# Properties

 Label 102.2.i.a Level $102$ Weight $2$ Character orbit 102.i Analytic conductor $0.814$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [102,2,Mod(5,102)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(102, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([8, 5]))

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

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

chi := DirichletCharacter("102.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$102 = 2 \cdot 3 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 102.i (of order $$16$$, degree $$8$$, 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: $$0.814474100617$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$3$$ over $$\Q(\zeta_{16})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q+O(q^{10})$$ 24 * q $$\operatorname{Tr}(f)(q) =$$ $$24 q - 16 q^{11} - 8 q^{12} + 16 q^{17} - 16 q^{18} - 48 q^{21} - 16 q^{25} + 16 q^{29} - 16 q^{31} - 16 q^{33} - 16 q^{37} + 24 q^{38} + 16 q^{39} - 32 q^{41} - 8 q^{43} + 80 q^{45} + 16 q^{46} + 16 q^{47} + 32 q^{49} - 8 q^{50} + 16 q^{52} - 64 q^{53} + 48 q^{54} + 32 q^{55} + 64 q^{57} + 16 q^{58} + 64 q^{59} + 16 q^{60} + 16 q^{61} - 16 q^{62} - 48 q^{63} + 8 q^{66} - 16 q^{69} - 16 q^{70} - 16 q^{72} - 64 q^{73} - 16 q^{74} + 32 q^{75} + 16 q^{77} + 16 q^{78} - 32 q^{79} + 16 q^{80} - 40 q^{81} - 64 q^{82} - 72 q^{83} + 16 q^{84} - 32 q^{85} + 80 q^{87} - 16 q^{88} + 48 q^{89} - 64 q^{91} - 16 q^{92} - 32 q^{93} - 32 q^{94} - 32 q^{95} - 16 q^{97} + 16 q^{98} + 24 q^{99}+O(q^{100})$$ 24 * q - 16 * q^11 - 8 * q^12 + 16 * q^17 - 16 * q^18 - 48 * q^21 - 16 * q^25 + 16 * q^29 - 16 * q^31 - 16 * q^33 - 16 * q^37 + 24 * q^38 + 16 * q^39 - 32 * q^41 - 8 * q^43 + 80 * q^45 + 16 * q^46 + 16 * q^47 + 32 * q^49 - 8 * q^50 + 16 * q^52 - 64 * q^53 + 48 * q^54 + 32 * q^55 + 64 * q^57 + 16 * q^58 + 64 * q^59 + 16 * q^60 + 16 * q^61 - 16 * q^62 - 48 * q^63 + 8 * q^66 - 16 * q^69 - 16 * q^70 - 16 * q^72 - 64 * q^73 - 16 * q^74 + 32 * q^75 + 16 * q^77 + 16 * q^78 - 32 * q^79 + 16 * q^80 - 40 * q^81 - 64 * q^82 - 72 * q^83 + 16 * q^84 - 32 * q^85 + 80 * q^87 - 16 * q^88 + 48 * q^89 - 64 * q^91 - 16 * q^92 - 32 * q^93 - 32 * q^94 - 32 * q^95 - 16 * q^97 + 16 * q^98 + 24 * q^99

