# Properties

 Label 90.3.j.a Level $90$ Weight $3$ Character orbit 90.j Analytic conductor $2.452$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [90,3,Mod(29,90)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(90, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([1, 3]))

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

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

chi := DirichletCharacter("90.29");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$90 = 2 \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 90.j (of order $$6$$, 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{2} + 3 \beta_1 q^{3} + (2 \beta_{2} - 2) q^{4} + ( - \beta_{7} - 2 \beta_{6} + \beta_{4} + \cdots - 1) q^{5}+ \cdots + 9 \beta_{2} q^{9}+O(q^{10})$$ q - b4 * q^2 + 3*b1 * q^3 + (2*b2 - 2) * q^4 + (-b7 - 2*b6 + b4 - 4*b3 - b2 + 2*b1 - 1) * q^5 + (-3*b6 + 3*b5) * q^6 + (4*b7 - 2*b4 - 7*b3 + 7*b1) * q^7 + 2*b7 * q^8 + 9*b2 * q^9 $$q - \beta_{4} q^{2} + 3 \beta_1 q^{3} + (2 \beta_{2} - 2) q^{4} + ( - \beta_{7} - 2 \beta_{6} + \beta_{4} + \cdots - 1) q^{5}+ \cdots + (63 \beta_{6} - 18 \beta_{2} - 18) q^{99}+O(q^{100})$$ q - b4 * q^2 + 3*b1 * q^3 + (2*b2 - 2) * q^4 + (-b7 - 2*b6 + b4 - 4*b3 - b2 + 2*b1 - 1) * q^5 + (-3*b6 + 3*b5) * q^6 + (4*b7 - 2*b4 - 7*b3 + 7*b1) * q^7 + 2*b7 * q^8 + 9*b2 * q^9 + (-b7 - 2*b6 - 2*b5 + 2*b4 + 4*b3 + 2) * q^10 + (7*b5 + 2*b2 - 4) * q^11 + (6*b3 - 6*b1) * q^12 + (7*b7 + 7*b4 - 6*b1) * q^13 + (-7*b6 - 4*b2 - 4) * q^14 + (-3*b5 - 6*b4 - 3*b3 - 6*b2 - 3*b1 + 12) * q^15 - 4*b2 * q^16 + (-17*b7 - 2*b3 + 4*b1) * q^17 + (9*b7 - 9*b4) * q^18 + (-b6 - b5 - 18) * q^19 + (4*b5 - 2*b4 + 4*b3 - 2*b2 + 4*b1 + 4) * q^20 + (6*b6 + 6*b5 + 21) * q^21 + (2*b7 + 2*b4 - 14*b1) * q^22 + (-2*b7 + 2*b4 - 14*b3 + 7*b1) * q^23 + 6*b6 * q^24 + (-12*b7 + 16*b6 - 8*b5 + 6*b4 + 4*b3 - 3*b2 - 4*b1) * q^25 + (6*b6 - 6*b5 - 28*b2 + 14) * q^26 + 27*b3 * q^27 + (-4*b7 + 8*b4 + 14*b3) * q^28 + (-6*b5 + 3*b2 - 6) * q^29 + (-6*b7 + 3*b6 - 6*b5 - 6*b4 + 12*b2 + 6*b1 - 12) * q^30 + (13*b6 - 26*b5 + 22*b2 - 22) * q^31 + (-4*b7 + 4*b4) * q^32 + (21*b7 + 6*b3 - 12*b1) * q^33 + (-4*b6 + 2*b5 + 34*b2) * q^34 + (-20*b7 - 5*b6 + 5*b5 + 15*b3 - 20*b2 - 30*b1 + 10) * q^35 - 18 * q^36 + (-3*b7 + 6*b4 - 26*b3) * q^37 + (18*b4 + 2*b3 + 2*b1) * q^38 + (42*b6 - 21*b5 - 18*b2) * q^39 + (-2*b7 - 4*b6 + 8*b5 - 2*b4 + 4*b2 - 8*b1 - 4) * q^40 + (-4*b6 + 3*b2 + 3) * q^41 + (-21*b4 - 12*b3 - 12*b1) * q^42 + (32*b7 - 16*b4 + 4*b3 - 4*b1) * q^43 + (14*b6 - 14*b5 - 8*b2 + 4) * q^44 + (-9*b7 - 18*b6 + 18*b5 - 18*b3 - 18*b2 + 36*b1 + 9) * q^45 + (-7*b6 - 7*b5 + 4) * q^46 + (10*b4 + 29*b3 + 29*b1) * q^47 - 12*b3 * q^48 + (-28*b6 + 56*b5 - 24*b2 + 24) * q^49 + (-3*b7 + 4*b6 + 3*b4 - 32*b3 + 12*b2 + 16*b1 + 12) * q^50 + (-51*b6 + 6*b2 + 6) * q^51 + (-28*b7 + 14*b4 - 12*b3 + 12*b1) * q^52 + (18*b7 + 16*b3 - 32*b1) * q^53 + 27*b5 * q^54 + (16*b7 - 3*b6 - 3*b5 - 32*b4 + 26*b3 - 22) * q^55 + (14*b5 - 8*b2 + 16) * q^56 + (-3*b7 - 3*b4 - 54*b1) * q^57 + (3*b7 + 3*b4 + 12*b1) * q^58 + (-52*b6 + 16*b2 + 16) * q^59 + (12*b7 - 6*b6 + 6*b5 - 6*b3 + 24*b2 + 12*b1 - 12) * q^60 + (-4*b6 + 2*b5 - 5*b2) * q^61 + (22*b7 - 26*b3 + 52*b1) * q^62 + (18*b7 + 18*b4 + 63*b1) * q^63 + 8 * q^64 + (48*b5 - 9*b4 - 22*b3 + 26*b2 - 22*b1 - 52) * q^65 + (12*b6 - 6*b5 - 42*b2) * q^66 + (-20*b7 - 20*b4 + 7*b1) * q^67 + (34*b7 - 34*b4 + 8*b3 - 4*b1) * q^68 + (-6*b5 - 21*b2 + 42) * q^69 + (-20*b7 + 30*b6 - 15*b5 + 10*b4 + 10*b3 + 40*b2 - 10*b1) * q^70 + (15*b6 - 15*b5 + 116*b2 - 58) * q^71 + 18*b4 * q^72 + (-2*b7 + 4*b4 - 16*b3) * q^73 + (-26*b5 - 6*b2 + 12) * q^74 + (-24*b7 - 18*b6 - 18*b5 + 48*b4 - 9*b3 - 12) * q^75 + (-2*b6 + 4*b5 - 36*b2 + 36) * q^76 + (37*b7 - 37*b4 - 28*b3 + 14*b1) * q^77 + (-18*b7 + 18*b4 - 84*b3 + 42*b1) * q^78 + (-60*b6 + 30*b5 - 72*b2) * q^79 + (4*b7 + 8*b6 - 8*b5 + 8*b3 + 8*b2 - 16*b1 - 4) * q^80 + (81*b2 - 81) * q^81 + (3*b7 - 6*b4 + 8*b3) * q^82 + (4*b4 + 9*b3 + 9*b1) * q^83 + (12*b6 - 24*b5 + 42*b2 - 42) * q^84 + (13*b7 + 36*b6 - 72*b5 + 13*b4 - 46*b2 + 62*b1 + 46) * q^85 + (4*b6 - 32*b2 - 32) * q^86 + (-18*b7 + 9*b3 - 18*b1) * q^87 + (-8*b7 + 4*b4 - 28*b3 + 28*b1) * q^88 + (80*b6 - 80*b5 - 46*b2 + 23) * q^89 + (-18*b7 - 36*b6 + 18*b5 + 9*b4 + 36*b3 + 18*b2 - 36*b1) * q^90 + (37*b6 + 37*b5 + 42) * q^91 + (-4*b4 + 14*b3 + 14*b1) * q^92 + (-78*b7 + 39*b4 + 66*b3 - 66*b1) * q^93 + (-29*b6 + 58*b5 - 20*b2 + 20) * q^94 + (12*b7 + 39*b6 - 12*b4 + 68*b3 + 22*b2 - 34*b1 + 22) * q^95 - 12*b5 * q^96 + (48*b7 - 24*b4 - 20*b3 + 20*b1) * q^97 + (-24*b7 + 56*b3 - 112*b1) * q^98 + (63*b6 - 18*b2 - 18) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{4} - 12 q^{5} + 36 q^{9}+O(q^{10})$$ 8 * q - 8 * q^4 - 12 * q^5 + 36 * q^9 $$8 q - 8 q^{4} - 12 q^{5} + 36 q^{9} + 16 q^{10} - 24 q^{11} - 48 q^{14} + 72 q^{15} - 16 q^{16} - 144 q^{19} + 24 q^{20} + 168 q^{21} - 12 q^{25} - 36 q^{29} - 48 q^{30} - 88 q^{31} + 136 q^{34} - 144 q^{36} - 72 q^{39} - 16 q^{40} + 36 q^{41} + 32 q^{46} + 96 q^{49} + 144 q^{50} + 72 q^{51} - 176 q^{55} + 96 q^{56} + 192 q^{59} - 20 q^{61} + 64 q^{64} - 312 q^{65} - 168 q^{66} + 252 q^{69} + 160 q^{70} + 72 q^{74} - 96 q^{75} + 144 q^{76} - 288 q^{79} - 324 q^{81} - 168 q^{84} + 184 q^{85} - 384 q^{86} + 72 q^{90} + 336 q^{91} + 80 q^{94} + 264 q^{95} - 216 q^{99}+O(q^{100})$$ 8 * q - 8 * q^4 - 12 * q^5 + 36 * q^9 + 16 * q^10 - 24 * q^11 - 48 * q^14 + 72 * q^15 - 16 * q^16 - 144 * q^19 + 24 * q^20 + 168 * q^21 - 12 * q^25 - 36 * q^29 - 48 * q^30 - 88 * q^31 + 136 * q^34 - 144 * q^36 - 72 * q^39 - 16 * q^40 + 36 * q^41 + 32 * q^46 + 96 * q^49 + 144 * q^50 + 72 * q^51 - 176 * q^55 + 96 * q^56 + 192 * q^59 - 20 * q^61 + 64 * q^64 - 312 * q^65 - 168 * q^66 + 252 * q^69 + 160 * q^70 + 72 * q^74 - 96 * q^75 + 144 * q^76 - 288 * q^79 - 324 * q^81 - 168 * q^84 + 184 * q^85 - 384 * q^86 + 72 * q^90 + 336 * q^91 + 80 * q^94 + 264 * q^95 - 216 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{3}$$ $$=$$ $$\zeta_{24}^{6}$$ v^6 $$\beta_{4}$$ $$=$$ $$\zeta_{24}^{7} + \zeta_{24}$$ v^7 + v $$\beta_{5}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}$$ -v^7 + v $$\beta_{6}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3}$$ -v^7 + v^5 + v^3 $$\beta_{7}$$ $$=$$ $$-\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ -v^5 + v^3 + v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{5} + \beta_{4} ) / 2$$ (b5 + b4) / 2 $$\zeta_{24}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{5} ) / 2$$ (b7 + b6 - b5) / 2 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2$$ (-b7 + b6 + b4) / 2 $$\zeta_{24}^{6}$$ $$=$$ $$\beta_{3}$$ b3 $$\zeta_{24}^{7}$$ $$=$$ $$( -\beta_{5} + \beta_{4} ) / 2$$ (-b5 + b4) / 2

