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Properties

Label 61.8.c.a
Level $61$
Weight $8$
Character orbit 61.c
Analytic conductor $19.055$
Analytic rank $0$
Dimension $72$
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(13,61)]
 
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([4]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.13");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 61.c (of order \(3\), 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: \(72\)
Relative dimension: \(36\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 9 q^{2} + 50 q^{3} - 2317 q^{4} - 446 q^{5} - 485 q^{6} - 1681 q^{7} + 3324 q^{8} + 52434 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 9 q^{2} + 50 q^{3} - 2317 q^{4} - 446 q^{5} - 485 q^{6} - 1681 q^{7} + 3324 q^{8} + 52434 q^{9} - 7337 q^{10} - 3866 q^{11} - 8441 q^{12} - 1243 q^{13} - 44912 q^{14} - 8576 q^{15} - 165985 q^{16} + 21184 q^{17} - 67503 q^{18} - 71559 q^{19} + 186236 q^{20} - 79578 q^{21} + 29795 q^{22} + 20104 q^{23} - 224366 q^{24} - 512348 q^{25} - 174339 q^{26} - 194950 q^{27} + 440184 q^{28} - 163228 q^{29} + 1127668 q^{30} + 365683 q^{31} - 931075 q^{32} + 10266 q^{33} + 12444 q^{34} + 539629 q^{35} - 1919422 q^{36} - 579948 q^{37} + 1853064 q^{38} + 355952 q^{39} - 608406 q^{40} + 288796 q^{41} - 301585 q^{42} + 446142 q^{43} + 1637674 q^{44} - 2179491 q^{45} - 2770601 q^{46} - 1074684 q^{47} + 1092194 q^{48} - 4137825 q^{49} + 3965926 q^{50} + 1792744 q^{51} - 3869552 q^{52} + 800942 q^{53} + 5379244 q^{54} + 1114748 q^{55} - 8619974 q^{56} - 474864 q^{57} + 10323788 q^{58} - 366455 q^{59} - 16699468 q^{60} + 761690 q^{61} + 78650 q^{62} - 5390394 q^{63} + 33841632 q^{64} - 7967056 q^{65} - 5093160 q^{66} + 1829007 q^{67} + 9500823 q^{68} - 18510062 q^{69} - 19264452 q^{70} + 5595189 q^{71} + 60712228 q^{72} - 168820 q^{73} + 11378843 q^{74} + 7107049 q^{75} - 10547467 q^{76} - 4173601 q^{77} - 6426607 q^{78} + 6566724 q^{79} - 7727465 q^{80} + 47113000 q^{81} - 29364391 q^{82} - 658460 q^{83} + 44395390 q^{84} - 38576432 q^{85} - 18320580 q^{86} - 6493109 q^{87} - 22729356 q^{88} + 27203432 q^{89} - 59200268 q^{90} + 3903932 q^{91} - 29138301 q^{92} + 36178528 q^{93} + 30977998 q^{94} + 12727214 q^{95} + 23403950 q^{96} - 25242303 q^{97} + 25381120 q^{98} - 47207746 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −11.1238 + 19.2670i −85.0493 −183.478 − 317.793i −198.704 + 344.165i 946.072 − 1638.64i −281.899 + 488.264i 5316.20 5046.39 −4420.68 − 7656.84i
13.2 −11.0337 + 19.1110i 59.3100 −179.487 − 310.880i 193.228 − 334.680i −654.411 + 1133.47i −693.836 + 1201.76i 5097.00 1330.67 4264.05 + 7385.54i
13.3 −10.4592 + 18.1159i 22.2276 −154.790 − 268.105i −50.7888 + 87.9688i −232.484 + 402.674i 639.780 − 1108.13i 3798.39 −1692.93 −1062.42 − 1840.17i
