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Properties

Label 55.6.g.a
Level $55$
Weight $6$
Character orbit 55.g
Analytic conductor $8.821$
Analytic rank $0$
Dimension $40$
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] = [55,6,Mod(16,55)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("55.16"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(55, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 55.g (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,-6] 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: \(8.82111008971\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 6 q^{2} - 11 q^{3} - 100 q^{4} - 250 q^{5} - 408 q^{6} - 207 q^{7} - 658 q^{8} - 579 q^{9} + 850 q^{10} + 835 q^{11} + 2618 q^{12} - 4094 q^{13} + 7 q^{14} - 275 q^{15} - 5268 q^{16} - 601 q^{17}+ \cdots + 267844 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.50844 − 7.72018i 8.99967 + 6.53864i −27.4204 + 19.9221i 7.72542 − 23.7764i 27.9044 − 85.8809i −75.9034 + 55.1470i 12.4351 + 9.03463i −36.8510 − 113.416i −202.937
16.2 −2.43891 − 7.50620i −16.7798 − 12.1913i −24.5062 + 17.8048i 7.72542 − 23.7764i −50.5855 + 155.686i −147.025 + 106.820i −10.9098 − 7.92643i 57.8446 + 178.027i −197.312
16.3 −1.51650 − 4.66730i −9.46811 − 6.87898i 6.40463 − 4.65324i 7.72542 − 23.7764i −17.7479 + 54.6224i 176.027 − 127.891i −158.478 − 115.141i −32.7665 − 100.845i −122.687
16.4 −1.18122 − 3.63543i 18.7455 + 13.6194i 14.0675 − 10.2206i 7.72542 − 23.7764i 27.3698 − 84.2357i 50.5409 − 36.7201i −152.733 − 110.967i 90.8151 + 279.500i −95.5630
16.5 0.0343569 + 0.105740i −3.72925 − 2.70946i 25.8785 − 18.8019i 7.72542 − 23.7764i 0.158372 − 0.487419i −49.6675 + 36.0856i 5.75553 + 4.18164i −68.5250 − 210.898i 2.77953
16.6 1.13337 + 3.48814i −20.0120 − 14.5396i 15.0059 − 10.9024i 7.72542 − 23.7764i 28.0352 − 86.2833i −137.506 + 99.9039i 149.987 + 108.972i 113.990 + 350.824i 91.6913
16.7 1.37638 + 4.23606i 2.00022 + 1.45324i 9.83876 − 7.14828i 7.72542 − 23.7764i −3.40297 + 10.4733i 66.7141 − 48.4706i 159.131 + 115.616i −73.2022 − 225.293i 111.351
16.8 2.00085 + 6.15800i 23.8828 + 17.3519i −8.02898 + 5.83339i 7.72542 − 23.7764i −59.0668 + 181.789i 13.2126 − 9.59949i 115.639 + 84.0166i 194.210 + 597.717i 161.873
16.9 2.99889 + 9.22963i 6.11686 + 4.44416i −50.3041 + 36.5481i 7.72542 − 23.7764i −22.6742 + 69.7839i −161.218 + 117.132i −236.944 − 172.150i −57.4257 − 176.738i 242.615
16.10 3.07336 + 9.45884i −18.6551 − 13.5537i −54.1356 + 39.3318i 7.72542 − 23.7764i 70.8687 − 218.112i 64.9350 − 47.1780i −280.934 − 204.111i 89.2187 + 274.587i 248.640
26.1 −8.99247 − 6.53341i 1.00782 − 3.10175i 28.2905 + 87.0693i −20.2254 + 14.6946i −29.3278 + 21.3079i −8.46834 − 26.0629i 204.544 − 629.520i 187.986 + 136.580i 277.883
26.2 −6.44853 − 4.68513i 8.60626 − 26.4874i 9.74455 + 29.9906i −20.2254 + 14.6946i −179.595 + 130.483i 68.2337 + 210.002i −1.14773 + 3.53234i −430.921 − 313.083i 199.271
26.3 −5.58705 − 4.05923i −0.316192 + 0.973140i 4.84925 + 14.9245i −20.2254 + 14.6946i 5.71678 − 4.15349i −1.13732 − 3.50032i −34.8012 + 107.107i 195.744 + 142.216i 172.649
26.4 −3.74064 − 2.71773i −8.37488 + 25.7752i −3.28224 − 10.1017i −20.2254 + 14.6946i 101.378 − 73.6551i 69.7290 + 214.604i −60.8975 + 187.423i −397.633 − 288.897i 115.592
26.5 −2.35490 − 1.71093i 5.78535 − 17.8055i −7.27029 − 22.3757i −20.2254 + 14.6946i −44.0879 + 32.0317i −64.4266 − 198.285i −49.9462 + 153.719i −86.9735 − 63.1900i 72.7704
26.6 0.508209 + 0.369235i −1.41785 + 4.36369i −9.76660 − 30.0585i −20.2254 + 14.6946i −2.33179 + 1.69414i 35.9361 + 110.600i 12.3470 − 38.0001i 179.560 + 130.458i −15.7045
26.7 2.22018 + 1.61306i −3.96667 + 12.2081i −7.56128 − 23.2712i −20.2254 + 14.6946i −28.4992 + 20.7059i −49.9522 − 153.737i 47.8875 − 147.383i 63.2869 + 45.9806i −68.6075
26.8 3.65492 + 2.65545i 7.08382 − 21.8018i −3.58156 − 11.0229i −20.2254 + 14.6946i 83.7843 − 60.8729i 17.4321 + 53.6504i 60.8542 − 187.290i −228.545 − 166.048i −112.943
26.9 7.11412 + 5.16871i −6.95884 + 21.4171i 14.0066 + 43.1079i −20.2254 + 14.6946i −160.205 + 116.396i −11.7729 − 36.2332i −36.2123 + 111.450i −213.676 − 155.244i −219.838
26.10 7.65403 + 5.56098i 1.95035 − 6.00257i 17.7711 + 54.6939i −20.2254 + 14.6946i 48.3082 − 35.0980i 40.8160 + 125.619i −74.5763 + 229.522i 164.364 + 119.418i −236.522
See all 40 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 16.10
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.6.g.a ✓ 40
11.c even 5 1 inner 55.6.g.a ✓ 40
11.c even 5 1 605.6.a.q 20
11.d odd 10 1 605.6.a.n 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.6.g.a ✓ 40 1.a even 1 1 trivial
55.6.g.a ✓ 40 11.c even 5 1 inner
605.6.a.n 20 11.d odd 10 1
605.6.a.q 20 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 6 T_{2}^{39} + 228 T_{2}^{38} + 1750 T_{2}^{37} + 38765 T_{2}^{36} + 217534 T_{2}^{35} + \cdots + 52\!\cdots\!96 \) acting on \(S_{6}^{\mathrm{new}}(55, [\chi])\). Copy content Toggle raw display