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Properties

Label 41.5.h.a
Level $41$
Weight $5$
Character orbit 41.h
Analytic conductor $4.238$
Analytic rank $0$
Dimension $208$
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] = [41,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.6");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(4.23816848644\)
Analytic rank: \(0\)
Dimension: \(208\)
Relative dimension: \(13\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 208 q - 16 q^{2} - 20 q^{4} - 16 q^{5} - 148 q^{6} - 16 q^{7} - 80 q^{8} + 256 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 208 q - 16 q^{2} - 20 q^{4} - 16 q^{5} - 148 q^{6} - 16 q^{7} - 80 q^{8} + 256 q^{9} - 12 q^{10} - 16 q^{11} - 1000 q^{12} - 420 q^{13} + 548 q^{14} + 332 q^{15} + 2548 q^{16} - 1900 q^{17} - 12 q^{18} + 80 q^{19} - 220 q^{20} + 900 q^{21} - 3672 q^{22} - 20 q^{23} + 3304 q^{24} - 20 q^{25} + 176 q^{26} + 2460 q^{27} - 1040 q^{28} - 3400 q^{29} + 13828 q^{30} + 3400 q^{31} + 1724 q^{32} - 1096 q^{33} - 4540 q^{34} - 13624 q^{35} - 34068 q^{36} - 5072 q^{37} - 10504 q^{38} - 4944 q^{39} + 5912 q^{41} + 30832 q^{42} + 3776 q^{43} + 21232 q^{44} + 25980 q^{45} - 4008 q^{46} + 15764 q^{47} + 14796 q^{48} + 21680 q^{49} + 3188 q^{50} + 4504 q^{51} - 51948 q^{52} - 26980 q^{53} - 54120 q^{54} + 24952 q^{55} + 22444 q^{56} - 4372 q^{57} - 3948 q^{58} + 14964 q^{59} + 19832 q^{60} - 10676 q^{61} - 28892 q^{62} - 33544 q^{63} - 20 q^{64} + 53924 q^{65} + 111580 q^{66} + 51248 q^{67} + 21940 q^{68} + 12864 q^{69} - 33512 q^{70} - 53224 q^{71} - 102080 q^{72} - 41716 q^{73} - 34056 q^{74} - 106904 q^{75} - 121776 q^{76} - 52060 q^{77} - 65932 q^{78} - 37708 q^{79} + 6880 q^{80} + 53936 q^{82} - 22040 q^{83} + 41800 q^{84} + 76696 q^{85} + 146980 q^{86} + 72764 q^{87} + 173360 q^{88} + 3500 q^{89} + 272200 q^{90} + 98188 q^{91} + 105124 q^{92} + 460 q^{93} + 8076 q^{94} - 42772 q^{95} - 33080 q^{96} - 94600 q^{97} - 58320 q^{98} - 113872 q^{99}+O(q^{100}) \) 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 −3.39041 + 6.65405i 2.98776 + 7.21309i −23.3770 − 32.1756i −2.69165 + 16.9944i −58.1260 − 4.57461i −66.1036 + 5.20247i 175.338 − 27.7709i 14.1737 − 14.1737i −103.956 − 75.5284i
6.2 −3.07119 + 6.02756i −5.46382 − 13.1908i −17.4947 − 24.0793i −3.89907 + 24.6178i 96.2890 + 7.57811i 87.1203 − 6.85652i 91.9636 − 14.5656i −86.8692 + 86.8692i −136.410 − 99.1078i
6.3 −2.56363 + 5.03141i −1.07800 − 2.60252i −9.33829 − 12.8531i 6.43315 − 40.6173i 15.8580 + 1.24805i 4.23602 − 0.333382i −0.628814 + 0.0995944i 51.6646 − 51.6646i 187.870 + 136.496i
6.4 −1.86561 + 3.66146i 3.26950 + 7.89328i −0.521249 − 0.717438i −2.46934 + 15.5908i −35.0006 − 2.75460i 40.1407 − 3.15915i −61.3409 + 9.71544i 5.66141 − 5.66141i −52.4783 − 38.1277i
6.5 −1.37975 + 2.70791i −4.28905 − 10.3547i 3.97549 + 5.47179i −1.54737 + 9.76969i 33.9574 + 2.67251i −80.7153 + 6.35243i −68.3302 + 10.8224i −31.5480 + 31.5480i −24.3205 − 17.6699i
