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Properties

Label 61.8.e.a
Level $61$
Weight $8$
Character orbit 61.e
Analytic conductor $19.055$
Analytic rank $0$
Dimension $140$
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(9,61)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(61, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.9");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 61.e (of order \(5\), degree \(4\), 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: \(140\)
Relative dimension: \(35\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 140 q - 11 q^{2} - 57 q^{3} - 2063 q^{4} - 833 q^{5} + 1019 q^{6} + 1978 q^{7} - 1283 q^{8} - 21798 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 140 q - 11 q^{2} - 57 q^{3} - 2063 q^{4} - 833 q^{5} + 1019 q^{6} + 1978 q^{7} - 1283 q^{8} - 21798 q^{9} - 2253 q^{10} - 3870 q^{11} + 23440 q^{12} + 22162 q^{13} + 58870 q^{14} - 19797 q^{15} - 142135 q^{16} + 64950 q^{17} - 132614 q^{18} - 157673 q^{19} + 40957 q^{20} - 30376 q^{21} + 168121 q^{22} - 107215 q^{23} - 220970 q^{24} - 305464 q^{25} + 494108 q^{26} + 111396 q^{27} + 444095 q^{28} + 26176 q^{29} - 830643 q^{30} - 114205 q^{31} + 718970 q^{32} - 73241 q^{33} - 67398 q^{34} + 282838 q^{35} - 2947979 q^{36} - 520305 q^{37} + 956121 q^{38} - 2336659 q^{39} + 2899050 q^{40} - 1568192 q^{41} + 1992943 q^{42} - 2090591 q^{43} - 738680 q^{44} - 1727558 q^{45} - 2667641 q^{46} + 751634 q^{47} + 9942292 q^{48} - 896337 q^{49} + 2340236 q^{50} - 3691036 q^{51} + 997636 q^{52} - 3970979 q^{53} - 1559274 q^{54} + 5551664 q^{55} + 1387818 q^{56} - 13629227 q^{57} - 8617483 q^{58} + 2313659 q^{59} + 23345562 q^{60} + 11902458 q^{61} + 17839352 q^{62} + 17571027 q^{63} - 18563595 q^{64} - 13157576 q^{65} - 14531201 q^{66} - 2973745 q^{67} - 16232820 q^{68} - 13053137 q^{69} - 4627075 q^{70} - 5915244 q^{71} - 17609190 q^{72} + 23845139 q^{73} + 31693132 q^{74} + 13254696 q^{75} + 42557889 q^{76} - 346067 q^{77} - 9511282 q^{78} + 754426 q^{79} + 381433 q^{80} - 4420094 q^{81} + 42506352 q^{82} - 45895725 q^{83} + 11785385 q^{84} - 22445158 q^{85} - 13692413 q^{86} + 4318933 q^{87} - 664823 q^{88} - 3178901 q^{89} + 69927948 q^{90} - 19273423 q^{91} - 4226783 q^{92} - 66791678 q^{93} - 8394565 q^{94} + 40137952 q^{95} - 79353435 q^{96} + 36277737 q^{97} + 66779516 q^{98} - 28757769 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −17.3754 − 12.6240i −12.2655 − 8.91139i 102.986 + 316.960i −94.8506 − 291.920i 100.621 + 309.678i 101.293 − 73.5936i 1362.35 − 4192.88i −604.791 − 1861.36i −2037.13 + 6269.64i
9.2 −17.2881 − 12.5605i −72.1502 − 52.4202i 101.557 + 312.561i 94.3955 + 290.519i 588.914 + 1812.49i −441.470 + 320.747i 1324.96 − 4077.80i 1781.96 + 5484.29i 2017.16 − 6208.19i
9.3 −16.6665 − 12.1089i 41.5121 + 30.1603i 91.5926 + 281.893i 93.7981 + 288.681i −326.653 − 1005.33i 614.687 − 446.596i 1072.04 − 3299.40i 137.790 + 424.073i 1932.33 − 5947.11i
9.4 −15.0441 − 10.9302i 24.0218 + 17.4529i 67.3015 + 207.133i −13.9298 − 42.8714i −170.623 − 525.124i −1091.83 + 793.262i 515.975 − 1588.01i −403.376 − 1241.46i −259.031 + 797.214i
9.5 −13.9278 − 10.1191i −32.0962 − 23.3193i 52.0324 + 160.139i −24.5854 − 75.6661i 211.059 + 649.572i −189.720 + 137.839i 214.821 − 661.150i −189.442 − 583.041i −423.255 + 1302.64i
