# Properties

 Label 90.3.k.b Level $90$ Weight $3$ Character orbit 90.k Analytic conductor $2.452$ Analytic rank $0$ Dimension $24$ 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(7,90)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(90, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([8, 3]))

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

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

chi := DirichletCharacter("90.7");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$90 = 2 \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 90.k (of order $$12$$, degree $$4$$, 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: $$24$$ Relative dimension: $$6$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 12 q^{2} + 4 q^{3} + 16 q^{6} + 6 q^{7} + 48 q^{8}+O(q^{10})$$ 24 * q + 12 * q^2 + 4 * q^3 + 16 * q^6 + 6 * q^7 + 48 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 12 q^{2} + 4 q^{3} + 16 q^{6} + 6 q^{7} + 48 q^{8} + 12 q^{10} - 12 q^{11} + 8 q^{12} - 38 q^{15} + 48 q^{16} - 36 q^{17} - 20 q^{18} + 12 q^{20} - 128 q^{21} + 12 q^{22} - 66 q^{23} - 42 q^{25} + 10 q^{27} + 24 q^{28} - 48 q^{30} + 72 q^{31} - 48 q^{32} - 122 q^{33} - 240 q^{35} - 8 q^{36} + 36 q^{37} - 36 q^{38} - 12 q^{40} - 24 q^{41} - 16 q^{42} - 108 q^{43} + 158 q^{45} - 264 q^{46} - 36 q^{47} + 32 q^{48} + 42 q^{50} + 380 q^{51} + 384 q^{53} + 288 q^{55} + 24 q^{56} + 154 q^{57} - 108 q^{58} - 56 q^{60} + 360 q^{61} + 144 q^{62} - 68 q^{63} + 144 q^{65} - 80 q^{66} + 144 q^{67} - 36 q^{68} - 174 q^{70} + 216 q^{71} + 32 q^{72} + 432 q^{73} - 184 q^{75} + 72 q^{76} - 48 q^{77} - 240 q^{78} + 416 q^{81} - 48 q^{82} - 378 q^{83} - 228 q^{85} + 216 q^{86} + 52 q^{87} - 24 q^{88} + 124 q^{90} - 1560 q^{91} - 132 q^{92} + 580 q^{93} + 264 q^{95} + 32 q^{96} - 294 q^{97} + 264 q^{98}+O(q^{100})$$ 24 * q + 12 * q^2 + 4 * q^3 + 16 * q^6 + 6 * q^7 + 48 * q^8 + 12 * q^10 - 12 * q^11 + 8 * q^12 - 38 * q^15 + 48 * q^16 - 36 * q^17 - 20 * q^18 + 12 * q^20 - 128 * q^21 + 12 * q^22 - 66 * q^23 - 42 * q^25 + 10 * q^27 + 24 * q^28 - 48 * q^30 + 72 * q^31 - 48 * q^32 - 122 * q^33 - 240 * q^35 - 8 * q^36 + 36 * q^37 - 36 * q^38 - 12 * q^40 - 24 * q^41 - 16 * q^42 - 108 * q^43 + 158 * q^45 - 264 * q^46 - 36 * q^47 + 32 * q^48 + 42 * q^50 + 380 * q^51 + 384 * q^53 + 288 * q^55 + 24 * q^56 + 154 * q^57 - 108 * q^58 - 56 * q^60 + 360 * q^61 + 144 * q^62 - 68 * q^63 + 144 * q^65 - 80 * q^66 + 144 * q^67 - 36 * q^68 - 174 * q^70 + 216 * q^71 + 32 * q^72 + 432 * q^73 - 184 * q^75 + 72 * q^76 - 48 * q^77 - 240 * q^78 + 416 * q^81 - 48 * q^82 - 378 * q^83 - 228 * q^85 + 216 * q^86 + 52 * q^87 - 24 * q^88 + 124 * q^90 - 1560 * q^91 - 132 * q^92 + 580 * q^93 + 264 * q^95 + 32 * q^96 - 294 * q^97 + 264 * q^98

