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Properties

Label 82.5.e.a
Level $82$
Weight $5$
Character orbit 82.e
Analytic conductor $8.476$
Analytic rank $0$
Dimension $28$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [82,5,Mod(3,82)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(82, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([3])) N = Newforms(chi, 5, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("82.3"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Level: \( N \) \(=\) \( 82 = 2 \cdot 41 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 82.e (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [28,-56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.47633697288\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 56 q^{2} + 16 q^{3} - 16 q^{6} + 448 q^{8} + 24 q^{9} + 328 q^{11} - 64 q^{12} - 348 q^{13} + 192 q^{14} + 1036 q^{15} - 1792 q^{16} - 1832 q^{17} - 96 q^{18} - 504 q^{19} + 2004 q^{21} - 856 q^{22}+ \cdots + 11544 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −2.00000 + 2.00000i −6.17854 + 14.9163i − 8.00000i −15.3528 − 15.3528i −17.4755 − 42.1897i 22.0538 − 53.2426i 16.0000 + 16.0000i −127.046 − 127.046i 61.4114
3.2 −2.00000 + 2.00000i −3.50051 + 8.45098i − 8.00000i 13.5876 + 13.5876i −9.90093 − 23.9030i −23.8235 + 57.5149i 16.0000 + 16.0000i −1.88977 − 1.88977i −54.3502
3.3 −2.00000 + 2.00000i −1.67187 + 4.03626i − 8.00000i 22.2467 + 22.2467i −4.72877 − 11.4163i 30.2176 − 72.9518i 16.0000 + 16.0000i 43.7794 + 43.7794i −88.9870
3.4 −2.00000 + 2.00000i 0.493125 − 1.19051i − 8.00000i −21.5675 − 21.5675i 1.39477 + 3.36727i −12.4355 + 30.0219i 16.0000 + 16.0000i 56.1015 + 56.1015i 86.2700
3.5 −2.00000 + 2.00000i 2.85757 − 6.89878i − 8.00000i 3.97800 + 3.97800i 8.08243 + 19.5127i 9.68798 − 23.3888i 16.0000 + 16.0000i 17.8481 + 17.8481i −15.9120
3.6 −2.00000 + 2.00000i 5.22888 − 12.6236i − 8.00000i 25.8818 + 25.8818i 14.7895 + 35.7050i −20.1271 + 48.5912i 16.0000 + 16.0000i −74.7392 − 74.7392i −103.527
3.7 −2.00000 + 2.00000i 5.35713 − 12.9333i − 8.00000i −14.6317 − 14.6317i 15.1523 + 36.5808i 11.3972 − 27.5153i 16.0000 + 16.0000i −81.2948 − 81.2948i 58.5267
27.1 −2.00000 − 2.00000i −14.6503 + 6.06835i 8.00000i −28.4451 + 28.4451i 41.4373 + 17.1639i −67.9576 + 28.1489i 16.0000 − 16.0000i 120.530 − 120.530i 113.780
27.2 −2.00000 − 2.00000i −9.30077 + 3.85250i 8.00000i 16.4694 − 16.4694i 26.3065 + 10.8965i 32.0596 − 13.2795i 16.0000 − 16.0000i 14.3868 − 14.3868i −65.8775
27.3 −2.00000 − 2.00000i −4.49014 + 1.85988i 8.00000i −21.9230 + 21.9230i 12.7000 + 5.26053i 50.6726 − 20.9893i 16.0000 − 16.0000i −40.5734 + 40.5734i 87.6922
27.4 −2.00000 − 2.00000i 2.35294 − 0.974620i 8.00000i −5.07070 + 5.07070i −6.65512 − 2.75664i 13.7461 − 5.69383i 16.0000 − 16.0000i −52.6892 + 52.6892i 20.2828
27.5 −2.00000 − 2.00000i 7.72425 − 3.19949i 8.00000i −2.80585 + 2.80585i −21.8475 − 9.04953i −85.3865 + 35.3682i 16.0000 − 16.0000i −7.84828 + 7.84828i 11.2234
27.6 −2.00000 − 2.00000i 7.94391 − 3.29047i 8.00000i 35.0704 − 35.0704i −22.4688 − 9.30687i −0.0786306 + 0.0325699i 16.0000 − 16.0000i −4.99718 + 4.99718i −140.282
27.7 −2.00000 − 2.00000i 15.8343 − 6.55878i 8.00000i −7.43719 + 7.43719i −44.7862 − 18.5510i 39.9737 − 16.5577i 16.0000 − 16.0000i 150.432 − 150.432i 29.7488
55.1 −2.00000 − 2.00000i −6.17854 − 14.9163i 8.00000i −15.3528 + 15.3528i −17.4755 + 42.1897i 22.0538 + 53.2426i 16.0000 − 16.0000i −127.046 + 127.046i 61.4114
55.2 −2.00000 − 2.00000i −3.50051 − 8.45098i 8.00000i 13.5876 − 13.5876i −9.90093 + 23.9030i −23.8235 − 57.5149i 16.0000 − 16.0000i −1.88977 + 1.88977i −54.3502
55.3 −2.00000 − 2.00000i −1.67187 − 4.03626i 8.00000i 22.2467 − 22.2467i −4.72877 + 11.4163i 30.2176 + 72.9518i 16.0000 − 16.0000i 43.7794 − 43.7794i −88.9870
55.4 −2.00000 − 2.00000i 0.493125 + 1.19051i 8.00000i −21.5675 + 21.5675i 1.39477 − 3.36727i −12.4355 − 30.0219i 16.0000 − 16.0000i 56.1015 − 56.1015i 86.2700
55.5 −2.00000 − 2.00000i 2.85757 + 6.89878i 8.00000i 3.97800 − 3.97800i 8.08243 − 19.5127i 9.68798 + 23.3888i 16.0000 − 16.0000i 17.8481 − 17.8481i −15.9120
55.6 −2.00000 − 2.00000i 5.22888 + 12.6236i 8.00000i 25.8818 − 25.8818i 14.7895 − 35.7050i −20.1271 − 48.5912i 16.0000 − 16.0000i −74.7392 + 74.7392i −103.527
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.7
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.e odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 82.5.e.a ✓ 28
41.e odd 8 1 inner 82.5.e.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
82.5.e.a ✓ 28 1.a even 1 1 trivial
82.5.e.a ✓ 28 41.e odd 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} - 16 T_{3}^{27} + 116 T_{3}^{26} - 380 T_{3}^{25} - 504 T_{3}^{24} + 127668 T_{3}^{23} + \cdots + 83\!\cdots\!68 \) acting on \(S_{5}^{\mathrm{new}}(82, [\chi])\). Copy content Toggle raw display