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Properties

Label 6043.2.a.b
Level $6043$
Weight $2$
Character orbit 6043.a
Self dual yes
Analytic conductor $48.254$
Analytic rank $1$
Dimension $243$
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] = [6043,2,Mod(1,6043)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6043, 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("6043.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6043 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6043.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: \(48.2535979415\)
Analytic rank: \(1\)
Dimension: \(243\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 243 q - 40 q^{2} - 27 q^{3} + 232 q^{4} - 85 q^{5} - 20 q^{6} - 28 q^{7} - 114 q^{8} + 210 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 243 q - 40 q^{2} - 27 q^{3} + 232 q^{4} - 85 q^{5} - 20 q^{6} - 28 q^{7} - 114 q^{8} + 210 q^{9} - 24 q^{10} - 37 q^{11} - 74 q^{12} - 113 q^{13} - 35 q^{14} - 34 q^{15} + 218 q^{16} - 125 q^{17} - 108 q^{18} - 46 q^{19} - 157 q^{20} - 113 q^{21} - 16 q^{22} - 60 q^{23} - 49 q^{24} + 208 q^{25} - 52 q^{26} - 90 q^{27} - 70 q^{28} - 137 q^{29} - 26 q^{30} - 36 q^{31} - 258 q^{32} - 153 q^{33} - 23 q^{34} - 77 q^{35} + 180 q^{36} - 108 q^{37} - 122 q^{38} - 32 q^{39} - 57 q^{40} - 186 q^{41} - 28 q^{42} - 54 q^{43} - 90 q^{44} - 233 q^{45} - 42 q^{46} - 188 q^{47} - 149 q^{48} + 189 q^{49} - 146 q^{50} - 34 q^{51} - 195 q^{52} - 196 q^{53} - 36 q^{54} - 57 q^{55} - 63 q^{56} - 76 q^{57} - 24 q^{58} - 137 q^{59} - 73 q^{60} - 96 q^{61} - 167 q^{62} - 113 q^{63} + 224 q^{64} - 131 q^{65} - 11 q^{66} - 71 q^{67} - 260 q^{68} - 162 q^{69} - 48 q^{70} - 77 q^{71} - 290 q^{72} - 160 q^{73} - 34 q^{74} - 100 q^{75} - 84 q^{76} - 416 q^{77} - 59 q^{78} - 17 q^{79} - 268 q^{80} + 147 q^{81} - 28 q^{82} - 238 q^{83} - 184 q^{84} - 108 q^{85} - 61 q^{86} - 127 q^{87} - 47 q^{88} - 183 q^{89} - 56 q^{90} - 14 q^{91} - 109 q^{92} - 206 q^{93} + q^{94} - 84 q^{95} - 54 q^{96} - 127 q^{97} - 294 q^{98} - 66 q^{99}+O(q^{100}) \) 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 −2.82300 0.597654 5.96933 −3.72082 −1.68718 −3.33856 −11.2054 −2.64281 10.5039
1.2 −2.80165 −0.611503 5.84923 2.20797 1.71322 −5.04593 −10.7842 −2.62606 −6.18596
1.3 −2.79141 −3.08096 5.79195 −0.513295 8.60020 −1.76306 −10.5849 6.49230 1.43282
1.4 −2.78744 0.620034 5.76982 0.534830 −1.72831 −0.323304 −10.5081 −2.61556 −1.49081
1.5 −2.77909 1.09974 5.72334 2.87165 −3.05629 2.58870 −10.3475 −1.79056 −7.98057
1.6 −2.76632 −1.35744 5.65252 −2.91911 3.75510 1.53004 −10.1040 −1.15737 8.07518
1.7 −2.76265 1.82603 5.63224 −3.05466 −5.04467 4.94747 −10.0346 0.334370 8.43896
1.8 −2.75980 −3.38147 5.61647 1.70872 9.33216 5.00632 −9.98072 8.43432 −4.71572
1.9 −2.72028 −2.17990 5.39995 −3.45469 5.92994 1.90963 −9.24882 1.75196 9.39775
1.10 −2.71539 −2.22479 5.37337 1.62014 6.04118 −3.35178 −9.16003 1.94968 −4.39933
1.11 −2.71243 2.12328 5.35728 1.35121 −5.75926 0.0653501 −9.10638 1.50834 −3.66505
1.12 −2.70932 3.28141 5.34043 −1.89583 −8.89039 0.209771 −9.05031 7.76762 5.13641
1.13 −2.64143 −1.28560 4.97716 2.15419 3.39581 3.78416 −7.86396 −1.34724 −5.69015
1.14 −2.64136 2.66871 4.97680 3.42573 −7.04902 −3.32847 −7.86280 4.12199 −9.04859
1.15 −2.63490 3.27029 4.94268 −2.75610 −8.61687 −4.50836 −7.75365 7.69478 7.26204
1.16 −2.56275 2.38749 4.56769 −3.37811 −6.11854 0.794528 −6.58035 2.70010 8.65725
1.17 −2.55204 −3.15051 4.51292 −4.10943 8.04024 −3.50763 −6.41307 6.92573 10.4874
1.18 −2.55150 0.112762 4.51017 −1.14524 −0.287713 −1.83176 −6.40472 −2.98728 2.92209
1.19 −2.55008 −1.24259 4.50293 −2.88473 3.16871 −3.99948 −6.38267 −1.45597 7.35631
1.20 −2.53808 −2.96165 4.44184 −3.22634 7.51690 0.844603 −6.19757 5.77138 8.18871
See next 80 embeddings (of 243 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.243
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(6043\) \(1\)

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 6043.2.a.b ✓ 243
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6043.2.a.b ✓ 243 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{243} + 40 T_{2}^{242} + 441 T_{2}^{241} - 3602 T_{2}^{240} - 113328 T_{2}^{239} + \cdots - 25998097516697 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6043))\). Copy content Toggle raw display