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Properties

Label 869.2.f.c
Level $869$
Weight $2$
Character orbit 869.f
Analytic conductor $6.939$
Analytic rank $0$
Dimension $176$
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] = [869,2,Mod(80,869)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(869, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("869.80");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 869 = 11 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 869.f (of order \(5\), degree \(4\), 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: \(6.93899993565\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(44\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 176 q - q^{2} - 9 q^{3} - 49 q^{4} - 19 q^{5} + 3 q^{6} + 3 q^{7} - 3 q^{8} - 71 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 176 q - q^{2} - 9 q^{3} - 49 q^{4} - 19 q^{5} + 3 q^{6} + 3 q^{7} - 3 q^{8} - 71 q^{9} + 8 q^{10} - 4 q^{11} + 82 q^{12} - 6 q^{13} - 26 q^{14} - 17 q^{15} - 63 q^{16} + 3 q^{17} - 3 q^{19} - 41 q^{20} + 2 q^{21} + 78 q^{23} + 11 q^{24} - 51 q^{25} - 17 q^{26} - 15 q^{27} - 34 q^{28} - 24 q^{30} + 17 q^{31} - 22 q^{32} + 11 q^{33} + 62 q^{34} + 21 q^{35} - 58 q^{36} + 25 q^{37} - 60 q^{38} - 22 q^{39} - 75 q^{40} + 13 q^{41} + 157 q^{42} - 20 q^{43} - 26 q^{44} + 228 q^{45} - 32 q^{46} - 48 q^{47} + 64 q^{48} - 55 q^{49} - 34 q^{50} - 77 q^{51} - 10 q^{52} - 30 q^{53} - 52 q^{54} - 15 q^{55} + 352 q^{56} - 13 q^{57} - 18 q^{58} - 5 q^{59} - 56 q^{60} - 25 q^{61} - 37 q^{62} + 38 q^{63} + 33 q^{64} - 20 q^{65} - 89 q^{66} + 30 q^{67} + 41 q^{68} - 41 q^{69} + 184 q^{70} - 37 q^{71} - 114 q^{72} - 51 q^{73} - 8 q^{74} + 55 q^{75} - 96 q^{76} - 26 q^{77} + 114 q^{78} - 44 q^{79} - 116 q^{80} - 27 q^{81} - 51 q^{82} + 29 q^{83} - 154 q^{84} + 31 q^{85} + 93 q^{86} - 118 q^{87} - 107 q^{88} + 430 q^{89} - 72 q^{90} - 56 q^{91} + 86 q^{92} - 47 q^{93} + 25 q^{94} + 34 q^{95} - 32 q^{96} + 46 q^{97} - 106 q^{98} - 121 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
80.1 −2.23682 + 1.62514i 0.950422 + 2.92510i 1.74422 − 5.36817i 1.68723 + 1.22585i −6.87962 − 4.99834i −0.471961 + 1.45255i 3.11376 + 9.58315i −5.22585 + 3.79680i −5.76620
80.2 −2.17412 + 1.57959i 0.480859 + 1.47993i 1.61367 − 4.96635i −0.923239 − 0.670773i −3.38314 − 2.45799i 0.228745 − 0.704006i 2.67562 + 8.23472i 0.468076 − 0.340077i 3.06678
80.3 −2.15416 + 1.56509i −1.02201 − 3.14542i 1.57286 − 4.84077i −2.94109 − 2.13683i 7.12443 + 5.17620i 1.30155 − 4.00577i 2.54240 + 7.82471i −6.42213 + 4.66595i 9.67989
80.4 −1.96008 + 1.42408i −0.839925 − 2.58502i 1.19587 − 3.68050i 2.88767 + 2.09802i 5.32759 + 3.87072i 0.668907 − 2.05868i 1.39996 + 4.30865i −3.54982 + 2.57909i −8.64781
80.5 −1.88315 + 1.36819i 0.417088 + 1.28367i 1.05628 − 3.25091i −1.85241 − 1.34586i −2.54174 − 1.84668i 0.738047 − 2.27148i 1.02012 + 3.13960i 0.953217 − 0.692553i 5.32976
80.6 −1.84890 + 1.34331i −0.169070 − 0.520343i 0.995933 − 3.06517i 0.795327 + 0.577839i 1.01157 + 0.734950i 1.19213 − 3.66901i 0.863638 + 2.65801i 2.18488 − 1.58741i −2.24670