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 1.36603 0.366025i −2.94504 + 0.571623i 1.73205 1.00000i 4.17929 2.74473i −3.81377 + 1.85881i 8.06163 2.16011i 2.00000 2.00000i 8.34650 3.36690i 4.70437 5.27910i
7.2 1.36603 0.366025i −0.900794 2.86157i 1.73205 1.00000i −3.46748 3.60230i −2.27791 3.57926i 1.76879 0.473946i 2.00000 2.00000i −7.37714 + 5.15537i −6.05520 3.65165i
7.3 1.36603 0.366025i −0.0929015 + 2.99856i 1.73205 1.00000i 3.41977 + 3.64763i 0.970644 + 4.13012i −6.01857 + 1.61267i 2.00000 2.00000i −8.98274 0.557142i 6.00661 + 3.73103i
7.4 1.36603 0.366025i 1.64244 2.51046i 1.73205 1.00000i 1.90856 + 4.62141i 1.32472 4.03052i −0.850119 + 0.227789i 2.00000 2.00000i −3.60480 8.24654i 4.29869 + 5.61438i
7.5 1.36603 0.366025i 2.04786 + 2.19232i 1.73205 1.00000i −4.80314 + 1.38918i 3.59987 + 2.24520i 11.2960 3.02677i 2.00000 2.00000i −0.612574 + 8.97913i −6.05274 + 3.65572i
7.6 1.36603 0.366025i 2.98049 + 0.341569i 1.73205 1.00000i 1.36109 4.81118i 4.19645 0.624344i −10.1597 + 2.72228i 2.00000 2.00000i 8.76666 + 2.03609i 0.0982721 7.07038i
13.1 1.36603 + 0.366025i −2.94504 0.571623i 1.73205 + 1.00000i 4.17929 + 2.74473i −3.81377 1.85881i 8.06163 + 2.16011i 2.00000 + 2.00000i 8.34650 + 3.36690i 4.70437 + 5.27910i
13.2 1.36603 + 0.366025i −0.900794 + 2.86157i 1.73205 + 1.00000i −3.46748 + 3.60230i −2.27791 + 3.57926i 1.76879 + 0.473946i 2.00000 + 2.00000i −7.37714 5.15537i −6.05520 + 3.65165i
13.3 1.36603 + 0.366025i −0.0929015 2.99856i 1.73205 + 1.00000i 3.41977 3.64763i 0.970644 4.13012i −6.01857 1.61267i 2.00000 + 2.00000i −8.98274 + 0.557142i 6.00661 3.73103i
13.4 1.36603 + 0.366025i 1.64244 + 2.51046i 1.73205 + 1.00000i 1.90856 4.62141i 1.32472 + 4.03052i −0.850119 0.227789i 2.00000 + 2.00000i −3.60480 + 8.24654i 4.29869 5.61438i
13.5 1.36603 + 0.366025i 2.04786 2.19232i 1.73205 + 1.00000i −4.80314 1.38918i 3.59987 2.24520i 11.2960 + 3.02677i 2.00000 + 2.00000i −0.612574 8.97913i −6.05274 3.65572i
13.6 1.36603 + 0.366025i 2.98049 0.341569i 1.73205 + 1.00000i 1.36109 + 4.81118i 4.19645 + 0.624344i −10.1597 2.72228i 2.00000 + 2.00000i 8.76666 2.03609i 0.0982721 + 7.07038i
43.1 −0.366025 1.36603i −2.99856 0.0929015i −1.73205 + 1.00000i 1.44906 + 4.78542i 0.970644 + 4.13012i 1.61267 + 6.01857i 2.00000 + 2.00000i 8.98274 + 0.557142i 6.00661 3.73103i
43.2 −0.366025 1.36603i −2.19232 + 2.04786i −1.73205 + 1.00000i 3.60463 3.46506i 3.59987 + 2.24520i −3.02677 11.2960i 2.00000 + 2.00000i 0.612574 8.97913i −6.05274 3.65572i
43.3 −0.366025 1.36603i −0.571623 2.94504i −1.73205 + 1.00000i −4.46665 + 2.24700i −3.81377 + 1.85881i −2.16011 8.06163i 2.00000 + 2.00000i −8.34650 + 3.36690i 4.70437 + 5.27910i
43.4 −0.366025 1.36603i −0.341569 + 2.98049i −1.73205 + 1.00000i −4.84715 1.22685i 4.19645 0.624344i 2.72228 + 10.1597i 2.00000 + 2.00000i −8.76666 2.03609i 0.0982721 + 7.07038i
43.5 −0.366025 1.36603i 2.51046 + 1.64244i −1.73205 + 1.00000i 3.04798 + 3.96356i 1.32472 4.03052i 0.227789 + 0.850119i 2.00000 + 2.00000i 3.60480 + 8.24654i 4.29869 5.61438i
43.6 −0.366025 1.36603i 2.86157 0.900794i −1.73205 + 1.00000i −1.38594 4.80408i −2.27791 3.57926i −0.473946 1.76879i 2.00000 + 2.00000i 7.37714 5.15537i −6.05520 + 3.65165i
67.1 −0.366025 + 1.36603i −2.99856 + 0.0929015i −1.73205 1.00000i 1.44906 4.78542i 0.970644 4.13012i 1.61267 6.01857i 2.00000 2.00000i 8.98274 0.557142i 6.00661 + 3.73103i
67.2 −0.366025 + 1.36603i −2.19232 2.04786i −1.73205 1.00000i 3.60463 + 3.46506i 3.59987 2.24520i −3.02677 + 11.2960i 2.00000 2.00000i 0.612574 + 8.97913i −6.05274 + 3.65572i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.6 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
9.c even 3 1 inner
45.k odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.3.k.b 24
3.b odd 2 1 270.3.l.a 24
5.c odd 4 1 inner 90.3.k.b 24
9.c even 3 1 inner 90.3.k.b 24
9.c even 3 1 810.3.g.h 12
9.d odd 6 1 270.3.l.a 24
9.d odd 6 1 810.3.g.j 12
15.e even 4 1 270.3.l.a 24
45.k odd 12 1 inner 90.3.k.b 24
45.k odd 12 1 810.3.g.h 12
45.l even 12 1 270.3.l.a 24
45.l even 12 1 810.3.g.j 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.3.k.b 24 1.a even 1 1 trivial
90.3.k.b 24 5.c odd 4 1 inner
90.3.k.b 24 9.c even 3 1 inner
90.3.k.b 24 45.k odd 12 1 inner
270.3.l.a 24 3.b odd 2 1
270.3.l.a 24 9.d odd 6 1
270.3.l.a 24 15.e even 4 1
270.3.l.a 24 45.l even 12 1
810.3.g.h 12 9.c even 3 1
810.3.g.h 12 45.k odd 12 1
810.3.g.j 12 9.d odd 6 1
810.3.g.j 12 45.l even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{24} - 6 T_{7}^{23} + 18 T_{7}^{22} - 772 T_{7}^{21} - 14187 T_{7}^{20} + 132888 T_{7}^{19} + \cdots + 11\!\cdots\!61$$ acting on $$S_{3}^{\mathrm{new}}(90, [\chi])$$.