↑

Properties

Label 869.2.f.b
Level $869$
Weight $2$
Character orbit 869.f
Analytic conductor $6.939$
Analytic rank $0$
Dimension $132$
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] = [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: \(132\)
Relative dimension: \(33\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 132 q + 2 q^{2} + 4 q^{3} - 32 q^{4} + 16 q^{5} - 4 q^{6} + 8 q^{7} + 2 q^{8} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 132 q + 2 q^{2} + 4 q^{3} - 32 q^{4} + 16 q^{5} - 4 q^{6} + 8 q^{7} + 2 q^{8} - 13 q^{9} + 2 q^{10} + 5 q^{11} - 60 q^{12} - 6 q^{13} + 17 q^{14} + 2 q^{15} - 32 q^{16} + 5 q^{17} - q^{18} - 6 q^{19} + 20 q^{20} + 28 q^{21} + 17 q^{22} - 68 q^{23} + 21 q^{24} + 23 q^{25} - 15 q^{26} + 28 q^{27} - q^{28} + 18 q^{29} - 25 q^{30} + 86 q^{32} + 14 q^{33} - 70 q^{34} + 31 q^{35} + 48 q^{36} - 35 q^{37} - 13 q^{38} + 7 q^{39} + 80 q^{40} - 5 q^{41} - 145 q^{42} - 16 q^{43} + 16 q^{44} - 182 q^{45} - 30 q^{46} + 29 q^{47} - 107 q^{48} + 23 q^{49} + 68 q^{50} + 21 q^{51} - 23 q^{53} + 106 q^{54} + 26 q^{55} - 234 q^{56} - q^{57} + 13 q^{58} + 42 q^{59} - 7 q^{60} - q^{61} + 93 q^{62} + 18 q^{63} - 96 q^{64} + 14 q^{65} + 86 q^{66} - 32 q^{67} + 50 q^{68} + 80 q^{69} - 145 q^{70} + 54 q^{71} + 68 q^{72} + 9 q^{73} - 13 q^{74} + 19 q^{75} + 148 q^{76} + 4 q^{77} - 56 q^{78} + 33 q^{79} + 68 q^{80} - 9 q^{81} + 45 q^{82} - 15 q^{83} + 67 q^{84} - 8 q^{85} - 94 q^{86} + 18 q^{87} + 14 q^{88} - 410 q^{89} - 33 q^{90} + 52 q^{91} - 56 q^{92} + 83 q^{93} + 88 q^{94} + 15 q^{95} - 34 q^{96} + 8 q^{97} + 110 q^{98} - 25 q^{99}+O(q^{100}) \) 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 −2.22723 + 1.61818i −0.270211 − 0.831625i 1.72402 − 5.30598i 0.944566 + 0.686267i 1.94754 + 1.41497i 0.344890 − 1.06146i 3.04478 + 9.37086i 1.80847 − 1.31393i −3.21426
80.2 −2.10113 + 1.52656i −0.690243 − 2.12435i 1.46634 − 4.51292i 0.761617 + 0.553347i 4.69325 + 3.40984i −1.51847 + 4.67337i 2.20316 + 6.78064i −1.60937 + 1.16928i −2.44498
80.3 −2.08118 + 1.51207i −0.334023 − 1.02802i 1.42694 − 4.39167i −1.58874 − 1.15429i 2.24960 + 1.63443i −0.507614 + 1.56228i 2.08090 + 6.40435i 1.48180 − 1.07659i 5.05183
80.4 −1.76635 + 1.28333i 0.483397 + 1.48774i 0.855025 − 2.63150i −0.902946 − 0.656029i −2.76311 − 2.00752i −0.361132 + 1.11145i 0.517430 + 1.59249i 0.447341 − 0.325012i 2.43682
80.5 −1.73229 + 1.25858i 0.586333 + 1.80455i 0.798763 − 2.45834i 2.64396 + 1.92095i −3.28687 − 2.38805i 1.14097 − 3.51155i 0.386982 + 1.19101i −0.485551 + 0.352774i −6.99778
80.6 −1.55479 + 1.12962i 0.846694 + 2.60586i 0.523298 − 1.61054i 2.03802 + 1.48071i −4.26007 − 3.09512i −0.395616 + 1.21758i −0.182067 − 0.560345i −3.64654 + 2.64937i −4.84134
80.7 −1.49463 + 1.08591i −0.866285 − 2.66615i 0.436674 − 1.34395i −0.866342 − 0.629434i 4.18997 + 3.04419i −0.912579 + 2.80863i −0.335054 − 1.03119i −3.93086 + 2.85594i 1.97837
