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Properties

Label 380.3.p.a.159.5
Level $380$
Weight $3$
Character 380.159
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(159,380)] 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("380.159"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 2])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.p (of order \(6\), 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 \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 159.5
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.5

$q$-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+(-1.99219 - 0.176550i) q^{2} +(1.30777 + 2.26512i) q^{3} +(3.93766 + 0.703443i) q^{4} +(-3.13801 + 3.89267i) q^{5} +(-2.20541 - 4.74343i) q^{6} -0.966370 q^{7} +(-7.72038 - 2.09659i) q^{8} +(1.07950 - 1.86975i) q^{9} +(6.93877 - 7.20093i) q^{10} +8.12124i q^{11} +(3.55616 + 9.83919i) q^{12} +(6.10392 + 3.52410i) q^{13} +(1.92519 + 0.170612i) q^{14} +(-12.9211 - 2.01726i) q^{15} +(15.0103 + 5.53984i) q^{16} +(-21.9043 + 12.6465i) q^{17} +(-2.48068 + 3.53432i) q^{18} +(-15.8231 + 10.5181i) q^{19} +(-15.0947 + 13.1206i) q^{20} +(-1.26378 - 2.18894i) q^{21} +(1.43380 - 16.1791i) q^{22} +(2.15738 - 3.73670i) q^{23} +(-5.34744 - 20.2294i) q^{24} +(-5.30576 - 24.4305i) q^{25} +(-11.5380 - 8.09834i) q^{26} +29.1867 q^{27} +(-3.80524 - 0.679786i) q^{28} +(-16.7987 + 29.0962i) q^{29} +(25.3852 + 6.30000i) q^{30} -58.2177i q^{31} +(-28.9254 - 13.6865i) q^{32} +(-18.3955 + 10.6207i) q^{33} +(45.8704 - 21.3270i) q^{34} +(3.03248 - 3.76176i) q^{35} +(5.56597 - 6.60307i) q^{36} +53.8794i q^{37} +(33.3796 - 18.1605i) q^{38} +18.4348i q^{39} +(32.3880 - 23.4738i) q^{40} +(-4.01029 - 6.94603i) q^{41} +(2.13124 + 4.58391i) q^{42} +(-37.0472 - 64.1677i) q^{43} +(-5.71283 + 31.9787i) q^{44} +(3.89083 + 10.0694i) q^{45} +(-4.95764 + 7.06334i) q^{46} +(4.99029 - 8.64343i) q^{47} +(7.08162 + 41.2450i) q^{48} -48.0661 q^{49} +(6.25690 + 49.6070i) q^{50} +(-57.2914 - 33.0772i) q^{51} +(21.5562 + 18.1705i) q^{52} +(-61.9022 - 35.7393i) q^{53} +(-58.1455 - 5.15291i) q^{54} +(-31.6133 - 25.4845i) q^{55} +(7.46074 + 2.02608i) q^{56} +(-44.5176 - 22.0858i) q^{57} +(38.6031 - 54.9993i) q^{58} +(32.5573 - 18.7970i) q^{59} +(-49.4600 - 17.0326i) q^{60} +(-44.1841 + 76.5291i) q^{61} +(-10.2783 + 115.981i) q^{62} +(-1.04320 + 1.80687i) q^{63} +(55.2086 + 32.3729i) q^{64} +(-32.8724 + 12.7019i) q^{65} +(38.5225 - 17.9107i) q^{66} +(-31.8085 + 55.0940i) q^{67} +(-95.1479 + 34.3891i) q^{68} +11.2854 q^{69} +(-6.70542 + 6.95876i) q^{70} +(-91.0583 + 52.5725i) q^{71} +(-12.2543 + 12.1719i) q^{72} +(112.445 - 64.9203i) q^{73} +(9.51241 - 107.338i) q^{74} +(48.3992 - 43.9675i) q^{75} +(-69.7047 + 30.2862i) q^{76} -7.84812i q^{77} +(3.25466 - 36.7257i) q^{78} +(-11.6822 + 6.74469i) q^{79} +(-68.6674 + 41.0462i) q^{80} +(28.4539 + 49.2835i) q^{81} +(6.76296 + 14.5459i) q^{82} +43.4290 q^{83} +(-3.43656 - 9.50830i) q^{84} +(19.5075 - 124.951i) q^{85} +(62.4764 + 134.375i) q^{86} -87.8749 q^{87} +(17.0269 - 62.6991i) q^{88} +(-53.4808 + 92.6314i) q^{89} +(-5.97353 - 20.7472i) q^{90} +(-5.89865 - 3.40559i) q^{91} +(11.1236 - 13.1963i) q^{92} +(131.870 - 76.1351i) q^{93} +(-11.4676 + 16.3383i) q^{94} +(8.70940 - 94.5999i) q^{95} +(-6.82616 - 83.4181i) q^{96} +(36.8345 - 21.2664i) q^{97} +(95.7570 + 8.48607i) q^{98} +(15.1847 + 8.76688i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 232 q - 2 q^{5} + 8 q^{6} - 328 q^{9} + 20 q^{14} + 12 q^{16} + 92 q^{20} - 40 q^{21} - 134 q^{24} - 2 q^{25} + 28 q^{26} - 4 q^{29} + 268 q^{30} - 70 q^{34} + 12 q^{36} - 42 q^{40} - 12 q^{41} + 98 q^{44}+ \cdots - 628 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.99219 − 0.176550i −0.996096 − 0.0882750i
\(3\) 1.30777 + 2.26512i 0.435922 + 0.755039i 0.997370 − 0.0724730i \(-0.0230891\pi\)
−0.561449 + 0.827512i \(0.689756\pi\)
\(4\) 3.93766 + 0.703443i 0.984415 + 0.175861i
\(5\) −3.13801 + 3.89267i −0.627602 + 0.778534i
\(6\) −2.20541 − 4.74343i −0.367569 − 0.790572i
\(7\) −0.966370 −0.138053 −0.0690264 − 0.997615i \(-0.521989\pi\)
−0.0690264 + 0.997615i \(0.521989\pi\)
\(8\) −7.72038 − 2.09659i −0.965048 − 0.262073i
\(9\) 1.07950 − 1.86975i 0.119945 − 0.207750i
\(10\) 6.93877 − 7.20093i 0.693877 − 0.720093i
\(11\) 8.12124i 0.738294i 0.929371 + 0.369147i \(0.120350\pi\)
−0.929371 + 0.369147i \(0.879650\pi\)
\(12\) 3.55616 + 9.83919i 0.296346 + 0.819933i
\(13\) 6.10392 + 3.52410i 0.469533 + 0.271085i 0.716044 − 0.698055i \(-0.245951\pi\)
−0.246511 + 0.969140i \(0.579284\pi\)
\(14\) 1.92519 + 0.170612i 0.137514 + 0.0121866i
\(15\) −12.9211 − 2.01726i −0.861409 − 0.134484i
\(16\) 15.0103 + 5.53984i 0.938146 + 0.346240i
\(17\) −21.9043 + 12.6465i −1.28849 + 0.743910i −0.978385 − 0.206792i \(-0.933698\pi\)
−0.310105 + 0.950702i \(0.600364\pi\)
\(18\) −2.48068 + 3.53432i −0.137815 + 0.196351i
\(19\) −15.8231 + 10.5181i −0.832793 + 0.553585i
\(20\) −15.0947 + 13.1206i −0.754735 + 0.656030i
\(21\) −1.26378 − 2.18894i −0.0601802 − 0.104235i
\(22\) 1.43380 − 16.1791i 0.0651729 − 0.735412i
\(23\) 2.15738 − 3.73670i 0.0937993 − 0.162465i −0.815308 − 0.579028i \(-0.803432\pi\)
0.909107 + 0.416563i \(0.136765\pi\)
\(24\) −5.34744 − 20.2294i −0.222810 − 0.842892i
\(25\) −5.30576 − 24.4305i −0.212230 − 0.977220i
\(26\) −11.5380 − 8.09834i −0.443770 − 0.311474i