9.6 −12.8229 − 9.31637i −25.3977 − 18.4525i 38.0774 + 117.190i 86.1502 + 265.143i 153.761 + 473.229i 1169.55 − 849.725i −23.4069 + 72.0389i −371.272 − 1142.66i 1365.48 − 4202.50i
9.7 −12.5714 − 9.13369i 45.4726 + 33.0378i 35.0627 + 107.912i −146.919 − 452.171i −269.899 − 830.665i 884.883 − 642.905i −69.7934 + 214.802i 300.443 + 924.667i −2283.00 + 7026.35i
9.8 −12.3235 − 8.95357i 69.8655 + 50.7602i 32.1489 + 98.9440i 9.06204 + 27.8901i −406.504 − 1251.09i −635.305 + 461.576i −112.803 + 347.172i 1628.76 + 5012.82i 138.040 − 424.842i
9.9 −11.4755 − 8.33743i −23.0182 − 16.7237i 22.6199 + 69.6169i 142.252 + 437.806i 124.712 + 383.825i −906.201 + 658.393i −240.204 + 739.272i −425.665 − 1310.06i 2017.77 − 6210.05i
9.10 −11.2180 − 8.15038i −64.0745 − 46.5529i 19.8615 + 61.1273i −128.750 − 396.252i 339.367 + 1044.46i 433.422 − 314.900i −273.064 + 840.405i 1262.56 + 3885.75i −1785.28 + 5494.53i
9.11 −8.67301 − 6.30131i −21.9398 − 15.9402i −4.03961 − 12.4326i −42.9209 − 132.097i 89.8400 + 276.499i −94.0913 + 68.3613i −467.344 + 1438.34i −448.555 − 1380.51i −460.130 + 1416.14i
9.12 −7.40883 − 5.38283i 32.9381 + 23.9309i −13.6383 − 41.9744i 44.5024 + 136.964i −115.216 − 354.600i 498.002 − 361.820i −487.127 + 1499.22i −163.592 − 503.485i 407.544 − 1254.29i
9.13 −5.80670 − 4.21881i 16.3094 + 11.8494i −23.6348 − 72.7405i −129.486 − 398.518i −44.7129 − 137.612i −1211.75 + 880.386i −453.537 + 1395.84i −550.234 − 1693.45i −929.384 + 2860.35i
9.14 −5.02252 − 3.64908i 35.4946 + 25.7884i −27.6442 − 85.0801i 144.957 + 446.133i −84.1689 − 259.045i −440.359 + 319.939i −417.180 + 1283.95i −80.9909 − 249.264i 899.921 − 2769.67i
9.15 −3.91173 − 2.84204i −57.3427 − 41.6619i −32.3297 − 99.5007i 14.5008 + 44.6287i 105.904 + 325.940i −832.905 + 605.141i −347.570 + 1069.71i 876.649 + 2698.05i 70.1135 − 215.787i
9.16 −2.26934 − 1.64877i −61.6698 − 44.8058i −37.1227 − 114.252i 102.607 + 315.793i 66.0754 + 203.359i 1230.82 − 894.239i −215.083 + 661.958i 1119.79 + 3446.37i 287.820 − 885.819i
9.17 −1.91271 − 1.38966i −4.84806 − 3.52232i −37.8269 − 116.419i −54.7830 − 168.605i 4.37808 + 13.4743i 822.060 − 597.262i −182.947 + 563.053i −664.723 − 2045.81i −129.520 + 398.621i
9.18 −0.356361 − 0.258912i 69.4000 + 50.4220i −39.4942 − 121.551i −16.0610 − 49.4308i −11.6766 − 35.9369i −257.263 + 186.913i −34.8198 + 107.164i 1598.15 + 4918.61i −7.07468 + 21.7736i
9.19 1.63355 + 1.18685i 51.7218 + 37.5781i −38.2943 − 117.858i −77.5523 − 238.681i 39.8909 + 122.772i 703.306 − 510.982i 157.190 − 483.782i 587.211 + 1807.25i 156.592 − 481.942i
9.20 2.78800 + 2.02560i −23.8765 − 17.3473i −35.8843 − 110.440i 115.267 + 354.756i −31.4291 − 96.7287i −116.958 + 84.9751i 259.973 − 800.114i −406.661 − 1251.57i −397.229 + 1222.55i
See next 80 embeddings (of 140 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.35
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.e even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 61.8.e.a ✓ 140
61.e even 5 1 inner 61.8.e.a ✓ 140
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.8.e.a ✓ 140 1.a even 1 1 trivial
61.8.e.a ✓ 140 61.e even 5 1 inner

Hecke kernels

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