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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 239.61
Character \(\chi\) \(=\) 380.239
Dual form 380.3.p.a.159.61

$q$-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+(0.213676 + 1.98855i) q^{2} +(-0.0580338 + 0.100518i) q^{3} +(-3.90868 + 0.849813i) q^{4} +(2.96569 - 4.02550i) q^{5} +(-0.212285 - 0.0939251i) q^{6} -7.81754 q^{7} +(-2.52509 - 7.59104i) q^{8} +(4.49326 + 7.78256i) q^{9} +(8.63862 + 5.03728i) q^{10} +3.02550i q^{11} +(0.141415 - 0.442209i) q^{12} +(-11.0968 + 6.40674i) q^{13} +(-1.67042 - 15.5456i) q^{14} +(0.232523 + 0.531719i) q^{15} +(14.5556 - 6.64330i) q^{16} +(-17.3218 - 10.0008i) q^{17} +(-14.5159 + 10.5980i) q^{18} +(-9.66050 - 16.3608i) q^{19} +(-8.17102 + 18.2547i) q^{20} +(0.453682 - 0.785800i) q^{21} +(-6.01637 + 0.646478i) q^{22} +(-17.7149 - 30.6831i) q^{23} +(0.909574 + 0.186721i) q^{24} +(-7.40936 - 23.8768i) q^{25} +(-15.1113 - 20.6976i) q^{26} -2.08765 q^{27} +(30.5563 - 6.64345i) q^{28} +(-20.2682 - 35.1055i) q^{29} +(-1.00767 + 0.576001i) q^{30} +31.8484i q^{31} +(16.3208 + 27.5251i) q^{32} +(-0.304116 - 0.175581i) q^{33} +(16.1858 - 36.5823i) q^{34} +(-23.1844 + 31.4695i) q^{35} +(-24.1765 - 26.6011i) q^{36} +47.3562i q^{37} +(30.4700 - 22.7063i) q^{38} -1.48723i q^{39} +(-38.0464 - 12.3479i) q^{40} +(15.6324 - 27.0762i) q^{41} +(1.65955 + 0.734263i) q^{42} +(16.3862 - 28.3818i) q^{43} +(-2.57111 - 11.8257i) q^{44} +(44.6544 + 4.99301i) q^{45} +(57.2296 - 41.7832i) q^{46} +(19.2417 + 33.3277i) q^{47} +(-0.176951 + 1.84863i) q^{48} +12.1139 q^{49} +(45.8971 - 19.8358i) q^{50} +(2.01051 - 1.16077i) q^{51} +(37.9293 - 34.4721i) q^{52} +(-62.0748 + 35.8389i) q^{53} +(-0.446082 - 4.15141i) q^{54} +(12.1792 + 8.97270i) q^{55} +(19.7400 + 59.3433i) q^{56} +(2.20518 - 0.0215721i) q^{57} +(65.4783 - 47.8055i) q^{58} +(-85.4045 - 49.3083i) q^{59} +(-1.36072 - 1.88072i) q^{60} +(17.0850 + 29.5921i) q^{61} +(-63.3322 + 6.80524i) q^{62} +(-35.1263 - 60.8405i) q^{63} +(-51.2478 + 38.3361i) q^{64} +(-7.11931 + 63.6706i) q^{65} +(0.284171 - 0.642268i) q^{66} +(-40.0811 - 69.4225i) q^{67} +(76.2044 + 24.3695i) q^{68} +4.11225 q^{69} +(-67.5328 - 39.3791i) q^{70} +(14.5276 + 8.38749i) q^{71} +(47.7318 - 53.7602i) q^{72} +(57.9890 + 33.4800i) q^{73} +(-94.1702 + 10.1189i) q^{74} +(2.83003 + 0.640891i) q^{75} +(51.6635 + 55.7395i) q^{76} -23.6520i q^{77} +(2.95744 - 0.317786i) q^{78} +(87.2930 + 50.3986i) q^{79} +(16.4249 - 78.2957i) q^{80} +(-40.3182 + 69.8332i) q^{81} +(57.1827 + 