# Properties

 Label 90.3.k.a 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 12 q^{2} - 6 q^{7} - 48 q^{8}+O(q^{10})$$ 24 * q - 12 * q^2 - 6 * q^7 - 48 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 12 q^{2} - 6 q^{7} - 48 q^{8} - 12 q^{10} - 12 q^{11} + 42 q^{15} + 48 q^{16} - 36 q^{17} - 12 q^{18} + 12 q^{20} + 96 q^{21} - 12 q^{22} - 54 q^{23} + 54 q^{25} + 162 q^{27} - 24 q^{28} - 48 q^{30} - 72 q^{31} + 48 q^{32} + 6 q^{33} - 336 q^{35} - 72 q^{36} + 132 q^{37} - 36 q^{38} + 12 q^{40} + 24 q^{41} - 144 q^{42} + 108 q^{43} - 186 q^{45} + 216 q^{46} + 48 q^{47} + 54 q^{50} + 108 q^{51} + 384 q^{53} - 552 q^{55} + 24 q^{56} + 186 q^{57} + 60 q^{58} + 144 q^{60} - 456 q^{61} + 144 q^{62} + 72 q^{63} + 264 q^{65} + 240 q^{66} + 12 q^{67} - 36 q^{68} + 174 q^{70} - 168 q^{71} + 96 q^{72} - 432 q^{73} - 468 q^{75} - 72 q^{76} - 48 q^{77} - 264 q^{78} - 480 q^{81} - 48 q^{82} - 246 q^{83} + 324 q^{85} + 216 q^{86} - 636 q^{87} + 24 q^{88} - 12 q^{90} + 1224 q^{91} - 108 q^{92} + 180 q^{93} + 432 q^{95} - 102 q^{97} + 24 q^{98}+O(q^{100})$$ 24 * q - 12 * q^2 - 6 * q^7 - 48 * q^8 - 12 * q^10 - 12 * q^11 + 42 * q^15 + 48 * q^16 - 36 * q^17 - 12 * q^18 + 12 * q^20 + 96 * q^21 - 12 * q^22 - 54 * q^23 + 54 * q^25 + 162 * q^27 - 24 * q^28 - 48 * q^30 - 72 * q^31 + 48 * q^32 + 6 * q^33 - 336 * q^35 - 72 * q^36 + 132 * q^37 - 36 * q^38 + 12 * q^40 + 24 * q^41 - 144 * q^42 + 108 * q^43 - 186 * q^45 + 216 * q^46 + 48 * q^47 + 54 * q^50 + 108 * q^51 + 384 * q^53 - 552 * q^55 + 24 * q^56 + 186 * q^57 + 60 * q^58 + 144 * q^60 - 456 * q^61 + 144 * q^62 + 72 * q^63 + 264 * q^65 + 240 * q^66 + 12 * q^67 - 36 * q^68 + 174 * q^70 - 168 * q^71 + 96 * q^72 - 432 * q^73 - 468 * q^75 - 72 * q^76 - 48 * q^77 - 264 * q^78 - 480 * q^81 - 48 * q^82 - 246 * q^83 + 324 * q^85 + 216 * q^86 - 636 * q^87 + 24 * q^88 - 12 * q^90 + 1224 * q^91 - 108 * q^92 + 180 * q^93 + 432 * q^95 - 102 * q^97 + 24 * 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 −2.54067 + 1.59530i 1.73205 1.00000i −4.00416 2.99444i 2.88670 3.10917i 6.73232 1.80392i −2.00000 + 2.00000i 3.91003 8.10627i 6.56583 + 2.62486i
7.2 −1.36603 + 0.366025i −2.17538 + 2.06584i 1.73205 1.00000i 4.54899 + 2.07525i 2.21548 3.61824i −9.78534 + 2.62197i −2.00000 + 2.00000i 0.464594 8.98800i −6.97363 1.16980i
7.3 −1.36603 + 0.366025i −1.07491 2.80082i 1.73205 1.00000i −4.68508 + 1.74643i 2.49352 + 3.43254i −1.30080 + 0.348547i −2.00000 + 2.00000i −6.68914 + 6.02125i 5.76070 4.10052i
7.4 −1.36603 + 0.366025i 1.11865 2.78364i 1.73205 1.00000i 3.61223 3.45714i −0.509217 + 4.21197i −8.95134 + 2.39851i −2.00000 + 2.00000i −6.49726 6.22781i −3.66899 + 6.04471i
7.5 −1.36603 + 0.366025i 1.72031 + 2.45775i 1.73205 1.00000i 3.16601 3.86994i −3.24958 2.72767i 5.91251 1.58425i −2.00000 + 2.00000i −3.08107 + 8.45618i −2.90835 + 6.44527i
7.6 −1.36603 + 0.366025i 2.95201 0.534441i 1.73205 1.00000i −0.0399035 + 4.99984i −3.83690 + 1.81057i 3.29458 0.882779i −2.00000 + 2.00000i 8.42875 3.15535i −1.77556 6.84452i
