# Properties

 Label 169.4.e.h Level $169$ Weight $4$ Character orbit 169.e Analytic conductor $9.971$ Analytic rank $0$ Dimension $36$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,Mod(23,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("169.23");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.e (of order $$6$$, degree $$2$$, not 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: $$9.97132279097$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$18$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36 q - 2 q^{3} + 74 q^{4} - 132 q^{9}+O(q^{10})$$ 36 * q - 2 * q^3 + 74 * q^4 - 132 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$36 q - 2 q^{3} + 74 q^{4} - 132 q^{9} - 294 q^{10} - 156 q^{12} - 588 q^{14} - 538 q^{16} - 110 q^{17} - 680 q^{22} - 408 q^{23} - 1228 q^{25} - 2672 q^{27} - 560 q^{29} + 1042 q^{30} - 40 q^{35} - 1818 q^{36} + 2956 q^{38} + 52 q^{40} + 8 q^{42} - 1066 q^{43} + 264 q^{48} + 806 q^{49} - 1880 q^{51} - 1112 q^{53} + 500 q^{55} + 500 q^{56} + 272 q^{61} + 4070 q^{62} - 1136 q^{64} + 13116 q^{66} + 3072 q^{68} - 4100 q^{69} + 3980 q^{74} + 4786 q^{75} + 2872 q^{77} + 1648 q^{79} + 1670 q^{81} + 5514 q^{82} + 1572 q^{87} - 1272 q^{88} + 5120 q^{90} + 16040 q^{92} + 5062 q^{94} - 3228 q^{95}+O(q^{100})$$ 36 * q - 2 * q^3 + 74 * q^4 - 132 * q^9 - 294 * q^10 - 156 * q^12 - 588 * q^14 - 538 * q^16 - 110 * q^17 - 680 * q^22 - 408 * q^23 - 1228 * q^25 - 2672 * q^27 - 560 * q^29 + 1042 * q^30 - 40 * q^35 - 1818 * q^36 + 2956 * q^38 + 52 * q^40 + 8 * q^42 - 1066 * q^43 + 264 * q^48 + 806 * q^49 - 1880 * q^51 - 1112 * q^53 + 500 * q^55 + 500 * q^56 + 272 * q^61 + 4070 * q^62 - 1136 * q^64 + 13116 * q^66 + 3072 * q^68 - 4100 * q^69 + 3980 * q^74 + 4786 * q^75 + 2872 * q^77 + 1648 * q^79 + 1670 * q^81 + 5514 * q^82 + 1572 * q^87 - 1272 * q^88 + 5120 * q^90 + 16040 * q^92 + 5062 * q^94 - 3228 * q^95

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1 −4.70109 + 2.71418i −0.837548 1.45068i 10.7335 18.5910i 7.70909i 7.87478 + 4.54651i 13.0120 + 7.51249i 73.1038i 12.0970 20.9527i 20.9238 + 36.2412i
23.2 −4.19172 + 2.42009i 3.09831 + 5.36643i 7.71365 13.3604i 15.2399i −25.9745 14.9964i 3.73794 + 2.15810i 35.9495i −5.69903 + 9.87102i −36.8818 63.8811i
23.3 −4.17905 + 2.41278i −2.22176 3.84820i 7.64299 13.2380i 12.7712i 18.5697 + 10.7212i −22.6787 13.0936i 35.1589i 3.62756 6.28312i −30.8140 53.3715i
23.4 −3.32067 + 1.91719i 0.139581 + 0.241762i 3.35124 5.80452i 11.3710i −0.927008 0.535209i 26.9007 + 15.5311i 4.97517i 13.4610 23.3152i −21.8004 37.7594i
23.5 −2.73781 + 1.58068i −3.54442 6.13911i 0.997073 1.72698i 13.6039i 19.4079 + 11.2051i 12.4114 + 7.16574i 18.9866i −11.6258 + 20.1364i 21.5034 + 37.2450i
23.6 −1.92949 + 1.11399i 4.87434 + 8.44260i −1.51803 + 2.62931i 8.20685i −18.8100 10.8600i −7.23560 4.17747i 24.5882i −34.0183 + 58.9214i −9.14239 15.8351i
23.7 −1.49617 + 0.863817i −3.44796 5.97204i −2.50764 + 4.34336i 20.8281i 10.3175 + 5.95681i 6.55206 + 3.78283i 22.4856i −10.2768 + 17.8000i −17.9916 31.1624i
