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Properties

Label 6034.2.a.p
Level $6034$
Weight $2$
Character orbit 6034.a
Self dual yes
Analytic conductor $48.182$
Analytic rank $0$
Dimension $27$
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] = [6034,2,Mod(1,6034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6034, 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("6034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6034 = 2 \cdot 7 \cdot 431 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6034.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: \(48.1817325796\)
Analytic rank: \(0\)
Dimension: \(27\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 27 q - 27 q^{2} + 4 q^{3} + 27 q^{4} + 9 q^{5} - 4 q^{6} + 27 q^{7} - 27 q^{8} + 35 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 27 q - 27 q^{2} + 4 q^{3} + 27 q^{4} + 9 q^{5} - 4 q^{6} + 27 q^{7} - 27 q^{8} + 35 q^{9} - 9 q^{10} + 24 q^{11} + 4 q^{12} - 13 q^{13} - 27 q^{14} + 16 q^{15} + 27 q^{16} - 5 q^{17} - 35 q^{18} + q^{19} + 9 q^{20} + 4 q^{21} - 24 q^{22} + 32 q^{23} - 4 q^{24} + 30 q^{25} + 13 q^{26} + q^{27} + 27 q^{28} + 26 q^{29} - 16 q^{30} + 21 q^{31} - 27 q^{32} + 7 q^{33} + 5 q^{34} + 9 q^{35} + 35 q^{36} + 4 q^{37} - q^{38} + 13 q^{39} - 9 q^{40} + 31 q^{41} - 4 q^{42} - 13 q^{43} + 24 q^{44} + 19 q^{45} - 32 q^{46} + 41 q^{47} + 4 q^{48} + 27 q^{49} - 30 q^{50} + 21 q^{51} - 13 q^{52} + 29 q^{53} - q^{54} + 9 q^{55} - 27 q^{56} - 26 q^{58} + 36 q^{59} + 16 q^{60} + q^{61} - 21 q^{62} + 35 q^{63} + 27 q^{64} + 46 q^{65} - 7 q^{66} - 2 q^{67} - 5 q^{68} + 43 q^{69} - 9 q^{70} + 70 q^{71} - 35 q^{72} - 21 q^{73} - 4 q^{74} + 37 q^{75} + q^{76} + 24 q^{77} - 13 q^{78} + 19 q^{79} + 9 q^{80} + 67 q^{81} - 31 q^{82} + 25 q^{83} + 4 q^{84} - 6 q^{85} + 13 q^{86} - 9 q^{87} - 24 q^{88} + 85 q^{89} - 19 q^{90} - 13 q^{91} + 32 q^{92} + 23 q^{93} - 41 q^{94} + 77 q^{95} - 4 q^{96} - 2 q^{97} - 27 q^{98} + 38 q^{99}+O(q^{100}) \) 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 −3.22794 1.00000 1.50843 3.22794 1.00000 −1.00000 7.41957 −1.50843
1.2 −1.00000 −3.20972 1.00000 −3.26323 3.20972 1.00000 −1.00000 7.30232 3.26323
1.3 −1.00000 −3.03412 1.00000 1.18387 3.03412 1.00000 −1.00000 6.20591 −1.18387
1.4 −1.00000 −2.59011 1.00000 1.58456 2.59011 1.00000 −1.00000 3.70867 −1.58456
1.5 −1.00000 −2.42517 1.00000 2.60497 2.42517 1.00000 −1.00000 2.88143 −2.60497
1.6 −1.00000 −1.90367 1.00000 −3.72741 1.90367 1.00000 −1.00000 0.623956 3.72741
1.7 −1.00000 −1.81149 1.00000 −1.26479 1.81149 1.00000 −1.00000 0.281479 1.26479
1.8 −1.00000 −1.32317 1.00000 −0.695345 1.32317 1.00000 −1.00000 −1.24923 0.695345
1.9 −1.00000 −0.743500 1.00000 −1.77052 0.743500 1.00000 −1.00000 −2.44721 1.77052
1.10 −1.00000 −0.679472 1.00000 −2.19065 0.679472 1.00000 −1.00000 −2.53832 2.19065
1.11 −1.00000 −0.394534 1.00000 4.36123 0.394534 1.00000 −1.00000 −2.84434 −4.36123
1.12 −1.00000 −0.302933 1.00000 3.48262 0.302933 1.00000 −1.00000 −2.90823 −3.48262
1.13 −1.00000 −0.252343 1.00000 2.45924 0.252343 1.00000 −1.00000 −2.93632 −2.45924
1.14 −1.00000 −0.222181 1.00000 0.181430 0.222181 1.00000 −1.00000 −2.95064 −0.181430
1.15 −1.00000 0.396413 1.00000 −1.85319 −0.396413 1.00000 −1.00000 −2.84286 1.85319
1.16 −1.00000 0.482352 1.00000 2.96654 −0.482352 1.00000 −1.00000 −2.76734 −2.96654
1.17 −1.00000 0.975695 1.00000 −1.65902 −0.975695 1.00000 −1.00000 −2.04802 1.65902
1.18 −1.00000 1.58330 1.00000 −2.70248 −1.58330 1.00000 −1.00000 −0.493160 2.70248
1.19 −1.00000 1.64919 1.00000 0.930561 −1.64919 1.00000 −1.00000 −0.280182 −0.930561
1.20 −1.00000 1.94651 1.00000 3.30450 −1.94651 1.00000 −1.00000 0.788913 −3.30450
See all 27 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.27
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(431\) \(1\)

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 6034.2.a.p ✓ 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6034.2.a.p ✓ 27 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6034))\):

\( T_{3}^{27} - 4 T_{3}^{26} - 50 T_{3}^{25} + 213 T_{3}^{24} + 1045 T_{3}^{23} - 4856 T_{3}^{22} + \cdots - 3008 \) Copy content Toggle raw display
\( T_{5}^{27} - 9 T_{5}^{26} - 42 T_{5}^{25} + 571 T_{5}^{24} + 347 T_{5}^{23} - 15567 T_{5}^{22} + \cdots - 17344112 \) Copy content Toggle raw display
\( T_{11}^{27} - 24 T_{11}^{26} + 127 T_{11}^{25} + 1378 T_{11}^{24} - 16830 T_{11}^{23} + \cdots - 328030013488 \) Copy content Toggle raw display