Properties

Label 29.10.e.a
Level $29$
Weight $10$
Character orbit 29.e
Analytic conductor $14.936$
Analytic rank $0$
Dimension $132$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 29.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.9360392488\)
Analytic rank: \(0\)
Dimension: \(132\)
Relative dimension: \(22\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 132q - 7q^{2} - 7q^{3} + 5797q^{4} + 1367q^{5} - 7607q^{6} + 4949q^{7} + 31766q^{8} + 112237q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 132q - 7q^{2} - 7q^{3} + 5797q^{4} + 1367q^{5} - 7607q^{6} + 4949q^{7} + 31766q^{8} + 112237q^{9} - 7q^{10} + 230503q^{11} + 370719q^{13} - 7q^{14} - 134953q^{15} - 942171q^{16} + 2518152q^{18} - 7q^{19} + 2630005q^{20} - 5671057q^{21} - 1956426q^{22} - 5580815q^{23} + 3549677q^{24} - 4029457q^{25} + 9649689q^{26} + 6835241q^{27} - 24789228q^{28} - 4235889q^{29} + 34444162q^{30} + 10383597q^{31} + 8347164q^{32} + 35912671q^{33} - 11257679q^{34} + 873325q^{35} - 125256554q^{36} - 32126087q^{37} + 131520700q^{38} + 192355527q^{39} - 158097415q^{40} + 127695676q^{42} - 78365203q^{43} - 238378098q^{44} + 57884904q^{45} + 6546071q^{47} - 400523802q^{48} - 221520647q^{49} + 154918442q^{50} + 278243026q^{51} - 52352040q^{52} + 352279862q^{53} + 155358204q^{54} + 143671997q^{55} - 575895495q^{56} - 298751838q^{57} - 1211966547q^{58} + 776098748q^{59} + 112101752q^{60} - 124049779q^{61} + 669065875q^{62} + 1929121403q^{63} + 1625889382q^{64} - 866654028q^{65} - 852661411q^{66} + 13275097q^{67} - 1504278412q^{68} - 1267751723q^{69} + 1776837051q^{71} + 6219206035q^{72} - 1507755424q^{73} - 4950184776q^{74} + 422070887q^{76} - 561272089q^{77} - 1228370591q^{78} - 973234717q^{79} + 1423181574q^{80} + 2306016039q^{81} - 1386574594q^{82} - 932915983q^{83} + 261177525q^{84} - 3402309596q^{85} - 3486024644q^{86} - 1597393079q^{87} + 5582534q^{88} + 2316401563q^{89} - 4748880458q^{90} + 659194133q^{91} + 11318573556q^{92} + 3911968919q^{93} + 3703837052q^{94} + 2797945913q^{95} - 1910264674q^{96} - 2438073624q^{97} - 1034416404q^{98} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −43.2850 9.87951i −99.6479 206.921i 1314.69 + 633.120i 467.851 2049.79i 2268.98 + 9941.04i 246.434 118.676i −32878.9 26220.0i −20614.4 + 25849.7i −40501.8 + 84102.9i
4.2 −39.6416 9.04794i 102.716 + 213.292i 1028.30 + 495.202i 153.222 671.311i −2141.98 9384.62i 7874.00 3791.92i −20006.3 15954.5i −22670.8 + 28428.3i −12148.0 + 25225.5i
4.3 −36.7996 8.39927i −9.86657 20.4882i 822.366 + 396.030i −323.078 + 1415.50i 191.000 + 836.827i −556.278 + 267.890i −11826.7 9431.50i 11949.7 14984.5i 23778.3 49376.1i
4.4 −31.7893 7.25570i 38.8499 + 80.6726i 496.618 + 239.158i 230.482 1009.81i −649.674 2846.41i −5320.98 + 2562.45i −999.429 797.018i 7273.39 9120.54i −14653.7 + 30428.7i
4.5 −23.4413 5.35033i −66.5806 138.256i 59.5740 + 28.6893i −62.4981 + 273.822i 821.022 + 3597.13i 9803.33 4721.04i 8381.83 + 6684.29i −2409.61 + 3021.55i 2930.08 6084.37i
