Properties

 Label 168.2.ba.c Level $168$ Weight $2$ Character orbit 168.ba Analytic conductor $1.341$ Analytic rank $0$ Dimension $48$ Inner twists $8$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,2,Mod(5,168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 3, 5]))

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

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

chi := DirichletCharacter("168.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$168 = 2^{3} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 168.ba (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: $$1.34148675396$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$48 q + 6 q^{4} - 4 q^{7} - 14 q^{9}+O(q^{10})$$ 48 * q + 6 * q^4 - 4 * q^7 - 14 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$48 q + 6 q^{4} - 4 q^{7} - 14 q^{9} - 30 q^{10} - 18 q^{12} + 4 q^{15} + 6 q^{16} - 36 q^{22} - 12 q^{24} - 8 q^{25} - 10 q^{28} + 22 q^{30} + 48 q^{31} - 42 q^{33} + 52 q^{36} - 8 q^{39} - 18 q^{40} + 12 q^{42} - 8 q^{46} - 36 q^{49} - 48 q^{52} + 60 q^{54} + 4 q^{57} + 30 q^{58} - 32 q^{60} - 6 q^{63} - 60 q^{64} + 54 q^{66} + 42 q^{70} + 42 q^{72} - 36 q^{73} - 68 q^{78} - 56 q^{79} + 42 q^{81} + 48 q^{82} + 48 q^{84} - 132 q^{87} + 6 q^{88} + 48 q^{94} + 90 q^{96}+O(q^{100})$$ 48 * q + 6 * q^4 - 4 * q^7 - 14 * q^9 - 30 * q^10 - 18 * q^12 + 4 * q^15 + 6 * q^16 - 36 * q^22 - 12 * q^24 - 8 * q^25 - 10 * q^28 + 22 * q^30 + 48 * q^31 - 42 * q^33 + 52 * q^36 - 8 * q^39 - 18 * q^40 + 12 * q^42 - 8 * q^46 - 36 * q^49 - 48 * q^52 + 60 * q^54 + 4 * q^57 + 30 * q^58 - 32 * q^60 - 6 * q^63 - 60 * q^64 + 54 * q^66 + 42 * q^70 + 42 * q^72 - 36 * q^73 - 68 * q^78 - 56 * q^79 + 42 * q^81 + 48 * q^82 + 48 * q^84 - 132 * q^87 + 6 * q^88 + 48 * q^94 + 90 * q^96

