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Properties

Label 29.10.e.a
Level $29$
Weight $10$
Character orbit 29.e
Analytic conductor $14.936$
Analytic rank $0$
Dimension $132$
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] = [29,10,Mod(4,29)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(29, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("29.4");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 29.e (of order \(14\), 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: \(14.9360392488\)
Analytic rank: \(0\)
Dimension: \(132\)
Relative dimension: \(22\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 132 q - 7 q^{2} - 7 q^{3} + 5797 q^{4} + 1367 q^{5} - 7607 q^{6} + 4949 q^{7} + 31766 q^{8} + 112237 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 132 q - 7 q^{2} - 7 q^{3} + 5797 q^{4} + 1367 q^{5} - 7607 q^{6} + 4949 q^{7} + 31766 q^{8} + 112237 q^{9} - 7 q^{10} + 230503 q^{11} + 370719 q^{13} - 7 q^{14} - 134953 q^{15} - 942171 q^{16} + 2518152 q^{18} - 7 q^{19} + 2630005 q^{20} - 5671057 q^{21} - 1956426 q^{22} - 5580815 q^{23} + 3549677 q^{24} - 4029457 q^{25} + 9649689 q^{26} + 6835241 q^{27} - 24789228 q^{28} - 4235889 q^{29} + 34444162 q^{30} + 10383597 q^{31} + 8347164 q^{32} + 35912671 q^{33} - 11257679 q^{34} + 873325 q^{35} - 125256554 q^{36} - 32126087 q^{37} + 131520700 q^{38} + 192355527 q^{39} - 158097415 q^{40} + 127695676 q^{42} - 78365203 q^{43} - 238378098 q^{44} + 57884904 q^{45} + 6546071 q^{47} - 400523802 q^{48} - 221520647 q^{49} + 154918442 q^{50} + 278243026 q^{51} - 52352040 q^{52} + 352279862 q^{53} + 155358204 q^{54} + 143671997 q^{55} - 575895495 q^{56} - 298751838 q^{57} - 1211966547 q^{58} + 776098748 q^{59} + 112101752 q^{60} - 124049779 q^{61} + 669065875 q^{62} + 1929121403 q^{63} + 1625889382 q^{64} - 866654028 q^{65} - 852661411 q^{66} + 13275097 q^{67} - 1504278412 q^{68} - 1267751723 q^{69} + 1776837051 q^{71} + 6219206035 q^{72} - 1507755424 q^{73} - 4950184776 q^{74} + 422070887 q^{76} - 561272089 q^{77} - 1228370591 q^{78} - 973234717 q^{79} + 1423181574 q^{80} + 2306016039 q^{81} - 1386574594 q^{82} - 932915983 q^{83} + 261177525 q^{84} - 3402309596 q^{85} - 3486024644 q^{86} - 1597393079 q^{87} + 5582534 q^{88} + 2316401563 q^{89} - 4748880458 q^{90} + 659194133 q^{91} + 11318573556 q^{92} + 3911968919 q^{93} + 3703837052 q^{94} + 2797945913 q^{95} - 1910264674 q^{96} - 2438073624 q^{97} - 1034416404 q^{98}+O(q^{100}) \) 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 −43.2850 − 9.87951i −99.6479 − 206.921i 1314.69 + 633.120i 467.851 − 2049.79i 2268.98 + 9941.04i 246.434 − 118.676i −32878.9 − 26220.0i −20614.4 + 25849.7i −40501.8 + 84102.9i
4.2 −39.6416 − 9.04794i 102.716 + 213.292i 1028.30 + 495.202i 153.222 − 671.311i −2141.98 − 9384.62i 7874.00 − 3791.92i −20006.3 − 15954.5i −22670.8 + 28428.3i −12148.0 + 25225.5i
4.3 −36.7996 − 8.39927i −9.86657 − 20.4882i 822.366 + 396.030i −323.078 + 1415.50i 191.000 + 836.827i −556.278 + 267.890i −11826.7 − 9431.50i 11949.7 − 14984.5i 23778.3 − 49376.1i
4.4 −31.7893 − 7.25570i 38.8499 + 80.6726i 496.618 + 239.158i 230.482 − 1009.81i −649.674 − 2846.41i −5320.98 + 2562.45i −999.429 − 797.018i 7273.39 − 9120.54i −14653.7 + 30428.7i