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −0.923880 + 0.382683i −1.72898 0.103064i 0.707107 0.707107i 2.09882 + 0.417480i 1.63681 0.566434i 0.745978 + 3.75028i −0.382683 + 0.923880i 2.97876 + 0.356391i −2.09882 + 0.417480i
5.2 −0.923880 + 0.382683i −0.317070 1.70278i 0.707107 0.707107i −2.04899 0.407570i 0.944561 + 1.45183i −0.635770 3.19623i −0.382683 + 0.923880i −2.79893 + 1.07980i 2.04899 0.407570i
5.3 −0.923880 + 0.382683i 1.33894 + 1.09874i 0.707107 0.707107i 0.715543 + 0.142330i −1.65749 0.502711i −0.110207 0.554049i −0.382683 + 0.923880i 0.585545 + 2.94230i −0.715543 + 0.142330i
11.1 0.382683 0.923880i −1.48785 + 0.886741i −0.707107 0.707107i 2.21277 3.31164i 0.249867 + 1.71393i 1.38649 0.926424i −0.923880 + 0.382683i 1.42738 2.63867i −2.21277 3.31164i
11.2 0.382683 0.923880i 0.464518 1.66860i −0.707107 0.707107i −0.995238 + 1.48948i −1.36382 1.06770i 2.77729 1.85573i −0.923880 + 0.382683i −2.56845 1.55019i 0.995238 + 1.48948i
11.3 0.382683 0.923880i 1.73044 + 0.0747517i −0.707107 0.707107i 0.630232 0.943208i 0.731271 1.57011i −4.16378 + 2.78215i −0.923880 + 0.382683i 2.98882 + 0.258706i −0.630232 0.943208i
23.1 −0.382683 + 0.923880i −1.20551 1.24368i −0.707107 0.707107i −3.29353 2.20066i 1.61034 0.637806i −0.750832 1.12370i 0.923880 0.382683i −0.0935025 + 2.99854i 3.29353 2.20066i
23.2 −0.382683 + 0.923880i 0.573016 + 1.63452i −0.707107 0.707107i 0.540978 + 0.361470i −1.72938 0.0961059i 0.676748 + 1.01283i 0.923880 0.382683i −2.34331 + 1.87321i −0.540978 + 0.361470i
23.3 −0.382683 + 0.923880i 1.33960 1.09794i −0.707107 0.707107i 0.904790 + 0.604561i 0.501723 + 1.65779i 0.0740832 + 0.110873i 0.923880 0.382683i 0.589049 2.94160i −0.904790 + 0.604561i
29.1 0.923880 0.382683i −1.27146 1.17618i 0.707107 0.707107i 0.284770 1.43164i −1.62478 0.600082i 0.146842 0.0292088i 0.382683 0.923880i 0.233206 + 2.99092i −0.284770 1.43164i
29.2 0.923880 0.382683i −0.799829 + 1.53632i 0.707107 0.707107i −0.456232 + 2.29363i −0.151023 + 1.72545i 3.60046 0.716176i 0.382683 0.923880i −1.72055 2.45758i 0.456232 + 2.29363i
29.3 0.923880 0.382683i 1.36418 1.06725i 0.707107 0.707107i −0.593905 + 2.98576i 0.851920 1.50806i −3.74730 + 0.745385i 0.382683 0.923880i 0.721973 2.91183i 0.593905 + 2.98576i
41.1 −0.923880 0.382683i −1.72898 + 0.103064i 0.707107 + 0.707107i 2.09882 0.417480i 1.63681 + 0.566434i 0.745978 3.75028i −0.382683 0.923880i 2.97876 0.356391i −2.09882 0.417480i
41.2 −0.923880 0.382683i −0.317070 + 1.70278i 0.707107 + 0.707107i −2.04899 + 0.407570i 0.944561 1.45183i −0.635770 + 3.19623i −0.382683 0.923880i −2.79893 1.07980i 2.04899 + 0.407570i
41.3 −0.923880 0.382683i 1.33894 1.09874i 0.707107 + 0.707107i 0.715543 0.142330i −1.65749 + 0.502711i −0.110207 + 0.554049i −0.382683 0.923880i 0.585545 2.94230i −0.715543 0.142330i
65.1 0.382683 + 0.923880i −1.48785 0.886741i −0.707107 + 0.707107i 2.21277 + 3.31164i 0.249867 1.71393i 1.38649 + 0.926424i −0.923880 0.382683i 1.42738 + 2.63867i −2.21277 + 3.31164i
65.2 0.382683 + 0.923880i 0.464518 + 1.66860i −0.707107 + 0.707107i −0.995238 1.48948i −1.36382 + 1.06770i 2.77729 + 1.85573i −0.923880 0.382683i −2.56845 + 1.55019i 0.995238 1.48948i
65.3 0.382683 + 0.923880i 1.73044 0.0747517i −0.707107 + 0.707107i 0.630232 + 0.943208i 0.731271 + 1.57011i −4.16378 2.78215i −0.923880 0.382683i 2.98882 0.258706i −0.630232 + 0.943208i
71.1 −0.382683 0.923880i −1.20551 + 1.24368i −0.707107 + 0.707107i −3.29353 + 2.20066i 1.61034 + 0.637806i −0.750832 + 1.12370i 0.923880 + 0.382683i −0.0935025 2.99854i 3.29353 + 2.20066i
71.2 −0.382683 0.923880i 0.573016 1.63452i −0.707107 + 0.707107i 0.540978 0.361470i −1.72938 + 0.0961059i 0.676748 1.01283i 0.923880 + 0.382683i −2.34331 1.87321i −0.540978 0.361470i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
51.i even 16 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 102.2.i.a 24
3.b odd 2 1 102.2.i.b yes 24
4.b odd 2 1 816.2.cj.b 24
12.b even 2 1 816.2.cj.a 24
17.e odd 16 1 102.2.i.b yes 24
51.i even 16 1 inner 102.2.i.a 24
68.i even 16 1 816.2.cj.a 24
204.t odd 16 1 816.2.cj.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.i.a 24 1.a even 1 1 trivial
102.2.i.a 24 51.i even 16 1 inner
102.2.i.b yes 24 3.b odd 2 1
102.2.i.b yes 24 17.e odd 16 1
816.2.cj.a 24 12.b even 2 1
816.2.cj.a 24 68.i even 16 1
816.2.cj.b 24 4.b odd 2 1
816.2.cj.b 24 204.t odd 16 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{24} + 8 T_{5}^{22} + 16 T_{5}^{21} + 156 T_{5}^{20} - 224 T_{5}^{19} + 1256 T_{5}^{18} + \cdots + 591872$$ acting on $$S_{2}^{\mathrm{new}}(102, [\chi])$$.