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Properties

Label 82.6.d.b
Level $82$
Weight $6$
Character orbit 82.d
Analytic conductor $13.151$
Analytic rank $0$
Dimension $36$
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(37,82)] 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([8])) 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("82.37"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: \( N \) \(=\) \( 82 = 2 \cdot 41 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 82.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36,36] 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: \(13.1514732247\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(9\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 36 q^{2} + 2 q^{3} - 144 q^{4} + 26 q^{5} + 112 q^{6} - 120 q^{7} + 576 q^{8} + 4006 q^{9} - 104 q^{10} + 747 q^{11} + 432 q^{12} + 907 q^{13} + 280 q^{14} - 4149 q^{15} - 2304 q^{16} - 3177 q^{17}+ \cdots + 276325 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 3.23607 − 2.35114i −23.2142 4.94427 − 15.2169i 2.84879 − 8.76768i −75.1228 + 54.5799i −116.783 − 84.8479i −19.7771 − 60.8676i 295.901 −11.3952 − 35.0707i
37.2 3.23607 − 2.35114i −20.9317 4.94427 − 15.2169i −11.1577 + 34.3400i −67.7364 + 49.2134i 105.237 + 76.4591i −19.7771 − 60.8676i 195.136 44.6310 + 137.360i
37.3 3.23607 − 2.35114i −20.7864 4.94427 − 15.2169i 24.6325 − 75.8110i −67.2663 + 48.8718i 84.4966 + 61.3903i −19.7771 − 60.8676i 189.076 −98.5300 − 303.244i
37.4 3.23607 − 2.35114i 1.86821 4.94427 − 15.2169i 28.8355 − 88.7465i 6.04567 − 4.39244i −47.2379 − 34.3203i −19.7771 − 60.8676i −239.510 −115.342 − 354.986i
37.5 3.23607 − 2.35114i 2.21783 4.94427 − 15.2169i −23.5211 + 72.3905i 7.17706 − 5.21444i 48.4929 + 35.2321i −19.7771 − 60.8676i −238.081 94.0843 + 289.562i
37.6 3.23607 − 2.35114i 2.27324 4.94427 − 15.2169i −9.23113 + 28.4105i 7.35635 − 5.34470i −115.841 − 84.1637i −19.7771 − 60.8676i −237.832 36.9245 + 113.642i
37.7 3.23607 − 2.35114i 19.5004 4.94427 − 15.2169i 10.8275 − 33.3236i 63.1046 − 45.8482i 164.744 + 119.693i −19.7771 − 60.8676i 137.266 −43.3100 − 133.294i
37.8 3.23607 − 2.35114i 24.2649 4.94427 − 15.2169i 13.8621 − 42.6631i 78.5227 − 57.0501i −139.296 − 101.205i −19.7771 − 60.8676i 345.784 −55.4484 − 170.653i
37.9 3.23607 − 2.35114i 27.6062 4.94427 − 15.2169i −30.5964 + 94.1661i 89.3355 − 64.9061i −19.4012 − 14.0958i −19.7771 − 60.8676i 519.102 122.386 + 376.664i
51.1 3.23607 + 2.35114i −23.2142 4.94427 + 15.2169i 2.84879 + 8.76768i −75.1228 − 54.5799i −116.783 + 84.8479i −19.7771 + 60.8676i 295.901 −11.3952 + 35.0707i
51.2 3.23607 + 2.35114i −20.9317 4.94427 + 15.2169i −11.1577 − 34.3400i −67.7364 − 49.2134i 105.237 − 76.4591i −19.7771 + 60.8676i 195.136 44.6310 − 137.360i
51.3 3.23607 + 2.35114i −20.7864 4.94427 + 15.2169i 24.6325 + 75.8110i −67.2663 − 48.8718i 84.4966 − 61.3903i −19.7771 + 60.8676i 189.076 −98.5300 + 303.244i
51.4 3.23607 + 2.35114i 1.86821 4.94427 + 15.2169i 28.8355 + 88.7465i 6.04567 + 4.39244i −47.2379 + 34.3203i −19.7771 + 60.8676i −239.510 −115.342 + 354.986i
51.5 3.23607 + 2.35114i 2.21783 4.94427 + 15.2169i −23.5211 − 72.3905i 7.17706 + 5.21444i 48.4929 − 35.2321i −19.7771 + 60.8676i −238.081 94.0843 − 289.562i
51.6 3.23607 + 2.35114i 2.27324 4.94427 + 15.2169i −9.23113 − 28.4105i 7.35635 + 5.34470i −115.841 + 84.1637i −19.7771 + 60.8676i −237.832 36.9245 − 113.642i
51.7 3.23607 + 2.35114i 19.5004 4.94427 + 15.2169i 10.8275 + 33.3236i 63.1046 + 45.8482i 164.744 − 119.693i −19.7771 + 60.8676i 137.266 −43.3100 + 133.294i
51.8 3.23607 + 2.35114i 24.2649 4.94427 + 15.2169i 13.8621 + 42.6631i 78.5227 + 57.0501i −139.296 + 101.205i −19.7771 + 60.8676i 345.784 −55.4484 + 170.653i
51.9 3.23607 + 2.35114i 27.6062 4.94427 + 15.2169i −30.5964 − 94.1661i 89.3355 + 64.9061i −19.4012 + 14.0958i −19.7771 + 60.8676i 519.102 122.386 − 376.664i
57.1 −1.23607 + 3.80423i −29.2964 −12.9443 − 9.40456i 72.0590 + 52.3539i 36.2123 − 111.450i −58.7271 − 180.743i 51.7771 − 37.6183i 615.279 −288.236 + 209.416i
57.2 −1.23607 + 3.80423i −20.2284 −12.9443 − 9.40456i −77.2940 − 56.1574i 25.0037 − 76.9535i −23.9851 − 73.8187i 51.7771 − 37.6183i 166.189 309.176 − 224.630i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.9
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 82.6.d.b ✓ 36
41.d even 5 1 inner 82.6.d.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
82.6.d.b ✓ 36 1.a even 1 1 trivial
82.6.d.b ✓ 36 41.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} - T_{3}^{17} - 3188 T_{3}^{16} + 1555 T_{3}^{15} + 4186687 T_{3}^{14} - 392118 T_{3}^{13} + \cdots + 76\!\cdots\!80 \) acting on \(S_{6}^{\mathrm{new}}(82, [\chi])\). Copy content Toggle raw display