Properties

Label 58.9.f.a
Level $58$
Weight $9$
Character orbit 58.f
Analytic conductor $23.628$
Analytic rank $0$
Dimension $120$
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] = [58,9,Mod(3,58)] 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("58.3"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(58, base_ring=CyclotomicField(28)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 58.f (of order \(28\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [120,-160] 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: \(23.6279593835\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 160 q^{2} - 32 q^{3} + 2576 q^{5} - 12048 q^{7} + 20480 q^{8} + 4992 q^{10} + 90908 q^{11} + 4096 q^{12} + 18304 q^{14} + 47654 q^{15} + 327680 q^{16} - 135312 q^{17} + 309472 q^{18} - 950 q^{19}+ \cdots + 273185148 q^{99}+O(q^{100}) \) 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 −9.57959 − 6.01926i −129.298 + 45.2434i 55.5371 + 115.324i −313.345 + 71.5191i 1510.96 + 344.866i −68.9177 − 33.1890i 162.142 − 1439.05i 9541.47 − 7609.07i 3432.21 + 1200.98i
3.2 −9.57959 − 6.01926i −106.701 + 37.3363i 55.5371 + 115.324i 1181.64 − 269.701i 1246.89 + 284.594i −1332.00 − 641.459i 162.142 − 1439.05i 4861.52 − 3876.93i −12943.0 − 4528.96i
3.3 −9.57959 − 6.01926i −42.2119 + 14.7706i 55.5371 + 115.324i −779.488 + 177.913i 493.280 + 112.588i −3687.66 − 1775.89i 162.142 − 1439.05i −3565.92 + 2843.73i 8538.09 + 2987.61i
3.4 −9.57959 − 6.01926i −36.2533 + 12.6856i 55.5371 + 115.324i 246.935 − 56.3612i 423.649 + 96.6951i −1250.39 − 602.155i 162.142 − 1439.05i −3976.22 + 3170.93i −2704.79 − 946.446i
3.5 −9.57959 − 6.01926i −35.6274 + 12.4666i 55.5371 + 115.324i −984.184 + 224.634i 416.335 + 95.0257i 2790.43 + 1343.80i 162.142 − 1439.05i −4015.70 + 3202.42i 10780.2 + 3772.16i
3.6 −9.57959 − 6.01926i −23.3020 + 8.15373i 55.5371 + 115.324i 458.723 − 104.700i 272.303 + 62.1514i 3469.94 + 1671.03i 162.142 − 1439.05i −4653.10 + 3710.72i −5024.60 − 1758.18i
3.7 −9.57959 − 6.01926i 53.4172 − 18.6915i 55.5371 + 115.324i 405.312 − 92.5098i −624.224 − 142.475i −1291.75 − 622.073i 162.142 − 1439.05i −2625.57 + 2093.82i −4439.56 − 1553.47i
3.8 −9.57959 − 6.01926i 88.9772 − 31.1345i 55.5371 + 115.324i −188.109 + 42.9347i −1039.77 − 237.321i 185.624 + 89.3920i 162.142 − 1439.05i 1817.99 − 1449.80i 2060.44 + 720.980i
3.9 −9.57959 − 6.01926i 96.7330 − 33.8483i 55.5371 + 115.324i −791.651 + 180.689i −1130.40 − 258.007i −1755.63 − 845.468i 162.142 − 1439.05i 3081.97 − 2457.79i 8671.31 + 3034.22i
3.10 −9.57959 − 6.01926i 139.019 − 48.6448i 55.5371 + 115.324i 950.372 − 216.916i −1624.55 − 370.793i 660.745 + 318.198i 162.142 − 1439.05i 11830.3 − 9434.38i −10409.8 − 3642.56i
11.1 −10.6788 + 3.73668i −15.4651 − 137.256i 100.074 − 79.8067i 422.277 − 876.868i 678.032 + 1407.95i −2312.49 + 2899.77i −770.465 + 1226.19i −12203.7 + 2785.40i −1232.85 + 10941.8i
11.2 −10.6788 + 3.73668i −12.6139 − 111.951i 100.074 − 79.8067i −69.1229 + 143.535i 553.027 + 1148.37i 258.106 − 323.655i −770.465 + 1226.19i −5977.46 + 1364.32i 201.806 − 1791.08i
11.3 −10.6788 + 3.73668i −9.95360 − 88.3406i 100.074 − 79.8067i −34.8510 + 72.3687i 436.393 + 906.180i 2270.63 − 2847.28i −770.465 + 1226.19i −1308.49 + 298.654i 101.748 − 903.040i
11.4 −10.6788 + 3.73668i −4.09624 − 36.3551i 100.074 − 79.8067i −461.142 + 957.571i 179.590 + 372.923i −2305.90 + 2891.51i −770.465 + 1226.19i 5091.59 − 1162.12i 1346.31 − 11948.9i
11.5 −10.6788 + 3.73668i −0.159200 − 1.41294i 100.074 − 79.8067i 106.375 − 220.891i 6.97978 + 14.4937i −874.009 + 1095.97i −770.465 + 1226.19i 6394.53 − 1459.51i −310.565 + 2756.34i
11.6 −10.6788 + 3.73668i 3.29165 + 29.2142i 100.074 − 79.8067i 492.356 − 1022.39i −144.315 − 299.673i 2466.98 − 3093.50i −770.465 + 1226.19i 5553.87 − 1267.63i −1437.44 + 12757.7i
11.7 −10.6788 + 3.73668i 4.07453 + 36.1624i 100.074 − 79.8067i −251.034 + 521.277i −178.639 − 370.947i 937.361 − 1175.41i −770.465 + 1226.19i 5105.38 − 1165.27i 732.899 − 6504.66i
11.8 −10.6788 + 3.73668i 8.39674 + 74.5231i 100.074 − 79.8067i 266.554 − 553.506i −368.136 − 764.443i −1737.44 + 2178.69i −770.465 + 1226.19i 913.312 − 208.458i −778.211 + 6906.82i
11.9 −10.6788 + 3.73668i 13.2034 + 117.184i 100.074 − 79.8067i −306.994 + 637.481i −578.875 − 1202.05i 928.277 − 1164.02i −770.465 + 1226.19i −7161.20 + 1634.50i 896.277 − 7954.68i
11.10 −10.6788 + 3.73668i 15.6042 + 138.491i 100.074 − 79.8067i 197.283 − 409.663i −684.132 − 1420.61i 1102.32 − 1382.27i −770.465 + 1226.19i −12539.8 + 2862.13i −575.973 + 5111.90i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 3.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.f odd 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 58.9.f.a ✓ 120
29.f odd 28 1 inner 58.9.f.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.9.f.a ✓ 120 1.a even 1 1 trivial
58.9.f.a ✓ 120 29.f odd 28 1 inner