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 −1.41416 0.0117470i 1.70617 0.298296i 1.99972 + 0.0332243i 0.337879 + 0.195075i −2.41631 + 0.401798i 1.39526 + 2.24795i −2.82755 0.0704754i 2.82204 1.01789i −0.475526 0.279837i
5.2 −1.37674 0.323405i −0.609093 + 1.62142i 1.79082 + 0.890488i 2.24840 + 1.29811i 1.36294 2.03529i −2.53678 + 0.751482i −2.17750 1.80513i −2.25801 1.97519i −2.67564 2.51430i
5.3 −1.33258 0.473539i −1.52791 0.815778i 1.55152 + 1.26205i 0.461663 + 0.266541i 1.64975 + 1.81061i 0.489180 2.60014i −1.46989 2.41649i 1.66901 + 2.49287i −0.488984 0.573802i
5.4 −1.30261 + 0.550645i −1.09218 1.34430i 1.39358 1.43455i −2.46958 1.42581i 2.16291 + 1.14970i −1.02032 + 2.44110i −1.02536 + 2.63603i −0.614290 + 2.93643i 4.00201 + 0.497414i
5.5 −1.26535 + 0.631567i 0.528554 1.64943i 1.20225 1.59831i 2.66818 + 1.54047i 0.372919 + 2.42094i −1.46307 2.20441i −0.511826 + 2.78173i −2.44126 1.74363i −4.34911 0.264112i
5.6 −1.07638 0.917276i 1.52791 + 0.815778i 0.317209 + 1.97468i −0.461663 0.266541i −0.896325 2.27961i 0.489180 2.60014i 1.46989 2.41649i 1.66901 + 2.49287i 0.252435 + 0.710374i
5.7 −0.998351 + 1.00165i 0.390548 + 1.68745i −0.00659077 1.99999i 1.54900 + 0.894317i −2.08013 1.29347i 2.63573 + 0.230049i 2.00986 + 1.99009i −2.69494 + 1.31806i −2.44224 + 0.658710i
5.8 −0.968446 1.03059i 0.609093 1.62142i −0.124224 + 1.99614i −2.24840 1.29811i −2.26089 + 0.942534i −2.53678 + 0.751482i 2.17750 1.80513i −2.25801 1.97519i 0.839632 + 3.57432i
5.9 −0.717256 1.21883i −1.70617 + 0.298296i −0.971089 + 1.74842i −0.337879 0.195075i 1.58733 + 1.86558i 1.39526 + 2.24795i 2.82755 0.0704754i 2.82204 1.01789i 0.00458308 + 0.551736i
5.10 −0.368276 + 1.36542i −1.26610 1.18195i −1.72875 1.00570i 1.54900 + 0.894317i 2.08013 1.29347i 2.63573 + 0.230049i 2.00986 1.99009i 0.206000 + 2.99292i −1.79158 + 1.78568i
5.11 −0.174432 1.40341i 1.09218 + 1.34430i −1.93915 + 0.489600i 2.46958 + 1.42581i 1.69610 1.76727i −1.02032 + 2.44110i 1.02536 + 2.63603i −0.614290 + 2.93643i 1.57023 3.71455i
5.12 −0.0857242 1.41161i −0.528554 + 1.64943i −1.98530 + 0.242019i −2.66818 1.54047i 2.37367 + 0.604718i −1.46307 2.20441i 0.511826 + 2.78173i −2.44126 1.74363i −1.94583 + 3.89849i
5.13 0.0857242 + 1.41161i 1.69273 + 0.366975i −1.98530 + 0.242019i 2.66818 + 1.54047i −0.372919 + 2.42094i −1.46307 2.20441i −0.511826 2.78173i 2.73066 + 1.24238i −1.94583 + 3.89849i
5.14 0.174432 + 1.40341i 0.618109 + 1.61801i −1.93915 + 0.489600i −2.46958 1.42581i −2.16291 + 1.14970i −1.02032 + 2.44110i −1.02536 2.63603i −2.23588 + 2.00021i 1.57023 3.71455i
5.15 0.368276 1.36542i −0.390548 1.68745i −1.72875 1.00570i −1.54900 0.894317i −2.44790 0.0881824i 2.63573 + 0.230049i −2.00986 + 1.99009i −2.69494 + 1.31806i −1.79158 + 1.78568i
5.16 0.717256 + 1.21883i 1.11142 1.32844i −0.971089 + 1.74842i 0.337879 + 0.195075i 2.41631 + 0.401798i 1.39526 + 2.24795i −2.82755 + 0.0704754i −0.529502 2.95290i 0.00458308 + 0.551736i
5.17 0.968446 + 1.03059i −1.70874 0.283220i −0.124224 + 1.99614i 2.24840 + 1.29811i −1.36294 2.03529i −2.53678 + 0.751482i −2.17750 + 1.80513i 2.83957 + 0.967897i 0.839632 + 3.57432i
5.18 0.998351 1.00165i 1.26610 + 1.18195i −0.00659077 1.99999i −1.54900 0.894317i 2.44790 0.0881824i 2.63573 + 0.230049i −2.00986 1.99009i 0.206000 + 2.99292i −2.44224 + 0.658710i
5.19 1.07638 + 0.917276i −0.0574700 + 1.73110i 0.317209 + 1.97468i 0.461663 + 0.266541i −1.64975 + 1.81061i 0.489180 2.60014i −1.46989 + 2.41649i −2.99339 0.198972i 0.252435 + 0.710374i
5.20 1.26535 0.631567i −1.69273 0.366975i 1.20225 1.59831i −2.66818 1.54047i −2.37367 + 0.604718i −1.46307 2.20441i 0.511826 2.78173i 2.73066 + 1.24238i −4.34911 0.264112i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.24 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
7.d odd 6 1 inner
8.b even 2 1 inner
21.g even 6 1 inner
24.h odd 2 1 inner
56.j odd 6 1 inner
168.ba even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.2.ba.c 48
3.b odd 2 1 inner 168.2.ba.c 48
4.b odd 2 1 672.2.bi.c 48
7.d odd 6 1 inner 168.2.ba.c 48
8.b even 2 1 inner 168.2.ba.c 48
8.d odd 2 1 672.2.bi.c 48
12.b even 2 1 672.2.bi.c 48
21.g even 6 1 inner 168.2.ba.c 48
24.f even 2 1 672.2.bi.c 48
24.h odd 2 1 inner 168.2.ba.c 48
28.f even 6 1 672.2.bi.c 48
56.j odd 6 1 inner 168.2.ba.c 48
56.m even 6 1 672.2.bi.c 48
84.j odd 6 1 672.2.bi.c 48
168.ba even 6 1 inner 168.2.ba.c 48
168.be odd 6 1 672.2.bi.c 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.ba.c 48 1.a even 1 1 trivial
168.2.ba.c 48 3.b odd 2 1 inner
168.2.ba.c 48 7.d odd 6 1 inner
168.2.ba.c 48 8.b even 2 1 inner
168.2.ba.c 48 21.g even 6 1 inner
168.2.ba.c 48 24.h odd 2 1 inner
168.2.ba.c 48 56.j odd 6 1 inner
168.2.ba.c 48 168.ba even 6 1 inner
672.2.bi.c 48 4.b odd 2 1
672.2.bi.c 48 8.d odd 2 1
672.2.bi.c 48 12.b even 2 1
672.2.bi.c 48 24.f even 2 1
672.2.bi.c 48 28.f even 6 1
672.2.bi.c 48 56.m even 6 1
672.2.bi.c 48 84.j odd 6 1
672.2.bi.c 48 168.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{24} - 28 T_{5}^{22} + 498 T_{5}^{20} - 5472 T_{5}^{18} + 44115 T_{5}^{16} - 241512 T_{5}^{14} + \cdots + 5184$$ acting on $$S_{2}^{\mathrm{new}}(168, [\chi])$$.