# Properties

 Label 90.3.b.a Level $90$ Weight $3$ Character orbit 90.b Analytic conductor $2.452$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [90,3,Mod(89,90)]

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([1, 1]))

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

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

chi := DirichletCharacter("90.89");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$90 = 2 \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 90.b (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: $$2.45232237924$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{8}^{3} + \zeta_{8}) q^{2} + 2 q^{4} + 5 \zeta_{8} q^{5} - 4 \zeta_{8}^{2} q^{7} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{8} +O(q^{10})$$ q + (-z^3 + z) * q^2 + 2 * q^4 + 5*z * q^5 - 4*z^2 * q^7 + (-2*z^3 + 2*z) * q^8 $$q + ( - \zeta_{8}^{3} + \zeta_{8}) q^{2} + 2 q^{4} + 5 \zeta_{8} q^{5} - 4 \zeta_{8}^{2} q^{7} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{8} + (5 \zeta_{8}^{2} + 5) q^{10} + (8 \zeta_{8}^{3} + 8 \zeta_{8}) q^{11} - 18 \zeta_{8}^{2} q^{13} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{14} + 4 q^{16} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{17} - 24 q^{19} + 10 \zeta_{8} q^{20} + 16 \zeta_{8}^{2} q^{22} + (28 \zeta_{8}^{3} - 28 \zeta_{8}) q^{23} + 25 \zeta_{8}^{2} q^{25} + ( - 18 \zeta_{8}^{3} - 18 \zeta_{8}) q^{26} - 8 \zeta_{8}^{2} q^{28} + ( - 27 \zeta_{8}^{3} - 27 \zeta_{8}) q^{29} + 4 q^{31} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8}) q^{32} + 2 q^{34} - 20 \zeta_{8}^{3} q^{35} + 56 \zeta_{8}^{2} q^{37} + (24 \zeta_{8}^{3} - 24 \zeta_{8}) q^{38} + (10 \zeta_{8}^{2} + 10) q^{40} + (17 \zeta_{8}^{3} + 17 \zeta_{8}) q^{41} - 80 \zeta_{8}^{2} q^{43} + (16 \zeta_{8}^{3} + 16 \zeta_{8}) q^{44} - 56 q^{46} + ( - 20 \zeta_{8}^{3} + 20 \zeta_{8}) q^{47} + 33 q^{49} + (25 \zeta_{8}^{3} + 25 \zeta_{8}) q^{50} - 36 \zeta_{8}^{2} q^{52} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{53} + (40 \zeta_{8}^{2} - 40) q^{55} + ( - 8 \zeta_{8}^{3} - 8 \zeta_{8}) q^{56} - 54 \zeta_{8}^{2} q^{58} + (44 \zeta_{8}^{3} + 44 \zeta_{8}) q^{59} + 110 q^{61} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8}) q^{62} + 8 q^{64} - 90 \zeta_{8}^{3} q^{65} - 32 \zeta_{8}^{2} q^{67} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{68} + ( - 20 \zeta_{8}^{2} + 20) q^{70} + (36 \zeta_{8}^{3} + 36 \zeta_{8}) q^{71} + 46 \zeta_{8}^{2} q^{73} + (56 \zeta_{8}^{3} + 56 \zeta_{8}) q^{74} - 48 