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Properties

Label 41.8.d.a
Level $41$
Weight $8$
Character orbit 41.d
Analytic conductor $12.808$
Analytic rank $0$
Dimension $96$
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] = [41,8,Mod(10,41)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(41, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.10");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 41.d (of order \(5\), degree \(4\), 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: \(12.8077860448\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(24\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 19 q^{2} + 18 q^{3} - 1539 q^{4} + 135 q^{5} + 1399 q^{6} - 3 q^{7} - 7718 q^{8} + 71822 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 19 q^{2} + 18 q^{3} - 1539 q^{4} + 135 q^{5} + 1399 q^{6} - 3 q^{7} - 7718 q^{8} + 71822 q^{9} - 294 q^{10} + 1164 q^{11} + 2002 q^{12} - 2621 q^{13} + 71978 q^{14} - 39732 q^{15} - 98619 q^{16} + 13586 q^{17} - 83530 q^{18} - 10368 q^{19} + 15624 q^{20} - 180851 q^{21} - 133239 q^{22} - 43359 q^{23} - 240024 q^{24} - 190151 q^{25} - 419499 q^{26} + 374622 q^{27} + 2900 q^{28} + 203211 q^{29} - 544262 q^{30} - 41222 q^{31} + 2374992 q^{32} + 390047 q^{33} - 303523 q^{34} - 877358 q^{35} + 62328 q^{36} - 645124 q^{37} - 1115476 q^{38} + 1186150 q^{39} + 4000422 q^{40} + 1434119 q^{41} + 5498008 q^{42} + 1234685 q^{43} - 1826102 q^{44} - 1585605 q^{45} - 4000130 q^{46} - 1711536 q^{47} - 4631329 q^{48} - 1756505 q^{49} + 2515560 q^{50} + 4862889 q^{51} - 12233881 q^{52} + 1441880 q^{53} - 2636946 q^{54} - 1769028 q^{55} - 7979107 q^{56} + 2760156 q^{57} - 2018462 q^{58} - 1213345 q^{59} + 21978464 q^{60} - 11564207 q^{61} - 5894814 q^{62} - 2744059 q^{63} - 6754818 q^{64} + 450746 q^{65} - 12109872 q^{66} + 8057858 q^{67} + 54614112 q^{68} - 7068878 q^{69} + 14201693 q^{70} + 8374478 q^{71} - 39644711 q^{72} - 3513136 q^{73} + 6038149 q^{74} - 14539635 q^{75} + 16162476 q^{76} + 16655746 q^{77} + 16889982 q^{78} + 395502 q^{79} + 37573541 q^{80} + 21289912 q^{81} + 43387911 q^{82} + 46805648 q^{83} - 39715132 q^{84} - 59920046 q^{85} + 31343907 q^{86} - 28338678 q^{87} - 90860020 q^{88} - 28594977 q^{89} - 82254382 q^{90} - 45571406 q^{91} + 19808325 q^{92} - 68614588 q^{93} - 20272304 q^{94} + 3140685 q^{95} + 103298828 q^{96} + 6445152 q^{97} - 11316398 q^{98} + 13842913 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −17.5697 − 12.7651i −14.3670 106.191 + 326.823i 165.761 + 510.159i 252.423 + 183.396i −289.978 + 210.681i 1447.18 − 4453.96i −1980.59 3599.88 − 11079.3i
10.2 −16.7030 − 12.1354i 87.0534 92.1668 + 283.660i −45.8098 − 140.988i −1454.05 − 1056.43i −998.032 + 725.113i 1086.24 − 3343.10i 5391.29 −945.790 + 2910.84i
10.3 −16.0489 − 11.6602i −72.6911 82.0519 + 252.530i −73.1912 − 225.259i 1166.61 + 847.591i 609.321 − 442.698i 843.047 − 2594.63i 3097.00 −1451.93 + 4468.58i
