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Properties

Label 41.7.h.a
Level $41$
Weight $7$
Character orbit 41.h
Analytic conductor $9.432$
Analytic rank $0$
Dimension $320$
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] = [41,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.6");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(9.43221742841\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 32 q^{2} + 24 q^{3} - 20 q^{4} - 16 q^{5} + 1004 q^{6} - 16 q^{7} + 1264 q^{8} - 5240 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q - 32 q^{2} + 24 q^{3} - 20 q^{4} - 16 q^{5} + 1004 q^{6} - 16 q^{7} + 1264 q^{8} - 5240 q^{9} - 12 q^{10} - 16 q^{11} - 7624 q^{12} - 4984 q^{13} + 12868 q^{14} - 7408 q^{15} + 69620 q^{16} + 49708 q^{17} - 12 q^{18} + 17120 q^{19} - 42940 q^{20} + 17616 q^{21} + 82856 q^{22} - 20 q^{23} - 96728 q^{24} - 20 q^{25} + 66592 q^{26} + 30504 q^{27} - 16400 q^{28} - 64740 q^{29} + 129700 q^{30} - 108020 q^{31} - 665316 q^{32} - 395896 q^{33} - 280636 q^{34} + 360224 q^{35} + 425772 q^{36} + 267156 q^{37} + 581336 q^{38} + 282096 q^{39} - 353676 q^{41} - 781328 q^{42} - 226096 q^{43} - 1600272 q^{44} - 880020 q^{45} + 606808 q^{46} + 333648 q^{47} + 1106796 q^{48} - 14560 q^{49} + 1300084 q^{50} + 1740868 q^{51} + 1061428 q^{52} - 231180 q^{53} - 1272936 q^{54} - 1032400 q^{55} - 98836 q^{56} + 261620 q^{57} + 644996 q^{58} - 1615692 q^{59} + 1090232 q^{60} - 1106020 q^{61} - 904924 q^{62} + 1718384 q^{63} - 20 q^{64} + 553488 q^{65} + 5151580 q^{66} + 192080 q^{67} - 5762220 q^{68} - 1914900 q^{69} - 5593320 q^{70} - 1546384 q^{71} - 1968320 q^{72} + 1114064 q^{73} - 498184 q^{74} + 6431308 q^{75} + 13598032 q^{76} + 5090000 q^{77} + 3439988 q^{78} + 2269712 q^{79} + 7920768 q^{80} - 4188640 q^{82} + 1185408 q^{83} - 11110328 q^{84} - 9048088 q^{85} - 14213020 q^{86} - 4044400 q^{87} - 6763280 q^{88} - 4465336 q^{89} - 9345752 q^{90} + 852704 q^{91} - 1281180 q^{92} + 7723084 q^{93} + 23470316 q^{94} + 12732560 q^{95} - 2333432 q^{96} + 901340 q^{97} + 6434128 q^{98} - 5325736 q^{99}+O(q^{100}) \) 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 −6.97690 + 13.6929i 4.57444 + 11.0437i −101.201 − 139.291i 7.13688 − 45.0605i −183.136 − 14.4131i 328.462 − 25.8505i 1641.94 − 260.058i 414.444 − 414.444i 567.217 + 412.108i
6.2 −6.55770 + 12.8702i −18.8487 − 45.5049i −85.0205 − 117.021i 19.7151 − 124.476i 709.261 + 55.8201i −611.809 + 48.1504i 1150.55 − 182.229i −1199.94 + 1199.94i 1472.75 + 1070.01i
6.3 −5.34751 + 10.4951i −5.62470 − 13.5792i −43.9324 − 60.4678i −38.0133 + 240.006i 172.593 + 13.5834i −238.647 + 18.7819i 124.975 − 19.7941i 362.723 − 362.723i −2315.61 − 1682.39i
6.4 −5.14682 + 10.1012i 17.1747 + 41.4633i −37.9262 − 52.2010i −6.16150 + 38.9022i −507.224 − 39.9194i −76.9173 + 6.05352i 5.86684 − 0.929215i −908.755 + 908.755i −361.247 − 262.461i
6.5 −5.10772 + 10.0245i −3.04146 − 7.34274i −36.7828 − 50.6271i 8.33145 − 52.6027i 89.1420 + 7.01562i 245.611 − 19.3300i −15.7954 + 2.50174i 470.815 − 470.815i 484.759 + 352.198i
