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Properties

Label 380.3.j.a.227.15
Level $380$
Weight $3$
Character 380.227
Analytic conductor $10.354$
Analytic rank $0$
Dimension $232$
Inner twists $8$

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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(227,380)] 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("380.227"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 2])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.j (of order \(4\), degree \(2\), 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: \(232\)
Relative dimension: \(116\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 227.15
Character \(\chi\) \(=\) 380.227
Dual form 380.3.j.a.303.15

$q$-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+(-1.87058 - 0.707774i) q^{2} +(3.26349 - 3.26349i) q^{3} +(2.99811 + 2.64789i) q^{4} +(4.82525 - 1.31032i) q^{5} +(-8.41441 + 3.79479i) q^{6} +(0.0356335 - 0.0356335i) q^{7} +(-3.73409 - 7.07506i) q^{8} -12.3007i q^{9} +(-9.95341 - 0.964142i) q^{10} +10.2555i q^{11} +(18.4257 - 1.14295i) q^{12} +(-14.5415 - 14.5415i) q^{13} +(-0.0918755 + 0.0414347i) q^{14} +(11.4710 - 20.0233i) q^{15} +(1.97737 + 15.8773i) q^{16} +(-18.4050 - 18.4050i) q^{17} +(-8.70610 + 23.0094i) q^{18} +(11.1642 - 15.3740i) q^{19} +(17.9362 + 8.84826i) q^{20} -0.232579i q^{21} +(7.25859 - 19.1838i) q^{22} +(-9.70362 - 9.70362i) q^{23} +(-35.2755 - 10.9032i) q^{24} +(21.5661 - 12.6452i) q^{25} +(16.9090 + 37.4932i) q^{26} +(-10.7718 - 10.7718i) q^{27} +(0.201187 - 0.0124797i) q^{28} +43.7357 q^{29} +(-35.6293 + 29.3364i) q^{30} +35.6361 q^{31} +(7.53875 - 31.0993i) q^{32} +(33.4688 + 33.4688i) q^{33} +(21.4014 + 47.4547i) q^{34} +(0.125249 - 0.218632i) q^{35} +(32.5709 - 36.8789i) q^{36} +(-39.6031 + 39.6031i) q^{37} +(-31.7649 + 20.8565i) q^{38} -94.9123 q^{39} +(-27.2885 - 29.2461i) q^{40} -43.8275i q^{41} +(-0.164613 + 0.435056i) q^{42} +(15.7276 + 15.7276i) q^{43} +(-27.1555 + 30.7472i) q^{44} +(-16.1178 - 59.3539i) q^{45} +(11.2834 + 25.0193i) q^{46} +(51.9384 - 51.9384i) q^{47} +(58.2686 + 45.3624i) q^{48} +48.9975i q^{49} +(-49.2911 + 8.38987i) q^{50} -120.129 q^{51} +(-5.09280 - 82.1016i) q^{52} +(-22.2272 - 22.2272i) q^{53} +(12.5254 + 27.7734i) q^{54} +(13.4380 + 49.4855i) q^{55} +(-0.385168 - 0.119050i) q^{56} +(-13.7385 - 86.6072i) q^{57} +(-81.8110 - 30.9550i) q^{58} +34.3865i q^{59} +(87.4108 - 29.6584i) q^{60} +15.3059 q^{61} +(-66.6600 - 25.2223i) q^{62} +(-0.438316 - 0.438316i) q^{63} +(-36.1131 + 52.8379i) q^{64} +(-89.2206 - 51.1126i) q^{65} +(-38.9176 - 86.2943i) q^{66} +(3.42853 + 3.42853i) q^{67} +(-6.44589 - 