# Properties

 Label 109.2.f.a Level 109 Weight 2 Character orbit 109.f Analytic conductor 0.870 Analytic rank 0 Dimension 42 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$109$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 109.f (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.870369382032$$ Analytic rank: $$0$$ Dimension: $$42$$ Relative dimension: $$7$$ over $$\Q(\zeta_{9})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$42q - 6q^{3} - 12q^{4} - 6q^{5} + 12q^{6} + 3q^{7} - 12q^{8} - 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$42q - 6q^{3} - 12q^{4} - 6q^{5} + 12q^{6} + 3q^{7} - 12q^{8} - 12q^{9} + 15q^{10} - 15q^{11} + 9q^{12} - 30q^{13} + 3q^{14} + 6q^{16} - 3q^{17} - 27q^{18} - 3q^{19} - 30q^{20} - 3q^{21} - 18q^{22} + 6q^{23} - 12q^{24} + 6q^{25} + 15q^{26} + 3q^{27} + 66q^{28} + 3q^{30} + 6q^{31} + 12q^{32} + 24q^{33} - 21q^{34} - 54q^{35} + 21q^{36} - 24q^{37} + 27q^{38} + 18q^{39} - 24q^{40} - 30q^{41} + 12q^{42} + 9q^{43} + 36q^{44} + 12q^{45} - 12q^{46} - 42q^{47} - 27q^{48} + 15q^{49} + 3q^{50} - 12q^{51} - 3q^{52} + 3q^{53} - 36q^{54} + 21q^{55} + 57q^{56} - 15q^{57} - 24q^{58} + 18q^{59} + 33q^{60} + 6q^{61} + 78q^{62} - 48q^{63} - 12q^{64} + 3q^{65} - 15q^{66} - 6q^{67} + 66q^{68} + 15q^{69} + 39q^{70} + 15q^{71} - 9q^{72} + 66q^{73} - 24q^{74} + 24q^{75} - 96q^{76} - 39q^{77} - 3q^{78} + 18q^{79} - 3q^{80} - 15q^{81} + 21q^{82} + 21q^{83} + 87q^{84} + 120q^{85} - 15q^{86} + 12q^{87} - 48q^{88} + 15q^{89} + 24q^{90} + 63q^{92} - 75q^{93} - 30q^{94} + 15q^{95} - 21q^{96} + 48q^{97} - 126q^{98} + 39q^{99} + O(q^{100})$$