6.6 −0.367304 + 0.720875i 5.29662 + 12.7872i 9.01982 + 12.4147i 2.72476 − 17.2035i −11.1634 − 0.878580i −40.2435 + 3.16723i −25.0480 + 3.96721i −78.1819 + 78.1819i 11.4007 + 8.28311i
6.7 0.0811793 − 0.159323i −2.18802 − 5.28235i 9.38577 + 12.9184i −3.24989 + 20.5190i −1.01922 − 0.0802147i 54.3060 − 4.27397i 5.64591 − 0.894225i 34.1598 − 34.1598i 3.00533 + 2.18350i
6.8 0.415870 − 0.816190i −1.29608 − 3.12901i 8.91134 + 12.2654i 4.48403 − 28.3110i −3.09287 − 0.243414i 28.9350 − 2.27724i 28.1929 − 4.46532i 49.1648 − 49.1648i −21.2424 − 15.4335i
6.9 1.43500 − 2.81635i 2.00316 + 4.83606i 3.53197 + 4.86133i −7.32803 + 46.2674i 16.4946 + 1.29815i −54.9800 + 4.32702i 68.7108 − 10.8827i 37.9009 − 37.9009i 119.789 + 87.0321i
6.10 1.98858 − 3.90280i −6.01041 − 14.5104i −1.87287 − 2.57778i 0.431123 − 2.72200i −68.5834 − 5.39763i 0.0165204 − 0.00130019i 55.4357 − 8.78016i −117.151 + 117.151i −9.76612 − 7.09550i
6.11 2.34351 − 4.59940i 4.83957 + 11.6837i −6.25790 − 8.61326i 0.728284 − 4.59820i 65.0799 + 5.12190i 59.4467 − 4.67855i 27.2944 − 4.32300i −55.8129 + 55.8129i −19.4422 − 14.1256i
6.12 2.51964 − 4.94507i 0.105357 + 0.254355i −8.70054 − 11.9753i 4.77928 − 30.1752i 1.52327 + 0.119884i −65.6538 + 5.16707i 6.56571 − 1.03991i 57.2221 − 57.2221i −137.176 − 99.6644i
6.13 3.44190 − 6.75511i −1.52150 − 3.67323i −24.3803 − 33.5566i −2.64005 + 16.6686i −30.0499 − 2.36498i 31.5071 − 2.47967i −190.783 + 30.2171i 46.0980 − 46.0980i 103.512 + 75.2056i
7.1 −3.39041 − 6.65405i 2.98776 − 7.21309i −23.3770 + 32.1756i −2.69165 − 16.9944i −58.1260 + 4.57461i −66.1036 − 5.20247i 175.338 + 27.7709i 14.1737 + 14.1737i −103.956 + 75.5284i
7.2 −3.07119 − 6.02756i −5.46382 + 13.1908i −17.4947 + 24.0793i −3.89907 − 24.6178i 96.2890 − 7.57811i 87.1203 + 6.85652i 91.9636 + 14.5656i −86.8692 − 86.8692i −136.410 + 99.1078i
7.3 −2.56363 − 5.03141i −1.07800 + 2.60252i −9.33829 + 12.8531i 6.43315 + 40.6173i 15.8580 − 1.24805i 4.23602 + 0.333382i −0.628814 − 0.0995944i 51.6646 + 51.6646i 187.870 − 136.496i
7.4 −1.86561 − 3.66146i 3.26950 − 7.89328i −0.521249 + 0.717438i −2.46934 − 15.5908i −35.0006 + 2.75460i 40.1407 + 3.15915i −61.3409 − 9.71544i 5.66141 + 5.66141i −52.4783 + 38.1277i
7.5 −1.37975 − 2.70791i −4.28905 + 10.3547i 3.97549 − 5.47179i −1.54737 − 9.76969i 33.9574 − 2.67251i −80.7153 − 6.35243i −68.3302 − 10.8224i −31.5480 − 31.5480i −24.3205 + 17.6699i
7.6 −0.367304 − 0.720875i 5.29662 − 12.7872i 9.01982 − 12.4147i 2.72476 + 17.2035i −11.1634 + 0.878580i −40.2435 − 3.16723i −25.0480 − 3.96721i −78.1819 − 78.1819i 11.4007 − 8.28311i
7.7 0.0811793 + 0.159323i −2.18802 + 5.28235i 9.38577 − 12.9184i −3.24989 − 20.5190i −1.01922 + 0.0802147i 54.3060 + 4.27397i 5.64591 + 0.894225i 34.1598 + 34.1598i 3.00533 − 2.18350i
See next 80 embeddings (of 208 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.13
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 41.5.h.a ✓ 208
41.h odd 40 1 inner 41.5.h.a ✓ 208
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.5.h.a ✓ 208 1.a even 1 1 trivial
41.5.h.a ✓ 208 41.h odd 40 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(41, [\chi])\).