# Properties

 Label 169.2.k.a Level $169$ Weight $2$ Character orbit 169.k Analytic conductor $1.349$ Analytic rank $0$ Dimension $360$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,2,Mod(4,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(78))

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

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

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

chi := DirichletCharacter("169.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 169.k (of order $$78$$, degree $$24$$, 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: $$1.34947179416$$ Analytic rank: $$0$$ Dimension: $$360$$ Relative dimension: $$15$$ over $$\Q(\zeta_{78})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{78}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$360 q - 23 q^{2} - 24 q^{3} - 41 q^{4} - 26 q^{5} - 32 q^{6} - 26 q^{7} - 26 q^{8} - 11 q^{9}+O(q^{10})$$ 360 * q - 23 * q^2 - 24 * q^3 - 41 * q^4 - 26 * q^5 - 32 * q^6 - 26 * q^7 - 26 * q^8 - 11 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$360 q - 23 q^{2} - 24 q^{3} - 41 q^{4} - 26 q^{5} - 32 q^{6} - 26 q^{7} - 26 q^{8} - 11 q^{9} - 27 q^{10} - 26 q^{11} - 14 q^{12} - 34 q^{13} - 30 q^{14} - 84 q^{15} - 13 q^{16} - 31 q^{17} + 52 q^{18} - 33 q^{19} - 29 q^{20} - 26 q^{21} - 33 q^{22} + 57 q^{23} + 58 q^{24} + 2 q^{25} - 29 q^{26} - 30 q^{27} - 26 q^{28} - 19 q^{29} + 178 q^{30} - 78 q^{31} - 30 q^{32} - 26 q^{33} - 91 q^{34} - 18 q^{35} - 51 q^{36} - 41 q^{37} + 25 q^{38} + 12 q^{39} - 134 q^{40} - 17 q^{41} + 250 q^{42} - 28 q^{43} + 42 q^{45} + 18 q^{46} + 117 q^{47} - 57 q^{48} - 117 q^{49} - 20 q^{50} - 59 q^{51} + 37 q^{52} - 75 q^{53} + 118 q^{54} + 64 q^{55} - 42 q^{56} - 104 q^{57} - 87 q^{58} + 170 q^{59} + 78 q^{60} - 15 q^{61} + 19 q^{62} + 39 q^{63} + 32 q^{64} - 17 q^{65} + 73 q^{66} + 20 q^{67} - 76 q^{68} - 11 q^{69} + 46 q^{71} - 198 q^{72} - 26 q^{73} + 29 q^{74} - 70 q^{75} + 58 q^{76} + 6 q^{77} + 128 q^{78} - 54 q^{79} - 24 q^{80} - 7 q^{81} + 81 q^{82} + 234 q^{83} - 273 q^{84} - 74 q^{85} + 52 q^{86} - 112 q^{87} + 256 q^{88} - 27 q^{89} + 28 q^{90} - 78 q^{91} - 34 q^{92} + 51 q^{93} + 28 q^{94} - 11 q^{95} + 143 q^{96} + 40 q^{97} - 47 q^{98}+O(q^{100})$$ 360 * q - 23 * q^2 - 24 * q^3 - 41 * q^4 - 26 * q^5 - 32 * q^6 - 26 * q^7 - 26 * q^8 - 11 * q^9 - 27 * q^10 - 26 * q^11 - 14 * q^12 - 34 * q^13 - 30 * q^14 - 84 * q^15 - 13 * q^16 - 31 * q^17 + 52 * q^18 - 33 * q^19 - 29 * q^20 - 26 * q^21 - 33 * q^22 + 57 * q^23 + 58 * q^24 + 2 * q^25 - 29 * q^26 - 30 * q^27 - 26 * q^28 - 19 * q^29 + 178 * q^30 - 78 * q^31 - 30 * q^32 - 26 * q^33 - 91 * q^34 - 18 * q^35 - 51 * q^36 - 41 * q^37 + 25 * q^38 + 12 * q^39 - 134 * q^40 - 17 * q^41 + 250 * q^42 - 28 * q^43 + 42 * q^45 + 18 * q^46 + 117 * q^47 - 57 * q^48 - 117 * q^49 - 20 * q^50 - 59 * q^51 + 37 * q^52 - 75 * q^53 + 118 * q^54 + 64 * q^55 - 42 * q^56 - 104 * q^57 - 87 * q^58 + 170 * q^59 + 78 * q^60 - 15 * q^61 + 19 * q^62 + 39 * q^63 + 32 * q^64 - 17 * q^65 + 73 * q^66 + 20 * q^67 - 76 * q^68 - 11 * q^69 + 46 * q^71 - 198 * q^72 - 26 * q^73 + 29 * q^74 - 70 * q^75 + 58 * q^76 + 6 * q^77 + 128 * q^78 - 54 * q^79 - 24 * q^80 - 7 * q^81 + 81 * q^82 + 234 * q^83 - 273 * q^84 - 74 * q^85 + 52 * q^86 - 112 * q^87 + 256 * q^88 - 27 * q^89 + 28 * q^90 - 78 * q^91 - 34 * q^92 + 51 * q^93 + 28 * q^94 - 11 * q^95 + 143 * q^96 + 40 * q^97 - 47 * q^98

