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Properties

Label 169.3.l.a
Level $169$
Weight $3$
Character orbit 169.l
Analytic conductor $4.605$
Analytic rank $0$
Dimension $1392$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,3,Mod(2,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(156))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.2");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 169.l (of order \(156\), degree \(48\), 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: \(4.60491646769\)
Analytic rank: \(0\)
Dimension: \(1392\)
Relative dimension: \(29\) over \(\Q(\zeta_{156})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{156}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1392 q - 50 q^{2} - 50 q^{3} - 46 q^{4} - 38 q^{5} - 62 q^{6} - 68 q^{7} - 46 q^{8} + 106 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1392 q - 50 q^{2} - 50 q^{3} - 46 q^{4} - 38 q^{5} - 62 q^{6} - 68 q^{7} - 46 q^{8} + 106 q^{9} - 76 q^{10} - 56 q^{11} - 52 q^{12} - 26 q^{13} - 28 q^{14} - 38 q^{15} - 234 q^{16} - 40 q^{17} - 50 q^{18} - 62 q^{19} - 78 q^{20} - 92 q^{21} - 54 q^{22} - 96 q^{23} - 406 q^{24} - 52 q^{25} - 104 q^{26} - 68 q^{27} - 8 q^{28} - 66 q^{29} + 834 q^{30} - 32 q^{31} + 618 q^{32} - 20 q^{33} - 34 q^{34} - 40 q^{35} + 2 q^{36} + 16 q^{37} - 962 q^{38} - 1066 q^{39} + 456 q^{40} - 152 q^{41} - 92 q^{42} - 232 q^{43} - 36 q^{44} + 80 q^{45} - 126 q^{46} - 374 q^{47} - 46 q^{48} + 20 q^{49} - 6 q^{50} - 52 q^{51} + 1118 q^{52} - 686 q^{53} - 68 q^{54} + 792 q^{55} + 32 q^{56} - 32 q^{57} - 12 q^{58} + 112 q^{59} - 1308 q^{60} + 88 q^{61} - 1488 q^{62} + 1326 q^{63} - 52 q^{64} - 104 q^{65} + 238 q^{66} + 246 q^{67} - 160 q^{68} - 358 q^{69} - 1048 q^{70} + 34 q^{71} - 124 q^{72} - 110 q^{73} + 1062 q^{74} - 1790 q^{75} - 66 q^{76} - 52 q^{77} + 832 q^{78} + 12 q^{79} + 88 q^{80} + 406 q^{81} - 1376 q^{82} + 136 q^{83} - 56 q^{84} - 1552 q^{85} + 1688 q^{86} - 154 q^{87} + 568 q^{88} + 448 q^{89} - 52 q^{90} - 104 q^{91} - 712 q^{92} - 682 q^{93} + 2742 q^{94} + 26 q^{95} + 3626 q^{96} - 2102 q^{97} - 38 q^{98} - 260 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −3.85288 − 0.0776014i −2.99370 + 2.24962i 10.8419 + 0.436915i −4.74183 − 0.868973i 11.7089 − 8.43520i −4.79241 + 7.25112i −26.3522 − 1.59402i 1.39749 − 4.82469i 18.2023 + 3.71602i
2.2 −3.81961 − 0.0769313i 3.71697 − 2.79312i 10.5868 + 0.426632i −4.13378 − 0.757543i −14.4123 + 10.3827i 7.43138 − 11.2440i −25.1509 − 1.52135i 3.51037 − 12.1192i 15.7312 + 3.21154i
2.3 −3.57387 − 0.0719817i −1.19121 + 0.895135i 8.77058 + 0.353442i 6.55330 + 1.20094i 4.32165 − 3.11335i 1.01073 − 1.52928i −17.0472 − 1.03116i −1.88625 + 6.51208i −23.3342 − 4.76371i
2.4 −3.12284 − 0.0628976i 2.75601 − 2.07101i 5.75143 + 0.231775i 0.175929 + 0.0322401i −8.73684 + 6.29408i −6.30564 + 9.54070i −5.47511 − 0.331183i 0.802560 − 2.77076i −0.547370 − 0.111746i
2.5 −3.05789 − 0.0615893i −0.481394 + 0.361744i 5.35013 + 0.215603i −2.10621 − 0.385977i 1.49433 − 1.07653i 2.01076 − 3.04236i −4.13510 − 0.250127i −2.40308 + 8.29638i 6.41678 + 1.31000i
2.6 −2.56551 − 0.0516722i −4.36662 + 3.28130i 2.58240 + 0.104067i 3.84380 + 0.704403i 11.3722 − 8.19257i 2.52883 − 3.82622i 3.62559 + 0.219308i 5.79648 − 20.0118i −9.82490 − 2.00577i
