↑

Properties

Label 31.8.g.a
Level $31$
Weight $8$
Character orbit 31.g
Analytic conductor $9.684$
Analytic rank $0$
Dimension $144$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [31,8,Mod(7,31)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([28])) 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("31.7"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 31.g (of order \(15\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.68393579001\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 6 q^{2} - 8 q^{3} - 2362 q^{4} - 75 q^{5} - 1637 q^{6} + 5518 q^{7} + 3401 q^{8} + 14542 q^{9} - 5312 q^{10} + 1361 q^{11} - 18281 q^{12} + 22457 q^{13} + 28470 q^{14} - 86918 q^{15} - 155730 q^{16}+ \cdots - 102755966 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −6.50086 − 20.0076i 12.4466 − 2.64560i −254.489 + 184.897i −49.9845 + 86.5756i −133.846 − 231.827i 1216.98 + 541.833i 3175.24 + 2306.95i −1850.01 + 823.676i 2057.11 + 437.253i
7.2 −6.02816 − 18.5528i −73.4236 + 15.6067i −204.312 + 148.441i −23.2656 + 40.2972i 732.156 + 1268.13i −1235.64 − 550.141i 1965.54 + 1428.04i 3149.53 − 1402.26i 887.873 + 188.723i
7.3 −5.07105 − 15.6071i 91.2249 − 19.3905i −114.312 + 83.0523i −24.8635 + 43.0649i −765.235 − 1325.43i −449.966 − 200.338i 176.534 + 128.260i 5948.07 − 2648.25i 798.203 + 169.663i
7.4 −5.03785 − 15.5049i 8.90236 − 1.89225i −111.468 + 80.9861i 146.446 − 253.652i −74.1879 − 128.497i −709.897 − 316.067i 129.014 + 93.7340i −1922.25 + 855.842i −4670.62 − 992.770i
7.5 −3.48119 − 10.7140i −57.4298 + 12.2071i 0.883320 − 0.641769i −149.805 + 259.470i 330.710 + 572.807i 948.617 + 422.351i −1176.53 − 854.796i 1151.24 − 512.566i 3301.45 + 701.746i
7.6 −3.14721 − 9.68611i 14.5381 − 3.09018i 19.6383 − 14.2681i −175.802 + 304.498i −75.6864 − 131.093i −506.982 − 225.723i −1254.66 − 911.567i −1796.12 + 799.682i 3502.68 + 744.518i
7.7 −3.04864 − 9.38274i −48.5446 + 10.3185i 24.8125 − 18.0273i 262.812 − 455.203i 244.811 + 424.024i 799.983 + 356.175i −1266.41 − 920.104i 252.185 − 112.280i −5072.27 − 1078.14i
7.8 −1.55903 − 4.79820i 51.7322 − 10.9960i 82.9620 − 60.2754i 69.7933 − 120.886i −133.413 − 231.078i 958.697 + 426.839i −940.998 − 683.675i 557.384 − 248.163i −688.843 − 146.418i
7.9 0.0902673 + 0.277814i −68.2722 + 14.5117i 103.485 − 75.1864i 30.5078 − 52.8410i −10.1943 − 17.6571i −598.793 − 266.600i 60.4785 + 43.9402i 2452.58 − 1091.96i 17.4338 + 3.70568i
7.10 0.634131 + 1.95166i 40.2459 − 8.55452i 100.147 − 72.7613i 154.916 − 268.322i 42.2167 + 73.1214i −1012.16 − 450.644i 418.014 + 303.705i −451.374 + 200.965i 621.909 + 132.191i
7.11 0.822068 + 2.53007i −31.4883 + 6.69304i 97.8287 − 71.0767i −80.8758 + 140.081i −42.8193 − 74.1652i −375.195 − 167.047i 535.733 + 389.233i −1051.21 + 468.029i −420.900 − 89.4650i
7.12 1.98393 + 6.10590i 65.6357 − 13.9513i 70.2081 − 51.0092i −258.562 + 447.842i 215.402 + 373.087i 364.514 + 162.292i 1115.58 + 810.513i 2115.48 − 941.873i −3247.45 − 690.267i
7.13 3.18005 + 9.78717i −14.7307 + 3.13111i 17.8781 − 12.9892i 43.8183 − 75.8955i −77.4890 − 134.215i 1230.52 + 547.864i 1249.64 + 907.917i −1790.73 + 797.286i 882.147 + 187.506i
7.14 4.89079 + 15.0523i 14.0858 − 2.99403i −99.0982 + 71.9990i −109.494 + 189.650i 113.958 + 197.381i −1077.80 − 479.865i 70.5248 + 51.2393i −1808.48 + 805.187i −3390.18 − 720.605i
7.15 4.89387 + 15.0618i 74.4930 − 15.8340i −99.3535 + 72.1845i 128.211 − 222.068i 603.048 + 1044.51i 141.206 + 62.8690i 66.5262 + 48.3341i 3300.57 − 1469.51i 3972.20 + 844.316i
7.16 5.12040 + 15.7590i −83.6064 + 17.7711i −118.572 + 86.1478i −218.479 + 378.417i −708.152 − 1226.56i 384.830 + 171.337i −248.849 − 180.799i 4676.30 − 2082.02i −7082.16 − 1505.36i
7.17 5.20562 + 16.0213i −55.0184 + 11.6945i −126.028 + 91.5648i 247.149 − 428.075i −473.766 − 820.587i −708.649 − 315.511i −378.590 − 275.062i 892.338 − 397.295i 8144.86 + 1731.24i
7.18 6.86187 + 21.1187i 10.3111 − 2.19169i −295.358 + 214.590i −40.5298 + 70.1998i 117.039 + 202.718i 629.172 + 280.125i −4259.11 − 3094.42i −1896.41 + 844.336i −1760.64 − 374.235i
9.1 −6.50086 + 20.0076i 12.4466 + 2.64560i −254.489 − 184.897i −49.9845 − 86.5756i −133.846 + 231.827i 1216.98 − 541.833i 3175.24 − 2306.95i −1850.01 − 823.676i 2057.11 − 437.253i
9.2 −6.02816 + 18.5528i −73.4236 − 15.6067i −204.312 − 148.441i −23.2656 − 40.2972i 732.156 − 1268.13i −1235.64 + 550.141i 1965.54 − 1428.04i 3149.53 + 1402.26i 887.873 − 188.723i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.8.g.a ✓ 144
31.g even 15 1 inner 31.8.g.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.8.g.a ✓ 144 1.a even 1 1 trivial
31.8.g.a ✓ 144 31.g even 15 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(31, [\chi])\).