# Properties

 Label 29.10.d.a Level $29$ Weight $10$ Character orbit 29.d Analytic conductor $14.936$ Analytic rank $0$ Dimension $126$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$29$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 29.d (of order $$7$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.9360392488$$ Analytic rank: $$0$$ Dimension: $$126$$ Relative dimension: $$21$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$126q - 23q^{2} - 5q^{3} - 4699q^{4} - 1031q^{5} + 12781q^{6} - 4959q^{7} - 43354q^{8} - 76050q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$126q - 23q^{2} - 5q^{3} - 4699q^{4} - 1031q^{5} + 12781q^{6} - 4959q^{7} - 43354q^{8} - 76050q^{9} + 8859q^{10} - 195479q^{11} + 141544q^{12} - 68475q^{13} - 271695q^{14} + 322691q^{15} - 916571q^{16} - 587320q^{17} - 3205692q^{18} + 172239q^{19} + 3022721q^{20} + 4240955q^{21} - 4806212q^{22} + 4136729q^{23} + 8763829q^{24} + 3929026q^{25} - 8916101q^{26} - 22233347q^{27} - 5636492q^{28} - 8783106q^{29} - 13875058q^{30} + 4440751q^{31} + 50266340q^{32} - 13013201q^{33} + 16498997q^{34} + 2015357q^{35} - 85629988q^{36} + 3514043q^{37} + 14330570q^{38} - 72230993q^{39} + 133179859q^{40} + 16544588q^{41} - 137066822q^{42} + 51166115q^{43} - 29984908q^{44} + 263764053q^{45} + 383616568q^{46} - 97798039q^{47} - 402814002q^{48} + 35450702q^{49} + 7192792q^{50} + 57784388q^{51} + 602315978q^{52} + 137607435q^{53} + 125382220q^{54} - 411109327q^{55} - 407148817q^{56} - 138815986q^{57} - 723336259q^{58} - 899263700q^{59} + 447938680q^{60} + 666491883q^{61} + 1037044473q^{62} - 190916493q^{63} + 65021266q^{64} + 1392512087q^{65} - 1056293177q^{66} - 903332019q^{67} - 1632042348q^{68} - 527752591q^{69} - 682041198q^{70} + 1107000159q^{71} + 430010561q^{72} + 657123873q^{73} - 198127908q^{74} - 2404866172q^{75} - 2839934913q^{76} - 45301525q^{77} + 2634791081q^{78} + 1783807493q^{79} + 3968800932q^{80} + 2649824196q^{81} + 3528483524q^{82} - 1900496455q^{83} - 3681471661q^{84} + 183497446q^{85} + 1213703072q^{86} - 4480302165q^{87} - 3448461606q^{88} - 2924525111q^{89} + 6284889126q^{90} + 3601999725q^{91} + 1893443670q^{92} - 961083481q^{93} + 1812573594q^{94} + 837629545q^{95} - 6708668090q^{96} + 4263408987q^{97} + 1842677548q^{98} - 1747477852q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −8.91229 39.0473i 127.724 + 61.5086i −983.966 + 473.853i −236.965 1038.21i 1263.43 5535.46i −2677.25 1289.29i 14486.6 + 18165.6i 257.944 + 323.451i −38427.5 + 18505.7i
7.2 −8.73749 38.2814i −154.771 74.5340i −927.828 + 446.819i 87.4849 + 383.296i −1500.95 + 6576.11i 1543.32 + 743.222i 12677.0 + 15896.5i 6126.74 + 7682.69i 13908.7 6698.09i
7.3 −7.97654 34.9475i 73.3338 + 35.3157i −696.407 + 335.372i 440.975 + 1932.04i 649.245 2844.53i −672.333 323.778i 5832.25 + 7313.41i −8141.50 10209.1i 64002.5 30822.0i
7.4 −5.87345 25.7333i 26.9577 + 12.9821i −166.408 + 80.1377i −430.743 1887.21i 175.738 769.959i 11130.5 + 5360.17i −5386.41 6754.35i −11714.0 14688.9i −46034.1 + 22168.8i