80.8 −1.45998 + 1.06073i −0.144822 − 0.445715i 0.388339 − 1.19518i −2.30822 − 1.67702i 0.684222 + 0.497116i 0.0181886 − 0.0559788i −0.414515 − 1.27575i 2.24936 − 1.63426i 5.14883
80.9 −1.19749 + 0.870025i 0.542834 + 1.67067i 0.0589974 − 0.181575i −0.521962 − 0.379227i −2.10356 − 1.52833i 0.421766 − 1.29806i −0.827472 − 2.54670i −0.0694200 + 0.0504366i 0.954980
80.10 −0.959862 + 0.697381i −0.521647 − 1.60546i −0.183038 + 0.563334i 2.85034 + 2.07089i 1.62033 + 1.17724i −0.813585 + 2.50396i −0.950436 − 2.92514i 0.121655 − 0.0883874i −4.18013
80.11 −0.861765 + 0.626109i 0.236240 + 0.727072i −0.267408 + 0.822996i 2.67924 + 1.94658i −0.658809 − 0.478653i 0.706463 − 2.17427i −0.943172 − 2.90279i 1.95423 − 1.41983i −3.52765
80.12 −0.824346 + 0.598922i −0.110626 − 0.340471i −0.297196 + 0.914675i 1.56947 + 1.14029i 0.295109 + 0.214410i 0.697740 − 2.14742i −0.932571 − 2.87016i 2.32337 − 1.68803i −1.97673
80.13 −0.538809 + 0.391468i −1.02963 − 3.16888i −0.480966 + 1.48026i 1.38554 + 1.00666i 1.79529 + 1.30435i 0.556864 − 1.71385i −0.731939 − 2.25268i −6.55461 + 4.76221i −1.14062
80.14 −0.357765 + 0.259931i 0.525421 + 1.61708i −0.557603 + 1.71612i 2.86287 + 2.08000i −0.608307 − 0.441961i −1.03566 + 3.18742i −0.519892 − 1.60006i 0.0881680 − 0.0640578i −1.56489
80.15 −0.301228 + 0.218855i 0.753900 + 2.32027i −0.575193 + 1.77026i −1.33472 − 0.969733i −0.734898 − 0.533935i −0.0821648 + 0.252877i −0.444285 − 1.36737i −2.38822 + 1.73514i 0.614287
80.16 −0.290840 + 0.211308i 0.412314 + 1.26897i −0.578097 + 1.77920i −2.79880 − 2.03344i −0.388061 − 0.281943i −0.435954 + 1.34173i −0.430007 − 1.32342i 0.986761 − 0.716924i 1.24368
80.17 0.189108 − 0.137395i −0.822865 − 2.53252i −0.601149 + 1.85015i 1.15111 + 0.836327i −0.503567 − 0.365863i 0.309499 − 0.952539i 0.284985 + 0.877094i −3.30950 + 2.40449i 0.332591
80.18 0.350200 − 0.254435i 0.0759467 + 0.233740i −0.560131 + 1.72391i 1.25165 + 0.909374i 0.0860681 + 0.0625321i 0.479856 − 1.47684i 0.509993 + 1.56960i 2.37818 − 1.72785i 0.669703
80.19 0.448718 − 0.326013i 0.137192 + 0.422234i −0.522970 + 1.60954i −1.20574 − 0.876019i 0.199214 + 0.144738i −1.13592 + 3.49601i 0.632854 + 1.94772i 2.26759 − 1.64750i −0.826630
80.20 0.494646 − 0.359381i −0.539458 − 1.66028i −0.502514 + 1.54658i −1.61138 − 1.17074i −0.863516 − 0.627381i 1.00052 − 3.07928i 0.685121 + 2.10859i −0.0384715 + 0.0279512i −1.21781
See next 80 embeddings (of 132 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 80.33
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.b ✓ 132
11.c even 5 1 inner 869.2.f.b ✓ 132
11.c even 5 1 9559.2.a.u 66
11.d odd 10 1 9559.2.a.t 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
869.2.f.b ✓ 132 1.a even 1 1 trivial
869.2.f.b ✓ 132 11.c even 5 1 inner
9559.2.a.t 66 11.d odd 10 1
9559.2.a.u 66 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{132} - 2 T_{2}^{131} + 51 T_{2}^{130} - 96 T_{2}^{129} + 1407 T_{2}^{128} - 2570 T_{2}^{127} + \cdots + 153036657601 \) acting on \(S_{2}^{\mathrm{new}}(869, [\chi])\). Copy content Toggle raw display