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Properties

Label 58.9.f.a
Level $58$
Weight $9$
Character orbit 58.f
Analytic conductor $23.628$
Analytic rank $0$
Dimension $120$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [58,9,Mod(3,58)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 58.f (of order \(28\), degree \(12\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 160 q^{2} - 32 q^{3} + 2576 q^{5} - 12048 q^{7} + 20480 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 160 q^{2} - 32 q^{3} + 2576 q^{5} - 12048 q^{7} + 20480 q^{8} + 4992 q^{10} + 90908 q^{11} + 4096 q^{12} + 18304 q^{14} + 47654 q^{15} + 327680 q^{16} - 135312 q^{17} + 309472 q^{18} - 950 q^{19} + 130048 q^{20} + 1117620 q^{21} - 607488 q^{22} - 838842 q^{23} - 163840 q^{24} + 1368448 q^{25} + 424032 q^{26} + 4399156 q^{27} - 2426150 q^{29} + 977728 q^{30} - 5081444 q^{31} + 2621440 q^{32} - 435806 q^{33} + 1622880 q^{34} + 8269044 q^{35} + 7216640 q^{36} + 1077562 q^{37} + 283360 q^{38} + 3644490 q^{39} + 3649536 q^{40} + 554288 q^{41} - 16988384 q^{42} + 10333124 q^{43} - 1457664 q^{44} + 23248020 q^{45} + 2712608 q^{46} + 17589660 q^{47} + 524288 q^{48} - 19962916 q^{49} - 2885760 q^{50} - 15479184 q^{51} + 9820160 q^{52} - 36988288 q^{53} - 23571872 q^{54} + 111893366 q^{55} - 2342912 q^{56} + 29897216 q^{58} + 58937972 q^{59} + 6099712 q^{60} - 70417960 q^{61} - 46351648 q^{62} - 96678736 q^{63} - 183996684 q^{65} + 78205056 q^{66} + 108369520 q^{67} + 17319936 q^{68} + 20707620 q^{69} - 108702176 q^{70} + 192370626 q^{71} + 101515264 q^{72} + 123234688 q^{73} - 121123808 q^{74} - 330413268 q^{75} - 28281600 q^{76} - 272026710 q^{77} - 102880800 q^{78} - 222074208 q^{79} - 42524044 q^{81} + 148378048 q^{82} + 86775932 q^{83} + 358473472 q^{84} + 369583916 q^{85} + 40363682 q^{87} - 23322624 q^{88} - 57249066 q^{89} - 438575776 q^{90} - 1062646620 q^{91} - 566277376 q^{92} - 482869352 q^{93} + 36425600 q^{94} + 1038052604 q^{95} + 18384204 q^{97} + 606425056 q^{98} + 273185148 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
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 990-1000
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