# Properties

 Label 171.2.g.c Level $171$ Weight $2$ Character orbit 171.g Analytic conductor $1.365$ Analytic rank $0$ Dimension $32$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,2,Mod(106,171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(171, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 4]))

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

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

chi := DirichletCharacter("171.106");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$171 = 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 171.g (of order $$3$$, 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: $$1.36544187456$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(\zeta_{3})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10})$$ 32 * q + q^2 - 2 * q^3 - 17 * q^4 - 6 * q^5 + 2 * q^6 + q^7 - 36 * q^8 - 10 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9} - 8 q^{10} + 7 q^{11} - 3 q^{12} - 4 q^{13} - 2 q^{14} + q^{15} - 11 q^{16} - 7 q^{17} + 6 q^{18} + 7 q^{19} - 3 q^{20} + 11 q^{21} + 16 q^{22} + 5 q^{23} + 27 q^{24} + 18 q^{25} - 4 q^{26} - 5 q^{27} - 10 q^{28} - 20 q^{29} - 5 q^{30} - 10 q^{31} + 17 q^{32} + 34 q^{33} + 26 q^{34} - 3 q^{35} - 16 q^{36} + 2 q^{37} + 38 q^{38} - 24 q^{40} - 12 q^{41} + 25 q^{42} + 7 q^{43} + 20 q^{44} - 35 q^{45} + 18 q^{47} - 33 q^{48} - 13 q^{49} + q^{50} - 28 q^{51} + 19 q^{52} + 16 q^{53} + 35 q^{54} + 15 q^{55} - 6 q^{56} + 6 q^{57} - 74 q^{59} + 50 q^{60} + 24 q^{61} + 54 q^{62} - 30 q^{63} - 64 q^{64} + 54 q^{65} + 4 q^{66} - 11 q^{67} - 2 q^{68} + 3 q^{69} - 48 q^{70} + 9 q^{71} - 10 q^{73} + 6 q^{74} - 76 q^{75} + 29 q^{76} + 46 q^{77} - 82 q^{78} - 8 q^{79} - 24 q^{80} + 26 q^{81} + 7 q^{82} + 3 q^{83} + 12 q^{84} - 27 q^{85} + 17 q^{86} - 9 q^{87} + 9 q^{88} + 30 q^{89} - 74 q^{90} - q^{91} - 17 q^{92} - 24 q^{93} - 18 q^{94} - 6 q^{95} - 5 q^{96} + 18 q^{98} - 10 q^{99}+O(q^{100})$$ 32 * q + q^2 - 2 * q^3 - 17 * q^4 - 6 * q^5 + 2 * q^6 + q^7 - 36 * q^8 - 10 * q^9 - 8 * q^10 + 7 * q^11 - 3 * q^12 - 4 * q^13 - 2 * q^14 + q^15 - 11 * q^16 - 7 * q^17 + 6 * q^18 + 7 * q^19 - 3 * q^20 + 11 * q^21 + 16 * q^22 + 5 * q^23 + 27 * q^24 + 18 * q^25 - 4 * q^26 - 5 * q^27 - 10 * q^28 - 20 * q^29 - 5 * q^30 - 10 * q^31 + 17 * q^32 + 34 * q^33 + 26 * q^34 - 3 * q^35 - 16 * q^36 + 2 * q^37 + 38 * q^38 - 24 * q^40 - 12 * q^41 + 25 * q^42 + 7 * q^43 + 20 * q^44 - 35 * q^45 + 18 * q^47 - 33 * q^48 - 13 * q^49 + q^50 - 28 * q^51 + 19 * q^52 + 16 * q^53 + 35 * q^54 + 15 * q^55 - 6 * q^56 + 6 * q^57 - 74 * q^59 + 50 * q^60 + 24 * q^61 + 54 * q^62 - 30 * q^63 - 64 * q^64 + 54 * q^65 + 4 * q^66 - 11 * q^67 - 2 * q^68 + 3 * q^69 - 48 * q^70 + 9 * q^71 - 10 * q^73 + 6 * q^74 - 76 * q^75 + 29 * q^76 + 46 * q^77 - 82 * q^78 - 8 * q^79 - 24 * q^80 + 26 * q^81 + 7 * q^82 + 3 * q^83 + 12 * q^84 - 27 * q^85 + 17 * q^86 - 9 * q^87 + 9 * q^88 + 30 * q^89 - 74 * q^90 - q^91 - 17 * q^92 - 24 * q^93 - 18 * q^94 - 6 * q^95 - 5 * q^96 + 18 * q^98 - 10 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
106.1 −1.23404 2.13742i 1.62292 + 0.605076i −2.04570 + 3.54325i 1.91524 −0.709450 4.21555i −0.324708 + 0.562412i 5.16173 2.26777 + 1.96399i −2.36348 4.09366i
106.2 −1.04884 1.81665i 0.154304 + 1.72516i −1.20014 + 2.07870i −2.89593 2.97217 2.08974i 0.116480 0.201749i 0.839660 −2.95238 + 0.532400i 3.03737 + 5.26088i
106.3 −0.978515 1.69484i −1.73022 0.0795352i −0.914982 + 1.58480i −0.196235 1.55825 + 3.01027i −2.23368 + 3.86885i −0.332766 2.98735 + 0.275227i 0.192018 + 0.332586i
