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Properties

Label 41.6.d.a
Level $41$
Weight $6$
Character orbit 41.d
Analytic conductor $6.576$
Analytic rank $0$
Dimension $64$
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] = [41,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.10");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(6.57573661233\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 3 q^{2} - 30 q^{3} - 227 q^{4} + 31 q^{5} - 77 q^{6} - 3 q^{7} + 946 q^{8} + 3434 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 3 q^{2} - 30 q^{3} - 227 q^{4} + 31 q^{5} - 77 q^{6} - 3 q^{7} + 946 q^{8} + 3434 q^{9} + 506 q^{10} - 492 q^{11} - 3302 q^{12} - 493 q^{13} - 1878 q^{14} + 948 q^{15} - 5051 q^{16} + 868 q^{17} + 926 q^{18} - 4680 q^{19} + 6000 q^{20} + 12367 q^{21} - 2431 q^{22} + 1857 q^{23} + 2160 q^{24} - 10635 q^{25} + 3521 q^{26} - 10938 q^{27} - 4732 q^{28} + 16323 q^{29} + 3418 q^{30} + 8354 q^{31} + 2944 q^{32} - 5251 q^{33} + 29625 q^{34} + 510 q^{35} - 57504 q^{36} - 44540 q^{37} + 33076 q^{38} + 9598 q^{39} - 40202 q^{40} - 1809 q^{41} - 130952 q^{42} + 37609 q^{43} + 10330 q^{44} - 32373 q^{45} + 65566 q^{46} + 33048 q^{47} - 16849 q^{48} + 33799 q^{49} + 41728 q^{50} - 2715 q^{51} + 225607 q^{52} + 71018 q^{53} + 8970 q^{54} + 1612 q^{55} - 87619 q^{56} - 16428 q^{57} + 19834 q^{58} - 8905 q^{59} - 96328 q^{60} + 19181 q^{61} + 148242 q^{62} - 109075 q^{63} - 122338 q^{64} - 118070 q^{65} + 111852 q^{66} - 150158 q^{67} + 333760 q^{68} - 188042 q^{69} + 61213 q^{70} - 29906 q^{71} - 4751 q^{72} + 263724 q^{73} + 24873 q^{74} + 214101 q^{75} - 545492 q^{76} - 135136 q^{77} - 184002 q^{78} - 292162 q^{79} + 92437 q^{80} + 321784 q^{81} - 370993 q^{82} - 596312 q^{83} + 506300 q^{84} - 467610 q^{85} - 513005 q^{86} + 591474 q^{87} + 394828 q^{88} + 77663 q^{89} + 575402 q^{90} + 505634 q^{91} - 437635 q^{92} + 385580 q^{93} + 244168 q^{94} + 860813 q^{95} + 309548 q^{96} - 473244 q^{97} - 646502 q^{98} - 709307 q^{99}+O(q^{100}) \) 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 −8.84192 − 6.42403i 9.66300 27.0229 + 83.1679i −17.1963 − 52.9248i −85.4395 − 62.0755i 142.898 − 103.821i 187.265 − 576.342i −149.626 −187.942 + 578.427i
10.2 −7.93513 − 5.76521i −22.4911 19.8401 + 61.0616i 2.90522 + 8.94135i 178.470 + 129.666i −152.611 + 110.878i 97.6087 − 300.409i 262.851 28.4955 − 87.7000i
10.3 −6.50229 − 4.72419i 14.7915 10.0733 + 31.0023i 9.57832 + 29.4790i −96.1790 − 69.8781i −124.956 + 90.7860i 1.48471 − 4.56946i −24.2101 76.9836 − 236.931i
10.4 −5.83263 − 4.23765i −8.51932 6.17329 + 18.9994i 22.1607 + 68.2035i 49.6900 + 36.1019i 133.290 − 96.8411i −26.7852 + 82.4363i −170.421 159.768 − 491.715i
10.5 −4.47610 − 3.25208i −14.3467 −0.429087 − 1.32059i −28.4148 − 87.4518i 64.2173 + 46.6566i 24.7477 − 17.9802i −57.0850 + 175.690i −37.1716 −157.212 + 483.850i
