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Properties

Label 6022.2.a.b
Level $6022$
Weight $2$
Character orbit 6022.a
Self dual yes
Analytic conductor $48.086$
Analytic rank $1$
Dimension $54$
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: \(1\)
Dimension: \(54\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 54 q + 54 q^{2} - 22 q^{3} + 54 q^{4} - 14 q^{5} - 22 q^{6} - 20 q^{7} + 54 q^{8} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q + 54 q^{2} - 22 q^{3} + 54 q^{4} - 14 q^{5} - 22 q^{6} - 20 q^{7} + 54 q^{8} + 32 q^{9} - 14 q^{10} - 22 q^{11} - 22 q^{12} - 24 q^{13} - 20 q^{14} - 32 q^{15} + 54 q^{16} - 67 q^{17} + 32 q^{18} - 54 q^{19} - 14 q^{20} - 18 q^{21} - 22 q^{22} - 52 q^{23} - 22 q^{24} + 20 q^{25} - 24 q^{26} - 82 q^{27} - 20 q^{28} - 30 q^{29} - 32 q^{30} - 78 q^{31} + 54 q^{32} - 48 q^{33} - 67 q^{34} - 71 q^{35} + 32 q^{36} - 15 q^{37} - 54 q^{38} - 39 q^{39} - 14 q^{40} - 61 q^{41} - 18 q^{42} - 53 q^{43} - 22 q^{44} - 14 q^{45} - 52 q^{46} - 96 q^{47} - 22 q^{48} + 6 q^{49} + 20 q^{50} - 13 q^{51} - 24 q^{52} - 39 q^{53} - 82 q^{54} - 75 q^{55} - 20 q^{56} - 17 q^{57} - 30 q^{58} - 78 q^{59} - 32 q^{60} - 23 q^{61} - 78 q^{62} - 52 q^{63} + 54 q^{64} - 43 q^{65} - 48 q^{66} - 54 q^{67} - 67 q^{68} + 7 q^{69} - 71 q^{70} - 41 q^{71} + 32 q^{72} - 62 q^{73} - 15 q^{74} - 81 q^{75} - 54 q^{76} - 85 q^{77} - 39 q^{78} - 45 q^{79} - 14 q^{80} + 22 q^{81} - 61 q^{82} - 117 q^{83} - 18 q^{84} - 53 q^{86} - 84 q^{87} - 22 q^{88} - 60 q^{89} - 14 q^{90} - 60 q^{91} - 52 q^{92} + 26 q^{93} - 96 q^{94} - 44 q^{95} - 22 q^{96} - 48 q^{97} + 6 q^{98} + 7 q^{99}+O(q^{100}) \) 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 −3.41232 1.00000 −1.85663 −3.41232 3.02838 1.00000 8.64394 −1.85663
1.2 1.00000 −3.40089 1.00000 3.63611 −3.40089 −2.39111 1.00000 8.56603 3.63611
1.3 1.00000 −3.18590 1.00000 −1.36832 −3.18590 −3.72265 1.00000 7.14997 −1.36832
1.4 1.00000 −3.17082 1.00000 0.413863 −3.17082 −1.62952 1.00000 7.05407 0.413863
1.5 1.00000 −3.06406 1.00000 3.64593 −3.06406 0.427453 1.00000 6.38849 3.64593
1.6 1.00000 −2.86198 1.00000 −0.548157 −2.86198 2.20032 1.00000 5.19094 −0.548157
1.7 1.00000 −2.75819 1.00000 0.593272 −2.75819 4.20000 1.00000 4.60761 0.593272
1.8 1.00000 −2.72056 1.00000 1.67549 −2.72056 −1.89176 1.00000 4.40146 1.67549
1.9 1.00000 −2.71927 1.00000 −3.84841 −2.71927 −3.35161 1.00000 4.39444 −3.84841
1.10 1.00000 −2.46224 1.00000 −1.05561 −2.46224 2.87810 1.00000 3.06261 −1.05561
1.11 1.00000 −2.44368 1.00000 2.22438 −2.44368 −3.70978 1.00000 2.97155 2.22438
1.12 1.00000 −2.44169 1.00000 −4.13245 −2.44169 3.61005 1.00000 2.96185 −4.13245
1.13 1.00000 −1.92077 1.00000 3.09656 −1.92077 −3.07737 1.00000 0.689347 3.09656
1.14 1.00000 −1.82742 1.00000 −0.665814 −1.82742 1.40743 1.00000 0.339448 −0.665814
1.15 1.00000 −1.65622 1.00000 3.86194 −1.65622 0.0747047 1.00000 −0.256946 3.86194
1.16 1.00000 −1.62746 1.00000 2.10353 −1.62746 −4.91169 1.00000 −0.351369 2.10353
1.17 1.00000 −1.61281 1.00000 1.01065 −1.61281 0.302486 1.00000 −0.398851 1.01065
1.18 1.00000 −1.59103 1.00000 −2.29870 −1.59103 −3.52840 1.00000 −0.468610 −2.29870
1.19 1.00000 −1.52904 1.00000 −3.77354 −1.52904 0.908453 1.00000 −0.662042 −3.77354
1.20 1.00000 −1.50713 1.00000 −3.56652 −1.50713 −0.922198 1.00000 −0.728548 −3.56652
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.54
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.b ✓ 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6022.2.a.b ✓ 54 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{54} + 22 T_{3}^{53} + 145 T_{3}^{52} - 288 T_{3}^{51} - 7760 T_{3}^{50} - 21976 T_{3}^{49} + \cdots - 50985 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6022))\). Copy content Toggle raw display