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Properties

Label 6037.2.a.a
Level $6037$
Weight $2$
Character orbit 6037.a
Self dual yes
Analytic conductor $48.206$
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] = [6037,2,Mod(1,6037)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6037, 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("6037.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6037 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6037.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: \(48.2056877002\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 243 q - 47 q^{2} - 31 q^{3} + 229 q^{4} - 40 q^{5} - 18 q^{6} - 42 q^{7} - 135 q^{8} + 214 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 243 q - 47 q^{2} - 31 q^{3} + 229 q^{4} - 40 q^{5} - 18 q^{6} - 42 q^{7} - 135 q^{8} + 214 q^{9} - 14 q^{10} - 112 q^{11} - 54 q^{12} - 45 q^{13} - 35 q^{14} - 56 q^{15} + 213 q^{16} - 71 q^{17} - 135 q^{18} - 69 q^{19} - 107 q^{20} - 36 q^{21} - 24 q^{22} - 162 q^{23} - 57 q^{24} + 203 q^{25} - 55 q^{26} - 115 q^{27} - 87 q^{28} - 76 q^{29} - 64 q^{30} - 35 q^{31} - 302 q^{32} - 77 q^{33} - 9 q^{34} - 264 q^{35} + 173 q^{36} - 61 q^{37} - 71 q^{38} - 123 q^{39} - 16 q^{40} - 74 q^{41} - 70 q^{42} - 178 q^{43} - 209 q^{44} - 107 q^{45} - 11 q^{46} - 191 q^{47} - 65 q^{48} + 211 q^{49} - 188 q^{50} - 175 q^{51} - 95 q^{52} - 122 q^{53} - 36 q^{54} - 47 q^{55} - 69 q^{56} - 103 q^{57} - 37 q^{58} - 212 q^{59} - 79 q^{60} - 14 q^{61} - 152 q^{62} - 203 q^{63} + 217 q^{64} - 159 q^{65} + 5 q^{66} - 202 q^{67} - 176 q^{68} - 34 q^{69} + 45 q^{70} - 170 q^{71} - 347 q^{72} - 57 q^{73} - 68 q^{74} - 124 q^{75} - 74 q^{76} - 166 q^{77} - 63 q^{78} - 48 q^{79} - 222 q^{80} + 159 q^{81} - 27 q^{82} - 434 q^{83} - 52 q^{84} - 57 q^{85} - 77 q^{86} - 184 q^{87} - 15 q^{88} - 62 q^{89} - 24 q^{90} - 81 q^{91} - 330 q^{92} - 164 q^{93} + 40 q^{94} - 182 q^{95} - 66 q^{96} - 21 q^{97} - 254 q^{98} - 306 q^{99}+O(q^{100}) \) 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 −2.81439 2.72661 5.92080 4.37726 −7.67375 −1.67878 −11.0347 4.43441 −12.3193
1.2 −2.80926 −0.794746 5.89194 −3.46470 2.23265 −2.15218 −10.9335 −2.36838 9.73325
1.3 −2.79753 −2.64112 5.82619 −2.23668 7.38863 −3.27355 −10.7039 3.97554 6.25720
1.4 −2.79471 3.05349 5.81042 −2.55663 −8.53363 4.45243 −10.6490 6.32380 7.14505
1.5 −2.76337 −0.302967 5.63621 1.94141 0.837209 −2.44102 −10.0482 −2.90821 −5.36484
1.6 −2.75691 −3.25303 5.60054 1.79806 8.96830 −3.80594 −9.92635 7.58219 −4.95708
1.7 −2.74252 −0.770641 5.52143 3.66315 2.11350 −4.05510 −9.65761 −2.40611 −10.0463
1.8 −2.73652 −0.570094 5.48855 −2.26504 1.56007 4.66290 −9.54649 −2.67499 6.19832
1.9 −2.72852 −2.79225 5.44481 1.20025 7.61871 1.91493 −9.39922 4.79667 −3.27490
1.10 −2.72738 0.874188 5.43862 −1.32205 −2.38425 −0.0787567 −9.37845 −2.23580 3.60574
1.11 −2.71870 0.776703 5.39133 1.67497 −2.11162 0.286183 −9.22001 −2.39673 −4.55375
1.12 −2.70464 1.55203 5.31508 0.654263 −4.19768 0.835894 −8.96611 −0.591204 −1.76955
1.13 −2.70152 −2.60067 5.29822 −1.66030 7.02577 3.17610 −8.91022 3.76349 4.48535
1.14 −2.63978 −0.462407 4.96844 1.16438 1.22065 1.07998 −7.83603 −2.78618 −3.07372
1.15 −2.63616 2.26078 4.94932 −3.62229 −5.95976 1.87932 −7.77485 2.11112 9.54892
1.16 −2.63608 3.35251 4.94891 0.139412 −8.83747 −3.57933 −7.77355 8.23930 −0.367500
1.17 −2.63128 2.77436 4.92361 −3.29020 −7.30010 −0.686595 −7.69283 4.69706 8.65743
1.18 −2.62743 0.461713 4.90341 −4.13985 −1.21312 1.40616 −7.62851 −2.78682 10.8772
1.19 −2.62641 −3.35140 4.89803 −4.35294 8.80215 2.67168 −7.61143 8.23188 11.4326
1.20 −2.59845 1.70782 4.75194 1.33764 −4.43768 4.30824 −7.15077 −0.0833526 −3.47580
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
\(6037\) \(1\)

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 6037.2.a.a ✓ 243
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6037.2.a.a ✓ 243 1.a even 1 1 trivial