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Properties

Label 6022.2.a.c
Level $6022$
Weight $2$
Character orbit 6022.a
Self dual yes
Analytic conductor $48.086$
Analytic rank $0$
Dimension $61$
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] = [6022,2,Mod(1,6022)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6022, 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("6022.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6022 = 2 \cdot 3011 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6022.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: \(48.0859120972\)
Analytic rank: \(0\)
Dimension: \(61\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 61 q - 61 q^{2} + 8 q^{3} + 61 q^{4} + 16 q^{5} - 8 q^{6} + 2 q^{7} - 61 q^{8} + 67 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 61 q - 61 q^{2} + 8 q^{3} + 61 q^{4} + 16 q^{5} - 8 q^{6} + 2 q^{7} - 61 q^{8} + 67 q^{9} - 16 q^{10} + 14 q^{11} + 8 q^{12} + 27 q^{13} - 2 q^{14} + 61 q^{16} + 60 q^{17} - 67 q^{18} - 29 q^{19} + 16 q^{20} + 30 q^{21} - 14 q^{22} + 39 q^{23} - 8 q^{24} + 61 q^{25} - 27 q^{26} + 32 q^{27} + 2 q^{28} + 36 q^{29} - 40 q^{31} - 61 q^{32} + 28 q^{33} - 60 q^{34} + 55 q^{35} + 67 q^{36} + 20 q^{37} + 29 q^{38} + 17 q^{39} - 16 q^{40} + 44 q^{41} - 30 q^{42} + 22 q^{43} + 14 q^{44} + 52 q^{45} - 39 q^{46} + 64 q^{47} + 8 q^{48} + 49 q^{49} - 61 q^{50} + 15 q^{51} + 27 q^{52} + 65 q^{53} - 32 q^{54} + 5 q^{55} - 2 q^{56} + 9 q^{57} - 36 q^{58} + 2 q^{59} + 45 q^{61} + 40 q^{62} + 28 q^{63} + 61 q^{64} + 41 q^{65} - 28 q^{66} - 20 q^{67} + 60 q^{68} + 21 q^{69} - 55 q^{70} - q^{71} - 67 q^{72} + 25 q^{73} - 20 q^{74} + 27 q^{75} - 29 q^{76} + 131 q^{77} - 17 q^{78} - 17 q^{79} + 16 q^{80} + 85 q^{81} - 44 q^{82} + 104 q^{83} + 30 q^{84} + 44 q^{85} - 22 q^{86} + 86 q^{87} - 14 q^{88} + 32 q^{89} - 52 q^{90} - 68 q^{91} + 39 q^{92} + 52 q^{93} - 64 q^{94} + 58 q^{95} - 8 q^{96} + 5 q^{97} - 49 q^{98} + 15 q^{99}+O(q^{100}) \) 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 −3.31019 1.00000 1.46037 3.31019 −0.428308 −1.00000 7.95734 −1.46037
1.2 −1.00000 −3.29982 1.00000 3.94529 3.29982 1.75060 −1.00000 7.88881 −3.94529
1.3 −1.00000 −2.88590 1.00000 −3.47728 2.88590 −4.98917 −1.00000 5.32845 3.47728
1.4 −1.00000 −2.85737 1.00000 −1.73685 2.85737 −2.09511 −1.00000 5.16453 1.73685
1.5 −1.00000 −2.74236 1.00000 0.141045 2.74236 −0.820556 −1.00000 4.52053 −0.141045
1.6 −1.00000 −2.71725 1.00000 0.844160 2.71725 4.53942 −1.00000 4.38347 −0.844160
1.7 −1.00000 −2.65789 1.00000 −1.96173 2.65789 1.75208 −1.00000 4.06439 1.96173
1.8 −1.00000 −2.60317 1.00000 2.24226 2.60317 −4.82749 −1.00000 3.77649 −2.24226
1.9 −1.00000 −2.51016 1.00000 1.21389 2.51016 4.01976 −1.00000 3.30092 −1.21389
1.10 −1.00000 −2.42852 1.00000 2.56237 2.42852 −2.36456 −1.00000 2.89773 −2.56237
1.11 −1.00000 −2.37453 1.00000 3.86248 2.37453 −2.00027 −1.00000 2.63837 −3.86248
1.12 −1.00000 −2.35686 1.00000 −2.52767 2.35686 0.0751824 −1.00000 2.55481 2.52767
1.13 −1.00000 −1.81947 1.00000 −2.66446 1.81947 1.39987 −1.00000 0.310478 2.66446
1.14 −1.00000 −1.69347 1.00000 −1.85965 1.69347 −0.879374 −1.00000 −0.132168 1.85965
1.15 −1.00000 −1.57891 1.00000 −2.16938 1.57891 −1.92539 −1.00000 −0.507029 2.16938
1.16 −1.00000 −1.42902 1.00000 1.04628 1.42902 2.47786 −1.00000 −0.957895 −1.04628
1.17 −1.00000 −1.39055 1.00000 1.34087 1.39055 −2.43238 −1.00000 −1.06637 −1.34087
1.18 −1.00000 −1.34981 1.00000 0.909179 1.34981 3.61244 −1.00000 −1.17801 −0.909179
1.19 −1.00000 −1.27129 1.00000 2.47111 1.27129 −1.62843 −1.00000 −1.38382 −2.47111
1.20 −1.00000 −1.13476 1.00000 3.93476 1.13476 4.57427 −1.00000 −1.71231 −3.93476
See all 61 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.61
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3011\) \(-1\)

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 6022.2.a.c ✓ 61
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6022.2.a.c ✓ 61 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{61} - 8 T_{3}^{60} - 93 T_{3}^{59} + 888 T_{3}^{58} + 3750 T_{3}^{57} - 46222 T_{3}^{56} + \cdots - 113498752 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6022))\). Copy content Toggle raw display