\(27\) 29.1867 1.08099
\(28\) −3.80524 − 0.679786i −0.135901 − 0.0242781i
\(29\) −16.7987 + 29.0962i −0.579265 + 1.00332i 0.416299 + 0.909228i \(0.363327\pi\)
−0.995564 + 0.0940883i \(0.970006\pi\)
\(30\) 25.3852 + 6.30000i 0.846174 + 0.210000i
\(31\) − 58.2177i − 1.87799i −0.343930 − 0.938995i \(-0.611758\pi\)
0.343930 − 0.938995i \(-0.388242\pi\)
\(32\) −28.9254 − 13.6865i −0.903919 − 0.427703i
\(33\) −18.3955 + 10.6207i −0.557441 + 0.321839i
\(34\) 45.8704 − 21.3270i 1.34913 − 0.627265i
\(35\) 3.03248 − 3.76176i 0.0866423 − 0.107479i
\(36\) 5.56597 − 6.60307i 0.154610 − 0.183419i
\(37\) 53.8794i 1.45620i 0.685470 + 0.728101i \(0.259597\pi\)
−0.685470 + 0.728101i \(0.740403\pi\)
\(38\) 33.3796 − 18.1605i 0.878409 − 0.477909i
\(39\) 18.4348i 0.472687i
\(40\) 32.3880 − 23.4738i 0.809699 − 0.586845i
\(41\) −4.01029 − 6.94603i −0.0978121 − 0.169415i 0.812967 − 0.582310i \(-0.197851\pi\)
−0.910779 + 0.412895i \(0.864518\pi\)
\(42\) 2.13124 + 4.58391i 0.0507439 + 0.109141i
\(43\) −37.0472 − 64.1677i −0.861564 − 1.49227i −0.870419 − 0.492311i \(-0.836152\pi\)
0.00885569 − 0.999961i \(-0.497181\pi\)
\(44\) −5.71283 + 31.9787i −0.129837 + 0.726788i
\(45\) 3.89083 + 10.0694i 0.0864630 + 0.223765i
\(46\) −4.95764 + 7.06334i −0.107775 + 0.153551i
\(47\) 4.99029 − 8.64343i 0.106176 − 0.183903i −0.808042 − 0.589125i \(-0.799473\pi\)
0.914218 + 0.405222i \(0.132806\pi\)
\(48\) 7.08162 + 41.2450i 0.147534 + 0.859270i
\(49\) −48.0661 −0.980941
\(50\) 6.25690 + 49.6070i 0.125138 + 0.992139i
\(51\) −57.2914 − 33.0772i −1.12336 − 0.648573i
\(52\) 21.5562 + 18.1705i 0.414542 + 0.349432i
\(53\) −61.9022 − 35.7393i −1.16797 − 0.674326i −0.214766 − 0.976665i \(-0.568899\pi\)
−0.953200 + 0.302339i \(0.902232\pi\)
\(54\) −58.1455 − 5.15291i −1.07677 − 0.0954243i
\(55\) −31.6133 − 25.4845i −0.574787 − 0.463355i
\(56\) 7.46074 + 2.02608i 0.133228 + 0.0361800i
\(57\) −44.5176 − 22.0858i −0.781010 − 0.387471i
\(58\) 38.6031 − 54.9993i 0.665571 − 0.948265i
\(59\) 32.5573 − 18.7970i 0.551819 − 0.318593i −0.198036 − 0.980195i \(-0.563456\pi\)
0.749855 + 0.661602i \(0.230123\pi\)
\(60\) −49.4600 − 17.0326i −0.824333 − 0.283876i
\(61\) −44.1841 + 76.5291i −0.724329 + 1.25457i 0.234921 + 0.972015i \(0.424517\pi\)
−0.959250 + 0.282560i \(0.908816\pi\)
\(62\) −10.2783 + 115.981i −0.165780 + 1.87066i
\(63\) −1.04320 + 1.80687i −0.0165587 + 0.0286805i
\(64\) 55.2086 + 32.3729i 0.862635 + 0.505827i
\(65\) −32.8724 + 12.7019i −0.505729 + 0.195414i
\(66\) 38.5225 − 17.9107i 0.583675 − 0.271374i
\(67\) −31.8085 + 55.0940i −0.474754 + 0.822299i −0.999582 − 0.0289097i \(-0.990796\pi\)
0.524828 + 0.851209i \(0.324130\pi\)
