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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(39,380)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("380.39"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(108\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 39.91
Character \(\chi\) \(=\) 380.39
Dual form 380.3.h.a.39.92

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.70952 - 1.03804i) q^{2} -1.20444 q^{3} +(1.84495 - 3.54911i) q^{4} +(3.17819 - 3.85994i) q^{5} +(-2.05903 + 1.25026i) q^{6} -6.44137 q^{7} +(-0.530130 - 7.98242i) q^{8} -7.54932 q^{9} +(1.42641 - 9.89774i) q^{10} +4.38090i q^{11} +(-2.22214 + 4.27470i) q^{12} -18.5880i q^{13} +(-11.0117 + 6.68640i) q^{14} +(-3.82795 + 4.64908i) q^{15} +(-9.19233 - 13.0958i) q^{16} +10.0468i q^{17} +(-12.9057 + 7.83649i) q^{18} +4.35890i q^{19} +(-7.83576 - 18.4011i) q^{20} +7.75827 q^{21} +(4.54754 + 7.48925i) q^{22} -16.8137 q^{23} +(0.638512 + 9.61437i) q^{24} +(-4.79828 - 24.5352i) q^{25} +(-19.2951 - 31.7766i) q^{26} +19.9327 q^{27} +(-11.8840 + 22.8611i) q^{28} +14.9419 q^{29} +(-1.71804 + 11.9213i) q^{30} -10.3268i q^{31} +(-29.3085 - 12.8457i) q^{32} -5.27654i q^{33} +(10.4289 + 17.1752i) q^{34} +(-20.4719 + 24.8633i) q^{35} +(-13.9281 + 26.7933i) q^{36} -32.8822i q^{37} +(4.52471 + 7.45164i) q^{38} +22.3882i q^{39} +(-32.4965 - 23.3233i) q^{40} +33.0801 q^{41} +(13.2630 - 8.05339i) q^{42} +77.3574 q^{43} +(15.5483 + 8.08253i) q^{44} +(-23.9931 + 29.1399i) q^{45} +(-28.7434 + 17.4532i) q^{46} -77.0608 q^{47} +(11.0716 + 15.7732i) q^{48} -7.50874 q^{49} +(-33.6713 - 36.9627i) q^{50} -12.1008i q^{51} +(-65.9707 - 34.2939i) q^{52} -77.3006i q^{53} +(34.0755 - 20.6909i) q^{54} +(16.9100 + 13.9233i) q^{55} +(3.41477 + 51.4177i) q^{56} -5.25005i q^{57} +(25.5436 - 15.5103i) q^{58} +41.4208i q^{59} +(9.43773 + 22.1631i) q^{60} -5.37481 q^{61} +(-10.7196 - 17.6539i) q^{62} +48.6279 q^{63} +(-63.4379 + 8.46344i) q^{64} +(-71.7485 - 59.0760i) q^{65} +(-5.47726 - 9.02038i) q^{66} +33.9875 q^{67} +(35.6571 + 18.5358i) q^{68} +20.2511 q^{69} +(-9.18807 + 63.7550i) q^{70} -65.9277i q^{71} +(4.00212 + 60.2618i) q^{72} -12.9194i q^{73} +(-34.1330 - 56.2129i) q^{74} +(5.77926 + 29.5513i) q^{75} +(15.4702 + 8.04194i) q^{76} -28.2190i q^{77} +(23.2398 + 38.2731i) q^{78} +49.5948i q^{79} +(-79.7641 - 6.13914i) q^{80} +43.9360 q^{81} +(56.5512 - 34.3384i) q^{82} +131.312 q^{83} +(14.3136 - 27.5349i) q^{84} +(38.7799 + 31.9305i) q^{85} +(132.244 - 80.3000i) q^{86} -17.9967 q^{87} +(34.9701 - 2.32245i) q^{88} +146.244 q^{89} +(-10.7685 + 74.7212i) q^{90} +119.732i q^{91} +(-31.0203 + 59.6735i) q^{92} +12.4381i q^{93} +(-131.737 + 79.9922i) q^{94} +(16.8251 + 13.8534i) q^{95} +(35.3005 + 15.4719i) q^{96} +58.5847i q^{97} +(-12.8364 + 7.79437i) q^{98} -33.0728i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q + 4 q^{5} + 324 q^{9} - 8 q^{10} + 8 q^{14} - 104 q^{16} - 16 q^{21} - 8 q^{24} - 76 q^{25} + 80 q^{26} - 88 q^{29} - 140 q^{30} - 88 q^{34} - 256 q^{36} + 44 q^{40} - 200 q^{41} - 8 q^{44} + 108 q^{45}+ \cdots + 720 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.70952 − 1.03804i 0.854762 − 0.519020i
