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Properties

Label 41.6.f.a
Level $41$
Weight $6$
Character orbit 41.f
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(4,41)]
 
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([3]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.4");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 41.f (of order \(10\), 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 \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 11 q^{2} - 227 q^{4} - 37 q^{5} + 135 q^{6} - 5 q^{7} - 1208 q^{8} - 4342 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 11 q^{2} - 227 q^{4} - 37 q^{5} + 135 q^{6} - 5 q^{7} - 1208 q^{8} - 4342 q^{9} - 1688 q^{10} + 1140 q^{11} + 1210 q^{12} - 5 q^{13} + 2200 q^{15} + 53 q^{16} - 1810 q^{17} + 820 q^{18} - 2240 q^{19} + 720 q^{20} - 3785 q^{21} + 4715 q^{22} + 7191 q^{23} - 17510 q^{24} - 13567 q^{25} + 11935 q^{26} - 2570 q^{28} + 37395 q^{29} - 17790 q^{30} + 8354 q^{31} + 53228 q^{32} + 10733 q^{33} - 39935 q^{34} + 18590 q^{35} + 56822 q^{36} + 42392 q^{37} - 43324 q^{39} + 16462 q^{40} + 25547 q^{41} - 82584 q^{42} + 10871 q^{43} + 44293 q^{45} - 31176 q^{46} - 12920 q^{47} - 61525 q^{48} - 29425 q^{49} + 134100 q^{50} - 301 q^{51} - 56955 q^{52} - 123130 q^{53} - 128450 q^{54} - 219835 q^{56} - 72730 q^{57} - 16250 q^{58} + 101479 q^{59} - 349500 q^{60} - 125391 q^{61} + 27700 q^{62} + 46335 q^{63} + 405298 q^{64} - 209500 q^{65} - 223598 q^{66} - 16820 q^{67} + 318800 q^{69} + 701815 q^{70} - 14890 q^{71} + 554619 q^{72} + 31416 q^{73} - 140667 q^{74} + 88055 q^{75} + 332310 q^{76} - 15988 q^{77} - 241076 q^{78} + 12169 q^{80} - 177268 q^{81} + 159925 q^{82} - 119808 q^{83} - 371258 q^{84} + 380419 q^{86} - 552916 q^{87} - 370060 q^{88} + 160175 q^{89} + 138120 q^{90} + 997890 q^{91} + 146175 q^{92} + 363110 q^{93} - 316690 q^{94} - 418865 q^{95} + 22160 q^{97} + 630282 q^{98} - 339895 q^{99}+O(q^{100}) \) 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 −7.84462 + 5.69945i 16.9098i 19.1658 − 58.9861i −28.9170 + 88.9975i −96.3765 − 132.651i −147.219 + 202.629i 89.9563 + 276.857i −42.9414 −280.393 − 862.962i
4.2 −7.55080 + 5.48598i − 20.3153i 17.0301 − 52.4132i 8.88512 − 27.3456i 111.449 + 153.397i −62.7153 + 86.3202i 66.6541 + 205.140i −169.711 82.9274 + 255.224i
4.3 −7.38437 + 5.36506i 1.20623i 15.8566 − 48.8015i −5.38106 + 16.5612i −6.47152 − 8.90728i 119.539 − 164.532i 54.4735 + 167.652i 241.545 −49.1161 − 151.164i
4.4 −7.20862 + 5.23737i 23.2559i 14.6456 − 45.0745i 29.1195 − 89.6205i −121.800 − 167.643i 40.1646 − 55.2818i 42.3869 + 130.454i −297.836 259.464 + 798.549i
4.5 −4.35748 + 3.16590i − 19.6567i −0.923788 + 2.84313i −25.6787 + 79.0308i 62.2312 + 85.6539i 84.0596 − 115.698i −58.2368 − 179.234i −143.387 −138.309 − 425.672i