13.1 −1.36603 0.366025i −2.54067 1.59530i 1.73205 + 1.00000i −4.00416 + 2.99444i 2.88670 + 3.10917i 6.73232 + 1.80392i −2.00000 2.00000i 3.91003 + 8.10627i 6.56583 2.62486i
13.2 −1.36603 0.366025i −2.17538 2.06584i 1.73205 + 1.00000i 4.54899 2.07525i 2.21548 + 3.61824i −9.78534 2.62197i −2.00000 2.00000i 0.464594 + 8.98800i −6.97363 + 1.16980i
13.3 −1.36603 0.366025i −1.07491 + 2.80082i 1.73205 + 1.00000i −4.68508 1.74643i 2.49352 3.43254i −1.30080 0.348547i −2.00000 2.00000i −6.68914 6.02125i 5.76070 + 4.10052i
13.4 −1.36603 0.366025i 1.11865 + 2.78364i 1.73205 + 1.00000i 3.61223 + 3.45714i −0.509217 4.21197i −8.95134 2.39851i −2.00000 2.00000i −6.49726 + 6.22781i −3.66899 6.04471i
13.5 −1.36603 0.366025i 1.72031 2.45775i 1.73205 + 1.00000i 3.16601 + 3.86994i −3.24958 + 2.72767i 5.91251 + 1.58425i −2.00000 2.00000i −3.08107 8.45618i −2.90835 6.44527i
13.6 −1.36603 0.366025i 2.95201 + 0.534441i 1.73205 + 1.00000i −0.0399035 4.99984i −3.83690 1.81057i 3.29458 + 0.882779i −2.00000 2.00000i 8.42875 + 3.15535i −1.77556 + 6.84452i
43.1 0.366025 + 1.36603i −2.45775 + 1.72031i −1.73205 + 1.00000i −4.93447 + 0.806874i −3.24958 2.72767i −1.58425 5.91251i −2.00000 2.00000i 3.08107 8.45618i −2.90835 6.44527i
43.2 0.366025 + 1.36603i −2.06584 2.17538i −1.73205 + 1.00000i −0.477275 + 4.97717i 2.21548 3.61824i 2.62197 + 9.78534i −2.00000 2.00000i −0.464594 + 8.98800i −6.97363 + 1.16980i
43.3 0.366025 + 1.36603i −1.59530 2.54067i −1.73205 + 1.00000i −0.591182 4.96493i 2.88670 3.10917i −1.80392 6.73232i −2.00000 2.00000i −3.91003 + 8.10627i 6.56583 2.62486i
43.4 0.366025 + 1.36603i 0.534441 + 2.95201i −1.73205 + 1.00000i 4.34994 + 2.46536i −3.83690 + 1.81057i −0.882779 3.29458i −2.00000 2.00000i −8.42875 + 3.15535i −1.77556 + 6.84452i
43.5 0.366025 + 1.36603i 2.78364 + 1.11865i −1.73205 + 1.00000i −4.80008 + 1.39971i −0.509217 + 4.21197i 2.39851 + 8.95134i −2.00000 2.00000i 6.49726 + 6.22781i −3.66899 6.04471i
43.6 0.366025 + 1.36603i 2.80082 1.07491i −1.73205 + 1.00000i 3.85499 3.18419i 2.49352 + 3.43254i 0.348547 + 1.30080i −2.00000 2.00000i 6.68914 6.02125i 5.76070 + 4.10052i
67.1 0.366025 1.36603i −2.45775 1.72031i −1.73205 1.00000i −4.93447 0.806874i −3.24958 + 2.72767i −1.58425 + 5.91251i −2.00000 + 2.00000i 3.08107 + 8.45618i −2.90835 + 6.44527i
67.2 0.366025 1.36603i −2.06584 + 2.17538i −1.73205 1.00000i −0.477275 4.97717i 2.21548 + 3.61824i 2.62197 9.78534i −2.00000 + 2.00000i −0.464594 8.98800i −6.97363 1.16980i
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.a 24
3.b odd 2 1 270.3.l.b 24
5.c odd 4 1 inner 90.3.k.a 24
9.c even 3 1 inner 90.3.k.a 24
9.c even 3 1 810.3.g.k 12
9.d odd 6 1 270.3.l.b 24
9.d odd 6 1 810.3.g.i 12
15.e even 4 1 270.3.l.b 24
45.k odd 12 1 inner 90.3.k.a 24
45.k odd 12 1 810.3.g.k 12
45.l even 12 1 270.3.l.b 24
45.l even 12 1 810.3.g.i 12

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{24} + 6 T_{7}^{23} + 18 T_{7}^{22} + 1196 T_{7}^{21} - 3891 T_{7}^{20} - 86784 T_{7}^{19} + \cdots + 11\!\cdots\!81$$ acting on $$S_{3}^{\mathrm{new}}(90, [\chi])$$.