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Properties

Label 1085.2.j.d
Level $1085$
Weight $2$
Character orbit 1085.j
Analytic conductor $8.664$
Analytic rank $0$
Dimension $42$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1085,2,Mod(156,1085)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1085, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1085.156");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1085 = 5 \cdot 7 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1085.j (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: \(8.66376861931\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(21\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 42 q + 4 q^{2} - 5 q^{3} - 22 q^{4} + 21 q^{5} + 12 q^{6} + 2 q^{7} - 24 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q + 4 q^{2} - 5 q^{3} - 22 q^{4} + 21 q^{5} + 12 q^{6} + 2 q^{7} - 24 q^{8} - 20 q^{9} - 4 q^{10} + 4 q^{11} - 21 q^{12} + 48 q^{13} + 4 q^{14} - 10 q^{15} - 20 q^{16} - 18 q^{17} + 16 q^{18} - 17 q^{19} - 44 q^{20} + 17 q^{21} + 18 q^{22} + 19 q^{23} - 21 q^{24} - 21 q^{25} - 16 q^{26} + 34 q^{27} - 8 q^{28} - 8 q^{29} + 6 q^{30} - 21 q^{31} + 18 q^{32} - 27 q^{33} - 4 q^{34} + q^{35} + 104 q^{36} + 19 q^{37} + 10 q^{38} - 21 q^{39} - 12 q^{40} + 10 q^{41} + 61 q^{42} - 30 q^{43} + 11 q^{44} + 20 q^{45} - 6 q^{46} - 4 q^{47} + 38 q^{48} + 10 q^{49} - 8 q^{50} - 35 q^{51} - 66 q^{52} + 19 q^{53} - 11 q^{54} + 8 q^{55} + 55 q^{56} - 54 q^{57} + q^{58} + q^{59} + 21 q^{60} - 60 q^{61} - 8 q^{62} + 34 q^{63} - 20 q^{64} + 24 q^{65} - 64 q^{66} - 2 q^{67} - 11 q^{68} + 36 q^{69} + 2 q^{70} - 2 q^{71} + 37 q^{72} - 51 q^{73} - 11 q^{74} - 5 q^{75} + 26 q^{76} + 74 q^{77} - 80 q^{78} - q^{79} + 20 q^{80} - 21 q^{81} + 2 q^{82} + 76 q^{83} + 115 q^{84} - 36 q^{85} - 26 q^{86} - 32 q^{87} - 5 q^{88} - 2 q^{89} + 32 q^{90} + 28 q^{91} - 172 q^{92} - 5 q^{93} - 67 q^{94} + 17 q^{95} + 56 q^{96} + 84 q^{97} + 72 q^{98} - 70 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
156.1 −1.33195 + 2.30700i −0.683827 − 1.18442i −2.54818 − 4.41358i 0.500000 − 0.866025i 3.64330 2.39166 − 1.13136i 8.24840 0.564760 − 0.978194i 1.33195 + 2.30700i
156.2 −1.13491 + 1.96571i −1.65498 − 2.86651i −1.57602 − 2.72975i 0.500000 − 0.866025i 7.51299 −0.890736 + 2.49130i 2.61493 −3.97793 + 6.88997i 1.13491 + 1.96571i
156.3 −1.02906 + 1.78238i −0.242586 − 0.420171i −1.11792 − 1.93629i 0.500000 − 0.866025i 0.998537 0.0964952 + 2.64399i 0.485366 1.38230 − 2.39422i 1.02906 + 1.78238i
156.4 −1.00349 + 1.73810i 0.976835 + 1.69193i −1.01400 − 1.75630i 0.500000 − 0.866025i −3.92099 −1.14469 − 2.38531i 0.0561989 −0.408414 + 0.707394i 1.00349 + 1.73810i
156.5 −0.770925 + 1.33528i −1.27701 − 2.21185i −0.188649 − 0.326750i 0.500000 − 0.866025i 3.93792 −1.93332 − 1.80618i −2.50196 −1.76151 + 3.05103i 0.770925 + 1.33528i
156.6 −0.710340 + 1.23034i −0.00548753 − 0.00950467i −0.00916471 − 0.0158737i 0.500000 − 0.866025i 0.0155920 1.73420 − 1.99814i −2.81532 1.49994 − 2.59797i 0.710340 + 1.23034i
