Properties

Label 43.9.d.a
Level 43
Weight 9
Character orbit 43.d
Analytic conductor 17.517
Analytic rank 0
Dimension 56
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 43 \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 43.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.5172802326\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56q - 84q^{3} - 6474q^{4} - 3q^{5} + 1791q^{6} - 5031q^{7} + 57858q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 56q - 84q^{3} - 6474q^{4} - 3q^{5} + 1791q^{6} - 5031q^{7} + 57858q^{9} + 12491q^{10} + 31928q^{11} + 56574q^{12} - 24868q^{13} + 106830q^{14} - 11989q^{15} + 655366q^{16} + 5963q^{17} + 326445q^{18} - 67476q^{19} - 713343q^{20} - 187232q^{21} - 210724q^{23} - 1651574q^{24} + 1650473q^{25} + 765q^{26} + 684492q^{28} - 5568699q^{29} - 3412593q^{30} + 2602959q^{31} - 554514q^{33} - 584967q^{34} + 8137398q^{35} - 3035428q^{36} - 3541812q^{37} + 6280356q^{38} - 2473841q^{40} + 7375370q^{41} + 15490227q^{43} - 2983894q^{44} + 3217584q^{46} - 7588210q^{47} + 5844987q^{48} + 33155723q^{49} - 9064422q^{50} + 3837506q^{52} - 5366122q^{53} - 45158560q^{54} + 40061478q^{55} - 39705477q^{56} + 13330347q^{57} + 19271684q^{58} + 18367244q^{59} - 36808121q^{60} - 6722484q^{61} - 15483591q^{62} - 40574925q^{63} - 87236036q^{64} - 33263013q^{66} - 8581935q^{67} - 123045082q^{68} - 26516709q^{69} - 132304098q^{71} - 100762446q^{72} + 278404467q^{73} + 182026410q^{74} + 179327130q^{76} + 143930505q^{77} - 143413472q^{78} - 62931368q^{79} + 121842717q^{80} - 65032540q^{81} - 29005780q^{83} + 370916864q^{84} + 73281798q^{86} + 137580504q^{87} - 124655943q^{89} + 63835252q^{90} + 287393496q^{91} - 215232271q^{92} - 124327923q^{93} + 413979855q^{95} + 51391714q^{96} - 446400270q^{97} - 351696195q^{98} - 146664065q^{99} + O(q^{100}) \)

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 31.1706i −22.2576 + 12.8504i −715.604 466.420 269.288i 400.555 + 693.781i 2878.38 + 1661.83i 14326.1i −2950.23 + 5109.95i −8393.86 14538.6i
7.2 29.0831i −96.3585 + 55.6326i −589.829 −695.509 + 401.552i 1617.97 + 2802.41i −1511.66 872.755i 9708.78i 2909.48 5039.36i 11678.4 + 20227.6i
7.3 26.9022i 74.3127 42.9044i −467.727 −575.075 + 332.020i −1154.22 1999.17i −1517.67 876.227i 5695.91i 401.081 694.692i 8932.05 + 15470.8i
7.4 25.0164i 105.358 60.8284i −369.821 539.881 311.700i −1521.71 2635.68i 1531.35 + 884.128i 2847.41i 4119.69 7135.51i −7797.62 13505.9i
7.5 23.8148i −40.9385 + 23.6358i −311.143 728.687 420.708i 562.882 + 974.940i −3233.30 1866.75i 1313.23i −2163.19 + 3746.76i −10019.1 17353.5i
7.6 21.4690i −45.9397 + 26.5233i −204.918 −326.905 + 188.739i 569.429 + 986.280i 2218.06 + 1280.60i 1096.68i −1873.53 + 3245.04i 4052.03 + 7018.32i
7.7 18.7653i −129.819 + 74.9511i −96.1366 343.803 198.495i 1406.48 + 2436.10i 1031.87 + 595.751i 2999.89i 7954.84 13778.2i −3724.82 6451.57i
7.8 15.0304i 10.2856 5.93838i 30.0883 −349.334 + 201.688i −89.2560 154.596i 764.498 + 441.383i 4300.01i −3209.97 + 5559.83i 3031.45 + 5250.62i
7.9 13.9154i 49.0781 28.3352i 62.3606 389.929 225.126i −394.297 682.943i −1302.81 752.179i 4430.13i −1674.73 + 2900.72i −3132.72 5426.03i
7.10 9.44319i −73.7010 + 42.5513i 166.826 −1017.04 + 587.188i 401.820 + 695.972i −3125.69 1804.62i 3992.83i 340.724 590.151i 5544.92 + 9604.09i
7.11 8.98674i 117.071 67.5912i 175.239 −869.912 + 502.244i −607.424 1052.09i 3700.56 + 2136.52i 3875.43i 5856.64 10144.0i 4513.54 + 7817.67i
7.12 7.02236i 126.337 72.9406i 206.686 353.829 204.284i −512.215 887.182i −2295.79 1325.47i 3249.15i 7360.15 12748.2i −1434.55 2484.72i
7.13 5.01466i 14.2615 8.23388i 230.853 929.674 536.748i −41.2901 71.5166i 3070.94 + 1773.01i 2441.40i −3144.91 + 5447.14i −2691.61 4662.00i
7.14 4.04858i −89.5347 + 51.6929i 239.609 −50.9246 + 29.4013i 209.283 + 362.488i 839.413 + 484.635i 2006.51i 2063.81 3574.63i 119.034 + 206.172i
7.15 0.0978549i 50.1238 28.9390i 255.990 −516.114 + 297.978i 2.83182 + 4.90486i −3014.43 1740.38i 50.1008i −1605.57 + 2780.93i −29.1587 50.5043i
7.16 0.360163i −80.5202 + 46.4884i 255.870 539.177 311.294i −16.7434 29.0004i −1661.31 959.156i 184.357i 1041.83 1804.51i 112.117 + 194.192i
7.17 7.28626i 47.2038 27.2531i 202.910 −237.272 + 136.989i 198.573 + 343.939i 742.931 + 428.932i 3343.74i −1795.04 + 3109.09i −998.137 1728.82i
7.18 9.49614i −65.9694 + 38.0875i 165.823 −622.087 + 359.162i −361.684 626.455i 2996.66 + 1730.12i 4005.69i −379.191 + 656.777i −3410.65 5907.42i
7.19 13.2055i 103.620 59.8253i 81.6138 541.475 312.621i 790.025 + 1368.36i 754.120 + 435.391i 4458.37i 3877.63 6716.26i 4128.32 + 7150.47i
7.20 13.4706i −22.2334 + 12.8365i 74.5432 377.208 217.781i −172.915 299.497i −2801.83 1617.64i 4452.61i −2950.95 + 5111.20i 2933.64 + 5081.22i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.9.d.a 56
43.d odd 6 1 inner 43.9.d.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.9.d.a 56 1.a even 1 1 trivial
43.9.d.a 56 43.d odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(43, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database