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Properties

Label 847.2.n.k
Level $847$
Weight $2$
Character orbit 847.n
Analytic conductor $6.763$
Analytic rank $0$
Dimension $48$
Inner twists $16$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [847,2,Mod(9,847)] 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("847.9"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(847, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([10, 18])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.n (of order \(15\), degree \(8\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,-2,4,-4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(6\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{3} + 4 q^{4} - 4 q^{5} + 8 q^{9} + 56 q^{12} - 20 q^{14} - 28 q^{15} - 4 q^{16} + 56 q^{20} + 16 q^{23} - 2 q^{25} + 6 q^{26} - 32 q^{27} - 14 q^{31} - 36 q^{36} - 4 q^{37} - 4 q^{38} - 24 q^{42}+ \cdots + 20 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −1.57025 − 1.74394i −2.08758 − 0.929451i −0.366583 + 3.48781i 3.43038 + 0.729150i 1.65712 + 5.10009i −1.84410 − 1.89718i 2.86111 − 2.07872i 1.48673 + 1.65118i −4.11497 − 7.12733i
9.2 −0.892261 − 0.990956i 2.29028 + 1.01970i 0.0231924 − 0.220661i −0.217028 − 0.0461306i −1.03305 − 3.17940i −0.899665 − 2.48809i −2.39695 + 1.74148i 2.19819 + 2.44134i 0.147932 + 0.256225i
9.3 −0.565745 − 0.628323i −1.11624 − 0.496982i 0.134334 − 1.27810i −1.25706 − 0.267196i 0.319241 + 0.982524i −0.993961 + 2.45195i −2.24710 + 1.63261i −1.00839 − 1.11993i 0.543289 + 0.941004i
9.4 0.565745 + 0.628323i −1.11624 − 0.496982i 0.134334 − 1.27810i −1.25706 − 0.267196i −0.319241 − 0.982524i 0.993961 − 2.45195i 2.24710 − 1.63261i −1.00839 − 1.11993i −0.543289 − 0.941004i
9.5 0.892261 + 0.990956i 2.29028 + 1.01970i 0.0231924 − 0.220661i −0.217028 − 0.0461306i 1.03305 + 3.17940i 0.899665 + 2.48809i 2.39695 − 1.74148i 2.19819 + 2.44134i −0.147932 − 0.256225i
9.6 1.57025 + 1.74394i −2.08758 − 0.929451i −0.366583 + 3.48781i 3.43038 + 0.729150i −1.65712 − 5.10009i 1.84410 + 1.89718i −2.86111 + 2.07872i 1.48673 + 1.65118i 4.11497 + 7.12733i
81.1 −0.245297 + 2.33385i −1.52906 − 1.69819i −3.43038 − 0.729150i −3.20382 − 1.42643i 4.33840 − 3.15203i −1.23447 − 2.34010i 1.09285 − 3.36344i −0.232249 + 2.20970i 4.11497 − 7.12733i
81.2 −0.139385 + 1.32616i 1.67752 + 1.86308i 0.217028 + 0.0461306i 0.202694 + 0.0902452i −2.70456 + 1.96498i −2.08830 − 1.62449i −0.915552 + 2.81778i −0.343391 + 3.26715i −0.147932 + 0.256225i
81.3 −0.0883780 + 0.840861i −0.817595 − 0.908031i 1.25706 + 0.267196i 1.17404 + 0.522715i 0.835785 − 0.607233i 2.63909 − 0.187620i −0.858314 + 2.64162i 0.157526 − 1.49876i −0.543289 + 0.941004i
81.4 0.0883780 − 0.840861i −0.817595 − 0.908031i 1.25706 + 0.267196i 1.17404 + 0.522715i −0.835785 + 0.607233i −2.63909 + 0.187620i 0.858314 − 2.64162i 0.157526 − 1.49876i 0.543289 − 0.941004i
81.5 0.139385 − 1.32616i 1.67752 + 1.86308i 0.217028 + 0.0461306i 0.202694 + 0.0902452i 2.70456 − 1.96498i 2.08830 + 1.62449i 0.915552 − 2.81778i −0.343391 + 3.26715i 0.147932 − 0.256225i
