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Properties

Label 859.2.a.b
Level $859$
Weight $2$
Character orbit 859.a
Self dual yes
Analytic conductor $6.859$
Analytic rank $0$
Dimension $42$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [859,2,Mod(1,859)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(859, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("859.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 859 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 859.a (trivial)

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: yes
Analytic conductor: \(6.85914953363\)
Analytic rank: \(0\)
Dimension: \(42\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42 q + 10 q^{2} + q^{3} + 48 q^{4} + 21 q^{5} + q^{6} + 2 q^{7} + 27 q^{8} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q + 10 q^{2} + q^{3} + 48 q^{4} + 21 q^{5} + q^{6} + 2 q^{7} + 27 q^{8} + 63 q^{9} + 3 q^{10} + 22 q^{11} - 8 q^{12} + 14 q^{13} + 26 q^{14} + 7 q^{15} + 52 q^{16} + 30 q^{17} + 19 q^{18} + q^{19} + 29 q^{20} + 49 q^{21} + 2 q^{22} + 9 q^{23} - 3 q^{24} + 55 q^{25} + 8 q^{26} - 17 q^{27} + 107 q^{29} - 11 q^{30} - 3 q^{31} + 64 q^{32} - 7 q^{33} - 2 q^{34} + 18 q^{35} + 65 q^{36} + 15 q^{37} + 10 q^{38} + 17 q^{39} - 10 q^{40} + 44 q^{41} - 21 q^{42} - 2 q^{43} + 39 q^{44} + 38 q^{45} - q^{46} + 14 q^{47} - 51 q^{48} + 48 q^{49} + 52 q^{50} + 27 q^{51} - 25 q^{52} + 89 q^{53} - 24 q^{54} - 23 q^{55} + 54 q^{56} + q^{57} - q^{58} + 29 q^{59} - 14 q^{60} + 42 q^{61} - 18 q^{62} - 18 q^{63} + 35 q^{64} + 44 q^{65} - 46 q^{66} - 14 q^{67} + 49 q^{68} + 53 q^{69} - 42 q^{70} + 18 q^{71} + 6 q^{72} - 11 q^{73} + 86 q^{74} - 35 q^{75} - 32 q^{76} + 56 q^{77} - 47 q^{78} + 33 q^{79} - 3 q^{80} + 82 q^{81} - 29 q^{82} + 5 q^{83} + 9 q^{84} + 4 q^{85} + 18 q^{86} - 22 q^{87} - 20 q^{88} + 63 q^{89} - 89 q^{90} - 42 q^{91} - 21 q^{92} - 35 q^{93} - 27 q^{94} + 41 q^{95} - 101 q^{96} - q^{97} - 39 q^{98} - 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.63139 −1.16127 4.92422 −1.01415 3.05576 −3.81074 −7.69476 −1.65144 2.66863
1.2 −2.50833 −2.75356 4.29173 2.56198 6.90685 −3.49225 −5.74842 4.58211 −6.42630
1.3 −2.44167 2.53203 3.96174 2.18158 −6.18236 2.64770 −4.78991 3.41115 −5.32670
1.4 −2.39243 0.929930 3.72373 3.81463 −2.22480 −1.17880 −4.12391 −2.13523 −9.12624
1.5 −2.19990 3.00238 2.83955 −0.915767 −6.60494 0.370418 −1.84693 6.01431 2.01459
1.6 −2.13988 −3.02925 2.57907 −1.54899 6.48223 1.60061 −1.23914 6.17638 3.31464
1.7 −1.95205 −0.352561 1.81051 0.588410 0.688218 4.42108 0.369892 −2.87570 −1.14861
1.8 −1.86825 0.805919 1.49036 −1.84284 −1.50566 −3.09719 0.952140 −2.35049 3.44289
1.9 −1.66068 −0.868667 0.757869 −0.869159 1.44258 0.868049 2.06279 −2.24542 1.44340
1.10 −1.48888 0.831203 0.216773 −4.09720 −1.23756 −3.04547 2.65502 −2.30910 6.10025
1.11 −1.30000 1.81233 −0.310001 2.95498 −2.35603 2.52751 3.00300 0.284549 −3.84147
1.12 −1.26489 −3.36123 −0.400048 0.315487 4.25160 −3.03106 3.03580 8.29788 −0.399057
1.13 −1.24119 2.63551 −0.459453 3.14475 −3.27117 −0.940601 3.05264 3.94593 −3.90322
1.14 −1.06408 −2.69968 −0.867728 3.52398 2.87269 2.11982 3.05150 4.28829 −3.74981
1.15 −0.618382 −0.410722 −1.61760 3.41550 0.253983 −5.12644 2.23706 −2.83131 −2.11208
1.16 −0.353447 −0.654623 −1.87507 1.93614 0.231375 4.76193 1.36963 −2.57147 −0.684323
1.17 −0.327352 2.93975 −1.89284 −0.866451 −0.962334 −3.05348 1.27433 5.64214 0.283635
1.18 −0.140862 −1.32420 −1.98016 −1.96304 0.186529 −3.48893 0.560653 −1.24650 0.276518
1.19 −0.0951469 3.39816 −1.99095 1.26109 −0.323325 4.51330 0.379726 8.54752 −0.119989
1.20 −0.0467311 1.76352 −1.99782 −2.63357 −0.0824112 2.66028 0.186822 0.110005 0.123069
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.42
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(859\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 859.2.a.b ✓ 42
3.b odd 2 1 7731.2.a.f 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
859.2.a.b ✓ 42 1.a even 1 1 trivial
7731.2.a.f 42 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{42} - 10 T_{2}^{41} - 16 T_{2}^{40} + 471 T_{2}^{39} - 651 T_{2}^{38} - 9662 T_{2}^{37} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(859))\). Copy content Toggle raw display