# Properties

 Label 169.2.i.a Level $169$ Weight $2$ Character orbit 169.i Analytic conductor $1.349$ Analytic rank $0$ Dimension $336$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 169.i (of order $$39$$, degree $$24$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.34947179416$$ Analytic rank: $$0$$ Dimension: $$336$$ Relative dimension: $$14$$ over $$\Q(\zeta_{39})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{39}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$336q - 26q^{2} - 26q^{3} - 12q^{4} - 26q^{5} - 26q^{6} - 26q^{7} - 26q^{8} - 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$336q - 26q^{2} - 26q^{3} - 12q^{4} - 26q^{5} - 26q^{6} - 26q^{7} - 26q^{8} - 12q^{9} - 22q^{10} - 26q^{11} - 34q^{12} - 13q^{13} - 30q^{14} + 26q^{15} - 8q^{16} - 24q^{17} - 104q^{18} - 13q^{19} - 26q^{20} - 26q^{21} + 19q^{22} + 67q^{23} + 52q^{24} - 58q^{25} - 26q^{26} - 38q^{27} - 26q^{28} - 22q^{29} - 120q^{30} + 26q^{31} + 117q^{32} - 26q^{33} + 39q^{34} - 18q^{35} - 2q^{36} - 26q^{37} + 41q^{38} + 26q^{39} + 67q^{40} - 26q^{41} - 270q^{42} - 16q^{43} + 39q^{45} - 39q^{47} + 31q^{48} + 72q^{49} - 26q^{50} + 19q^{51} + 39q^{52} + 36q^{53} - 182q^{54} + 116q^{55} - 10q^{56} + 52q^{57} + 26q^{58} - 234q^{59} + 78q^{60} - 16q^{61} + 53q^{62} + 39q^{63} - 82q^{64} - 26q^{65} - 135q^{66} + 130q^{67} + 51q^{68} + 25q^{69} + 156q^{70} - 104q^{71} + 143q^{72} - 26q^{73} + 79q^{74} + 152q^{75} - 104q^{76} - 58q^{77} + 104q^{78} - 46q^{79} - 13q^{80} + 4q^{81} + 118q^{82} - 286q^{83} + 221q^{84} + 130q^{85} + 52q^{86} + 90q^{87} - 204q^{88} + 117q^{89} - 86q^{90} + 26q^{91} - 22q^{92} + 39q^{93} + 126q^{94} - 47q^{95} + 143q^{96} + 52q^{97} - 26q^{98} - 52q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −2.19950 + 1.65282i −0.237088 + 1.16133i 1.54956 5.34971i 2.23759 + 1.17438i −1.39800 2.94622i 1.15088 + 0.187036i 3.48258 + 9.18281i 1.46745 + 0.625223i −6.86262 + 1.11528i
3.2 −2.03393 + 1.52840i 0.603355 2.95543i 1.24444 4.29630i −1.41359 0.741908i 3.28990 + 6.93331i −0.790704 0.128502i 2.23100 + 5.88267i −5.61059 2.39045i 4.00907 0.651538i
3.3 −1.59194 + 1.19627i −0.271241 + 1.32863i 0.546788 1.88773i −0.978405 0.513506i −1.15759 2.43957i −3.69364 0.600275i −0.0244856 0.0645632i 1.06827 + 0.455145i 2.17185 0.352960i
3.4 −1.33556 + 1.00361i 0.253347 1.24098i 0.220054 0.759713i 1.62091 + 0.850719i 0.907094 + 1.91166i −0.617530 0.100358i −0.716255 1.88861i 1.28410 + 0.547103i −3.01861 + 0.490571i
3.5 −1.20477 + 0.905325i 0.192647 0.943646i 0.0754199 0.260380i −3.09172 1.62266i 0.622212 + 1.31128i 4.19009 + 0.680956i −0.923922 2.43618i 1.90658 + 0.812319i 5.19384 0.844081i
3.6 −0.600398 + 0.451170i −0.605141 + 2.96418i −0.399511 + 1.37927i 2.19021 + 1.14951i −0.974023 2.05271i 0.324401 + 0.0527202i −0.915052 2.41279i −5.66022 2.41159i −1.83362 + 0.297993i