81.6 0.245297 − 2.33385i −1.52906 − 1.69819i −3.43038 − 0.729150i −3.20382 − 1.42643i −4.33840 + 3.15203i 1.23447 + 2.34010i −1.09285 + 3.36344i −0.232249 + 2.20970i −4.11497 + 7.12733i
130.1 −2.29542 + 0.487907i 0.238862 + 2.27262i 3.20382 − 1.42643i −2.34665 + 2.60622i −1.65712 − 5.10009i 2.37419 + 1.16758i −2.86111 + 2.07872i −2.17332 + 0.461954i 4.11497 − 7.12733i
130.2 −1.30432 + 0.277243i −0.262055 − 2.49328i −0.202694 + 0.0902452i 0.148464 − 0.164886i 1.03305 + 3.17940i 2.64433 + 0.0867694i 2.39695 − 1.74148i −3.21335 + 0.683020i −0.147932 + 0.256225i
130.3 −0.827016 + 0.175788i 0.127721 + 1.21518i −1.17404 + 0.522715i 0.859928 − 0.955047i −0.319241 − 0.982524i −2.02479 + 1.70301i 2.24710 − 1.63261i 1.47409 − 0.313327i −0.543289 + 0.941004i
130.4 0.827016 − 0.175788i 0.127721 + 1.21518i −1.17404 + 0.522715i 0.859928 − 0.955047i 0.319241 + 0.982524i 2.02479 − 1.70301i −2.24710 + 1.63261i 1.47409 − 0.313327i 0.543289 − 0.941004i
130.5 1.30432 − 0.277243i −0.262055 − 2.49328i −0.202694 + 0.0902452i 0.148464 − 0.164886i −1.03305 − 3.17940i −2.64433 − 0.0867694i −2.39695 + 1.74148i −3.21335 + 0.683020i 0.147932 − 0.256225i
130.6 2.29542 − 0.487907i 0.238862 + 2.27262i 3.20382 − 1.42643i −2.34665 + 2.60622i 1.65712 + 5.10009i −2.37419 − 1.16758i 2.86111 − 2.07872i −2.17332 + 0.461954i −4.11497 + 7.12733i
366.1 −0.245297 − 2.33385i −1.52906 + 1.69819i −3.43038 + 0.729150i −3.20382 + 1.42643i 4.33840 + 3.15203i −1.23447 + 2.34010i 1.09285 + 3.36344i −0.232249 − 2.20970i 4.11497 + 7.12733i
366.2 −0.139385 − 1.32616i 1.67752 − 1.86308i 0.217028 − 0.0461306i 0.202694 − 0.0902452i −2.70456 − 1.96498i −2.08830 + 1.62449i −0.915552 − 2.81778i −0.343391 − 3.26715i −0.147932 − 0.256225i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
77.h odd 6 1 inner
77.m even 15 3 inner
77.o odd 30 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.n.k 48
7.c even 3 1 inner 847.2.n.k 48
11.b odd 2 1 inner 847.2.n.k 48
11.c even 5 1 847.2.e.e ✓ 12
11.c even 5 3 inner 847.2.n.k 48
11.d odd 10 1 847.2.e.e ✓ 12
11.d odd 10 3 inner 847.2.n.k 48
77.h odd 6 1 inner 847.2.n.k 48
77.m even 15 1 847.2.e.e ✓ 12
77.m even 15 3 inner 847.2.n.k 48
77.m even 15 1 5929.2.a.bk 6
77.n even 30 1 5929.2.a.bl 6
77.o odd 30 1 847.2.e.e ✓ 12
77.o odd 30 3 inner 847.2.n.k 48
77.o odd 30 1 5929.2.a.bk 6
77.p odd 30 1 5929.2.a.bl 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.e.e ✓ 12 11.c even 5 1
847.2.e.e ✓ 12 11.d odd 10 1
847.2.e.e ✓ 12 77.m even 15 1
847.2.e.e ✓ 12 77.o odd 30 1
847.2.n.k 48 1.a even 1 1 trivial
847.2.n.k 48 7.c even 3 1 inner
847.2.n.k 48 11.b odd 2 1 inner
847.2.n.k 48 11.c even 5 3 inner
847.2.n.k 48 11.d odd 10 3 inner
847.2.n.k 48 77.h odd 6 1 inner
847.2.n.k 48 77.m even 15 3 inner
847.2.n.k 48 77.o odd 30 3 inner
5929.2.a.bk 6 77.m even 15 1
5929.2.a.bk 6 77.o odd 30 1
5929.2.a.bl 6 77.n even 30 1
5929.2.a.bl 6 77.p odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} - 8 T_{2}^{46} + 15 T_{2}^{44} + 166 T_{2}^{42} - 1384 T_{2}^{40} + 7678 T_{2}^{38} + \cdots + 5764801 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display