2.7 −2.53214 − 0.0510003i 2.53133 − 1.90217i 2.41240 + 0.0972164i 9.19956 + 1.68588i −6.50671 + 4.68748i 2.04693 − 3.09709i 4.00856 + 0.242473i 0.285426 − 0.985404i −23.2086 − 4.73808i
2.8 −2.39985 − 0.0483358i 0.856835 − 0.643870i 1.76020 + 0.0709336i −7.19950 − 1.31936i −2.08740 + 1.50378i −1.55257 + 2.34910i 5.36305 + 0.324405i −2.18436 + 7.54128i 17.2140 + 3.51426i
2.9 −1.90533 − 0.0383756i −2.37519 + 1.78484i −0.367929 − 0.0148270i 4.78634 + 0.877129i 4.59403 − 3.30957i −4.71740 + 7.13762i 8.30943 + 0.502628i −0.0480733 + 0.165968i −9.08592 − 1.85490i
2.10 −1.84248 − 0.0371097i 4.35910 − 3.27565i −0.603395 − 0.0243160i −0.825014 − 0.151189i −8.15312 + 5.87356i −0.469701 + 0.710677i 8.46881 + 0.512269i 5.76791 − 19.9131i 1.51446 + 0.309180i
2.11 −1.50572 − 0.0303270i −0.140185 + 0.105342i −1.73048 − 0.0697358i −1.54495 − 0.283123i 0.214274 − 0.154364i 5.94248 − 8.99122i 8.61662 + 0.521210i −2.49540 + 8.61513i 2.31769 + 0.473159i
2.12 −1.25430 − 0.0252629i −3.34492 + 2.51355i −2.42414 − 0.0976894i −7.94291 − 1.45559i 4.25902 − 3.06823i −3.57883 + 5.41491i 8.04716 + 0.486763i 2.36664 − 8.17059i 9.92598 + 2.02640i
2.13 −0.453646 − 0.00913694i 1.44207 − 1.08364i −3.79105 − 0.152774i 4.77979 + 0.875928i −0.664090 + 0.478415i 3.48787 − 5.27728i 3.53004 + 0.213528i −1.59868 + 5.51928i −2.16033 − 0.441034i
2.14 −0.273802 − 0.00551468i −2.64881 + 1.99045i −3.92182 − 0.158044i −2.06910 − 0.379177i 0.736226 − 0.530382i 4.45238 − 6.73663i 2.16636 + 0.131041i 0.550342 − 1.90000i 0.564433 + 0.115230i
2.15 −0.200611 − 0.00404053i 0.852849 − 0.640874i −3.95653 − 0.159443i 3.37132 + 0.617818i −0.173680 + 0.125120i −5.24689 + 7.93876i 1.59422 + 0.0964325i −2.18733 + 7.55153i −0.673827 − 0.137563i
2.16 −0.0881003 − 0.00177444i 2.65711 − 1.99669i −3.98900 − 0.160751i −6.22228 − 1.14028i −0.237635 + 0.171194i −3.01358 + 4.55968i 0.702976 + 0.0425222i 0.569504 − 1.96616i 0.546162 + 0.111500i
2.17 0.692541 + 0.0139486i −3.62315 + 2.72262i −3.51734 − 0.141744i 1.51837 + 0.278252i −2.54716 + 1.83499i 0.238697 − 0.361158i −5.19959 − 0.314517i 3.21060 − 11.0843i 1.04766 + 0.213880i
2.18 0.982116 + 0.0197809i −2.81182 + 2.11295i −3.03260 − 0.122209i 9.21096 + 1.68797i −2.80333 + 2.01954i 0.691982 − 1.04700i −6.89804 − 0.417254i 0.937846 − 3.23782i 9.01284 + 1.83998i
2.19 1.11545 + 0.0224665i 4.13892 − 3.11019i −2.75303 − 0.110943i 3.00078 + 0.549914i 4.68664 − 3.37629i 2.11360 − 3.19796i −7.52295 − 0.455055i 4.95337 − 17.1010i 3.33488 + 0.680820i
2.20 1.42167 + 0.0286341i 1.93850 − 1.45669i −1.97642 − 0.0796471i −8.64743 − 1.58470i 2.79762 − 2.01542i 6.31230 − 9.55076i −8.48501 − 0.513249i −0.868120 + 2.99710i −12.2484 − 2.50054i
See next 80 embeddings (of 1392 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.29
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.l odd 156 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.3.l.a ✓ 1392
169.l odd 156 1 inner 169.3.l.a ✓ 1392
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.3.l.a ✓ 1392 1.a even 1 1 trivial
169.3.l.a ✓ 1392 169.l odd 156 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(169, [\chi])\).