# 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [90,3,Mod(37,90)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(90, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("90.37");

S:= CuspForms(chi, 3);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$2.45232237924$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 $$S_{3}^{\mathrm{new}}(90, [\chi])$$:

 $$T_{7}^{2} - 4T_{7} + 8$$ T7^2 - 4*T7 + 8 $$T_{11} - 8$$ T11 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 25$$
$7$ $$T^{2} - 4T + 8$$
$11$ $$(T - 8)^{2}$$
$13$ $$T^{2} - 6T + 18$$
$17$ $$T^{2} + 14T + 98$$
$19$ $$T^{2} + 400$$
$23$ $$T^{2} - 4T + 8$$
$29$ $$T^{2} + 1600$$
$31$ $$(T - 52)^{2}$$
$37$ $$T^{2} + 6T + 18$$
$41$ $$(T - 8)^{2}$$
$43$ $$T^{2} + 84T + 3528$$
$47$ $$T^{2} - 36T + 648$$
$53$ $$T^{2} + 106T + 5618$$
$59$ $$T^{2} + 400$$
$61$ $$(T + 48)^{2}$$
$67$ $$T^{2} - 124T + 7688$$
$71$ $$(T - 28)^{2}$$
$73$ $$T^{2} + 94T + 4418$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 36T + 648$$
$89$ $$T^{2} + 6400$$
$97$ $$T^{2} + 126T + 7938$$