4.6 −21.8728 4.99233i −11.8733 24.6553i −7.79837 3.75550i 593.649 2600.95i 136.617 + 598.556i −1813.09 + 873.139i 9132.64 + 7283.04i 11805.2 14803.3i −25969.6 + 53926.4i
4.7 −20.8398 4.75655i −98.0825 203.670i −49.6235 23.8974i −133.957 + 586.902i 1075.25 + 4710.98i −5718.27 + 2753.77i 9477.13 + 7557.76i −19589.3 + 24564.2i 5583.26 11593.8i
4.8 −20.7241 4.73015i 108.635 + 225.583i −54.1809 26.0921i −432.228 + 1893.72i −1184.33 5188.87i −10119.8 + 4873.44i 9508.60 + 7582.85i −26814.0 + 33623.6i 17915.1 37201.1i
4.9 −14.0968 3.21750i 63.5414 + 131.945i −272.929 131.436i 30.5824 133.990i −471.197 2064.45i 5267.91 2536.89i 9212.56 + 7346.77i −1099.86 + 1379.18i −862.229 + 1790.44i
4.10 −4.08372 0.932084i 3.89931 + 8.09700i −445.488 214.536i −560.536 + 2455.87i −8.37663 36.7004i 1868.32 899.734i 3296.03 + 2628.50i 12221.8 15325.6i 4578.15 9506.62i
4.11 −0.729154 0.166425i −38.7761 80.5193i −460.792 221.906i 26.4117 115.717i 14.8733 + 65.1643i −8751.97 + 4214.73i 598.443 + 477.242i 7292.37 9144.34i −38.5163 + 79.9800i
4.12 6.64802 + 1.51737i −71.0184 147.471i −419.402 201.973i 345.044 1511.73i −248.364 1088.15i 4671.19 2249.53i −5211.35 4155.92i −4432.00 + 5557.56i 4587.72 9526.49i
4.13 8.86626 + 2.02367i 45.0762 + 93.6016i −386.781 186.264i 69.9428 306.440i 210.239 + 921.116i 1930.08 929.476i −6692.78 5337.31i 5542.75 6950.39i 1240.26 2575.43i
4.14 8.94046 + 2.04060i 101.023 + 209.777i −385.528 185.661i 456.096 1998.29i 475.124 + 2081.65i −6625.33 + 3190.59i −6738.82 5374.03i −21528.6 + 26996.0i 8155.42 16934.9i
4.15 16.7749 + 3.82877i −106.901 221.983i −194.557 93.6940i −480.857 + 2106.77i −943.342 4133.05i 3209.49 1545.61i −9792.61 7809.34i −25576.5 + 32071.9i −16132.7 + 33499.8i
4.16 25.0876 + 5.72609i 18.6267 + 38.6787i 135.306 + 65.1598i −277.936 + 1217.72i 245.822 + 1077.02i −7762.93 + 3738.43i −7279.40 5805.13i 11123.1 13947.9i −13945.5 + 28958.2i
4.17 25.1563 + 5.74177i 103.456 + 214.830i 138.577 + 66.7350i −384.822 + 1686.01i 1369.08 + 5998.34i 7325.43 3527.74i −7226.09 5762.61i −23176.3 + 29062.2i −19361.4 + 40204.4i
4.18 28.1428 + 6.42341i −39.5780 82.1846i 289.462 + 139.397i −3.26559 + 14.3075i −585.931 2567.13i 2325.76 1120.03i −4304.36 3432.61i 7084.26 8883.38i −183.805 + 381.676i
4.19 32.2233 + 7.35476i 27.7652 + 57.6551i 522.953 + 251.841i 507.417 2223.14i 470.648 + 2062.05i 5665.53 2728.37i 1768.42 + 1410.27i 9718.95 12187.2i 32701.3 67905.0i
4.20 35.1721 + 8.02781i −104.016 215.992i 711.338 + 342.562i 373.217 1635.17i −1924.53 8431.91i −9972.96 + 4802.72i 7827.86 + 6242.51i −23560.9 + 29544.4i 26253.7 54516.3i
See next 80 embeddings (of 132 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.e even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.10.e.a 132
29.e even 14 1 inner 29.10.e.a 132
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.10.e.a 132 1.a even 1 1 trivial
29.10.e.a 132 29.e even 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(29, [\chi])\).