\(3\) −1.20444 −0.401481 −0.200741 − 0.979644i \(-0.564335\pi\)
−0.200741 + 0.979644i \(0.564335\pi\)
\(4\) 1.84495 − 3.54911i 0.461237 − 0.887277i
\(5\) 3.17819 − 3.85994i 0.635637 − 0.771988i
\(6\) −2.05903 + 1.25026i −0.343171 + 0.208377i
\(7\) −6.44137 −0.920196 −0.460098 − 0.887868i \(-0.652186\pi\)
−0.460098 + 0.887868i \(0.652186\pi\)
\(8\) −0.530130 − 7.98242i −0.0662663 − 0.997802i
\(9\) −7.54932 −0.838813
\(10\) 1.42641 − 9.89774i 0.142641 − 0.989774i
\(11\) 4.38090i 0.398263i 0.979973 + 0.199132i \(0.0638122\pi\)
−0.979973 + 0.199132i \(0.936188\pi\)
\(12\) −2.22214 + 4.27470i −0.185178 + 0.356225i
\(13\) − 18.5880i − 1.42984i −0.699204 − 0.714922i \(-0.746462\pi\)
0.699204 − 0.714922i \(-0.253538\pi\)
\(14\) −11.0117 + 6.68640i −0.786549 + 0.477600i
\(15\) −3.82795 + 4.64908i −0.255196 + 0.309939i
\(16\) −9.19233 − 13.0958i −0.574521 − 0.818490i
\(17\) 10.0468i 0.590987i 0.955345 + 0.295493i \(0.0954840\pi\)
−0.955345 + 0.295493i \(0.904516\pi\)
\(18\) −12.9057 + 7.83649i −0.716986 + 0.435360i
\(19\) 4.35890i 0.229416i
\(20\) −7.83576 − 18.4011i −0.391788 − 0.920056i
\(21\) 7.75827 0.369441
\(22\) 4.54754 + 7.48925i 0.206707 + 0.340421i
\(23\) −16.8137 −0.731029 −0.365514 − 0.930806i \(-0.619107\pi\)
−0.365514 + 0.930806i \(0.619107\pi\)
\(24\) 0.638512 + 9.61437i 0.0266047 + 0.400599i
\(25\) −4.79828 − 24.5352i −0.191931 − 0.981408i
\(26\) −19.2951 − 31.7766i −0.742118 − 1.22218i
\(27\) 19.9327 0.738249
\(28\) −11.8840 + 22.8611i −0.424428 + 0.816469i
\(29\) 14.9419 0.515239 0.257619 − 0.966246i \(-0.417062\pi\)
0.257619 + 0.966246i \(0.417062\pi\)
\(30\) −1.71804 + 11.9213i −0.0572679 + 0.397376i
\(31\) − 10.3268i − 0.333123i −0.986031 − 0.166562i \(-0.946734\pi\)
0.986031 − 0.166562i \(-0.0532664\pi\)
\(32\) −29.3085 − 12.8457i −0.915891 − 0.401427i
\(33\) − 5.27654i − 0.159895i
\(34\) 10.4289 + 17.1752i 0.306734 + 0.505153i
\(35\) −20.4719 + 24.8633i −0.584911 + 0.710380i
\(36\) −13.9281 + 26.7933i −0.386892 + 0.744259i
\(37\) − 32.8822i − 0.888708i −0.895851 − 0.444354i \(-0.853433\pi\)
0.895851 − 0.444354i \(-0.146567\pi\)
\(38\) 4.52471 + 7.45164i 0.119071 + 0.196096i
\(39\) 22.3882i 0.574056i
\(40\) −32.4965 − 23.3233i −0.812413 − 0.583083i
\(41\) 33.0801 0.806832 0.403416 − 0.915017i \(-0.367823\pi\)
0.403416 + 0.915017i \(0.367823\pi\)
\(42\) 13.2630 − 8.05339i 0.315785 − 0.191747i
\(43\) 77.3574 1.79901 0.899505 − 0.436911i \(-0.143928\pi\)
0.899505 + 0.436911i \(0.143928\pi\)
\(44\) 15.5483 + 8.08253i 0.353370 + 0.183694i
\(45\) −23.9931 + 29.1399i −0.533180 + 0.647553i
\(46\) −28.7434 + 17.4532i −0.624856 + 0.379418i
\(47\) −77.0608 −1.63959 −0.819796 − 0.572655i \(-0.805913\pi\)
−0.819796 + 0.572655i \(0.805913\pi\)
\(48\) 11.0716 + 15.7732i 0.230659 + 0.328608i
\(49\) −7.50874 −0.153240
\(50\) −33.6713 − 36.9627i −0.673426 − 0.739255i
\(51\) − 12.1008i − 0.237270i
\(52\) −65.9707 − 34.2939i −1.26867 − 0.659497i
