Properties

Label 6002.2.a.d
Level $6002$
Weight $2$
Character orbit 6002.a
Self dual yes
Analytic conductor $47.926$
Analytic rank $0$
Dimension $79$
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: \(79\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 79 q + 79 q^{2} + 17 q^{3} + 79 q^{4} + 18 q^{5} + 17 q^{6} + 19 q^{7} + 79 q^{8} + 118 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 79 q + 79 q^{2} + 17 q^{3} + 79 q^{4} + 18 q^{5} + 17 q^{6} + 19 q^{7} + 79 q^{8} + 118 q^{9} + 18 q^{10} + 28 q^{11} + 17 q^{12} + 47 q^{13} + 19 q^{14} + 14 q^{15} + 79 q^{16} + 36 q^{17} + 118 q^{18} + 29 q^{19} + 18 q^{20} + 45 q^{21} + 28 q^{22} + 23 q^{23} + 17 q^{24} + 161 q^{25} + 47 q^{26} + 50 q^{27} + 19 q^{28} + 53 q^{29} + 14 q^{30} + 29 q^{31} + 79 q^{32} + 34 q^{33} + 36 q^{34} + 33 q^{35} + 118 q^{36} + 89 q^{37} + 29 q^{38} - 7 q^{39} + 18 q^{40} + 58 q^{41} + 45 q^{42} + 88 q^{43} + 28 q^{44} + 45 q^{45} + 23 q^{46} + 3 q^{47} + 17 q^{48} + 162 q^{49} + 161 q^{50} + 29 q^{51} + 47 q^{52} + 88 q^{53} + 50 q^{54} + 37 q^{55} + 19 q^{56} + 54 q^{57} + 53 q^{58} + 37 q^{59} + 14 q^{60} + 55 q^{61} + 29 q^{62} + 21 q^{63} + 79 q^{64} + 55 q^{65} + 34 q^{66} + 107 q^{67} + 36 q^{68} + 39 q^{69} + 33 q^{70} - 5 q^{71} + 118 q^{72} + 71 q^{73} + 89 q^{74} + 37 q^{75} + 29 q^{76} + 61 q^{77} - 7 q^{78} + 29 q^{79} + 18 q^{80} + 215 q^{81} + 58 q^{82} + 42 q^{83} + 45 q^{84} + 84 q^{85} + 88 q^{86} + 15 q^{87} + 28 q^{88} + 72 q^{89} + 45 q^{90} + 70 q^{91} + 23 q^{92} + 97 q^{93} + 3 q^{94} - 18 q^{95} + 17 q^{96} + 93 q^{97} + 162 q^{98} + 49 q^{99}+O(q^{100}) \) 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 1.00000 −3.35201 1.00000 1.77899 −3.35201 −4.18423 1.00000 8.23600 1.77899
1.2 1.00000 −3.34863 1.00000 −3.86481 −3.34863 −0.0235424 1.00000 8.21330 −3.86481
1.3 1.00000 −3.27666 1.00000 3.56247 −3.27666 2.04525 1.00000 7.73650 3.56247
1.4 1.00000 −3.13631 1.00000 1.68515 −3.13631 −2.68800 1.00000 6.83646 1.68515
1.5 1.00000 −3.08764 1.00000 −2.69580 −3.08764 −4.17911 1.00000 6.53354 −2.69580
1.6 1.00000 −2.95638 1.00000 −0.704147 −2.95638 −0.285333 1.00000 5.74016 −0.704147
1.7 1.00000 −2.83087 1.00000 4.29043 −2.83087 1.15444 1.00000 5.01383 4.29043
1.8 1.00000 −2.78890 1.00000 −1.90255 −2.78890 1.43646 1.00000 4.77794 −1.90255
1.9 1.00000 −2.71638 1.00000 −0.997028 −2.71638 0.847005 1.00000 4.37871 −0.997028
1.10 1.00000 −2.68976 1.00000 1.26880 −2.68976 4.90307 1.00000 4.23478 1.26880
1.11 1.00000 −2.51885 1.00000 −4.29358 −2.51885 3.16827 1.00000 3.34462 −4.29358
1.12 1.00000 −2.46943 1.00000 4.39243 −2.46943 −4.67132 1.00000 3.09808 4.39243
1.13 1.00000 −2.46614 1.00000 2.56513 −2.46614 4.43338 1.00000 3.08187 2.56513
1.14 1.00000 −2.38888 1.00000 −1.13195 −2.38888 −2.56197 1.00000 2.70673 −1.13195
1.15 1.00000 −2.13218 1.00000 −2.93673 −2.13218 2.83180 1.00000 1.54618 −2.93673
1.16 1.00000 −2.06582 1.00000 0.633869 −2.06582 −3.21505 1.00000 1.26759 0.633869
1.17 1.00000 −2.02789 1.00000 2.83925 −2.02789 −1.41231 1.00000 1.11233 2.83925
1.18 1.00000 −1.97835 1.00000 0.563154 −1.97835 3.00235 1.00000 0.913870 0.563154
1.19 1.00000 −1.93711 1.00000 −2.50082 −1.93711 −1.24216 1.00000 0.752399 −2.50082
1.20 1.00000 −1.66636 1.00000 2.51408 −1.66636 2.07406 1.00000 −0.223255 2.51408
See all 79 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.79
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.d 79
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6002.2.a.d 79 1.a even 1 1 trivial