# Properties

 Label 90.3.g.b Level 90 Weight 3 Character orbit 90.g Analytic conductor 2.452 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$90 = 2 \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 90.g (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.45232237924$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 10) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + i ) q^{2} + 2 i q^{4} + 5 i q^{5} + ( 2 + 2 i ) q^{7} + ( -2 + 2 i ) q^{8} +O(q^{10})$$ $$q + ( 1 + i ) q^{2} + 2 i q^{4} + 5 i q^{5} + ( 2 + 2 i ) q^{7} + ( -2 + 2 i ) q^{8} + ( -5 + 5 i ) q^{10} + 8 q^{11} + ( 3 - 3 i ) q^{13} + 4 i q^{14} -4 q^{16} + ( -7 - 7 i ) q^{17} -20 i q^{19} -10 q^{20} + ( 8 + 8 i ) q^{22} + ( 2 - 2 i ) q^{23} -25 q^{25} + 6 q^{26} + ( -4 + 4 i ) q^{28} -40 i q^{29} + 52 q^{31} + ( -4 - 4 i ) q^{32} -14 i q^{34} + ( -10 + 10 i ) q^{35} + ( -3 - 3 i ) q^{37} + ( 20 - 20 i ) q^{38} + ( -10 - 10 i ) q^{40} + 8 q^{41} + ( -42 + 42 i ) q^{43} + 16 i q^{44} + 4 q^{46} + ( 18 + 18 i ) q^{47} -41 i q^{49} + ( -25 - 25 i ) q^{50} + ( 6 + 6 i ) q^{52} + ( -53 + 53 i ) q^{53} + 40 i q^{55} -8 q^{56} + ( 40 - 40 i ) q^{58} + 20 i q^{59} -48 q^{61} + ( 52 + 52 i ) q^{62} -8 i q^{64} + ( 15 + 15 i ) q^{65} + ( 62 + 62 i ) q^{67} + ( 14 - 14 i ) q^{68} -20 q^{70} + 28 q^{71} + ( -47 + 47 i ) q^{73} -6 i q^{74} + 40 q^{76} + ( 16 + 16 i ) q^{77} -20 i q^{80} + ( 8 + 8 i ) q^{82} + ( -18 + 18 i ) q^{83} + ( 35 - 35 i ) q^{85} -84 q^{86} + ( -16 + 16 i ) q^{88} -80 i q^{89} + 12 q^{91} + ( 4 + 4 i ) q^{92} + 36 i q^{94} + 100 q^{95} + ( -63 - 63 i ) q^{97} + ( 41 - 41 i ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 4q^{7} - 4q^{8} + O(q^{10})$$ $$2q + 2q^{2} + 4q^{7} - 4q^{8} - 10q^{10} + 16q^{11} + 6q^{13} - 8q^{16} - 14q^{17} - 20q^{20} + 16q^{22} + 4q^{23} - 50q^{25} + 12q^{26} - 8q^{28} + 104q^{31} - 8q^{32} - 20q^{35} - 6q^{37} + 40q^{38} - 20q^{40} + 16q^{41} - 84q^{43} + 8q^{46} + 36q^{47} - 50q^{50} + 12q^{52} - 106q^{53} - 16q^{56} + 80q^{58} - 96q^{61} + 104q^{62} + 30q^{65} + 124q^{67} + 28q^{68} - 40q^{70} + 56q^{71} - 94q^{73} + 80q^{76} + 32q^{77} + 16q^{82} - 36q^{83} + 70q^{85} - 168q^{86} - 32q^{88} + 24q^{91} + 8q^{92} + 200q^{95} - 126q^{97} + 82q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/90\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$37$$ $$\chi(n)$$ $$1$$ $$i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 1.00000i − 1.00000i
1.00000 + 1.00000i 0 2.00000i 5.00000i 0 2.00000 + 2.00000i −2.00000 + 2.00000i 0 −5.00000 + 5.00000i
