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Properties

Label 71.8.d.a
Level $71$
Weight $8$
Character orbit 71.d
Analytic conductor $22.179$
Analytic rank $0$
Dimension $246$
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] = [71,8,Mod(20,71)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(71, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("71.20");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 71 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 71.d (of order \(7\), degree \(6\), 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: \(22.1793368094\)
Analytic rank: \(0\)
Dimension: \(246\)
Relative dimension: \(41\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 246 q + 9 q^{2} - 70 q^{3} - 2747 q^{4} - 154 q^{5} + 77 q^{6} - 691 q^{7} + 3065 q^{8} - 24189 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 246 q + 9 q^{2} - 70 q^{3} - 2747 q^{4} - 154 q^{5} + 77 q^{6} - 691 q^{7} + 3065 q^{8} - 24189 q^{9} - 20422 q^{10} + 1783 q^{11} + 37674 q^{12} - 9387 q^{13} + 85714 q^{14} - 28546 q^{15} - 90311 q^{16} + 159264 q^{17} - 13941 q^{18} + 25497 q^{19} - 110524 q^{20} - 109934 q^{21} + 31591 q^{22} - 86427 q^{23} - 41982 q^{24} + 3471108 q^{25} - 367454 q^{26} + 788081 q^{27} - 323317 q^{28} - 239147 q^{29} - 1119995 q^{30} - 442122 q^{31} - 483455 q^{32} + 448913 q^{33} - 662135 q^{34} - 1659836 q^{35} - 1940815 q^{36} + 840389 q^{37} + 1140576 q^{38} + 121773 q^{39} + 100090 q^{40} + 418960 q^{41} - 1385706 q^{42} - 3046339 q^{43} - 5160405 q^{44} + 2967360 q^{45} + 715266 q^{46} - 5597572 q^{47} + 1982697 q^{48} - 2201460 q^{49} + 3643622 q^{50} + 9711926 q^{51} - 16496431 q^{52} - 2075155 q^{53} - 1769890 q^{54} + 3068393 q^{55} + 2998287 q^{56} + 21814324 q^{57} + 13270408 q^{58} - 597222 q^{59} + 19962696 q^{60} - 4607550 q^{61} + 9926384 q^{62} - 6907193 q^{63} - 9933979 q^{64} - 2357266 q^{65} + 13175132 q^{66} - 5089787 q^{67} + 6851015 q^{68} + 15756898 q^{69} - 5505028 q^{70} + 11547823 q^{71} - 56636028 q^{72} - 10198373 q^{73} + 40993009 q^{74} + 4224890 q^{75} + 4654242 q^{76} - 71354626 q^{77} - 7475245 q^{78} + 1786823 q^{79} + 20182346 q^{80} - 19049697 q^{81} - 36366656 q^{82} + 4602765 q^{83} + 32901046 q^{84} + 39316138 q^{85} - 13570409 q^{86} - 22001468 q^{87} - 74441496 q^{88} - 7380488 q^{89} + 41110683 q^{90} + 46312491 q^{91} - 187189 q^{92} - 60535638 q^{93} - 42472595 q^{94} + 13875818 q^{95} - 93581318 q^{96} - 41203622 q^{97} - 95757755 q^{98} + 101250097 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 −4.94070 − 21.6466i 40.9336 − 19.7126i −328.842 + 158.362i 297.530 −628.952 − 788.681i −91.3689 + 400.313i 3280.73 + 4113.91i −76.5966 + 96.0491i −1470.00 − 6440.51i
20.2 −4.70545 − 20.6159i −2.31697 + 1.11579i −287.551 + 138.477i −402.108 33.9055 + 42.5161i 29.7932 − 130.533i 2520.29 + 3160.35i −1359.45 + 1704.69i 1892.10 + 8289.84i
20.3 −4.54526 − 19.9141i −54.8839 + 26.4307i −260.587 + 125.492i 137.296 775.804 + 972.827i 330.377 − 1447.48i 2053.35 + 2574.82i 950.085 − 1191.37i −624.047 − 2734.13i
20.4 −4.19089 − 18.3615i −44.3826 + 21.3736i −204.257 + 98.3649i 373.775 578.453 + 725.357i −352.094 + 1542.62i 1159.09 + 1453.45i 149.417 − 187.363i −1566.45 − 6863.07i