13.4 −9.03122 + 15.6425i −51.3235 −99.1259 − 171.691i 186.336 − 322.743i 463.514 − 802.829i 186.606 − 323.210i 1268.92 447.098 3365.68 + 5829.54i
13.5 −8.77597 + 15.2004i 82.0424 −90.0353 − 155.946i −266.765 + 462.051i −720.002 + 1247.08i −408.888 + 708.215i 913.941 4543.96 −4682.25 − 8109.89i
13.6 −8.62610 + 14.9408i −5.72847 −84.8191 − 146.911i −52.5388 + 90.9999i 49.4143 − 85.5881i −572.928 + 992.341i 718.349 −2154.18 −906.409 − 1569.95i
13.7 −8.37409 + 14.5044i −13.9341 −76.2508 − 132.070i −92.5306 + 160.268i 116.686 − 202.106i −84.6978 + 146.701i 410.358 −1992.84 −1549.72 − 2684.19i
13.8 −7.58599 + 13.1393i 63.8026 −51.0946 − 88.4984i 77.2546 − 133.809i −484.006 + 838.323i 522.854 − 905.610i −391.602 1883.77 1172.11 + 2030.15i
13.9 −5.73316 + 9.93013i −41.1001 −1.73830 − 3.01082i −234.825 + 406.730i 235.634 − 408.130i 735.946 − 1274.70i −1427.83 −497.778 −2692.59 − 4663.69i
13.10 −5.67046 + 9.82152i 31.1513 −0.308179 − 0.533782i 230.636 − 399.473i −176.642 + 305.953i −522.665 + 905.282i −1444.65 −1216.60 2615.62 + 4530.39i
13.11 −5.55165 + 9.61573i 61.7227 2.35847 + 4.08499i 31.7679 − 55.0236i −342.662 + 593.509i −3.10222 + 5.37320i −1473.59 1622.69 352.728 + 610.943i
13.12 −5.21013 + 9.02420i −78.6311 9.70915 + 16.8167i −29.4467 + 51.0032i 409.678 − 709.584i 221.727 − 384.043i −1536.14 3995.86 −306.842 − 531.466i
13.13 −4.46668 + 7.73651i −68.2159 24.0976 + 41.7383i 42.6931 − 73.9467i 304.698 − 527.753i −878.630 + 1521.83i −1574.01 2466.41 381.393 + 660.591i
13.14 −3.26259 + 5.65098i 40.9657 42.7110 + 73.9775i −156.665 + 271.352i −133.655 + 231.496i −171.002 + 296.185i −1392.62 −508.809 −1022.27 − 1770.62i
13.15 −3.00105 + 5.19797i −1.12438 45.9874 + 79.6525i 195.559 − 338.717i 3.37433 − 5.84451i 530.309 − 918.523i −1320.31 −2185.74 1173.76 + 2033.01i
13.16 −2.93289 + 5.07992i 1.37807 46.7963 + 81.0536i −131.537 + 227.829i −4.04173 + 7.00048i −129.081 + 223.575i −1299.81 −2185.10 −771.569 − 1336.40i
13.17 −1.26675 + 2.19407i −52.1369 60.7907 + 105.293i 103.504 − 179.275i 66.0441 − 114.392i 30.9641 − 53.6314i −632.312 531.254 262.227 + 454.190i
13.18 −0.142716 + 0.247191i 90.5972 63.9593 + 110.781i 92.2217 − 159.733i −12.9297 + 22.3948i −373.712 + 647.288i −73.0473 6020.85 26.3230 + 45.5928i
13.19 0.426981 − 0.739552i 63.6968 63.6354 + 110.220i −117.591 + 203.673i 27.1973 − 47.1071i 841.524 − 1457.56i 217.991 1870.28 100.418 + 173.929i
13.20 0.928315 − 1.60789i 15.9899 62.2765 + 107.866i 49.7971 − 86.2511i 14.8437 − 25.7100i −704.428 + 1220.11i 468.897 −1931.32 −92.4548 − 160.136i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 61.8.c.a ✓ 72
61.c even 3 1 inner 61.8.c.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.8.c.a ✓ 72 1.a even 1 1 trivial
61.8.c.a ✓ 72 61.c even 3 1 inner

Hecke kernels

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