Properties

Label 29.8.e.a
Level $29$
Weight $8$
Character orbit 29.e
Analytic conductor $9.059$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 29.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.05916573904\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96q - 7q^{2} - 7q^{3} + 709q^{4} + 191q^{5} + 1801q^{6} - 667q^{7} + 9086q^{8} + 9103q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 96q - 7q^{2} - 7q^{3} + 709q^{4} + 191q^{5} + 1801q^{6} - 667q^{7} + 9086q^{8} + 9103q^{9} - 7q^{10} + 4207q^{11} + 377q^{13} - 7q^{14} - 74473q^{15} - 125851q^{16} - 194880q^{18} - 7q^{19} + 162901q^{20} + 86933q^{21} + 252614q^{22} + 238675q^{23} - 490819q^{24} - 107349q^{25} + 367017q^{26} + 75593q^{27} + 1397540q^{28} + 379863q^{29} - 175646q^{30} - 143885q^{31} + 473172q^{32} - 1419437q^{33} - 2314463q^{34} - 1018325q^{35} + 1720462q^{36} + 662991q^{37} - 2763860q^{38} + 42189q^{39} + 580601q^{40} - 3296492q^{42} - 1587425q^{43} + 2344398q^{44} + 6508254q^{45} + 526673q^{47} + 6175806q^{48} - 1391517q^{49} - 4194526q^{50} - 68204q^{51} - 7720808q^{52} + 1762640q^{53} - 2032428q^{54} + 2655709q^{55} + 17091081q^{56} + 4906842q^{57} + 13231229q^{58} - 8638324q^{59} - 9957136q^{60} + 2797697q^{61} - 884069q^{62} - 20232301q^{63} - 17528026q^{64} - 7454088q^{65} - 17541475q^{66} - 4890949q^{67} + 22029140q^{68} + 13204429q^{69} - 1091301q^{71} + 25137595q^{72} + 14363636q^{73} + 3185328q^{74} - 41019881q^{76} - 25204627q^{77} - 15753743q^{78} + 31186057q^{79} + 24146022q^{80} - 29435757q^{81} - 51438506q^{82} - 8367235q^{83} + 78906093q^{84} + 41540198q^{85} + 65794492q^{86} + 11089375q^{87} + 39260390q^{88} + 6628699q^{89} + 32127214q^{90} + 24315293q^{91} - 39469980q^{92} - 31050649q^{93} - 102445588q^{94} - 75684679q^{95} - 95069050q^{96} + 38131380q^{97} + 19581828q^{98} + O(q^{100}) \)

