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Properties

Label 869.2.u.a
Level $869$
Weight $2$
Character orbit 869.u
Analytic conductor $6.939$
Analytic rank $0$
Dimension $792$
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(45,869)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(869, base_ring=CyclotomicField(78))
 
chi = DirichletCharacter(H, H._module([0, 64]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("869.45");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 869 = 11 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 869.u (of order \(39\), 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: \(6.93899993565\)
Analytic rank: \(0\)
Dimension: \(792\)
Relative dimension: \(33\) over \(\Q(\zeta_{39})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{39}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 792 q - q^{3} + 36 q^{4} + 11 q^{7} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 792 q - q^{3} + 36 q^{4} + 11 q^{7} + 32 q^{9} - 18 q^{10} - 33 q^{11} - 188 q^{12} - 7 q^{13} - 59 q^{14} + 14 q^{15} + 46 q^{16} + 20 q^{17} - 71 q^{18} - 5 q^{19} + 26 q^{20} - 20 q^{21} - 18 q^{23} + 116 q^{24} - 65 q^{25} + 7 q^{26} - 25 q^{27} - 13 q^{28} - 14 q^{29} + 4 q^{30} - 9 q^{31} - 20 q^{32} - 2 q^{33} - 14 q^{34} - 31 q^{35} + 12 q^{36} - 34 q^{37} - 134 q^{38} - 77 q^{39} + 60 q^{40} + 75 q^{41} - 93 q^{42} - 9 q^{43} - 36 q^{44} - 31 q^{45} - 140 q^{46} - 8 q^{47} + 195 q^{48} + 30 q^{49} - 82 q^{50} - 107 q^{51} + 58 q^{52} - 12 q^{53} - 120 q^{54} + 9 q^{56} - 134 q^{57} + 138 q^{58} - 24 q^{59} + 238 q^{60} + 2 q^{61} + 98 q^{62} - 84 q^{63} + 24 q^{64} - 106 q^{65} - 23 q^{67} + 97 q^{68} + 192 q^{69} - 92 q^{70} - 16 q^{71} + 162 q^{72} - 7 q^{73} + 28 q^{74} + 279 q^{75} + 201 q^{76} + 2 q^{77} - 8 q^{78} + 19 q^{79} - 150 q^{80} + 89 q^{81} + 9 q^{82} + 18 q^{83} + 150 q^{84} + 80 q^{85} + 453 q^{86} - 104 q^{87} + 46 q^{89} - 99 q^{90} - 93 q^{92} + 45 q^{93} + 76 q^{94} + 26 q^{95} + 48 q^{96} - 31 q^{97} + 83 q^{98} - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1 −1.70621 − 2.08973i −0.827212 + 0.523097i −1.05576 + 5.17147i −0.346083 + 0.147452i 2.50453 + 0.836135i −0.0532473 − 1.32132i 7.83075 − 4.10990i −0.875429 + 1.84493i 0.898626 + 0.471635i
45.2 −1.69757 − 2.07915i 2.05990 − 1.30260i −1.04106 + 5.09943i 3.34581 − 1.42552i −6.20512 − 2.07157i −0.175705 − 4.36008i 7.61638 − 3.99738i 1.26033 − 2.65609i −8.64362 − 4.53652i
45.3 −1.61949 − 1.98352i 2.23944 − 1.41614i −0.911541 + 4.46502i −0.651810 + 0.277710i −6.43570 − 2.14855i 0.0894491 + 2.21966i 5.79795 − 3.04300i 1.72357 − 3.63235i 1.60645 + 0.843129i
45.4 −1.51413 − 1.85448i 0.361036 − 0.228306i −0.746431 + 3.65626i −3.96564 + 1.68960i −0.970044 − 0.323848i 0.124772 + 3.09620i 3.67091 − 1.92664i −1.20785 + 2.54550i 9.13782 + 4.79590i
45.5 −1.46424 − 1.79336i −0.709687 + 0.448779i −0.672116 + 3.29224i 2.33085 − 0.993083i 1.84397 + 0.615609i −0.0213747 − 0.530408i 2.78831 − 1.46342i −0.983824 + 2.07337i −5.19388 − 2.72596i
45.6 −1.27099 − 1.55668i −1.89484 + 1.19823i −0.407789 + 1.99748i −1.96319 + 0.836436i 4.27359 + 1.42673i 0.0554407 + 1.37575i 0.0688377 − 0.0361288i 0.868604 − 1.83054i 3.79726 + 1.99296i
