Properties

Label 41.5.e.a
Level $41$
Weight $5$
Character orbit 41.e
Analytic conductor $4.238$
Analytic rank $0$
Dimension $52$
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(3,41)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(41, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.3");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 41.e (of order \(8\), 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: \(4.23816848644\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(13\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 4 q^{2} - 20 q^{3} - 4 q^{5} + 128 q^{6} - 4 q^{7} + 60 q^{8} - 276 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q - 4 q^{2} - 20 q^{3} - 4 q^{5} + 128 q^{6} - 4 q^{7} + 60 q^{8} - 276 q^{9} - 8 q^{10} - 4 q^{11} + 980 q^{12} + 400 q^{13} - 568 q^{14} - 352 q^{15} - 2568 q^{16} + 1880 q^{17} - 8 q^{18} - 100 q^{19} + 200 q^{20} - 920 q^{21} + 3652 q^{22} - 3324 q^{24} - 196 q^{26} - 2480 q^{27} + 1020 q^{28} + 3380 q^{29} + 2152 q^{30} - 1744 q^{32} - 6064 q^{33} - 5080 q^{34} + 4604 q^{35} + 2688 q^{36} - 6648 q^{37} + 4724 q^{38} - 1236 q^{39} - 4312 q^{41} - 5272 q^{42} + 824 q^{43} + 13308 q^{44} + 17428 q^{46} + 3116 q^{47} + 21024 q^{48} - 15580 q^{49} - 3208 q^{50} - 17064 q^{51} + 18328 q^{52} + 10580 q^{53} + 4820 q^{54} - 24972 q^{55} - 22464 q^{56} + 4352 q^{57} + 3928 q^{58} - 14984 q^{59} - 19852 q^{60} + 10656 q^{61} + 28872 q^{62} + 33524 q^{63} - 33784 q^{65} + 812 q^{67} + 20940 q^{68} - 7124 q^{69} - 8308 q^{70} + 19604 q^{71} - 10784 q^{73} - 49124 q^{74} - 2556 q^{75} - 68744 q^{76} + 7880 q^{77} - 22288 q^{78} + 13048 q^{79} - 38380 q^{80} - 14836 q^{82} + 47920 q^{83} + 72340 q^{84} + 28004 q^{85} - 6544 q^{87} + 4420 q^{88} + 69440 q^{89} - 4920 q^{90} + 32992 q^{91} - 21984 q^{92} + 39840 q^{93} + 36364 q^{94} + 36032 q^{95} - 65940 q^{96} + 15220 q^{97} - 99800 q^{98} - 22228 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −5.29291 + 5.29291i −3.41101 + 8.23490i 40.0297i −15.5092 15.5092i −25.5324 61.6407i −7.15724 + 17.2791i 127.187 + 127.187i 1.09702 + 1.09702i 164.177
3.2 −4.22452 + 4.22452i 2.48954 6.01029i 19.6931i 20.2159 + 20.2159i 14.8735 + 35.9077i −16.4789 + 39.7837i 15.6016 + 15.6016i 27.3499 + 27.3499i −170.805
3.3 −3.77357 + 3.77357i 1.75756 4.24312i 12.4797i −6.75991 6.75991i 9.37944 + 22.6440i 28.3432 68.4265i −13.2842 13.2842i 42.3606 + 42.3606i 51.0180
3.4 −2.94669 + 2.94669i −5.64860 + 13.6369i 1.36599i 29.0808 + 29.0808i −23.5391 56.8285i −4.25338 + 10.2686i −43.1219 43.1219i −96.7833 96.7833i −171.384
3.5 −2.30381 + 2.30381i 5.64187 13.6207i 5.38492i −29.1694 29.1694i 18.3817 + 44.3772i −29.0115 + 70.0400i −49.2668 49.2668i −96.4165 96.4165i 134.401