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 −19.9614 4.55605i 16.7140 + 34.7070i 262.375 + 126.353i 46.2709 202.726i −175.507 768.948i −856.134 + 412.292i −2612.69 2083.55i 438.356 549.681i −1847.26 + 3835.88i
4.2 −17.1849 3.92235i −28.6166 59.4230i 164.613 + 79.2734i −53.9771 + 236.489i 258.697 + 1133.42i −694.864 + 334.629i −753.925 601.235i −1348.61 + 1691.11i 1855.19 3852.33i
4.3 −16.3184 3.72456i 24.7323 + 51.3572i 137.092 + 66.0202i −116.379 + 509.891i −212.308 930.182i 1113.60 536.279i −316.180 252.145i −662.302 + 830.500i 3798.24 7887.12i
4.4 −16.0299 3.65873i −13.6888 28.4250i 128.249 + 61.7614i 57.5812 252.280i 115.431 + 505.735i 1251.46 602.670i −184.410 147.062i 742.974 931.660i −1846.05 + 3833.35i
4.5 −9.35936 2.13621i 15.2659 + 31.7000i −32.2898 15.5499i −15.7357 + 68.9427i −75.1613 329.303i −676.409 + 325.741i 1229.71 + 980.664i 591.731 742.007i 294.553 611.645i
4.6 −6.62931 1.51310i 34.2559 + 71.1332i −73.6657 35.4756i 94.1794 412.627i −119.462 523.396i 849.727 409.207i 1115.16 + 889.311i −2522.89 + 3163.60i −1248.69 + 2592.93i
4.7 −5.36405 1.22431i −9.60762 19.9504i −88.0499 42.4026i 41.6869 182.642i 27.1103 + 118.778i −510.272 + 245.734i 971.000 + 774.347i 1057.86 1326.51i −447.222 + 928.666i
4.8 −3.46579 0.791044i −14.7361 30.5998i −103.938 50.0539i −91.2666 + 399.865i 26.8664 + 117.709i 200.782 96.6916i 676.389 + 539.402i 644.377 808.023i 632.622 1313.65i
4.9 −1.04671 0.238905i −39.6898 82.4167i −114.285 55.0370i 73.7799 323.251i 21.8540 + 95.7484i 216.705 104.360i 213.918 + 170.594i −3853.65 + 4832.33i −154.452 + 320.724i
4.10 5.20280 + 1.18750i 29.1385 + 60.5068i −89.6651 43.1804i −44.8992 + 196.716i 79.7498 + 349.407i −458.849 + 220.970i −949.289 757.033i −1448.45 + 1816.30i −467.202 + 970.156i
4.11 6.40796 + 1.46258i 1.99118 + 4.13473i −76.4012 36.7929i −7.61716 + 33.3729i 6.71206 + 29.4074i 1563.59 752.987i −1093.53 872.060i 1350.44 1693.40i −97.6209 + 202.712i
4.12 9.92390 + 2.26507i 1.31301 + 2.72650i −21.9707 10.5805i 100.667 441.052i 6.85452 + 30.0316i −925.876 + 445.878i −1212.74 967.127i 1357.86 1702.71i 1998.02 4148.94i
4.13 10.9235 + 2.49321i −26.6467 55.3324i −2.21759 1.06794i −50.3658 + 220.667i −153.119 670.858i −425.854 + 205.081i −1142.84 911.381i −988.053 + 1238.98i −1100.34 + 2284.88i
4.14 17.9262 + 4.09153i 6.16655 + 12.8050i 189.283 + 91.1538i −94.6542 + 414.707i 58.1506 + 254.775i −762.411 + 367.158i 1180.07 + 941.076i 1237.63 1551.94i −3393.57 + 7046.83i
4.15 17.9566 + 4.09849i 26.3233 + 54.6609i 190.320 + 91.6531i 30.2961 132.736i 248.651 + 1089.41i 459.975 221.512i 1198.65 + 955.891i −931.324 + 1167.84i 1088.03 2259.33i
4.16 19.1958 + 4.38132i −23.2918 48.3658i 233.959 + 112.669i 36.5819 160.276i −235.198 1030.47i 541.526 260.785i 2026.99 + 1616.47i −433.175 + 543.185i 1404.44 2916.35i
5.1 −16.8298 + 13.4213i −72.7388 + 16.6021i 74.6272 326.963i 216.453 + 271.423i 1001.35 1255.66i −204.467 895.829i 1936.81 + 4021.83i 3044.88 1466.34i −7285.70 1662.91i
5.2 −15.7422 + 12.5540i 39.5051 9.01678i 61.7319 270.465i −50.2510 63.0128i −508.702 + 637.892i 190.445 + 834.393i 1305.38 + 2710.65i −491.067 + 236.485i 1582.13 + 361.110i
5.3 −11.8009 + 9.41091i 31.2296 7.12795i 22.2135 97.3238i 62.6921 + 78.6134i −301.457 + 378.015i −274.053 1200.70i −184.508 383.134i −1045.94 + 503.698i −1479.65 337.720i
5.4 −10.9983 + 8.77088i −44.4854 + 10.1535i 15.5523 68.1393i −208.135 260.993i 400.210 501.847i −9.74527 42.6968i −354.671 736.482i −94.5645 + 45.5399i 4578.28 + 1044.96i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.16
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 29.8.e.a 96
29.e even 14 1 inner 29.8.e.a 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.8.e.a 96 1.a even 1 1 trivial
29.8.e.a 96 29.e even 14 1 inner

Hecke kernels

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