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Properties

Label 61.8.f.a
Level $61$
Weight $8$
Character orbit 61.f
Analytic conductor $19.055$
Analytic rank $0$
Dimension $70$
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] = [61,8,Mod(14,61)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(61, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.14");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 61.f (of order \(6\), degree \(2\), 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: \(19.0554865545\)
Analytic rank: \(0\)
Dimension: \(70\)
Relative dimension: \(35\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 70 q - 3 q^{2} + 50 q^{3} + 2227 q^{4} - 27 q^{5} - 1779 q^{6} + 495 q^{7} + 48160 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 70 q - 3 q^{2} + 50 q^{3} + 2227 q^{4} - 27 q^{5} - 1779 q^{6} + 495 q^{7} + 48160 q^{9} - 6687 q^{10} + 4065 q^{12} - 18421 q^{13} - 41808 q^{14} + 18424 q^{15} - 150369 q^{16} - 50154 q^{17} + 101133 q^{18} + 16685 q^{19} + 4860 q^{20} - 59118 q^{21} - 29797 q^{22} - 561628 q^{25} - 110235 q^{26} + 307058 q^{27} - 358329 q^{29} + 1635486 q^{30} + 584475 q^{31} + 1306659 q^{32} - 1788196 q^{34} - 520107 q^{35} + 2410022 q^{36} - 281666 q^{39} - 704508 q^{40} + 1084554 q^{41} - 1632251 q^{42} - 2777646 q^{43} + 106872 q^{44} + 214580 q^{45} + 961531 q^{46} - 91644 q^{47} - 2697338 q^{48} + 1479514 q^{49} - 1772508 q^{51} - 5460628 q^{52} - 9218028 q^{54} + 8943780 q^{55} + 10259226 q^{56} + 7321792 q^{57} - 9556932 q^{58} + 3231999 q^{59} + 21803792 q^{60} + 2349121 q^{61} - 464190 q^{62} + 415170 q^{63} - 17978832 q^{64} - 6774402 q^{65} - 21920594 q^{66} + 708771 q^{67} - 3767217 q^{68} - 23291552 q^{70} + 9778029 q^{71} - 3377955 q^{73} + 12388293 q^{74} - 202015 q^{75} - 3598631 q^{76} + 18072507 q^{77} + 31449957 q^{78} + 3741012 q^{79} + 23937045 q^{80} - 1508354 q^{81} - 9864267 q^{82} + 9137268 q^{83} + 15928764 q^{86} - 5010063 q^{87} + 6288528 q^{88} - 6928932 q^{90} - 22886160 q^{91} - 39209487 q^{92} + 30121476 q^{93} - 50532006 q^{95} + 5884356 q^{96} + 21317280 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −19.4645 + 11.2379i −46.6377 188.579 − 326.628i 131.259 + 227.347i 907.780 − 524.107i 941.041 − 543.310i 5599.98i −11.9268 −5109.78 − 2950.13i
14.2 −18.7084 + 10.8013i 78.6903 169.335 − 293.298i −91.2654 − 158.076i −1472.17 + 849.957i 795.325 − 459.181i 4551.03i 4005.17 3414.85 + 1971.57i
14.3 −17.2552 + 9.96230i 10.4323 134.495 − 232.952i −74.3008 − 128.693i −180.011 + 103.930i −1230.61 + 710.494i 2809.17i −2078.17 2564.15 + 1480.41i
14.4 −15.7465 + 9.09123i 51.9209 101.301 − 175.459i 259.429 + 449.344i −817.571 + 472.025i −221.145 + 127.678i 1356.45i 508.778 −8170.18 − 4717.06i
14.5 −15.1622 + 8.75392i −73.1886 89.2622 − 154.607i −43.1730 − 74.7778i 1109.70 − 640.687i −543.951 + 314.050i 884.573i 3169.57 1309.20 + 755.866i
14.6 −14.8308 + 8.56259i −38.5696 82.6358 − 143.129i −232.206 − 402.193i 572.019 − 330.255i 898.709 − 518.870i 638.284i −699.387 6887.62 + 3976.57i
14.7 −13.4989 + 7.79359i 25.3491 57.4802 − 99.5587i −34.1484 − 59.1468i −342.185 + 197.560i 594.515 − 343.243i − 203.249i −1544.42 921.933 + 532.278i
14.8 −12.3863 + 7.15122i −17.3170 38.2798 − 66.3025i 141.970 + 245.899i 214.493 − 123.837i 517.598 − 298.836i − 735.724i −1887.12 −3516.95 − 2030.51i
14.9 −11.1753 + 6.45209i −62.0620 19.2589 − 33.3573i 193.927 + 335.891i 693.564 − 400.429i −561.092 + 323.946i − 1154.69i 1664.69 −4334.40 − 2502.47i
14.10 −10.9899 + 6.34502i 77.0972 16.5184 − 28.6108i −56.1135 − 97.1915i −847.289 + 489.183i −141.115 + 81.4730i − 1205.09i 3756.97 1233.36 + 712.082i
14.11 −8.07576 + 4.66254i 56.3582 −20.5214 + 35.5441i −257.140 − 445.380i −455.136 + 262.773i −226.724 + 130.899i − 1576.34i 989.251 4153.20 + 2397.85i
14.12 −6.47541 + 3.73858i 51.3397 −36.0460 + 62.4336i 117.091 + 202.807i −332.446 + 191.938i −1156.01 + 667.424i − 1496.12i 448.767 −1516.42 − 875.507i
14.13 −5.99963 + 3.46389i −31.6300 −40.0029 + 69.2871i −216.878 − 375.643i 189.768 − 109.563i −939.051 + 542.161i − 1441.02i −1186.54 2602.37 + 1502.48i
14.14 −5.17453 + 2.98752i 2.58373 −46.1495 + 79.9333i 32.3249 + 55.9885i −13.3696 + 7.71894i 1105.96 − 638.529i − 1316.29i −2180.32 −334.533 − 193.143i
14.15 −4.73880 + 2.73595i −80.3086 −49.0292 + 84.9210i −66.2919 − 114.821i 380.567 − 219.720i 740.926 − 427.774i − 1236.97i 4262.48 628.288 + 362.742i
14.16 −4.39178 + 2.53559i −16.3218 −51.1415 + 88.5797i 65.0485 + 112.667i 71.6818 − 41.3855i −790.642 + 456.478i − 1167.81i −1920.60 −571.357 − 329.873i
14.17 −2.77061 + 1.59961i 80.3809 −58.8825 + 101.987i 178.936 + 309.926i −222.704 + 128.578i 1223.35 − 706.300i − 786.257i 4274.08 −991.522 − 572.456i
14.18 2.07160 − 1.19604i 43.6266 −61.1390 + 105.896i −122.722 − 212.561i 90.3768 − 52.1791i 15.3003 − 8.83363i 598.683i −283.717 −508.463 − 293.561i
14.19 2.28928 − 1.32172i −62.2280 −60.5061 + 104.800i 246.819 + 427.503i −142.457 + 82.2478i 784.706 − 453.050i 658.247i 1685.33 1130.08 + 652.449i
14.20 2.81153 − 1.62324i 26.2790 −58.7302 + 101.724i −171.459 − 296.976i 73.8841 − 42.6570i 994.612 − 574.240i 796.882i −1496.42 −964.125 − 556.638i
See all 70 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.35
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 61.8.f.a ✓ 70
61.f even 6 1 inner 61.8.f.a ✓ 70
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.8.f.a ✓ 70 1.a even 1 1 trivial
61.8.f.a ✓ 70 61.f even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(61, [\chi])\).