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Properties

Label 71.3.f.a
Level $71$
Weight $3$
Character orbit 71.f
Analytic conductor $1.935$
Analytic rank $0$
Dimension $66$
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] = [71,3,Mod(23,71)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(71, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("71.23");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 71 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 71.f (of order \(14\), degree \(6\), 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: \(1.93460987696\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(11\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 3 q^{2} - 10 q^{3} - 31 q^{4} - 6 q^{5} - 3 q^{6} - 7 q^{7} - 11 q^{8} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q - 3 q^{2} - 10 q^{3} - 31 q^{4} - 6 q^{5} - 3 q^{6} - 7 q^{7} - 11 q^{8} - 11 q^{9} + 28 q^{10} - 7 q^{11} + 8 q^{12} - 7 q^{13} - 128 q^{15} + 117 q^{16} - 229 q^{18} - 19 q^{19} + 74 q^{20} + 98 q^{21} - 91 q^{22} - 7 q^{23} + 134 q^{24} + 132 q^{25} + 28 q^{26} + 173 q^{27} + 21 q^{28} + 49 q^{29} + 55 q^{30} + 154 q^{31} - 295 q^{32} - 133 q^{33} + 21 q^{34} - 378 q^{35} - 139 q^{36} + 17 q^{37} - 82 q^{38} - 35 q^{39} + 110 q^{40} + 56 q^{42} + 95 q^{43} + 329 q^{44} - 74 q^{45} - 490 q^{47} + 49 q^{48} - 138 q^{49} - 318 q^{50} - 392 q^{51} + 567 q^{52} - 7 q^{53} - 74 q^{54} - 553 q^{55} + 357 q^{56} + 1128 q^{57} - 14 q^{58} + 252 q^{59} + 596 q^{60} - 182 q^{61} + 98 q^{62} + 525 q^{63} - 191 q^{64} + 98 q^{65} + 357 q^{67} - 49 q^{68} + 140 q^{69} + 373 q^{71} + 632 q^{72} + 275 q^{73} + 13 q^{74} - 534 q^{75} - 1746 q^{76} + 568 q^{77} - 903 q^{78} - 325 q^{79} - 110 q^{80} - 1069 q^{81} + 392 q^{82} - 405 q^{83} - 1134 q^{84} - 149 q^{86} + 658 q^{87} - 176 q^{89} - 705 q^{90} + 157 q^{91} - 161 q^{92} - 378 q^{93} + 1477 q^{94} - 666 q^{95} - 282 q^{96} + 168 q^{97} + 585 q^{98} + 889 q^{99}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −2.07321 − 2.59972i −0.565462 + 2.47745i −1.57027 + 6.87980i 2.58721 7.61300 − 3.66623i 5.17779 + 4.12915i 9.15755 − 4.41004i 2.29070 + 1.10314i −5.36382 − 6.72602i
23.2 −1.73422 − 2.17464i 0.0310597 − 0.136081i −0.831473 + 3.64292i −4.59838 −0.349793 + 0.168452i −7.92451 − 6.31959i −0.660074 + 0.317875i 8.09117 + 3.89650i 7.97461 + 9.99984i
23.3 −1.43374 − 1.79786i 0.942120 − 4.12770i −0.286587 + 1.25562i 9.11661 −8.77177 + 4.22426i −1.03036 − 0.821685i −5.61896 + 2.70595i −8.04157 − 3.87262i −13.0709 − 16.3904i
23.4 −0.970698 − 1.21722i 0.916004 − 4.01328i 0.350721 − 1.53661i −7.45794 −5.77419 + 2.78070i 8.92623 + 7.11843i −7.82163 + 3.76670i −7.15860 − 3.44740i 7.23941 + 9.07793i
23.5 −0.782507 − 0.981232i −1.29850 + 5.68910i 0.539583 − 2.36407i −7.08500 6.59841 − 3.17763i −0.969711 − 0.773319i −7.26495 + 3.49862i −22.5710 − 10.8696i 5.54406 + 6.95204i
