Properties

Label 29.11.f.a
Level $29$
Weight $11$
Character orbit 29.f
Analytic conductor $18.425$
Analytic rank $0$
Dimension $288$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 29.f (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.4253603275\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 288q - 22q^{2} - 12q^{3} - 14q^{4} - 14q^{5} - 14q^{6} - 10q^{7} - 76784q^{8} - 14q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 288q - 22q^{2} - 12q^{3} - 14q^{4} - 14q^{5} - 14q^{6} - 10q^{7} - 76784q^{8} - 14q^{9} - 109580q^{10} + 263632q^{11} - 729308q^{12} - 14q^{13} - 502630q^{14} + 721736q^{15} + 16030218q^{16} + 4537990q^{17} - 3653920q^{18} - 43706q^{19} - 28125194q^{20} + 8318690q^{21} + 24431218q^{22} - 1804156q^{23} - 132227942q^{24} + 76826206q^{25} + 37018580q^{26} + 57957318q^{27} - 4091366q^{29} - 201956616q^{30} - 140893664q^{31} + 121061292q^{32} + 310830436q^{33} + 244927186q^{34} - 14588490q^{35} - 559552838q^{36} + 133858412q^{37} + 21929586q^{38} + 966545612q^{39} - 137299672q^{40} + 408348734q^{41} - 14q^{42} - 600101864q^{43} - 52997658q^{44} - 1241198886q^{45} - 102035336q^{46} + 830850184q^{47} + 1136468440q^{48} - 739457742q^{49} - 3509687658q^{50} - 2771976830q^{51} + 70578506q^{52} - 1225542446q^{53} + 5404425756q^{54} + 5929963944q^{55} + 4468933076q^{56} - 7683766382q^{58} - 2316484772q^{59} - 15584506756q^{60} - 4735872558q^{61} + 1072705536q^{62} + 10136272594q^{63} + 18488581888q^{64} + 8688930046q^{65} + 17014292728q^{66} - 3314978814q^{67} - 19937284744q^{68} - 6050719742q^{69} - 8553971346q^{70} - 15734676412q^{71} + 10135621864q^{72} + 13472460778q^{73} - 2860394408q^{74} - 2553693192q^{75} + 1533964122q^{76} - 18826817864q^{77} - 10732341438q^{78} - 13300227084q^{79} - 20977281038q^{80} + 55998578950q^{81} + 12348943522q^{82} + 31177470234q^{83} + 107438251148q^{84} + 3272565918q^{85} - 49229617522q^{87} - 63399171248q^{88} - 34988176660q^{89} - 129126675228q^{90} - 26793083114q^{91} + 4799950834q^{92} + 23488122938q^{93} - 28591378914q^{94} + 138203606358q^{95} + 240366548688q^{96} - 54535391692q^{97} - 114107920008q^{98} - 79431838892q^{99} + O(q^{100}) \)

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 −60.4165 6.80730i 298.044 + 187.273i 2605.48 + 594.685i −931.971 + 743.222i −16731.9 13343.3i −2936.78 12866.9i −94601.8 33102.6i 28138.4 + 58429.9i 61365.7 38558.6i
2.2 −54.5116 6.14198i −196.863 123.697i 1935.46 + 441.757i 2084.33 1662.19i 9971.57 + 7952.06i 595.728 + 2610.06i −49771.1 17415.7i −2166.38 4498.53i −123829. + 77807.0i
2.3 −53.3751 6.01393i −211.856 133.118i 1814.41 + 414.128i −3826.59 + 3051.60i 10507.3 + 8379.28i −3164.18 13863.2i −42438.5 14849.9i 1542.20 + 3202.42i 222597. 139867.i
2.4 −47.2252 5.32100i 125.327 + 78.7484i 1203.58 + 274.709i −564.100 + 449.855i −5499.59 4385.77i 7130.90 + 31242.5i −9443.84 3304.54i −16114.8 33462.7i 29033.4 18242.9i
2.5 −43.8958 4.94587i 168.989 + 106.183i 904.055 + 206.345i 4440.32 3541.03i −6892.76 5496.79i −2779.11 12176.1i 4031.70 + 1410.75i −8337.85 17313.7i −212425. + 133475.i
