# Properties

 Label 90.2.c.a Level $90$ Weight $2$ Character orbit 90.c Analytic conductor $0.719$ 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] = [90,2,Mod(19,90)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(90, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("90.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$90 = 2 \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 90.c (of order $$2$$, degree $$1$$, 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: $$0.718653618192$$ 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 30) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} + (i + 2) q^{5} + 2 i q^{7} - i q^{8} +O(q^{10})$$ q + i * q^2 - q^4 + (i + 2) * q^5 + 2*i * q^7 - i * q^8 $$q + i q^{2} - q^{4} + (i + 2) q^{5} + 2 i q^{7} - i q^{8} + (2 i - 1) q^{10} - 2 q^{11} - 6 i q^{13} - 2 q^{14} + q^{16} - 2 i q^{17} + ( - i - 2) q^{20} - 2 i q^{22} - 4 i q^{23} + (4 i + 3) q^{25} + 6 q^{26} - 2 i q^{28} - 8 q^{31} + i q^{32} + 2 q^{34} + (4 i - 2) q^{35} + 2 i q^{37} + ( - 2 i + 1) q^{40} - 2 q^{41} + 4 i q^{43} + 2 q^{44} + 4 q^{46} + 8 i q^{47} + 3 q^{49} + (3 i - 4) q^{50} + 6 i q^{52} + 6 i q^{53} + ( - 2 i - 4) q^{55} + 2 q^{56} + 10 q^{59} + 2 q^{61} - 8 i q^{62} - q^{64} + ( - 12 i + 6) q^{65} - 8 i q^{67} + 2 i q^{68} + ( - 2 i - 4) q^{70} - 12 q^{71} + 4 i q^{73} - 2 q^{74} - 4 i q^{77} + (i + 2) q^{80} - 2 i q^{82} - 4 i q^{83} + ( - 4 i + 2) q^{85} - 4 q^{86} + 2 i q^{88} - 10 q^{89} + 12 q^{91} + 4 i q^{92} - 8 q^{94} - 8 i q^{97} + 3 i q^{98} +O(q^{100})$$ q + i * q^2 - q^4 + (i + 2) * q^5 + 2*i * q^7 - i * q^8 + (2*i - 1) * q^10 - 2 * q^11 - 6*i * q^13 - 2 * q^14 + q^16 - 2*i * q^17 + (-i - 2) * q^20 - 2*i * q^22 - 4*i * q^23 + (4*i + 3) * q^25 + 6 * q^26 - 2*i * q^28 - 8 * q^31 + i * q^32 + 2 * q^34 + (4*i - 2) * q^35 + 2*i * q^37 + (-2*i + 1) * q^40 - 2 * q^41 + 4*i * q^43 + 2 * q^44 + 4 * q^46 + 8*i * q^47 + 3 * q^49 + (3*i - 4) * q^50 + 6*i * q^52 + 6*i * q^53 + (-2*i - 4) * q^55 + 2 * q^56 + 10 * q^59 + 2 * q^61 - 8*i * q^62 - q^64 + (-12*i + 6) * q^65 - 8*i * q^67 + 2*i * q^68 + (-2*i - 4) * q^70 - 12 * q^71 + 4*i * q^73 - 2 * q^74 - 4*i * q^77 + (i + 2) * q^80 - 2*i * q^82 - 4*i * q^83 + (-4*i + 2) * q^85 - 4 * q^86 + 2*i * q^88 - 10 * q^89 + 12 * q^91 + 4*i * q^92 - 8 * q^94 - 8*i * q^97 + 3*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 4 q^{5}+O(q^{10})$$ 2 * q - 2 * q^4 + 4 * q^5 $$2 q - 2 q^{4} + 4 q^{5} - 2 q^{10} - 4 q^{11} - 4 q^{14} + 2 q^{16} - 4 q^{20} + 6 q^{25} + 12 q^{26} - 16 q^{31} + 4 q^{34} - 4 q^{35} + 2 q^{40} - 4 q^{41} + 4 q^{44} + 8 q^{46} + 6 q^{49} - 8 q^{50} - 8 q^{55} + 4 q^{56} + 20 q^{59} + 4 q^{61} - 2 q^{64} + 12 q^{65} - 8 q^{70} - 24 q^{71} - 4 q^{74} + 4 q^{80} + 4 q^{85} - 8 q^{86} - 20 q^{89} + 24 q^{91} - 16 q^{94}+O(q^{100})$$ 2 * q - 2 * q^4 + 4 * q^5 - 2 * q^10 - 4 * q^11 - 4 * q^14 + 2 * q^16 - 4 * q^20 + 6 * q^25 + 12 * q^26 - 16 * q^31 + 4 * q^34 - 4 * q^35 + 2 * q^40 - 4 * q^41 + 4 * q^44 + 8 * q^46 + 6 * q^49 - 8 * q^50 - 8 * q^55 + 4 * q^56 + 20 * q^59 + 4 * q^61 - 2 * q^64 + 12 * q^65 - 8 * q^70 - 24 * q^71 - 4 * q^74 + 4 * q^80 + 4 * q^85 - 8 * q^86 - 20 * q^89 + 24 * q^91 - 16 * q^94