## Character values

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

 $$n$$ $$11$$ $$37$$ $$\chi(n)$$ $$1 - \beta_{2}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 −0.258819 − 0.965926i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.965926 + 0.258819i −0.258819 + 0.965926i 0.965926 + 0.258819i 0.258819 − 0.965926i −0.965926 − 0.258819i
−0.707107 + 1.22474i −2.59808 + 1.50000i −1.00000 1.73205i −1.48967 4.77293i 4.24264i −1.81954 1.05051i 2.82843 4.50000 7.79423i 6.89898 + 1.55051i
29.2 −0.707107 + 1.22474i 2.59808 1.50000i −1.00000 1.73205i −2.92455 + 4.05549i 4.24264i 10.3048 + 5.94949i 2.82843 4.50000 7.79423i −2.89898 6.44949i
29.3 0.707107 1.22474i −2.59808 + 1.50000i −1.00000 1.73205i −4.97443 + 0.504984i 4.24264i −10.3048 5.94949i −2.82843 4.50000 7.79423i −2.89898 + 6.44949i
29.4 0.707107 1.22474i 2.59808 1.50000i −1.00000 1.73205i 3.38865 + 3.67656i 4.24264i 1.81954 + 1.05051i −2.82843 4.50000 7.79423i 6.89898 1.55051i
59.1 −0.707107 1.22474i −2.59808 1.50000i −1.00000 + 1.73205i −1.48967 + 4.77293i 4.24264i −1.81954 + 1.05051i 2.82843 4.50000 + 7.79423i 6.89898 1.55051i
59.2 −0.707107 1.22474i 2.59808 + 1.50000i −1.00000 + 1.73205i −2.92455 4.05549i 4.24264i 10.3048 5.94949i 2.82843 4.50000 + 7.79423i −2.89898 + 6.44949i
59.3 0.707107 + 1.22474i −2.59808 1.50000i −1.00000 + 1.73205i −4.97443 0.504984i 4.24264i −10.3048 + 5.94949i −2.82843 4.50000 + 7.79423i −2.89898 6.44949i
59.4 0.707107 + 1.22474i 2.59808 + 1.50000i −1.00000 + 1.73205i 3.38865 3.67656i 4.24264i 1.81954 1.05051i −2.82843 4.50000 + 7.79423i 6.89898 + 1.55051i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.4 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
9.d odd 6 1 inner
45.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.3.j.a 8
3.b odd 2 1 270.3.j.a 8
5.b even 2 1 inner 90.3.j.a 8
5.c odd 4 1 450.3.i.a 4
5.c odd 4 1 450.3.i.c 4
9.c even 3 1 270.3.j.a 8
9.c even 3 1 810.3.b.a 8
9.d odd 6 1 inner 90.3.j.a 8
9.d odd 6 1 810.3.b.a 8
15.d odd 2 1 270.3.j.a 8
15.e even 4 1 1350.3.i.a 4
15.e even 4 1 1350.3.i.c 4
45.h odd 6 1 inner 90.3.j.a 8
45.h odd 6 1 810.3.b.a 8
45.j even 6 1 270.3.j.a 8
45.j even 6 1 810.3.b.a 8
45.k odd 12 1 1350.3.i.a 4
45.k odd 12 1 1350.3.i.c 4
45.l even 12 1 450.3.i.a 4
45.l even 12 1 450.3.i.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.3.j.a 8 1.a even 1 1 trivial
90.3.j.a 8 5.b even 2 1 inner
90.3.j.a 8 9.d odd 6 1 inner
90.3.j.a 8 45.h odd 6 1 inner
270.3.j.a 8 3.b odd 2 1
270.3.j.a 8 9.c even 3 1
270.3.j.a 8 15.d odd 2 1
270.3.j.a 8 45.j even 6 1
450.3.i.a 4 5.c odd 4 1
450.3.i.a 4 45.l even 12 1
450.3.i.c 4 5.c odd 4 1
450.3.i.c 4 45.l even 12 1
810.3.b.a 8 9.c even 3 1
810.3.b.a 8 9.d odd 6 1
810.3.b.a 8 45.h odd 6 1
810.3.b.a 8 45.j even 6 1
1350.3.i.a 4 15.e even 4 1
1350.3.i.a 4 45.k odd 12 1
1350.3.i.c 4 15.e even 4 1
1350.3.i.c 4 45.k odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} - 146T_{7}^{6} + 20691T_{7}^{4} - 91250T_{7}^{2} + 390625$$ acting on $$S_{3}^{\mathrm{new}}(90, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + 2 T^{2} + 4)^{2}$$
$3$ $$(T^{4} - 9 T^{2} + 81)^{2}$$
$5$ $$T^{8} + 12 T^{7} + \cdots + 390625$$
$7$ $$T^{8} - 146 T^{6} + \cdots + 390625$$
$11$ $$(T^{4} + 12 T^{3} + \cdots + 7396)^{2}$$
$13$ $$T^{8} + \cdots + 4430766096$$
$17$ $$(T^{4} - 1180 T^{2} + 320356)^{2}$$
$19$ $$(T^{2} + 36 T + 318)^{4}$$
$23$ $$T^{8} + 310 T^{6} + \cdots + 373301041$$
$29$ $$(T^{4} + 18 T^{3} + \cdots + 2025)^{2}$$
$31$ $$(T^{4} + 44 T^{3} + \cdots + 280900)^{2}$$
$37$ $$(T^{4} + 1460 T^{2} + 386884)^{2}$$
$41$ $$(T^{4} - 18 T^{3} + \cdots + 25)^{2}$$
$43$ $$T^{8} + \cdots + 5337948160000$$
$47$ $$T^{8} + \cdots + 29120366676241$$
$53$ $$(T^{4} - 2832 T^{2} + 14400)^{2}$$
$59$ $$(T^{4} - 96 T^{3} + \cdots + 21529600)^{2}$$
$61$ $$(T^{4} + 10 T^{3} + 99 T^{2} + \cdots + 1)^{2}$$
$67$ $$T^{8} + \cdots + 30549950894401$$
$71$ $$(T^{4} + 21084 T^{2} + 92968164)^{2}$$
$73$ $$(T^{4} + 560 T^{2} + 53824)^{2}$$
$79$ $$(T^{4} + 144 T^{3} + \cdots + 46656)^{2}$$
$83$ $$T^{8} + \cdots + 1982119441$$
$89$ $$(T^{4} + 28774 T^{2} + 125731369)^{2}$$
$97$ $$T^{8} + \cdots + 87219461226496$$