↑

Properties

Label 169.3.j.a
Level $169$
Weight $3$
Character orbit 169.j
Analytic conductor $4.605$
Analytic rank $0$
Dimension $720$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,3,Mod(5,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(52))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.5");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 169.j (of order \(52\), 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: \(4.60491646769\)
Analytic rank: \(0\)
Dimension: \(720\)
Relative dimension: \(30\) over \(\Q(\zeta_{52})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{52}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 720 q - 22 q^{2} - 22 q^{3} - 26 q^{4} - 34 q^{5} - 10 q^{6} - 14 q^{7} - 62 q^{8} - 202 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 720 q - 22 q^{2} - 22 q^{3} - 26 q^{4} - 34 q^{5} - 10 q^{6} - 14 q^{7} - 62 q^{8} - 202 q^{9} - 26 q^{10} - 22 q^{11} - 26 q^{12} - 34 q^{13} - 38 q^{14} + 2 q^{15} + 242 q^{16} - 26 q^{17} - 58 q^{18} - 26 q^{19} - 42 q^{20} - 58 q^{21} - 108 q^{22} + 262 q^{24} - 26 q^{25} + 62 q^{26} + 2 q^{27} - 30 q^{28} - 54 q^{29} - 858 q^{30} - 66 q^{31} - 696 q^{32} + 50 q^{33} + 82 q^{34} - 2 q^{35} - 26 q^{36} - 66 q^{37} + 884 q^{38} + 922 q^{39} - 522 q^{40} - 58 q^{41} - 106 q^{42} - 26 q^{43} + 42 q^{44} - 212 q^{45} - 54 q^{46} + 368 q^{47} + 94 q^{48} - 26 q^{49} + 102 q^{50} - 26 q^{51} - 1276 q^{52} + 620 q^{53} - 178 q^{54} - 834 q^{55} - 26 q^{56} + 14 q^{57} + 114 q^{58} - 82 q^{59} + 1338 q^{60} + 254 q^{61} + 1404 q^{62} - 1392 q^{63} - 26 q^{64} - 82 q^{65} - 304 q^{66} - 358 q^{67} - 434 q^{68} + 208 q^{69} + 838 q^{70} - 310 q^{71} + 22 q^{72} - 26 q^{73} - 1308 q^{74} + 1664 q^{75} - 106 q^{76} - 26 q^{77} - 574 q^{78} - 78 q^{79} + 62 q^{80} - 262 q^{81} + 1274 q^{82} + 26 q^{83} + 158 q^{84} + 1522 q^{85} - 1562 q^{86} - 122 q^{87} - 442 q^{88} - 616 q^{89} - 26 q^{90} - 182 q^{91} + 646 q^{92} + 544 q^{93} - 2394 q^{94} - 26 q^{95} - 4064 q^{96} + 1914 q^{97} - 250 q^{98} - 178 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −3.89687 − 0.714127i 2.81830 + 1.47916i 10.9355 + 4.14730i 0.339194 − 5.60754i −9.92622 − 7.77670i −4.28826 − 2.59234i −26.0910 − 15.7726i 0.642307 + 0.930542i −5.32629 + 21.6096i
5.2 −3.78845 − 0.694260i −1.17333 − 0.615809i 10.1303 + 3.84193i −0.572829 + 9.46999i 4.01756 + 3.14756i 8.28718 + 5.00977i −22.5267 − 13.6179i −4.11511 − 5.96177i 8.74477 − 35.4789i
5.3 −3.42537 − 0.627723i −4.84261 − 2.54160i 7.59907 + 2.88195i 0.344854 − 5.70112i 14.9923 + 11.7457i 2.55543 + 1.54481i −12.2998 − 7.43551i 11.8786 + 17.2091i −4.75997 + 19.3120i
5.4 −3.10651 − 0.569289i −1.62486 − 0.852791i 5.58624 + 2.11858i 0.0346066 − 0.572115i 4.56215 + 3.57422i −3.36545 − 2.03449i −5.33658 − 3.22607i −3.19967 − 4.63553i −0.433204 + 1.75758i
5.5 −2.93123 − 0.537168i 5.07171 + 2.66184i 4.56351 + 1.73071i −0.310582 + 5.13453i −13.4365 − 10.5268i 3.68439 + 2.22729i −2.24597 − 1.35774i 13.5243 + 19.5933i 3.66850 − 14.8837i
5.6 −2.69019 − 0.492995i 2.56341 + 1.34538i 3.25399 + 1.23408i −0.00435722 + 0.0720334i −6.23279 − 4.88308i 3.00776 + 1.81826i 1.21674 + 0.735545i −0.351552 − 0.509311i 0.0472339 − 0.191635i