\(53\) − 77.3006i − 1.45850i −0.684246 − 0.729251i \(-0.739869\pi\)
0.684246 − 0.729251i \(-0.260131\pi\)
\(54\) 34.0755 − 20.6909i 0.631027 − 0.383166i
\(55\) 16.9100 + 13.9233i 0.307455 + 0.253151i
\(56\) 3.41477 + 51.4177i 0.0609780 + 0.918173i
\(57\) − 5.25005i − 0.0921061i
\(58\) 25.5436 − 15.5103i 0.440407 − 0.267419i
\(59\) 41.4208i 0.702047i 0.936367 + 0.351023i \(0.114166\pi\)
−0.936367 + 0.351023i \(0.885834\pi\)
\(60\) 9.43773 + 22.1631i 0.157295 + 0.369385i
\(61\) −5.37481 −0.0881116 −0.0440558 − 0.999029i \(-0.514028\pi\)
−0.0440558 + 0.999029i \(0.514028\pi\)
\(62\) −10.7196 − 17.6539i −0.172897 − 0.284741i
\(63\) 48.6279 0.771872
\(64\) −63.4379 + 8.46344i −0.991218 + 0.132241i
\(65\) −71.7485 − 59.0760i −1.10382 − 0.908862i
\(66\) −5.47726 − 9.02038i −0.0829888 − 0.136672i
\(67\) 33.9875 0.507276 0.253638 − 0.967299i \(-0.418373\pi\)
0.253638 + 0.967299i \(0.418373\pi\)
\(68\) 35.6571 + 18.5358i 0.524369 + 0.272585i
\(69\) 20.2511 0.293494
\(70\) −9.18807 + 63.7550i −0.131258 + 0.910786i
\(71\) − 65.9277i − 0.928560i −0.885689 − 0.464280i \(-0.846313\pi\)
0.885689 − 0.464280i \(-0.153687\pi\)
\(72\) 4.00212 + 60.2618i 0.0555850 + 0.836969i
\(73\) − 12.9194i − 0.176979i −0.996077 − 0.0884893i \(-0.971796\pi\)
0.996077 − 0.0884893i \(-0.0282039\pi\)
\(74\) −34.1330 − 56.2129i −0.461257 − 0.759634i
\(75\) 5.77926 + 29.5513i 0.0770567 + 0.394017i
\(76\) 15.4702 + 8.04194i 0.203555 + 0.105815i
\(77\) − 28.2190i − 0.366480i
\(78\) 23.2398 + 38.2731i 0.297946 + 0.490681i
\(79\) 49.5948i 0.627782i 0.949459 + 0.313891i \(0.101633\pi\)
−0.949459 + 0.313891i \(0.898367\pi\)
\(80\) −79.7641 − 6.13914i −0.997051 − 0.0767393i
\(81\) 43.9360 0.542420
\(82\) 56.5512 − 34.3384i 0.689649 − 0.418762i
\(83\) 131.312 1.58207 0.791037 − 0.611768i \(-0.209541\pi\)
0.791037 + 0.611768i \(0.209541\pi\)
\(84\) 14.3136 − 27.5349i 0.170400 − 0.327797i
\(85\) 38.7799 + 31.9305i 0.456235 + 0.375653i
\(86\) 132.244 − 80.3000i 1.53773 − 0.933721i
\(87\) −17.9967 −0.206859
\(88\) 34.9701 − 2.32245i 0.397388 − 0.0263914i
\(89\) 146.244 1.64319 0.821596 − 0.570070i \(-0.193084\pi\)
0.821596 + 0.570070i \(0.193084\pi\)
\(90\) −10.7685 + 74.7212i −0.119650 + 0.830235i
\(91\) 119.732i 1.31574i
\(92\) −31.0203 + 59.6735i −0.337177 + 0.648625i
\(93\) 12.4381i 0.133743i
\(94\) −131.737 + 79.9922i −1.40146 + 0.850981i
\(95\) 16.8251 + 13.8534i 0.177106 + 0.145825i
\(96\) 35.3005 + 15.4719i 0.367713 + 0.161165i
\(97\) 58.5847i 0.603966i 0.953313 + 0.301983i \(0.0976485\pi\)
−0.953313 + 0.301983i \(0.902351\pi\)
\(98\) −12.8364 + 7.79437i −0.130983 + 0.0795344i
\(99\) − 33.0728i − 0.334068i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 380.3.h.a.39.91 yes 108
4.3 odd 2 inner 380.3.h.a.39.17 ✓ 108
5.4 even 2 inner 380.3.h.a.39.18 yes 108
20.19 odd 2 inner 380.3.h.a.39.92 yes 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.h.a.39.17 ✓ 108 4.3 odd 2 inner
380.3.h.a.39.18 yes 108 5.4 even 2 inner
380.3.h.a.39.91 yes 108 1.1 even 1 trivial
380.3.h.a.39.92 yes 108 20.19 odd 2 inner