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −2.81837 0.113576i 0.467697 1.61468i 5.93678 + 0.479267i 1.59753 + 0.605865i −1.50153 + 4.49764i 1.15667 + 2.71481i −11.0774 1.34504i 0.147123 + 0.0930351i −4.43363 1.88899i
4.2 −2.41817 0.0974488i −0.437408 + 1.51011i 3.84452 + 0.310362i −2.71784 1.03074i 1.20488 3.60907i −0.958962 2.25077i −4.46148 0.541722i 0.446473 + 0.282332i 6.47174 + 2.75735i
4.3 −2.00062 0.0806224i −0.731230 + 2.52450i 2.00248 + 0.161657i 3.39858 + 1.28891i 1.66645 4.99162i 0.132610 + 0.311247i −0.0178867 0.00217183i −3.30283 2.08858i −6.69537 2.85263i
4.4 −1.59712 0.0643616i 0.883856 3.05143i 0.553124 + 0.0446528i −1.36888 0.519147i −1.60802 + 4.81660i −0.861700 2.02248i 2.29299 + 0.278419i −5.99443 3.79065i 2.15285 + 0.917242i
4.5 −1.55556 0.0626871i 0.386039 1.33276i 0.422335 + 0.0340944i 1.77605 + 0.673567i −0.684055 + 2.04899i −0.0926507 0.217459i 2.43612 + 0.295798i 0.908347 + 0.574404i −2.72054 1.15911i
4.6 −1.34677 0.0542729i 0.0769569 0.265686i −0.182676 0.0147471i −2.82623 1.07185i −0.118063 + 0.353641i 1.59063 + 3.73334i 2.92129 + 0.354709i 2.47090 + 1.56250i 3.74810 + 1.59692i
4.7 −0.522698 0.0210640i −0.684753 + 2.36404i −1.72074 0.138913i −1.25186 0.474767i 0.407715 1.22126i −0.851571 1.99871i 1.93512 + 0.234966i −2.58423 1.63417i 0.644343 + 0.274529i
4.8 −0.0643979 0.00259515i −0.109583 + 0.378323i −1.98937 0.160599i 1.64339 + 0.623256i 0.00803870 0.0240788i 0.909238 + 2.13406i 0.255655 + 0.0310422i 2.40445 + 1.52048i −0.104214 0.0444013i
4.9 0.397859 + 0.0160332i 0.320025 1.10485i −1.83548 0.148175i −2.52701 0.958369i 0.145039 0.434446i −1.61420 3.78868i −1.51844 0.184373i 1.41728 + 0.896235i −0.990029 0.421812i
4.10 0.596805 + 0.0240504i 0.703900 2.43014i −1.63792 0.132226i 3.70001 + 1.40323i 0.478537 1.43339i −0.324840 0.762428i −2.16021 0.262296i −2.87456 1.81776i 2.17443 + 0.926440i
4.11 1.10078 + 0.0443599i −0.581947 + 2.00912i −0.783766 0.0632721i 1.05138 + 0.398737i −0.729720 + 2.18578i 1.23850 + 2.90687i −3.04723 0.370001i −1.16231 0.735002i 1.13965 + 0.485561i
4.12 1.69738 + 0.0684019i 0.785283 2.71111i 0.882895 + 0.0712746i −1.38138 0.523888i 1.51837 4.54806i 1.73151 + 4.06400i −1.87901 0.228153i −4.19788 2.65458i −2.30888 0.983724i
4.13 1.95970 + 0.0789734i 0.177516 0.612858i 1.84069 + 0.148596i 0.0713672 + 0.0270660i 0.396279 1.18700i −0.605333 1.42077i −0.298517 0.0362466i 2.19149 + 1.38581i 0.137721 + 0.0586775i
4.14 1.99581 + 0.0804286i −0.751408 + 2.59416i 1.98329 + 0.160108i 1.41011 + 0.534785i −1.70832 + 5.11703i −1.63799 3.84451i −0.0203394 0.00246965i −3.62949 2.29515i 2.77131 + 1.18074i
4.15 2.57861 + 0.103914i −0.381537 + 1.31722i 4.64490 + 0.374975i −3.32428 1.26073i −1.12071 + 3.35694i 0.633807 + 1.48760i 6.81465 + 0.827448i 0.946076 + 0.598262i −8.44101 3.59638i
10.1 −2.60774 0.532374i −2.66189 + 0.432599i 4.67692 + 1.99265i −0.498005 + 2.02048i 7.17182 + 0.289014i 1.97372 0.936542i −6.75456 4.66233i 4.05291 1.35306i 2.37432 5.00377i
10.2 −2.52063 0.514590i 2.70717 0.439959i 4.24882 + 1.81025i 0.847165 3.43708i −7.05019 0.284113i 0.693293 0.328971i −5.54370 3.82654i 4.28962 1.43209i −3.90408 + 8.22767i
10.3 −1.99584 0.407454i 1.34046 0.217845i 1.97740 + 0.842491i −0.647035 + 2.62512i −2.76410 0.111389i −3.12689 + 1.48373i −0.250449 0.172873i −1.09624 + 0.365979i 2.36099 4.97569i
10.4 −1.98141 0.404507i −0.359410 + 0.0584099i 1.92239 + 0.819053i 0.199530 0.809524i 0.735766 + 0.0296503i 0.512743 0.243300i −0.149120 0.102930i −2.71985 + 0.908018i −0.722808 + 1.52328i
10.5 −1.44039 0.294058i −2.46995 + 0.401407i 0.148295 + 0.0631826i 0.720721 2.92408i 3.67574 + 0.148127i −2.39377 + 1.13586i 2.22471 + 1.53561i 3.09394 1.03291i −1.89797 + 3.99988i
See next 80 embeddings (of 360 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4.15 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.k even 78 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.2.k.a 360
169.k even 78 1 inner 169.2.k.a 360

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.2.k.a 360 1.a even 1 1 trivial
169.2.k.a 360 169.k even 78 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(169, [\chi])$$.