20.5 −4.17211 − 18.2792i −70.8179 + 34.1041i −201.399 + 96.9888i −258.636 918.856 + 1152.21i −204.488 + 895.921i 1116.82 + 1400.45i 2488.51 − 3120.50i 1079.06 + 4727.66i
20.6 −3.99318 − 17.4953i 77.0358 − 37.0985i −174.815 + 84.1865i −165.930 −956.666 − 1199.62i 273.852 − 1199.82i 738.787 + 926.409i 3194.64 − 4005.95i 662.587 + 2902.98i
20.7 −3.76621 − 16.5008i 36.1839 − 17.4252i −142.769 + 68.7539i 146.843 −423.806 − 531.437i 50.5249 − 221.364i 321.450 + 403.086i −357.939 + 448.841i −553.042 − 2423.04i
20.8 −3.56514 − 15.6199i −21.3332 + 10.2735i −115.947 + 55.8372i 284.122 236.528 + 296.597i 175.538 − 769.080i 6.90979 + 8.66461i −1014.01 + 1271.53i −1012.93 − 4437.96i
20.9 −3.46855 − 15.1967i 41.7461 − 20.1039i −103.585 + 49.8840i −198.329 −450.311 − 564.672i −325.512 + 1426.16i −126.626 − 158.784i −25.0007 + 31.3499i 687.913 + 3013.94i
20.10 −3.29306 − 14.4278i −21.6240 + 10.4136i −81.9942 + 39.4863i −246.616 221.454 + 277.695i 5.98099 − 26.2044i −341.335 − 428.021i −1004.42 + 1259.50i 812.123 + 3558.14i
20.11 −2.42518 − 10.6254i −77.3586 + 37.2539i 8.30657 − 4.00023i −134.882 583.446 + 731.618i 174.700 − 765.412i −932.434 − 1169.23i 3232.93 − 4053.96i 327.113 + 1433.18i
20.12 −2.27345 − 9.96062i 80.5744 − 38.8026i 21.2786 − 10.2472i 471.382 −569.679 − 714.355i −288.491 + 1263.96i −965.812 − 1211.09i 3623.02 − 4543.12i −1071.66 − 4695.26i
20.13 −2.13690 − 9.36236i 24.3624 − 11.7323i 32.2366 − 15.5243i 411.473 −161.902 − 203.018i 98.2210 − 430.335i −980.625 − 1229.66i −907.695 + 1138.21i −879.276 − 3852.36i
20.14 −1.96381 − 8.60400i 50.8938 − 24.5092i 45.1518 − 21.7440i −413.592 −310.822 − 389.759i 11.9291 − 52.2650i −980.069 − 1228.97i 625.909 − 784.865i 812.215 + 3558.54i
20.15 −1.92388 − 8.42907i −11.0616 + 5.32699i 47.9761 − 23.1041i −470.518 66.1828 + 82.9906i 322.860 − 1414.54i −977.042 − 1225.17i −1269.59 + 1592.02i 905.219 + 3966.02i
20.16 −1.82515 − 7.99649i −43.2185 + 20.8129i 54.7113 − 26.3476i −101.833 245.310 + 307.609i −204.914 + 897.787i −965.130 − 1210.23i 71.0863 − 89.1393i 185.861 + 814.309i
20.17 −1.41780 − 6.21179i −10.5957 + 5.10260i 78.7478 − 37.9230i 73.7854 46.7188 + 58.5835i −175.653 + 769.587i −855.709 − 1073.03i −1277.34 + 1601.73i −104.613 − 458.340i
20.18 −1.36077 − 5.96192i −66.2780 + 31.9178i 81.6312 − 39.3115i 509.829 280.480 + 351.711i −40.0567 + 175.500i −833.491 − 1045.16i 2010.45 − 2521.03i −693.760 − 3039.56i
20.19 −1.18731 − 5.20193i 40.0603 − 19.2920i 89.6737 − 43.1846i 112.387 −147.920 − 185.485i 322.014 − 1410.83i −756.938 − 949.170i −130.925 + 164.175i −133.437 − 584.627i
20.20 −0.162049 − 0.709985i 55.6149 − 26.7827i 114.846 − 55.3070i −324.719 −28.0277 − 35.1456i −146.561 + 642.126i −115.997 − 145.455i 1012.13 − 1269.17i 52.6206 + 230.546i
See next 80 embeddings (of 246 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.41
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 71.8.d.a ✓ 246
71.d even 7 1 inner 71.8.d.a ✓ 246
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.8.d.a ✓ 246 1.a even 1 1 trivial
71.8.d.a ✓ 246 71.d even 7 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(71, [\chi])\).