# Properties

 Label 169.8.a.h Level $169$ Weight $8$ Character orbit 169.a Self dual yes Analytic conductor $52.793$ Analytic rank $1$ Dimension $21$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,8,Mod(1,169)]

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([0]))

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

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

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 169.a (trivial)

## 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: yes Analytic conductor: $$52.7930693068$$ Analytic rank: $$1$$ Dimension: $$21$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$21 q - 31 q^{2} - 26 q^{3} + 1409 q^{4} - 680 q^{5} - 1470 q^{6} - 2929 q^{7} - 4716 q^{8} + 15465 q^{9}+O(q^{10})$$ 21 * q - 31 * q^2 - 26 * q^3 + 1409 * q^4 - 680 * q^5 - 1470 * q^6 - 2929 * q^7 - 4716 * q^8 + 15465 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$21 q - 31 q^{2} - 26 q^{3} + 1409 q^{4} - 680 q^{5} - 1470 q^{6} - 2929 q^{7} - 4716 q^{8} + 15465 q^{9} - 5167 q^{10} - 14824 q^{11} + 21795 q^{12} - 179 q^{14} - 36398 q^{15} + 113205 q^{16} + 45016 q^{17} - 72967 q^{18} - 119316 q^{19} - 131130 q^{20} - 94178 q^{21} - 274174 q^{22} - 128105 q^{23} - 316902 q^{24} + 366537 q^{25} - 106643 q^{27} - 170352 q^{28} - 62329 q^{29} + 9761 q^{30} - 467367 q^{31} - 677464 q^{32} - 587690 q^{33} - 578614 q^{34} + 848860 q^{35} - 274245 q^{36} - 697064 q^{37} + 225831 q^{38} - 2288827 q^{40} - 2351315 q^{41} + 2736836 q^{42} + 894489 q^{43} + 263025 q^{44} + 1938345 q^{45} + 3170392 q^{46} - 1070934 q^{47} + 6302388 q^{48} + 131530 q^{49} - 3970474 q^{50} - 1184341 q^{51} - 4550027 q^{53} - 7916233 q^{54} + 5524030 q^{55} + 2334386 q^{56} - 5590305 q^{57} - 7510959 q^{58} - 10746233 q^{59} - 15186203 q^{60} + 5717012 q^{61} - 13593653 q^{62} - 22347508 q^{63} + 9767924 q^{64} - 17806977 q^{66} - 11305923 q^{67} + 12252492 q^{68} + 2772206 q^{69} - 28670666 q^{70} - 23267025 q^{71} - 49774322 q^{72} - 4763727 q^{73} + 9625078 q^{74} - 43212397 q^{75} - 63359948 q^{76} + 9446036 q^{77} - 18613281 q^{79} - 57904093 q^{80} + 41055449 q^{81} + 24190327 q^{82} - 21455789 q^{83} - 60768429 q^{84} - 26216249 q^{85} - 35632347 q^{86} + 9027894 q^{87} - 75678180 q^{88} - 54037507 q^{89} + 3335560 q^{90} - 69339390 q^{92} - 20085986 q^{93} + 28955713 q^{94} + 17660031 q^{95} - 66667956 q^{96} - 10916609 q^{97} - 41738840 q^{98} - 37605493 q^{99}+O(q^{100})$$ 21 * q - 31 * q^2 - 26 * q^3 + 1409 * q^4 - 680 * q^5 - 1470 * q^6 - 2929 * q^7 - 4716 * q^8 + 15465 * q^9 - 5167 * q^10 - 14824 * q^11 + 21795 * q^12 - 179 * q^14 - 36398 * q^15 + 113205 * q^16 + 45016 * q^17 - 72967 * q^18 - 119316 * q^19 - 131130 * q^20 - 94178 * q^21 - 274174 * q^22 - 128105 * q^23 - 316902 * q^24 + 366537 * q^25 - 106643 * q^27 - 170352 * q^28 - 62329 * q^29 + 9761 * q^30 - 467367 * q^31 - 677464 * q^32 - 587690 * q^33 - 578614 * q^34 + 848860 * q^35 - 274245 * q^36 - 697064 * q^37 + 