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Properties

Label 41.9.h.a
Level $41$
Weight $9$
Character orbit 41.h
Analytic conductor $16.703$
Analytic rank $0$
Dimension $432$
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] = [41,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.6");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 9 \)
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: \(16.7025230125\)
Analytic rank: \(0\)
Dimension: \(432\)
Relative dimension: \(27\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 432 q - 16 q^{2} - 240 q^{3} - 20 q^{4} - 16 q^{5} - 6484 q^{6} - 16 q^{7} - 1040 q^{8} + 21520 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 432 q - 16 q^{2} - 240 q^{3} - 20 q^{4} - 16 q^{5} - 6484 q^{6} - 16 q^{7} - 1040 q^{8} + 21520 q^{9} - 12 q^{10} - 16 q^{11} + 1880 q^{12} + 162580 q^{13} - 164188 q^{14} + 136052 q^{15} + 1572852 q^{16} - 230140 q^{17} - 12 q^{18} - 68944 q^{19} - 804700 q^{20} - 953508 q^{21} - 1470872 q^{22} - 20 q^{23} + 1757032 q^{24} - 20 q^{25} + 1841648 q^{26} + 2069940 q^{27} - 262160 q^{28} + 1290656 q^{29} - 3823292 q^{30} - 5338640 q^{31} + 8242364 q^{32} + 12690584 q^{33} + 14911748 q^{34} + 2001656 q^{35} - 8790612 q^{36} - 11099272 q^{37} - 20417224 q^{38} - 14227152 q^{39} - 1410544 q^{41} + 24960112 q^{42} - 5989024 q^{43} + 50505712 q^{44} + 4549980 q^{45} - 9893992 q^{46} - 45295636 q^{47} - 71425524 q^{48} - 15518704 q^{49} + 16411508 q^{50} + 67483792 q^{51} + 181968212 q^{52} - 21642580 q^{53} - 111252072 q^{54} + 25747752 q^{55} - 89098196 q^{56} + 8422268 q^{57} - 66110828 q^{58} - 94139148 q^{59} + 7301432 q^{60} + 36326316 q^{61} - 36257372 q^{62} - 160336024 q^{63} - 20 q^{64} + 139173884 q^{65} - 7030820 q^{66} - 218860432 q^{67} - 240503180 q^{68} - 50351208 q^{69} + 244854488 q^{70} + 229713416 q^{71} + 653081920 q^{72} + 183845204 q^{73} + 127168440 q^{74} - 46831664 q^{75} - 1013654320 q^{76} - 192980740 q^{77} + 51505268 q^{78} - 197540980 q^{79} - 8871200 q^{80} + 568452656 q^{82} + 61932760 q^{83} + 1196704456 q^{84} + 1138476696 q^{85} + 1002098980 q^{86} - 326061436 q^{87} - 378949200 q^{88} - 225621532 q^{89} - 898274360 q^{90} - 1332862380 q^{91} - 1065389276 q^{92} - 549986420 q^{93} - 1233221492 q^{94} - 390994732 q^{95} + 2313527176 q^{96} + 1245803840 q^{97} + 608261040 q^{98} + 439342880 q^{99}+O(q^{100}) \) 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 −13.5394 + 26.5726i 37.8701 + 91.4264i −372.314 − 512.446i 166.740 − 1052.76i −2942.17 − 231.554i −2361.13 + 185.825i 11117.2 − 1760.79i −2285.32 + 2285.32i 25716.9 + 18684.4i
6.2 −13.3929 + 26.2850i −30.9483 − 74.7158i −361.059 − 496.954i −45.3827 + 286.535i 2378.39 + 187.183i 1289.14 − 101.457i 10438.9 − 1653.37i 14.6753 − 14.6753i −6923.77 − 5030.41i
6.3 −12.6353 + 24.7982i 47.4093 + 114.456i −304.826 − 419.558i −166.814 + 1053.22i −3437.34 − 270.525i 1444.41 − 113.678i 7218.66 − 1143.32i −6213.27 + 6213.27i −24010.2 − 17444.5i
6.4 −11.0649 + 21.7161i −2.33873 − 5.64620i −198.685 − 273.467i −28.9021 + 182.481i 148.491 + 11.6865i −3324.60 + 261.652i 1974.52 − 312.732i 4612.92 − 4612.92i −3642.97 − 2646.78i