103.915i) q^{68} -63.3352 q^{69} +(-0.389030 + 0.320319i) q^{70} -17.1730 q^{71} +(-87.0282 + 45.9319i) q^{72} +(-6.89725 + 6.89725i) q^{73} +(102.111 - 46.0505i) q^{74} +(29.1134 - 111.648i) q^{75} +(74.1803 - 16.5313i) q^{76} +(0.365440 + 0.365440i) q^{77} +(177.541 + 67.1764i) q^{78} +61.2804i q^{79} +(30.3456 + 74.0212i) q^{80} +40.3992 q^{81} +(-31.0199 + 81.9827i) q^{82} +(85.7234 + 85.7234i) q^{83} +(0.615842 - 0.697297i) q^{84} +(-112.925 - 64.6926i) q^{85} +(-18.2881 - 40.5512i) q^{86} +(142.731 - 142.731i) q^{87} +(72.5585 - 38.2951i) q^{88} -33.9414 q^{89} +(-11.8596 + 122.434i) q^{90} -1.03633 q^{91} +(-3.39844 - 54.7866i) q^{92} +(116.298 - 116.298i) q^{93} +(-133.915 + 60.3942i) q^{94} +(33.7255 - 88.8121i) q^{95} +(-76.8896 - 126.095i) q^{96} +(-79.7895 + 79.7895i) q^{97} +(34.6791 - 91.6535i) q^{98} +126.150 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 232 q - 8 q^{5} - 32 q^{6} - 64 q^{16} - 8 q^{17} + 80 q^{20} - 8 q^{25} + 40 q^{26} + 40 q^{28} - 104 q^{30} + 104 q^{36} + 184 q^{38} + 192 q^{42} + 192 q^{45} + 280 q^{58} + 112 q^{61} - 280 q^{62} + 216 q^{66}+ \cdots + 504 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.87058 − 0.707774i −0.935288 − 0.353887i
\(3\) 3.26349 − 3.26349i 1.08783 − 1.08783i 0.0920770 − 0.995752i \(-0.470649\pi\)
0.995752 − 0.0920770i \(-0.0293506\pi\)
\(4\) 2.99811 + 2.64789i 0.749528 + 0.661972i
\(5\) 4.82525 − 1.31032i 0.965051 − 0.262063i
\(6\) −8.41441 + 3.79479i −1.40240 + 0.632465i
\(7\) 0.0356335 − 0.0356335i 0.00509049 − 0.00509049i −0.704557 − 0.709647i \(-0.748854\pi\)
0.709647 + 0.704557i \(0.248854\pi\)
\(8\) −3.73409 − 7.07506i −0.466762 − 0.884383i
\(9\) − 12.3007i − 1.36674i
\(10\) −9.95341 − 0.964142i −0.995341 − 0.0964142i
\(11\) 10.2555i 0.932321i 0.884700 + 0.466161i \(0.154363\pi\)
−0.884700 + 0.466161i \(0.845637\pi\)
\(12\) 18.4257 − 1.14295i 1.53547 − 0.0952460i
\(13\) −14.5415 − 14.5415i −1.11858 − 1.11858i −0.991950 − 0.126630i \(-0.959584\pi\)
−0.126630 − 0.991950i \(-0.540416\pi\)
\(14\) −0.0918755 + 0.0414347i −0.00656254 + 0.00295962i
\(15\) 11.4710 − 20.0233i 0.764730 − 1.33489i
\(16\) 1.97737 + 15.8773i 0.123585 + 0.992334i
\(17\) −18.4050 − 18.4050i −1.08265 − 1.08265i −0.996262 − 0.0863882i \(-0.972467\pi\)
−0.0863882 − 0.996262i \(-0.527533\pi\)
\(18\) −8.70610 + 23.0094i −0.483672 + 1.27830i
\(19\) 11.1642 − 15.3740i 0.587591 − 0.809158i
\(20\) 17.9362 + 8.84826i 0.896811 + 0.442413i
\(21\) − 0.232579i − 0.0110752i
\(22\) 7.25859 − 19.1838i 0.329936 − 0.871989i
\(23\) −9.70362 − 9.70362i −0.421896 − 0.421896i 0.463960 − 0.885856i \(-0.346428\pi\)
−0.885856 + 0.463960i \(0.846428\pi\)
