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Properties

Label 6034.2.a.n
Level $6034$
Weight $2$
Character orbit 6034.a
Self dual yes
Analytic conductor $48.182$
Analytic rank $0$
Dimension $24$
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: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{2} + 7 q^{3} + 24 q^{4} + 8 q^{5} - 7 q^{6} - 24 q^{7} - 24 q^{8} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{2} + 7 q^{3} + 24 q^{4} + 8 q^{5} - 7 q^{6} - 24 q^{7} - 24 q^{8} + 19 q^{9} - 8 q^{10} + 15 q^{11} + 7 q^{12} - 7 q^{13} + 24 q^{14} + 13 q^{15} + 24 q^{16} - 5 q^{17} - 19 q^{18} + 6 q^{19} + 8 q^{20} - 7 q^{21} - 15 q^{22} + 3 q^{23} - 7 q^{24} + 12 q^{25} + 7 q^{26} + 22 q^{27} - 24 q^{28} + 5 q^{29} - 13 q^{30} + 13 q^{31} - 24 q^{32} - 8 q^{33} + 5 q^{34} - 8 q^{35} + 19 q^{36} + 2 q^{37} - 6 q^{38} + 7 q^{39} - 8 q^{40} + 25 q^{41} + 7 q^{42} - 15 q^{43} + 15 q^{44} + 41 q^{45} - 3 q^{46} + 35 q^{47} + 7 q^{48} + 24 q^{49} - 12 q^{50} + 31 q^{51} - 7 q^{52} + 2 q^{53} - 22 q^{54} + 14 q^{55} + 24 q^{56} - 13 q^{57} - 5 q^{58} + 35 q^{59} + 13 q^{60} - 7 q^{61} - 13 q^{62} - 19 q^{63} + 24 q^{64} - 4 q^{65} + 8 q^{66} + 10 q^{67} - 5 q^{68} + 6 q^{69} + 8 q^{70} + 58 q^{71} - 19 q^{72} + 9 q^{73} - 2 q^{74} + 7 q^{75} + 6 q^{76} - 15 q^{77} - 7 q^{78} + 31 q^{79} + 8 q^{80} + 16 q^{81} - 25 q^{82} - q^{83} - 7 q^{84} - 4 q^{85} + 15 q^{86} + 30 q^{87} - 15 q^{88} + 45 q^{89} - 41 q^{90} + 7 q^{91} + 3 q^{92} + 25 q^{93} - 35 q^{94} - 10 q^{95} - 7 q^{96} - 9 q^{97} - 24 q^{98} + 64 q^{99}+O(q^{100}) \) 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 −2.83864 1.00000 0.330019 2.83864 −1.00000 −1.00000 5.05788 −0.330019
1.2 −1.00000 −2.67063 1.00000 3.15601 2.67063 −1.00000 −1.00000 4.13228 −3.15601
1.3 −1.00000 −2.58057 1.00000 −2.90281 2.58057 −1.00000 −1.00000 3.65935 2.90281
1.4 −1.00000 −1.92027 1.00000 −2.53143 1.92027 −1.00000 −1.00000 0.687441 2.53143
1.5 −1.00000 −1.91791 1.00000 0.374562 1.91791 −1.00000 −1.00000 0.678384 −0.374562
1.6 −1.00000 −1.69422 1.00000 3.76480 1.69422 −1.00000 −1.00000 −0.129635 −3.76480
1.7 −1.00000 −1.44122 1.00000 −2.06061 1.44122 −1.00000 −1.00000 −0.922894 2.06061
1.8 −1.00000 −0.920246 1.00000 2.96551 0.920246 −1.00000 −1.00000 −2.15315 −2.96551
1.9 −1.00000 −0.797155 1.00000 1.40867 0.797155 −1.00000 −1.00000 −2.36454 −1.40867
1.10 −1.00000 −0.0752307 1.00000 −2.78876 0.0752307 −1.00000 −1.00000 −2.99434 2.78876
1.11 −1.00000 0.0287183 1.00000 1.06252 −0.0287183 −1.00000 −1.00000 −2.99918 −1.06252
1.12 −1.00000 0.430787 1.00000 1.34011 −0.430787 −1.00000 −1.00000 −2.81442 −1.34011
1.13 −1.00000 0.632516 1.00000 −0.321740 −0.632516 −1.00000 −1.00000 −2.59992 0.321740
1.14 −1.00000 0.685916 1.00000 1.44966 −0.685916 −1.00000 −1.00000 −2.52952 −1.44966
1.15 −1.00000 1.20386 1.00000 −2.97794 −1.20386 −1.00000 −1.00000 −1.55072 2.97794
1.16 −1.00000 1.44853 1.00000 −1.26403 −1.44853 −1.00000 −1.00000 −0.901751 1.26403
1.17 −1.00000 1.52105 1.00000 −3.19164 −1.52105 −1.00000 −1.00000 −0.686407 3.19164
1.18 −1.00000 1.56085 1.00000 −1.28710 −1.56085 −1.00000 −1.00000 −0.563752 1.28710
1.19 −1.00000 2.10114 1.00000 4.16694 −2.10114 −1.00000 −1.00000 1.41477 −4.16694
1.20 −1.00000 2.39691 1.00000 1.79254 −2.39691 −1.00000 −1.00000 2.74520 −1.79254
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
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.n ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6034.2.a.n ✓ 24 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6034))\):

\( T_{3}^{24} - 7 T_{3}^{23} - 21 T_{3}^{22} + 240 T_{3}^{21} + 15 T_{3}^{20} - 3403 T_{3}^{19} + 3352 T_{3}^{18} + \cdots + 80 \) Copy content Toggle raw display
\( T_{5}^{24} - 8 T_{5}^{23} - 34 T_{5}^{22} + 403 T_{5}^{21} + 238 T_{5}^{20} - 8597 T_{5}^{19} + \cdots + 268111 \) Copy content Toggle raw display
\( T_{11}^{24} - 15 T_{11}^{23} - 28 T_{11}^{22} + 1462 T_{11}^{21} - 4044 T_{11}^{20} - 51868 T_{11}^{19} + \cdots - 27387055 \) Copy content Toggle raw display