25.3004i) q^{82} +38.3764 q^{83} +(-1.10552 + 3.45699i) q^{84} +(-91.6293 + 40.0699i) q^{85} +(59.9400 + 26.5204i) q^{86} +4.70496 q^{87} +(22.9667 - 7.63966i) q^{88} +(-22.1989 - 38.4497i) q^{89} +(-0.387296 + 89.8644i) q^{90} +(86.7496 - 50.0849i) q^{91} +(95.3167 + 104.876i) q^{92} +(-3.20132 - 1.84828i) q^{93} +(-62.1623 + 45.3845i) q^{94} +(-94.5104 - 9.63257i) q^{95} +(-3.71392 + 0.0431331i) q^{96} +(-102.346 - 59.0898i) q^{97} +(2.58845 + 24.0891i) q^{98} +(-23.5461 + 13.5944i) 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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.213676 + 1.98855i 0.106838 + 0.994276i
\(3\) −0.0580338 + 0.100518i −0.0193446 + 0.0335059i −0.875536 − 0.483154i \(-0.839491\pi\)
0.856191 + 0.516659i \(0.172825\pi\)
\(4\) −3.90868 + 0.849813i −0.977171 + 0.212453i
\(5\) 2.96569 − 4.02550i 0.593138 − 0.805101i
\(6\) −0.212285 − 0.0939251i −0.0353808 − 0.0156542i
\(7\) −7.81754 −1.11679 −0.558396 − 0.829575i \(-0.688583\pi\)
−0.558396 + 0.829575i \(0.688583\pi\)
\(8\) −2.52509 − 7.59104i −0.315636 − 0.948880i
\(9\) 4.49326 + 7.78256i 0.499252 + 0.864729i
\(10\) 8.63862 + 5.03728i 0.863862 + 0.503728i
\(11\) 3.02550i 0.275045i 0.990499 + 0.137523i \(0.0439140\pi\)
−0.990499 + 0.137523i \(0.956086\pi\)
\(12\) 0.141415 − 0.442209i 0.0117846 − 0.0368508i
\(13\) −11.0968 + 6.40674i −0.853600 + 0.492826i −0.861864 − 0.507140i \(-0.830703\pi\)
0.00826419 + 0.999966i \(0.497369\pi\)
\(14\) −1.67042 − 15.5456i −0.119316 − 1.11040i
\(15\) 0.232523 + 0.531719i 0.0155016 + 0.0354480i
\(16\) 14.5556 − 6.64330i 0.909727 − 0.415206i
\(17\) −17.3218 − 10.0008i −1.01893 − 0.588280i −0.105137 − 0.994458i \(-0.533528\pi\)
−0.913794 + 0.406177i \(0.866862\pi\)
\(18\) −14.5159 + 10.5980i −0.806441 + 0.588780i
\(19\) −9.66050 − 16.3608i −0.508448 − 0.861093i
\(20\) −8.17102 + 18.2547i −0.408551 + 0.912735i
\(21\) 0.453682 − 0.785800i 0.0216039 − 0.0374190i
\(22\) −6.01637 + 0.646478i −0.273471 + 0.0293853i
\(23\) −17.7149 − 30.6831i −0.770212 − 1.33405i −0.937447 − 0.348129i \(-0.886817\pi\)
0.167235 − 0.985917i \(-0.446516\pi\)
\(24\) 0.909574 + 0.186721i 0.0378989 + 0.00778005i
\(25\) −7.40936 − 23.8768i −0.296375 − 0.955072i
\(26\) −15.1113 − 20.6976i −0.581202 − 0.796061i
\(27\) −2.08765 −0.0773205
\(28\) 30.5563 − 6.64345i 1.09130 − 0.237266i
\(29\) −20.2682 − 35.1055i −0.698902 − 1.21053i −0.968848 − 0.247658i \(-0.920339\pi\)
0.269946 − 0.962876i \(-0.412994\pi\)
\(30\) −1.00767 + 0.576001i −0.0335889 + 0.0192000i
\(31\) 31.8484i 1.02737i 0.857980 + 0.513684i \(0.171719\pi\)
