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Properties

Label 51.6.h.a
Level $51$
Weight $6$
Character orbit 51.h
Analytic conductor $8.180$
Analytic rank $0$
Dimension $56$
Inner twists $2$

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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] = [51,6,Mod(19,51)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(51, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 7])) N = Newforms(chi, 6, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("51.19"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: \( N \) \(=\) \( 51 = 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 51.h (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.17957481046\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 88 q^{5} + 360 q^{6} + 1280 q^{10} - 632 q^{11} - 4256 q^{14} - 8192 q^{16} + 6248 q^{17} + 4152 q^{19} - 16384 q^{20} - 19592 q^{22} - 3608 q^{23} + 10440 q^{24} + 30384 q^{25} - 19936 q^{26} + 13624 q^{28}+ \cdots - 51192 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −7.76922 − 7.76922i −8.31492 + 3.44415i 88.7215i 1.79786 + 4.34041i 91.3588 + 37.8420i −2.94505 + 7.10998i 440.682 − 440.682i 57.2756 − 57.2756i 19.7537 − 47.6896i
19.2 −6.70687 − 6.70687i 8.31492 − 3.44415i 57.9642i 14.8469 + 35.8437i −78.8665 − 32.6676i 90.1287 − 217.590i 174.139 − 174.139i 57.2756 − 57.2756i 140.822 − 339.975i
19.3 −4.62537 − 4.62537i −8.31492 + 3.44415i 10.7882i 6.51557 + 15.7300i 54.3901 + 22.5291i −31.5050 + 76.0599i −98.1126 + 98.1126i 57.2756 − 57.2756i 42.6201 − 102.894i
19.4 −4.34108 − 4.34108i −8.31492 + 3.44415i 5.69003i −29.2662 − 70.6549i 51.0471 + 21.1444i 43.1831 − 104.253i −114.214 + 114.214i 57.2756 − 57.2756i −179.672 + 433.766i
19.5 −3.44799 − 3.44799i 8.31492 − 3.44415i − 8.22272i 33.9684 + 82.0071i −40.5452 − 16.7944i −45.3741 + 109.543i −138.688 + 138.688i 57.2756 − 57.2756i 165.637 − 399.883i
19.6 −1.90101 − 1.90101i 8.31492 − 3.44415i − 24.7723i −15.9904 − 38.6041i −22.3541 − 9.25937i −7.02861 + 16.9686i −107.925 + 107.925i 57.2756 − 57.2756i −42.9890 + 103.785i
19.7 −1.06332 − 1.06332i −8.31492 + 3.44415i − 29.7387i 29.0694 + 70.1798i 12.5037 + 5.17918i 48.3729 − 116.783i −65.6480 + 65.6480i 57.2756 − 57.2756i 43.7135 − 105.534i
19.8 0.568402 + 0.568402i −8.31492 + 3.44415i − 31.3538i 14.2339 + 34.3638i −6.68388 − 2.76855i −45.3488 + 109.482i 36.0105 − 36.0105i 57.2756 − 57.2756i −11.4418 + 27.6231i
19.9 1.70560 + 1.70560i 8.31492 − 3.44415i − 26.1818i 18.8073 + 45.4050i 20.0563 + 8.30758i 31.6187 − 76.3343i 99.2350 − 99.2350i 57.2756 − 57.2756i −45.3649 + 109.521i
19.10 3.88932 + 3.88932i −8.31492 + 3.44415i − 1.74633i −17.4864 − 42.2159i −45.7348 − 18.9440i 37.7648 − 91.1724i 131.250 − 131.250i 57.2756 − 57.2756i 96.1811 − 232.202i
19.11 4.00033 + 4.00033i 8.31492 − 3.44415i 0.00529928i −26.4070 − 63.7521i 47.0402 + 19.4847i 33.4286 − 80.7039i 127.989 − 127.989i 57.2756 − 57.2756i 149.393 − 360.666i
19.12 5.68094 + 5.68094i 8.31492 − 3.44415i 32.5461i 5.40158 + 13.0406i 66.8025 + 27.6705i −93.5505 + 225.851i −3.10250 + 3.10250i 57.2756 − 57.2756i −43.3966 + 104.769i
19.13 6.80846 + 6.80846i −8.31492 + 3.44415i 60.7102i 3.21855 + 7.77028i −80.0611 − 33.1624i −45.3056 + 109.377i −195.472 + 195.472i 57.2756 − 57.2756i −30.9902 + 74.8170i
19.14 7.20182 + 7.20182i 8.31492 − 3.44415i 71.7323i 24.1431 + 58.2867i 84.6866 + 35.0784i 69.9994 − 168.994i −286.145 + 286.145i 57.2756 − 57.2756i −245.895 + 593.644i
25.1 −7.65585 + 7.65585i 3.44415 − 8.31492i − 85.2240i −89.6080 − 37.1168i 37.2898 + 90.0256i −8.69145 + 3.60012i 407.474 + 407.474i −57.2756 − 57.2756i 970.186 − 401.864i
25.2 −5.79991 + 5.79991i −3.44415 + 8.31492i − 35.2780i −42.7660 − 17.7142i −28.2500 − 68.2015i 151.688 − 62.8311i 19.0118 + 19.0118i −57.2756 − 57.2756i 350.780 − 145.298i
25.3 −5.30075 + 5.30075i 3.44415 − 8.31492i − 24.1960i 28.6013 + 11.8471i 25.8187 + 62.3319i −117.456 + 48.6520i −41.3672 − 41.3672i −57.2756 − 57.2756i −214.407 + 88.8103i
25.4 −4.75794 + 4.75794i −3.44415 + 8.31492i − 13.2760i −13.0733 − 5.41515i −23.1748 − 55.9489i −112.019 + 46.3996i −89.0879 − 89.0879i −57.2756 − 57.2756i 87.9671 − 36.4371i
25.5 −2.64592 + 2.64592i −3.44415 + 8.31492i 17.9983i 96.6645 + 40.0397i −12.8876 − 31.1135i 132.408 − 54.8452i −132.291 − 132.291i −57.2756 − 57.2756i −361.708 + 149.824i
25.6 −1.51861 + 1.51861i 3.44415 − 8.31492i 27.3876i 21.2252 + 8.79177i 7.39681 + 17.8575i −38.4004 + 15.9060i −90.1869 − 90.1869i −57.2756 − 57.2756i −45.5842 + 18.8816i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 51.6.h.a ✓ 56
17.d even 8 1 inner 51.6.h.a ✓ 56
17.e odd 16 1 867.6.a.t 28
17.e odd 16 1 867.6.a.u 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.6.h.a ✓ 56 1.a even 1 1 trivial
51.6.h.a ✓ 56 17.d even 8 1 inner
867.6.a.t 28 17.e odd 16 1
867.6.a.u 28 17.e odd 16 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(51, [\chi])\).