106.4 −0.888985 1.53977i 1.15573 1.29008i −0.580589 + 1.00561i −1.27957 −3.01384 0.632688i 0.657761 1.13928i −1.49140 −0.328599 2.98195i 1.13752 + 1.97024i
106.5 −0.803309 1.39137i −1.24014 1.20915i −0.290611 + 0.503353i 3.75880 −0.686165 + 2.69682i 2.27973 3.94861i −2.27943 0.0758986 + 2.99904i −3.01948 5.22989i
106.6 −0.395929 0.685769i 0.659141 + 1.60173i 0.686481 1.18902i 2.59093 0.837442 1.08619i −0.373088 + 0.646207i −2.67091 −2.13107 + 2.11153i −1.02582 1.77678i
106.7 −0.269545 0.466866i −1.40907 + 1.00723i 0.854691 1.48037i −0.947325 0.850050 + 0.386354i 1.18430 2.05126i −1.99969 0.970970 2.83852i 0.255347 + 0.442274i
106.8 −0.185445 0.321199i −0.894876 1.48297i 0.931221 1.61292i −3.55521 −0.310379 + 0.562442i −0.124876 + 0.216291i −1.43254 −1.39839 + 2.65415i 0.659294 + 1.14193i
106.9 −0.0732670 0.126902i 0.157437 1.72488i 0.989264 1.71346i 2.57004 −0.230426 + 0.106398i −1.73898 + 3.01201i −0.582990 −2.95043 0.543121i −0.188299 0.326143i
106.10 0.0973467 + 0.168609i 1.55267 + 0.767609i 0.981047 1.69922i −1.90563 0.0217210 + 0.336519i 1.69446 2.93489i 0.771394 1.82155 + 2.38368i −0.185507 0.321308i
106.11 0.616796 + 1.06832i 0.506944 + 1.65620i 0.239126 0.414178i −0.551543 −1.45668 + 1.56312i −1.62156 + 2.80862i 3.05715 −2.48602 + 1.67920i −0.340189 0.589225i
106.12 0.847114 + 1.46725i 0.0521113 1.73127i −0.435206 + 0.753799i 0.0882176 2.58434 1.39012i 1.84695 3.19901i 1.91378 −2.99457 0.180437i 0.0747304 + 0.129437i
106.13 1.01016 + 1.74964i −1.70377 + 0.311691i −1.04083 + 1.80277i −4.18739 −2.26643 2.66614i −0.976107 + 1.69067i −0.164982 2.80570 1.06210i −4.22991 7.32643i
106.14 1.19971 + 2.07797i −0.340918 + 1.69817i −1.87863 + 3.25388i 0.719678 −3.93774 + 1.32890i 1.65862 2.87282i −4.21641 −2.76755 1.15787i 0.863408 + 1.49547i
106.15 1.30160 + 2.25443i −1.17382 1.27363i −2.38830 + 4.13666i 2.87794 1.34348 4.30405i −1.80240 + 3.12185i −7.22804 −0.244287 + 2.99004i 3.74592 + 6.48812i
106.16 1.30515 + 2.26059i 1.63157 0.581354i −2.40684 + 4.16877i −2.00201 3.44365 + 2.92956i 0.257107 0.445323i −7.34455 2.32405 1.89704i −2.61292 4.52572i
121.1 −1.23404 + 2.13742i 1.62292 0.605076i −2.04570 3.54325i 1.91524 −0.709450 + 4.21555i −0.324708 0.562412i 5.16173 2.26777 1.96399i −2.36348 + 4.09366i
121.2 −1.04884 + 1.81665i 0.154304 1.72516i −1.20014 2.07870i −2.89593 2.97217 + 2.08974i 0.116480 + 0.201749i 0.839660 −2.95238 0.532400i 3.03737 5.26088i
121.3 −0.978515 + 1.69484i −1.73022 + 0.0795352i −0.914982 1.58480i −0.196235 1.55825 3.01027i −2.23368 3.86885i −0.332766 2.98735 0.275227i 0.192018 0.332586i
121.4 −0.888985 + 1.53977i 1.15573 + 1.29008i −0.580589 1.00561i −1.27957 −3.01384 + 0.632688i 0.657761 + 1.13928i −1.49140 −0.328599 + 2.98195i 1.13752 1.97024i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 106.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.2.g.c 32
3.b odd 2 1 513.2.g.c 32
9.c even 3 1 171.2.h.c yes 32
9.d odd 6 1 513.2.h.c 32
19.c even 3 1 171.2.h.c yes 32
57.h odd 6 1 513.2.h.c 32
171.g even 3 1 inner 171.2.g.c 32
171.n odd 6 1 513.2.g.c 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.g.c 32 1.a even 1 1 trivial
171.2.g.c 32 171.g even 3 1 inner
171.2.h.c yes 32 9.c even 3 1
171.2.h.c yes 32 19.c even 3 1
513.2.g.c 32 3.b odd 2 1
513.2.g.c 32 171.n odd 6 1
513.2.h.c 32 9.d odd 6 1
513.2.h.c 32 57.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(171, [\chi])$$:

