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Properties

Label 6041.2.a.d
Level $6041$
Weight $2$
Character orbit 6041.a
Self dual yes
Analytic conductor $48.238$
Analytic rank $1$
Dimension $101$
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] = [6041,2,Mod(1,6041)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6041, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6041.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6041 = 7 \cdot 863 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6041.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: \(48.2376278611\)
Analytic rank: \(1\)
Dimension: \(101\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 101 q + 3 q^{2} - 17 q^{3} + 85 q^{4} - 12 q^{5} - 17 q^{6} - 101 q^{7} - 3 q^{8} + 88 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 101 q + 3 q^{2} - 17 q^{3} + 85 q^{4} - 12 q^{5} - 17 q^{6} - 101 q^{7} - 3 q^{8} + 88 q^{9} - 23 q^{10} - 13 q^{11} - 31 q^{12} - 35 q^{13} - 3 q^{14} - 20 q^{15} + 45 q^{16} - 19 q^{17} + 3 q^{18} - 59 q^{19} - 31 q^{20} + 17 q^{21} - 13 q^{22} - 29 q^{23} - 59 q^{24} + 103 q^{25} - 18 q^{26} - 47 q^{27} - 85 q^{28} - 26 q^{29} - 8 q^{30} - 125 q^{31} + 12 q^{32} - 18 q^{33} - 66 q^{34} + 12 q^{35} + 40 q^{36} + 22 q^{37} - 31 q^{38} - 94 q^{39} - 79 q^{40} - 39 q^{41} + 17 q^{42} - 5 q^{43} - 53 q^{44} - 50 q^{45} - 37 q^{46} - 47 q^{47} - 81 q^{48} + 101 q^{49} + 2 q^{50} - 23 q^{51} - 56 q^{52} - 5 q^{53} - 77 q^{54} - 155 q^{55} + 3 q^{56} + 61 q^{57} - 31 q^{58} - 33 q^{59} - 48 q^{60} - 96 q^{61} - 38 q^{62} - 88 q^{63} - 33 q^{64} - 8 q^{65} - 91 q^{66} + 8 q^{67} - 41 q^{68} - 91 q^{69} + 23 q^{70} - 116 q^{71} - 5 q^{72} - 62 q^{73} - 23 q^{74} - 94 q^{75} - 112 q^{76} + 13 q^{77} + 17 q^{78} - 127 q^{79} - 87 q^{80} + 37 q^{81} - 118 q^{82} - 58 q^{83} + 31 q^{84} - 6 q^{85} - 26 q^{86} - 82 q^{87} - 40 q^{88} - 57 q^{89} - 123 q^{90} + 35 q^{91} - 28 q^{92} - 10 q^{93} - 107 q^{94} - 70 q^{95} - 76 q^{96} - 69 q^{97} + 3 q^{98} - 67 q^{99}+O(q^{100}) \) 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 −2.71484 1.06768 5.37033 2.50113 −2.89858 −1.00000 −9.14989 −1.86006 −6.79016
1.2 −2.68338 −0.794633 5.20055 −0.431207 2.13231 −1.00000 −8.58829 −2.36856 1.15709
1.3 −2.67103 −2.02602 5.13442 3.05924 5.41156 −1.00000 −8.37215 1.10475 −8.17132
1.4 −2.59007 −1.31514 4.70847 −2.80668 3.40631 −1.00000 −7.01513 −1.27040 7.26950
1.5 −2.44864 −2.39074 3.99581 −0.961081 5.85406 −1.00000 −4.88702 2.71565 2.35334
1.6 −2.44717 0.719096 3.98863 −3.39767 −1.75975 −1.00000 −4.86651 −2.48290 8.31466
1.7 −2.41856 2.09832 3.84945 −1.02982 −5.07493 −1.00000 −4.47302 1.40297 2.49069
1.8 −2.41637 −2.07802 3.83883 1.82876 5.02127 −1.00000 −4.44330 1.31818 −4.41896
1.9 −2.41440 0.0560859 3.82935 0.0819711 −0.135414 −1.00000 −4.41678 −2.99685 −0.197911
1.10 −2.41346 2.83250 3.82477 1.74211 −6.83611 −1.00000 −4.40400 5.02306 −4.20452
1.11 −2.37665 1.96222 3.64847 −3.56190 −4.66352 −1.00000 −3.91784 0.850318 8.46540
1.12 −2.34195 3.16467 3.48474 −1.79579 −7.41150 −1.00000 −3.47719 7.01511 4.20566
1.13 −2.15616 −2.99968 2.64901 −1.14985 6.46779 −1.00000 −1.39937 5.99811 2.47926
1.14 −2.14147 −2.96762 2.58591 4.08817 6.35507 −1.00000 −1.25470 5.80675 −8.75470
1.15 −2.08521 1.06538 2.34812 1.41345 −2.22154 −1.00000 −0.725898 −1.86497 −2.94734
1.16 −2.00569 2.11059 2.02278 3.12491 −4.23318 −1.00000 −0.0456843 1.45460 −6.26760
1.17 −1.98854 −0.0463309 1.95428 −0.0496669 0.0921307 −1.00000 0.0909182 −2.99785 0.0987645
1.18 −1.87092 −1.43026 1.50033 1.61115 2.67590 −1.00000 0.934843 −0.954356 −3.01432
1.19 −1.83237 1.94542 1.35759 1.21630 −3.56474 −1.00000 1.17714 0.784666 −2.22871
1.20 −1.79653 2.70133 1.22753 2.96874 −4.85303 −1.00000 1.38776 4.29718 −5.33345
See next 80 embeddings (of 101 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.101
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(863\) \(1\)

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 6041.2.a.d ✓ 101
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6041.2.a.d ✓ 101 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{101} - 3 T_{2}^{100} - 139 T_{2}^{99} + 423 T_{2}^{98} + 9322 T_{2}^{97} - 28815 T_{2}^{96} + \cdots + 403759 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6041))\). Copy content Toggle raw display