10.6 −3.55207 − 2.58073i 24.9826 −3.93150 − 12.0999i −13.1520 − 40.4777i −88.7399 − 64.4733i 50.5459 − 36.7238i −60.6783 + 186.749i 381.129 −57.7452 + 177.722i
10.7 −1.46873 − 1.06710i −2.02729 −8.87007 − 27.2993i 6.44975 + 19.8503i 2.97754 + 2.16331i −12.6698 + 9.20517i −34.0554 + 104.812i −238.890 11.7092 − 36.0372i
10.8 −0.554525 − 0.402886i −29.4839 −9.74336 − 29.9870i 8.61103 + 26.5020i 16.3495 + 11.8786i −20.1644 + 14.6503i −13.4563 + 41.4142i 626.300 5.90226 − 18.1653i
10.9 1.23018 + 0.893779i 20.4019 −9.17404 − 28.2348i 31.8552 + 98.0401i 25.0980 + 18.2348i 58.9793 − 42.8510i 28.9863 − 89.2107i 173.238 −48.4386 + 149.079i
10.10 1.45818 + 1.05943i 3.25488 −8.88464 − 27.3441i −13.1460 − 40.4593i 4.74622 + 3.44833i −178.716 + 129.845i 33.8371 − 104.140i −232.406 23.6946 − 72.9245i
10.11 3.32647 + 2.41682i 12.9799 −4.66417 − 14.3549i −24.6325 − 75.8111i 43.1772 + 31.3701i 116.895 − 84.9293i 59.8370 − 184.159i −74.5224 101.283 − 311.716i
10.12 3.96897 + 2.88363i −14.7616 −2.45112 − 7.54377i −0.646298 − 1.98910i −58.5885 − 42.5670i 150.927 − 109.655i 60.5374 − 186.315i −25.0940 3.17068 − 9.75836i
10.13 5.76002 + 4.18490i −12.5241 5.77591 + 17.7764i 24.1085 + 74.1983i −72.1393 − 52.4123i −117.508 + 85.3742i 29.2811 − 90.1178i −86.1458 −171.647 + 528.276i
10.14 6.53458 + 4.74765i 21.7107 10.2720 + 31.6141i −0.323693 − 0.996225i 141.870 + 103.075i −52.7574 + 38.3305i −3.09756 + 9.53330i 228.353 2.61453 − 8.04670i
10.15 7.04327 + 5.11724i −23.1409 13.5330 + 41.6504i −32.0853 − 98.7483i −162.988 − 118.418i −107.903 + 78.3963i −31.7288 + 97.6511i 292.503 279.333 − 859.699i
10.16 8.53270 + 6.19937i −0.287791 24.4863 + 75.3610i 4.28646 + 13.1924i −2.45564 − 1.78412i 87.6941 − 63.7135i −153.962 + 473.847i −242.917 −45.2093 + 139.140i
16.1 −3.35143 + 10.3146i 1.45753 −69.2713 − 50.3286i 22.2922 + 16.1963i −4.88480 + 15.0339i −62.9085 − 193.612i 470.506 − 341.843i −240.876 −241.770 + 175.656i
16.2 −2.75259 + 8.47161i 28.9990 −38.3029 − 27.8287i −9.24038 − 6.71353i −79.8224 + 245.668i 45.7847 + 140.911i 110.582 − 80.3426i 597.941 82.3094 − 59.8013i
16.3 −2.67422 + 8.23040i −23.5156 −34.6994 − 25.2106i −19.8235 − 14.4026i 62.8859 − 193.543i 21.1481 + 65.0870i 76.2490 − 55.3982i 309.984 171.551 − 124.639i
16.4 −2.27904 + 7.01417i 2.57990 −18.1161 − 13.1621i 47.1293 + 34.2414i −5.87971 + 18.0959i 44.1095 + 135.755i −57.3232 + 41.6477i −236.344 −347.585 + 252.535i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.16
Significant digits:
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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 41.6.d.a ✓ 64
41.d even 5 1 inner 41.6.d.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.6.d.a ✓ 64 1.a even 1 1 trivial
41.6.d.a ✓ 64 41.d even 5 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(41, [\chi])\).