73.1 1.00000 1.00000i 0 2.00000i 5.00000i 0 2.00000 2.00000i −2.00000 2.00000i 0 −5.00000 5.00000i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.3.g.b 2
3.b odd 2 1 10.3.c.a 2
4.b odd 2 1 720.3.bh.c 2
5.b even 2 1 450.3.g.b 2
5.c odd 4 1 inner 90.3.g.b 2
5.c odd 4 1 450.3.g.b 2
12.b even 2 1 80.3.p.c 2
15.d odd 2 1 50.3.c.c 2
15.e even 4 1 10.3.c.a 2
15.e even 4 1 50.3.c.c 2
20.e even 4 1 720.3.bh.c 2
21.c even 2 1 490.3.f.b 2
24.f even 2 1 320.3.p.a 2
24.h odd 2 1 320.3.p.h 2
60.h even 2 1 400.3.p.b 2
60.l odd 4 1 80.3.p.c 2
60.l odd 4 1 400.3.p.b 2
105.k odd 4 1 490.3.f.b 2
120.q odd 4 1 320.3.p.a 2
120.w even 4 1 320.3.p.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.3.c.a 2 3.b odd 2 1
10.3.c.a 2 15.e even 4 1
50.3.c.c 2 15.d odd 2 1
50.3.c.c 2 15.e even 4 1
80.3.p.c 2 12.b even 2 1
80.3.p.c 2 60.l odd 4 1
90.3.g.b 2 1.a even 1 1 trivial
90.3.g.b 2 5.c odd 4 1 inner
320.3.p.a 2 24.f even 2 1
320.3.p.a 2 120.q odd 4 1
320.3.p.h 2 24.h odd 2 1
320.3.p.h 2 120.w even 4 1
400.3.p.b 2 60.h even 2 1
400.3.p.b 2 60.l odd 4 1
450.3.g.b 2 5.b even 2 1
450.3.g.b 2 5.c odd 4 1
490.3.f.b 2 21.c even 2 1
490.3.f.b 2 105.k odd 4 1
720.3.bh.c 2 4.b odd 2 1
720.3.bh.c 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(90, [\chi])$$:

 $$T_{7}^{2} - 4 T_{7} + 8$$ $$T_{11} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 2 T^{2}$$
$3$ 1
$5$ $$1 + 25 T^{2}$$
$7$ $$1 - 4 T + 8 T^{2} - 196 T^{3} + 2401 T^{4}$$
$11$ $$( 1 - 8 T + 121 T^{2} )^{2}$$
$13$ $$1 - 6 T + 18 T^{2} - 1014 T^{3} + 28561 T^{4}$$
$17$ $$( 1 - 16 T + 289 T^{2} )( 1 + 30 T + 289 T^{2} )$$
$19$ $$1 - 322 T^{2} + 130321 T^{4}$$
$23$ $$1 - 4 T + 8 T^{2} - 2116 T^{3} + 279841 T^{4}$$
$29$ $$( 1 - 42 T + 841 T^{2} )( 1 + 42 T + 841 T^{2} )$$
$31$ $$( 1 - 52 T + 961 T^{2} )^{2}$$
$37$ $$1 + 6 T + 18 T^{2} + 8214 T^{3} + 1874161 T^{4}$$
$41$ $$( 1 - 8 T + 1681 T^{2} )^{2}$$
$43$ $$1 + 84 T + 3528 T^{2} + 155316 T^{3} + 3418801 T^{4}$$
$47$ $$1 - 36 T + 648 T^{2} - 79524 T^{3} + 4879681 T^{4}$$
$53$ $$( 1 + 53 T )^{2}( 1 + 2809 T^{2} )$$
$59$ $$1 - 6562 T^{2} + 12117361 T^{4}$$
$61$ $$( 1 + 48 T + 3721 T^{2} )^{2}$$
$67$ $$1 - 124 T + 7688 T^{2} - 556636 T^{3} + 20151121 T^{4}$$
$71$ $$( 1 - 28 T + 5041 T^{2} )^{2}$$
$73$ $$1 + 94 T + 4418 T^{2} + 500926 T^{3} + 28398241 T^{4}$$
$79$ $$( 1 - 79 T )^{2}( 1 + 79 T )^{2}$$
$83$ $$1 + 36 T + 648 T^{2} + 248004 T^{3} + 47458321 T^{4}$$
$89$ $$1 - 9442 T^{2} + 62742241 T^{4}$$
$97$ $$1 + 126 T + 7938 T^{2} + 1185534 T^{3} + 88529281 T^{4}$$