q^{76} + ( - 32 \zeta_{8}^{3} + 32 \zeta_{8}) q^{77} + 36 q^{79} + 20 \zeta_{8} q^{80} + 34 \zeta_{8}^{2} q^{82} + (4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{83} + (5 \zeta_{8}^{2} + 5) q^{85} + ( - 80 \zeta_{8}^{3} - 80 \zeta_{8}) q^{86} + 32 \zeta_{8}^{2} q^{88} + ( - 41 \zeta_{8}^{3} - 41 \zeta_{8}) q^{89} - 72 q^{91} + (56 \zeta_{8}^{3} - 56 \zeta_{8}) q^{92} + 40 q^{94} - 120 \zeta_{8} q^{95} - 14 \zeta_{8}^{2} q^{97} + ( - 33 \zeta_{8}^{3} + 33 \zeta_{8}) q^{98} +O(q^{100})$$ q + (-z^3 + z) * q^2 + 2 * q^4 + 5*z * q^5 - 4*z^2 * q^7 + (-2*z^3 + 2*z) * q^8 + (5*z^2 + 5) * q^10 + (8*z^3 + 8*z) * q^11 - 18*z^2 * q^13 + (-4*z^3 - 4*z) * q^14 + 4 * q^16 + (-z^3 + z) * q^17 - 24 * q^19 + 10*z * q^20 + 16*z^2 * q^22 + (28*z^3 - 28*z) * q^23 + 25*z^2 * q^25 + (-18*z^3 - 18*z) * q^26 - 8*z^2 * q^28 + (-27*z^3 - 27*z) * q^29 + 4 * q^31 + (-4*z^3 + 4*z) * q^32 + 2 * q^34 - 20*z^3 * q^35 + 56*z^2 * q^37 + (24*z^3 - 24*z) * q^38 + (10*z^2 + 10) * q^40 + (17*z^3 + 17*z) * q^41 - 80*z^2 * q^43 + (16*z^3 + 16*z) * q^44 - 56 * q^46 + (-20*z^3 + 20*z) * q^47 + 33 * q^49 + (25*z^3 + 25*z) * q^50 - 36*z^2 * q^52 + (-3*z^3 + 3*z) * q^53 + (40*z^2 - 40) * q^55 + (-8*z^3 - 8*z) * q^56 - 54*z^2 * q^58 + (44*z^3 + 44*z) * q^59 + 110 * q^61 + (-4*z^3 + 4*z) * q^62 + 8 * q^64 - 90*z^3 * q^65 - 32*z^2 * q^67 + (-2*z^3 + 2*z) * q^68 + (-20*z^2 + 20) * q^70 + (36*z^3 + 36*z) * q^71 + 46*z^2 * q^73 + (56*z^3 + 56*z) * q^74 - 48 * q^76 + (-32*z^3 + 32*z) * q^77 + 36 * q^79 + 20*z * q^80 + 34*z^2 * q^82 + (4*z^3 - 4*z) * q^83 + (5*z^2 + 5) * q^85 + (-80*z^3 - 80*z) * q^86 + 32*z^2 * q^88 + (-41*z^3 - 41*z) * q^89 - 72 * q^91 + (56*z^3 - 56*z) * q^92 + 40 * q^94 - 120*z * q^95 - 14*z^2 * q^97 + (-33*z^3 + 33*z) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{4}+O(q^{10})$$ 4 * q + 8 * q^4 $$4 q + 8 q^{4} + 20 q^{10} + 16 q^{16} - 96 q^{19} + 16 q^{31} + 8 q^{34} + 40 q^{40} - 224 q^{46} + 132 q^{49} - 160 q^{55} + 440 q^{61} + 32 q^{64} + 80 q^{70} - 192 q^{76} + 144 q^{79} + 20 q^{85} - 288 q^{91} + 160 q^{94}+O(q^{100})$$ 4 * q + 8 * q^4 + 20 * q^10 + 16 * q^16 - 96 * q^19 + 16 * q^31 + 8 * q^34 + 40 * q^40 - 224 * q^46 + 132 * q^49 - 160 * q^55 + 440 * q^61 + 32 * q^64 + 80 * q^70 - 192 * q^76 + 144 * q^79 + 20 * q^85 - 288 * q^91 + 160 * 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
89.1
 −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
−1.41421 0 2.00000 −3.53553 3.53553i 0 4.00000i −2.82843 0 5.00000 + 5.00000i
