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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(179,380)] 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("380.179"); S:= CuspForms(chi, 2); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.s (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: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(56\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 179.2
Character \(\chi\) \(=\) 380.179
Dual form 380.2.s.a.259.2

$q$-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+(-1.40034 + 0.197594i) q^{2} +(0.952596 + 0.549982i) q^{3} +(1.92191 - 0.553399i) q^{4} +(-2.22480 + 0.224228i) q^{5} +(-1.44263 - 0.581934i) q^{6} -1.22694 q^{7} +(-2.58199 + 1.15471i) q^{8} +(-0.895041 - 1.55026i) q^{9} +(3.07117 - 0.753603i) q^{10} -0.0812753i q^{11} +(2.13517 + 0.529851i) q^{12} +(-1.99480 - 3.45509i) q^{13} +(1.71814 - 0.242437i) q^{14} +(-2.24265 - 1.01000i) q^{15} +(3.38750 - 2.12717i) q^{16} +(-4.46708 - 2.57907i) q^{17} +(1.55968 + 1.99403i) q^{18} +(-2.32718 - 3.68568i) q^{19} +(-4.15178 + 1.66215i) q^{20} +(-1.16878 - 0.674796i) q^{21} +(0.0160596 + 0.113813i) q^{22} +(3.94419 + 6.83155i) q^{23} +(-3.09466 - 0.320075i) q^{24} +(4.89944 - 0.997722i) q^{25} +(3.47611 + 4.44415i) q^{26} -5.26891i q^{27} +(-2.35808 + 0.678990i) q^{28} +(-0.920899 + 0.531682i) q^{29} +(3.34005 + 0.971208i) q^{30} -0.146201 q^{31} +(-4.32334 + 3.64812i) q^{32} +(0.0446999 - 0.0774226i) q^{33} +(6.76505 + 2.72891i) q^{34} +(2.72970 - 0.275115i) q^{35} +(-2.57810 - 2.48414i) q^{36} -2.88865 q^{37} +(3.98711 + 4.70137i) q^{38} -4.38841i q^{39} +(5.48548 - 3.14794i) q^{40} +(-10.5269 - 6.07769i) q^{41} +(1.77003 + 0.714001i) q^{42} +(3.50785 - 6.07577i) q^{43} +(-0.0449777 - 0.156204i) q^{44} +(2.33889 + 3.24831i) q^{45} +(-6.87310 - 8.78715i) q^{46} +(-2.81704 - 4.87926i) q^{47} +(4.39682 - 0.163273i) q^{48} -5.49461 q^{49} +(-6.66375 + 2.36525i) q^{50} +(-2.83688 - 4.91363i) q^{51} +(-5.74587 - 5.53647i) q^{52} +(3.68541 + 6.38332i) q^{53} +(1.04111 + 7.37828i) q^{54} +(0.0182242 + 0.180821i) q^{55} +(3.16795 - 1.41676i) q^{56} +(-0.189805 - 4.79087i) q^{57} +(1.18452 - 0.926500i) q^{58} +(-1.76709 + 3.06069i) q^{59} +(-4.86912 - 0.700047i) q^{60} +(1.88854 + 3.27105i) q^{61} +(0.204732 - 0.0288886i) q^{62} +(1.09816 + 1.90208i) q^{63} +(5.33330 - 5.96288i) q^{64} +(5.21275 + 7.23959i) q^{65} +(-0.0472969 + 0.117250i) q^{66} +(8.93896 - 5.16091i) q^{67} +(-10.0126 - 2.48467i) q^{68} +8.67694i q^{69} +(-3.76815 + 0.924628i) q^{70} +(0.376619 - 0.652324i) q^{71} +(4.10107 + 2.96923i) q^{72} +(1.44429 + 0.833861i) q^{73} +(4.04510 - 0.570781i) q^{74} +(5.21592 + 1.74418i) q^{75} +(-6.51229 - 5.79570i) q^{76} +0.0997203i q^{77} +(0.867125 + 6.14527i) q^{78} +(-7.08430 + 12.2704i) q^{79} +(-7.05953 + 5.49209i) q^{80} +(0.212682 - 0.368377i) q^{81} +(15.9421 + 6.43079i) q^{82} -2.81532 q^{83} +(-2.61973 - 0.650097i) q^{84} +(10.5167 + 4.73627i) q^{85} +(-3.71165 + 9.20129i) q^{86} -1.16966 q^{87} +(0.0938492 + 0.209852i) q^{88} +(-0.325562 + 0.187964i) q^{89} +(-3.91710 - 4.08659i) q^{90} +(2.44751 + 4.23920i) q^{91} +(11.3610 + 10.9469i) q^{92} +(-0.139271 - 0.0804080i) q^{93} +(4.90893 + 6.27600i) q^{94} +(6.00393 + 7.67807i) q^{95} +(-6.12479 + 1.09743i) q^{96} +(-2.65080 + 4.59131i) q^{97} +(7.69433 - 1.08570i) q^{98} +(-0.125998 + 0.0727447i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 112 q - 2 q^{5} - 8 q^{6} + 44 q^{9} + 6 q^{10} - 36 q^{14} - 4 q^{16} + 44 q^{20} - 48 q^{21} + 2 q^{24} - 2 q^{25} - 36 q^{26} - 12 q^{29} - 32 q^{30} - 30 q^{34} + 20 q^{36} - 24 q^{40} - 24 q^{41} - 14 q^{44}+ \cdots - 84 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.40034 + 0.197594i −0.990191 + 0.139720i
\(3\) 0.952596 + 0.549982i 0.549982 + 0.317532i 0.749115 − 0.662440i \(-0.230479\pi\)
−0.199133 + 0.979972i \(0.563813\pi\)
\(4\) 1.92191 − 0.553399i 0.960956 − 0.276700i
\(5\) −2.22480 + 0.224228i −0.994959 + 0.100278i
\(6\) −1.44263 − 0.581934i −0.588952 − 0.237574i
\(7\) −1.22694 −0.463741 −0.231871 − 0.972747i \(-0.574485\pi\)
−0.231871 + 0.972747i \(0.574485\pi\)
\(8\) −2.58199 + 1.15471i −0.912870 + 0.408251i
\(9\) −0.895041 − 1.55026i −0.298347 − 0.516752i
\(10\) 3.07117 − 0.753603i 0.971189 − 0.238310i
\(11\) − 0.0812753i − 0.0245054i −0.999925 − 0.0122527i \(-0.996100\pi\)
0.999925 − 0.0122527i \(-0.00390026\pi\)
\(12\) 2.13517 + 0.529851i 0.616369 + 0.152955i
\(13\) −1.99480 − 3.45509i −0.553258 − 0.958270i −0.998037 − 0.0626299i \(-0.980051\pi\)
0.444779 − 0.895640i \(-0.353282\pi\)
\(14\) 1.71814 − 0.242437i 0.459192 − 0.0647941i
\(15\) −2.24265 − 1.01000i −0.579051 − 0.260781i
\(16\) 3.38750 − 2.12717i 0.846875 − 0.531793i
\(17\) −4.46708 − 2.57907i −1.08343 − 0.625517i −0.151608 − 0.988441i \(-0.548445\pi\)
−0.931819 + 0.362924i \(0.881778\pi\)
\(18\) 1.55968 + 1.99403i 0.367621 + 0.469998i
\(19\) −2.32718 − 3.68568i −0.533891 − 0.845553i
\(20\) −4.15178 + 1.66215i −0.928366 + 0.371667i
\(21\) −1.16878 − 0.674796i −0.255049 − 0.147253i
\(22\) 0.0160596 + 0.113813i 0.00342391 + 0.0242651i
\(23\) 3.94419 + 6.83155i 0.822421 + 1.42448i 0.903874 + 0.427799i \(0.140711\pi\)
−0.0814526 + 0.996677i \(0.525956\pi\)
\(24\) −3.09466 − 0.320075i −0.631694 − 0.0653350i
\(25\) 4.89944 − 0.997722i 0.979889 − 0.199544i
\(26\) 3.47611 + 4.44415i 0.681720 + 0.871569i
\(27\) − 5.26891i − 1.01400i
