# Properties

 Label 169.8.b.f Level $169$ Weight $8$ Character orbit 169.b Analytic conductor $52.793$ Analytic rank $0$ Dimension $42$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,8,Mod(168,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("169.168");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 169.b (of order $$2$$, degree $$1$$, 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: $$52.7930693068$$ Analytic rank: $$0$$ Dimension: $$42$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$42 q - 52 q^{3} - 2818 q^{4} + 30930 q^{9}+O(q^{10})$$ 42 * q - 52 * q^3 - 2818 * q^4 + 30930 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$42 q - 52 q^{3} - 2818 q^{4} + 30930 q^{9} + 10334 q^{10} - 43590 q^{12} - 358 q^{14} + 226410 q^{16} - 90032 q^{17} - 548348 q^{22} + 256210 q^{23} - 733074 q^{25} - 213286 q^{27} - 124658 q^{29} - 19522 q^{30} + 1697720 q^{35} + 548490 q^{36} - 451662 q^{38} - 4577654 q^{40} + 5473672 q^{42} - 1788978 q^{43} + 12604776 q^{48} - 263060 q^{49} + 2368682 q^{51} - 9100054 q^{53} + 11048060 q^{55} - 4668772 q^{56} + 11434024 q^{61} + 27187306 q^{62} - 19535848 q^{64} - 35613954 q^{66} + 24504984 q^{68} - 5544412 q^{69} + 19250156 q^{74} + 86424794 q^{75} - 18892072 q^{77} - 37226562 q^{79} + 82110898 q^{81} - 48380654 q^{82} + 18055788 q^{87} + 151356360 q^{88} - 6671120 q^{90} - 138678780 q^{92} + 57911426 q^{94} - 35320062 q^{95}+O(q^{100})$$ 42 * q - 52 * q^3 - 2818 * q^4 + 30930 * q^9 + 10334 * q^10 - 43590 * q^12 - 358 * q^14 + 226410 * q^16 - 90032 * q^17 - 548348 * q^22 + 256210 * q^23 - 733074 * q^25 - 213286 * q^27 - 124658 * q^29 - 19522 * q^30 + 1697720 * q^35 + 548490 * q^36 - 451662 * q^38 - 4577654 * q^40 + 5473672 * q^42 - 1788978 * q^43 + 12604776 * q^48 - 263060 * q^49 + 2368682 * q^51 - 9100054 * q^53 + 11048060 * q^55 - 4668772 * q^56 + 11434024 * q^61 + 27187306 * q^62 - 19535848 * q^64 - 35613954 * q^66 + 24504984 * q^68 - 5544412 * q^69 + 19250156 * q^74 + 86424794 * q^75 - 18892072 * q^77 - 37226562 * q^79 + 82110898 * q^81 - 48380654 * q^82 + 18055788 * q^87 + 151356360 * q^88 - 6671120 * q^90 - 138678780 * q^92 + 57911426 * q^94 - 35320062 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
168.1 21.6353i 79.0768 −340.087 120.676i 1710.85i 240.114i 4588.56i 4066.15 2610.86
168.2 21.5931i 28.3750 −338.261 473.706i 612.705i 99.1966i 4540.19i −1381.86 −10228.8
168.3 20.8456i 28.0386 −306.538 273.942i 584.480i 1317.98i 3721.73i −1400.84 5710.48
168.4 20.4423i −14.6497 −289.887 478.639i 299.473i 1197.76i 3309.33i −1972.39 9784.47
168.5 19.6237i −72.4635 −257.091 413.760i 1422.00i 1005.88i 2533.24i 3063.96 8119.51
168.6 17.7683i 58.1116 −187.714 15.5333i 1032.55i 1688.45i 1061.02i 1189.96 −276.002
168.7 16.7260i −13.7721 −151.758 0.583029i 230.352i 274.333i 397.369i −1997.33 9.75172
168.8 15.3948i −46.3939 −108.998 391.457i 714.223i 1742.39i 292.525i −34.6060 −6026.39
168.9 15.0231i −85.3884 −97.6921 439.379i 1282.79i 536.518i 455.317i 5104.17 −6600.82
168.10 13.2906i 38.7978 −48.6390 274.097i 515.644i 774.181i 1054.75i −681.732 −3642.90
168.11 13.1620i −21.4559 −45.2387 344.184i 282.403i 168.957i 1089.31i −1726.64 4530.16
168.12 11.8381i −26.1495 −12.1398 422.131i 309.560i 1215.99i 1371.56i −1503.20 4997.21
168.13 10.5911i −6.51000 15.8290 397.099i 68.9480i 1187.35i 1523.30i −2144.62 −4205.71
168.14 8.99747i 87.5208 47.0455 190.033i 787.466i 44.6495i 1574.97i 5472.89 1709.81
168.15 8.58294i 18.1259 54.3331 37.6136i 155.574i 711.860i 1564.95i −1858.45 −322.835
168.16 6.93821i −64.5145 79.8613 33.5113i 447.615i 160.658i 1442.18i 1975.12 −232.508
168.17 5.77104i 82.1608 94.6950 17.6329i 474.154i 888.558i 1285.18i 4563.40 101.761
168.18 3.19657i −57.0706 117.782 431.499i 182.430i 225.139i 785.660i 1070.05 −1379.32
168.19 2.91821i 44.2582 119.484 197.474i 129.155i 303.075i 722.210i −228.212 −576.269
168.20 2.90053i −90.5117 119.587 240.795i 262.532i 1391.07i 718.132i 6005.38 698.433
See all 42 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 168.42 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.8.b.f 42
13.b even 2 1 inner 169.8.b.f 42
13.d odd 4 1 169.8.a.h 21
13.d odd 4 1 169.8.a.i yes 21

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.8.a.h 21 13.d odd 4 1
169.8.a.i yes 21 13.d odd 4 1
169.8.b.f 42 1.a even 1 1 trivial
169.8.b.f 42 13.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{42} + 4097 T_{2}^{40} + 7721510 T_{2}^{38} + 8881814775 T_{2}^{36} + 6976509369092 T_{2}^{34} + \cdots + 38\!\cdots\!24$$ acting on $$S_{8}^{\mathrm{new}}(169, [\chi])$$.