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Properties

Label 43.10.e.a
Level $43$
Weight $10$
Character orbit 43.e
Analytic conductor $22.147$
Analytic rank $0$
Dimension $192$
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] = [43,10,Mod(4,43)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(43, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("43.4");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 43.e (of order \(7\), degree \(6\), 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: \(22.1465409550\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 192 q - 23 q^{2} + 141 q^{3} - 7771 q^{4} + 677 q^{5} - 6572 q^{6} - 23872 q^{7} + 40269 q^{8} - 322935 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q - 23 q^{2} + 141 q^{3} - 7771 q^{4} + 677 q^{5} - 6572 q^{6} - 23872 q^{7} + 40269 q^{8} - 322935 q^{9} + 12467 q^{10} + 26107 q^{11} - 8535 q^{12} - 124789 q^{13} + 402537 q^{14} + 536701 q^{15} + 470873 q^{16} + 157321 q^{17} - 126108 q^{18} + 1524020 q^{19} - 3731811 q^{20} - 1204984 q^{21} + 5705415 q^{22} - 1812193 q^{23} + 1656287 q^{24} - 6908369 q^{25} + 11484147 q^{26} + 4196853 q^{27} + 13689789 q^{28} + 14198685 q^{29} - 43510444 q^{30} + 2623633 q^{31} + 68431179 q^{32} + 35109256 q^{33} - 31068733 q^{34} - 25357310 q^{35} + 251714416 q^{36} - 19142308 q^{37} + 119772015 q^{38} - 31427027 q^{39} - 33090881 q^{40} - 50107991 q^{41} - 162738222 q^{42} - 73848259 q^{43} - 321658712 q^{44} + 91481051 q^{45} + 137200254 q^{46} + 17832081 q^{47} + 196320169 q^{48} + 845134600 q^{49} - 440263230 q^{50} - 49676244 q^{51} - 317591839 q^{52} - 69882873 q^{53} - 60806391 q^{54} - 120195607 q^{55} - 926966829 q^{56} + 282620123 q^{57} + 1032073799 q^{58} - 5290157 q^{59} - 1271653181 q^{60} - 217057103 q^{61} - 185133921 q^{62} + 802790658 q^{63} - 1352549483 q^{64} + 503543227 q^{65} - 234740783 q^{66} - 688857817 q^{67} + 1407702672 q^{68} - 234230557 q^{69} - 207245722 q^{70} + 315865545 q^{71} + 2050452844 q^{72} - 1188540917 q^{73} - 580770182 q^{74} - 1096370858 q^{75} - 1030976936 q^{76} + 126981666 q^{77} + 712275668 q^{78} - 355387036 q^{79} + 1135667026 q^{80} + 816824016 q^{81} - 1149421181 q^{82} + 554073368 q^{83} - 5832481562 q^{84} - 6971027678 q^{85} + 3317069529 q^{86} - 3939884094 q^{87} + 5966424645 q^{88} + 172293025 q^{89} + 11626974152 q^{90} + 852733752 q^{91} + 7930102584 q^{92} + 1786213690 q^{93} + 3419396521 q^{94} - 439143565 q^{95} - 7908382084 q^{96} - 619685773 q^{97} - 4402632007 q^{98} - 5258298740 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −26.2907 + 32.9675i −118.868 − 149.056i −281.723 − 1234.31i 2176.31 + 1048.06i 8039.13 −8174.81 28647.4 + 13795.8i −3708.17 + 16246.5i −91768.4 + 44193.3i
4.2 −26.2638 + 32.9338i −77.4797 − 97.1565i −280.916 − 1230.77i −1723.13 − 829.815i 5234.65 4745.47 28480.4 + 13715.4i 943.600 − 4134.18i 72584.9 − 34955.1i
4.3 −26.1677 + 32.8132i 147.714 + 185.228i −278.030 − 1218.13i 460.421 + 221.727i −9943.26 −4982.64 27885.8 + 13429.1i −8109.94 + 35532.0i −19323.7 + 9305.83i
