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Properties

Label 869.2.u.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 792 q + 3 q^{3} + 36 q^{4} + 8 q^{6} - 9 q^{7} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 792 q + 3 q^{3} + 36 q^{4} + 8 q^{6} - 9 q^{7} + 32 q^{9} + 2 q^{10} + 33 q^{11} + 140 q^{12} + 5 q^{13} - 67 q^{14} - 2 q^{15} + 38 q^{16} - 44 q^{17} - 43 q^{18} - 5 q^{19} + 2 q^{20} - 20 q^{21} + 6 q^{23} - 116 q^{24} + 135 q^{25} - 17 q^{26} - 57 q^{27} + 3 q^{28} + 10 q^{29} - 26 q^{30} - q^{31} - 6 q^{33} - 2 q^{34} - 53 q^{35} + 76 q^{36} - 6 q^{37} + 74 q^{38} + 71 q^{39} - 12 q^{40} - 67 q^{41} + 27 q^{42} - 5 q^{43} + 36 q^{44} + 25 q^{45} - 24 q^{46} - 145 q^{48} + 22 q^{49} - 82 q^{50} + 79 q^{51} - 30 q^{52} + 32 q^{53} - 104 q^{54} - 291 q^{56} - 142 q^{57} - 138 q^{58} - 44 q^{59} - 442 q^{60} - 58 q^{61} - 46 q^{62} - 70 q^{63} - 80 q^{64} + 134 q^{65} + 8 q^{66} + 25 q^{67} - 95 q^{68} - 220 q^{69} - 80 q^{70} + 40 q^{71} - 162 q^{72} - 37 q^{73} + 4 q^{74} + 83 q^{75} - 9 q^{76} + 4 q^{77} + 368 q^{78} + 31 q^{79} - 166 q^{80} + 25 q^{81} + 197 q^{82} - 8 q^{83} + 148 q^{84} + 100 q^{85} - 255 q^{86} - 144 q^{87} + 14 q^{89} - 227 q^{90} + 96 q^{91} + 79 q^{92} + 37 q^{93} - 12 q^{94} - 4 q^{95} - 300 q^{96} - 83 q^{97} + 21 q^{98} + 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1 −1.70835 − 2.09235i 0.139652 − 0.0883105i −1.05941 + 5.18936i 2.03775 − 0.868204i −0.423351 − 0.141335i 0.141368 + 3.50802i 7.88425 − 4.13797i −1.27437 + 2.68569i −5.29778 − 2.78049i
45.2 −1.63742 − 2.00547i −2.64987 + 1.67568i −0.940735 + 4.60802i 3.49433 − 1.48879i 7.69948 + 2.57046i −0.135539 − 3.36338i 6.19671 − 3.25229i 2.92785 − 6.17031i −8.70742 − 4.57001i
45.3 −1.62802 − 1.99396i 1.85414 − 1.17249i −0.925381 + 4.53281i −2.77818 + 1.18367i −5.35646 − 1.78825i −0.172403 − 4.27814i 5.98617 − 3.14178i 0.777026 − 1.63755i 6.88312 + 3.61254i
45.4 −1.55429 − 1.90366i −0.761070 + 0.481272i −0.808052 + 3.95810i −1.49101 + 0.635260i 2.09910 + 0.700783i −0.115798 − 2.87350i 4.43865 − 2.32958i −0.938472 + 1.97779i 3.52678 + 1.85100i
45.5 −1.37921 − 1.68923i 1.47374 − 0.931936i −0.551217 + 2.70004i −1.22513 + 0.521981i −3.60686 − 1.20415i 0.0338595 + 0.840214i 1.45929 − 0.765895i 0.0173239 − 0.0365092i 2.57147 + 1.34961i
45.6 −1.35385 − 1.65816i 1.92507 − 1.21734i −0.516545 + 2.53020i 2.52057 − 1.07392i −4.62480 − 1.54399i 0.0540547 + 1.34135i 1.10390 − 0.579370i 0.937906 − 1.97660i −5.19319 − 2.72560i
45.7 −1.18753 − 1.45446i −2.62423 + 1.65946i −0.305172 + 1.49483i −3.25235 + 1.38570i 5.52996 + 1.84617i −0.0956318 − 2.37308i −0.788628 + 0.413904i 2.84668 − 5.99926i 5.87770 + 3.08485i
45.8 −1.08855 − 1.33323i −0.715522 + 0.452469i −0.192516 + 0.943005i −0.967398 + 0.412170i 1.38213 + 0.461423i 0.127230 + 3.15717i −1.58125 + 0.829902i −0.978834 + 2.06285i 1.60258 + 0.841099i