6.5 −10.2958 + 20.2067i −37.0111 − 89.3528i −151.833 − 208.980i 179.231 − 1131.62i 2186.58 + 172.088i 1736.92 − 136.698i 51.8204 − 8.20754i −1974.77 + 1974.77i 21021.0 + 15272.6i
6.6 −9.86322 + 19.3577i 24.1272 + 58.2483i −126.963 − 174.750i 55.4893 − 350.346i −1365.52 − 107.469i 3820.63 − 300.690i −858.273 + 135.937i 1828.59 − 1828.59i 6234.58 + 4529.68i
6.7 −8.08503 + 15.8678i 19.4835 + 47.0373i −35.9450 − 49.4741i −23.6901 + 149.573i −903.901 − 71.1386i −145.214 + 11.4286i −3427.26 + 542.825i 2806.43 − 2806.43i −2181.85 − 1585.21i
6.8 −7.33470 + 14.3952i −51.3979 − 124.085i −2.94982 − 4.06007i −63.0861 + 398.310i 2163.22 + 170.249i −866.465 + 68.1923i −4004.95 + 634.322i −8116.12 + 8116.12i −5271.02 − 3829.62i
6.9 −5.04836 + 9.90797i 55.9277 + 135.021i 77.7911 + 107.070i 81.8231 − 516.611i −1620.13 − 127.507i −236.318 + 18.5986i −4265.23 + 675.546i −10463.5 + 10463.5i 4705.49 + 3418.74i
6.10 −4.86100 + 9.54025i −14.4938 − 34.9912i 83.0859 + 114.358i −170.053 + 1073.67i 404.279 + 31.8175i 3834.69 − 301.797i −4202.20 + 665.563i 3625.02 − 3625.02i −9416.48 − 6841.47i
6.11 −4.04080 + 7.93051i −24.7802 − 59.8247i 103.908 + 143.017i 141.732 − 894.863i 574.572 + 45.2198i −2932.60 + 230.801i −3804.58 + 602.586i 1674.39 − 1674.39i 6524.00 + 4739.97i
6.12 −3.68742 + 7.23697i 37.9696 + 91.6667i 111.696 + 153.737i −132.649 + 837.515i −803.399 − 63.2289i −3765.52 + 296.353i −3578.15 + 566.724i −2321.77 + 2321.77i −5571.94 − 4048.25i
6.13 −2.27349 + 4.46198i −0.119232 − 0.287852i 135.733 + 186.820i 69.6351 − 439.659i 1.55546 + 0.122418i 1241.91 − 97.7405i −2408.39 + 381.451i 4639.26 − 4639.26i 1803.43 + 1310.27i
6.14 0.596723 − 1.17113i −16.0212 − 38.6786i 149.458 + 205.711i −109.654 + 692.330i −54.8581 − 4.31742i −1970.71 + 155.098i 662.442 − 104.921i 3399.97 − 3399.97i 745.378 + 541.549i
6.15 2.29523 − 4.50464i −46.7938 − 112.970i 135.449 + 186.430i 58.2846 − 367.994i −616.293 − 48.5033i 4352.68 − 342.563i 2429.01 − 384.717i −5933.28 + 5933.28i −1523.91 − 1107.18i
6.16 3.06278 − 6.01104i −49.1180 − 118.581i 123.721 + 170.287i 9.45339 − 59.6863i −863.235 − 67.9381i −2284.77 + 179.815i 3108.34 − 492.312i −7009.65 + 7009.65i −329.823 − 239.631i
6.17 3.08386 − 6.05241i 29.9456 + 72.2950i 123.352 + 169.779i 44.1378 − 278.675i 529.907 + 41.7046i 868.765 − 68.3733i 3125.51 − 495.032i 309.497 − 309.497i −1550.54 − 1126.54i
6.18 3.80077 − 7.45943i 44.5773 + 107.619i 109.276 + 150.405i −108.558 + 685.408i 972.206 + 76.5142i 2347.49 − 184.752i 3654.09 − 578.752i −4955.41 + 4955.41i 4700.15 + 3414.86i
6.19 4.98313 − 9.77994i 23.3745 + 56.4310i 79.6574 + 109.639i 159.611 − 1007.75i 668.370 + 52.6019i −3727.39 + 293.352i 4244.54 − 672.269i 2001.23 − 2001.23i −9060.33 − 6582.71i
6.20 7.29761 − 14.3224i −16.5462 − 39.9460i −1.40193 − 1.92959i −97.2849 + 614.233i −692.869 − 54.5300i −647.115 + 50.9291i 4026.51 − 637.736i 3317.42 − 3317.42i 8087.32 + 5875.78i
See next 80 embeddings (of 432 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.27
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 41.9.h.a ✓ 432
41.h odd 40 1 inner 41.9.h.a ✓ 432
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.9.h.a ✓ 432 1.a even 1 1 trivial
41.9.h.a ✓ 432 41.h odd 40 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(41, [\chi])\).