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 − 1.00000i 1.00000i
1.00000i 0 −1.00000 2.00000 1.00000i 0 2.00000i 1.00000i 0 −1.00000 2.00000i
19.2 1.00000i 0 −1.00000 2.00000 + 1.00000i 0 2.00000i 1.00000i 0 −1.00000 + 2.00000i
 $$n$$: 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.2.c.a 2
3.b odd 2 1 30.2.c.a 2
4.b odd 2 1 720.2.f.f 2
5.b even 2 1 inner 90.2.c.a 2
5.c odd 4 1 450.2.a.b 1
5.c odd 4 1 450.2.a.f 1
8.b even 2 1 2880.2.f.e 2
8.d odd 2 1 2880.2.f.c 2
9.c even 3 2 810.2.i.b 4
9.d odd 6 2 810.2.i.e 4
12.b even 2 1 240.2.f.a 2
15.d odd 2 1 30.2.c.a 2
15.e even 4 1 150.2.a.a 1
15.e even 4 1 150.2.a.c 1
20.d odd 2 1 720.2.f.f 2
20.e even 4 1 3600.2.a.o 1
20.e even 4 1 3600.2.a.bg 1
21.c even 2 1 1470.2.g.g 2
21.g even 6 2 1470.2.n.a 4
21.h odd 6 2 1470.2.n.h 4
24.f even 2 1 960.2.f.i 2
24.h odd 2 1 960.2.f.h 2
40.e odd 2 1 2880.2.f.c 2
40.f even 2 1 2880.2.f.e 2
45.h odd 6 2 810.2.i.e 4
45.j even 6 2 810.2.i.b 4
48.i odd 4 1 3840.2.d.g 2
48.i odd 4 1 3840.2.d.y 2
48.k even 4 1 3840.2.d.j 2
48.k even 4 1 3840.2.d.x 2
60.h even 2 1 240.2.f.a 2
60.l odd 4 1 1200.2.a.g 1
60.l odd 4 1 1200.2.a.m 1
105.g even 2 1 1470.2.g.g 2
105.k odd 4 1 7350.2.a.bg 1
105.k odd 4 1 7350.2.a.cc 1
105.o odd 6 2 1470.2.n.h 4
105.p even 6 2 1470.2.n.a 4
120.i odd 2 1 960.2.f.h 2
120.m even 2 1 960.2.f.i 2
120.q odd 4 1 4800.2.a.m 1
120.q odd 4 1 4800.2.a.cj 1
120.w even 4 1 4800.2.a.l 1
120.w even 4 1 4800.2.a.cg 1
240.t even 4 1 3840.2.d.j 2
240.t even 4 1 3840.2.d.x 2
240.bm odd 4 1 3840.2.d.g 2
240.bm odd 4 1 3840.2.d.y 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.c.a 2 3.b odd 2 1
30.2.c.a 2 15.d odd 2 1
90.2.c.a 2 1.a even 1 1 trivial
90.2.c.a 2 5.b even 2 1 inner
150.2.a.a 1 15.e even 4 1
150.2.a.c 1 15.e even 4 1
240.2.f.a 2 12.b even 2 1
240.2.f.a 2 60.h even 2 1
450.2.a.b 1 5.c odd 4 1
450.2.a.f 1 5.c odd 4 1
720.2.f.f 2 4.b odd 2 1
720.2.f.f 2 20.d odd 2 1
810.2.i.b 4 9.c even 3 2
810.2.i.b 4 45.j even 6 2
810.2.i.e 4 9.d odd 6 2
810.2.i.e 4 45.h odd 6 2
960.2.f.h 2 24.h odd 2 1
960.2.f.h 2 120.i odd 2 1
960.2.f.i 2 24.f even 2 1
960.2.f.i 2 120.m even 2 1
1200.2.a.g 1 60.l odd 4 1
1200.2.a.m 1 60.l odd 4 1
1470.2.g.g 2 21.c even 2 1
1470.2.g.g 2 105.g even 2 1
1470.2.n.a 4 21.g even 6 2
1470.2.n.a 4 105.p even 6 2
1470.2.n.h 4 21.h odd 6 2
1470.2.n.h 4 105.o odd 6 2
2880.2.f.c 2 8.d odd 2 1
2880.2.f.c 2 40.e odd 2 1
2880.2.f.e 2 8.b even 2 1
2880.2.f.e 2 40.f even 2 1
3600.2.a.o 1 20.e even 4 1
3600.2.a.bg 1 20.e even 4 1
3840.2.d.g 2 48.i odd 4 1
3840.2.d.g 2 240.bm odd 4 1
3840.2.d.j 2 48.k even 4 1
3840.2.d.j 2 240.t even 4 1
3840.2.d.x 2 48.k even 4 1
3840.2.d.x 2 240.t even 4 1
3840.2.d.y 2 48.i odd 4 1
3840.2.d.y 2 240.bm odd 4 1
4800.2.a.l 1 120.w even 4 1
4800.2.a.m 1 120.q odd 4 1
4800.2.a.cg 1 120.w even 4 1
4800.2.a.cj 1 120.q odd 4 1
7350.2.a.bg 1 105.k odd 4 1
7350.2.a.cc 1 105.k odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(90, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4T + 5$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$T^{2}$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 10)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} + 64$$