 $$T_{2}^{32} - T_{2}^{31} + 25 T_{2}^{30} - 10 T_{2}^{29} + 358 T_{2}^{28} - 34 T_{2}^{27} + 3447 T_{2}^{26} + \cdots + 81$$ T2^32 - T2^31 + 25*T2^30 - 10*T2^29 + 358*T2^28 - 34*T2^27 + 3447*T2^26 + 717*T2^25 + 24681*T2^24 + 10382*T2^23 + 135298*T2^22 + 82109*T2^21 + 580960*T2^20 + 420887*T2^19 + 1945837*T2^18 + 1581978*T2^17 + 5070162*T2^16 + 4256568*T2^15 + 9953691*T2^14 + 8450307*T2^13 + 14418975*T2^12 + 11385480*T2^11 + 14005971*T2^10 + 10297053*T2^9 + 8939259*T2^8 + 4637646*T2^7 + 2245509*T2^6 + 590571*T2^5 + 154773*T2^4 + 18333*T2^3 + 5562*T2^2 + 567*T2 + 81 $$T_{5}^{16} + 3 T_{5}^{15} - 40 T_{5}^{14} - 115 T_{5}^{13} + 600 T_{5}^{12} + 1690 T_{5}^{11} + \cdots - 189$$ T5^16 + 3*T5^15 - 40*T5^14 - 115*T5^13 + 600*T5^12 + 1690*T5^11 - 4152*T5^10 - 12111*T5^9 + 12405*T5^8 + 43371*T5^7 - 6663*T5^6 - 67191*T5^5 - 25815*T5^4 + 23535*T5^3 + 13887*T5^2 + 756*T5 - 189