7.5 −5.54005 24.2726i −131.756 63.4505i −97.1687 + 46.7940i −324.293 1420.82i −810.169 + 3549.58i −5845.17 2814.89i −6273.59 7866.83i 1061.61 + 1331.22i −32690.3 + 15742.8i
7.6 −4.51238 19.7700i 207.546 + 99.9488i 90.8034 43.7286i 242.704 + 1063.35i 1039.46 4554.19i 7696.37 + 3706.38i −7747.68 9715.29i 20813.3 + 26099.1i 19927.4 9596.52i
7.7 −3.60417 15.7909i 194.806 + 93.8135i 224.933 108.322i −195.891 858.256i 779.288 3414.28i −8317.05 4005.28i −7691.72 9645.11i 16876.1 + 21162.0i −12846.6 + 6186.61i
7.8 −3.55763 15.5870i −34.2508 16.4943i 230.999 111.243i 376.920 + 1651.40i −135.245 + 592.548i −3361.64 1618.88i −7659.50 9604.70i −11371.1 14258.9i 24399.3 11750.1i
7.9 −2.47449 10.8414i −225.022 108.365i 349.882 168.494i 219.969 + 963.748i −618.017 + 2707.71i 8189.04 + 3943.63i −6242.39 7827.70i 26619.8 + 33380.1i 9904.11 4769.57i
7.10 0.385822 + 1.69040i 56.2085 + 27.0686i 458.587 220.844i −382.587 1676.22i −24.0702 + 105.458i −3478.73 1675.27i 1103.74 + 1384.05i −9845.46 12345.8i 2685.87 1293.45i
7.11 0.627749 + 2.75035i 11.2588 + 5.42194i 454.126 218.695i −3.74500 16.4079i −7.84454 + 34.3692i 4557.29 + 2194.68i 1787.13 + 2240.99i −12174.8 15266.7i 42.7766 20.6001i
7.12 1.74711 + 7.65458i −184.828 89.0086i 405.756 195.402i −360.877 1581.11i 358.408 1570.29i −2780.15 1338.85i 4711.00 + 5907.41i 13966.8 + 17513.8i 11472.2 5524.73i
7.13 3.16487 + 13.8662i 167.309 + 80.5715i 279.040 134.379i 294.188 + 1288.92i −587.712 + 2574.94i 197.795 + 95.2529i 7286.76 + 9137.30i 9228.22 + 11571.8i −16941.4 + 8158.56i
7.14 3.51960 + 15.4204i −156.258 75.2501i 235.896 113.601i 407.649 + 1786.03i 610.417 2674.41i −9946.82 4790.14i 7631.23 + 9569.25i 6481.96 + 8128.12i −26106.5 + 12572.2i
7.15 5.60294 + 24.5481i 216.248 + 104.140i −109.919 + 52.9341i −599.659 2627.28i −1344.80 + 5891.96i 6165.53 + 2969.16i 6122.63 + 7677.54i 23646.0 + 29651.2i 61134.7 29440.9i
7.16 5.62456 + 24.6428i −142.401 68.5766i −114.336 + 55.0612i −253.940 1112.59i 888.978 3894.87i 4795.80 + 2309.54i 6069.00 + 7610.28i 3303.10 + 4141.95i 25988.9 12515.6i
7.17 6.40812 + 28.0758i −74.0782 35.6742i −285.891 + 137.678i 487.167 + 2134.42i 526.879 2308.41i 7791.48 + 3752.18i 3495.59 + 4383.34i −8057.22 10103.4i −56803.7 + 27355.2i
7.18 6.98240 + 30.5919i 109.509 + 52.7366i −425.814 + 205.061i 98.6572 + 432.245i −848.680 + 3718.31i −9851.07 4744.03i 770.482 + 966.154i −3061.13 3838.54i −12534.3 + 6036.22i
7.19 7.92629 + 34.7274i −14.5980 7.03001i −681.868 + 328.370i −247.282 1083.41i 128.426 562.671i −1483.34 714.340i −5437.11 6817.92i −12108.5 15183.5i 35664.0 17174.9i
7.20 9.66013 + 42.3238i 159.134 + 76.6349i −1236.69 + 595.558i 242.294 + 1061.56i −1706.22 + 7475.46i 6300.89 + 3034.35i −23294.5 29210.4i 7178.56 + 9001.63i −42588.6 + 20509.6i
See next 80 embeddings (of 126 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 25.21 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 29.10.d.a 126
29.d even 7 1 inner 29.10.d.a 126

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.10.d.a 126 1.a even 1 1 trivial
29.10.d.a 126 29.d even 7 1 inner

## Hecke kernels

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