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Properties

Label 539.2.j.b
Level $539$
Weight $2$
Character orbit 539.j
Analytic conductor $4.304$
Analytic rank $0$
Dimension $144$
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] = [539,2,Mod(78,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([6, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.78");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.j (of order \(7\), degree \(6\), 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: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(24\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 22 q^{4} + 2 q^{5} + 24 q^{6} - 2 q^{7} - 6 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 22 q^{4} + 2 q^{5} + 24 q^{6} - 2 q^{7} - 6 q^{8} - 16 q^{9} + 24 q^{11} + 25 q^{12} - 15 q^{13} + 35 q^{14} - 6 q^{15} - 54 q^{16} + 6 q^{17} + 20 q^{18} - 70 q^{19} + 16 q^{20} - 41 q^{21} + 12 q^{23} + 10 q^{24} - 48 q^{25} - 10 q^{26} - 6 q^{27} - 6 q^{28} + 28 q^{29} - 12 q^{30} - 164 q^{31} - 65 q^{32} + 80 q^{34} + 6 q^{36} + 43 q^{37} - 20 q^{38} + 34 q^{39} + 56 q^{40} - 7 q^{41} + 66 q^{42} - 14 q^{43} + 36 q^{44} + 24 q^{45} + 22 q^{46} + 3 q^{47} + 10 q^{48} - 46 q^{49} - 30 q^{50} - 122 q^{51} + 76 q^{52} + 21 q^{53} + 62 q^{54} - 2 q^{55} - 105 q^{56} + 27 q^{57} - 8 q^{58} + 18 q^{59} + 20 q^{60} + 36 q^{61} - 108 q^{62} + 9 q^{63} + 56 q^{64} - 94 q^{65} + 25 q^{66} - 30 q^{67} - 72 q^{68} + 6 q^{69} - 70 q^{70} - q^{71} + 125 q^{72} - 6 q^{73} - 112 q^{74} + 112 q^{75} + 85 q^{76} + 2 q^{77} + 87 q^{78} - 42 q^{79} - 320 q^{80} - 14 q^{81} + 87 q^{82} + 49 q^{83} + 86 q^{84} + 74 q^{85} + 63 q^{86} + 13 q^{87} + 6 q^{88} - 126 q^{89} - 152 q^{90} + 26 q^{91} - 41 q^{92} + 123 q^{93} + 73 q^{94} - 14 q^{95} + 75 q^{96} - 142 q^{97} - 189 q^{98} - 152 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
78.1 −2.40563 − 1.15849i −0.491424 − 2.15307i 3.19797 + 4.01013i 0.713147 + 3.12450i −1.31212 + 5.74879i −2.03490 − 1.69091i −1.85916 − 8.14550i −1.69130 + 0.814488i 1.90413 − 8.34256i
78.2 −2.29683 − 1.10610i 0.692481 + 3.03396i 2.80501 + 3.51737i 0.520261 + 2.27941i 1.76534 − 7.73444i −2.53372 + 0.761734i −1.41754 − 6.21067i −6.02246 + 2.90027i 1.32630 − 5.81089i
78.3 −2.24736 − 1.08227i 0.422502 + 1.85110i 2.63233 + 3.30084i −0.569563 − 2.49542i 1.05388 − 4.61735i 0.887235 + 2.49255i −1.23329 − 5.40339i −0.545158 + 0.262534i −1.42071 + 6.22453i
78.4 −2.10775 − 1.01504i −0.192048 − 0.841416i 2.16533 + 2.71524i −0.327654 − 1.43554i −0.449282 + 1.96843i 1.70912 − 2.01963i −0.766762 − 3.35940i 2.03181 − 0.978467i −0.766522 + 3.35835i
78.5 −1.72100 − 0.828791i −0.317465 − 1.39090i 1.02798 + 1.28904i 0.125358 + 0.549230i −0.606411 + 2.65686i −1.78992 + 1.94838i 0.149302 + 0.654133i 0.869079 − 0.418526i 0.239455 − 1.04912i
78.6 −1.66850 − 0.803507i 0.0994369 + 0.435661i 0.891290 + 1.11764i 0.930726 + 4.07778i 0.184147 − 0.806800i 1.99599 + 1.73667i 0.235087 + 1.02999i 2.52299 − 1.21501i 1.72361 − 7.55162i
78.7 −1.34039 − 0.645498i 0.379927 + 1.66457i 0.132998 + 0.166775i −0.832022 − 3.64533i 0.565225 − 2.47641i 2.15655 − 1.53274i 0.591481 + 2.59145i 0.0764638 − 0.0368230i −1.23782 + 5.42323i
