# Properties

 Label 90.6.a.f Level 90 Weight 6 Character orbit 90.a Self dual yes Analytic conductor 14.435 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$90 = 2 \cdot 3^{2} \cdot 5$$ Weight: $$k$$ = $$6$$ Character orbit: $$[\chi]$$ = 90.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.4345437832$$ 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

 $$f(q)$$ $$=$$ $$q + 4q^{2} + 16q^{4} + 25q^{5} - 22q^{7} + 64q^{8} + O(q^{10})$$ $$q + 4q^{2} + 16q^{4} + 25q^{5} - 22q^{7} + 64q^{8} + 100q^{10} + 768q^{11} - 46q^{13} - 88q^{14} + 256q^{16} - 378q^{17} + 1100q^{19} + 400q^{20} + 3072q^{22} + 1986q^{23} + 625q^{25} - 184q^{26} - 352q^{28} + 5610q^{29} - 3988q^{31} + 1024q^{32} - 1512q^{34} - 550q^{35} - 142q^{37} + 4400q^{38} + 1600q^{40} - 1542q^{41} - 5026q^{43} + 12288q^{44} + 7944q^{46} - 24738q^{47} - 16323q^{49} + 2500q^{50} - 736q^{52} + 14166q^{53} + 19200q^{55} - 1408q^{56} + 22440q^{58} - 28380q^{59} + 5522q^{61} - 15952q^{62} + 4096q^{64} - 1150q^{65} - 24742q^{67} - 6048q^{68} - 2200q^{70} - 42372q^{71} - 52126q^{73} - 568q^{74} + 17600q^{76} - 16896q^{77} - 39640q^{79} + 6400q^{80} - 6168q^{82} + 59826q^{83} - 9450q^{85} - 20104q^{86} + 49152q^{88} - 57690q^{89} + 1012q^{91} + 31776q^{92} - 98952q^{94} + 27500q^{95} - 144382q^{97} - 65292q^{98} + O(q^{100})$$

## 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 25.0000 0 −22.0000 64.0000 0 100.000
 $$n$$: 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 90.6.a.f 1
3.b odd 2 1 10.6.a.a 1
4.b odd 2 1 720.6.a.r 1
5.b even 2 1 450.6.a.h 1
5.c odd 4 2 450.6.c.o 2
12.b even 2 1 80.6.a.h 1
15.d odd 2 1 50.6.a.g 1
15.e even 4 2 50.6.b.d 2
21.c even 2 1 490.6.a.j 1
24.f even 2 1 320.6.a.a 1
24.h odd 2 1 320.6.a.p 1
60.h even 2 1 400.6.a.a 1
60.l odd 4 2 400.6.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.a 1 3.b odd 2 1
50.6.a.g 1 15.d odd 2 1
50.6.b.d 2 15.e even 4 2
80.6.a.h 1 12.b even 2 1
90.6.a.f 1 1.a even 1 1 trivial
320.6.a.a 1 24.f even 2 1
320.6.a.p 1 24.h odd 2 1
400.6.a.a 1 60.h even 2 1
400.6.c.a 2 60.l odd 4 2
450.6.a.h 1 5.b even 2 1
450.6.c.o 2 5.c odd 4 2
490.6.a.j 1 21.c even 2 1
720.6.a.r 1 4.b odd 2 1

## 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(90))$$:

 $$T_{7} + 22$$ $$T_{11} - 768$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T$$
$3$ 1
$5$ $$1 - 25 T$$
$7$ $$1 + 22 T + 16807 T^{2}$$
$11$ $$1 - 768 T + 161051 T^{2}$$
$13$ $$1 + 46 T + 371293 T^{2}$$
$17$ $$1 + 378 T + 1419857 T^{2}$$
$19$ $$1 - 1100 T + 2476099 T^{2}$$
$23$ $$1 - 1986 T + 6436343 T^{2}$$
$29$ $$1 - 5610 T + 20511149 T^{2}$$
$31$ $$1 + 3988 T + 28629151 T^{2}$$
$37$ $$1 + 142 T + 69343957 T^{2}$$
$41$ $$1 + 1542 T + 115856201 T^{2}$$
$43$ $$1 + 5026 T + 147008443 T^{2}$$
$47$ $$1 + 24738 T + 229345007 T^{2}$$
$53$ $$1 - 14166 T + 418195493 T^{2}$$
$59$ $$1 + 28380 T + 714924299 T^{2}$$
$61$ $$1 - 5522 T + 844596301 T^{2}$$
$67$ $$1 + 24742 T + 1350125107 T^{2}$$
$71$ $$1 + 42372 T + 1804229351 T^{2}$$
$73$ $$1 + 52126 T + 2073071593 T^{2}$$
$79$ $$1 + 39640 T + 3077056399 T^{2}$$
$83$ $$1 - 59826 T + 3939040643 T^{2}$$
$89$ $$1 + 57690 T + 5584059449 T^{2}$$
$97$ $$1 + 144382 T + 8587340257 T^{2}$$