Properties

Label 29.9.f.a
Level $29$
Weight $9$
Character orbit 29.f
Analytic conductor $11.814$
Analytic rank $0$
Dimension $228$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 29.f (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.8139796918\)
Analytic rank: \(0\)
Dimension: \(228\)
Relative dimension: \(19\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 228q - 6q^{2} - 12q^{3} - 14q^{4} - 14q^{5} - 14q^{6} - 10q^{7} + 3664q^{8} - 14q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 228q - 6q^{2} - 12q^{3} - 14q^{4} - 14q^{5} - 14q^{6} - 10q^{7} + 3664q^{8} - 14q^{9} - 15884q^{10} + 8376q^{11} - 5036q^{12} - 14q^{13} - 12742q^{14} + 35516q^{15} + 245898q^{16} - 76248q^{17} + 465008q^{18} + 164474q^{19} - 907274q^{20} - 628270q^{21} + 1214962q^{22} + 173408q^{23} + 1858618q^{24} + 43976q^{25} - 4604204q^{26} - 1727586q^{27} + 1079922q^{29} + 8909208q^{30} + 3657848q^{31} + 1160364q^{32} - 4279184q^{33} - 8655374q^{34} - 8269058q^{35} - 3366854q^{36} - 5622890q^{37} + 15536626q^{38} + 16110008q^{39} - 23544696q^{40} - 9561672q^{41} - 14q^{42} + 5181008q^{43} + 51041750q^{44} - 35245194q^{45} - 43159224q^{46} - 12806808q^{47} + 83954488q^{48} + 10429400q^{49} + 80809398q^{50} + 18760882q^{51} - 93050550q^{52} - 30960322q^{53} - 106220916q^{54} - 126174004q^{55} - 40136268q^{56} + 159368354q^{58} + 11188020q^{59} + 255893612q^{60} + 139657128q^{61} + 102501616q^{62} + 9797746q^{63} - 135941792q^{64} - 14207858q^{65} - 447597656q^{66} - 148930894q^{67} - 167094904q^{68} + 66986722q^{69} + 586255934q^{70} - 15272096q^{71} + 41230600q^{72} - 219415072q^{73} - 347464696q^{74} + 391507416q^{75} + 271512490q^{76} + 240416780q^{77} + 141747906q^{78} + 112667220q^{79} - 375642638q^{80} - 358099268q^{81} - 262154910q^{82} - 109463350q^{83} - 694921012q^{84} - 356627386q^{85} + 199703522q^{87} + 251202768q^{88} + 839849154q^{89} + 675826020q^{90} + 533256766q^{91} + 742339570q^{92} + 78998570q^{93} - 209116610q^{94} - 633414122q^{95} - 431412576q^{96} - 402046736q^{97} - 2192087384q^{98} - 1966368188q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −31.1717 3.51221i 2.26708 + 1.42450i 709.757 + 161.997i 40.2894 32.1297i −65.6655 52.3665i 235.726 + 1032.78i −13975.6 4890.26i −2843.60 5904.80i −1368.74 + 860.034i
2.2 −24.5490 2.76601i −135.176 84.9370i 345.422 + 78.8403i −399.043 + 318.226i 3083.51 + 2459.02i 40.6060 + 177.907i −2292.28 802.105i 8211.67 + 17051.7i 10676.3 6708.39i
2.3 −22.9178 2.58222i 95.4072 + 59.9483i 268.977 + 61.3923i 748.172 596.647i −2031.73 1620.25i −79.6660 349.039i −433.075 151.540i 2662.02 + 5527.74i −18687.1 + 11741.9i
2.4 −21.5736 2.43076i −17.4548 10.9676i 209.928 + 47.9147i −119.794 + 95.5326i 349.902 + 279.037i −857.743 3758.02i 833.466 + 291.642i −2662.33 5528.38i 2816.60 1769.79i
2.5 −21.4136 2.41273i 91.7346 + 57.6407i 203.138 + 46.3649i −731.114 + 583.044i −1825.29 1455.62i 246.757 + 1081.11i 968.946 + 339.049i 2246.08 + 4664.03i 17062.5 10721.1i