\(68\) −95.1479 + 34.3891i −1.39923 + 0.505722i
\(69\) 11.2854 0.163557
\(70\) −6.70542 + 6.95876i −0.0957917 + 0.0994109i
\(71\) −91.0583 + 52.5725i −1.28251 + 0.740458i −0.977307 − 0.211829i \(-0.932058\pi\)
−0.305204 + 0.952287i \(0.598725\pi\)
\(72\) −12.2543 + 12.1719i −0.170198 + 0.169054i
\(73\) 112.445 − 64.9203i 1.54035 − 0.889319i 0.541530 − 0.840682i \(-0.317845\pi\)
0.998816 − 0.0486376i \(-0.0154879\pi\)
\(74\) 9.51241 − 107.338i 0.128546 − 1.45052i
\(75\) 48.3992 − 43.9675i 0.645323 − 0.586233i
\(76\) −69.7047 + 30.2862i −0.917167 + 0.398502i
\(77\) − 7.84812i − 0.101924i
\(78\) 3.25466 − 36.7257i 0.0417264 − 0.470842i
\(79\) −11.6822 + 6.74469i −0.147875 + 0.0853759i −0.572112 − 0.820175i \(-0.693876\pi\)
0.424237 + 0.905551i \(0.360542\pi\)
\(80\) −68.6674 + 41.0462i −0.858342 + 0.513078i
\(81\) 28.4539 + 49.2835i 0.351282 + 0.608438i
\(82\) 6.76296 + 14.5459i 0.0824751 + 0.177388i
\(83\) 43.4290 0.523241 0.261621 − 0.965171i \(-0.415743\pi\)
0.261621 + 0.965171i \(0.415743\pi\)
\(84\) −3.43656 − 9.50830i −0.0409114 − 0.113194i
\(85\) 19.5075 − 124.951i 0.229500 − 1.47001i
\(86\) 62.4764 + 134.375i 0.726470 + 1.56250i
\(87\) −87.8749 −1.01006
\(88\) 17.0269 − 62.6991i 0.193487 − 0.712490i
\(89\) −53.4808 + 92.6314i −0.600908 + 1.04080i 0.391776 + 0.920060i \(0.371861\pi\)
−0.992684 + 0.120742i \(0.961473\pi\)
\(90\) −5.97353 − 20.7472i −0.0663726 − 0.230524i
\(91\) −5.89865 − 3.40559i −0.0648203 − 0.0374240i
\(92\) 11.1236 − 13.1963i 0.120909 − 0.143438i
\(93\) 131.870 − 76.1351i 1.41796 − 0.818657i
\(94\) −11.4676 + 16.3383i −0.121996 + 0.173812i
\(95\) 8.70940 − 94.5999i 0.0916779 − 0.995789i
\(96\) −6.82616 − 83.4181i −0.0711059 − 0.868939i
\(97\) 36.8345 − 21.2664i 0.379737 − 0.219241i −0.297967 − 0.954576i \(-0.596308\pi\)
0.677704 + 0.735335i \(0.262975\pi\)
\(98\) 95.7570 + 8.48607i 0.977112 + 0.0865926i
\(99\) 15.1847 + 8.76688i 0.153381 + 0.0885544i
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.p.a.159.5 ✓ 232
4.3 odd 2 inner 380.3.p.a.159.43 yes 232
5.4 even 2 inner 380.3.p.a.159.112 yes 232
19.11 even 3 inner 380.3.p.a.239.74 yes 232
20.19 odd 2 inner 380.3.p.a.159.74 yes 232
76.11 odd 6 inner 380.3.p.a.239.112 yes 232
95.49 even 6 inner 380.3.p.a.239.43 yes 232
380.239 odd 6 inner 380.3.p.a.239.5 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.5 ✓ 232 1.1 even 1 trivial
380.3.p.a.159.43 yes 232 4.3 odd 2 inner
380.3.p.a.159.74 yes 232 20.19 odd 2 inner
380.3.p.a.159.112 yes 232 5.4 even 2 inner
380.3.p.a.239.5 yes 232 380.239 odd 6 inner
380.3.p.a.239.43 yes 232 95.49 even 6 inner
380.3.p.a.239.74 yes 232 19.11 even 3 inner
380.3.p.a.239.112 yes 232 76.11 odd 6 inner