\(24\) −35.2755 − 10.9032i −1.46981 − 0.454300i
\(25\) 21.5661 − 12.6452i 0.862646 − 0.505808i
\(26\) 16.9090 + 37.4932i 0.650344 + 1.44205i
\(27\) −10.7718 − 10.7718i −0.398954 − 0.398954i
\(28\) 0.201187 − 0.0124797i 0.00718524 − 0.000445704i
\(29\) 43.7357 1.50813 0.754064 − 0.656800i \(-0.228091\pi\)
0.754064 + 0.656800i \(0.228091\pi\)
\(30\) −35.6293 + 29.3364i −1.18764 + 0.977879i
\(31\) 35.6361 1.14955 0.574776 − 0.818311i \(-0.305089\pi\)
0.574776 + 0.818311i \(0.305089\pi\)
\(32\) 7.53875 − 31.0993i 0.235586 − 0.971854i
\(33\) 33.4688 + 33.4688i 1.01421 + 1.01421i
\(34\) 21.4014 + 47.4547i 0.629454 + 1.39573i
\(35\) 0.125249 − 0.218632i 0.00357856 − 0.00624662i
\(36\) 32.5709 − 36.8789i 0.904746 − 1.02441i
\(37\) −39.6031 + 39.6031i −1.07035 + 1.07035i −0.0730224 + 0.997330i \(0.523264\pi\)
−0.997330 + 0.0730224i \(0.976736\pi\)
\(38\) −31.7649 + 20.8565i −0.835918 + 0.548855i
\(39\) −94.9123 −2.43365
\(40\) −27.2885 − 29.2461i −0.682213 − 0.731154i
\(41\) − 43.8275i − 1.06896i −0.845180 − 0.534481i \(-0.820507\pi\)
0.845180 − 0.534481i \(-0.179493\pi\)
\(42\) −0.164613 + 0.435056i −0.00391936 + 0.0103585i
\(43\) 15.7276 + 15.7276i 0.365758 + 0.365758i 0.865927 − 0.500170i \(-0.166729\pi\)
−0.500170 + 0.865927i \(0.666729\pi\)
\(44\) −27.1555 + 30.7472i −0.617171 + 0.698801i
\(45\) −16.1178 − 59.3539i −0.358173 − 1.31898i
\(46\) 11.2834 + 25.0193i 0.245291 + 0.543898i
\(47\) 51.9384 − 51.9384i 1.10507 − 1.10507i 0.111284 − 0.993789i \(-0.464504\pi\)
0.993789 − 0.111284i \(-0.0354965\pi\)
\(48\) 58.2686 + 45.3624i 1.21393 + 0.945050i
\(49\) 48.9975i 0.999948i
\(50\) −49.2911 + 8.38987i −0.985821 + 0.167797i
\(51\) −120.129 −2.35548
\(52\) −5.09280 − 82.1016i −0.0979385 − 1.57888i
\(53\) −22.2272 − 22.2272i −0.419380 − 0.419380i 0.465610 − 0.884990i \(-0.345835\pi\)
−0.884990 + 0.465610i \(0.845835\pi\)
\(54\) 12.5254 + 27.7734i 0.231952 + 0.514321i
\(55\) 13.4380 + 49.4855i 0.244327 + 0.899737i
\(56\) −0.385168 − 0.119050i −0.00687800 − 0.00212590i
\(57\) −13.7385 − 86.6072i −0.241026 − 1.51942i
\(58\) −81.8110 − 30.9550i −1.41054 − 0.533707i
\(59\) 34.3865i 0.582822i 0.956598 + 0.291411i \(0.0941248\pi\)
−0.956598 + 0.291411i \(0.905875\pi\)
\(60\) 87.4108 − 29.6584i 1.45685 − 0.494307i
\(61\) 15.3059 0.250916 0.125458 − 0.992099i \(-0.459960\pi\)
0.125458 + 0.992099i \(0.459960\pi\)
\(62\) −66.6600 − 25.2223i −1.07516 − 0.406811i
\(63\) −0.438316 − 0.438316i −0.00695740 − 0.00695740i
\(64\) −36.1131 + 52.8379i −0.564267 + 0.825592i
\(65\) −89.2206 − 51.1126i −1.37263 − 0.786348i
\(66\) −38.9176 − 86.2943i −0.589661 − 1.30749i
\(67\) 3.42853 + 3.42853i 0.0511722 + 0.0511722i 0.732230 − 0.681058i \(-0.238480\pi\)