−0.857980 + 0.513684i \(0.828281\pi\)
\(32\) 16.3208 + 27.5251i 0.510024 + 0.860160i
\(33\) −0.304116 − 0.175581i −0.00921563 − 0.00532065i
\(34\) 16.1858 − 36.5823i 0.476053 − 1.07595i
\(35\) −23.1844 + 31.4695i −0.662411 + 0.899129i
\(36\) −24.1765 − 26.6011i −0.671569 − 0.738921i
\(37\) 47.3562i 1.27990i 0.768418 + 0.639948i \(0.221044\pi\)
−0.768418 + 0.639948i \(0.778956\pi\)
\(38\) 30.4700 − 22.7063i 0.801843 − 0.597535i
\(39\) − 1.48723i − 0.0381341i
\(40\) −38.0464 − 12.3479i −0.951160 − 0.308698i
\(41\) 15.6324 − 27.0762i 0.381279 − 0.660395i −0.609966 − 0.792427i \(-0.708817\pi\)
0.991245 + 0.132032i \(0.0421503\pi\)
\(42\) 1.65955 + 0.734263i 0.0395130 + 0.0174825i
\(43\) 16.3862 − 28.3818i 0.381075 − 0.660041i −0.610141 − 0.792293i \(-0.708887\pi\)
0.991216 + 0.132251i \(0.0422206\pi\)
\(44\) −2.57111 − 11.8257i −0.0584343 − 0.268766i
\(45\) 44.6544 + 4.99301i 0.992319 + 0.110956i
\(46\) 57.2296 − 41.7832i 1.24412 − 0.908331i
\(47\) 19.2417 + 33.3277i 0.409399 + 0.709099i 0.994822 − 0.101628i \(-0.0324052\pi\)
−0.585424 + 0.810727i \(0.699072\pi\)
\(48\) −0.176951 + 1.84863i −0.00368647 + 0.0385132i
\(49\) 12.1139 0.247222
\(50\) 45.8971 − 19.8358i 0.917941 − 0.396716i
\(51\) 2.01051 − 1.16077i 0.0394217 − 0.0227601i
\(52\) 37.9293 − 34.4721i 0.729410 − 0.662925i
\(53\) −62.0748 + 35.8389i −1.17122 + 0.676205i −0.953968 − 0.299909i \(-0.903044\pi\)
−0.217255 + 0.976115i \(0.569710\pi\)
\(54\) −0.446082 − 4.15141i −0.00826078 − 0.0768780i
\(55\) 12.1792 + 8.97270i 0.221439 + 0.163140i
\(56\) 19.7400 + 59.3433i 0.352500 + 1.05970i
\(57\) 2.20518 − 0.0215721i 0.0386874 − 0.000378457i
\(58\) 65.4783 − 47.8055i 1.12894 − 0.824233i
\(59\) −85.4045 − 49.3083i −1.44753 − 0.835734i −0.449200 − 0.893431i \(-0.648291\pi\)
−0.998334 + 0.0576968i \(0.981624\pi\)
\(60\) −1.36072 − 1.88072i −0.0226787 − 0.0313454i
\(61\) 17.0850 + 29.5921i 0.280082 + 0.485116i 0.971405 − 0.237430i \(-0.0763050\pi\)
−0.691323 + 0.722546i \(0.742972\pi\)
\(62\) −63.3322 + 6.80524i −1.02149 + 0.109762i
\(63\) −35.1263 − 60.8405i −0.557560 − 0.965722i
\(64\) −51.2478 + 38.3361i −0.800747 + 0.599002i
\(65\) −7.11931 + 63.6706i −0.109528 + 0.979547i
\(66\) 0.284171 − 0.642268i 0.00430561 − 0.00973133i
\(67\) −40.0811 − 69.4225i −0.598225 − 1.03616i −0.993083 − 0.117414i \(-0.962539\pi\)
0.394858 − 0.918742i \(-0.370794\pi\)
\(68\) 76.2044 + 24.3695i 1.12065 + 0.358375i
\(69\) 4.11225 0.0595978
\(70\) −67.5328 − 39.3791i −0.964754 − 0.562559i
