Properties

Label 71.5.e.a
Level $71$
Weight $5$
Character orbit 71.e
Analytic conductor $7.339$
Analytic rank $0$
Dimension $92$
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] = [71,5,Mod(14,71)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(71, base_ring=CyclotomicField(10))
 
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("71.14");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 71 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 71.e (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: \(7.33926737895\)
Analytic rank: \(0\)
Dimension: \(92\)
Relative dimension: \(23\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 92 q + 6 q^{2} - 13 q^{3} - 170 q^{4} - 9 q^{5} - 70 q^{6} - 5 q^{7} + 329 q^{8} - 742 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 92 q + 6 q^{2} - 13 q^{3} - 170 q^{4} - 9 q^{5} - 70 q^{6} - 5 q^{7} + 329 q^{8} - 742 q^{9} + 75 q^{10} - 395 q^{11} + 1306 q^{12} - 5 q^{13} - 1025 q^{14} + 303 q^{15} - 1430 q^{16} - 560 q^{17} - 420 q^{18} + 83 q^{19} + 1796 q^{20} - 5 q^{21} - 2380 q^{22} + 849 q^{24} - 968 q^{25} + 4880 q^{27} - 1230 q^{28} + 9 q^{29} + 4312 q^{30} + 2770 q^{31} - 7780 q^{32} - 1680 q^{33} + 355 q^{35} + 6392 q^{36} - 4722 q^{37} + 1703 q^{38} - 2741 q^{40} + 5485 q^{42} + 683 q^{43} + 11095 q^{44} - 18856 q^{45} + 20465 q^{46} - 12005 q^{47} - 23780 q^{48} + 5632 q^{49} + 12166 q^{50} + 19325 q^{52} - 1430 q^{53} + 23768 q^{54} - 10460 q^{55} - 14645 q^{56} + 1140 q^{57} - 17583 q^{58} + 3955 q^{59} + 5490 q^{60} + 13630 q^{61} - 21935 q^{62} - 5585 q^{63} + 12481 q^{64} + 3250 q^{65} + 66740 q^{66} - 46680 q^{67} - 4575 q^{68} - 5435 q^{69} - 12829 q^{71} - 29794 q^{72} + 4327 q^{73} + 4533 q^{74} - 8579 q^{75} - 22813 q^{76} + 39664 q^{77} + 13120 q^{78} - 34081 q^{79} + 39762 q^{80} - 41491 q^{81} - 60875 q^{82} - 17571 q^{83} + 122015 q^{84} - 33245 q^{85} - 32869 q^{86} - 34233 q^{87} - 113910 q^{88} - 36432 q^{89} - 9308 q^{90} + 68114 q^{91} + 68490 q^{92} + 24470 q^{93} + 57539 q^{95} + 36869 q^{96} + 117096 q^{98} - 24650 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −2.33798 + 7.19556i −5.29880 + 16.3080i −33.3656 24.2415i 22.9444 16.6701i −104.957 76.2556i 28.6513 + 9.30937i 154.505 112.254i −172.344 125.215i 66.3071 + 204.072i
14.2 −2.21238 + 6.80899i −0.750926 + 2.31111i −28.5235 20.7235i −38.4378 + 27.9267i −14.0750 10.2261i 18.6593 + 6.06278i 111.538 81.0369i 60.7530 + 44.1397i −105.114 323.507i
14.3 −2.21097 + 6.80468i 1.63361 5.02773i −28.4709 20.6853i 16.3761 11.8979i 30.6002 + 22.2324i −21.7377 7.06300i 111.091 80.7124i 42.9210 + 31.1839i 44.7543 + 137.740i
14.4 −1.78313 + 5.48790i 4.10332 12.6287i −13.9932 10.1667i −10.6624 + 7.74667i 61.9883 + 45.0372i −11.6810 3.79537i 6.05266 4.39751i −77.1168 56.0286i −23.5006 72.3274i
14.5 −1.48247 + 4.56257i −2.26569 + 6.97309i −5.67503 4.12315i 11.4019 8.28397i −28.4564 20.6748i −88.7423 28.8341i −34.8732 + 25.3369i 22.0398 + 16.0128i 20.8932 + 64.3027i
