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Properties

Label 6009.2.a.a
Level $6009$
Weight $2$
Character orbit 6009.a
Self dual yes
Analytic conductor $47.982$
Analytic rank $1$
Dimension $74$
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: \(1\)
Dimension: \(74\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 74 q - 15 q^{2} + 74 q^{3} + 57 q^{4} - 34 q^{5} - 15 q^{6} - 24 q^{7} - 39 q^{8} + 74 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 74 q - 15 q^{2} + 74 q^{3} + 57 q^{4} - 34 q^{5} - 15 q^{6} - 24 q^{7} - 39 q^{8} + 74 q^{9} - 9 q^{10} - 34 q^{11} + 57 q^{12} - 16 q^{13} - 41 q^{14} - 34 q^{15} + 31 q^{16} - 77 q^{17} - 15 q^{18} - 42 q^{19} - 71 q^{20} - 24 q^{21} - 20 q^{22} - 73 q^{23} - 39 q^{24} + 46 q^{25} - 38 q^{26} + 74 q^{27} - 20 q^{28} - 91 q^{29} - 9 q^{30} - 42 q^{31} - 78 q^{32} - 34 q^{33} + 2 q^{34} - 68 q^{35} + 57 q^{36} - 14 q^{37} - 30 q^{38} - 16 q^{39} - 7 q^{40} - 63 q^{41} - 41 q^{42} - 42 q^{43} - 59 q^{44} - 34 q^{45} - 11 q^{46} - 65 q^{47} + 31 q^{48} + 36 q^{49} - 64 q^{50} - 77 q^{51} - 13 q^{52} - 107 q^{53} - 15 q^{54} - 54 q^{55} - 94 q^{56} - 42 q^{57} - 86 q^{59} - 71 q^{60} - 43 q^{61} - 66 q^{62} - 24 q^{63} - 11 q^{64} - 90 q^{65} - 20 q^{66} - 46 q^{67} - 157 q^{68} - 73 q^{69} + 29 q^{70} - 118 q^{71} - 39 q^{72} - 3 q^{73} - 73 q^{74} + 46 q^{75} - 73 q^{76} - 124 q^{77} - 38 q^{78} - 169 q^{79} - 91 q^{80} + 74 q^{81} - 14 q^{82} - 89 q^{83} - 20 q^{84} - 8 q^{85} - 60 q^{86} - 91 q^{87} - 28 q^{88} - 102 q^{89} - 9 q^{90} - 71 q^{91} - 102 q^{92} - 42 q^{93} - 26 q^{94} - 109 q^{95} - 78 q^{96} + 4 q^{97} - 41 q^{98} - 34 q^{99}+O(q^{100}) \) 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 −2.77334 1.00000 5.69142 −2.86594 −2.77334 1.39621 −10.2376 1.00000 7.94822
1.2 −2.73842 1.00000 5.49897 2.33287 −2.73842 −1.43409 −9.58166 1.00000 −6.38839
1.3 −2.62474 1.00000 4.88928 1.20941 −2.62474 4.21658 −7.58361 1.00000 −3.17438
1.4 −2.59810 1.00000 4.75013 −3.44165 −2.59810 4.57778 −7.14513 1.00000 8.94177
1.5 −2.59552 1.00000 4.73672 −0.667625 −2.59552 0.166398 −7.10320 1.00000 1.73283
1.6 −2.56637 1.00000 4.58626 −2.16434 −2.56637 −3.01232 −6.63731 1.00000 5.55451
1.7 −2.52365 1.00000 4.36882 2.64577 −2.52365 −0.383454 −5.97808 1.00000 −6.67700
1.8 −2.49404 1.00000 4.22024 −2.73101 −2.49404 0.905065 −5.53738 1.00000 6.81126
1.9 −2.36670 1.00000 3.60127 −3.59054 −2.36670 −0.0750605 −3.78973 1.00000 8.49774
1.10 −2.19562 1.00000 2.82075 1.49340 −2.19562 0.838552 −1.80204 1.00000 −3.27894
1.11 −2.19184 1.00000 2.80415 1.63232 −2.19184 −3.56778 −1.76257 1.00000 −3.57777
1.12 −2.17449 1.00000 2.72842 −0.915481 −2.17449 2.88508 −1.58394 1.00000 1.99071
1.13 −2.15543 1.00000 2.64587 −3.73001 −2.15543 −4.51139 −1.39212 1.00000 8.03977
1.14 −2.13910 1.00000 2.57575 3.20421 −2.13910 −4.87621 −1.23159 1.00000 −6.85413
1.15 −2.00831 1.00000 2.03332 2.87361 −2.00831 3.47808 −0.0669129 1.00000 −5.77111
1.16 −1.86945 1.00000 1.49486 −3.51711 −1.86945 4.32252 0.944337 1.00000 6.57508
1.17 −1.79865 1.00000 1.23512 0.910992 −1.79865 −0.511700 1.37574 1.00000 −1.63855
1.18 −1.76995 1.00000 1.13274 0.338435 −1.76995 −2.77174 1.53501 1.00000 −0.599015
1.19 −1.63218 1.00000 0.663999 −1.51481 −1.63218 1.75829 2.18059 1.00000 2.47244
1.20 −1.62634 1.00000 0.644983 −3.49223 −1.62634 2.70733 2.20372 1.00000 5.67955
See all 74 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.74
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.a ✓ 74
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6009.2.a.a ✓ 74 1.a even 1 1 trivial