a 2
60.h even 2 1 400.6.a.k 1
60.l odd 4 2 80.6.c.c 2
120.q odd 4 2 320.6.c.a 2
120.w even 4 2 320.6.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.b.a 2 15.e even 4 2
50.6.a.c 1 3.b odd 2 1
50.6.a.e 1 15.d odd 2 1
80.6.c.c 2 60.l odd 4 2
90.6.c.a 2 5.c odd 4 2
320.6.c.a 2 120.q odd 4 2
320.6.c.b 2 120.w even 4 2
400.6.a.c 1 12.b even 2 1
400.6.a.k 1 60.h even 2 1
450.6.a.c 1 5.b even 2 1
450.6.a.w 1 1.a even 1 1 trivial
720.6.f.a 2 20.e even 4 2

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(450))$$:

 $$T_{7} - 158$$ $$T_{11} - 148$$ $$T_{17} + 2048$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T$$
$3$ 1
$5$ 1
$7$ $$1 - 158 T + 16807 T^{2}$$
$11$ $$1 - 148 T + 161051 T^{2}$$
$13$ $$1 - 684 T + 371293 T^{2}$$
$17$ $$1 + 2048 T + 1419857 T^{2}$$
$19$ $$1 - 2220 T + 2476099 T^{2}$$
$23$ $$1 - 1246 T + 6436343 T^{2}$$
$29$ $$1 - 270 T + 20511149 T^{2}$$
$31$ $$1 + 2048 T + 28629151 T^{2}$$
$37$ $$1 + 4372 T + 69343957 T^{2}$$
$41$ $$1 - 2398 T + 115856201 T^{2}$$
$43$ $$1 - 2294 T + 147008443 T^{2}$$
$47$ $$1 - 10682 T + 229345007 T^{2}$$
$53$ $$1 + 2964 T + 418195493 T^{2}$$
$59$ $$1 - 39740 T + 714924299 T^{2}$$
$61$ $$1 + 42298 T + 844596301 T^{2}$$
$67$ $$1 - 32098 T + 1350125107 T^{2}$$
$71$ $$1 - 4248 T + 1804229351 T^{2}$$
$73$ $$1 - 30104 T + 2073071593 T^{2}$$
$79$ $$1 - 35280 T + 3077056399 T^{2}$$
$83$ $$1 - 27826 T + 3939040643 T^{2}$$
$89$ $$1 - 85210 T + 5584059449 T^{2}$$
$97$ $$1 + 97232 T + 8587340257 T^{2}$$