6.6 −3.86136 + 7.57834i 7.79327 + 18.8146i −4.90294 − 6.74832i 26.2242 − 165.573i −172.676 − 13.5899i −524.563 + 41.2840i −467.569 + 74.0557i 222.226 − 222.226i 1153.51 + 838.073i
6.7 −3.29125 + 6.45944i −15.0619 − 36.3627i 6.72617 + 9.25778i 2.10747 − 13.3061i 284.456 + 22.3872i 362.914 − 28.5619i −540.200 + 85.5593i −579.907 + 579.907i 79.0135 + 57.4067i
6.8 −1.59802 + 3.13630i 10.9468 + 26.4280i 30.3356 + 41.7533i −23.0745 + 145.687i −100.379 − 7.90002i 337.833 − 26.5880i −401.931 + 63.6596i −63.1246 + 63.1246i −420.044 − 305.180i
6.9 −1.55617 + 3.05416i −7.80101 − 18.8333i 30.7120 + 42.2715i 7.36195 − 46.4815i 69.6596 + 5.48233i −70.1529 + 5.52115i −393.573 + 62.3358i 221.643 − 221.643i 130.505 + 94.8178i
6.10 −0.611157 + 1.19946i 11.7231 + 28.3020i 36.5531 + 50.3110i 38.0136 − 240.008i −41.1119 − 3.23557i 624.450 − 49.1453i −167.781 + 26.5739i −148.091 + 148.091i 264.649 + 192.279i
6.11 −0.0841703 + 0.165193i 3.44828 + 8.32488i 37.5981 + 51.7493i −12.4199 + 78.4162i −1.66546 − 0.131074i −426.356 + 33.5550i −23.4329 + 3.71140i 458.068 − 458.068i −11.9085 − 8.65200i
6.12 1.16276 − 2.28205i −18.7882 − 45.3587i 33.7625 + 46.4701i −32.0109 + 202.109i −125.357 − 9.86581i −105.725 + 8.32078i 307.204 − 48.6563i −1188.94 + 1188.94i 424.002 + 308.055i
6.13 2.14969 − 4.21901i −13.0610 − 31.5320i 24.4394 + 33.6380i 30.9849 − 195.631i −161.111 − 12.6797i −193.543 + 15.2322i 493.772 − 78.2057i −308.197 + 308.197i −758.761 − 551.272i
6.14 2.60765 − 5.11779i 19.0994 + 46.1100i 18.2263 + 25.0863i −0.900568 + 5.68597i 285.786 + 22.4918i −215.283 + 16.9431i 538.994 − 85.3682i −1245.86 + 1245.86i 26.7512 + 19.4359i
6.15 3.18169 − 6.24443i −3.69392 − 8.91792i 8.74858 + 12.0414i −13.8263 + 87.2961i −67.4402 − 5.30766i 484.192 − 38.1068i 546.035 − 86.4834i 449.597 − 449.597i 501.123 + 364.087i
6.16 3.64754 − 7.15871i 5.74724 + 13.8751i −0.324289 − 0.446345i 11.9853 − 75.6724i 120.291 + 9.46709i −89.0098 + 7.00522i 503.493 − 79.7455i 355.994 − 355.994i −498.000 − 361.818i
6.17 5.73067 − 11.2471i −9.05828 − 21.8686i −56.0379 − 77.1296i −14.7536 + 93.1507i −297.868 − 23.4427i −501.768 + 39.4900i −390.698 + 61.8805i 119.296 − 119.296i 963.125 + 699.752i
6.18 6.04754 − 11.8690i 12.6522 + 30.5451i −66.6814 − 91.7791i −36.0021 + 227.308i 439.053 + 34.5542i 164.671 − 12.9599i −650.543 + 103.036i −257.442 + 257.442i 2480.19 + 1801.96i
6.19 6.18022 − 12.1294i −15.0795 − 36.4050i −71.3081 − 98.1472i 14.7083 − 92.8643i −534.764 − 42.0869i 486.433 − 38.2831i −770.651 + 122.059i −582.456 + 582.456i −1035.49 − 752.324i
6.20 6.45062 − 12.6601i 8.97640 + 21.6709i −81.0483 − 111.553i 22.4910 − 142.002i 332.259 + 26.1493i 17.2193 − 1.35519i −1036.92 + 164.233i 126.427 − 126.427i −1652.68 − 1200.74i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.20
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 41.7.h.a ✓ 320
41.h odd 40 1 inner 41.7.h.a ✓ 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.7.h.a ✓ 320 1.a even 1 1 trivial
41.7.h.a ✓ 320 41.h odd 40 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(41, [\chi])\).