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Properties

Label 29.8.d.a
Level $29$
Weight $8$
Character orbit 29.d
Analytic conductor $9.059$
Analytic rank $0$
Dimension $102$
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] = [29,8,Mod(7,29)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(29, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([6]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("29.7");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 29.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: \(9.05916573904\)
Analytic rank: \(0\)
Dimension: \(102\)
Relative dimension: \(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 102 q + q^{2} - 5 q^{3} - 1275 q^{4} + 133 q^{5} - 2675 q^{6} + 657 q^{7} - 6562 q^{8} - 15054 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 102 q + q^{2} - 5 q^{3} - 1275 q^{4} + 133 q^{5} - 2675 q^{6} + 657 q^{7} - 6562 q^{8} - 15054 q^{9} + 5611 q^{10} + 2893 q^{11} - 7400 q^{12} + 5655 q^{13} - 5103 q^{14} + 55181 q^{15} - 37531 q^{16} + 66488 q^{17} + 111588 q^{18} + 25497 q^{19} - 69535 q^{20} + 221033 q^{21} - 184180 q^{22} - 357991 q^{23} - 279611 q^{24} - 401432 q^{25} - 304829 q^{26} + 777205 q^{27} - 317084 q^{28} + 345978 q^{29} + 1233614 q^{30} - 191155 q^{31} - 1142836 q^{32} - 446051 q^{33} + 854365 q^{34} - 421273 q^{35} - 915292 q^{36} + 1109067 q^{37} + 3198170 q^{38} + 1816963 q^{39} - 1864493 q^{40} - 68548 q^{41} + 4917730 q^{42} + 1733909 q^{43} - 5516860 q^{44} - 6926991 q^{45} + 7670856 q^{46} + 3560867 q^{47} - 3047946 q^{48} - 6860794 q^{49} - 6723464 q^{50} - 2911378 q^{51} - 9704310 q^{52} - 8343921 q^{53} + 11251300 q^{54} + 6948055 q^{55} - 963169 q^{56} - 679450 q^{57} + 19338341 q^{58} + 10876120 q^{59} + 2379472 q^{60} - 2230165 q^{61} - 11842431 q^{62} - 10985505 q^{63} - 14037070 q^{64} - 11815483 q^{65} - 387401 q^{66} + 3903123 q^{67} + 5327748 q^{68} + 9379769 q^{69} + 51913122 q^{70} + 9175293 q^{71} + 42974969 q^{72} - 22305785 q^{73} - 31357308 q^{74} + 5262752 q^{75} + 27826047 q^{76} - 9623065 q^{77} - 13423495 q^{78} - 43328225 q^{79} - 69999516 q^{80} - 14867886 q^{81} + 39940476 q^{82} + 11503001 q^{83} - 58649317 q^{84} - 493576 q^{85} + 17849216 q^{86} + 28282227 q^{87} + 80429114 q^{88} + 14809627 q^{89} + 43848318 q^{90} - 41942203 q^{91} - 94433178 q^{92} + 3218315 q^{93} + 52700970 q^{94} - 22341683 q^{95} - 153425906 q^{96} - 74993043 q^{97} - 24094580 q^{98} + 258723776 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −4.88848 − 21.4178i −32.9981 − 15.8910i −319.502 + 153.864i −92.6629 − 405.983i −179.041 + 784.430i −185.793 − 89.4731i 3104.08 + 3892.39i −527.225 − 661.119i −8242.29 + 3969.28i
7.2 −4.52236 − 19.8138i 80.5781 + 38.8044i −256.810 + 123.673i 30.7072 + 134.537i 404.457 − 1772.04i 638.605 + 307.536i 1989.88 + 2495.23i 3623.47 + 4543.69i 2526.82 − 1216.85i
7.3 −3.86456 − 16.9317i −23.9092 − 11.5141i −156.425 + 75.3302i 75.3962 + 330.333i −102.555 + 449.321i 1430.73 + 689.005i 493.968 + 619.416i −924.496 − 1159.28i 5301.73 − 2553.18i
7.4 −3.27276 − 14.3389i 23.7734 + 11.4487i −79.5692 + 38.3185i 14.1711 + 62.0877i 86.3567 − 378.354i −1112.14 − 535.580i −363.913 − 456.333i −929.469 − 1165.52i 843.890 − 406.396i
