Properties

Label 169.2.g.a
Level $169$
Weight $2$
Character orbit 169.g
Analytic conductor $1.349$
Analytic rank $0$
Dimension $156$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 169.g (of order \(13\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.34947179416\)
Analytic rank: \(0\)
Dimension: \(156\)
Relative dimension: \(13\) over \(\Q(\zeta_{13})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{13}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 156q - 10q^{2} - 9q^{3} - 20q^{4} - 7q^{5} - q^{6} - 5q^{7} + 2q^{8} - 14q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 156q - 10q^{2} - 9q^{3} - 20q^{4} - 7q^{5} - q^{6} - 5q^{7} + 2q^{8} - 14q^{9} + q^{10} - q^{11} + 11q^{12} - 65q^{13} + 9q^{14} - 41q^{15} + 3q^{17} - 13q^{18} - 6q^{19} + 29q^{20} + 19q^{21} - 22q^{22} - 82q^{23} - 31q^{24} + 2q^{25} + 26q^{26} + 21q^{27} + 43q^{28} + 13q^{29} - 81q^{30} - 33q^{31} - 93q^{32} + 35q^{33} - 24q^{34} + 27q^{35} + 54q^{36} + 25q^{37} - 56q^{38} - 13q^{39} - 52q^{40} + 29q^{41} - 63q^{42} + 21q^{43} + 45q^{44} + 33q^{46} - 69q^{47} + 54q^{48} - 54q^{49} + 80q^{50} - 16q^{51} + 13q^{52} - 45q^{53} + 29q^{54} - 83q^{55} + 91q^{56} - 11q^{57} + 25q^{58} - 57q^{59} + 51q^{60} + 39q^{61} + 4q^{62} + 26q^{63} + 86q^{64} + 65q^{65} - 138q^{66} - 101q^{67} + 36q^{68} + 32q^{69} - 90q^{70} + 20q^{71} + 13q^{72} + 61q^{73} - 4q^{74} - 67q^{75} - 107q^{76} + 67q^{77} + 13q^{78} + 57q^{79} + 160q^{80} + 78q^{81} - 31q^{82} - 59q^{83} - 36q^{84} - 61q^{85} + 41q^{86} - 9q^{87} - 45q^{88} - 66q^{89} + 191q^{90} + 39q^{91} + 79q^{92} - 80q^{93} - 21q^{94} - 28q^{95} + 70q^{96} + 7q^{97} + 158q^{98} + 130q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −1.55724 + 2.25606i −1.88671 + 0.990222i −1.95558 5.15643i −0.621291 0.550416i 0.704070 5.79854i −0.145399 0.0358377i 9.35521 + 2.30585i 0.874940 1.26757i 2.20927 0.544537i
14.2 −1.29190 + 1.87164i 1.09718 0.575847i −1.12482 2.96591i 0.226698 + 0.200837i −0.339674 + 2.79747i 2.40457 + 0.592674i 2.58801 + 0.637888i −0.831980 + 1.20533i −0.668766 + 0.164836i
14.3 −0.958835 + 1.38911i 0.0179783 0.00943576i −0.301058 0.793826i −2.19225 1.94216i −0.00413093 + 0.0340212i −2.78022 0.685263i −1.88632 0.464936i −1.70396 + 2.46861i 4.79989 1.18307i
14.4 −0.885893 + 1.28344i −1.70070 + 0.892595i −0.153196 0.403946i 2.77189 + 2.45568i 0.361046 2.97348i 3.26613 + 0.805028i −2.37420 0.585188i 0.391453 0.567117i −5.60732 + 1.38208i
14.5 −0.707334 + 1.02475i 2.32028 1.21778i 0.159418 + 0.420350i 1.61135 + 1.42753i −0.393297 + 3.23909i −1.03213 0.254398i −2.96148 0.729939i 2.19654 3.18224i −2.60263 + 0.641490i
14.6 −0.479816 + 0.695133i −2.71090 + 1.42279i 0.456223 + 1.20296i −1.82534 1.61711i 0.311704 2.56712i −0.220084 0.0542458i −2.69533 0.664340i 3.62047 5.24515i 1.99993 0.492940i