45.9 −1.06131 − 1.29987i −1.70680 + 1.07931i −0.163230 + 0.799554i 0.964537 − 0.410951i 3.21441 + 1.07313i −0.0220691 − 0.547639i −1.75923 + 0.923313i 0.462162 − 0.973985i −1.55786 − 0.817627i
45.10 −0.989523 − 1.21195i 2.74973 − 1.73883i −0.0896055 + 0.438917i 1.45260 − 0.618896i −4.82829 − 1.61192i −0.0670673 − 1.66426i −2.15016 + 1.12849i 3.25144 − 6.85226i −2.18745 − 1.14806i
45.11 −0.589305 − 0.721768i 0.966185 − 0.610979i 0.226383 − 1.10890i −3.08224 + 1.31322i −1.01036 − 0.337309i −0.0287526 − 0.713489i −2.58389 + 1.35613i −0.725858 + 1.52971i 2.76422 + 1.45077i
45.12 −0.519627 − 0.636427i 0.940105 − 0.594486i 0.265024 − 1.29817i 3.81091 − 1.62368i −0.866851 − 0.289397i −0.0915270 − 2.27122i −2.41891 + 1.26954i −0.755694 + 1.59259i −3.01360 − 1.58166i
45.13 −0.501586 − 0.614331i −1.78998 + 1.13191i 0.274237 − 1.34330i −3.44244 + 1.46669i 1.59320 + 0.531887i 0.175382 + 4.35206i −2.36728 + 1.24244i 0.636712 − 1.34184i 2.62771 + 1.37913i
45.14 −0.441060 − 0.540201i 0.242425 − 0.153300i 0.302769 − 1.48306i −0.100237 + 0.0427070i −0.189737 − 0.0633435i −0.0456557 − 1.13294i −2.16970 + 1.13875i −1.25081 + 2.63602i 0.0672810 + 0.0353118i
45.15 −0.404342 − 0.495229i −0.972600 + 0.615035i 0.318292 − 1.55910i 1.41891 − 0.604543i 0.697846 + 0.232975i 0.0331761 + 0.823256i −2.03301 + 1.06700i −0.718395 + 1.51398i −0.873113 − 0.458245i
45.16 −0.149598 − 0.183224i 2.32033 − 1.46729i 0.388860 − 1.90476i −1.57637 + 0.671630i −0.615960 − 0.205638i −0.103164 − 2.56000i −0.826061 + 0.433551i 1.94492 − 4.09883i 0.358881 + 0.188355i
45.17 0.109779 + 0.134455i −2.38662 + 1.50921i 0.394025 − 1.93006i −0.704852 + 0.300309i −0.464922 − 0.155214i −0.171178 − 4.24775i 0.610155 − 0.320234i 2.13218 − 4.49347i −0.117756 − 0.0618032i
45.18 0.191585 + 0.234649i 1.52830 − 0.966439i 0.381696 − 1.86967i 1.83687 − 0.782616i 0.519574 + 0.173459i 0.123405 + 3.06227i 1.04830 − 0.550192i 0.115621 − 0.243666i 0.535557 + 0.281082i
45.19 0.297062 + 0.363835i −0.559223 + 0.353631i 0.355921 − 1.74342i −3.77264 + 1.60737i −0.294787 − 0.0984145i 0.00978448 + 0.242799i 1.57185 − 0.824972i −1.09840 + 2.31483i −1.70553 − 0.895129i
45.20 0.356487 + 0.436617i −2.63244 + 1.66466i 0.336500 − 1.64828i −0.791796 + 0.337353i −1.66525 − 0.555942i 0.173997 + 4.31768i 1.83783 − 0.964566i 2.87260 − 6.05387i −0.429559 − 0.225450i
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.b ✓ 792
79.g even 39 1 inner 869.2.u.b ✓ 792
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
869.2.u.b ✓ 792 1.a even 1 1 trivial
869.2.u.b ✓ 792 79.g even 39 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{792} - 51 T_{2}^{790} + 1171 T_{2}^{788} - 14492 T_{2}^{786} + 67889 T_{2}^{784} + \cdots + 20\!\cdots\!61 \) acting on \(S_{2}^{\mathrm{new}}(869, [\chi])\). Copy content Toggle raw display