45.7 −1.05655 − 1.29404i 0.0815544 − 0.0515719i −0.158188 + 0.774858i 3.79338 − 1.61621i −0.152902 − 0.0510463i 0.119224 + 2.95851i −1.78862 + 0.938740i −1.28209 + 2.70194i −6.09933 − 3.20118i
45.8 −1.04474 − 1.27958i 1.75989 − 1.11289i −0.145782 + 0.714087i −0.734183 + 0.312806i −3.26267 − 1.08924i −0.0547117 − 1.35766i −1.85936 + 0.975866i 0.572627 − 1.20678i 1.16729 + 0.612643i
45.9 −1.02732 − 1.25824i 1.00759 − 0.637164i −0.127730 + 0.625663i 0.903690 − 0.385026i −1.83683 − 0.613225i 0.0816845 + 2.02698i −1.95816 + 1.02772i −0.676809 + 1.42634i −1.41284 − 0.741515i
45.10 −0.926982 − 1.13535i −0.557380 + 0.352466i −0.0296654 + 0.145311i −3.12398 + 1.33100i 0.916852 + 0.306090i −0.173418 − 4.30332i −2.40317 + 1.26128i −1.09964 + 2.31744i 4.40702 + 2.31298i
45.11 −0.861397 − 1.05502i −1.66235 + 1.05121i 0.0289891 − 0.141998i 2.55100 − 1.08688i 2.54100 + 0.848309i −0.147113 − 3.65057i −2.58678 + 1.35765i 0.372304 − 0.784615i −3.34411 − 1.75512i
45.12 −0.587446 − 0.719491i 0.620794 − 0.392566i 0.227477 − 1.11426i −1.59027 + 0.677553i −0.647131 − 0.216044i 0.0631771 + 1.56772i −2.58024 + 1.35421i −1.05480 + 2.22295i 1.42169 + 0.746163i
45.13 −0.566143 − 0.693399i −2.78733 + 1.76260i 0.239767 − 1.17446i 1.50121 − 0.639604i 2.80022 + 0.934849i 0.00739721 + 0.183560i −2.53537 + 1.33066i 3.37638 − 7.11558i −1.29340 − 0.678827i
45.14 −0.287388 − 0.351987i −2.06576 + 1.30631i 0.358749 − 1.75727i −1.89764 + 0.808509i 1.05348 + 0.351702i −0.0297392 − 0.737970i −1.52635 + 0.801091i 1.27484 − 2.68667i 0.829943 + 0.435588i
45.15 −0.260956 − 0.319613i −0.912508 + 0.577035i 0.365997 − 1.79277i 0.694407 − 0.295859i 0.422552 + 0.141069i 0.162395 + 4.02979i −1.39920 + 0.734359i −0.786377 + 1.65725i −0.275770 − 0.144735i
45.16 −0.234973 − 0.287789i 2.37301 − 1.50060i 0.372441 − 1.82434i 2.71802 − 1.15804i −0.989449 − 0.330326i 0.0983674 + 2.44096i −1.27049 + 0.666802i 2.09329 − 4.41152i −0.971932 − 0.510109i
45.17 −0.210675 − 0.258029i 1.76075 − 1.11343i 0.377856 − 1.85086i 0.983534 − 0.419045i −0.658244 − 0.219754i −0.176814 − 4.38760i −1.14709 + 0.602040i 0.574437 − 1.21060i −0.315332 − 0.165499i
45.18 0.283268 + 0.346940i 0.876128 − 0.554030i 0.359925 − 1.76303i −2.08711 + 0.889235i 0.440394 + 0.147025i 0.134803 + 3.34509i 1.50680 − 0.790829i −0.825426 + 1.73955i −0.899723 − 0.472211i
45.19 0.294182 + 0.360307i −0.293912 + 0.185858i 0.356773 − 1.74759i −1.84395 + 0.785633i −0.153429 − 0.0512223i −0.119489 − 2.96509i 1.55836 − 0.817892i −1.23424 + 2.60110i −0.825525 − 0.433269i
45.20 0.306423 + 0.375300i −0.978995 + 0.619079i 0.353096 − 1.72958i 3.26016 − 1.38902i −0.532328 − 0.177717i −0.0835394 − 2.07301i 1.61533 − 0.847788i −0.710905 + 1.49820i 1.52029 + 0.797910i
See next 80 embeddings (of 792 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.33
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
79.g even 39 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 869.2.u.a ✓ 792
79.g even 39 1 inner 869.2.u.a ✓ 792
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
869.2.u.a ✓ 792 1.a even 1 1 trivial
869.2.u.a ✓ 792 79.g even 39 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{792} - 51 T_{2}^{790} + 1169 T_{2}^{788} + 4 T_{2}^{787} - 14554 T_{2}^{786} + \cdots + 73\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(869, [\chi])\). Copy content Toggle raw display