\(28\) −2.35808 + 0.678990i −0.445635 + 0.128317i
\(29\) −0.920899 + 0.531682i −0.171007 + 0.0987308i −0.583060 − 0.812429i \(-0.698145\pi\)
0.412054 + 0.911160i \(0.364812\pi\)
\(30\) 3.34005 + 0.971208i 0.609807 + 0.177317i
\(31\) −0.146201 −0.0262585 −0.0131293 − 0.999914i \(-0.504179\pi\)
−0.0131293 + 0.999914i \(0.504179\pi\)
\(32\) −4.32334 + 3.64812i −0.764265 + 0.644902i
\(33\) 0.0446999 − 0.0774226i 0.00778126 − 0.0134775i
\(34\) 6.76505 + 2.72891i 1.16020 + 0.468004i
\(35\) 2.72970 − 0.275115i 0.461404 − 0.0465029i
\(36\) −2.57810 − 2.48414i −0.429683 − 0.414024i
\(37\) −2.88865 −0.474891 −0.237446 − 0.971401i \(-0.576310\pi\)
−0.237446 + 0.971401i \(0.576310\pi\)
\(38\) 3.98711 + 4.70137i 0.646795 + 0.762664i
\(39\) − 4.38841i − 0.702708i
\(40\) 5.48548 − 3.14794i 0.867330 − 0.497733i
\(41\) −10.5269 − 6.07769i −1.64402 − 0.949175i −0.979384 − 0.202008i \(-0.935253\pi\)
−0.664636 − 0.747167i \(-0.731413\pi\)
\(42\) 1.77003 + 0.714001i 0.273122 + 0.110173i
\(43\) 3.50785 − 6.07577i 0.534942 − 0.926547i −0.464224 − 0.885718i \(-0.653667\pi\)
0.999166 − 0.0408291i \(-0.0129999\pi\)
\(44\) −0.0449777 − 0.156204i −0.00678065 − 0.0235487i
\(45\) 2.33889 + 3.24831i 0.348662 + 0.484230i
\(46\) −6.87310 − 8.78715i −1.01338 − 1.29559i
\(47\) −2.81704 − 4.87926i −0.410908 − 0.711713i 0.584082 − 0.811695i \(-0.301455\pi\)
−0.994989 + 0.0999819i \(0.968122\pi\)
\(48\) 4.39682 − 0.163273i 0.634627 − 0.0235664i
\(49\) −5.49461 −0.784944
\(50\) −6.66375 + 2.36525i −0.942397 + 0.334498i
\(51\) −2.83688 − 4.91363i −0.397243 − 0.688045i
\(52\) −5.74587 − 5.53647i −0.796809 − 0.767770i
\(53\) 3.68541 + 6.38332i 0.506230 + 0.876816i 0.999974 + 0.00720886i \(0.00229467\pi\)
−0.493744 + 0.869607i \(0.664372\pi\)
\(54\) 1.04111 + 7.37828i 0.141677 + 1.00406i
\(55\) 0.0182242 + 0.180821i 0.00245735 + 0.0243819i
\(56\) 3.16795 − 1.41676i 0.423335 − 0.189323i
\(57\) −0.189805 − 4.79087i −0.0251402 − 0.634566i
\(58\) 1.18452 − 0.926500i 0.155535 − 0.121655i
\(59\) −1.76709 + 3.06069i −0.230055 + 0.398468i −0.957824 − 0.287355i \(-0.907224\pi\)
0.727769 + 0.685823i \(0.240557\pi\)
\(60\) −4.86912 − 0.700047i −0.628600 − 0.0903756i
\(61\) 1.88854 + 3.27105i 0.241803 + 0.418815i 0.961228 − 0.275755i \(-0.0889280\pi\)
−0.719425 + 0.694570i \(0.755595\pi\)
\(62\) 0.204732 − 0.0288886i 0.0260010 − 0.00366885i
\(63\) 1.09816 + 1.90208i 0.138356 + 0.239639i
\(64\) 5.33330 − 5.96288i 0.666663 − 0.745360i
\(65\) 5.21275 + 7.23959i 0.646562 + 0.897961i
\(66\) −0.0472969 + 0.117250i −0.00582185 + 0.0144325i
\(67\) 8.93896 − 5.16091i 1.09207 − 0.630506i 0.157942 − 0.987448i \(-0.449514\pi\)
0.934126 + 0.356943i \(0.116181\pi\)
\(68\) −10.0126 − 2.48467i −1.21421 − 0.301311i
