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Properties

Label 82.6.f.b
Level $82$
Weight $6$
Character orbit 82.f
Analytic conductor $13.151$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [82,6,Mod(23,82)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(82, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([9])) 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("82.23"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: \( N \) \(=\) \( 82 = 2 \cdot 41 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 82.f (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40] 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: \(13.1514732247\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 40 q^{2} - 160 q^{4} + 24 q^{5} - 100 q^{6} + 640 q^{8} - 4576 q^{9} - 96 q^{10} + 20 q^{11} + 400 q^{12} - 275 q^{13} + 1385 q^{15} - 2560 q^{16} - 105 q^{17} - 2476 q^{18} + 4720 q^{19} + 1424 q^{20}+ \cdots - 537690 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −1.23607 − 3.80423i − 30.8481i −12.9443 + 9.40456i −44.9680 + 32.6711i −117.353 + 38.1303i 185.924 + 60.4103i 51.7771 + 37.6183i −708.603 179.872 + 130.685i
23.2 −1.23607 − 3.80423i − 28.1115i −12.9443 + 9.40456i 68.7458 − 49.9467i −106.942 + 34.7477i −165.128 − 53.6532i 51.7771 + 37.6183i −547.254 −274.983 − 199.787i
23.3 −1.23607 − 3.80423i − 12.8019i −12.9443 + 9.40456i 0.908820 − 0.660296i −48.7014 + 15.8241i −131.516 − 42.7322i 51.7771 + 37.6183i 79.1107 −3.63528 − 2.64118i
23.4 −1.23607 − 3.80423i − 10.4652i −12.9443 + 9.40456i 16.4447 − 11.9478i −39.8119 + 12.9357i 166.594 + 54.1298i 51.7771 + 37.6183i 133.480 −65.7788 − 47.7911i
23.5 −1.23607 − 3.80423i − 7.71384i −12.9443 + 9.40456i −79.0566 + 57.4380i −29.3452 + 9.53483i −36.5570 − 11.8781i 51.7771 + 37.6183i 183.497 316.226 + 229.752i
23.6 −1.23607 − 3.80423i − 5.05453i −12.9443 + 9.40456i 58.6886 − 42.6398i −19.2286 + 6.24774i 76.6909 + 24.9184i 51.7771 + 37.6183i 217.452 −234.754 − 170.559i
23.7 −1.23607 − 3.80423i 12.1022i −12.9443 + 9.40456i 13.1781 − 9.57448i 46.0394 − 14.9591i −42.6653 − 13.8628i 51.7771 + 37.6183i 96.5377 −52.7126 − 38.2979i
23.8 −1.23607 − 3.80423i 18.1311i −12.9443 + 9.40456i −14.5751 + 10.5894i 68.9748 − 22.4113i −225.745 − 73.3489i 51.7771 + 37.6183i −85.7366 58.3003 + 42.3577i
23.9 −1.23607 − 3.80423i 22.0026i −12.9443 + 9.40456i −65.6913 + 47.7275i 83.7030 − 27.1968i 125.713 + 40.8467i 51.7771 + 37.6183i −241.116 262.765 + 190.910i
23.10 −1.23607 − 3.80423i 27.3707i −12.9443 + 9.40456i 80.2757 − 58.3237i 104.124 − 33.8320i 108.180 + 35.1500i 51.7771 + 37.6183i −506.154 −321.103 − 233.295i
25.1 −1.23607 + 3.80423i − 27.3707i −12.9443 − 9.40456i 80.2757 + 58.3237i 104.124 + 33.8320i 108.180 − 35.1500i 51.7771 − 37.6183i −506.154 −321.103 + 233.295i
25.2 −1.23607 + 3.80423i − 22.0026i −12.9443 − 9.40456i −65.6913 − 47.7275i 83.7030 + 27.1968i 125.713 − 40.8467i 51.7771 − 37.6183i −241.116 262.765 − 190.910i
25.3 −1.23607 + 3.80423i − 18.1311i −12.9443 − 9.40456i −14.5751 − 10.5894i 68.9748 + 22.4113i −225.745 + 73.3489i 51.7771 − 37.6183i −85.7366 58.3003 − 42.3577i
25.4 −1.23607 + 3.80423i − 12.1022i −12.9443 − 9.40456i 13.1781 + 9.57448i 46.0394 + 14.9591i −42.6653 + 13.8628i 51.7771 − 37.6183i 96.5377 −52.7126 + 38.2979i
25.5 −1.23607 + 3.80423i 5.05453i −12.9443 − 9.40456i 58.6886 + 42.6398i −19.2286 − 6.24774i 76.6909 − 24.9184i 51.7771 − 37.6183i 217.452 −234.754 + 170.559i
25.6 −1.23607 + 3.80423i 7.71384i −12.9443 − 9.40456i −79.0566 − 57.4380i −29.3452 − 9.53483i −36.5570 + 11.8781i 51.7771 − 37.6183i 183.497 316.226 − 229.752i
25.7 −1.23607 + 3.80423i 10.4652i −12.9443 − 9.40456i 16.4447 + 11.9478i −39.8119 − 12.9357i 166.594 − 54.1298i 51.7771 − 37.6183i 133.480 −65.7788 + 47.7911i
25.8 −1.23607 + 3.80423i 12.8019i −12.9443 − 9.40456i 0.908820 + 0.660296i −48.7014 − 15.8241i −131.516 + 42.7322i 51.7771 − 37.6183i 79.1107 −3.63528 + 2.64118i
25.9 −1.23607 + 3.80423i 28.1115i −12.9443 − 9.40456i 68.7458 + 49.9467i −106.942 − 34.7477i −165.128 + 53.6532i 51.7771 − 37.6183i −547.254 −274.983 + 199.787i
25.10 −1.23607 + 3.80423i 30.8481i −12.9443 − 9.40456i −44.9680 − 32.6711i −117.353 − 38.1303i 185.924 − 60.4103i 51.7771 − 37.6183i −708.603 179.872 − 130.685i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.10
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 82.6.f.b ✓ 40
41.f even 10 1 inner 82.6.f.b ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
82.6.f.b ✓ 40 1.a even 1 1 trivial
82.6.f.b ✓ 40 41.f even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} + 7148 T_{3}^{38} + 23240104 T_{3}^{36} + 45532216346 T_{3}^{34} + 60043359870554 T_{3}^{32} + \cdots + 12\!\cdots\!00 \) acting on \(S_{6}^{\mathrm{new}}(82, [\chi])\). Copy content Toggle raw display