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Properties

Label 380.3.h.a.39.18
Level $380$
Weight $3$
Character 380.39
Analytic conductor $10.354$
Analytic rank $0$
Dimension $108$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

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 39.18
Character \(\chi\) \(=\) 380.39
Dual form 380.3.h.a.39.17

$q$-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} +(-9.43996 - 3.29958i) 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} +(19.5629 - 4.15833i) 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} +(-11.3699 - 3.97416i) 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} +(-29.1268 + 27.4159i) 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} +(-17.2657 - 46.9243i) 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} +(23.5624 - 5.00848i) 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} +(-60.8063 - 21.2538i) 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} +(21.3342 - 77.1029i) 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} +(71.2652 + 24.9096i) 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 \(-0.435665\pi\)
0.200741 + 0.979644i \(0.435665\pi\)
\(4\) 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 \(-0.347814\pi\)
0.460098 + 0.887868i \(0.347814\pi\)
\(8\) 0.530130 + 7.98242i 0.0662663 + 0.997802i
\(9\) −7.54932 −0.838813
\(10\) −9.43996 − 3.29958i −0.943996 − 0.329958i
\(11\) 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 \(0.253538\pi\)
−0.699204 + 0.714922i \(0.746462\pi\)
\(14\) −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 \(-0.904516\pi\)
0.955345 − 0.295493i \(-0.0954840\pi\)
\(18\) 12.9057 − 7.83649i 0.716986 − 0.435360i
\(19\) 4.35890i 0.229416i
\(20\) 19.5629 − 4.15833i 0.978147 − 0.207917i
\(21\) 7.75827 0.369441
\(22\) −4.54754 − 7.48925i −0.206707 − 0.340421i
\(23\) 16.8137 0.731029 0.365514 − 0.930806i \(-0.380893\pi\)
0.365514 + 0.930806i \(0.380893\pi\)
\(24\) 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 \(0.417062\pi\)
\(30\) −11.3699 − 3.97416i −0.378997 − 0.132472i
\(31\) − 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 \(0.146567\pi\)
−0.895851 + 0.444354i \(0.853433\pi\)
\(38\) −4.52471 − 7.45164i −0.119071 − 0.196096i
\(39\) 22.3882i 0.574056i
\(40\) −29.1268 + 27.4159i −0.728170 + 0.685397i
\(41\) 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 \(-0.856072\pi\)
−0.899505 + 0.436911i \(0.856072\pi\)
\(44\) 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 \(-0.194087\pi\)
0.819796 + 0.572655i \(0.194087\pi\)
\(48\) −11.0716 − 15.7732i −0.230659 − 0.328608i
\(49\) −7.50874 −0.153240
\(50\) −17.2657 − 46.9243i −0.345315 − 0.938487i
\(51\) − 12.1008i − 0.237270i
\(52\) 65.9707 + 34.2939i 1.26867 + 0.659497i
\(53\) 77.3006i 1.45850i 0.684246 + 0.729251i \(0.260131\pi\)
−0.684246 + 0.729251i \(0.739869\pi\)
\(54\) 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 \(0.885834\pi\)
\(60\) 23.5624 − 5.00848i 0.392707 − 0.0834746i
\(61\) −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 \(-0.581627\pi\)
−0.253638 + 0.967299i \(0.581627\pi\)
\(68\) −35.6571 − 18.5358i −0.524369 − 0.272585i
\(69\) 20.2511 0.293494
\(70\) −60.8063 − 21.2538i −0.868661 − 0.303626i
\(71\) − 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 \(0.0282039\pi\)
−0.996077 + 0.0884893i \(0.971796\pi\)
\(74\) −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 \(0.898367\pi\)
\(80\) 21.3342 − 77.1029i 0.266678 − 0.963786i
\(81\) 43.9360 0.542420
\(82\) −56.5512 + 34.3384i −0.689649 + 0.418762i
\(83\) −131.312 −1.58207 −0.791037 − 0.611768i \(-0.790459\pi\)
−0.791037 + 0.611768i \(0.790459\pi\)
\(84\) 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 \(0.193084\pi\)
\(90\) 71.2652 + 24.9096i 0.791836 + 0.276773i
\(91\) 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 \(-0.902351\pi\)
0.953313 − 0.301983i \(-0.0976485\pi\)
\(98\) 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.39.18 yes 108
4.3 odd 2 inner 380.3.h.a.39.92 yes 108
5.4 even 2 inner 380.3.h.a.39.91 yes 108
20.19 odd 2 inner 380.3.h.a.39.17 ✓ 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.h.a.39.17 ✓ 108 20.19 odd 2 inner
380.3.h.a.39.18 yes 108 1.1 even 1 trivial
380.3.h.a.39.91 yes 108 5.4 even 2 inner
380.3.h.a.39.92 yes 108 4.3 odd 2 inner