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Properties

Label 6002.2.a.c
Level $6002$
Weight $2$
Character orbit 6002.a
Self dual yes
Analytic conductor $47.926$
Analytic rank $0$
Dimension $69$
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] = [6002,2,Mod(1,6002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6002, 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("6002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6002 = 2 \cdot 3001 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6002.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: \(47.9262112932\)
Analytic rank: \(0\)
Dimension: \(69\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 69 q - 69 q^{2} + 11 q^{3} + 69 q^{4} - 2 q^{5} - 11 q^{6} + 23 q^{7} - 69 q^{8} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 69 q - 69 q^{2} + 11 q^{3} + 69 q^{4} - 2 q^{5} - 11 q^{6} + 23 q^{7} - 69 q^{8} + 72 q^{9} + 2 q^{10} - 14 q^{11} + 11 q^{12} + 31 q^{13} - 23 q^{14} + 34 q^{15} + 69 q^{16} - 4 q^{17} - 72 q^{18} + 17 q^{19} - 2 q^{20} - 11 q^{21} + 14 q^{22} + 33 q^{23} - 11 q^{24} + 119 q^{25} - 31 q^{26} + 44 q^{27} + 23 q^{28} - 25 q^{29} - 34 q^{30} + 49 q^{31} - 69 q^{32} + 10 q^{33} + 4 q^{34} - 11 q^{35} + 72 q^{36} + 73 q^{37} - 17 q^{38} + 31 q^{39} + 2 q^{40} - 46 q^{41} + 11 q^{42} + 76 q^{43} - 14 q^{44} + 9 q^{45} - 33 q^{46} + 23 q^{47} + 11 q^{48} + 100 q^{49} - 119 q^{50} + 25 q^{51} + 31 q^{52} + 30 q^{53} - 44 q^{54} + 81 q^{55} - 23 q^{56} + 12 q^{57} + 25 q^{58} - 3 q^{59} + 34 q^{60} + 13 q^{61} - 49 q^{62} + 65 q^{63} + 69 q^{64} - 27 q^{65} - 10 q^{66} + 105 q^{67} - 4 q^{68} + 19 q^{69} + 11 q^{70} + 51 q^{71} - 72 q^{72} + 43 q^{73} - 73 q^{74} + 77 q^{75} + 17 q^{76} - 19 q^{77} - 31 q^{78} + 89 q^{79} - 2 q^{80} + 73 q^{81} + 46 q^{82} - 10 q^{83} - 11 q^{84} + 44 q^{85} - 76 q^{86} + 57 q^{87} + 14 q^{88} - 28 q^{89} - 9 q^{90} + 76 q^{91} + 33 q^{92} + 59 q^{93} - 23 q^{94} + 72 q^{95} - 11 q^{96} + 89 q^{97} - 100 q^{98} - 17 q^{99}+O(q^{100}) \) 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 −1.00000 −3.40047 1.00000 −0.810151 3.40047 1.45096 −1.00000 8.56323 0.810151
1.2 −1.00000 −3.06382 1.00000 −3.70565 3.06382 4.91357 −1.00000 6.38702 3.70565
1.3 −1.00000 −2.95145 1.00000 1.54937 2.95145 1.05407 −1.00000 5.71107 −1.54937
1.4 −1.00000 −2.80010 1.00000 1.44998 2.80010 −0.125244 −1.00000 4.84055 −1.44998
1.5 −1.00000 −2.79134 1.00000 2.80110 2.79134 2.82300 −1.00000 4.79155 −2.80110
1.6 −1.00000 −2.68289 1.00000 −4.14764 2.68289 −3.25782 −1.00000 4.19789 4.14764
1.7 −1.00000 −2.66325 1.00000 −0.772436 2.66325 4.62727 −1.00000 4.09290 0.772436
1.8 −1.00000 −2.63767 1.00000 0.590977 2.63767 −1.00146 −1.00000 3.95729 −0.590977
1.9 −1.00000 −2.45213 1.00000 −3.61679 2.45213 0.0579543 −1.00000 3.01293 3.61679
1.10 −1.00000 −2.40560 1.00000 −3.03439 2.40560 −1.41996 −1.00000 2.78689 3.03439
1.11 −1.00000 −2.33178 1.00000 −0.767442 2.33178 −0.132499 −1.00000 2.43720 0.767442
1.12 −1.00000 −2.31615 1.00000 2.10238 2.31615 −3.46770 −1.00000 2.36454 −2.10238
1.13 −1.00000 −2.29088 1.00000 2.54707 2.29088 4.39407 −1.00000 2.24813 −2.54707
1.14 −1.00000 −2.19747 1.00000 −2.90019 2.19747 −0.534047 −1.00000 1.82886 2.90019
1.15 −1.00000 −2.02205 1.00000 3.59068 2.02205 0.941965 −1.00000 1.08869 −3.59068
1.16 −1.00000 −1.90243 1.00000 −0.692359 1.90243 −4.50666 −1.00000 0.619255 0.692359
1.17 −1.00000 −1.43088 1.00000 1.76140 1.43088 −2.51027 −1.00000 −0.952590 −1.76140
1.18 −1.00000 −1.36851 1.00000 0.212949 1.36851 −1.32734 −1.00000 −1.12718 −0.212949
1.19 −1.00000 −1.30456 1.00000 −4.01452 1.30456 3.40012 −1.00000 −1.29813 4.01452
1.20 −1.00000 −1.29953 1.00000 −2.92449 1.29953 −2.47331 −1.00000 −1.31122 2.92449
See all 69 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.69
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3001\) \(-1\)

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 6002.2.a.c ✓ 69
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6002.2.a.c ✓ 69 1.a even 1 1 trivial