4.5 −23.4413 − 5.35033i −66.5806 − 138.256i 59.5740 + 28.6893i −62.4981 + 273.822i 821.022 + 3597.13i 9803.33 − 4721.04i 8381.83 + 6684.29i −2409.61 + 3021.55i 2930.08 − 6084.37i
4.6 −21.8728 − 4.99233i −11.8733 − 24.6553i −7.79837 − 3.75550i 593.649 − 2600.95i 136.617 + 598.556i −1813.09 + 873.139i 9132.64 + 7283.04i 11805.2 − 14803.3i −25969.6 + 53926.4i
4.7 −20.8398 − 4.75655i −98.0825 − 203.670i −49.6235 − 23.8974i −133.957 + 586.902i 1075.25 + 4710.98i −5718.27 + 2753.77i 9477.13 + 7557.76i −19589.3 + 24564.2i 5583.26 − 11593.8i
4.8 −20.7241 − 4.73015i 108.635 + 225.583i −54.1809 − 26.0921i −432.228 + 1893.72i −1184.33 − 5188.87i −10119.8 + 4873.44i 9508.60 + 7582.85i −26814.0 + 33623.6i 17915.1 − 37201.1i
4.9 −14.0968 − 3.21750i 63.5414 + 131.945i −272.929 − 131.436i 30.5824 − 133.990i −471.197 − 2064.45i 5267.91 − 2536.89i 9212.56 + 7346.77i −1099.86 + 1379.18i −862.229 + 1790.44i
4.10 −4.08372 − 0.932084i 3.89931 + 8.09700i −445.488 − 214.536i −560.536 + 2455.87i −8.37663 − 36.7004i 1868.32 − 899.734i 3296.03 + 2628.50i 12221.8 − 15325.6i 4578.15 − 9506.62i
4.11 −0.729154 − 0.166425i −38.7761 − 80.5193i −460.792 − 221.906i 26.4117 − 115.717i 14.8733 + 65.1643i −8751.97 + 4214.73i 598.443 + 477.242i 7292.37 − 9144.34i −38.5163 + 79.9800i
4.12 6.64802 + 1.51737i −71.0184 − 147.471i −419.402 − 201.973i 345.044 − 1511.73i −248.364 − 1088.15i 4671.19 − 2249.53i −5211.35 − 4155.92i −4432.00 + 5557.56i 4587.72 − 9526.49i
4.13 8.86626 + 2.02367i 45.0762 + 93.6016i −386.781 − 186.264i 69.9428 − 306.440i 210.239 + 921.116i 1930.08 − 929.476i −6692.78 − 5337.31i 5542.75 − 6950.39i 1240.26 − 2575.43i
4.14 8.94046 + 2.04060i 101.023 + 209.777i −385.528 − 185.661i 456.096 − 1998.29i 475.124 + 2081.65i −6625.33 + 3190.59i −6738.82 − 5374.03i −21528.6 + 26996.0i 8155.42 − 16934.9i
4.15 16.7749 + 3.82877i −106.901 − 221.983i −194.557 − 93.6940i −480.857 + 2106.77i −943.342 − 4133.05i 3209.49 − 1545.61i −9792.61 − 7809.34i −25576.5 + 32071.9i −16132.7 + 33499.8i
4.16 25.0876 + 5.72609i 18.6267 + 38.6787i 135.306 + 65.1598i −277.936 + 1217.72i 245.822 + 1077.02i −7762.93 + 3738.43i −7279.40 − 5805.13i 11123.1 − 13947.9i −13945.5 + 28958.2i
4.17 25.1563 + 5.74177i 103.456 + 214.830i 138.577 + 66.7350i −384.822 + 1686.01i 1369.08 + 5998.34i 7325.43 − 3527.74i −7226.09 − 5762.61i −23176.3 + 29062.2i −19361.4 + 40204.4i
4.18 28.1428 + 6.42341i −39.5780 − 82.1846i 289.462 + 139.397i −3.26559 + 14.3075i −585.931 − 2567.13i 2325.76 − 1120.03i −4304.36 − 3432.61i 7084.26 − 8883.38i −183.805 + 381.676i
4.19 32.2233 + 7.35476i 27.7652 + 57.6551i 522.953 + 251.841i 507.417 − 2223.14i 470.648 + 2062.05i 5665.53 − 2728.37i 1768.42 + 1410.27i 9718.95 − 12187.2i 32701.3 − 67905.0i
4.20 35.1721 + 8.02781i −104.016 − 215.992i 711.338 + 342.562i 373.217 − 1635.17i −1924.53 − 8431.91i −9972.96 + 4802.72i 7827.86 + 6242.51i −23560.9 + 29544.4i 26253.7 − 54516.3i
See next 80 embeddings (of 132 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.e even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.10.e.a ✓ 132
29.e even 14 1 inner 29.10.e.a ✓ 132
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.10.e.a ✓ 132 1.a even 1 1 trivial
29.10.e.a ✓ 132 29.e even 14 1 inner

Hecke kernels

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