156.7 −0.587774 + 1.01805i 0.989210 + 1.71336i 0.309044 + 0.535279i 0.500000 − 0.866025i −2.32573 −1.81747 + 1.92270i −3.07769 −0.457072 + 0.791672i 0.587774 + 1.01805i
156.8 −0.331045 + 0.573387i −0.386113 − 0.668768i 0.780818 + 1.35242i 0.500000 − 0.866025i 0.511284 2.47081 + 0.946095i −2.35813 1.20183 − 2.08164i 0.331045 + 0.573387i
156.9 −0.119785 + 0.207473i 0.851462 + 1.47478i 0.971303 + 1.68235i 0.500000 − 0.866025i −0.407968 0.663521 − 2.56120i −0.944527 0.0500242 − 0.0866444i 0.119785 + 0.207473i
156.10 0.0421816 − 0.0730607i −0.783476 − 1.35702i 0.996441 + 1.72589i 0.500000 − 0.866025i −0.132193 −2.58973 + 0.541556i 0.336853 0.272330 − 0.471690i −0.0421816 − 0.0730607i
156.11 0.0487364 − 0.0844139i 1.40478 + 2.43316i 0.995250 + 1.72382i 0.500000 − 0.866025i 0.273856 −2.57650 − 0.601355i 0.388965 −2.44684 + 4.23804i −0.0487364 − 0.0844139i
156.12 0.314226 − 0.544255i −0.762561 − 1.32079i 0.802524 + 1.39001i 0.500000 − 0.866025i −0.958466 2.64570 + 0.0171482i 2.26560 0.337001 − 0.583702i −0.314226 − 0.544255i
156.13 0.343434 − 0.594845i 0.712603 + 1.23426i 0.764106 + 1.32347i 0.500000 − 0.866025i 0.978928 2.26275 − 1.37112i 2.42342 0.484395 − 0.838996i −0.343434 − 0.594845i
156.14 0.552963 − 0.957759i −1.29834 − 2.24879i 0.388465 + 0.672841i 0.500000 − 0.866025i −2.87173 0.0293321 + 2.64559i 3.07108 −1.87137 + 3.24130i −0.552963 − 0.957759i
156.15 0.773945 − 1.34051i −1.14005 − 1.97462i −0.197982 − 0.342915i 0.500000 − 0.866025i −3.52934 −2.50486 − 0.851866i 2.48287 −1.09941 + 1.90424i −0.773945 − 1.34051i
156.16 0.881830 − 1.52737i 0.756156 + 1.30970i −0.555248 − 0.961717i 0.500000 − 0.866025i 2.66720 2.42433 + 1.05954i 1.56878 0.356456 − 0.617400i −0.881830 − 1.52737i
156.17 1.04051 − 1.80221i −0.253233 − 0.438612i −1.16531 − 2.01838i 0.500000 − 0.866025i −1.05396 0.106549 + 2.64360i −0.688024 1.37175 − 2.37593i −1.04051 − 1.80221i
156.18 1.17661 − 2.03795i 0.691083 + 1.19699i −1.76881 − 3.06368i 0.500000 − 0.866025i 3.25254 −1.78677 + 1.95127i −3.61837 0.544807 − 0.943634i −1.17661 − 2.03795i
156.19 1.22080 − 2.11448i 1.56664 + 2.71350i −1.98070 − 3.43066i 0.500000 − 0.866025i 7.65019 2.62870 + 0.299879i −4.78892 −3.40871 + 5.90405i −1.22080 − 2.11448i
156.20 1.30265 − 2.25626i −1.64921 − 2.85651i −2.39381 − 4.14619i 0.500000 − 0.866025i −8.59339 0.642033 − 2.56667i −7.26258 −3.93978 + 6.82391i −1.30265 − 2.25626i
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 156.21
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1085.2.j.d ✓ 42
7.c even 3 1 inner 1085.2.j.d ✓ 42
7.c even 3 1 7595.2.a.bg 21
7.d odd 6 1 7595.2.a.bf 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1085.2.j.d ✓ 42 1.a even 1 1 trivial
1085.2.j.d ✓ 42 7.c even 3 1 inner
7595.2.a.bf 21 7.d odd 6 1
7595.2.a.bg 21 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{42} - 4 T_{2}^{41} + 40 T_{2}^{40} - 120 T_{2}^{39} + 784 T_{2}^{38} - 2010 T_{2}^{37} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(1085, [\chi])\). Copy content Toggle raw display