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Properties

Label 6009.2.a.d
Level $6009$
Weight $2$
Character orbit 6009.a
Self dual yes
Analytic conductor $47.982$
Analytic rank $0$
Dimension $93$
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: \(93\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 93 q + 2 q^{2} - 93 q^{3} + 114 q^{4} - 20 q^{5} - 2 q^{6} + 28 q^{7} + 6 q^{8} + 93 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 93 q + 2 q^{2} - 93 q^{3} + 114 q^{4} - 20 q^{5} - 2 q^{6} + 28 q^{7} + 6 q^{8} + 93 q^{9} + 19 q^{10} + 10 q^{11} - 114 q^{12} + 20 q^{13} + 13 q^{14} + 20 q^{15} + 148 q^{16} - 43 q^{17} + 2 q^{18} + 50 q^{19} - 31 q^{20} - 28 q^{21} + 36 q^{22} + 21 q^{23} - 6 q^{24} + 137 q^{25} + 2 q^{26} - 93 q^{27} + 62 q^{28} - q^{29} - 19 q^{30} + 58 q^{31} + 19 q^{32} - 10 q^{33} + 30 q^{34} + 30 q^{35} + 114 q^{36} + 42 q^{37} - 6 q^{38} - 20 q^{39} + 53 q^{40} - 7 q^{41} - 13 q^{42} + 60 q^{43} + 25 q^{44} - 20 q^{45} + 57 q^{46} + 9 q^{47} - 148 q^{48} + 145 q^{49} + 41 q^{50} + 43 q^{51} + 71 q^{52} - 45 q^{53} - 2 q^{54} + 78 q^{55} + 44 q^{56} - 50 q^{57} + 40 q^{58} + 42 q^{59} + 31 q^{60} + 69 q^{61} - 42 q^{62} + 28 q^{63} + 230 q^{64} - 4 q^{65} - 36 q^{66} + 76 q^{67} - 91 q^{68} - 21 q^{69} + 57 q^{70} + 92 q^{71} + 6 q^{72} + 29 q^{73} + 59 q^{74} - 137 q^{75} + 131 q^{76} - 98 q^{77} - 2 q^{78} + 215 q^{79} - 37 q^{80} + 93 q^{81} + 50 q^{82} - 27 q^{83} - 62 q^{84} + 52 q^{85} + 82 q^{86} + q^{87} + 136 q^{88} - 14 q^{89} + 19 q^{90} + 101 q^{91} - 14 q^{92} - 58 q^{93} + 112 q^{94} + 59 q^{95} - 19 q^{96} + 38 q^{97} - 16 q^{98} + 10 q^{99}+O(q^{100}) \) 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 −2.81022 −1.00000 5.89735 −2.74818 2.81022 1.73585 −10.9524 1.00000 7.72298
1.2 −2.77500 −1.00000 5.70062 0.612657 2.77500 3.28614 −10.2692 1.00000 −1.70012
1.3 −2.71219 −1.00000 5.35600 0.0136473 2.71219 −2.53463 −9.10211 1.00000 −0.0370143
1.4 −2.70585 −1.00000 5.32163 −3.73335 2.70585 −4.22653 −8.98782 1.00000 10.1019
1.5 −2.66976 −1.00000 5.12761 3.59325 2.66976 3.20381 −8.34997 1.00000 −9.59311
1.6 −2.59266 −1.00000 4.72188 −2.86941 2.59266 1.94917 −7.05690 1.00000 7.43940
1.7 −2.58147 −1.00000 4.66401 −0.715328 2.58147 −2.52841 −6.87707 1.00000 1.84660
1.8 −2.57633 −1.00000 4.63748 2.33581 2.57633 −3.18098 −6.79503 1.00000 −6.01782
1.9 −2.46607 −1.00000 4.08152 −3.84806 2.46607 3.41335 −5.13317 1.00000 9.48961
1.10 −2.41534 −1.00000 3.83389 2.77205 2.41534 2.36070 −4.42946 1.00000 −6.69545
1.11 −2.40646 −1.00000 3.79107 −1.64760 2.40646 4.01941 −4.31013 1.00000 3.96488
1.12 −2.31785 −1.00000 3.37244 −1.66098 2.31785 0.0321142 −3.18111 1.00000 3.84991
1.13 −2.29477 −1.00000 3.26596 1.38948 2.29477 −4.16968 −2.90508 1.00000 −3.18854
1.14 −2.26328 −1.00000 3.12245 3.34447 2.26328 −1.47338 −2.54043 1.00000 −7.56947
1.15 −2.15308 −1.00000 2.63577 −3.48527 2.15308 4.31227 −1.36887 1.00000 7.50407
1.16 −2.07852 −1.00000 2.32023 −3.95771 2.07852 −1.61443 −0.665606 1.00000 8.22618
1.17 −1.99887 −1.00000 1.99550 2.90160 1.99887 2.75939 0.00900131 1.00000 −5.79994
1.18 −1.98067 −1.00000 1.92305 0.325740 1.98067 2.23155 0.152420 1.00000 −0.645183
1.19 −1.89273 −1.00000 1.58242 −0.0965283 1.89273 5.12594 0.790373 1.00000 0.182702
1.20 −1.84702 −1.00000 1.41149 −1.45421 1.84702 −3.40926 1.08699 1.00000 2.68596
See all 93 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.93
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.d ✓ 93
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6009.2.a.d ✓ 93 1.a even 1 1 trivial