\(71\) 14.5276 + 8.38749i 0.204614 + 0.118134i 0.598806 − 0.800894i \(-0.295642\pi\)
−0.394192 + 0.919028i \(0.628976\pi\)
\(72\) 47.7318 − 53.7602i 0.662942 − 0.746670i
\(73\) 57.9890 + 33.4800i 0.794370 + 0.458630i 0.841499 − 0.540259i \(-0.181674\pi\)
−0.0471287 + 0.998889i \(0.515007\pi\)
\(74\) −94.1702 + 10.1189i −1.27257 + 0.136742i
\(75\) 2.83003 + 0.640891i 0.0377337 + 0.00854521i
\(76\) 51.6635 + 55.7395i 0.679782 + 0.733414i
\(77\) − 23.6520i − 0.307168i
\(78\) 2.95744 − 0.317786i 0.0379158 − 0.00407418i
\(79\) 87.2930 + 50.3986i 1.10497 + 0.637957i 0.937523 − 0.347923i \(-0.113113\pi\)
0.167452 + 0.985880i \(0.446446\pi\)
\(80\) 16.4249 − 78.2957i 0.205311 − 0.978697i
\(81\) −40.3182 + 69.8332i −0.497756 + 0.862138i
\(82\) 57.1827 + 25.3004i 0.697350 + 0.308542i
\(83\) 38.3764 0.462366 0.231183 − 0.972910i \(-0.425740\pi\)
0.231183 + 0.972910i \(0.425740\pi\)
\(84\) −1.10552 + 3.45699i −0.0131609 + 0.0411546i
\(85\) −91.6293 + 40.0699i −1.07799 + 0.471411i
\(86\) 59.9400 + 26.5204i 0.696977 + 0.308376i
\(87\) 4.70496 0.0540799
\(88\) 22.9667 − 7.63966i 0.260985 − 0.0868144i
\(89\) −22.1989 − 38.4497i −0.249426 − 0.432019i 0.713941 − 0.700206i \(-0.246909\pi\)
−0.963367 + 0.268187i \(0.913575\pi\)
\(90\) −0.387296 + 89.8644i −0.00430329 + 0.998494i
\(91\) 86.7496 − 50.0849i 0.953292 − 0.550384i
\(92\) 95.3167 + 104.876i 1.03605 + 1.13996i
\(93\) −3.20132 − 1.84828i −0.0344228 − 0.0198740i
\(94\) −62.1623 + 45.3845i −0.661301 + 0.482814i
\(95\) −94.5104 − 9.63257i −0.994846 − 0.101396i
\(96\) −3.71392 + 0.0431331i −0.0386866 + 0.000449303i
\(97\) −102.346 − 59.0898i −1.05512 − 0.609173i −0.131040 − 0.991377i \(-0.541832\pi\)
−0.924078 + 0.382204i \(0.875165\pi\)
\(98\) 2.58845 + 24.0891i 0.0264128 + 0.245807i
\(99\) −23.5461 + 13.5944i −0.237840 + 0.137317i
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.239.61 yes 232
4.3 odd 2 inner 380.3.p.a.239.101 yes 232
5.4 even 2 inner 380.3.p.a.239.56 yes 232
19.7 even 3 inner 380.3.p.a.159.16 ✓ 232
20.19 odd 2 inner 380.3.p.a.239.16 yes 232
76.7 odd 6 inner 380.3.p.a.159.56 yes 232
95.64 even 6 inner 380.3.p.a.159.101 yes 232
380.159 odd 6 inner 380.3.p.a.159.61 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.16 ✓ 232 19.7 even 3 inner
380.3.p.a.159.56 yes 232 76.7 odd 6 inner
380.3.p.a.159.61 yes 232 380.159 odd 6 inner
380.3.p.a.159.101 yes 232 95.64 even 6 inner
380.3.p.a.239.16 yes 232 20.19 odd 2 inner
380.3.p.a.239.56 yes 232 5.4 even 2 inner
380.3.p.a.239.61 yes 232 1.1 even 1 trivial
380.3.p.a.239.101 yes 232 4.3 odd 2 inner