2.6 −35.2938 3.97666i 164.902 + 103.615i 231.515 + 52.8418i −2509.40 + 2001.18i −5408.00 4312.74i −2684.83 11763.0i 26367.7 + 9226.46i −9163.66 19028.5i 96524.5 60650.4i
2.7 −25.6712 2.89245i −357.922 224.897i −347.681 79.3559i 1823.02 1453.81i 8537.79 + 6808.66i −131.712 577.070i 33665.0 + 11779.9i 51908.9 + 107790.i −51004.1 + 32048.0i
2.8 −24.2302 2.73009i −222.660 139.906i −418.676 95.5600i −3285.43 + 2620.04i 5013.14 + 3997.84i 5541.19 + 24277.5i 33451.3 + 11705.1i 4383.12 + 9101.64i 86759.7 54514.7i
2.9 −23.5047 2.64834i −49.6686 31.2088i −452.869 103.365i 494.026 393.973i 1084.79 + 865.093i −3650.42 15993.5i 33232.7 + 11628.6i −24127.4 50101.1i −12655.3 + 7951.86i
2.10 −19.3066 2.17533i 391.127 + 245.761i −630.313 143.865i 593.623 473.399i −7016.72 5595.65i 1021.49 + 4475.44i 30634.9 + 10719.6i 66961.1 + 139046.i −12490.7 + 7848.40i
2.11 −4.75908 0.536219i 86.5441 + 54.3792i −975.965 222.758i 3659.28 2918.18i −382.711 305.202i 5076.39 + 22241.1i 9154.17 + 3203.18i −21087.6 43788.9i −18979.6 + 11925.7i
2.12 0.757424 + 0.0853412i 157.893 + 99.2110i −997.760 227.732i −3734.49 + 2978.16i 111.125 + 88.6196i −313.352 1372.88i −1473.00 515.426i −10532.9 21871.8i −3082.75 + 1937.02i
2.13 7.04486 + 0.793765i −103.202 64.8458i −949.326 216.678i 1386.51 1105.71i −675.568 538.747i −710.542 3113.09i −13368.1 4677.69i −19174.8 39816.9i 10645.5 6688.99i
2.14 12.3156 + 1.38763i −317.080 199.234i −848.579 193.683i −3318.67 + 2646.55i −3628.55 2893.67i −7010.74 30716.1i −22160.7 7754.37i 35224.8 + 73145.1i −44543.8 + 27988.7i
2.15 19.8291 + 2.23421i −232.445 146.055i −610.123 139.257i −1193.27 + 951.598i −4282.87 3415.48i 5593.25 + 24505.6i −31073.9 10873.2i 7078.35 + 14698.3i −25787.5 + 16203.4i
2.16 21.2097 + 2.38976i 242.606 + 152.439i −554.184 126.489i −1573.54 + 1254.86i 4781.31 + 3812.97i 932.676 + 4086.32i −32081.5 11225.8i 9999.49 + 20764.1i −36373.3 + 22854.9i
2.17 23.5437 + 2.65273i 243.217 + 152.824i −451.060 102.951i 3023.75 2411.36i 5320.82 + 4243.21i −7168.45 31407.0i −33246.3 11633.4i 10179.2 + 21137.3i 77586.7 48751.0i
2.18 34.3912 + 3.87495i −318.343 200.028i 169.411 + 38.6669i 4257.16 3394.97i −10173.1 8112.76i 622.677 + 2728.12i −27774.2 9718.63i 35710.5 + 74153.6i 159564. 100261.i
2.19 44.0608 + 4.96446i 271.259 + 170.443i 918.379 + 209.614i 1792.39 1429.39i 11105.7 + 8856.50i 4523.66 + 19819.4i −3431.97 1200.90i 18910.0 + 39267.0i 86070.4 54081.6i
2.20 44.2102 + 4.98130i −5.41800 3.40436i 931.406 + 212.587i −3133.77 + 2499.10i −222.573 177.496i 2520.68 + 11043.8i −2882.49 1008.63i −25602.6 53164.4i −150993. + 94875.4i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.24
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 29.11.f.a 288
29.f odd 28 1 inner 29.11.f.a 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.11.f.a 288 1.a even 1 1 trivial
29.11.f.a 288 29.f odd 28 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(29, [\chi])\).