80.7 −1.84338 + 1.33929i 0.171738 + 0.528556i 0.986307 − 3.03554i 2.30780 + 1.67671i −1.02447 − 0.744322i −0.642153 + 1.97634i 0.839126 + 2.58257i 2.17717 − 1.58181i −6.49976
80.8 −1.83778 + 1.33522i 1.02990 + 3.16969i 0.976569 − 3.00557i −2.96610 − 2.15500i −6.12497 − 4.45005i 0.566266 − 1.74279i 0.814455 + 2.50663i −6.55923 + 4.76556i 8.32845
80.9 −1.79384 + 1.30330i −0.430931 − 1.32627i 0.901237 − 2.77372i −3.17716 − 2.30834i 2.50155 + 1.81748i −0.277439 + 0.853871i 0.627950 + 1.93263i 0.853761 − 0.620294i 8.70780
80.10 −1.56876 + 1.13977i −0.614883 − 1.89241i 0.543902 − 1.67396i 2.65004 + 1.92537i 3.12153 + 2.26792i 0.205208 − 0.631566i −0.143750 − 0.442417i −0.776100 + 0.563869i −6.35178
80.11 −1.37837 + 1.00144i 0.723136 + 2.22558i 0.278979 − 0.858610i −3.00674 − 2.18452i −3.22555 − 2.34350i −1.51429 + 4.66051i −0.577667 − 1.77788i −2.00325 + 1.45544i 6.33208
80.12 −1.23863 + 0.899918i −0.0623386 − 0.191859i 0.106320 − 0.327218i 0.599186 + 0.435334i 0.249871 + 0.181542i −0.886010 + 2.72686i −0.783451 − 2.41121i 2.39413 − 1.73944i −1.13394
80.13 −1.09001 + 0.791942i 0.0538291 + 0.165669i −0.0570739 + 0.175655i −0.717310 − 0.521156i −0.189875 − 0.137952i −1.19144 + 3.66687i −0.909595 − 2.79945i 2.40250 − 1.74552i 1.19460
80.14 −1.00345 + 0.729049i −0.590941 − 1.81873i −0.142634 + 0.438983i −1.36163 − 0.989283i 1.91892 + 1.39418i 1.52616 − 4.69703i −0.943482 − 2.90374i −0.531512 + 0.386166i 2.08757
80.15 −0.815640 + 0.592597i 0.612455 + 1.88494i −0.303937 + 0.935421i −0.132713 − 0.0964216i −1.61655 − 1.17450i 1.27900 − 3.93635i −0.929518 − 2.86076i −0.750858 + 0.545531i 0.165385
80.16 −0.679625 + 0.493776i 0.893537 + 2.75002i −0.399959 + 1.23095i 1.36606 + 0.992504i −1.96517 − 1.42778i −0.921006 + 2.83456i −0.855178 − 2.63197i −4.33717 + 3.15114i −1.41849
80.17 −0.657155 + 0.477451i −0.506417 − 1.55859i −0.414140 + 1.27459i −1.86235 − 1.35308i 1.07695 + 0.782448i 0.0885070 − 0.272396i −0.838424 − 2.58040i 0.254299 − 0.184759i 1.86988
80.18 −0.580330 + 0.421635i −0.920314 − 2.83244i −0.459026 + 1.41274i −3.27308 − 2.37803i 1.72834 + 1.25571i −1.33987 + 4.12369i −0.772605 − 2.37784i −4.74866 + 3.45011i 2.90213
80.19 −0.494595 + 0.359344i 0.905500 + 2.78684i −0.502538 + 1.54665i −1.85393 − 1.34696i −1.44929 − 1.05297i 0.600837 − 1.84919i −0.685065 − 2.10841i −4.51952 + 3.28362i 1.40097
80.20 −0.267953 + 0.194680i −0.245649 − 0.756030i −0.584135 + 1.79778i −0.491858 − 0.357356i 0.213006 + 0.154758i 0.0997449 − 0.306983i −0.398169 − 1.22544i 1.91581 − 1.39192i 0.201365
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 80.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 869.2.f.c ✓ 176
11.c even 5 1 inner 869.2.f.c ✓ 176
11.c even 5 1 9559.2.a.x 88
11.d odd 10 1 9559.2.a.w 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
869.2.f.c ✓ 176 1.a even 1 1 trivial
869.2.f.c ✓ 176 11.c even 5 1 inner
9559.2.a.w 88 11.d odd 10 1
9559.2.a.x 88 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{176} + T_{2}^{175} + 69 T_{2}^{174} + 71 T_{2}^{173} + 2560 T_{2}^{172} + 2665 T_{2}^{171} + \cdots + 2364390625 \) acting on \(S_{2}^{\mathrm{new}}(869, [\chi])\). Copy content Toggle raw display