89.2 −1.41421 0 2.00000 −3.53553 + 3.53553i 0 4.00000i −2.82843 0 5.00000 5.00000i
89.3 1.41421 0 2.00000 3.53553 3.53553i 0 4.00000i 2.82843 0 5.00000 5.00000i
89.4 1.41421 0 2.00000 3.53553 + 3.53553i 0 4.00000i 2.82843 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.3.b.a 4
3.b odd 2 1 inner 90.3.b.a 4
4.b odd 2 1 720.3.c.c 4
5.b even 2 1 inner 90.3.b.a 4
5.c odd 4 1 450.3.d.b 2
5.c odd 4 1 450.3.d.e 2
8.b even 2 1 2880.3.c.h 4
8.d odd 2 1 2880.3.c.a 4
9.c even 3 2 810.3.j.d 8
9.d odd 6 2 810.3.j.d 8
12.b even 2 1 720.3.c.c 4
15.d odd 2 1 inner 90.3.b.a 4
15.e even 4 1 450.3.d.b 2
15.e even 4 1 450.3.d.e 2
20.d odd 2 1 720.3.c.c 4
20.e even 4 1 3600.3.l.c 2
20.e even 4 1 3600.3.l.i 2
24.f even 2 1 2880.3.c.a 4
24.h odd 2 1 2880.3.c.h 4
40.e odd 2 1 2880.3.c.a 4
40.f even 2 1 2880.3.c.h 4
45.h odd 6 2 810.3.j.d 8
45.j even 6 2 810.3.j.d 8
60.h even 2 1 720.3.c.c 4
60.l odd 4 1 3600.3.l.c 2
60.l odd 4 1 3600.3.l.i 2
120.i odd 2 1 2880.3.c.h 4
120.m even 2 1 2880.3.c.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.3.b.a 4 1.a even 1 1 trivial
90.3.b.a 4 3.b odd 2 1 inner
90.3.b.a 4 5.b even 2 1 inner
90.3.b.a 4 15.d odd 2 1 inner
450.3.d.b 2 5.c odd 4 1
450.3.d.b 2 15.e even 4 1
450.3.d.e 2 5.c odd 4 1
450.3.d.e 2 15.e even 4 1
720.3.c.c 4 4.b odd 2 1
720.3.c.c 4 12.b even 2 1
720.3.c.c 4 20.d odd 2 1
720.3.c.c 4 60.h even 2 1
810.3.j.d 8 9.c even 3 2
810.3.j.d 8 9.d odd 6 2
810.3.j.d 8 45.h odd 6 2
810.3.j.d 8 45.j even 6 2
2880.3.c.a 4 8.d odd 2 1
2880.3.c.a 4 24.f even 2 1
2880.3.c.a 4 40.e odd 2 1
2880.3.c.a 4 120.m even 2 1
2880.3.c.h 4 8.b even 2 1
2880.3.c.h 4 24.h odd 2 1
2880.3.c.h 4 40.f even 2 1
2880.3.c.h 4 120.i odd 2 1
3600.3.l.c 2 20.e even 4 1
3600.3.l.c 2 60.l odd 4 1
3600.3.l.i 2 20.e even 4 1
3600.3.l.i 2 60.l odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(90, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 625$$
$7$ $$(T^{2} + 16)^{2}$$
$11$ $$(T^{2} + 128)^{2}$$
$13$ $$(T^{2} + 324)^{2}$$
$17$ $$(T^{2} - 2)^{2}$$
$19$ $$(T + 24)^{4}$$
$23$ $$(T^{2} - 1568)^{2}$$
$29$ $$(T^{2} + 1458)^{2}$$
$31$ $$(T - 4)^{4}$$
$37$ $$(T^{2} + 3136)^{2}$$
$41$ $$(T^{2} + 578)^{2}$$
$43$ $$(T^{2} + 6400)^{2}$$
$47$ $$(T^{2} - 800)^{2}$$
$53$ $$(T^{2} - 18)^{2}$$
$59$ $$(T^{2} + 3872)^{2}$$
$61$ $$(T - 110)^{4}$$
$67$ $$(T^{2} + 1024)^{2}$$
$71$ $$(T^{2} + 2592)^{2}$$
$73$ $$(T^{2} + 2116)^{2}$$
$79$ $$(T - 36)^{4}$$
$83$ $$(T^{2} - 32)^{2}$$
$89$ $$(T^{2} + 3362)^{2}$$
$97$ $$(T^{2} + 196)^{2}$$
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