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Properties

Label 869.2.e.a
Level $869$
Weight $2$
Character orbit 869.e
Analytic conductor $6.939$
Analytic rank $0$
Dimension $66$
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(23,869)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(869, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("869.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 869 = 11 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 869.e (of order \(3\), degree \(2\), 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: \(66\)
Relative dimension: \(33\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 3 q^{3} - 36 q^{4} - 8 q^{6} - 4 q^{7} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q - 3 q^{3} - 36 q^{4} - 8 q^{6} - 4 q^{7} - 32 q^{9} - 2 q^{10} - 33 q^{11} + 68 q^{12} - 5 q^{13} + 2 q^{14} + 2 q^{15} - 38 q^{16} + 44 q^{17} - 22 q^{18} + 5 q^{19} - 2 q^{20} + 20 q^{21} - 6 q^{23} - 40 q^{24} - 31 q^{25} + 17 q^{26} + 18 q^{27} - 3 q^{28} - 10 q^{29} + 26 q^{30} + q^{31} + 6 q^{33} + 2 q^{34} + 14 q^{35} - 76 q^{36} - 20 q^{37} + 30 q^{38} - 19 q^{39} + 12 q^{40} + 2 q^{41} - 53 q^{42} + 5 q^{43} - 36 q^{44} - 25 q^{45} - 80 q^{46} - 13 q^{47} - 63 q^{48} - 35 q^{49} + 30 q^{50} + 25 q^{51} - 22 q^{52} - 32 q^{53} - 21 q^{56} - 14 q^{57} + 60 q^{58} - 8 q^{59} - 65 q^{60} + 58 q^{61} + 46 q^{62} - 21 q^{63} + 80 q^{64} + 126 q^{65} - 8 q^{66} - 38 q^{67} - 48 q^{68} + 90 q^{69} - 24 q^{70} + 12 q^{71} + 6 q^{72} - 15 q^{73} + 9 q^{74} - 83 q^{75} + 74 q^{76} - 4 q^{77} - 56 q^{78} - 31 q^{79} + 36 q^{80} - 25 q^{81} - 93 q^{82} + 8 q^{83} - 18 q^{84} + 4 q^{85} + 47 q^{86} + 66 q^{87} + 38 q^{89} + 97 q^{90} + 8 q^{91} - 53 q^{92} - 24 q^{93} + 12 q^{94} + 56 q^{95} + 222 q^{96} - 8 q^{97} - 21 q^{98} - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −1.36330 − 2.36130i −1.27687 − 2.21160i −2.71716 + 4.70626i −1.34179 + 2.32405i −3.48150 + 6.03013i 0.936173 − 1.62150i 9.36398 −1.76078 + 3.04976i 7.31703
23.2 −1.30604 − 2.26213i −1.52976 − 2.64962i −2.41149 + 4.17682i 1.91283 − 3.31312i −3.99585 + 6.92102i −2.36024 + 4.08806i 7.37385 −3.18031 + 5.50847i −9.99294
23.3 −1.26559 − 2.19207i 0.828059 + 1.43424i −2.20343 + 3.81646i −0.234964 + 0.406970i 2.09597 − 3.63032i −0.417774 + 0.723606i 6.09222 0.128636 − 0.222805i 1.18947
23.4 −1.22662 − 2.12457i −0.187262 − 0.324347i −2.00919 + 3.48002i 1.26046 − 2.18319i −0.459398 + 0.795700i 1.14742 − 1.98739i 4.95156 1.42987 − 2.47660i −6.18444
23.5 −1.11805 − 1.93653i −0.637946 − 1.10495i −1.50009 + 2.59824i −0.426999 + 0.739584i −1.42652 + 2.47080i 1.40772 − 2.43823i 2.23653 0.686050 − 1.18827i 1.90963
23.6 −1.06830 − 1.85034i −0.707651 − 1.22569i −1.28251 + 2.22137i −1.64066 + 2.84170i −1.51196 + 2.61879i −2.34741 + 4.06584i 1.20721 0.498459 − 0.863356i 7.01083