14.6 −1.42593 + 4.38858i −0.426217 + 1.31176i −4.28203 3.11108i 15.9930 11.6196i −5.14900 3.74097i 71.2142 + 23.1389i −39.9712 + 29.0408i 63.9913 + 46.4924i 28.1885 + 86.7552i
14.7 −1.38037 + 4.24834i −3.77624 + 11.6221i −3.19872 2.32401i −17.7367 + 12.8865i −44.1620 32.0856i −1.89994 0.617330i −43.5331 + 31.6286i −55.2825 40.1651i −30.2630 93.1399i
14.8 −0.903467 + 2.78059i 3.64933 11.2315i 6.02886 + 4.38023i −21.7356 + 15.7918i 27.9331 + 20.2946i 36.5121 + 11.8635i −55.4714 + 40.3023i −47.2983 34.3642i −24.2731 74.7051i
14.9 −0.750757 + 2.31059i 4.38800 13.5049i 8.16908 + 5.93518i 34.8891 25.3484i 27.9099 + 20.2777i −32.0797 10.4233i −51.2949 + 37.2679i −97.5967 70.9081i 32.3766 + 99.6450i
14.10 −0.395056 + 1.21586i 0.615850 1.89539i 11.6220 + 8.44390i −20.6238 + 14.9841i 2.06123 + 1.49757i −51.4846 16.7284i −31.4063 + 22.8180i 62.3171 + 45.2761i −10.0710 30.9952i
14.11 −0.0189892 + 0.0584427i −3.28390 + 10.1068i 12.9412 + 9.40234i 33.4497 24.3027i −0.528310 0.383839i −1.40375 0.456106i −1.59067 + 1.15569i −25.8329 18.7687i 0.785130 + 2.41638i
14.12 0.00803474 0.0247284i −4.10998 + 12.6492i 12.9437 + 9.40417i −17.3417 + 12.5995i 0.279772 + 0.203266i 44.2278 + 14.3705i 0.673113 0.489045i −77.5801 56.3652i 0.172229 + 0.530066i
14.13 0.0929586 0.286097i 1.01964 3.13814i 12.8711 + 9.35137i 5.77327 4.19453i −0.803027 0.583433i −0.0874858 0.0284258i 7.76577 5.64216i 56.7221 + 41.2110i −0.663367 2.04163i
14.14 0.681743 2.09819i 3.85088 11.8518i 9.00665 + 6.54371i 7.43329 5.40060i −22.2420 16.1597i 84.8852 + 27.5809i 48.4274 35.1845i −60.1053 43.6691i −6.26389 19.2783i
14.15 1.13409 3.49038i 4.67733 14.3953i 2.04766 + 1.48771i −34.6068 + 25.1433i −44.9407 32.6513i −52.0590 16.9150i 55.0205 39.9747i −119.818 87.0528i 48.5124 + 149.306i
14.16 1.22495 3.77000i −1.45133 + 4.46673i 0.231852 + 0.168450i 1.12547 0.817703i 15.0618 + 10.9430i 1.80874 + 0.587695i 52.2304 37.9476i 47.6851 + 34.6452i −1.70410 5.24468i
14.17 1.24405 3.82878i 3.15116 9.69828i −0.167663 0.121814i 18.0016 13.0789i −33.2124 24.1302i −71.2215 23.1413i 51.4364 37.3707i −18.5964 13.5111i −27.6815 85.1950i
14.18 1.28524 3.95557i −4.83631 + 14.8846i −1.05042 0.763175i −4.79770 + 3.48573i 52.6613 + 38.2607i −85.4311 27.7582i 49.4681 35.9407i −132.632 96.3625i 7.62186 + 23.4577i
14.19 1.30403 4.01340i −0.422249 + 1.29955i −1.46264 1.06267i −37.2304 + 27.0495i 4.66499 + 3.38931i 56.2511 + 18.2771i 48.4518 35.2023i 64.0198 + 46.5131i 60.0108 + 184.694i
14.20 1.78194 5.48424i −0.705335 + 2.17080i −13.9573 10.1406i 36.0884 26.2198i 10.6483 + 7.73646i 16.7832 + 5.45319i −5.84178 + 4.24430i 61.3155 + 44.5483i −79.4882 244.639i
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.23
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.e odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 71.5.e.a 92
71.e odd 10 1 inner 71.5.e.a 92
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.5.e.a 92 1.a even 1 1 trivial
71.5.e.a 92 71.e odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(71, [\chi])\).