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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [43,9,Mod(7,43)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(43, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("43.7");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 43.d (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.5172802326\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 7.28
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 \(S_{9}^{\mathrm{new}}(43, [\chi])\).