7.5 −3.19096 − 13.9805i −78.1597 − 37.6397i −69.9485 + 33.6854i 64.7588 + 283.727i −276.818 + 1212.82i −1342.76 − 646.638i −450.290 − 564.645i 3328.62 + 4173.95i 3760.00 − 1810.72i
7.6 −2.97504 − 13.0345i 16.5267 + 7.95883i −45.7228 + 22.0190i −31.3343 − 137.285i 54.5717 − 239.094i −44.2003 − 21.2858i −643.958 − 807.498i −1153.78 − 1446.80i −1696.21 + 816.853i
7.7 −1.47079 − 6.44396i −55.4601 − 26.7082i 75.9626 − 36.5816i −99.5091 − 435.978i −90.5362 + 396.665i 407.385 + 196.186i −874.953 − 1097.16i 998.922 + 1252.61i −2663.07 + 1282.47i
7.8 −0.870510 − 3.81395i 60.7856 + 29.2728i 101.536 − 48.8970i −67.6714 − 296.488i 58.7306 − 257.315i 489.647 + 235.801i −587.085 − 736.181i 1474.42 + 1848.86i −1071.88 + 516.191i
7.9 −0.592907 − 2.59770i −23.8286 − 11.4752i 108.928 − 52.4567i 39.3242 + 172.291i −15.6811 + 68.7032i 535.061 + 257.672i −413.496 − 518.507i −927.452 − 1162.99i 424.243 − 204.305i
7.10 −0.240160 − 1.05221i 53.2557 + 25.6466i 114.275 − 55.0317i 117.842 + 516.297i 14.1957 − 62.1954i −631.923 − 304.318i −171.482 − 215.031i 814.854 + 1021.79i 514.951 − 247.988i
7.11 1.06847 + 4.68128i −48.2964 − 23.2583i 94.5513 − 45.5335i 40.3356 + 176.722i 57.2754 − 250.940i −546.208 − 263.040i 697.385 + 874.493i 428.024 + 536.725i −784.187 + 377.644i
7.12 1.76623 + 7.73834i 6.60622 + 3.18139i 58.5616 − 28.2018i −74.2479 − 325.301i −12.9506 + 56.7403i −1381.45 − 665.273i 955.122 + 1197.68i −1330.05 − 1667.83i 2386.16 − 1149.11i
7.13 2.40722 + 10.5467i 28.3934 + 13.6735i 9.88560 − 4.76065i 6.04429 + 26.4818i −75.8617 + 332.372i 1128.60 + 543.507i 937.350 + 1175.40i −744.355 − 933.391i −264.746 + 127.495i
7.14 3.24238 + 14.2058i −68.0840 − 32.7875i −75.9671 + 36.5838i −25.8780 − 113.379i 245.018 − 1073.50i 448.514 + 215.993i 396.856 + 497.641i 2196.83 + 2754.74i 1526.73 − 735.234i
7.15 3.76445 + 16.4931i 74.9493 + 36.0937i −142.528 + 68.6378i 7.11367 + 31.1670i −313.155 + 1372.02i −610.316 − 293.913i −318.479 − 399.360i 2951.07 + 3700.53i −487.262 + 234.653i
7.16 4.20910 + 18.4413i −14.7322 − 7.09466i −207.039 + 99.7048i 96.3879 + 422.303i 68.8251 − 301.543i −427.839 − 206.037i −1200.55 − 1505.44i −1196.87 − 1500.83i −7382.09 + 3555.03i
7.17 4.74471 + 20.7879i −1.02360 − 0.492942i −294.303 + 141.729i −102.675 − 449.849i 5.39054 − 23.6175i 314.144 + 151.284i −2640.95 − 3311.64i −1362.77 − 1708.86i 8864.28 − 4268.81i
16.1 −19.7275 − 9.50024i −11.3098 + 14.1820i 219.111 + 274.757i −151.482 − 72.9499i 357.845 − 172.329i 922.971 − 1157.37i −1088.60 − 4769.46i 413.435 + 1811.38i 2295.31 + 2878.23i
16.2 −17.2138 − 8.28972i 41.7608 − 52.3664i 147.788 + 185.321i 140.632 + 67.7250i −1152.96 + 555.239i −797.740 + 1000.33i −463.553 − 2030.96i −511.622 − 2241.56i −1859.39 − 2331.61i
16.3 −13.8603 − 6.67478i −39.9604 + 50.1088i 67.7493 + 84.9550i 281.850 + 135.732i 888.330 − 427.797i −39.6382 + 49.7048i 66.1999 + 290.041i −427.401 − 1872.57i −3000.56 − 3762.58i
See next 80 embeddings (of 102 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.17
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.8.d.a ✓ 102
29.d even 7 1 inner 29.8.d.a ✓ 102
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.8.d.a ✓ 102 1.a even 1 1 trivial
29.8.d.a ✓ 102 29.d even 7 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(29, [\chi])\).