3.6 −1.51281 + 1.51281i −3.63807 + 8.78307i 11.4228i −15.0929 15.0929i −7.78341 18.7908i 9.07206 21.9019i −41.4855 41.4855i −6.63110 6.63110i 45.6653
3.7 0.00440217 0.00440217i 6.27012 15.1374i 16.0000i 21.5970 + 21.5970i −0.0390353 0.0942396i 32.8790 79.3770i 0.140869 + 0.140869i −132.551 132.551i 0.190148
3.8 0.289313 0.289313i 0.442941 1.06935i 15.8326i 5.11819 + 5.11819i −0.181230 0.437527i −12.6159 + 30.4574i 9.20959 + 9.20959i 56.3283 + 56.3283i 2.96152
3.9 2.97521 2.97521i −6.14728 + 14.8408i 1.70370i −22.5131 22.5131i 25.8651 + 62.4440i −31.7980 + 76.7671i 42.5344 + 42.5344i −125.186 125.186i −133.962
3.10 2.98094 2.98094i 1.87477 4.52610i 1.77196i −31.6311 31.6311i −7.90344 19.0806i 23.9981 57.9365i 42.4129 + 42.4129i 40.3048 + 40.3048i −188.580
3.11 3.08327 3.08327i −3.25836 + 7.86638i 3.01309i 21.5585 + 21.5585i 14.2078 + 34.3006i 13.4709 32.5217i 40.0421 + 40.0421i 6.01260 + 6.01260i 132.941
3.12 4.13918 4.13918i 4.26929 10.3070i 18.2657i 7.17275 + 7.17275i −24.9911 60.3338i −17.9434 + 43.3193i −9.37801 9.37801i −30.7311 30.7311i 59.3787
3.13 5.58200 5.58200i −2.10724 + 5.08733i 46.3174i −2.03816 2.03816i 16.6349 + 40.1601i 11.2022 27.0444i −169.232 169.232i 35.8352 + 35.8352i −22.7540
14.1 −5.29291 5.29291i −3.41101 8.23490i 40.0297i −15.5092 + 15.5092i −25.5324 + 61.6407i −7.15724 17.2791i 127.187 127.187i 1.09702 1.09702i 164.177
14.2 −4.22452 4.22452i 2.48954 + 6.01029i 19.6931i 20.2159 20.2159i 14.8735 35.9077i −16.4789 39.7837i 15.6016 15.6016i 27.3499 27.3499i −170.805
14.3 −3.77357 3.77357i 1.75756 + 4.24312i 12.4797i −6.75991 + 6.75991i 9.37944 22.6440i 28.3432 + 68.4265i −13.2842 + 13.2842i 42.3606 42.3606i 51.0180
14.4 −2.94669 2.94669i −5.64860 13.6369i 1.36599i 29.0808 29.0808i −23.5391 + 56.8285i −4.25338 10.2686i −43.1219 + 43.1219i −96.7833 + 96.7833i −171.384
14.5 −2.30381 2.30381i 5.64187 + 13.6207i 5.38492i −29.1694 + 29.1694i 18.3817 44.3772i −29.0115 70.0400i −49.2668 + 49.2668i −96.4165 + 96.4165i 134.401
14.6 −1.51281 1.51281i −3.63807 8.78307i 11.4228i −15.0929 + 15.0929i −7.78341 + 18.7908i 9.07206 + 21.9019i −41.4855 + 41.4855i −6.63110 + 6.63110i 45.6653
14.7 0.00440217 + 0.00440217i 6.27012 + 15.1374i 16.0000i 21.5970 21.5970i −0.0390353 + 0.0942396i 32.8790 + 79.3770i 0.140869 0.140869i −132.551 + 132.551i 0.190148
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.e odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 41.5.e.a 52
41.e odd 8 1 inner 41.5.e.a 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.5.e.a 52 1.a even 1 1 trivial
41.5.e.a 52 41.e odd 8 1 inner

Hecke kernels

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