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1 −1.22244 2.11733i −1.03698 + 0.870129i −1.98872 + 3.44456i 0.575007 + 3.26103i 3.10999 + 1.13194i 0.0336150 + 0.190640i 4.83459 −0.202743 + 1.14981i 6.20175 5.20389i
16.2 −0.603099 1.04460i −1.82067 + 1.52773i 0.272543 0.472059i −0.272178 1.54360i 2.69391 + 0.980502i −0.703227 3.98820i −3.06988 0.459961 2.60857i −1.44829 + 1.21526i
16.3 −0.529087 0.916406i 1.30845 1.09792i 0.440133 0.762333i 0.109661 + 0.621921i −1.69843 0.618177i 0.171837 + 0.974534i −3.04783 −0.0143299 + 0.0812691i 0.511912 0.429545i
16.4 0.219432 + 0.380068i −0.0615666 + 0.0516605i 0.903699 1.56525i −0.640473 3.63230i −0.0331442 0.0120635i 0.228918 + 1.29826i 1.67093 −0.519823 + 2.94806i 1.23998 1.04047i
16.5 0.503663 + 0.872370i −0.352094 + 0.295442i 0.492647 0.853289i 0.517671 + 2.93586i −0.435072 0.158353i −0.307807 1.74566i 3.00717 −0.484260 + 2.74638i −2.30042 + 1.93028i
16.6 1.05944 + 1.83500i −1.78471 + 1.49755i −1.24481 + 2.15608i −0.222139 1.25981i −4.63879 1.68838i 0.247600 + 1.40421i −1.03746 0.421589 2.39095i 2.07641 1.74232i
16.7 1.33814 + 2.31772i 1.04183 0.874202i −2.58123 + 4.47082i −0.127857 0.725111i 3.42028 + 1.24488i −0.876671 4.97185i −8.46360 −0.199757 + 1.13288i 1.50952 1.26663i
27.1 −1.13355 1.96336i 1.99711 + 0.726890i −1.56986 + 2.71908i 0.184738 0.155013i −0.836675 4.74502i 2.60110 2.18258i 2.58386 1.16196 + 0.974999i −0.513756 0.186992i
27.2 −0.754385 1.30663i −0.821044 0.298836i −0.138193 + 0.239358i 1.83302 1.53808i 0.228915 + 1.29824i −0.324966 + 0.272679i −2.60054 −1.71332 1.43765i −3.39251 1.23477i
27.3 −0.641080 1.11038i −2.04885 0.745722i 0.178032 0.308361i −2.94251 + 2.46906i 0.485442 + 2.75308i 0.512859 0.430340i −3.02085 1.34357 + 1.12739i 4.62799 + 1.68445i
27.4 −0.304565 0.527522i 1.79458 + 0.653175i 0.814480 1.41072i 0.460326 0.386260i −0.202003 1.14562i −2.95565 + 2.48009i −2.21051 0.495757 + 0.415990i −0.343960 0.125191i
27.5 0.368017 + 0.637423i 1.41819 + 0.516179i 0.729128 1.26289i −2.11121 + 1.77151i 0.192893 + 1.09395i 1.03302 0.866806i 2.54539 −0.553313 0.464285i −1.90616 0.693787i
27.6 0.384791 + 0.666477i −2.14187 0.779577i 0.703872 1.21914i 1.33144 1.11721i −0.304602 1.72748i 0.847872 0.711449i 2.62254 1.68174 + 1.41114i 1.25692 + 0.457482i
27.7 1.14108 + 1.97641i −0.0847756 0.0308558i −1.60412 + 2.77841i 0.0705498 0.0591983i −0.0357520 0.202760i −0.100894 + 0.0846602i −2.75738 −2.29190 1.92313i 0.197503 + 0.0718851i
38.1 −1.09927 + 1.90400i −0.271232 + 1.53823i −1.41681 2.45399i −1.85814 + 0.676309i −2.63064 2.20737i −0.0942741 + 0.0343130i 1.83275 0.526478 + 0.191622i 0.754918 4.28135i
38.2 −0.892046 + 1.54507i 0.172558 0.978624i −0.591492 1.02449i 3.13823 1.14222i 1.35811 + 1.13959i −0.0641258 + 0.0233399i −1.45763 1.89115 + 0.688322i −1.03463 + 5.86769i
38.3 −0.253598 + 0.439245i 0.420069 2.38233i 0.871376 + 1.50927i −1.20157 + 0.437336i 0.939898 + 0.788668i 4.19577 1.52714i −1.89831 −2.67995 0.975422i 0.112619 0.638692i
38.4 −0.0775572 + 0.134333i −0.124689 + 0.707146i 0.987970 + 1.71121i −0.326444 + 0.118816i −0.0853226 0.0715941i −1.80571 + 0.657225i −0.616725 2.33457 + 0.849714i 0.00935720 0.0530673i
38.5 0.682493 1.18211i −0.498285 + 2.82591i 0.0684068 + 0.118484i −3.58556 + 1.30504i 3.00047 + 2.51769i 4.26974 1.55406i 2.91672 −4.91842 1.79016i −0.904418 + 5.12921i
38.6 0.809610 1.40229i 0.309749 1.75667i −0.310937 0.538559i −1.08704 + 0.395650i −2.21258 1.85658i −1.51205 + 0.550341i 2.23149 −0.170882 0.0621961i −0.325263 + 1.84466i
See all 42 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 105.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
109.f even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 109.2.f.a 42
3.b odd 2 1 981.2.w.a 42
109.f even 9 1 inner 109.2.f.a 42
327.o odd 18 1 981.2.w.a 42

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
109.2.f.a 42 1.a even 1 1 trivial
109.2.f.a 42 109.f even 9 1 inner
981.2.w.a 42 3.b odd 2 1
981.2.w.a 42 327.o odd 18 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(109, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database