2.6 −18.5588 2.09108i −58.4072 36.6997i 90.4766 + 20.6507i 495.290 394.980i 1007.23 + 803.238i 641.791 + 2811.87i 2876.87 + 1006.66i −782.176 1624.21i −10017.9 + 6294.69i
2.7 −7.14136 0.804638i −31.1173 19.5523i −199.230 45.4729i −619.444 + 493.990i 206.487 + 164.668i 232.502 + 1018.66i 3122.70 + 1092.68i −2260.72 4694.43i 4821.16 3029.33i
2.8 −5.51607 0.621512i 67.5754 + 42.4604i −219.541 50.1088i 264.780 211.155i −346.361 276.213i 606.153 + 2655.73i 2521.16 + 882.193i −83.1675 172.699i −1591.78 + 1000.18i
2.9 −4.14353 0.466864i −78.5126 49.3327i −232.631 53.0964i 799.859 637.866i 302.287 + 241.066i −699.825 3066.14i 1946.68 + 681.171i 883.797 + 1835.22i −3612.03 + 2269.59i
2.10 −3.38682 0.381603i 91.1302 + 57.2609i −238.257 54.3805i 116.314 92.7574i −286.791 228.708i −926.307 4058.42i 1609.73 + 563.269i 2179.19 + 4525.14i −429.332 + 269.767i
2.11 5.20158 + 0.586077i −110.135 69.2024i −222.869 50.8683i −222.021 + 177.056i −532.318 424.509i −45.3512 198.696i −2394.29 837.798i 4494.03 + 9331.94i −1258.63 + 790.849i
2.12 9.08550 + 1.02369i 24.7393 + 15.5448i −168.083 38.3639i −38.0474 + 30.3418i 208.856 + 166.557i −85.6032 375.052i −3697.10 1293.67i −2476.32 5142.12i −376.740 + 236.721i
2.13 14.2835 + 1.60937i 128.283 + 80.6053i −48.1524 10.9905i −467.828 + 373.081i 1702.60 + 1357.78i 469.147 + 2055.47i −4143.33 1449.81i 7112.49 + 14769.2i −7282.66 + 4576.00i
2.14 14.6914 + 1.65532i 1.69586 + 1.06558i −36.4848 8.32741i 784.225 625.399i 23.1506 + 18.4620i 313.531 + 1373.67i −4094.63 1432.77i −2844.97 5907.64i 12556.6 7889.83i
2.15 18.1222 + 2.04189i 3.92609 + 2.46693i 74.6647 + 17.0417i −848.114 + 676.349i 66.1123 + 52.7228i −652.501 2858.79i −3088.37 1080.67i −2837.38 5891.89i −16750.8 + 10525.2i
2.16 19.6004 + 2.20843i −83.4552 52.4384i 129.715 + 29.6067i 14.4052 11.4877i −1519.95 1212.12i 772.012 + 3382.40i −2289.00 800.955i 1368.27 + 2841.25i 307.716 193.351i
2.17 25.7386 + 2.90004i 82.6049 + 51.9041i 404.482 + 92.3203i 418.219 333.519i 1975.61 + 1575.49i −324.689 1422.56i 3884.38 + 1359.20i 1282.82 + 2663.80i 11731.6 7371.44i
2.18 26.7386 + 3.01272i −87.9248 55.2468i 456.297 + 104.147i 197.846 157.777i −2184.55 1742.12i −752.858 3298.49i 5385.14 + 1884.34i 1831.85 + 3803.87i 5765.47 3622.69i
2.19 29.9840 + 3.37838i 11.3862 + 7.15440i 638.043 + 145.629i −500.927 + 399.476i 317.232 + 252.984i 838.383 + 3673.20i 11348.1 + 3970.86i −2768.25 5748.33i −16369.4 + 10285.5i
3.1 −25.9849 16.3274i 93.5645 32.7396i 297.558 + 617.886i −652.275 + 148.877i −2965.82 676.929i 2496.53 + 1202.27i 1476.82 13107.1i 2552.84 2035.82i 19380.1 + 6781.39i
See next 80 embeddings (of 228 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.19
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.f odd 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.9.f.a 228
29.f odd 28 1 inner 29.9.f.a 228
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.9.f.a 228 1.a even 1 1 trivial
29.9.f.a 228 29.f odd 28 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(29, [\chi])\).