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Properties

Label 41.11.h.a
Level $41$
Weight $11$
Character orbit 41.h
Analytic conductor $26.050$
Analytic rank $0$
Dimension $544$
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] = [41,11,Mod(6,41)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(41, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.6");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 41.h (of order \(40\), degree \(16\), 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: \(26.0496473596\)
Analytic rank: \(0\)
Dimension: \(544\)
Relative dimension: \(34\) over \(\Q(\zeta_{40})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{40}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 544 q + 48 q^{2} + 72 q^{3} - 20 q^{4} - 16 q^{5} + 34412 q^{6} - 16 q^{7} - 102416 q^{8} + 242968 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 544 q + 48 q^{2} + 72 q^{3} - 20 q^{4} - 16 q^{5} + 34412 q^{6} - 16 q^{7} - 102416 q^{8} + 242968 q^{9} - 12 q^{10} - 16 q^{11} - 258184 q^{12} - 1717356 q^{13} - 1175676 q^{14} - 2282788 q^{15} + 32505844 q^{16} - 5033040 q^{17} - 12 q^{18} - 6048256 q^{19} + 1636100 q^{20} - 3986748 q^{21} + 24419304 q^{22} - 20 q^{23} + 36267688 q^{24} - 20 q^{25} + 9974064 q^{26} - 12095484 q^{27} - 4194320 q^{28} - 20806556 q^{29} - 370268060 q^{30} + 2354200 q^{31} + 225754204 q^{32} + 112152944 q^{33} - 239826172 q^{34} - 287329576 q^{35} + 216837420 q^{36} + 345130576 q^{37} + 333684632 q^{38} + 666397104 q^{39} - 17675108 q^{41} - 2288879120 q^{42} + 160994720 q^{43} + 63885296 q^{44} + 1486249980 q^{45} - 351255720 q^{46} + 1261437508 q^{47} - 780578388 q^{48} - 299825056 q^{49} - 2437962156 q^{50} - 88484936 q^{51} - 916699788 q^{52} + 2497052608 q^{53} + 182213592 q^{54} - 1637216360 q^{55} - 1542812052 q^{56} + 4986495020 q^{57} + 4931743380 q^{58} + 3176564532 q^{59} - 2876106568 q^{60} + 1788511480 q^{61} - 318795228 q^{62} + 12784962680 q^{63} - 20 q^{64} + 3665220388 q^{65} - 23094234020 q^{66} - 10199930992 q^{67} + 1354212500 q^{68} + 8819561856 q^{69} + 27828283480 q^{70} - 3223393672 q^{71} - 30940495040 q^{72} - 22624244596 q^{73} - 19082264840 q^{74} - 38815221512 q^{75} + 39458020176 q^{76} + 450833732 q^{77} + 53913683828 q^{78} + 24627394244 q^{79} + 29183336768 q^{80} - 53652343696 q^{82} - 24512340376 q^{83} - 107893342904 q^{84} - 69328940048 q^{85} - 42189205020 q^{86} + 14959139660 q^{87} + 86708151920 q^{88} + 39718954628 q^{89} + 160730175208 q^{90} + 91832715148 q^{91} + 83998530788 q^{92} - 21156146036 q^{93} - 80645750676 q^{94} - 28258469380 q^{95} - 179042416376 q^{96} + 41491216292 q^{97} + 45143638544 q^{98} + 164011815320 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −27.1728 + 53.3296i 19.1837 + 46.3135i −1503.79 − 2069.80i −608.644 + 3842.82i −2991.16 − 235.409i 2482.56 − 195.382i 90708.6 − 14366.8i 39977.0 − 39977.0i −188398. − 136879.i
6.2 −26.8729 + 52.7411i −55.4086 − 133.768i −1457.58 − 2006.18i 527.094 − 3327.94i 8544.07 + 672.433i −8201.73 + 645.490i 85110.5 − 13480.2i 26930.1 − 26930.1i 161355. + 117231.i
6.3 −24.7390 + 48.5529i −166.663 − 402.360i −1143.48 − 1573.86i −713.257 + 4503.33i 23658.8 + 1861.99i −9886.38 + 778.075i 49591.1 − 7854.46i −92363.3 + 92363.3i −201005. − 146038.i
6.4 −24.0935 + 47.2862i 71.0556 + 171.543i −1053.59 − 1450.15i 422.274 − 2666.13i −9823.61 − 773.135i 27377.7 − 2154.67i 40281.6 − 6379.98i 17375.7 − 17375.7i 115897. + 84204.3i
