↑

Properties

Label 6046.2.a.f
Level $6046$
Weight $2$
Character orbit 6046.a
Self dual yes
Analytic conductor $48.278$
Analytic rank $0$
Dimension $67$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6046,2,Mod(1,6046)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6046, 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("6046.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6046 = 2 \cdot 3023 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6046.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: \(48.2775530621\)
Analytic rank: \(0\)
Dimension: \(67\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9} + 21 q^{10} + 56 q^{11} + 21 q^{12} + 33 q^{13} + 38 q^{14} + 25 q^{15} + 67 q^{16} + 30 q^{17} + 90 q^{18} + 36 q^{19} + 21 q^{20} + 20 q^{21} + 56 q^{22} + 65 q^{23} + 21 q^{24} + 72 q^{25} + 33 q^{26} + 57 q^{27} + 38 q^{28} + 84 q^{29} + 25 q^{30} + 52 q^{31} + 67 q^{32} - 9 q^{33} + 30 q^{34} + 30 q^{35} + 90 q^{36} + 52 q^{37} + 36 q^{38} + 41 q^{39} + 21 q^{40} + 46 q^{41} + 20 q^{42} + 61 q^{43} + 56 q^{44} + 23 q^{45} + 65 q^{46} + 51 q^{47} + 21 q^{48} + 81 q^{49} + 72 q^{50} + 33 q^{51} + 33 q^{52} + 72 q^{53} + 57 q^{54} + 14 q^{55} + 38 q^{56} - 26 q^{57} + 84 q^{58} + 71 q^{59} + 25 q^{60} + 42 q^{61} + 52 q^{62} + 63 q^{63} + 67 q^{64} - 2 q^{65} - 9 q^{66} + 70 q^{67} + 30 q^{68} + 21 q^{69} + 30 q^{70} + 104 q^{71} + 90 q^{72} - 31 q^{73} + 52 q^{74} + 69 q^{75} + 36 q^{76} + 48 q^{77} + 41 q^{78} + 79 q^{79} + 21 q^{80} + 123 q^{81} + 46 q^{82} + 41 q^{83} + 20 q^{84} + 6 q^{85} + 61 q^{86} + 19 q^{87} + 56 q^{88} + 58 q^{89} + 23 q^{90} + 31 q^{91} + 65 q^{92} + 13 q^{93} + 51 q^{94} + 77 q^{95} + 21 q^{96} - 8 q^{97} + 81 q^{98} + 129 q^{99}+O(q^{100}) \) 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 −3.37105 1.00000 3.37148 −3.37105 1.10661 1.00000 8.36395 3.37148
1.2 1.00000 −3.14488 1.00000 −1.06232 −3.14488 −3.74953 1.00000 6.89025 −1.06232
1.3 1.00000 −3.02893 1.00000 −3.61003 −3.02893 0.772328 1.00000 6.17443 −3.61003
1.4 1.00000 −3.02873 1.00000 −2.84681 −3.02873 −0.956843 1.00000 6.17323 −2.84681
1.5 1.00000 −2.76774 1.00000 −2.60035 −2.76774 3.45506 1.00000 4.66041 −2.60035
1.6 1.00000 −2.74369 1.00000 0.0184852 −2.74369 0.788378 1.00000 4.52782 0.0184852
1.7 1.00000 −2.73291 1.00000 −0.132829 −2.73291 3.97016 1.00000 4.46882 −0.132829
1.8 1.00000 −2.66695 1.00000 2.34176 −2.66695 3.40715 1.00000 4.11263 2.34176
1.9 1.00000 −2.41094 1.00000 0.848657 −2.41094 −1.16640 1.00000 2.81262 0.848657
1.10 1.00000 −2.39039 1.00000 0.762400 −2.39039 −3.68654 1.00000 2.71398 0.762400
1.11 1.00000 −2.34396 1.00000 3.11002 −2.34396 1.16030 1.00000 2.49414 3.11002
1.12 1.00000 −2.14148 1.00000 3.02625 −2.14148 4.90651 1.00000 1.58595 3.02625
1.13 1.00000 −2.08665 1.00000 −0.513266 −2.08665 0.880996 1.00000 1.35412 −0.513266
1.14 1.00000 −1.75304 1.00000 −0.582580 −1.75304 −2.67925 1.00000 0.0731519 −0.582580
1.15 1.00000 −1.55218 1.00000 3.12516 −1.55218 2.04511 1.00000 −0.590743 3.12516
1.16 1.00000 −1.49577 1.00000 −3.91198 −1.49577 2.85048 1.00000 −0.762675 −3.91198
1.17 1.00000 −1.35974 1.00000 1.23410 −1.35974 −3.50749 1.00000 −1.15110 1.23410
1.18 1.00000 −1.34907 1.00000 0.876974 −1.34907 −1.47730 1.00000 −1.18002 0.876974
1.19 1.00000 −1.25114 1.00000 3.81116 −1.25114 −1.42946 1.00000 −1.43464 3.81116
1.20 1.00000 −1.23522 1.00000 −0.854459 −1.23522 4.26002 1.00000 −1.47423 −0.854459
See all 67 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.67
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3023\) \(1\)

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 6046.2.a.f ✓ 67
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6046.2.a.f ✓ 67 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6046))\):

\( T_{3}^{67} - 21 T_{3}^{66} + 75 T_{3}^{65} + 1451 T_{3}^{64} - 12648 T_{3}^{63} - 25428 T_{3}^{62} + \cdots + 12878086144 \) Copy content Toggle raw display
\( T_{11}^{67} - 56 T_{11}^{66} + 1153 T_{11}^{65} - 6491 T_{11}^{64} - 133984 T_{11}^{63} + \cdots + 66\!\cdots\!26 \) Copy content Toggle raw display