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Properties

Label 6002.2.a.a
Level $6002$
Weight $2$
Character orbit 6002.a
Self dual yes
Analytic conductor $47.926$
Analytic rank $1$
Dimension $47$
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: \(1\)
Dimension: \(47\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 47 q + 47 q^{2} - 13 q^{3} + 47 q^{4} - 14 q^{5} - 13 q^{6} - 17 q^{7} + 47 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 47 q + 47 q^{2} - 13 q^{3} + 47 q^{4} - 14 q^{5} - 13 q^{6} - 17 q^{7} + 47 q^{8} + 12 q^{9} - 14 q^{10} - 30 q^{11} - 13 q^{12} - 39 q^{13} - 17 q^{14} - 18 q^{15} + 47 q^{16} - 26 q^{17} + 12 q^{18} - 23 q^{19} - 14 q^{20} - 39 q^{21} - 30 q^{22} - 25 q^{23} - 13 q^{24} - 19 q^{25} - 39 q^{26} - 46 q^{27} - 17 q^{28} - 53 q^{29} - 18 q^{30} - 23 q^{31} + 47 q^{32} - 26 q^{33} - 26 q^{34} - 31 q^{35} + 12 q^{36} - 83 q^{37} - 23 q^{38} - 9 q^{39} - 14 q^{40} - 48 q^{41} - 39 q^{42} - 78 q^{43} - 30 q^{44} - 27 q^{45} - 25 q^{46} - 15 q^{47} - 13 q^{48} - 12 q^{49} - 19 q^{50} - 47 q^{51} - 39 q^{52} - 76 q^{53} - 46 q^{54} - 39 q^{55} - 17 q^{56} - 44 q^{57} - 53 q^{58} - 33 q^{59} - 18 q^{60} - 33 q^{61} - 23 q^{62} - 7 q^{63} + 47 q^{64} - 67 q^{65} - 26 q^{66} - 85 q^{67} - 26 q^{68} - 33 q^{69} - 31 q^{70} - 17 q^{71} + 12 q^{72} - 59 q^{73} - 83 q^{74} - 21 q^{75} - 23 q^{76} - 59 q^{77} - 9 q^{78} - 49 q^{79} - 14 q^{80} - 41 q^{81} - 48 q^{82} - 30 q^{83} - 39 q^{84} - 84 q^{85} - 78 q^{86} + 9 q^{87} - 30 q^{88} - 50 q^{89} - 27 q^{90} - 42 q^{91} - 25 q^{92} - 43 q^{93} - 15 q^{94} + 8 q^{95} - 13 q^{96} - 49 q^{97} - 12 q^{98} - 49 q^{99}+O(q^{100}) \) 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 −3.27359 1.00000 0.938843 −3.27359 3.32454 1.00000 7.71641 0.938843
1.2 1.00000 −3.25356 1.00000 −1.47632 −3.25356 0.915630 1.00000 7.58563 −1.47632
1.3 1.00000 −2.97652 1.00000 2.57540 −2.97652 −1.79126 1.00000 5.85967 2.57540
1.4 1.00000 −2.82438 1.00000 −1.67949 −2.82438 4.44281 1.00000 4.97711 −1.67949
1.5 1.00000 −2.77545 1.00000 1.33751 −2.77545 −0.156144 1.00000 4.70313 1.33751
1.6 1.00000 −2.69765 1.00000 −1.67926 −2.69765 −4.07202 1.00000 4.27732 −1.67926
1.7 1.00000 −2.53917 1.00000 0.927459 −2.53917 −2.39755 1.00000 3.44739 0.927459
1.8 1.00000 −2.32207 1.00000 −3.20826 −2.32207 3.53388 1.00000 2.39199 −3.20826
1.9 1.00000 −2.18587 1.00000 0.199820 −2.18587 0.0442522 1.00000 1.77803 0.199820
1.10 1.00000 −2.03423 1.00000 −3.81435 −2.03423 −2.54300 1.00000 1.13810 −3.81435
1.11 1.00000 −1.93355 1.00000 2.56764 −1.93355 0.413652 1.00000 0.738635 2.56764
1.12 1.00000 −1.68417 1.00000 −2.87245 −1.68417 −3.82153 1.00000 −0.163575 −2.87245
1.13 1.00000 −1.63738 1.00000 3.71883 −1.63738 2.15943 1.00000 −0.318981 3.71883
1.14 1.00000 −1.60528 1.00000 1.61064 −1.60528 −2.90280 1.00000 −0.423077 1.61064
1.15 1.00000 −1.49159 1.00000 1.25928 −1.49159 −0.926965 1.00000 −0.775168 1.25928
1.16 1.00000 −1.48674 1.00000 1.68773 −1.48674 3.31084 1.00000 −0.789607 1.68773
1.17 1.00000 −1.41473 1.00000 −1.78933 −1.41473 1.98770 1.00000 −0.998550 −1.78933
1.18 1.00000 −1.29467 1.00000 −3.90714 −1.29467 −0.238571 1.00000 −1.32383 −3.90714
1.19 1.00000 −0.985702 1.00000 −1.57345 −0.985702 −1.72023 1.00000 −2.02839 −1.57345
1.20 1.00000 −0.879988 1.00000 −1.86923 −0.879988 3.94056 1.00000 −2.22562 −1.86923
See all 47 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.47
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.a ✓ 47
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6002.2.a.a ✓ 47 1.a even 1 1 trivial