10.4 −14.3070 − 10.3946i 6.17813 57.0869 + 175.695i −86.5394 − 266.341i −88.3902 − 64.2193i 46.3911 − 33.7051i 310.055 − 954.252i −2148.83 −1530.39 + 4710.07i
10.5 −12.9001 − 9.37244i 24.3124 39.0146 + 120.075i 51.9882 + 160.003i −313.631 − 227.866i 338.641 − 246.037i −8.60326 + 26.4781i −1595.91 828.969 − 2551.30i
10.6 −10.6155 − 7.71261i 70.2921 13.6503 + 42.0112i 45.2559 + 139.283i −746.185 − 542.135i 1334.35 − 969.461i −339.898 + 1046.10i 2753.97 593.825 − 1827.60i
10.7 −10.5329 − 7.65261i −51.6297 12.8257 + 39.4733i −26.1720 − 80.5490i 543.811 + 395.102i −1179.24 + 856.766i −347.989 + 1071.00i 478.628 −340.743 + 1048.70i
10.8 −8.68783 − 6.31208i −78.3681 −3.91814 − 12.0588i 109.951 + 338.394i 680.849 + 494.666i 667.936 − 485.284i −466.838 + 1436.78i 3954.56 1180.73 − 3633.92i
10.9 −6.33427 − 4.60212i −19.9876 −20.6107 − 63.4331i 50.0323 + 153.984i 126.607 + 91.9854i −305.258 + 221.783i −471.066 + 1449.79i −1787.50 391.732 − 1205.63i
10.10 −5.12065 − 3.72037i 62.0240 −27.1743 − 83.6338i 134.210 + 413.055i −317.603 − 230.752i −1138.62 + 827.255i −422.356 + 1299.88i 1659.98 849.476 − 2614.42i
10.11 −4.64175 − 3.37243i 48.0274 −29.3816 − 90.4273i −129.925 − 399.869i −222.931 − 161.969i −374.274 + 271.926i −395.520 + 1217.29i 119.632 −745.448 + 2294.25i
10.12 −3.12399 − 2.26971i −39.6848 −34.9465 − 107.554i −110.875 − 341.237i 123.975 + 90.0730i 1399.27 − 1016.63i −287.681 + 885.392i −612.117 −428.138 + 1317.67i
10.13 1.14387 + 0.831072i 23.1200 −38.9364 − 119.834i 41.9517 + 129.114i 26.4463 + 19.2144i 560.710 − 407.379i 110.978 − 341.556i −1652.47 −59.3158 + 182.555i
10.14 1.94432 + 1.41263i −70.4343 −37.7693 − 116.242i −88.4039 − 272.079i −136.947 − 99.4975i −681.343 + 495.024i 185.832 − 571.933i 2773.99 212.462 − 653.890i
10.15 3.04340 + 2.21116i −19.4315 −35.1811 − 108.276i 86.0117 + 264.717i −59.1377 − 42.9660i 326.046 − 236.886i 281.143 − 865.268i −1809.42 −323.563 + 995.823i
10.16 5.88707 + 4.27720i 78.6491 −23.1911 − 71.3749i −17.8768 − 55.0192i 463.013 + 336.398i 293.859 − 213.501i 456.586 − 1405.23i 3998.69 130.086 − 400.365i
10.17 7.46034 + 5.42026i −59.6296 −13.2766 − 40.8613i 149.541 + 460.239i −444.857 − 323.208i −432.561 + 314.274i 487.179 − 1499.38i 1368.69 −1378.99 + 4244.09i
10.18 7.69168 + 5.58834i 6.71135 −11.6217 − 35.7678i −42.5705 − 131.018i 51.6216 + 37.5053i −1081.19 + 785.532i 486.551 − 1497.45i −2141.96 404.736 − 1245.65i
10.19 11.6008 + 8.42846i 0.557033 23.9850 + 73.8184i −118.045 − 363.305i 6.46202 + 4.69493i 375.838 − 273.063i 223.251 − 687.097i −2186.69 1692.69 − 5209.56i
10.20 11.9094 + 8.65267i −75.9227 27.4104 + 84.3604i −12.2006 − 37.5495i −904.192 − 656.934i 722.598 − 524.998i 178.767 − 550.187i 3577.26 179.602 − 552.758i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 41.8.d.a ✓ 96
41.d even 5 1 inner 41.8.d.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.8.d.a ✓ 96 1.a even 1 1 trivial
41.8.d.a ✓ 96 41.d even 5 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(41, [\chi])\).