\(69\) 8.67694i 1.04458i
\(70\) −3.76815 + 0.924628i −0.450380 + 0.110514i
\(71\) 0.376619 − 0.652324i 0.0446965 − 0.0774166i −0.842812 − 0.538209i \(-0.819101\pi\)
0.887508 + 0.460792i \(0.152435\pi\)
\(72\) 4.10107 + 2.96923i 0.483316 + 0.349927i
\(73\) 1.44429 + 0.833861i 0.169041 + 0.0975960i 0.582134 − 0.813093i \(-0.302218\pi\)
−0.413092 + 0.910689i \(0.635551\pi\)
\(74\) 4.04510 − 0.570781i 0.470233 − 0.0663520i
\(75\) 5.21592 + 1.74418i 0.602282 + 0.201400i
\(76\) −6.51229 − 5.79570i −0.747011 − 0.664812i
\(77\) 0.0997203i 0.0113642i
\(78\) 0.867125 + 6.14527i 0.0981826 + 0.695815i
\(79\) −7.08430 + 12.2704i −0.797046 + 1.38052i 0.124486 + 0.992221i \(0.460272\pi\)
−0.921532 + 0.388303i \(0.873061\pi\)
\(80\) −7.05953 + 5.49209i −0.789279 + 0.614035i
\(81\) 0.212682 − 0.368377i 0.0236314 − 0.0409307i
\(82\) 15.9421 + 6.43079i 1.76051 + 0.710162i
\(83\) −2.81532 −0.309021 −0.154511 − 0.987991i \(-0.549380\pi\)
−0.154511 + 0.987991i \(0.549380\pi\)
\(84\) −2.61973 − 0.650097i −0.285836 − 0.0709314i
\(85\) 10.5167 + 4.73627i 1.14069 + 0.513720i
\(86\) −3.71165 + 9.20129i −0.400237 + 0.992201i
\(87\) −1.16966 −0.125401
\(88\) 0.0938492 + 0.209852i 0.0100044 + 0.0223703i
\(89\) −0.325562 + 0.187964i −0.0345095 + 0.0199241i −0.517156 − 0.855891i \(-0.673009\pi\)
0.482646 + 0.875816i \(0.339676\pi\)
\(90\) −3.91710 − 4.08659i −0.412898 − 0.430765i
\(91\) 2.44751 + 4.23920i 0.256568 + 0.444389i
\(92\) 11.3610 + 10.9469i 1.18446 + 1.14130i
\(93\) −0.139271 − 0.0804080i −0.0144417 − 0.00833792i
\(94\) 4.90893 + 6.27600i 0.506318 + 0.647320i
\(95\) 6.00393 + 7.67807i 0.615990 + 0.787754i
\(96\) −6.12479 + 1.09743i −0.625109 + 0.112005i
\(97\) −2.65080 + 4.59131i −0.269148 + 0.466177i −0.968642 − 0.248461i \(-0.920075\pi\)
0.699494 + 0.714638i \(0.253409\pi\)
\(98\) 7.69433 − 1.08570i 0.777245 − 0.109673i
\(99\) −0.125998 + 0.0727447i −0.0126632 + 0.00731112i
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 380.2.s.a.179.2 ✓ 112
4.3 odd 2 inner 380.2.s.a.179.36 yes 112
5.4 even 2 inner 380.2.s.a.179.55 yes 112
19.12 odd 6 inner 380.2.s.a.259.21 yes 112
20.19 odd 2 inner 380.2.s.a.179.21 yes 112
76.31 even 6 inner 380.2.s.a.259.55 yes 112
95.69 odd 6 inner 380.2.s.a.259.36 yes 112
380.259 even 6 inner 380.2.s.a.259.2 yes 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.s.a.179.2 ✓ 112 1.1 even 1 trivial
380.2.s.a.179.21 yes 112 20.19 odd 2 inner
380.2.s.a.179.36 yes 112 4.3 odd 2 inner
380.2.s.a.179.55 yes 112 5.4 even 2 inner
380.2.s.a.259.2 yes 112 380.259 even 6 inner
380.2.s.a.259.21 yes 112 19.12 odd 6 inner
380.2.s.a.259.36 yes 112 95.69 odd 6 inner
380.2.s.a.259.55 yes 112 76.31 even 6 inner