4.6 −3.67481 + 2.66991i − 3.85522i −3.51271 + 10.8110i 19.2678 − 59.3002i 10.2931 + 14.1672i −47.4706 + 65.3377i −60.8727 − 187.347i 228.137 87.5204 + 269.360i
4.7 −2.85883 + 2.07706i 10.4325i −6.02983 + 18.5579i −5.05043 + 15.5436i −21.6690 − 29.8248i −27.4430 + 37.7721i −56.2508 − 173.122i 134.162 −17.8467 − 54.9265i
4.8 −1.33702 + 0.971400i 27.9541i −9.04455 + 27.8362i −16.1080 + 49.5755i −27.1547 − 37.3752i 98.3321 − 135.343i −31.2897 − 96.2997i −538.434 −26.6209 − 81.9306i
4.9 −0.345180 + 0.250788i − 28.7734i −9.83229 + 30.2607i 16.3294 − 50.2566i 7.21603 + 9.93201i 26.2129 − 36.0790i −8.41421 − 25.8963i −584.909 6.96719 + 21.4428i
4.10 1.76846 − 1.28486i − 11.5405i −8.41197 + 25.8894i −17.5603 + 54.0450i −14.8279 − 20.4089i −61.6873 + 84.9053i 40.0037 + 123.119i 109.817 38.3856 + 118.139i
4.11 2.48584 − 1.80607i 24.3511i −6.97103 + 21.4546i 24.9903 − 76.9123i 43.9798 + 60.5330i −94.4960 + 130.063i 51.8038 + 159.436i −349.976 −76.7869 − 236.326i
4.12 2.81801 − 2.04741i 0.415051i −6.13922 + 18.8946i 17.2643 − 53.1340i 0.849777 + 1.16962i 122.097 − 168.052i 55.8288 + 171.823i 242.828 −60.1359 − 185.079i
4.13 5.57021 − 4.04699i 15.5446i 4.76052 − 14.6514i −17.8914 + 55.0641i 62.9089 + 86.5867i −16.4499 + 22.6413i 35.3072 + 108.664i 1.36529 123.185 + 379.125i
4.14 6.40444 − 4.65310i − 16.8231i 9.47699 − 29.1672i 20.7888 − 63.9814i −78.2796 − 107.743i −120.306 + 165.587i 3.25799 + 10.0271i −40.0170 −164.571 − 506.498i
4.15 7.30502 − 5.30741i − 19.1899i 15.3062 − 47.1077i −13.1147 + 40.3630i −101.849 − 140.183i 76.4503 − 105.225i −48.9188 − 150.557i −125.254 118.419 + 364.458i
4.16 8.42859 − 6.12373i 13.0161i 23.6525 − 72.7949i 12.3163 − 37.9058i 79.7071 + 109.707i 10.2398 − 14.0938i −143.397 − 441.331i 73.5811 −128.315 − 394.914i
23.1 −3.13379 − 9.64482i 11.6475i −57.3133 + 41.6406i −8.33708 + 6.05724i 112.338 − 36.5007i 103.891 + 33.7562i 318.684 + 231.537i 107.337 84.5476 + 61.4275i
23.2 −2.67523 − 8.23350i − 18.7873i −34.7451 + 25.2438i −23.8795 + 17.3495i −154.686 + 50.2604i −48.6094 − 15.7941i 76.6730 + 55.7062i −109.964 206.730 + 150.198i
23.3 −2.46188 − 7.57689i 6.40820i −25.4599 + 18.4977i 86.1272 − 62.5751i 48.5543 − 15.7762i −174.972 − 56.8517i −3.41489 − 2.48107i 201.935 −686.160 − 498.524i
23.4 −1.79067 − 5.51110i 30.5883i −1.27722 + 0.927958i −15.9554 + 11.5922i 168.575 − 54.7734i −104.403 − 33.9227i −142.616 − 103.616i −692.644 92.4568 + 67.1738i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 41.6.f.a ✓ 64
41.f even 10 1 inner 41.6.f.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.6.f.a ✓ 64 1.a even 1 1 trivial
41.6.f.a ✓ 64 41.f even 10 1 inner

Hecke kernels

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