225831 * q^38 - 2288827 * q^40 - 2351315 * q^41 + 2736836 * q^42 + 894489 * q^43 + 263025 * q^44 + 1938345 * q^45 + 3170392 * q^46 - 1070934 * q^47 + 6302388 * q^48 + 131530 * q^49 - 3970474 * q^50 - 1184341 * q^51 - 4550027 * q^53 - 7916233 * q^54 + 5524030 * q^55 + 2334386 * q^56 - 5590305 * q^57 - 7510959 * q^58 - 10746233 * q^59 - 15186203 * q^60 + 5717012 * q^61 - 13593653 * q^62 - 22347508 * q^63 + 9767924 * q^64 - 17806977 * q^66 - 11305923 * q^67 + 12252492 * q^68 + 2772206 * q^69 - 28670666 * q^70 - 23267025 * q^71 - 49774322 * q^72 - 4763727 * q^73 + 9625078 * q^74 - 43212397 * q^75 - 63359948 * q^76 + 9446036 * q^77 - 18613281 * q^79 - 57904093 * q^80 + 41055449 * q^81 + 24190327 * q^82 - 21455789 * q^83 - 60768429 * q^84 - 26216249 * q^85 - 35632347 * q^86 + 9027894 * q^87 - 75678180 * q^88 - 54037507 * q^89 + 3335560 * q^90 - 69339390 * q^92 - 20085986 * q^93 + 28955713 * q^94 + 17660031 * q^95 - 66667956 * q^96 - 10916609 * q^97 - 41738840 * q^98 - 37605493 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −21.6353 79.0768 340.087 120.676 −1710.85 −240.114 −4588.56 4066.15 −2610.86
1.2 −21.5931 28.3750 338.261 −473.706 −612.705 99.1966 −4540.19 −1381.86 10228.8
1.3 −19.6237 −72.4635 257.091 413.760 1422.00 1005.88 −2533.24 3063.96 −8119.51
1.4 −17.7683 58.1116 187.714 −15.5333 −1032.55 −1688.45 −1061.02 1189.96 276.002
1.5 −15.3948 −46.3939 108.998 −391.457 714.223 −1742.39 292.525 −34.6060 6026.39
1.6 −13.1620 −21.4559 45.2387 344.184 282.403 168.957 1089.31 −1726.64 −4530.16
1.7 −11.8381 −26.1495 12.1398 422.131 309.560 1215.99 1371.56 −1503.20 −4997.21
1.8 −10.5911 −6.51000 −15.8290 −397.099 68.9480 1187.35 1523.30 −2144.62 4205.71
1.9 −5.77104 82.1608 −94.6950 17.6329 −474.154 −888.558 1285.18 4563.40 −101.761
1.10 −3.19657 −57.0706 −117.782 −431.499 182.430 −225.139 785.660 1070.05 1379.32
1.11 −2.91821 44.2582 −119.484 −197.474 −129.155 303.075 722.210 −228.212 576.269
1.12 −1.25436 8.41428 −126.427 307.776 −10.5545 −83.6974 319.143 −2116.20 −386.063
1.13 2.90053 −90.5117 −119.587 −240.795 −262.532 −1391.07 −718.132 6005.38 −698.433
1.14 6.93821 −64.5145 −79.8613 33.5113 −447.615 −160.658 −1442.18 1975.12 232.508
1.15 8.58294 18.1259 −54.3331 37.6136 155.574 711.860 −1564.95 −1858.45 322.835
1.16 8.99747 87.5208 −47.0455 −190.033 787.466 −44.6495 −1574.97 5472.89 −1709.81
1.17 13.2906 38.7978 48.6390 274.097 515.644 −774.181 −1054.75 −681.732 3642.90
1.18 15.0231 −85.3884 97.6921 439.379 −1282.79 −536.518 −455.317 5104.17 6600.82
1.19 16.7260 −13.7721 151.758 −0.583029 −230.352 274.333 397.369 −1997.33 −9.75172
1.20 20.4423 −14.6497 289.887 −478.639 −299.473 1197.76 3309.33 −1972.39 −9784.47
See all 21 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.21 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{21} + 31 T_{2}^{20} - 1568 T_{2}^{19} - 54321 T_{2}^{18} + 947492 T_{2}^{17} + \cdots - 61\!\cdots\!68$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(169))$$.