23.7 −0.874088 − 1.51396i 1.58568 + 2.74647i −0.528060 + 0.914626i 0.119778 − 0.207461i 2.77204 − 4.80132i 1.79863 − 3.11532i −1.65007 −3.52875 + 6.11197i −0.418785
23.8 −0.855240 − 1.48132i 0.693703 + 1.20153i −0.462870 + 0.801714i −0.984601 + 1.70538i 1.18657 − 2.05519i 0.608438 − 1.05385i −1.83750 0.537551 − 0.931066i 3.36828
23.9 −0.781473 − 1.35355i 0.683968 + 1.18467i −0.221401 + 0.383478i 0.583426 − 1.01052i 1.06901 − 1.85157i −2.25054 + 3.89804i −2.43382 0.564375 − 0.977526i −1.82373
23.10 −0.749825 − 1.29874i 0.294930 + 0.510834i −0.124476 + 0.215599i 1.20283 − 2.08337i 0.442292 − 0.766073i 1.72405 − 2.98614i −2.62596 1.32603 − 2.29676i −3.60765
23.11 −0.704541 − 1.22030i −1.51428 − 2.62280i 0.00724389 − 0.0125468i 0.00233661 − 0.00404713i −2.13374 + 3.69575i −0.395989 + 0.685874i −2.83858 −3.08607 + 5.34523i −0.00658496
23.12 −0.478763 − 0.829242i 0.0109441 + 0.0189558i 0.541572 − 0.938030i −1.07644 + 1.86444i 0.0104793 − 0.0181506i −0.530055 + 0.918082i −2.95219 1.49976 − 2.59766i 2.06143
23.13 −0.323118 − 0.559657i −1.06585 − 1.84611i 0.791189 − 1.37038i 0.345089 − 0.597712i −0.688791 + 1.19302i 1.74486 − 3.02219i −2.31506 −0.772071 + 1.33727i −0.446019
23.14 −0.293106 − 0.507674i 1.32336 + 2.29213i 0.828178 − 1.43445i −0.404872 + 0.701258i 0.775770 − 1.34367i −1.42402 + 2.46647i −2.14340 −2.00258 + 3.46857i 0.474681
23.15 −0.256851 − 0.444879i 0.207046 + 0.358615i 0.868055 − 1.50352i 1.82127 − 3.15453i 0.106360 − 0.184221i −0.132236 + 0.229039i −1.91925 1.41426 − 2.44958i −1.87118
23.16 −0.189056 − 0.327455i −0.525550 − 0.910280i 0.928516 − 1.60824i −1.86353 + 3.22773i −0.198717 + 0.344188i −0.314928 + 0.545471i −1.45839 0.947594 − 1.64128i 1.40925
23.17 0.0357824 + 0.0619769i −0.885799 − 1.53425i 0.997439 − 1.72762i 1.70343 − 2.95042i 0.0633920 − 0.109798i −1.99459 + 3.45473i 0.285893 −0.0692790 + 0.119995i 0.243811
23.18 0.101838 + 0.176389i 1.11836 + 1.93706i 0.979258 − 1.69612i 1.44180 − 2.49728i −0.227784 + 0.394533i 0.912690 − 1.58083i 0.806257 −1.00146 + 1.73458i 0.587323
23.19 0.219951 + 0.380966i 1.50443 + 2.60574i 0.903243 − 1.56446i −1.53433 + 2.65754i −0.661800 + 1.14627i −1.11776 + 1.93602i 1.67448 −3.02660 + 5.24223i −1.34991
23.20 0.348422 + 0.603484i 0.329418 + 0.570568i 0.757204 − 1.31152i −1.73142 + 2.99891i −0.229553 + 0.397597i 1.80360 − 3.12392i 2.44899 1.28297 − 2.22217i −2.41306
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.33
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
79.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 869.2.e.a ✓ 66
79.c even 3 1 inner 869.2.e.a ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
869.2.e.a ✓ 66 1.a even 1 1 trivial
869.2.e.a ✓ 66 79.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{66} + 51 T_{2}^{64} + 1430 T_{2}^{62} + 27701 T_{2}^{60} + 409450 T_{2}^{58} - 2 T_{2}^{57} + \cdots + 912025 \) acting on \(S_{2}^{\mathrm{new}}(869, [\chi])\). Copy content Toggle raw display