23.8 −0.337850 + 0.195058i −1.80483 3.12606i −3.92391 + 6.79640i 7.52136i 1.21953 + 0.704093i 16.9261 + 9.77228i 6.18247i 6.98515 12.0986i 1.46710 + 2.54109i
23.9 −0.129088 + 0.0745292i 3.24429 + 5.61927i −3.98889 + 6.90896i 10.2526i −0.837600 0.483588i 25.6987 + 14.8372i 2.38162i −7.55083 + 13.0784i −0.764114 1.32348i
23.10 0.129088 0.0745292i 3.24429 + 5.61927i −3.98889 + 6.90896i 10.2526i 0.837600 + 0.483588i −25.6987 14.8372i 2.38162i −7.55083 + 13.0784i −0.764114 1.32348i
23.11 0.337850 0.195058i −1.80483 3.12606i −3.92391 + 6.79640i 7.52136i −1.21953 0.704093i −16.9261 9.77228i 6.18247i 6.98515 12.0986i 1.46710 + 2.54109i
23.12 1.49617 0.863817i −3.44796 5.97204i −2.50764 + 4.34336i 20.8281i −10.3175 5.95681i −6.55206 3.78283i 22.4856i −10.2768 + 17.8000i −17.9916 31.1624i
23.13 1.92949 1.11399i 4.87434 + 8.44260i −1.51803 + 2.62931i 8.20685i 18.8100 + 10.8600i 7.23560 + 4.17747i 24.5882i −34.0183 + 58.9214i −9.14239 15.8351i
23.14 2.73781 1.58068i −3.54442 6.13911i 0.997073 1.72698i 13.6039i −19.4079 11.2051i −12.4114 7.16574i 18.9866i −11.6258 + 20.1364i 21.5034 + 37.2450i
23.15 3.32067 1.91719i 0.139581 + 0.241762i 3.35124 5.80452i 11.3710i 0.927008 + 0.535209i −26.9007 15.5311i 4.97517i 13.4610 23.3152i −21.8004 37.7594i
23.16 4.17905 2.41278i −2.22176 3.84820i 7.64299 13.2380i 12.7712i −18.5697 10.7212i 22.6787 + 13.0936i 35.1589i 3.62756 6.28312i −30.8140 53.3715i
23.17 4.19172 2.42009i 3.09831 + 5.36643i 7.71365 13.3604i 15.2399i 25.9745 + 14.9964i −3.73794 2.15810i 35.9495i −5.69903 + 9.87102i −36.8818 63.8811i
23.18 4.70109 2.71418i −0.837548 1.45068i 10.7335 18.5910i 7.70909i −7.87478 4.54651i −13.0120 7.51249i 73.1038i 12.0970 20.9527i 20.9238 + 36.2412i
147.1 −4.70109 2.71418i −0.837548 + 1.45068i 10.7335 + 18.5910i 7.70909i 7.87478 4.54651i 13.0120 7.51249i 73.1038i 12.0970 + 20.9527i 20.9238 36.2412i
147.2 −4.19172 2.42009i 3.09831 5.36643i 7.71365 + 13.3604i 15.2399i −25.9745 + 14.9964i 3.73794 2.15810i 35.9495i −5.69903 9.87102i −36.8818 + 63.8811i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 23.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.e.h 36
13.b even 2 1 inner 169.4.e.h 36
13.c even 3 1 169.4.b.g 18
13.c even 3 1 inner 169.4.e.h 36
13.d odd 4 1 169.4.c.k 18
13.d odd 4 1 169.4.c.l 18
13.e even 6 1 169.4.b.g 18
13.e even 6 1 inner 169.4.e.h 36
13.f odd 12 1 169.4.a.k 9
13.f odd 12 1 169.4.a.l yes 9
13.f odd 12 1 169.4.c.k 18
13.f odd 12 1 169.4.c.l 18
39.k even 12 1 1521.4.a.bg 9
39.k even 12 1 1521.4.a.bh 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.4.a.k 9 13.f odd 12 1
169.4.a.l yes 9 13.f odd 12 1
169.4.b.g 18 13.c even 3 1
169.4.b.g 18 13.e even 6 1
169.4.c.k 18 13.d odd 4 1
169.4.c.k 18 13.f odd 12 1
169.4.c.l 18 13.d odd 4 1
169.4.c.l 18 13.f odd 12 1
169.4.e.h 36 1.a even 1 1 trivial
169.4.e.h 36 13.b even 2 1 inner
169.4.e.h 36 13.c even 3 1 inner
169.4.e.h 36 13.e even 6 1 inner
1521.4.a.bg 9 39.k even 12 1
1521.4.a.bh 9 39.k even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{36} - 109 T_{2}^{34} + 7095 T_{2}^{32} - 304748 T_{2}^{30} + 9732537 T_{2}^{28} + \cdots + 14003408896$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.