−0.681058 + 0.732230i \(0.738480\pi\)
\(68\) −6.44589 − 103.915i −0.0947925 − 1.52816i
\(69\) −63.3352 −0.917902
\(70\) −0.389030 + 0.320319i −0.00555758 + 0.00457598i
\(71\) −17.1730 −0.241874 −0.120937 − 0.992660i \(-0.538590\pi\)
−0.120937 + 0.992660i \(0.538590\pi\)
\(72\) −87.0282 + 45.9319i −1.20872 + 0.637944i
\(73\) −6.89725 + 6.89725i −0.0944828 + 0.0944828i −0.752768 − 0.658286i \(-0.771282\pi\)
0.658286 + 0.752768i \(0.271282\pi\)
\(74\) 102.111 − 46.0505i 1.37987 − 0.622305i
\(75\) 29.1134 − 111.648i 0.388178 − 1.48864i
\(76\) 74.1803 − 16.5313i 0.976056 − 0.217517i
\(77\) 0.365440 + 0.365440i 0.00474598 + 0.00474598i
\(78\) 177.541 + 67.1764i 2.27616 + 0.861236i
\(79\) 61.2804i 0.775702i 0.921722 + 0.387851i \(0.126782\pi\)
−0.921722 + 0.387851i \(0.873218\pi\)
\(80\) 30.3456 + 74.0212i 0.379320 + 0.925265i
\(81\) 40.3992 0.498756
\(82\) −31.0199 + 81.9827i −0.378292 + 0.999789i
\(83\) 85.7234 + 85.7234i 1.03281 + 1.03281i 0.999443 + 0.0333687i \(0.0106236\pi\)
0.0333687 + 0.999443i \(0.489376\pi\)
\(84\) 0.615842 − 0.697297i 0.00733146 − 0.00830116i
\(85\) −112.925 − 64.6926i −1.32853 − 0.761090i
\(86\) −18.2881 − 40.5512i −0.212652 − 0.471526i
\(87\) 142.731 − 142.731i 1.64059 − 1.64059i
\(88\) 72.5585 − 38.2951i 0.824529 − 0.435172i
\(89\) −33.9414 −0.381364 −0.190682 − 0.981652i \(-0.561070\pi\)
−0.190682 + 0.981652i \(0.561070\pi\)
\(90\) −11.8596 + 122.434i −0.131774 + 1.36038i
\(91\) −1.03633 −0.0113883
\(92\) −3.39844 − 54.7866i −0.0369396 − 0.595507i
\(93\) 116.298 − 116.298i 1.25051 − 1.25051i
\(94\) −133.915 + 60.3942i −1.42463 + 0.642491i
\(95\) 33.7255 − 88.8121i 0.355005 − 0.934864i
\(96\) −76.8896 − 126.095i −0.800933 − 1.31349i
\(97\) −79.7895 + 79.7895i −0.822572 + 0.822572i −0.986476 − 0.163904i \(-0.947591\pi\)
0.163904 + 0.986476i \(0.447591\pi\)
\(98\) 34.6791 − 91.6535i 0.353868 − 0.935240i
\(99\) 126.150 1.27424
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.j.a.227.15 ✓ 232
4.3 odd 2 inner 380.3.j.a.227.73 yes 232
5.3 odd 4 inner 380.3.j.a.303.44 yes 232
19.18 odd 2 inner 380.3.j.a.227.102 yes 232
20.3 even 4 inner 380.3.j.a.303.102 yes 232
76.75 even 2 inner 380.3.j.a.227.44 yes 232
95.18 even 4 inner 380.3.j.a.303.73 yes 232
380.303 odd 4 inner 380.3.j.a.303.15 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.15 ✓ 232 1.1 even 1 trivial
380.3.j.a.227.44 yes 232 76.75 even 2 inner
380.3.j.a.227.73 yes 232 4.3 odd 2 inner
380.3.j.a.227.102 yes 232 19.18 odd 2 inner
380.3.j.a.303.15 yes 232 380.303 odd 4 inner
380.3.j.a.303.44 yes 232 5.3 odd 4 inner
380.3.j.a.303.73 yes 232 95.18 even 4 inner
380.3.j.a.303.102 yes 232 20.3 even 4 inner