3.7 −0.108893 + 0.0818278i 0.662843 3.24682i −0.551273 + 1.90322i 2.52336 + 1.32436i 0.193501 + 0.407795i 3.39567 + 0.551850i −0.192308 0.507076i −7.34253 3.12836i −0.383146 + 0.0622673i
3.8 −0.0796839 + 0.0598785i −0.377738 + 1.85028i −0.553671 + 1.91149i −2.61855 1.37432i −0.0806927 0.170056i −1.55454 0.252638i −0.141029 0.371862i −0.520928 0.221947i 0.290949 0.0472838i
3.9 0.282507 0.212290i −0.0177954 + 0.0871674i −0.521692 + 1.80109i −0.416761 0.218733i 0.0134775 + 0.0284032i 2.24768 + 0.365284i 0.485593 + 1.28040i 2.75266 + 1.17280i −0.164173 + 0.0266807i
3.10 0.470185 0.353321i 0.0357404 0.175068i −0.460197 + 1.58878i 3.13432 + 1.64502i −0.0450506 0.0949422i −4.24484 0.689853i 0.762089 + 2.00946i 2.73057 + 1.16339i 2.05493 0.333958i
3.11 1.18774 0.892529i 0.500302 2.45064i 0.0576840 0.199148i −1.46041 0.766482i −1.59304 3.35726i −0.935624 0.152054i 0.944448 + 2.49030i −2.99541 1.27623i −2.41869 + 0.393076i
3.12 1.43066 1.07507i −0.532083 + 2.60631i 0.334579 1.15510i −0.657900 0.345292i 2.04075 + 4.30078i 2.44433 + 0.397242i 0.506036 + 1.33431i −3.74982 1.59765i −1.31245 + 0.213294i
3.13 1.62971 1.22465i 0.199780 0.978589i 0.599756 2.07060i −0.493428 0.258971i −0.872842 1.83948i −0.169663 0.0275730i −0.112560 0.296796i 1.84221 + 0.784894i −1.12129 + 0.182228i
3.14 2.07621 1.56017i −0.286924 + 1.40545i 1.32008 4.55745i 1.11471 + 0.585047i 1.59702 + 3.36565i −4.08991 0.664676i −2.52777 6.66517i 0.866985 + 0.369388i 3.22715 0.524463i
9.1 −0.714316 + 2.46611i 2.17505 + 0.926703i −3.88105 2.45423i 1.06811 + 1.54743i −3.83902 + 4.70195i −2.68431 0.896153i 4.98112 4.41289i 1.79390 + 1.86765i −4.57909 + 1.52872i
9.2 −0.710076 + 2.45147i −1.50435 0.640943i −3.81511 2.41253i −0.878874 1.27327i 2.63945 3.23274i 1.06738 + 0.356344i 4.80250 4.25464i −0.225916 0.235203i 3.74545 1.25041i
9.3 −0.516682 + 1.78379i −2.17665 0.927386i −1.22458 0.774377i 2.29226 + 3.32092i 2.77890 3.40354i −3.94229 1.31613i −0.766097 + 0.678703i 1.79961 + 1.87359i −7.10820 + 2.37307i
9.4 −0.431904 + 1.49111i 2.71969 + 1.15875i −0.346475 0.219098i −2.18511 3.16567i −2.90247 + 3.55488i 2.71121 + 0.905134i −1.84763 + 1.63686i 3.97585 + 4.13930i 5.66411 1.89096i
9.5 −0.386628 + 1.33479i 1.08858 + 0.463801i 0.0581875 + 0.0367956i 0.442938 + 0.641707i −1.03995 + 1.27371i 0.633960 + 0.211647i −2.15196 + 1.90647i −1.10828 1.15384i −1.02780 + 0.343129i
9.6 −0.318349 + 1.09907i −2.14880 0.915520i 0.583776 + 0.369158i −0.125099 0.181237i 1.69029 2.07023i 4.07007 + 1.35879i −2.30453 + 2.04164i 1.70101 + 1.77094i 0.239017 0.0797956i
See next 80 embeddings (of 336 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 165.14 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.i even 39 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.2.i.a 336
169.i even 39 1 inner 169.2.i.a 336

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.2.i.a 336 1.a even 1 1 trivial
169.2.i.a 336 169.i even 39 1 inner

## Hecke kernels

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