# Properties

 Label 90.6.a.b Level 90 Weight 6 Character orbit 90.a Self dual yes Analytic conductor 14.435 Analytic rank 1 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - 4q^{2} + 16q^{4} + 25q^{5} - 118q^{7} - 64q^{8} + O(q^{10})$$ $$q - 4q^{2} + 16q^{4} + 25q^{5} - 118q^{7} - 64q^{8} - 100q^{10} - 192q^{11} + 1106q^{13} + 472q^{14} + 256q^{16} - 762q^{17} - 2740q^{19} + 400q^{20} + 768q^{22} - 1566q^{23} + 625q^{25} - 4424q^{26} - 1888q^{28} - 5910q^{29} - 6868q^{31} - 1024q^{32} + 3048q^{34} - 2950q^{35} - 5518q^{37} + 10960q^{38} - 1600q^{40} + 378q^{41} - 2434q^{43} - 3072q^{44} + 6264q^{46} - 13122q^{47} - 2883q^{49} - 2500q^{50} + 17696q^{52} + 9174q^{53} - 4800q^{55} + 7552q^{56} + 23640q^{58} + 34980q^{59} - 9838q^{61} + 27472q^{62} + 4096q^{64} + 27650q^{65} + 33722q^{67} - 12192q^{68} + 11800q^{70} - 70212q^{71} + 21986q^{73} + 22072q^{74} - 43840q^{76} + 22656q^{77} + 4520q^{79} + 6400q^{80} - 1512q^{82} + 109074q^{83} - 19050q^{85} + 9736q^{86} + 12288q^{88} - 38490q^{89} - 130508q^{91} - 25056q^{92} + 52488q^{94} - 68500q^{95} - 1918q^{97} + 11532q^{98} + O(q^{100})$$

## 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 −118.000 −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.b 1
3.b odd 2 1 10.6.a.c 1
4.b odd 2 1 720.6.a.v 1
5.b even 2 1 450.6.a.u 1
5.c odd 4 2 450.6.c.f 2
12.b even 2 1 80.6.a.c 1
15.d odd 2 1 50.6.a.b 1
15.e even 4 2 50.6.b.b 2
21.c even 2 1 490.6.a.k 1
24.f even 2 1 320.6.a.k 1
24.h odd 2 1 320.6.a.f 1
60.h even 2 1 400.6.a.i 1
60.l odd 4 2 400.6.c.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.c 1 3.b odd 2 1
50.6.a.b 1 15.d odd 2 1
50.6.b.b 2 15.e even 4 2
80.6.a.c 1 12.b even 2 1
90.6.a.b 1 1.a even 1 1 trivial
320.6.a.f 1 24.h odd 2 1
320.6.a.k 1 24.f even 2 1
400.6.a.i 1 60.h even 2 1
400.6.c.i 2 60.l odd 4 2
450.6.a.u 1 5.b even 2 1
450.6.c.f 2 5.c odd 4 2
490.6.a.k 1 21.c even 2 1
720.6.a.v 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(90))$$:

 $$T_{7} + 118$$ $$T_{11} + 192$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T$$
$3$ 1
$5$ $$1 - 25 T$$
$7$ $$1 + 118 T + 16807 T^{2}$$
$11$ $$1 + 192 T + 161051 T^{2}$$
$13$ $$1 - 1106 T + 371293 T^{2}$$
$17$ $$1 + 762 T + 1419857 T^{2}$$
$19$ $$1 + 2740 T + 2476099 T^{2}$$
$23$ $$1 + 1566 T + 6436343 T^{2}$$
$29$ $$1 + 5910 T + 20511149 T^{2}$$
$31$ $$1 + 6868 T + 28629151 T^{2}$$
$37$ $$1 + 5518 T + 69343957 T^{2}$$
$41$ $$1 - 378 T + 115856201 T^{2}$$
$43$ $$1 + 2434 T + 147008443 T^{2}$$
$47$ $$1 + 13122 T + 229345007 T^{2}$$
$53$ $$1 - 9174 T + 418195493 T^{2}$$
$59$ $$1 - 34980 T + 714924299 T^{2}$$
$61$ $$1 + 9838 T + 844596301 T^{2}$$
$67$ $$1 - 33722 T + 1350125107 T^{2}$$
$71$ $$1 + 70212 T + 1804229351 T^{2}$$
$73$ $$1 - 21986 T + 2073071593 T^{2}$$
$79$ $$1 - 4520 T + 3077056399 T^{2}$$
$83$ $$1 - 109074 T + 3939040643 T^{2}$$
$89$ $$1 + 38490 T + 5584059449 T^{2}$$
$97$ $$1 + 1918 T + 8587340257 T^{2}$$