5.7 −2.68961 − 0.492890i 1.35223 + 0.709706i 3.25102 + 1.23295i −0.320821 + 5.30380i −3.28717 − 2.57534i −11.1706 − 6.75288i 1.22394 + 0.739895i −3.78774 − 5.48748i 3.47707 − 14.1070i
5.8 −2.57743 − 0.472331i 1.36177 + 0.714710i 2.67997 + 1.01638i 0.593540 − 9.81239i −3.17227 − 2.48532i 6.35016 + 3.83881i 2.54243 + 1.53695i −3.76899 − 5.46032i −6.16450 + 25.0104i
5.9 −2.20853 − 0.404728i −5.21056 − 2.73471i 0.973721 + 0.369284i −0.505290 + 8.35344i 10.4009 + 8.14855i −5.98887 − 3.62040i 5.68493 + 3.43666i 14.5587 + 21.0920i 4.49681 − 18.2443i
5.10 −2.19267 − 0.401821i −1.90733 − 1.00104i 0.906258 + 0.343698i −0.00411255 + 0.0679886i 3.77990 + 2.96136i 8.11874 + 4.90795i 5.78175 + 3.49519i −2.47676 − 3.58821i 0.0363367 − 0.147424i
5.11 −1.23702 − 0.226692i −2.71137 − 1.42303i −2.26124 − 0.857574i 0.283366 − 4.68460i 3.03142 + 2.37497i −6.51442 − 3.93811i 6.90778 + 4.17590i 0.213894 + 0.309879i −1.41249 + 5.73071i
5.12 −0.992467 − 0.181876i 3.21951 + 1.68973i −2.78815 − 1.05741i 0.233707 − 3.86364i −2.88794 − 2.26255i −10.1385 − 6.12897i 6.02875 + 3.64450i 2.39748 + 3.47334i −0.934651 + 3.79203i
5.13 −0.721008 − 0.132130i −2.08690 − 1.09529i −3.23767 − 1.22789i −0.294570 + 4.86982i 1.35995 + 1.06546i 1.95877 + 1.18412i 4.68135 + 2.82997i −1.95708 − 2.83532i 0.855834 − 3.47226i
5.14 −0.701607 − 0.128574i 1.48553 + 0.779668i −3.26434 − 1.23800i −0.293027 + 4.84432i −0.942016 − 0.738022i 3.36050 + 2.03150i 4.57280 + 2.76435i −3.51366 − 5.09041i 0.828446 − 3.36114i
5.15 −0.667348 − 0.122296i 3.56324 + 1.87013i −3.30967 − 1.25519i 0.182452 − 3.01630i −2.14921 − 1.68380i 4.18348 + 2.52900i 4.37765 + 2.64638i 4.08670 + 5.92061i −0.490640 + 1.99061i
5.16 0.0434563 + 0.00796367i −4.38060 − 2.29911i −3.73824 − 1.41773i 0.460887 − 7.61938i −0.172055 − 0.134797i 4.16575 + 2.51828i −0.302394 − 0.182803i 8.79112 + 12.7361i 0.0807067 − 0.327440i
5.17 0.660471 + 0.121036i 4.71303 + 2.47359i −3.31849 − 1.25854i −0.433924 + 7.17362i 2.81343 + 2.20418i −2.64517 − 1.59906i −4.33797 − 2.62239i 10.9814 + 15.9093i −1.15486 + 4.68545i
5.18 0.819752 + 0.150225i 0.116058 + 0.0609121i −3.09064 − 1.17212i 0.237708 − 3.92979i 0.0859885 + 0.0673677i 2.52537 + 1.52664i −5.21032 − 3.14975i −5.10282 − 7.39272i 0.785215 − 3.18574i
5.19 1.03775 + 0.190175i −0.355968 − 0.186826i −2.69930 − 1.02371i −0.450909 + 7.45441i −0.333877 − 0.261576i −8.43109 − 5.09677i −6.21803 − 3.75893i −5.02077 − 7.27385i −1.88557 + 7.65007i
5.20 1.21813 + 0.223230i −4.33208 − 2.27365i −2.30606 − 0.874574i −0.247110 + 4.08522i −4.76948 − 3.73665i 8.44262 + 5.10374i −6.85309 − 4.14284i 8.48485 + 12.2924i −1.21296 + 4.92116i
See next 80 embeddings (of 720 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.j odd 52 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.3.j.a ✓ 720
169.j odd 52 1 inner 169.3.j.a ✓ 720
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.3.j.a ✓ 720 1.a even 1 1 trivial
169.3.j.a ✓ 720 169.j odd 52 1 inner

Hecke kernels

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