23.6 −0.227195 − 0.284893i 0.182110 − 0.797877i 0.860537 − 3.77026i 0.978655 −0.268684 + 0.129392i −6.28969 − 5.01586i −2.58285 + 1.24384i 7.50528 + 3.61435i −0.222345 − 0.278812i
23.7 0.0359468 + 0.0450758i −0.500010 + 2.19069i 0.889344 − 3.89647i 3.84308 −0.116721 + 0.0562097i 5.29029 + 4.21887i 0.415384 − 0.200038i 3.55962 + 1.71422i 0.138146 + 0.173230i
23.8 1.06381 + 1.33398i 0.836918 − 3.66678i 0.242281 − 1.06150i −1.38116 5.78173 − 2.78434i 0.126351 + 0.100762i 7.82278 − 3.76725i −4.63612 − 2.23264i −1.46930 − 1.84244i
23.9 1.57767 + 1.97833i −0.852792 + 3.73633i −0.534684 + 2.34260i 3.85116 −8.73713 + 4.20758i −7.78266 − 6.20647i 3.64119 − 1.75350i −5.12416 − 2.46767i 6.07585 + 7.61888i
23.10 1.71893 + 2.15547i −0.471597 + 2.06620i −0.801242 + 3.51047i −7.06965 −5.26427 + 2.53514i 6.72247 + 5.36099i 0.991709 − 0.477582i 4.06193 + 1.95612i −12.1522 − 15.2384i
23.11 2.32521 + 2.91573i 0.582086 − 2.55029i −2.20476 + 9.65967i 1.47448 8.78942 − 4.23276i −3.46871 − 2.76620i −19.8513 + 9.55990i 1.94358 + 0.935978i 3.42847 + 4.29917i
26.1 −3.30342 + 1.59084i −0.247110 + 0.309866i 5.88786 − 7.38314i −4.34217 0.323360 − 1.41673i 5.09184 − 10.5733i −4.44114 + 19.4579i 1.96773 + 8.62121i 14.3440 − 6.90772i
26.2 −2.70627 + 1.30327i −0.0764317 + 0.0958423i 3.13143 − 3.92669i 5.85528 0.0819363 − 0.358986i −3.40360 + 7.06764i −0.683380 + 2.99408i 1.99934 + 8.75970i −15.8460 + 7.63102i
26.3 −2.15213 + 1.03641i −3.08981 + 3.87450i 1.06356 − 1.33366i −0.0562760 2.63410 − 11.5407i −1.54196 + 3.20191i 1.21943 − 5.34269i −3.46212 − 15.1685i 0.121113 − 0.0583251i
26.4 −1.49930 + 0.722023i 2.58296 − 3.23893i −0.767386 + 0.962271i 2.51761 −1.53404 + 6.72108i 2.66416 − 5.53219i 1.93694 − 8.48629i −1.81630 − 7.95775i −3.77465 + 1.81777i
26.5 −1.06218 + 0.511518i 0.733756 − 0.920101i −1.62739 + 2.04068i −8.50231 −0.308731 + 1.35264i −1.67072 + 3.46929i 1.73408 − 7.59749i 1.69450 + 7.42409i 9.03097 − 4.34908i
26.6 0.188461 − 0.0907581i −0.565562 + 0.709193i −2.46668 + 3.09312i 7.26947 −0.0422215 + 0.184985i −1.82316 + 3.78583i −0.370332 + 1.62253i 1.81959 + 7.97217i 1.37001 − 0.659763i
26.7 0.890369 − 0.428779i −2.26035 + 2.83438i −1.88505 + 2.36378i −4.08904 −0.797217 + 3.49284i 0.0396541 − 0.0823427i −1.54446 + 6.76674i −0.921881 − 4.03903i −3.64076 + 1.75330i
26.8 1.53694 − 0.740150i 3.41140 − 4.27775i −0.679605 + 0.852198i −0.471725 2.07692 − 9.09958i −4.29854 + 8.92602i −1.93212 + 8.46519i −4.65888 − 20.4119i −0.725011 + 0.349147i
26.9 1.66922 − 0.803853i 1.13552 − 1.42390i −0.353852 + 0.443716i 1.73750 0.750825 − 3.28958i 5.22052 − 10.8405i −1.88302 + 8.25007i 1.26461 + 5.54063i 2.90026 − 1.39669i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.f odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 71.3.f.a ✓ 66
71.f odd 14 1 inner 71.3.f.a ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.3.f.a ✓ 66 1.a even 1 1 trivial
71.3.f.a ✓ 66 71.f odd 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(71, [\chi])\).