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Properties

Label 62.8.d.a
Level $62$
Weight $8$
Character orbit 62.d
Analytic conductor $19.368$
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] = [62,8,Mod(33,62)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(62, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("62.33"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: \( N \) \(=\) \( 62 = 2 \cdot 31 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 62.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,-80] 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: \(19.3678715800\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 80 q^{2} + 28 q^{3} - 640 q^{4} + 168 q^{5} - 1936 q^{6} + 1402 q^{7} - 5120 q^{8} - 6164 q^{9} - 1496 q^{10} - 3402 q^{11} + 1792 q^{12} + 6413 q^{13} + 11216 q^{14} + 46465 q^{15} - 40960 q^{16}+ \cdots + 13336614 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1 −6.47214 − 4.70228i −63.6525 + 46.2463i 19.7771 + 60.8676i −282.324 629.431 −452.897 − 1393.87i 158.217 − 486.941i 1237.11 − 3807.42i 1827.24 + 1327.57i
33.2 −6.47214 − 4.70228i −62.8734 + 45.6802i 19.7771 + 60.8676i 155.004 621.727 296.473 + 912.449i 158.217 − 486.941i 1190.57 − 3664.18i −1003.21 − 728.874i
33.3 −6.47214 − 4.70228i −28.7770 + 20.9077i 19.7771 + 60.8676i −353.423 284.563 378.018 + 1163.42i 158.217 − 486.941i −284.836 + 876.636i 2287.40 + 1661.90i
33.4 −6.47214 − 4.70228i −27.0752 + 19.6713i 19.7771 + 60.8676i 383.739 267.735 −272.247 − 837.889i 158.217 − 486.941i −329.712 + 1014.75i −2483.61 − 1804.45i
33.5 −6.47214 − 4.70228i 6.77917 − 4.92536i 19.7771 + 60.8676i 35.5870 −67.0362 −136.422 − 419.862i 158.217 − 486.941i −654.122 + 2013.18i −230.324 − 167.340i
33.6 −6.47214 − 4.70228i 11.4662 − 8.33068i 19.7771 + 60.8676i 363.071 −113.384 367.182 + 1130.07i 158.217 − 486.941i −613.747 + 1888.92i −2349.84 − 1707.26i
33.7 −6.47214 − 4.70228i 21.3101 − 15.4827i 19.7771 + 60.8676i −413.368 −210.725 −233.484 − 718.591i 158.217 − 486.941i −461.415 + 1420.09i 2675.37 + 1943.77i
33.8 −6.47214 − 4.70228i 47.3215 − 34.3811i 19.7771 + 60.8676i −206.240 −467.941 47.7422 + 146.935i 158.217 − 486.941i 381.447 − 1173.97i 1334.82 + 969.801i
33.9 −6.47214 − 4.70228i 66.2571 − 48.1386i 19.7771 + 60.8676i 520.167 −655.187 −313.887 − 966.043i 158.217 − 486.941i 1396.86 − 4299.10i −3366.59 − 2445.97i
33.10 −6.47214 − 4.70228i 66.4310 − 48.2650i 19.7771 + 60.8676i −95.3661 −656.906 418.464 + 1287.90i 158.217 − 486.941i 1407.75 − 4332.62i 617.222 + 448.438i
35.1 2.47214 + 7.60845i −27.4416 + 84.4565i −51.7771 + 37.6183i −131.008 −710.422 1040.49 − 755.962i −414.217 − 300.946i −4610.54 − 3349.75i −323.870 − 996.768i
35.2 2.47214 + 7.60845i −23.1547 + 71.2627i −51.7771 + 37.6183i 456.392 −599.441 −1202.00 + 873.305i −414.217 − 300.946i −2772.92 − 2014.64i 1128.26 + 3472.44i
35.3 2.47214 + 7.60845i −18.8921 + 58.1438i −51.7771 + 37.6183i −370.754 −489.088 −699.088 + 507.917i −414.217 − 300.946i −1254.47 − 911.423i −916.555 − 2820.87i
35.4 2.47214 + 7.60845i −10.2355 + 31.5015i −51.7771 + 37.6183i 307.829 −264.981 767.579 − 557.679i −414.217 − 300.946i 881.740 + 640.622i 760.996 + 2342.11i
35.5 2.47214 + 7.60845i −3.50721 + 10.7941i −51.7771 + 37.6183i 115.605 −90.7965 −125.169 + 90.9405i −414.217 − 300.946i 1665.11 + 1209.77i 285.791 + 879.575i
35.6 2.47214 + 7.60845i −2.70695 + 8.33115i −51.7771 + 37.6183i −344.363 −70.0791 −605.274 + 439.758i −414.217 − 300.946i 1707.24 + 1240.38i −851.311 − 2620.07i
35.7 2.47214 + 7.60845i 9.33796 − 28.7393i −51.7771 + 37.6183i −490.921 241.746 1157.34 − 840.856i −414.217 − 300.946i 1030.57 + 748.754i −1213.62 − 3735.15i
35.8 2.47214 + 7.60845i 13.8902 − 42.7498i −51.7771 + 37.6183i 131.384 359.598 −895.544 + 650.651i −414.217 − 300.946i 134.715 + 97.8762i 324.798 + 999.626i
35.9 2.47214 + 7.60845i 17.9591 − 55.2724i −51.7771 + 37.6183i −22.0653 464.935 −104.095 + 75.6294i −414.217 − 300.946i −963.190 − 699.798i −54.5484 − 167.883i
35.10 2.47214 + 7.60845i 21.5637 − 66.3662i −51.7771 + 37.6183i 325.055 558.253 1267.82 − 921.124i −414.217 − 300.946i −2170.16 − 1576.72i 803.581 + 2473.17i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.10
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 62.8.d.a ✓ 40
31.d even 5 1 inner 62.8.d.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
62.8.d.a ✓ 40 1.a even 1 1 trivial
62.8.d.a ✓ 40 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} - 28 T_{3}^{39} + 14409 T_{3}^{38} - 732825 T_{3}^{37} + 163121057 T_{3}^{36} + \cdots + 43\!\cdots\!61 \) acting on \(S_{8}^{\mathrm{new}}(62, [\chi])\). Copy content Toggle raw display