6.5 −23.9513 + 47.0071i 148.406 + 358.284i −1034.11 − 1423.33i 5.42351 − 34.2427i −20396.4 − 1605.23i −11446.5 + 900.862i 38316.6 − 6068.75i −64588.9 + 64588.9i 1479.75 + 1075.10i
6.6 −20.7850 + 40.7928i −143.206 − 345.731i −630.146 − 867.322i 391.311 − 2470.64i 17079.9 + 1344.22i 16445.6 − 1294.30i 2173.65 − 344.272i −57267.8 + 57267.8i 92651.1 + 67315.0i
6.7 −18.5818 + 36.4688i −52.5651 − 126.903i −382.799 − 526.878i 213.046 − 1345.12i 5604.77 + 441.105i −29600.3 + 2329.59i −15068.5 + 2386.62i 28412.6 − 28412.6i 45096.2 + 32764.3i
6.8 −18.2034 + 35.7261i −34.4492 − 83.1677i −343.101 − 472.237i −515.612 + 3255.45i 3598.35 + 283.196i 24276.1 − 1910.57i −17436.4 + 2761.65i 36023.8 − 36023.8i −106919. − 77680.9i
6.9 −16.9106 + 33.1890i 64.2239 + 155.050i −213.648 − 294.061i −622.009 + 3927.21i −6232.03 − 490.471i −9509.79 + 748.436i −24300.7 + 3848.86i 21838.1 − 21838.1i −119822. − 87055.4i
6.10 −14.3889 + 28.2399i 90.5169 + 218.527i 11.4437 + 15.7509i 711.666 − 4493.28i −7473.61 − 588.186i −2806.89 + 220.907i −32664.9 + 5173.61i 2193.18 − 2193.18i 116650. + 84750.9i
6.11 −12.1378 + 23.8217i −94.4492 − 228.021i 181.743 + 250.147i −272.088 + 1717.89i 6578.25 + 517.720i −1399.58 + 110.150i −35205.3 + 5575.96i −1318.79 + 1318.79i −37620.7 − 27333.0i
6.12 −10.2158 + 20.0497i 175.868 + 424.583i 304.265 + 418.785i −340.651 + 2150.79i −10309.4 − 811.367i 31484.9 − 2477.91i −34263.5 + 5426.80i −107587. + 107587.i −39642.6 − 28802.0i
6.13 −7.81605 + 15.3399i −22.2917 − 53.8169i 427.671 + 588.639i 564.316 − 3562.95i 999.777 + 78.6841i 8299.25 − 653.165i −29784.8 + 4717.45i 39354.6 − 39354.6i 50244.5 + 36504.7i
6.14 −6.33902 + 12.4410i 102.551 + 247.579i 487.296 + 670.706i −474.299 + 2994.61i −3730.21 − 293.574i −13165.3 + 1036.13i −25555.2 + 4047.55i −9024.94 + 9024.94i −34249.4 − 24883.6i
6.15 −4.20317 + 8.24919i −172.960 − 417.563i 551.510 + 759.088i 676.138 − 4268.97i 4171.54 + 328.307i −18358.1 + 1444.82i −17943.7 + 2842.00i −102690. + 102690.i 32373.6 + 23520.8i
6.16 −2.41607 + 4.74181i −112.649 − 271.960i 585.245 + 805.520i −818.253 + 5166.24i 1561.75 + 122.912i −17262.9 + 1358.62i −10616.1 + 1681.43i −19518.2 + 19518.2i −22520.4 − 16362.0i
6.17 −1.43127 + 2.80903i −133.122 − 321.386i 596.050 + 820.392i −54.3698 + 343.277i 1093.31 + 86.0457i 20009.7 − 1574.80i −6346.17 + 1005.13i −43813.3 + 43813.3i −886.457 − 644.048i
6.18 1.42273 − 2.79227i 40.2889 + 97.2660i 596.120 + 820.488i −151.761 + 958.180i 328.913 + 25.8860i 13987.4 − 1100.83i 6308.68 − 999.196i 33916.5 − 33916.5i 2459.58 + 1786.99i
6.19 1.64989 − 3.23809i 142.689 + 344.483i 594.129 + 817.748i 478.797 − 3023.00i 1350.89 + 106.317i −19395.2 + 1526.44i 7303.79 − 1156.81i −56554.1 + 56554.1i −8998.80 − 6538.01i
6.20 5.80585 − 11.3946i −21.3712 − 51.5946i 505.763 + 696.123i 80.0068 − 505.143i −711.978 − 56.0339i −28356.3 + 2231.69i 23802.6 − 3769.96i 39548.7 − 39548.7i −5291.40 − 3844.43i
See next 80 embeddings (of 544 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.h odd 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 41.11.h.a ✓ 544
41.h odd 40 1 inner 41.11.h.a ✓ 544
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.11.h.a ✓ 544 1.a even 1 1 trivial
41.11.h.a ✓ 544 41.h odd 40 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(41, [\chi])\).