4.4 −24.3918 + 30.5863i 35.3753 + 44.3592i −226.634 − 992.949i 417.253 + 200.939i −2219.65 5951.78 17852.1 + 8597.14i 3663.55 − 16051.1i −16323.5 + 7861.00i
4.5 −21.3847 + 26.8156i 52.4290 + 65.7439i −147.838 − 647.722i 678.011 + 326.513i −2884.14 1232.33 4708.81 + 2267.64i 2806.42 − 12295.7i −23254.7 + 11198.9i
4.6 −19.7891 + 24.8147i −46.9187 − 58.8342i −110.232 − 482.956i −600.197 − 289.040i 2388.43 −12373.3 −475.376 − 228.929i 3119.78 − 13668.6i 19049.8 − 9173.89i
4.7 −19.0176 + 23.8473i 111.115 + 139.334i −93.0949 − 407.875i −2441.31 − 1175.67i −5435.90 −2551.51 −2573.21 − 1239.19i −2687.54 + 11774.9i 74464.4 − 35860.2i
4.8 −17.2044 + 21.5737i −168.740 − 211.593i −55.5006 − 243.164i −169.052 − 81.4112i 7467.91 4297.73 −6528.11 − 3143.77i −11918.6 + 52218.7i 4664.79 − 2246.44i
4.9 −15.4630 + 19.3900i −85.3477 − 107.023i −22.9363 − 100.491i −767.251 − 369.489i 3394.90 −181.232 −9137.29 − 4400.29i 210.268 − 921.243i 19028.4 − 9163.58i
4.10 −15.4429 + 19.3648i −54.0674 − 67.7984i −22.5810 − 98.9339i 2199.98 + 1059.45i 2147.86 8325.76 −9161.06 − 4411.73i 2706.54 − 11858.1i −54490.1 + 26241.0i
4.11 −11.6015 + 14.5478i 67.3089 + 84.4027i 36.8863 + 161.609i −323.716 − 155.893i −2008.76 2183.78 −11362.5 − 5471.90i 1786.55 − 7827.40i 6023.51 − 2900.77i
4.12 −11.0261 + 13.8263i 166.166 + 208.366i 44.3393 + 194.263i 324.564 + 156.302i −4713.08 9013.73 −11332.6 − 5457.50i −11425.2 + 50057.0i −5739.75 + 2764.12i
4.13 −10.8792 + 13.6421i 99.6054 + 124.901i 46.1810 + 202.332i 2039.80 + 982.318i −2787.54 −10145.3 −11311.8 − 5447.46i −1299.20 + 5692.15i −35592.4 + 17140.4i
4.14 −5.99579 + 7.51849i −60.3022 − 75.6165i 93.3526 + 409.004i 174.955 + 84.2538i 930.081 −3051.46 −8070.88 − 3886.73i 2298.37 − 10069.8i −1682.45 + 810.227i
4.15 −4.59349 + 5.76005i −30.1741 − 37.8371i 101.853 + 446.246i −2012.89 − 969.358i 356.548 12089.1 −6436.80 − 3099.80i 3858.71 − 16906.1i 14829.7 − 7141.63i
4.16 −2.04805 + 2.56818i 38.4554 + 48.2216i 111.530 + 488.644i −1030.76 − 496.387i −202.600 −4771.50 −2998.62 − 1444.06i 3533.38 − 15480.7i 3385.86 − 1630.54i
4.17 −0.689453 + 0.864546i −111.926 − 140.350i 113.659 + 497.971i 1280.32 + 616.570i 198.507 −2707.51 −1018.98 − 490.715i −2790.97 + 12228.0i −1415.77 + 681.801i
4.18 2.75125 − 3.44996i −150.990 − 189.336i 109.598 + 480.180i −2399.77 − 1155.67i −1068.61 −8095.40 3993.68 + 1923.26i −8670.08 + 37986.1i −10589.4 + 5099.58i
4.19 4.12806 − 5.17643i 85.9143 + 107.733i 104.176 + 456.426i 1027.10 + 494.625i 912.333 8194.69 5846.90 + 2815.72i 154.716 − 677.855i 6800.33 − 3274.86i
4.20 5.28774 − 6.63062i 132.521 + 166.177i 97.9258 + 429.041i −992.851 − 478.132i 1802.59 −5295.49 7274.81 + 3503.36i −5672.86 + 24854.4i −8420.25 + 4054.98i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.10.e.a ✓ 192
43.e even 7 1 inner 43.10.e.a ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.10.e.a ✓ 192 1.a even 1 1 trivial
43.10.e.a ✓ 192 43.e even 7 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(43, [\chi])\).