Properties

Label 58.6.d.b
Level $58$
Weight $6$
Character orbit 58.d
Analytic conductor $9.302$
Analytic rank $0$
Dimension $42$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [58,6,Mod(7,58)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("58.7"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(58, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([6])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 58.d (of order \(7\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.30226154915\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(7\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 42 q + 28 q^{2} - q^{3} - 112 q^{4} + 156 q^{5} - 136 q^{6} + 85 q^{7} + 448 q^{8} - 840 q^{9} + 244 q^{10} - 242 q^{11} - 1024 q^{12} + 2047 q^{13} + 332 q^{14} - 1593 q^{15} - 1792 q^{16} - 3102 q^{17}+ \cdots - 1418124 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 0.890084 + 3.89971i −23.7231 − 11.4245i −14.4155 + 6.94214i −12.2274 − 53.5718i 23.4365 − 102.682i 83.2407 + 40.0866i −39.9033 − 50.0372i 280.760 + 352.062i 198.031 − 95.3667i
7.2 0.890084 + 3.89971i −18.5736 − 8.94458i −14.4155 + 6.94214i 15.2700 + 66.9022i 18.3492 − 80.3932i −79.9181 − 38.4865i −39.9033 − 50.0372i 113.466 + 142.282i −247.308 + 119.097i
7.3 0.890084 + 3.89971i −3.29761 − 1.58805i −14.4155 + 6.94214i −7.89025 − 34.5695i 3.25777 − 14.2732i 59.7909 + 28.7938i −39.9033 − 50.0372i −143.156 − 179.512i 127.788 − 61.5394i
7.4 0.890084 + 3.89971i −0.787677 − 0.379325i −14.4155 + 6.94214i 13.5822 + 59.5075i 0.778160 − 3.40934i 86.5987 + 41.7037i −39.9033 − 50.0372i −151.031 − 189.387i −219.973 + 105.933i
7.5 0.890084 + 3.89971i 8.75305 + 4.21524i −14.4155 + 6.94214i −14.6826 − 64.3287i −8.64729 + 37.8863i −184.088 − 88.6522i −39.9033 − 50.0372i −92.6605 − 116.193i 237.795 − 114.516i
7.6 0.890084 + 3.89971i 22.8815 + 11.0191i −14.4155 + 6.94214i 20.0449 + 87.8225i −22.6050 + 99.0392i −154.953 − 74.6214i −39.9033 − 50.0372i 250.633 + 314.284i −324.641 + 156.339i
7.7 0.890084 + 3.89971i 24.8611 + 11.9725i −14.4155 + 6.94214i −12.9827 − 56.8809i −24.5607 + 107.607i 202.187 + 97.3679i −39.9033 − 50.0372i 323.224 + 405.311i 210.263 − 101.257i
23.1 −2.49396 + 3.12733i −6.16274 + 27.0007i −3.56033 − 15.5988i 51.4577 − 64.5259i −69.0704 − 86.6116i 45.3465 − 198.676i 57.6620 + 27.7686i −472.124 − 227.363i 73.4601 + 321.850i
23.2 −2.49396 + 3.12733i −5.10634 + 22.3723i −3.56033 − 15.5988i −27.4330 + 34.3999i −57.2306 − 71.7649i −32.5049 + 142.413i 57.6620 + 27.7686i −255.512 − 123.048i −39.1629 − 171.584i
23.3 −2.49396 + 3.12733i −1.35661 + 5.94371i −3.56033 − 15.5988i 29.3005 − 36.7416i −15.2046 − 19.0659i −36.8157 + 161.300i 57.6620 + 27.7686i 185.448 + 89.3071i 41.8289 + 183.264i
23.4 −2.49396 + 3.12733i −0.657084 + 2.87887i −3.56033 − 15.5988i −3.09939 + 3.88652i −7.36443 − 9.23470i 27.5648 − 120.769i 57.6620 + 27.7686i 211.079 + 101.650i −4.42464 − 19.3856i
23.5 −2.49396 + 3.12733i 3.97982 − 17.4367i −3.56033 − 15.5988i −37.2328 + 46.6885i 44.6048 + 55.9326i −26.6089 + 116.581i 57.6620 + 27.7686i −69.2647 − 33.3561i −53.1530 − 232.878i
23.6 −2.49396 + 3.12733i 3.99269 − 17.4931i −3.56033 − 15.5988i −19.3448 + 24.2576i 44.7491 + 56.1135i 33.1561 − 145.267i 57.6620 + 27.7686i −71.1319 − 34.2553i −27.6163 − 120.995i
23.7 −2.49396 + 3.12733i 5.92556 − 25.9616i −3.56033 − 15.5988i 60.6463 − 76.0481i 66.4122 + 83.2783i −8.78135 + 38.4736i 57.6620 + 27.7686i −419.956 − 202.240i 86.5777 + 379.322i
25.1 0.890084 − 3.89971i −23.7231 + 11.4245i −14.4155 − 6.94214i −12.2274 + 53.5718i 23.4365 + 102.682i 83.2407 − 40.0866i −39.9033 + 50.0372i 280.760 − 352.062i 198.031 + 95.3667i
25.2 0.890084 − 3.89971i −18.5736 + 8.94458i −14.4155 − 6.94214i 15.2700 − 66.9022i 18.3492 + 80.3932i −79.9181 + 38.4865i −39.9033 + 50.0372i 113.466 − 142.282i −247.308 − 119.097i
25.3 0.890084 − 3.89971i −3.29761 + 1.58805i −14.4155 − 6.94214i −7.89025 + 34.5695i 3.25777 + 14.2732i 59.7909 − 28.7938i −39.9033 + 50.0372i −143.156 + 179.512i 127.788 + 61.5394i
25.4 0.890084 − 3.89971i −0.787677 + 0.379325i −14.4155 − 6.94214i 13.5822 − 59.5075i 0.778160 + 3.40934i 86.5987 − 41.7037i −39.9033 + 50.0372i −151.031 + 189.387i −219.973 − 105.933i
25.5 0.890084 − 3.89971i 8.75305 − 4.21524i −14.4155 − 6.94214i −14.6826 + 64.3287i −8.64729 − 37.8863i −184.088 + 88.6522i −39.9033 + 50.0372i −92.6605 + 116.193i 237.795 + 114.516i
25.6 0.890084 − 3.89971i 22.8815 − 11.0191i −14.4155 − 6.94214i 20.0449 − 87.8225i −22.6050 − 99.0392i −154.953 + 74.6214i −39.9033 + 50.0372i 250.633 − 314.284i −324.641 − 156.339i
See all 42 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 7.7
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 58.6.d.b ✓ 42
29.d even 7 1 inner 58.6.d.b ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.6.d.b ✓ 42 1.a even 1 1 trivial
58.6.d.b ✓ 42 29.d even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{42} + T_{3}^{41} + 1271 T_{3}^{40} - 13739 T_{3}^{39} + 1058140 T_{3}^{38} + \cdots + 26\!\cdots\!56 \) acting on \(S_{6}^{\mathrm{new}}(58, [\chi])\). Copy content Toggle raw display