78.8 −1.09546 − 0.527547i 0.340787 + 1.49308i −0.325248 − 0.407848i 0.274342 + 1.20197i 0.414353 − 1.81540i −1.52702 − 2.16060i 0.682252 + 2.98914i 0.589742 − 0.284005i 0.333564 − 1.46144i
78.9 −1.09426 − 0.526968i −0.705374 − 3.09044i −0.327271 − 0.410385i −0.246778 − 1.08120i −0.856702 + 3.75346i −1.79566 − 1.94309i 0.682379 + 2.98970i −6.35038 + 3.05818i −0.299721 + 1.31316i
78.10 −0.333170 − 0.160446i −0.136126 − 0.596409i −1.16172 − 1.45675i 0.169579 + 0.742973i −0.0503383 + 0.220546i 1.69986 − 2.02743i 0.317893 + 1.39278i 2.36573 − 1.13928i 0.0627086 − 0.274744i
78.11 −0.244688 − 0.117835i 0.754832 + 3.30713i −1.20099 − 1.50600i −0.234189 − 1.02605i 0.204999 − 0.898161i 2.53087 + 0.771158i 0.237274 + 1.03957i −7.66446 + 3.69101i −0.0636018 + 0.278658i
78.12 −0.199065 − 0.0958645i −0.0476338 − 0.208697i −1.21654 − 1.52550i −0.845035 − 3.70234i −0.0105245 + 0.0461107i −2.64079 + 0.161890i 0.194260 + 0.851107i 2.66162 − 1.28177i −0.186706 + 0.818014i
78.13 −0.134333 − 0.0646912i 0.251008 + 1.09974i −1.23312 − 1.54628i −0.182768 − 0.800759i 0.0374247 − 0.163968i 0.629805 + 2.56970i 0.131972 + 0.578208i 1.55649 − 0.749567i −0.0272503 + 0.119391i
78.14 0.531984 + 0.256190i −0.421002 − 1.84453i −1.02961 − 1.29109i 0.742378 + 3.25257i 0.248584 − 1.08912i 0.435707 + 2.60963i −0.479749 − 2.10192i −0.522140 + 0.251449i −0.438343 + 1.92051i
78.15 0.819895 + 0.394840i −0.672726 − 2.94741i −0.730651 − 0.916208i −0.596851 − 2.61497i 0.612190 − 2.68218i 2.46853 − 0.952039i −0.642296 − 2.81408i −5.53173 + 2.66394i 0.543143 − 2.37966i
78.16 1.00163 + 0.482360i 0.608299 + 2.66513i −0.476385 − 0.597368i 0.851736 + 3.73170i −0.676263 + 2.96290i −2.64572 − 0.0135489i −0.683781 − 2.99584i −4.03000 + 1.94075i −0.946899 + 4.14863i
78.17 1.14630 + 0.552031i −0.0305531 − 0.133862i −0.237703 − 0.298071i 0.212456 + 0.930829i 0.0388727 − 0.170313i 0.354945 − 2.62183i −0.674164 − 2.95371i 2.68592 − 1.29347i −0.270308 + 1.18430i
78.18 1.24279 + 0.598495i −0.519588 − 2.27646i −0.0606537 − 0.0760573i −0.165215 − 0.723855i 0.716715 − 3.14013i −2.31834 + 1.27488i −0.643746 − 2.82044i −2.20941 + 1.06399i 0.227896 − 0.998480i
78.19 1.57487 + 0.758417i 0.440021 + 1.92786i 0.658037 + 0.825152i 0.432906 + 1.89669i −0.769144 + 3.36984i 2.41733 + 1.07542i −0.367409 − 1.60972i −0.820109 + 0.394944i −0.756708 + 3.31536i
78.20 1.58845 + 0.764959i 0.297594 + 1.30384i 0.691044 + 0.866542i −0.823119 − 3.60632i −0.524673 + 2.29874i −0.211219 − 2.63731i −0.349809 − 1.53261i 1.09146 − 0.525618i 1.45120 − 6.35813i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 78.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.j.b ✓ 144
49.e even 7 1 inner 539.2.j.b ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
539.2.j.b ✓ 144 1.a even 1 1 trivial
539.2.j.b ✓ 144 49.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{144} + 35 T_{2}^{142} + 2 T_{2}^{141} + 728 T_{2}^{140} + 59 T_{2}^{139} + 11493 T_{2}^{138} + \cdots + 33\!\cdots\!09 \) acting on \(S_{2}^{\mathrm{new}}(539, [\chi])\). Copy content Toggle raw display