Properties

Label 6041.2.a.c
Level $6041$
Weight $2$
Character orbit 6041.a
Self dual yes
Analytic conductor $48.238$
Analytic rank $1$
Dimension $83$
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] = [6041,2,Mod(1,6041)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6041, 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("6041.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6041 = 7 \cdot 863 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6041.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: \(48.2376278611\)
Analytic rank: \(1\)
Dimension: \(83\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 83 q - 8 q^{2} - 12 q^{3} + 48 q^{4} - 11 q^{5} - 8 q^{6} + 83 q^{7} - 18 q^{8} + 39 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 83 q - 8 q^{2} - 12 q^{3} + 48 q^{4} - 11 q^{5} - 8 q^{6} + 83 q^{7} - 18 q^{8} + 39 q^{9} - 20 q^{10} - 26 q^{11} - 14 q^{12} - 22 q^{13} - 8 q^{14} - 37 q^{15} - 10 q^{16} - 9 q^{17} - 27 q^{18} - 42 q^{19} - 22 q^{20} - 12 q^{21} - 44 q^{22} - 46 q^{23} - 24 q^{24} - 20 q^{25} - 9 q^{26} - 39 q^{27} + 48 q^{28} - 36 q^{29} - 11 q^{30} - 107 q^{31} - 19 q^{32} - 25 q^{33} - 24 q^{34} - 11 q^{35} - 32 q^{36} - 75 q^{37} - 16 q^{38} - 78 q^{39} - 34 q^{40} - 17 q^{41} - 8 q^{42} - 87 q^{43} - 32 q^{44} - 17 q^{45} - 56 q^{46} - 39 q^{47} - 16 q^{48} + 83 q^{49} - 26 q^{50} - 71 q^{51} - 53 q^{52} - 28 q^{53} - 25 q^{54} - 94 q^{55} - 18 q^{56} - 79 q^{57} - 69 q^{58} - 26 q^{59} - 43 q^{60} - 56 q^{61} - 6 q^{62} + 39 q^{63} - 108 q^{64} - 26 q^{65} + 10 q^{66} - 123 q^{67} - 11 q^{68} + 2 q^{69} - 20 q^{70} - 96 q^{71} - 11 q^{72} - 53 q^{73} - 26 q^{74} - 27 q^{75} - 65 q^{76} - 26 q^{77} - 43 q^{78} - 160 q^{79} + 12 q^{80} - 53 q^{81} - 20 q^{82} - 2 q^{83} - 14 q^{84} - 110 q^{85} + 24 q^{86} - 52 q^{87} - 79 q^{88} - 5 q^{89} - 4 q^{90} - 22 q^{91} - 51 q^{92} - 30 q^{93} - 9 q^{94} - 76 q^{95} - 3 q^{96} - 44 q^{97} - 8 q^{98} - 82 q^{99}+O(q^{100}) \) 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 −2.73002 −0.0142784 5.45300 −0.484021 0.0389802 1.00000 −9.42677 −2.99980 1.32139
1.2 −2.58676 −0.792673 4.69133 0.0189688 2.05046 1.00000 −6.96183 −2.37167 −0.0490676
1.3 −2.50552 2.64911 4.27764 −1.70463 −6.63740 1.00000 −5.70669 4.01777 4.27099
1.4 −2.50484 1.86042 4.27422 2.20949 −4.66004 1.00000 −5.69655 0.461144 −5.53442
1.5 −2.49510 −2.54767 4.22552 0.711578 6.35668 1.00000 −5.55291 3.49060 −1.77546
1.6 −2.36693 2.60578 3.60235 0.153258 −6.16771 1.00000 −3.79266 3.79011 −0.362751
1.7 −2.28673 0.00653898 3.22911 3.50344 −0.0149528 1.00000 −2.81064 −2.99996 −8.01141
1.8 −2.27758 −1.23058 3.18737 1.22240 2.80274 1.00000 −2.70432 −1.48568 −2.78412
1.9 −2.25188 −3.15195 3.07097 1.48598 7.09783 1.00000 −2.41170 6.93481 −3.34625
1.10 −2.19298 0.436268 2.80917 2.07310 −0.956728 1.00000 −1.77450 −2.80967 −4.54628
1.11 −2.16622 −1.36958 2.69252 −2.62137 2.96681 1.00000 −1.50014 −1.12425 5.67847
1.12 −2.15056 −0.709425 2.62493 −2.42733 1.52566 1.00000 −1.34395 −2.49672 5.22013
1.13 −2.14754 −2.08287 2.61192 −2.63815 4.47305 1.00000 −1.31412 1.33837 5.66553
1.14 −2.14224 0.933242 2.58920 −2.33430 −1.99923 1.00000 −1.26222 −2.12906 5.00064
1.15 −1.96601 2.65332 1.86520 0.0208201 −5.21646 1.00000 0.265017 4.04013 −0.0409325
1.16 −1.90567 −1.35998 1.63156 3.50685 2.59167 1.00000 0.702124 −1.15045 −6.68289
1.17 −1.79680 0.970719 1.22848 −1.22425 −1.74419 1.00000 1.38627 −2.05770 2.19973
1.18 −1.78752 2.42627 1.19524 −3.07648 −4.33701 1.00000 1.43852 2.88678 5.49928
1.19 −1.77588 −2.45572 1.15374 3.79745 4.36107 1.00000 1.50285 3.03058 −6.74380
1.20 −1.71609 −2.62448 0.944964 −3.91107 4.50383 1.00000 1.81054 3.88787 6.71175
See all 83 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.83
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(863\) \(-1\)

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 6041.2.a.c 83
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6041.2.a.c 83 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{83} + 8 T_{2}^{82} - 75 T_{2}^{81} - 754 T_{2}^{80} + 2369 T_{2}^{79} + 33869 T_{2}^{78} - 34220 T_{2}^{77} - 964839 T_{2}^{76} - 103069 T_{2}^{75} + 19565838 T_{2}^{74} + 16244910 T_{2}^{73} - 300488895 T_{2}^{72} + \cdots - 299 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6041))\). Copy content Toggle raw display