14.7 −0.0917982 + 0.132993i −0.752776 + 0.395087i 0.699950 + 1.84562i 1.16867 + 1.03535i 0.0165598 0.136382i −2.31458 0.570492i −0.623512 0.153682i −1.29362 + 1.87413i −0.244976 + 0.0603811i
14.8 0.0291084 0.0421708i 2.28283 1.19812i 0.708279 + 1.86758i −2.44658 2.16748i 0.0159238 0.131144i 1.95389 + 0.481590i 0.198879 + 0.0490192i 2.07162 3.00127i −0.162620 + 0.0400823i
14.9 0.481559 0.697658i 1.04345 0.547645i 0.454382 + 1.19811i 0.674379 + 0.597448i 0.120414 0.991695i −0.515253 0.126998i 2.70085 + 0.665700i −0.915320 + 1.32607i 0.741567 0.182780i
14.10 0.711778 1.03119i −1.78709 + 0.937940i 0.152488 + 0.402078i −0.453990 0.402200i −0.304822 + 2.51044i 4.78395 + 1.17914i 2.95631 + 0.728665i 0.609782 0.883421i −0.737885 + 0.181872i
14.11 1.00858 1.46118i −2.49701 + 1.31053i −0.408612 1.07742i 2.96132 + 2.62350i −0.603512 + 4.97037i −1.91869 0.472914i 1.46133 + 0.360186i 2.81336 4.07585i 6.82015 1.68102i
14.12 1.13519 1.64460i 0.811870 0.426102i −0.706856 1.86382i −2.38631 2.11409i 0.220855 1.81891i −0.234049 0.0576880i 0.0128813 + 0.00317496i −1.22662 + 1.77707i −6.18573 + 1.52465i
14.13 1.39315 2.01832i 0.422612 0.221804i −1.42355 3.75360i 1.09172 + 0.967184i 0.141089 1.16197i 0.0247635 + 0.00610365i −4.79682 1.18231i −1.57479 + 2.28148i 3.47302 0.856022i
27.1 −0.952850 2.51246i −0.297229 + 0.430610i −3.90751 + 3.46175i 0.365684 + 3.01168i 1.36511 + 0.336468i −2.17334 1.14066i 7.66220 + 4.02143i 0.966734 + 2.54907i 7.21828 3.78845i
27.2 −0.814971 2.14890i 0.722310 1.04645i −2.45658 + 2.17634i −0.370802 3.05383i −2.83737 0.699350i 0.780099 + 0.409428i 2.60879 + 1.36920i 0.490496 + 1.29333i −6.26019 + 3.28560i
27.3 −0.568511 1.49904i −1.77394 + 2.57000i −0.426895 + 0.378196i 0.00472042 + 0.0388761i 4.86104 + 1.19814i −3.29128 1.72740i −2.02954 1.06519i −2.39421 6.31302i 0.0555933 0.0291776i
27.4 −0.553905 1.46053i −0.577862 + 0.837177i −0.329307 + 0.291741i 0.268400 + 2.21047i 1.54280 + 0.380266i 3.47659 + 1.82465i −2.15772 1.13246i 0.696873 + 1.83750i 3.07979 1.61640i
27.5 −0.235915 0.622057i 1.17724 1.70553i 1.16572 1.03274i 0.100994 + 0.831764i −1.33867 0.329951i 1.56980 + 0.823895i −2.09560 1.09986i −0.459118 1.21059i 0.493578 0.259050i
27.6 −0.168406 0.444051i 0.0744765 0.107898i 1.32820 1.17668i −0.265542 2.18694i −0.0604545 0.0149007i −3.17143 1.66450i −1.58721 0.833034i 1.05772 + 2.78898i −0.926393 + 0.486209i
27.7 −0.000492135 0.00129765i −1.38011 + 1.99943i 1.49702 1.32624i −0.444348 3.65953i 0.00327377 0.000806912i 3.94802 + 2.07208i −0.00491548 0.00257984i −1.02922 2.71382i −0.00453012 + 0.00237759i
See next 80 embeddings (of 156 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.g even 13 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.2.g.a 156
169.g even 13 1 inner 169.2.g.a 156
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.2.g.a 156 1.a even 1 1 trivial
169.2.g.a 156 169.g even 13 1 inner

Hecke kernels

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