Properties

Label 6009.2.a.c
Level $6009$
Weight $2$
Character orbit 6009.a
Self dual yes
Analytic conductor $47.982$
Analytic rank $0$
Dimension $92$
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] = [6009,2,Mod(1,6009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6009, 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("6009.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6009 = 3 \cdot 2003 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6009.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: \(47.9821065746\)
Analytic rank: \(0\)
Dimension: \(92\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 92 q + 17 q^{2} + 92 q^{3} + 107 q^{4} + 34 q^{5} + 17 q^{6} + 22 q^{7} + 51 q^{8} + 92 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 92 q + 17 q^{2} + 92 q^{3} + 107 q^{4} + 34 q^{5} + 17 q^{6} + 22 q^{7} + 51 q^{8} + 92 q^{9} + 13 q^{10} + 40 q^{11} + 107 q^{12} + 6 q^{13} + 37 q^{14} + 34 q^{15} + 133 q^{16} + 77 q^{17} + 17 q^{18} + 34 q^{19} + 55 q^{20} + 22 q^{21} + 8 q^{22} + 83 q^{23} + 51 q^{24} + 110 q^{25} + 22 q^{26} + 92 q^{27} + 32 q^{28} + 97 q^{29} + 13 q^{30} + 44 q^{31} + 104 q^{32} + 40 q^{33} + 20 q^{34} + 80 q^{35} + 107 q^{36} + 12 q^{37} + 54 q^{38} + 6 q^{39} + 23 q^{40} + 67 q^{41} + 37 q^{42} + 30 q^{43} + 87 q^{44} + 34 q^{45} + 33 q^{46} + 69 q^{47} + 133 q^{48} + 112 q^{49} + 58 q^{50} + 77 q^{51} - 3 q^{52} + 113 q^{53} + 17 q^{54} + 42 q^{55} + 92 q^{56} + 34 q^{57} - 30 q^{58} + 72 q^{59} + 55 q^{60} + 19 q^{61} + 60 q^{62} + 22 q^{63} + 147 q^{64} + 74 q^{65} + 8 q^{66} + 26 q^{67} + 171 q^{68} + 83 q^{69} - 35 q^{70} + 134 q^{71} + 51 q^{72} - 17 q^{73} + 95 q^{74} + 110 q^{75} + 27 q^{76} + 108 q^{77} + 22 q^{78} + 159 q^{79} + 79 q^{80} + 92 q^{81} - 64 q^{82} + 73 q^{83} + 32 q^{84} - 4 q^{85} + 22 q^{86} + 97 q^{87} - 16 q^{88} + 50 q^{89} + 13 q^{90} + 17 q^{91} + 154 q^{92} + 44 q^{93} + 8 q^{94} + 155 q^{95} + 104 q^{96} - 20 q^{97} + 63 q^{98} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −2.78023 1.00000 5.72970 −1.51058 −2.78023 −3.00947 −10.3695 1.00000 4.19978
1.2 −2.66639 1.00000 5.10965 3.96550 −2.66639 2.39934 −8.29154 1.00000 −10.5736
1.3 −2.62844 1.00000 4.90869 −1.09828 −2.62844 −4.21500 −7.64532 1.00000 2.88675
1.4 −2.60458 1.00000 4.78383 2.52259 −2.60458 −2.57820 −7.25072 1.00000 −6.57030
1.5 −2.49751 1.00000 4.23755 1.03851 −2.49751 3.59057 −5.58831 1.00000 −2.59368
1.6 −2.48563 1.00000 4.17837 2.55956 −2.48563 1.79588 −5.41463 1.00000 −6.36212
1.7 −2.40248 1.00000 3.77189 0.0504655 −2.40248 −0.217870 −4.25692 1.00000 −0.121242
1.8 −2.37626 1.00000 3.64660 −0.379121 −2.37626 2.10974 −3.91276 1.00000 0.900889
1.9 −2.31255 1.00000 3.34790 −0.928530 −2.31255 −3.24971 −3.11708 1.00000 2.14727
1.10 −2.28351 1.00000 3.21443 −3.38813 −2.28351 0.924728 −2.77317 1.00000 7.73684
1.11 −2.23454 1.00000 2.99319 2.17127 −2.23454 −3.00622 −2.21933 1.00000 −4.85180
1.12 −2.22934 1.00000 2.96995 −1.56663 −2.22934 2.33278 −2.16235 1.00000 3.49254
1.13 −2.22794 1.00000 2.96373 3.46015 −2.22794 3.74184 −2.14714 1.00000 −7.70902
1.14 −2.15336 1.00000 2.63696 −3.93365 −2.15336 −2.88832 −1.37159 1.00000 8.47056
1.15 −1.90307 1.00000 1.62167 −1.77542 −1.90307 −0.953587 0.719990 1.00000 3.37874
1.16 −1.89797 1.00000 1.60229 4.27518 −1.89797 −0.912076 0.754847 1.00000 −8.11416
1.17 −1.84272 1.00000 1.39563 −0.293700 −1.84272 0.646833 1.11368 1.00000 0.541209
1.18 −1.77532 1.00000 1.15178 0.669700 −1.77532 2.59999 1.50587 1.00000 −1.18893
1.19 −1.73440 1.00000 1.00813 1.28297 −1.73440 −0.642029 1.72030 1.00000 −2.22518
1.20 −1.66735 1.00000 0.780065 −0.964493 −1.66735 4.83846 2.03406 1.00000 1.60815
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.92
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(2003\) \(1\)

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 6009.2.a.c 92
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6009.2.a.c 92 1.a even 1 1 trivial